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3.34M
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variables (real : Type) [ordered_ring real]
variables (log exp : real → real)
variable log_exp_eq : ∀ x, log (exp x) = x
variable exp_log_eq : ∀ {x}, x > 0 → exp (log x) = x
variable exp_pos : ∀ x, exp x > 0
variable exp_add : ∀ x y, exp (x + y) = exp x * exp y
-- this ensures the assumptions are available in tactic proofs
include log_exp_eq exp_log_eq exp_pos exp_add
example (x y z : real) :
exp (x + y + z) = exp x * exp y * exp z :=
by rw [exp_add, exp_add]
example (y : real) (h : y > 0) : exp (log y) = y :=
exp_log_eq h
theorem log_mul {x y : real} (hx : x > 0) (hy : y > 0) :
log (x * y) = log x + log y :=
calc log (x * y)
= log (exp (log x) * exp (log y)) : by rw [exp_log_eq hx, exp_log_eq hy]
... = log (exp (log x + log y)) : by rw exp_add
... = log x + log y : by rw log_exp_eq
|
theory ForExperiments
(* Trying to explore/clarify the conditions under which
we can obtain a while rule *)
imports "./ImpCtl" "../Hoare/Hoare_Core"
begin
lemma WhileHelper :
assumes H0 : "gs = \<lparr> s_sem = pcomps' fs \<rparr>"
(*assumes H1 : "seq_sem_l_gen lfts \<in> set fs" *)
assumes HF : "f = imp_sem_l_gen lfts"
assumes Hpres : "sups_pres (set fs)"
assumes Hnemp : "g \<in> set fs"
assumes Hdom_while : "(f \<downharpoonleft> (set fs) SwhileC')"
assumes Hdom_skip : "(f \<downharpoonleft> (set fs) Sskip')"
assumes Hwhile : "lfts SwhileC' = SwhileC"
assumes Hskip : "lfts Sskip' = ImpCtl.Sskip"
(* TODO: generalize. these should be expressed using valid-sets *)
assumes Cond : "get_cond (payload st) = Some True"
assumes Cont : "s_cont st = Inl ((G SwhileC' [G Sskip' l]) # c') \<or>
s_cont st = Inl ((G Sskip' l) # (G SwhileC' [G Sskip' l]) # c')"
assumes Exec : "sem_exec_p gs st st'"
shows "payload st' = payload st \<and>
(s_cont st' = Inl ((G SwhileC' [G Sskip' l]) # c') \<or>
s_cont st' = Inl ((G Sskip' l) # (G SwhileC' [G Sskip' l]) # c'))"
using Exec Cond Cont unfolding sem_exec_p_def
proof(induction arbitrary: c' rule:converse_rtranclp_induct)
case base
then show ?case by auto
next
case (step y z)
show ?case
proof(cases "sem_step gs y")
case (Inr bad)
then have False using H0 step.prems
by(auto simp add: sem_step_def)
then show ?thesis by auto
next
case (Inl y')
have Zeq : "y' = z"
using Inl step.hyps unfolding sem_step_p_eq by auto
consider
(CaseA) "s_cont y = Inl (G SwhileC' [G Sskip' l] # c')" |
(CaseB) "s_cont y = Inl (G Sskip' l # G SwhileC' [G Sskip' l] # c')"
using step.prems by auto
then show ?thesis
proof cases
case CaseA
then have F_eq : "sem_step \<lparr> s_sem = f \<rparr> y = Inl y'"
using sym[OF dominant_pcomps'[OF Hpres Hnemp Hdom_while]] step.prems Inl H0
by(auto simp add: sem_step_def)
have C1 : "payload z = payload y"
using Hwhile H0 F_eq CaseA step.prems(1) unfolding HF Zeq
by(cases y; cases y';
auto simp add: s_cont_def sem_step_def imp_sem_l_gen_def imp_ctl_sem_def imp_sem_lifting_gen_def
schem_lift_defs
merge_l_def fst_l_def snd_l_def prio_l_def triv_l_def option_l_def LNew_def
get_cond_def
split: md_prio.splits md_triv.splits option.splits)
hence C2 : "get_cond (payload z) = Some True" using step.prems by auto
have "s_cont z = Inl (G Sskip' l # G SwhileC' [G Sskip' l] # c')"
using Hwhile H0 F_eq CaseA step.prems(1) unfolding HF Zeq
by(cases y; cases y';
auto simp add: s_cont_def sem_step_def imp_sem_l_gen_def imp_ctl_sem_def imp_sem_lifting_gen_def
schem_lift_defs
merge_l_def fst_l_def snd_l_def prio_l_def triv_l_def option_l_def LNew_def
get_cond_def
split: md_prio.splits md_triv.splits option.splits)
hence C3 : "s_cont z = Inl (G SwhileC' [G Sskip' l] # c') \<or>
s_cont z = Inl (G Sskip' l # G SwhileC' [G Sskip' l] # c')"
by auto
then show ?thesis using step.IH[OF C2 C3] CaseA unfolding C1
by auto
next
case CaseB
then have F_eq : "sem_step \<lparr> s_sem = f \<rparr> y = Inl y'"
using sym[OF dominant_pcomps'[OF Hpres Hnemp Hdom_skip]] step.prems Inl H0
by(auto simp add: sem_step_def)
have C1 : "payload z = payload y"
using Hwhile H0 F_eq CaseB step.prems(1) unfolding HF Zeq
by(cases y;
auto simp add: s_cont_def sem_step_def imp_sem_l_gen_def imp_ctl_sem_def imp_sem_lifting_gen_def
schem_lift_defs
merge_l_def fst_l_def snd_l_def prio_l_def triv_l_def option_l_def LNew_def
get_cond_def
split: md_prio.splits md_triv.splits option.splits)
hence C2 : "get_cond (payload z) = Some True" using step.prems by auto
have C3' : "s_cont z = Inl (G SwhileC' [G Sskip' l] # c')"
using Hskip H0 F_eq CaseB step.prems(1) unfolding HF Zeq
by(cases y;
auto simp add: s_cont_def sem_step_def imp_sem_l_gen_def imp_ctl_sem_def imp_sem_lifting_gen_def
schem_lift_defs
merge_l_def fst_l_def snd_l_def prio_l_def triv_l_def option_l_def LNew_def
get_cond_def
split: md_prio.splits md_triv.splits option.splits)
hence C3 : "s_cont z = Inl (G SwhileC' [G Sskip' l] # c') \<or>
s_cont z = Inl (G Sskip' l # G SwhileC' [G Sskip' l] # c')"
by auto
then show ?thesis using step.IH[OF C2 C3] CaseB C3' unfolding C1
by(auto)
qed
qed
qed
lemma WhileHelper1 :
assumes H0 : "gs = \<lparr> s_sem = pcomps' fs \<rparr>"
(*assumes H1 : "seq_sem_l_gen lfts \<in> set fs" *)
assumes HF : "f = imp_sem_l_gen lfts"
assumes Hpres : "sups_pres (set fs)"
assumes Hnemp : "g \<in> set fs"
assumes Hdom_while : "(f \<downharpoonleft> (set fs) SwhileC')"
assumes Hdom_skip : "(f \<downharpoonleft> (set fs) Sskip')"
assumes Hwhile : "lfts SwhileC' = SwhileC"
assumes Hskip : "lfts Sskip' = ImpCtl.Sskip"
(* TODO: generalize. these should be expressed using valid-sets *)
assumes Cond : "\<And> st . P1 st \<Longrightarrow> get_cond st = Some True"
assumes Htrue : "|gs| {- P1 -} [body]
{- P1 -}"
shows "|gs| {- P1 -} replicate n body @ [(G SwhileC' [G Sskip' l])] {- (\<lambda> st . False ) -}"
proof(induction n arbitrary: )
case 0
have "|gs| {-P1-} [G SwhileC' [G Sskip' l]] {-C False-}"
proof
fix c'
assume Hc' : "|gs| {\<lambda> x. False} c'"
show "|gs| {P1} [G SwhileC' [G Sskip' l]] @ c'"
proof
fix m :: "('a, 'b) state"
assume Pay : "P1 (payload m)"
assume Cont : "s_cont m = Inl ([G SwhileC' [G Sskip' l]] @ c')"
hence Cont' : "s_cont m = Inl (G SwhileC' [G Sskip' l] # c')" by auto
have Cond : "get_cond (payload m) = Some True"
using Cond[OF Pay] by auto
show "safe gs m"
proof
fix m' :: "('a, 'b) state"
assume Exec : "sem_exec_p gs m m'"
have Cont'' : "s_cont m = Inl (G SwhileC' [G Sskip' l] # c') \<or>
s_cont m = Inl (G Sskip' l # G SwhileC' [G Sskip' l] # c')"
using disjI1[OF Cont'] by auto
have Cases : "(s_cont m' = Inl (G SwhileC' [G Sskip' l] # c') \<or>
s_cont m' = Inl (G Sskip' l # G SwhileC' [G Sskip' l] # c'))"
using WhileHelper[OF H0 HF Hpres Hnemp Hdom_while Hdom_skip Hwhile Hskip Cond Cont'' Exec]
by auto
consider (CaseA) "s_cont m' = Inl (G SwhileC' [G Sskip' l] # c')" |
(CaseB) "s_cont m' = Inl (G Sskip' l # G SwhileC' [G Sskip' l] # c')"
using Cases by auto
then show "imm_safe gs m'"
proof cases
case CaseA
then show ?thesis
proof(cases "sem_step gs m'")
case (Inr bad)
then have False using H0 CaseA
by(auto simp add: sem_step_def)
then show ?thesis by auto
next
case (Inl m'')
then show ?thesis
unfolding imm_safe_def sem_step_p_eq by blast
qed
next
case CaseB
then show ?thesis
proof(cases "sem_step gs m'")
case (Inr bad)
then have False using H0 CaseB
by(auto simp add: sem_step_def)
then show ?thesis by auto
next
case (Inl m'')
then show ?thesis
unfolding imm_safe_def sem_step_p_eq by blast
qed
qed
qed
qed
qed
thus ?case by auto
next
case (Suc n)
have "|gs| {-P1-} body # replicate ( n) body @ [G SwhileC' [G Sskip' l]] {-C False-}"
using HCat[OF Htrue Suc.IH] by auto
then show ?case by auto
qed
(* use notion of absorption, again from Appel paper.
"separation logic for small step cminor"
*)
type_synonym 'full only_control =
"'full gensyn list md_triv option md_prio * String.literal md_triv option md_prio"
(* do we need to partition out the continuation from the state here? *)
definition absorbs :: "('syn, 'mstate) semc \<Rightarrow> 'syn gensyn \<Rightarrow> 'mstate \<Rightarrow> nat \<Rightarrow> bool" where
"absorbs gs st m n =
(\<forall> j . j \<le> n \<longrightarrow>
(\<exists> prefix st' .
(\<forall> k m_full .
s_cont m_full = Inl (st # k) \<longrightarrow>
payload m_full = m \<longrightarrow>
(\<exists> m_full' .
sem_exec_c_p gs m_full j m_full' \<and>
payload m_full' = st' \<and>
s_cont m_full' = Inl (prefix @ k)))))"
lemma absorbs_0 :
"absorbs gs st m 0"
proof-
have 0: "(\<And> a aa k . s_cont (a, aa, m) = Inl (st # k) \<Longrightarrow>
\<exists>ab ac.
sem_exec_c_p gs (a, aa, m) 0 (ab, ac, m) \<and>
s_cont (ab, ac, m) = Inl (st # k)) \<Longrightarrow> absorbs gs st m 0"
unfolding absorbs_def
apply(auto)
apply(rule_tac x = "[st]" in exI)
apply(rule_tac x = m in exI)
apply(auto)
done
have 1: "\<And> a aa k . s_cont (a, aa, m) = Inl (st # k) \<Longrightarrow>
\<exists>ab ac.
sem_exec_c_p gs (a, aa, m) 0 (ab, ac, m) \<and>
s_cont (ab, ac, m) = Inl (st # k)"
proof-
fix a aa k
assume H: "s_cont (a, aa, m) = Inl (st# k)"
show "\<exists>ab ac.
sem_exec_c_p gs (a, aa, m) 0 (ab, ac, m) \<and>
s_cont (ab, ac, m) = Inl (st # k)"
using sem_exec_c_p.intros(1)[of gs "(a, aa, m)"] H
by auto
qed
show "absorbs gs st m 0" using 0 1
by auto
qed
lemma absorbs_less :
assumes H : "absorbs gs st m (Suc n)"
shows "absorbs gs st m n" using H
unfolding absorbs_def
proof(step)
fix j
assume H0 : "\<forall>j\<le>Suc n.
\<exists>prefix st'.
\<forall>k m_full.
s_cont m_full = Inl (st # k) \<longrightarrow>
payload m_full = m \<longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full j m_full' \<and> payload m_full' = st' \<and> s_cont m_full' = Inl (prefix @ k))"
assume H1: "j \<le> n"
have H0' : "\<exists>prefix st'.
\<forall>k m_full.
s_cont m_full = Inl (st # k) \<longrightarrow>
payload m_full = m \<longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full j m_full' \<and> payload m_full' = st' \<and> s_cont m_full' = Inl (prefix @ k))"
using H0 H1
by auto
then obtain prefix st' where Hpfx :
"\<And> k m_full . s_cont m_full = Inl (st # k) \<Longrightarrow>
payload m_full = m \<Longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full j m_full' \<and> payload m_full' = st' \<and> s_cont m_full' = Inl (prefix @ k))"
by blast
then show " \<exists>prefix st'.
\<forall>k m_full.
s_cont m_full = Inl (st # k) \<longrightarrow> payload m_full = m \<longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full j m_full' \<and> payload m_full' = st' \<and> s_cont m_full' = Inl (prefix @ k))"
by blast
qed
lemma absorbs_at_most :
assumes H: "\<not> absorbs gs st m n"
shows "\<exists> i . i < n \<and> absorbs gs st m i \<and> \<not> absorbs gs st m (Suc i)" using H
proof(induction n arbitrary: m)
case 0
then have False using absorbs_0[of gs st m] by auto
then show ?case by auto
next
case (Suc n)
then show ?case
proof(cases "absorbs gs st m n")
case False
have "\<exists>i<n. absorbs gs st m i \<and> \<not> absorbs gs st m (Suc i)"
using Suc.IH[OF False] by auto
then obtain i where Li : "i < n" and Ai : "absorbs gs st m i" and Nai : "\<not> absorbs gs st m (Suc i)"
by blast
have C1 : "i < Suc n" using Li by auto
show ?thesis using C1 Ai Nai by blast
next
case True
have C1 : "n < Suc n" by auto
show ?thesis using Suc.prems C1 True by blast
qed
qed
lemma HWhile_gen0 :
assumes H0 : "gs = \<lparr> s_sem = pcomps' fs \<rparr>"
(*assumes H1 : "seq_sem_l_gen lfts \<in> set fs" *)
assumes HF : "f = imp_sem_l_gen lfts"
assumes Hpres : "sups_pres (set fs)"
assumes Hnemp : "g \<in> set fs"
assumes Hdom_while : "(f \<downharpoonleft> (set fs) SwhileC')"
assumes Hdom_skip : "(f \<downharpoonleft> (set fs) Sskip')"
assumes Hwhile : "lfts SwhileC' = SwhileC"
assumes Hskip : "lfts Sskip' = ImpCtl.Sskip"
(* TODO: generalize. these should be expressed using valid-sets *)
assumes P1_valid : "\<And> st. P1 st \<Longrightarrow> get_cond st = Some True"
assumes Htrue : "|gs| {- P1 -} [body]
{- P1 -}"
shows "|gs| {- P1 -} [(G Swhile'C [body])] {- (\<lambda> st . False ) -}"
proof
fix c'
assume Guard : "|gs| {\<lambda>st. False } c'"
show "|gs| {P1} [(G Swhile'C [body])] @ c'"
proof
fix m :: "('a, 'b) state"
assume P : "P1 (payload m)"
assume Cont : "s_cont m = Inl (([G Swhile'C [body]]) @ c')"
hence Cont' : "s_cont m = Inl ((G Swhile'C [body]) # c')"
by auto
show "safe gs m"
proof
fix m' :: "('a, 'b) state"
assume Exec : "sem_exec_p gs m m'"
obtain n where N : "sem_exec_c_p gs m n m'"
using exec_p_imp_exec_c_p[OF Exec]
by auto
have Unfold : "|gs| {-P1-} replicate n body @
[G SwhileC' [G Sskip' []]] {-C False-}"
using WhileHelper1[OF H0 HF Hpres Hnemp Hdom_while Hdom_skip Hwhile Hskip
P1_valid _ ] Htrue
by fastforce
have Guard' : "|gs| {P1} (replicate n body @ [G SwhileC' [G Sskip' []]]) @ c'"
using HTE[OF Unfold Guard] by auto
(* gross low-level deconstruction of m to build a new state *)
obtain m1 m2xx m3 m4 m5 where Mcomp : "m = (m1, m2xx, m3, m4, m5)"
by(cases m; auto)
obtain malt :: "('a, 'b) state" where Malt :
"malt = (mdp 0 (Some (mdt (replicate n body @ [G SwhileC' [G Sskip' []]] @ c'))),
mdp 0 None,
m3,
m4,
m5)" by simp
have Malt_pl : "payload malt = payload m" using Malt Mcomp by auto
hence Malt_P : "P1 (payload malt)" using P unfolding Malt_pl by auto
have Malt_cont : "s_cont malt = Inl (((replicate n body @ [G SwhileC' [G Sskip' []]]) @ c'))"
using Malt by (auto simp add: s_cont_def)
have Malt_safe : "safe gs malt" using guardedD[OF Guard' Malt_P Malt_cont] by auto
obtain m2 where M2 : "sem_step_p gs m m2" and M2_cont : "s_cont m2 = Inl (body # ([G Swhile'C [body]] @ c'))" and M2_pay : "payload m2 = payload m"
sorry
obtain maltb :: "('a, 'b) state" where Maltb :
"maltb = (mdp 0 (Some (mdt (body # []))),
mdp 0 None,
m3,
m4,
m5)" by simp
have Maltb_pl : "payload maltb = payload m" using Maltb Mcomp by auto
hence Maltb_P : "P1 (payload maltb)" using P unfolding Maltb_pl by auto
have Maltb_cont : "s_cont maltb = Inl (body # [])"
using Maltb by (auto simp add: s_cont_def)
show "imm_safe gs m'" using N
proof(cases rule: sem_exec_c_p.cases)
case Excp_0
have "imm_safe gs m" sorry
then show ?thesis using Excp_0 by auto
next
(* We are just doing this to get the fact that n > 0. or should be we doing an actual induction? *)
case (Excp_Suc mz2 n')
have Malt_cont' : "s_cont malt = Inl (body # ((replicate n' body @ [G SwhileC' [G Sskip' []]]) @ c'))"
using Excp_Suc Malt_cont
by auto
show "imm_safe gs m'"
proof(cases "absorbs gs body (payload m) n")
case True
have "\<exists>prefix st'.
\<forall>k m_full.
s_cont m_full = Inl (body # k) \<longrightarrow>
payload m_full = payload m \<longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full n m_full' \<and> payload m_full' = st' \<and> s_cont m_full' = Inl (prefix @ k))"
using True unfolding absorbs_def by auto
then obtain prefix and m'x :: "(bool md_triv option md_prio * int md_triv option md_prio * 'b)" where
Absorbed : "\<forall>k m_full.
s_cont m_full = Inl (body # k) \<longrightarrow>
payload m_full = payload m \<longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full n m_full' \<and> payload m_full' = m'x \<and> s_cont m_full' = Inl (prefix @ k))"
by blast
then obtain malt' ::"('a, 'b) state" where
"(sem_exec_c_p gs malt n malt' \<and> payload malt' = m'x \<and> s_cont malt' = Inl (prefix @ ((replicate n' body @ [G SwhileC' [G Sskip' []]]) @ c')))"
using Malt_cont' Malt_pl
by blast
have "imm_safe gs malt'" sorry (* since malt is known to be safe *)
obtain m2' ::"('a, 'b) state" where "sem_exec_c_p gs m2 n m2' \<and> payload m2' = m'x \<and> s_cont m2' = Inl (prefix @ ([G Swhile'C [body]] @ c'))"
using M2_cont M2_pay Absorbed
by blast
hence M2'_exec : "sem_exec_c_p gs m2 n m2' " by auto
have "imm_safe gs m2'" (* since prefix will be enough to guarantee safety *) sorry
have Exec_m2' : "sem_exec_c_p gs m (n+1) m2'" using sem_exec_c_p.intros(2)[OF M2 M2'_exec] by auto
have Duh : "n \<le> n + 1" by simp
obtain m'_copy where M'_copy1 : "sem_exec_c_p gs m n m'_copy" and M'_copy2 : "sem_exec_c_p gs m'_copy (n + 1 - n) m2'" using exec_c_p_split[OF Exec_m2' Duh]
by auto
have M'_copy_m' : "m'_copy = m'" using exec_c_p_determ[OF N M'_copy1]
by auto
have Last_step : "sem_exec_c_p gs m'_copy (1) m2'" using M'_copy2 by auto
hence "sem_step_p gs m' m2'"
using Excp_1'[OF Last_step] unfolding M'_copy_m' by auto
then show ?thesis unfolding imm_safe_def by blast
next
case False
obtain i where
Ilt : "i < n" and Iabs : "absorbs gs body (payload m) i" and Imax : "\<not> absorbs gs body (payload m) (Suc i)"
using absorbs_at_most[OF False] by blast
have M2_cont' : "s_cont m2 = Inl (body # ([G Swhile'C [body]] @ c'))" using M2_cont by auto
have "\<exists>prefix st'.
\<forall>k m_full.
s_cont m_full = Inl (body # k) \<longrightarrow>
payload m_full = payload m \<longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full i m_full' \<and> payload m_full' = st' \<and> s_cont m_full' = Inl (prefix @ k))"
using Iabs unfolding absorbs_def by auto
then obtain prefix and m'x :: "(bool md_triv option md_prio * int md_triv option md_prio * 'b)" where
Absorbed : "\<forall>k m_full.
s_cont m_full = Inl (body # k) \<longrightarrow>
payload m_full = payload m \<longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full i m_full' \<and> payload m_full' = m'x \<and> s_cont m_full' = Inl (prefix @ k))"
by blast
hence Absorbed' :
"\<And> k m_full.
s_cont m_full = Inl (body # k) \<Longrightarrow>
payload m_full = payload m \<Longrightarrow> (\<exists>m_full'. sem_exec_c_p gs m_full i m_full' \<and> payload m_full' = m'x \<and> s_cont m_full' = Inl (prefix @ k))"
by blast
obtain m2' ::"('a, 'b) state" where M2_exec :
"(sem_exec_c_p gs m2 i m2' \<and> payload m2' = m'x \<and> s_cont m2' = Inl (prefix @ (([G Swhile'C [body]] @ c'))))"
using Absorbed'[OF M2_cont' M2_pay]
by blast
obtain malt' ::"('a, 'b) state" where Malt_exec :
"(sem_exec_c_p gs malt i malt' \<and> payload malt' = m'x \<and> s_cont malt' = Inl (prefix @ ((replicate n' body @ [G SwhileC' [G Sskip' []]]) @ c')))"
using Absorbed' Malt_cont' Malt_pl
by blast
have "imm_safe gs malt'" sorry (* since malt is known to be safe *)
obtain maltb' ::"('a, 'b) state" where Maltb' :
"(sem_exec_c_p gs maltb i maltb' \<and> payload maltb' = m'x \<and> s_cont maltb' = Inl (prefix @ []))"
using Absorbed'[OF Maltb_cont Maltb_pl] Maltb_cont
by blast
obtain j where
Jleq : "j \<le> Suc i"
and Imax' : "\<not> (\<exists>prefix st'.
\<forall>k m_full.
s_cont m_full = Inl (body # k) \<longrightarrow>
payload m_full = payload m \<longrightarrow>
(\<exists>m_full'.
sem_exec_c_p gs m_full j m_full' \<and>
payload m_full' = st' \<and>
s_cont m_full' = Inl (prefix @ k)))"
using Imax unfolding absorbs_def
by auto
show "imm_safe gs m'"
proof(cases "j = Suc i")
case False' : False
hence Jleq' : "j \<le> i" using Jleq by auto
have Uhoh : "(\<exists>prefix st'.
\<forall>k m_full.
s_cont m_full = Inl (body # k) \<longrightarrow>
payload m_full = payload m \<longrightarrow>
(\<exists>m_full'.
sem_exec_c_p gs m_full j m_full' \<and>
payload m_full' = st' \<and>
s_cont m_full' = Inl (prefix @ k)))"
using Iabs Jleq' unfolding absorbs_def
by blast
have False
using Imax' Uhoh by blast
then show ?thesis by auto
next
case True' : True
(* case split on whether prefix is [].
if it is nil, "apply induction on the case n-j" (i.e., spill over into next iteration *)
(* if prefix is non-nil, we need some fact about how the resulting continuation
will be the same between malt' and m2'
*)
show "imm_safe gs m'"
proof(cases prefix)
case Nil
then show ?thesis (* do an inductive thingy, n - j < n *) sorry
next
case (Cons preh pret)
obtain mfin1 mfin1p kfin1 where "sem_exec_c_p gs m2' 1 mfin1 \<and> payload mfin1 = mfin1p \<and> s_cont mfin1 = kfin1" sorry
(* we know we are safe here (malt') *)
obtain mfin2 mfin2p kfin2 where "sem_exec_c_p gs malt' 1 mfin2 \<and> payload mfin2 = mfin2p \<and> s_cont mfin2 = kfin2"
sorry
show "imm_safe gs m'"
using HTE[OF Htrue]
(* idea: we are looking at some kind of function f where
f ((ph#pt) @ ((replicate n' body @ [G SwhileC' [G Sskip' []]]) @ c'))) =
f ((ph#pt) @ (([G Swhile'C [body]] @ c')))
*)
(* executing preh should be imm. safe - regardless of whether this happens in maltb' or malt' *)
(* we need to be able to use the fact that lack of suffix implies escape happens. but unlike appel's system,
our continuations might be too free-form to make this work.
*)
then show ?thesis sorry
qed
(*
"\<And>prefix a aa b.
(\<And> k m_full.
s_cont m_full = Inl (body # k) \<Longrightarrow>
payload m_full = payload m \<Longrightarrow>
(\<exists>m_full'.
sem_exec_c_p gs m_full j m_full' \<and>
payload m_full' = (a, aa, b) \<and> s_cont m_full' = Inl (prefix @ k))) \<Longrightarrow> False"
*)
(* idea: 2 possibilities at this point
- stepping into prefix results in an empty continuation \<Rightarrow> we are done
- stepping into the prefix results in "escape" beyond the loop.
however, can we cheat at this point? use information from the continuation list to do
different things in different cases in a way that is bad?
*)
(* we still have n-i steps to go ... we need to know something about the behavior of the prefix. *)
(* one idea: prefix cannot choose whether to fail based on data (or shape) of continuation *)
(* another idea: just based on data of continuation *)
(* prefix should be safe for any state that was produced from running body starting at P1 *)
(* {-?-} prefix {-P1-} *)
have "|gs| {-(\<lambda> st . st = m'x)-} prefix {-P1-}"
proof
fix cx
assume G : "|gs| {P1} cx"
show "|gs| {\<lambda>st. st = m'x} prefix @ cx"
proof
fix mp :: "('a, 'b) state"
assume Hmp : "(payload mp) = m'x"
assume Hmpc : "s_cont mp = Inl (prefix @ cx)"
have "|gs| {P1} [body] @ cx" using HTE[OF Htrue G] by auto
show "safe gs mp"
proof
fix mp'
assume "sem_exec_p gs mp mp'"
show "imm_safe gs mp'"
then show ?thesis
qed
have False using Imax Iabs unfolding absorbs_def
(*
exists Suc i where
forall prefix st'
exists k, m_full where
s_cont m_full \<noteq> Inl (body # k) \<and>
payload m_full
*)
(* idea: we know prefix must be something that throws away the continuation. the appel paper does a specific case analysis on "non-local"
control flow at this point. perhaps we need to do the same? this would require a "closed world" of non-local control flow when we instantiate
this theorem, which isn't unreasonable. *)
then show ?thesis using Imax unfolding absorbs_def
qed
qed
qed
qed
qed
(*
lemma HWhile_gen0 :
assumes H0 : "gs = \<lparr> s_sem = pcomps' fs \<rparr>"
(*assumes H1 : "seq_sem_l_gen lfts \<in> set fs" *)
assumes HF : "f = imp_sem_l_gen lfts"
assumes Hpres : "sups_pres (set fs)"
assumes Hnemp : "g \<in> set fs"
assumes Hdom_while : "(f \<downharpoonleft> (set fs) SwhileC')"
assumes Hdom_skip : "(f \<downharpoonleft> (set fs) Sskip')"
assumes Hwhile : "lfts SwhileC' = SwhileC"
assumes Hskip : "lfts Sskip' = ImpCtl.Sskip"
(* TODO: generalize. these should be expressed using valid-sets *)
assumes P1_valid : "\<And> st. P1 st \<Longrightarrow> get_cond st = Some True"
assumes Htrue : "|gs| {- P1 -} [body]
{- P1 -}"
shows "|gs| {- P1 -} [(G Swhile'C [body])] {- (\<lambda> st . False ) -}"
proof
fix c'
assume Guard : " |gs| {C False} c'"
show "|gs| {P1} ([G Swhile'C [body]]) @ c'"
proof
fix m :: "('a, 'b) state"
assume Mp : "P1 (payload m)"
assume Mc : "s_cont m = Inl (([G Swhile'C [body]]) @ c')"
show "safe gs m"
proof
fix m' :: "('a, 'b) state"
assume Exec : "sem_exec_p gs m m'"
obtain cnt where Exec' : "sem_exec_c_p gs m cnt m'"
using exec_p_imp_exec_c_p[OF Exec] by auto
have "|gs| {- P1 -} replicate cnt body @ [(G Swhile'C [body])] {- (\<lambda> st . False ) -}"
then show "imm_safe gs m'" using Guard Mp Mc
proof(induction arbitrary: c' rule: sem_exec_c_p.induct)
case (Excp_0 m0)
then show ?case unfolding imm_safe_def sorry
next
case (Excp_Suc m1 m2 n m3)
then show ?case
qed
*)
end |
theory ExF018
imports Main
begin
lemma "(\<forall>x. \<forall>y. (P x \<longleftrightarrow> P y) \<longrightarrow> (x \<longleftrightarrow> y)) \<Longrightarrow> (\<forall>x. \<forall>y. (x \<longleftrightarrow> y) \<longrightarrow> (P x \<longleftrightarrow> P y)) \<Longrightarrow> (\<forall>x. P (P x) \<longleftrightarrow> x)"
proof -
{
assume "(\<forall>x. \<forall>y. (P x \<longleftrightarrow> P y) \<longrightarrow> (x \<longleftrightarrow> y))"
{
assume "(\<forall>x. \<forall>y. (x \<longleftrightarrow> y) \<longrightarrow> (P x \<longleftrightarrow> P y))"
{
assume "\<forall>x. P (P x)"
{
|
lemma homotopic_paths: "homotopic_paths s p q \<longleftrightarrow> (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and> h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and> (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and> (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and> (\<forall>t \<in> {0..1::real}. pathstart(h \<circ> Pair t) = pathstart p \<and> pathfinish(h \<circ> Pair t) = pathfinish p))" |
[STATEMENT]
lemma Deq_enabled:
"\<And>q. basevars (ic,q,oc) \<Longrightarrow> \<turnstile> Enabled (<Deq ic q oc>_(ic,q,oc)) = (q \<noteq> #[])"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<And>q. basevars (ic, q, oc) \<Longrightarrow> \<turnstile> Enabled (<Deq ic q oc>_(ic, q, oc)) = (q \<noteq> #[])
[PROOF STEP]
apply (unfold Deq_visible [temp_rewrite])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<And>q. basevars (ic, q, oc) \<Longrightarrow> \<turnstile> Enabled Deq ic q oc = (q \<noteq> #[])
[PROOF STEP]
apply (force elim!: base_enabled [temp_use] enabledE [temp_use] simp: Deq_def)
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done |
lemma fract_poly_0 [simp]: "fract_poly 0 = 0" |
(** Formal Reasoning About Programs <http://adam.chlipala.net/frap/>
* Chapter 7: Model Checking
* Author: Adam Chlipala
* License: https://creativecommons.org/licenses/by-nc-nd/4.0/ *)
Set Warnings "-notation-overridden". (* <-- needed while we play with defining one
* of the book's notations ourselves locally *)
Require Import Frap TransitionSystems.
Set Implicit Arguments.
(* Coming up with invariants ourselves can be tedious! Let's investigate how we
* can automate the choice of invariants, for systems with only finitely many
* reachable states. This style is known as model checking. *)
Definition oneStepClosure_current {state} (sys : trsys state)
(invariant1 invariant2 : state -> Prop) :=
forall st, invariant1 st
-> invariant2 st.
Definition oneStepClosure_new {state} (sys : trsys state)
(invariant1 invariant2 : state -> Prop) :=
forall st st', invariant1 st
-> sys.(Step) st st'
-> invariant2 st'.
Definition oneStepClosure {state} (sys : trsys state)
(invariant1 invariant2 : state -> Prop) :=
oneStepClosure_current sys invariant1 invariant2
/\ oneStepClosure_new sys invariant1 invariant2.
Theorem prove_oneStepClosure : forall state (sys : trsys state) (inv1 inv2 : state -> Prop),
(forall st, inv1 st -> inv2 st)
-> (forall st st', inv1 st -> sys.(Step) st st' -> inv2 st')
-> oneStepClosure sys inv1 inv2.
Proof.
unfold oneStepClosure.
propositional.
Qed.
Theorem oneStepClosure_done : forall state (sys : trsys state) (invariant : state -> Prop),
(forall st, sys.(Initial) st -> invariant st)
-> oneStepClosure sys invariant invariant
-> invariantFor sys invariant.
Proof.
unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new.
propositional.
apply invariant_induction.
assumption.
simplify.
eapply H2.
eassumption.
assumption.
Qed.
Inductive multiStepClosure {state} (sys : trsys state)
: (state -> Prop) -> (state -> Prop) -> Prop :=
(* We might be done, if one-step closure has no effect. *)
| MscDone : forall inv,
oneStepClosure sys inv inv
-> multiStepClosure sys inv inv
(* Or we might need to run another one-step closure and recurse. *)
| MscStep : forall inv inv' inv'',
oneStepClosure sys inv inv'
-> multiStepClosure sys inv' inv''
-> multiStepClosure sys inv inv''.
Lemma multiStepClosure_ok' : forall state (sys : trsys state) (inv inv' : state -> Prop),
multiStepClosure sys inv inv'
-> (forall st, sys.(Initial) st -> inv st)
-> invariantFor sys inv'.
Proof.
induct 1; simplify.
apply oneStepClosure_done.
assumption.
assumption.
apply IHmultiStepClosure.
simplify.
unfold oneStepClosure, oneStepClosure_current in *.
propositional.
apply H3.
apply H1.
assumption.
Qed.
Theorem multiStepClosure_ok : forall state (sys : trsys state) (inv : state -> Prop),
multiStepClosure sys sys.(Initial) inv
-> invariantFor sys inv.
Proof.
simplify.
eapply multiStepClosure_ok'.
eassumption.
propositional.
Qed.
Theorem oneStepClosure_empty : forall state (sys : trsys state),
oneStepClosure sys (constant nil) (constant nil).
Proof.
unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new; propositional.
Qed.
Theorem oneStepClosure_split : forall state (sys : trsys state) st sts (inv1 inv2 : state -> Prop),
(forall st', sys.(Step) st st' -> inv1 st')
-> oneStepClosure sys (constant sts) inv2
-> oneStepClosure sys (constant (st :: sts)) ({st} \cup inv1 \cup inv2).
Proof.
unfold oneStepClosure, oneStepClosure_current, oneStepClosure_new; propositional.
invert H0.
left.
(* [left] and [right]: prove a disjunction by proving the left or right case,
* respectively. Note that here, we are using the fact that set union
* [\cup] is defined in terms of disjunction. *)
left.
simplify.
propositional.
right.
apply H1.
assumption.
simplify.
propositional.
right.
left.
apply H.
equality.
right.
right.
eapply H2.
eassumption.
assumption.
Qed.
Theorem singleton_in : forall {A} (x : A) rest,
({x} \cup rest) x.
Proof.
simplify.
left.
simplify.
equality.
Qed.
(* OK, back to our example from last chapter, of factorial as a transition
* system. Here's a good overall correctness condition, which we didn't bother
* to state before. *)
Definition fact_correct (original_input : nat) (st : fact_state) : Prop :=
match st with
| AnswerIs ans => fact original_input = ans
| WithAccumulator _ _ => True
end.
(* Let's also restate the initial-states set using a singleton set. *)
Theorem fact_init_is : forall original_input,
fact_init original_input = {WithAccumulator original_input 1}.
Proof.
simplify.
apply sets_equal; simplify.
propositional.
invert H.
equality.
rewrite <- H0.
constructor.
Qed.
(* Now we will prove that factorial is correct, for the input 2, without needing
* to write out an inductive invariant ourselves. *)
Theorem factorial_ok_2 :
invariantFor (factorial_sys 2) (fact_correct 2).
Proof.
simplify.
eapply invariant_weaken.
apply multiStepClosure_ok.
simplify.
rewrite fact_init_is.
eapply MscStep.
apply oneStepClosure_split.
simplify.
invert H.
simplify.
apply singleton_in.
apply oneStepClosure_empty.
simplify.
eapply MscStep.
apply oneStepClosure_split.
simplify.
invert H.
simplify.
apply singleton_in.
apply oneStepClosure_split.
simplify.
invert H.
simplify.
apply singleton_in.
apply oneStepClosure_empty.
simplify.
eapply MscStep.
apply oneStepClosure_split.
simplify.
invert H.
simplify.
apply singleton_in.
apply oneStepClosure_split.
simplify.
invert H.
simplify.
apply singleton_in.
apply oneStepClosure_split.
simplify.
invert H.
simplify.
apply singleton_in.
apply oneStepClosure_empty.
simplify.
apply MscDone.
apply prove_oneStepClosure; simplify.
assumption.
propositional; subst; invert H0; simplify; propositional.
simplify.
unfold fact_correct.
propositional; subst; trivial.
Qed.
(* BEGIN CODE THAT WILL NOT BE EXPLAINED IN DETAIL! *)
Local Hint Rewrite fact_init_is.
Ltac model_check_done :=
apply MscDone; apply prove_oneStepClosure; simplify; propositional; subst;
repeat match goal with
| [ H : _ |- _ ] => invert H
end; simplify; equality.
Theorem singleton_in_other : forall {A} (x : A) (s1 s2 : set A),
s2 x
-> (s1 \cup s2) x.
Proof.
simplify.
right.
right.
assumption.
Qed.
Ltac singletoner :=
repeat match goal with
| _ => apply singleton_in
| [ |- (_ \cup _) _ ] => apply singleton_in_other
end.
Ltac model_check_step0 :=
eapply MscStep; [
repeat ((apply oneStepClosure_empty; simplify)
|| (apply oneStepClosure_split; [ simplify;
repeat match goal with
| [ H : _ |- _ ] => invert H; try congruence
end; solve [ singletoner ] | ]))
| simplify ].
Ltac model_check_step :=
match goal with
| [ |- multiStepClosure _ ?inv1 _ ] =>
model_check_step0;
match goal with
| [ |- multiStepClosure _ ?inv2 _ ] =>
(assert (inv1 = inv2) by compare_sets; fail 3)
|| idtac
end
end.
Ltac model_check_steps1 := model_check_step || model_check_done.
Ltac model_check_steps := repeat model_check_steps1.
Ltac model_check_finish := simplify; propositional; subst; simplify; equality.
Ltac model_check_infer :=
apply multiStepClosure_ok; simplify; model_check_steps.
Ltac model_check_find_invariant :=
simplify; eapply invariant_weaken; [ model_check_infer | ]; cbv beta in *.
Ltac model_check := model_check_find_invariant; model_check_finish.
(* END CODE THAT WILL NOT BE EXPLAINED IN DETAIL! *)
(* Now watch this. We can check various instances of factorial
* automatically. *)
Theorem factorial_ok_2_snazzy :
invariantFor (factorial_sys 2) (fact_correct 2).
Proof.
model_check.
Qed.
Theorem factorial_ok_3 :
invariantFor (factorial_sys 3) (fact_correct 3).
Proof.
model_check.
Qed.
Theorem factorial_ok_5 :
invariantFor (factorial_sys 5) (fact_correct 5).
Proof.
model_check.
Qed.
(* Let's see that last one broken into two steps, so that we get a look at the
* inferred invariant. *)
Theorem factorial_ok_5_again :
invariantFor (factorial_sys 5) (fact_correct 5).
Proof.
model_check_find_invariant.
model_check_finish.
Qed.
(** * Abstraction *)
Inductive isEven : nat -> Prop :=
| EvenO : isEven 0
| EvenSS : forall n, isEven n -> isEven (S (S n)).
Inductive add2_thread :=
| Read
| Write (local : nat).
Inductive add2_init : threaded_state nat add2_thread -> Prop :=
| Add2Init : add2_init {| Shared := 0; Private := Read |}.
Inductive add2_step : threaded_state nat add2_thread -> threaded_state nat add2_thread -> Prop :=
| StepRead : forall global,
add2_step {| Shared := global; Private := Read |}
{| Shared := global; Private := Write global |}
| StepWrite : forall global local,
add2_step {| Shared := global; Private := Write local |}
{| Shared := S (S local); Private := Read |}.
Definition add2_sys1 := {|
Initial := add2_init;
Step := add2_step
|}.
Definition add2_sys := parallel add2_sys1 add2_sys1.
Definition add2_correct (st : threaded_state nat (add2_thread * add2_thread)) :=
isEven st.(Shared).
Inductive simulates state1 state2 (R : state1 -> state2 -> Prop)
(* [R] is a relation connecting the states of the two systems. *)
(sys1 : trsys state1) (sys2 : trsys state2) : Prop :=
| Simulates :
(* Every initial state of [sys1] has some matching initial state of [sys2]. *)
(forall st1, sys1.(Initial) st1
-> exists st2, R st1 st2
/\ sys2.(Initial) st2)
(* Starting from a pair of related states, every step in [sys1] can be matched
* in [sys2], to destinations that are also related. *)
-> (forall st1 st2, R st1 st2
-> forall st1', sys1.(Step) st1 st1'
-> exists st2', R st1' st2'
/\ sys2.(Step) st2 st2')
-> simulates R sys1 sys2.
(* Given an invariant for [sys2], we now have a generic way of defining an
* invariant for [sys1], by composing with [R]. *)
Inductive invariantViaSimulation state1 state2 (R : state1 -> state2 -> Prop)
(inv2 : state2 -> Prop)
: state1 -> Prop :=
| InvariantViaSimulation : forall st1 st2, R st1 st2
-> inv2 st2
-> invariantViaSimulation R inv2 st1.
(* By way of a lemma, let's prove that, given a simulation, any
* invariant-via-simulation really is an invariant for the original system. *)
Lemma invariant_simulates' : forall state1 state2 (R : state1 -> state2 -> Prop)
(sys1 : trsys state1) (sys2 : trsys state2),
(forall st1 st2, R st1 st2
-> forall st1', sys1.(Step) st1 st1'
-> exists st2', R st1' st2'
/\ sys2.(Step) st2 st2')
-> forall st1 st1', sys1.(Step)^* st1 st1'
-> forall st2, R st1 st2
-> exists st2', R st1' st2'
/\ sys2.(Step)^* st2 st2'.
Proof.
induct 2.
simplify.
exists st2.
(* [exists E]: prove [exists x, P(x)] by proving [P(E)]. *)
propositional.
constructor.
simplify.
eapply H in H2.
first_order.
(* [first_order]: simplify first-order logic structure. Be forewarned: this
* one is especially likely to run forever! *)
apply IHtrc in H2.
first_order.
exists x1.
propositional.
econstructor.
eassumption.
assumption.
assumption.
Qed.
Theorem invariant_simulates : forall state1 state2 (R : state1 -> state2 -> Prop)
(sys1 : trsys state1) (sys2 : trsys state2) (inv2 : state2 -> Prop),
simulates R sys1 sys2
-> invariantFor sys2 inv2
-> invariantFor sys1 (invariantViaSimulation R inv2).
Proof.
simplify.
invert H.
unfold invariantFor; simplify.
apply H1 in H.
first_order.
apply invariant_simulates' with (sys2 := sys2) (R := R) (st2 := x) in H3; try assumption.
first_order.
unfold invariantFor in H0.
apply H0 with (s' := x0) in H4; try assumption.
econstructor.
eassumption.
assumption.
Qed.
(*Theorem add2_ok :
invariantFor add2_sys add2_correct.
Proof.
Admitted.*)
Inductive add2_bthread :=
| BRead
| BWrite (local : bool).
Inductive add2_binit : threaded_state bool add2_bthread -> Prop :=
| Add2BInit : add2_binit {| Shared := true; Private := BRead |}.
Inductive add2_bstep : threaded_state bool add2_bthread -> threaded_state bool add2_bthread -> Prop :=
| StepBRead : forall global,
add2_bstep {| Shared := global; Private := BRead |}
{| Shared := global; Private := BWrite global |}
| StepBWrite : forall global local,
add2_bstep {| Shared := global; Private := BWrite local |}
{| Shared := local; Private := BRead |}.
Definition add2_bsys1 := {|
Initial := add2_binit;
Step := add2_bstep
|}.
Definition add2_bsys := parallel add2_bsys1 add2_bsys1.
(* This invariant formalizes the connection between local states of threads, in
* the original and abstracted systems. *)
Inductive R_private1 : add2_thread -> add2_bthread -> Prop :=
| RpRead : R_private1 Read BRead
| RpWrite : forall n b, (b = true <-> isEven n)
-> R_private1 (Write n) (BWrite b).
(* We lift [R_private1] to a relation over whole states. *)
Inductive add2_R : threaded_state nat (add2_thread * add2_thread)
-> threaded_state bool (add2_bthread * add2_bthread)
-> Prop :=
| Add2_R : forall n b th1 th2 th1' th2',
(b = true <-> isEven n)
-> R_private1 th1 th1'
-> R_private1 th2 th2'
-> add2_R {| Shared := n; Private := (th1, th2) |}
{| Shared := b; Private := (th1', th2') |}.
(* Let's also recharacterize the initial states via a singleton set. *)
Theorem add2_init_is :
parallel_init add2_binit add2_binit = { {| Shared := true; Private := (BRead, BRead) |} }.
Proof.
simplify.
apply sets_equal; simplify.
propositional.
invert H.
invert H2.
invert H4.
equality.
invert H0.
constructor.
constructor.
constructor.
Qed.
(* We ask Coq to remember this lemma as a hint, which will be used by the
* model-checking tactics that we refrain from explaining in detail. *)
Local Hint Rewrite add2_init_is.
(* Now, let's verify the original system. *)
Theorem add2_ok :
invariantFor add2_sys add2_correct.
Proof.
(* First step: strengthen the invariant. We leave an underscore for the
* unknown invariant, to be found by model checking. *)
eapply invariant_weaken with (invariant1 := invariantViaSimulation add2_R _).
(* One way to find an invariant-by-simulation is to find an invariant for the
* abstracted system, as this step asks to do. *)
apply invariant_simulates with (sys2 := add2_bsys).
(* Now we must prove that the simulation via [add2_R] is valid, which is
* routine. *)
constructor; simplify.
invert H.
invert H0.
invert H1.
exists {| Shared := true; Private := (BRead, BRead) |}; simplify.
propositional.
constructor.
propositional.
constructor.
constructor.
constructor.
invert H.
invert H0; simplify.
invert H7.
invert H2.
exists {| Shared := b; Private := (BWrite b, th2') |}.
propositional.
constructor.
propositional.
constructor.
propositional.
assumption.
constructor.
constructor.
invert H2.
exists {| Shared := b0; Private := (BRead, th2') |}.
propositional.
constructor.
propositional.
constructor.
assumption.
invert H0.
propositional.
constructor.
assumption.
constructor.
constructor.
invert H7.
invert H3.
exists {| Shared := b; Private := (th1', BWrite b) |}.
propositional.
constructor.
propositional.
assumption.
constructor.
propositional.
constructor.
constructor.
invert H3.
exists {| Shared := b0; Private := (th1', BRead) |}.
propositional.
constructor.
propositional.
constructor.
assumption.
invert H0.
propositional.
assumption.
constructor.
constructor.
constructor.
(* OK, we're glad to have that over with! Such a process could also be
* automated, but we won't bother doing so here. However, we are now in a
* good state, where our model checker can find the invariant
* automatically. *)
model_check_infer.
(* It finds exactly four reachable states. We finish by showing that they all
* obey the original invariant. *)
invert 1.
invert H0.
simplify.
unfold add2_correct.
simplify.
propositional; subst.
invert H.
propositional.
invert H1.
propositional.
invert H.
propositional.
invert H1.
propositional.
Qed.
(** * Another abstraction example *)
Inductive pc :=
| i_gets_0
| j_gets_0
| Loop
| i_add_n
| j_add_n
| n_sub_1
| Done.
Record vars := {
N : nat;
I : nat;
J : nat
}.
Record state := {
Pc : pc;
Vars : vars
}.
Inductive initial : state -> Prop :=
| Init : forall vs, initial {| Pc := i_gets_0; Vars := vs |}.
Inductive step : state -> state -> Prop :=
| Step_i_gets_0 : forall n i j,
step {| Pc := i_gets_0; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := j_gets_0; Vars := {| N := n;
I := 0;
J := j |} |}
| Step_j_gets_0 : forall n i j,
step {| Pc := j_gets_0; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := Loop; Vars := {| N := n;
I := i;
J := 0 |} |}
| Step_Loop_done : forall i j,
step {| Pc := Loop; Vars := {| N := 0;
I := i;
J := j |} |}
{| Pc := Done; Vars := {| N := 0;
I := i;
J := j |} |}
| Step_Loop_enter : forall n i j,
step {| Pc := Loop; Vars := {| N := S n;
I := i;
J := j |} |}
{| Pc := i_add_n; Vars := {| N := S n;
I := i;
J := j |} |}
| Step_i_add_n : forall n i j,
step {| Pc := i_add_n; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := j_add_n; Vars := {| N := n;
I := i + n;
J := j |} |}
| Step_j_add_n : forall n i j,
step {| Pc := j_add_n; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := n_sub_1; Vars := {| N := n;
I := i;
J := j + n |} |}
| Step_n_sub_1 : forall n i j,
step {| Pc := n_sub_1; Vars := {| N := n;
I := i;
J := j |} |}
{| Pc := Loop; Vars := {| N := n - 1;
I := i;
J := j |} |}.
Definition loopy_sys := {|
Initial := initial;
Step := step
|}.
Definition loopy_correct (st : state) :=
st.(Pc) = Done -> st.(Vars).(I) = st.(Vars).(J).
(* Spoiler alert: here's a good abstraction! *)
Inductive absvars :=
| Unknown
(* We don't know anything about the values of the variables. *)
| i_is_0
(* We know [i == 0]. *)
| i_eq_j
(* We know [i == j]. *)
| i_eq_j_plus_n.
(* We know [i == j + n]. *)
(* To get our abstract states, we keep the same program counters and just change
* out the variable state. *)
Record absstate := {
APc : pc;
AVars : absvars
}.
(* Here's the rather boring new abstract step relation. Note the clever state
* transformations, in terms of our new abstraction. *)
Inductive absstep : absstate -> absstate -> Prop :=
| AStep_i_gets_0 : forall vs,
absstep {| APc := i_gets_0; AVars := vs |}
{| APc := j_gets_0; AVars := i_is_0 |}
| AStep_j_gets_0_i_is_0 :
absstep {| APc := j_gets_0; AVars := i_is_0 |}
{| APc := Loop; AVars := i_eq_j |}
| AStep_j_gets_0_Other : forall vs,
vs <> i_is_0
-> absstep {| APc := j_gets_0; AVars := vs |}
{| APc := Loop; AVars := Unknown |}
| AStep_Loop_done : forall vs,
absstep {| APc := Loop; AVars := vs |}
{| APc := Done; AVars := vs |}
| AStep_Loop_enter : forall vs,
absstep {| APc := Loop; AVars := vs |}
{| APc := i_add_n; AVars := vs |}
| AStep_i_add_n_i_eq_j :
absstep {| APc := i_add_n; AVars := i_eq_j |}
{| APc := j_add_n; AVars := i_eq_j_plus_n |}
| AStep_i_add_n_Other : forall vs,
vs <> i_eq_j
-> absstep {| APc := i_add_n; AVars := vs |}
{| APc := j_add_n; AVars := Unknown |}
| AStep_j_add_n_i_eq_j_plus_n :
absstep {| APc := j_add_n; AVars := i_eq_j_plus_n |}
{| APc := n_sub_1; AVars := i_eq_j |}
| AStep_j_add_n_i_Other : forall vs,
vs <> i_eq_j_plus_n
-> absstep {| APc := j_add_n; AVars := vs |}
{| APc := n_sub_1; AVars := Unknown |}
| AStep_n_sub_1_bad :
absstep {| APc := n_sub_1; AVars := i_eq_j_plus_n |}
{| APc := Loop; AVars := Unknown |}
| AStep_n_sub_1_good : forall vs,
vs <> i_eq_j_plus_n
-> absstep {| APc := n_sub_1; AVars := vs |}
{| APc := Loop; AVars := vs |}.
Definition absloopy_sys := {|
Initial := { {| APc := i_gets_0; AVars := Unknown |} };
Step := absstep
|}.
(* Now we need our simulation relation. First, we define one just at the level
* of local-variable state. It formalizes our intuition about those values. *)
Inductive Rvars : vars -> absvars -> Prop :=
| Rv_Unknown : forall vs, Rvars vs Unknown
| Rv_i_is_0 : forall vs, vs.(I) = 0 -> Rvars vs i_is_0
| Rv_i_eq_j : forall vs, vs.(I) = vs.(J) -> Rvars vs i_eq_j
| Rv_i_eq_j_plus_n : forall vs, vs.(I) = vs.(J) + vs.(N) -> Rvars vs i_eq_j_plus_n.
(* We lift to full states in the obvious way. *)
Inductive R : state -> absstate -> Prop :=
| Rcon : forall pc vs avs, Rvars vs avs -> R {| Pc := pc; Vars := vs |}
{| APc := pc; AVars := avs |}.
(* Now we are ready to prove the original system correct. *)
Theorem loopy_ok :
invariantFor loopy_sys loopy_correct.
Proof.
eapply invariant_weaken with (invariant1 := invariantViaSimulation R _).
apply invariant_simulates with (sys2 := absloopy_sys).
(* Here comes another boring simulation proof. *)
constructor; simplify.
invert H.
exists {| APc := i_gets_0; AVars := Unknown |}.
propositional.
constructor.
constructor.
invert H0.
invert H.
exists {| APc := j_gets_0; AVars := i_is_0 |}.
propositional; repeat constructor.
invert H.
invert H3.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := i_eq_j |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
exists {| APc := Done; AVars := st2.(AVars) |}.
invert H; simplify; propositional; repeat constructor; equality.
exists {| APc := i_add_n; AVars := st2.(AVars) |}.
invert H; simplify; propositional; repeat constructor; equality.
invert H.
invert H3.
exists {| APc := j_add_n; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := j_add_n; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := j_add_n; AVars := i_eq_j_plus_n |}; repeat constructor; simplify; equality.
exists {| APc := j_add_n; AVars := Unknown |}; repeat constructor; equality.
invert H.
invert H3.
exists {| APc := n_sub_1; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := n_sub_1; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := n_sub_1; AVars := Unknown |}; repeat constructor; equality.
exists {| APc := n_sub_1; AVars := i_eq_j |}; repeat constructor; simplify; equality.
invert H.
invert H3.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := i_is_0 |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := i_eq_j |}; propositional; repeat constructor; equality.
exists {| APc := Loop; AVars := Unknown |}; propositional; repeat constructor; equality.
(* Finally, we can call the model checker to find an invariant of the abstract
* system. *)
model_check_infer.
(* We get 7 neat little states, one per program counter. Next, we prove that
* each of them implies the original invariant. *)
invert 1. (* Note that this [1] means "first premise below the double
* line." *)
invert H0.
unfold loopy_correct.
simplify.
propositional; subst.
(* Most of the hypotheses we invert are contradictory, implying that distinct
* program counters are equal. *)
invert H2.
invert H1.
invert H2.
invert H1.
invert H.
assumption.
invert H2.
invert H1.
invert H2.
Qed.
(** * Modularity *)
Inductive stepWithInterference shared private (inv : shared -> Prop)
(step : threaded_state shared private -> threaded_state shared private -> Prop)
: threaded_state shared private -> threaded_state shared private -> Prop :=
(* First kind of step: this thread runs in the normal way. *)
| StepSelf : forall st st',
step st st'
-> stepWithInterference inv step st st'
(* Second kind of step: other threads change shared state to some new value
* satisfying [inv]. *)
| StepEnvironment : forall sh pr sh',
inv sh'
-> stepWithInterference inv step
{| Shared := sh; Private := pr |}
{| Shared := sh'; Private := pr |}.
(* Via this relation, we have an operator to build a new transition system from
* an old one, given [inv]. *)
Definition withInterference shared private (inv : shared -> Prop)
(sys : trsys (threaded_state shared private))
: trsys (threaded_state shared private) := {|
Initial := sys.(Initial);
Step := stepWithInterference inv sys.(Step)
|}.
(* Tired of simulation proofs yet? Then you'll love this theorem, which shows
* a free simulation for any use of [withInterference]! We even get to pick the
* trivial simulation relation, state equality. *)
Theorem withInterference_abstracts : forall shared private (inv : shared -> Prop)
(sys : trsys (threaded_state shared private)),
simulates (fun st st' => st = st') sys (withInterference inv sys).
Proof.
simplify.
constructor; simplify.
exists st1; propositional.
exists st1'; propositional.
constructor.
equality.
Qed.
Lemma withInterference_parallel_init : forall shared private1 private2
(invs : shared -> Prop)
(sys1 : trsys (threaded_state shared private1))
(sys2 : trsys (threaded_state shared private2))
st st',
(withInterference invs (parallel sys1 sys2)).(Step)^* st st'
-> forall st1 st2,
(forall st1', (withInterference invs sys1).(Step)^* st1 st1' -> invs st1'.(Shared))
-> (forall st2', (withInterference invs sys2).(Step)^* st2 st2' -> invs st2'.(Shared))
-> (withInterference invs sys1).(Step)^* st1
{| Shared := st.(Shared);
Private := fst st.(Private) |}
-> (withInterference invs sys2).(Step)^* st2
{| Shared := st.(Shared);
Private := snd st.(Private) |}
-> (withInterference invs sys1).(Step)^* st1
{| Shared := st'.(Shared);
Private := fst st'.(Private) |}.
Proof.
induct 1; simplify.
assumption.
invert H; simplify.
invert H5; simplify.
apply IHtrc with (st2 := {| Shared := sh'; Private := pr2 |}).
simplify.
apply H1.
assumption.
simplify.
eapply H2.
eapply trc_trans.
eassumption.
eapply TrcFront.
apply StepEnvironment with (sh' := sh').
apply H1 with (st1' := {| Shared := sh'; Private := pr1' |}).
eapply trc_trans.
eassumption.
eapply TrcFront.
econstructor.
eassumption.
constructor.
assumption.
eapply trc_trans.
eassumption.
eapply TrcFront.
econstructor.
eassumption.
constructor.
constructor.
apply IHtrc with (st2 := {| Shared := sh'; Private := pr2' |}).
assumption.
simplify.
apply H2.
eapply trc_trans.
eassumption.
eapply TrcFront.
constructor.
eassumption.
eassumption.
eapply trc_trans.
eassumption.
eapply TrcFront.
apply StepEnvironment with (sh' := sh').
apply H2 with (st2' := {| Shared := sh'; Private := pr2' |}).
eapply trc_trans.
eassumption.
eapply TrcFront.
econstructor.
eassumption.
constructor.
constructor.
constructor.
apply IHtrc with (st2 := {| Shared := sh'; Private := snd pr |}).
assumption.
simplify.
eapply H2.
eapply trc_trans.
eassumption.
eapply TrcFront.
apply StepEnvironment with (sh' := sh').
assumption.
assumption.
eapply trc_trans.
eassumption.
eapply TrcFront.
apply StepEnvironment with (sh' := sh').
assumption.
constructor.
constructor.
Qed.
Lemma withInterference_parallel_step : forall shared private1 private2
(invs : shared -> Prop)
(sys1 : trsys (threaded_state shared private1))
(sys2 : trsys (threaded_state shared private2))
st st',
(withInterference invs (parallel sys1 sys2)).(Step)^* st st'
-> forall st1 st2,
(forall st1', (withInterference invs sys1).(Step)^* st1 st1' -> invs st1'.(Shared))
-> (forall st2', (withInterference invs sys2).(Step)^* st2 st2' -> invs st2'.(Shared))
-> (withInterference invs sys1).(Step)^* st1
{| Shared := st.(Shared);
Private := fst st.(Private) |}
-> (withInterference invs sys2).(Step)^* st2
{| Shared := st.(Shared);
Private := snd st.(Private) |}
-> (withInterference invs sys2).(Step)^* st2
{| Shared := st'.(Shared);
Private := snd st'.(Private) |}.
Proof.
induct 1; simplify.
assumption.
invert H; simplify.
invert H5; simplify.
apply IHtrc with (st1 := {| Shared := sh'; Private := pr1' |}).
simplify.
apply H1.
eapply trc_trans.
eassumption.
eapply TrcFront.
econstructor.
eassumption.
assumption.
assumption.
constructor.
eapply trc_trans.
eassumption.
eapply TrcFront.
apply StepEnvironment with (sh' := sh').
apply H1 with (st1' := {| Shared := sh'; Private := pr1' |}).
eapply trc_trans.
eassumption.
eapply TrcFront.
econstructor.
eassumption.
constructor.
constructor.
apply IHtrc with (st1 := {| Shared := sh'; Private := pr1 |}).
simplify.
apply H1.
eapply trc_trans.
eassumption.
eapply TrcFront.
apply StepEnvironment with (sh' := sh').
apply H2 with (st2' := {| Shared := sh'; Private := pr2' |}).
eapply trc_trans.
eassumption.
eapply TrcFront.
constructor.
eassumption.
constructor.
assumption.
assumption.
constructor.
eapply trc_trans.
eassumption.
eapply TrcFront.
constructor.
eassumption.
constructor.
apply IHtrc with (st1 := {| Shared := sh'; Private := fst pr |}).
simplify.
eapply H1.
eapply trc_trans.
eassumption.
eapply TrcFront.
apply StepEnvironment with (sh' := sh').
assumption.
assumption.
assumption.
constructor.
eapply trc_trans.
eassumption.
eapply TrcFront.
apply StepEnvironment with (sh' := sh').
assumption.
constructor.
Qed.
Theorem withInterference_parallel : forall shared private1 private2
(invs : shared -> Prop)
(sys1 : trsys (threaded_state shared private1))
(sys2 : trsys (threaded_state shared private2)),
invariantFor (withInterference invs sys1)
(fun st => invs st.(Shared))
-> invariantFor (withInterference invs sys2)
(fun st => invs st.(Shared))
-> invariantFor (withInterference invs (parallel sys1 sys2))
(fun st => invs st.(Shared)).
Proof.
unfold invariantFor.
simplify.
invert H1.
(* [assert P]: first prove proposition [P], then continue with it as a new
* hypothesis. *)
assert ((withInterference invs sys1).(Step)^*
{| Shared := sh; Private := pr1 |}
{| Shared := s'.(Shared); Private := fst s'.(Private) |}).
apply withInterference_parallel_init with (sys2 := sys2)
(st := {| Shared := sh; Private := (pr1, pr2) |})
(st2 := {| Shared := sh; Private := pr2 |});
simplify; propositional.
apply H in H1; propositional.
apply H0 in H1; propositional.
constructor.
constructor.
assert ((withInterference invs sys2).(Step)^*
{| Shared := sh; Private := pr2 |}
{| Shared := s'.(Shared); Private := snd s'.(Private) |}).
apply withInterference_parallel_step with (sys1 := sys1)
(st := {| Shared := sh; Private := (pr1, pr2) |})
(st1 := {| Shared := sh; Private := pr1 |});
simplify; propositional.
apply H in H5; propositional.
apply H0 in H5; propositional.
constructor.
constructor.
apply H in H1; try assumption.
Qed.
Inductive twoadd_pc := ReadIt | Add3 | Add7 | WriteIt.
Definition twoadd_initial := { {| Shared := 0; Private := (ReadIt, 0) |} }.
Inductive twoadd_step : threaded_state nat (twoadd_pc * nat)
-> threaded_state nat (twoadd_pc * nat) -> Prop :=
| Step_ReadIt : forall g l,
twoadd_step {| Shared := g; Private := (ReadIt, l) |}
{| Shared := g; Private := (Add3, g) |}
| Step_Add3 : forall g l,
twoadd_step {| Shared := g; Private := (Add3, l) |}
{| Shared := g; Private := (Add7, 3 + l) |}
| Step_Add7 : forall g l,
twoadd_step {| Shared := g; Private := (Add7, l) |}
{| Shared := g; Private := (WriteIt, 7 + l) |}
| Step_WriteIt : forall g l,
twoadd_step {| Shared := g; Private := (WriteIt, l) |}
{| Shared := l; Private := (ReadIt, l) |}.
Definition twoadd_sys := {|
Initial := twoadd_initial;
Step := twoadd_step
|}.
(* Invariant to prove: the global variable is always even, again. *)
Definition twoadd_correct private (st : threaded_state nat private) :=
isEven st.(Shared).
(* Here's an abstract version of the system where, much like before, we model
* integers as Booleans, recording whether they are even or not. *)
Definition twoadd_ainitial := { {| Shared := true; Private := (ReadIt, true) |} }.
Inductive twoadd_astep : threaded_state bool (twoadd_pc * bool)
-> threaded_state bool (twoadd_pc * bool) -> Prop :=
| AStep_ReadIt : forall g l,
twoadd_astep {| Shared := g; Private := (ReadIt, l) |}
{| Shared := g; Private := (Add3, g) |}
| AStep_Add3 : forall g l,
twoadd_astep {| Shared := g; Private := (Add3, l) |}
{| Shared := g; Private := (Add7, negb l) |}
| AStep_Add7 : forall g l,
twoadd_astep {| Shared := g; Private := (Add7, l) |}
{| Shared := g; Private := (WriteIt, negb l) |}
| AStep_WriteIt : forall g l,
twoadd_astep {| Shared := g; Private := (WriteIt, l) |}
{| Shared := l; Private := (ReadIt, l) |}
| AStep_Someone_Made_It_Even : forall g pr,
twoadd_astep {| Shared := g; Private := pr |}
{| Shared := true; Private := pr |}.
Definition twoadd_asys := {|
Initial := twoadd_ainitial;
Step := twoadd_astep
|}.
(* Here's a simulation relation at the level of integers and their Boolean
* counterparts. *)
Definition even_R (n : nat) (b : bool) :=
isEven n <-> b = true.
(* A few unsurprising properties hold of [even_R]. *)
Lemma even_R_0 : even_R 0 true.
Proof.
unfold even_R; propositional.
constructor.
Qed.
Lemma even_R_forward : forall n, isEven n -> even_R n true.
Proof.
unfold even_R; propositional.
Qed.
Lemma even_R_backward : forall n, even_R n true -> isEven n.
Proof.
unfold even_R; propositional.
Qed.
Lemma even_R_add2 : forall n b,
even_R n b
-> even_R (S (S n)) b.
Proof.
unfold even_R; propositional.
invert H; propositional.
constructor; assumption.
Qed.
(* The cases for evenness of an integer and its successor *)
Lemma isEven_decide : forall n,
(isEven n /\ ~isEven (S n)) \/ (~isEven n /\ isEven (S n)).
Proof.
induct n; simplify; propositional.
left; propositional.
constructor.
invert H.
right; propositional.
constructor; assumption.
left; propositional.
invert H.
propositional.
Qed.
Lemma even_R_add1 : forall n b,
even_R n b
-> even_R (S n) (negb b).
Proof.
unfold even_R; simplify.
assert ((isEven n /\ ~isEven (S n)) \/ (~isEven n /\ isEven (S n))).
apply isEven_decide.
cases b; simplify; propositional.
equality.
equality.
Qed.
(* Here's the top-level simulation relation for our choice of abstraction. *)
Inductive twoadd_R : threaded_state nat (twoadd_pc * nat)
-> threaded_state bool (twoadd_pc * bool) -> Prop :=
| Twoadd_R : forall pc gn ln gb lb,
even_R gn gb
-> even_R ln lb
-> twoadd_R {| Shared := gn; Private := (pc, ln) |}
{| Shared := gb; Private := (pc, lb) |}.
(* Step 1 of main proof: model-check an individual thread. *)
Lemma twoadd_ok :
invariantFor (withInterference isEven twoadd_sys)
(fun st => isEven (Shared st)).
Proof.
eapply invariant_weaken.
apply invariant_simulates with (sys2 := twoadd_asys) (R := twoadd_R).
(* Boring simulation proof begins here. *)
constructor; simplify.
invert H.
exists {| Shared := true; Private := (ReadIt, true) |}; propositional.
constructor; propositional.
apply even_R_0.
apply even_R_0.
constructor.
equality.
simplify.
propositional.
invert H0.
invert H1.
invert H.
exists {| Shared := gb; Private := (Add3, gb) |}; propositional.
constructor; propositional.
constructor.
invert H.
exists {| Shared := gb; Private := (Add7, negb lb) |}; propositional.
constructor; propositional.
apply even_R_add2.
apply even_R_add1.
assumption.
constructor.
invert H.
exists {| Shared := gb; Private := (WriteIt, negb lb) |}; propositional.
constructor; propositional.
repeat apply even_R_add2.
apply even_R_add1.
assumption.
constructor.
invert H.
exists {| Shared := lb; Private := (ReadIt, lb) |}; propositional.
constructor; propositional.
constructor.
invert H.
exists {| Shared := true; Private := (pc0, lb) |}; propositional.
constructor; propositional.
apply even_R_forward.
assumption.
constructor.
(* Now find an invariant automatically. *)
model_check_infer.
(* Now prove that the invariant implies the correctness condition. *)
invert 1.
invert H0.
simplify.
propositional.
invert H0.
apply even_R_backward.
assumption.
invert H1.
apply even_R_backward.
assumption.
invert H0.
apply even_R_backward.
assumption.
invert H1.
apply even_R_backward.
assumption.
Qed.
(* Step 2: lift that result to the two-thread system, with no new model
* checking. *)
Theorem twoadd2_ok :
invariantFor (parallel twoadd_sys twoadd_sys) (twoadd_correct (private := _)).
Proof.
eapply invariant_weaken.
eapply invariant_simulates.
apply withInterference_abstracts.
apply withInterference_parallel.
apply twoadd_ok.
apply twoadd_ok.
unfold twoadd_correct.
invert 1.
assumption.
Qed.
(* In fact, this modularity technique is so powerful that we now get correctness
* for any number of threads, "for free"! Here's a tactic definition, which we
* won't explain, but which is able to derive correctness for any number of
* threads, just by repeating use of [withInterference_parallel] and
* [twoadd_ok]. *)
Ltac twoadd := eapply invariant_weaken; [ eapply invariant_simulates; [
apply withInterference_abstracts
| repeat (apply withInterference_parallel
|| apply twoadd_ok) ]
| unfold twoadd_correct; invert 1; assumption ].
(* For instance, let's verify the three-thread version. *)
Theorem twoadd3_ok :
invariantFor (parallel twoadd_sys (parallel twoadd_sys twoadd_sys)) (twoadd_correct (private := _)).
Proof.
twoadd.
Qed.
(* To save us time defining versions with many threads, here's a recursive
* function, creating exponentially many threads with respect to its
* parameter. *)
Fixpoint manyadds_state (n : nat) : Type :=
match n with
| O => twoadd_pc * nat
| S n' => manyadds_state n' * manyadds_state n'
end%type.
Fixpoint manyadds (n : nat) : trsys (threaded_state nat (manyadds_state n)) :=
match n with
| O => twoadd_sys
| S n' => parallel (manyadds n') (manyadds n')
end.
(* Here are some examples of the systems we produce. *)
Eval simpl in manyadds 0.
Eval simpl in manyadds 1.
Eval simpl in manyadds 2.
Eval simpl in manyadds 3.
Theorem twoadd4_ok :
invariantFor (manyadds 4) (twoadd_correct (private := _)).
Proof.
twoadd.
Qed.
Theorem twoadd6_ok :
invariantFor (manyadds 6) (twoadd_correct (private := _)).
Proof.
twoadd.
Qed.
|
Formal statement is: lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z" Informal statement is: The real part of a real number is the number itself. |
import Mt.Reservation
import Mt.Task.Basic
namespace Mt.TaskM
variable {spec : Spec}
local instance : IsReservation spec.Reservation :=spec.is_reservation
/-- Validation primitive for reasoning for composable tasks.
Validation may assume `assuming` and must ensure the following:
* The task has to behave conform to the specification at all times
* The task never panics
* Finally, `motive` holds after the task completes
### Proving `valid`
The definition is rather cumbersome to work with. You should use
helper theorems like `valid_pure`, `valid_bind`, `valid_rmr`, ...
They are designed to be used with the `apply` tactic.
### Blocking predicate: `assuming`
When validating our task, we can assume `assuming state = true`. However,
in most cases we have `assuming = λ _ => true`, i.e. our hypothesis does
not provide anything useful.
There is one important exception: Blocking threads. If a thread
waits for a certain condition before it continues its task, we can
safely assume that this condition holds when the task is excuting.
### Final goal: `motive`
A valid thread must drop its reservations in the end. Therefore, the
final goal on those tasks is `λ _ r => r = IsReservation.empty`.
However, intermediate tasks (i.e. single operations) do not need to
share this goal.
In fact, they usally do not. If one operation
prepares the next operation, it usually creates some reservation to
ensure that no other thread undos this preparation. In this example,
the motive should encode that the preparation has been made.
`motive` is the only way to pass facts from one iteration to the next.
See `valid_bind` for more information.
-/
def valid {T : Type} (p : TaskM spec T) (r : spec.Reservation)
(assuming : spec.State -> Bool)
(motive : T -> spec.Reservation -> Prop)
: Prop :=∀ env_r s,
assuming s →
spec.validate (env_r + r) s → ∃ r' : spec.Reservation,
match h : p.iterate s with
| IterationResult.Done s' t => spec.validate (env_r + r') s' ∧ motive t r'
| IterationResult.Panic .. => False
| IterationResult.Running s' block_until cont =>
spec.validate (env_r + r') s' ∧
cont.valid r' block_until motive
termination_by valid => p
decreasing_by simp_wf ; exact is_direct_cont.running h
/-- To prove that `pure t` is valid you need to prove that the `motive` holds -/
theorem valid_pure {T : Type} {t : T} {r assuming motive}
(is_valid : motive t r)
: valid (spec :=spec) (pure t) r assuming motive :=by
rw [valid]
intro env_r s _ initial_valid
exists r
/-- To prove that `a >>= f` is valid you need to prove that both `a` and `f u`
for all results `u` are valid.
In many cases, `a` does something to prepare `f u`. Since only `f u` needs
to fulfil the final motive, we can choose an arbitrary motive to validate `a`
as "intermediate goal".
To validate `f u` with the original `motive`, we can use the fact that the
intermediate goal `motive_u` has been ensured by `a`. Motives use only results
and reservations, which cannot be changed by other threads. Therefore, they
stay valid even if other threads become active between `a` and `f u`.
-/
theorem valid_bind {U V : Type}
{mu : TaskM spec U}
{f : U -> TaskM spec V}
{r assuming motive}
(motive_u : U -> spec.Reservation -> Prop)
(mu_valid : mu.valid r assuming motive_u)
(f_valid : ∀ r' u,
motive_u u r' →
(f u).valid r' (λ _ => true) motive)
: valid (mu >>= f) r assuming motive :=by
rw [valid]
intro env_r s assuming_true initial_valid
rw [iterate_bind]
rw [valid] at mu_valid
cases iteration : iterate mu s
all_goals (
simp only []
have mu_valid :=mu_valid env_r s assuming_true initial_valid
rw [iteration] at mu_valid
simp only [] at mu_valid
cases mu_valid ; rename_i r' mu_valid
exists r'
)
. exact ⟨mu_valid.left, f_valid _ _ mu_valid.right⟩
. constructor
. exact mu_valid.left
. have :=is_direct_cont.running iteration
exact valid_bind motive_u mu_valid.right f_valid
termination_by valid_bind => mu
theorem valid_rmr {T : Type}
{f : spec.State -> T × spec.State}
{r assuming motive}
(f_valid : ∀ env_r s,
assuming s →
spec.validate (env_r + r) s → ∃ r' : spec.Reservation,
match f s with
| ⟨t, s'⟩ => spec.validate (env_r + r') s' ∧ motive t r'
)
: (atomic_read_modify_read f).valid r assuming motive :=by
rw [valid]
intro env_r s assuming_true initial_valid
rw [iterate_rmr]
exact f_valid env_r s assuming_true initial_valid
theorem valid_rm
{f : spec.State -> spec.State}
{r assuming motive}
(f_valid : ∀ env_r s,
assuming s →
spec.validate (env_r + r) s → ∃ r' : spec.Reservation,
match f s with
| s' => spec.validate (env_r + r') s' ∧ motive ⟨⟩ r')
: (atomic_read_modify f).valid r assuming motive :=valid_rmr f_valid
theorem valid_read
{f : spec.State -> T}
{r assuming motive}
(f_valid : ∀ env_r s,
assuming s →
spec.validate (env_r + r) s → ∃ r' : spec.Reservation,
match f s with
| t => spec.validate (env_r + r') s ∧ motive t r')
: (atomic_read f).valid r assuming motive :=valid_rmr f_valid
theorem valid_assert
{cond : spec.State -> Bool}
{r assuming motive}
(motive_holds : motive ⟨⟩ r)
(assertion_succeeds : ∀ env_r s,
assuming s →
spec.validate (env_r + r) s →
cond s)
: (atomic_assert cond).valid r assuming motive :=by
rw [valid]
intro env_r s assuming_true initial_valid
rw [iterate_assert]
have cond_true :=assertion_succeeds env_r s assuming_true initial_valid
rw [cond_true]
exists r
theorem valid_blocking_rmr
{block_until : spec.State -> Bool}
{f : spec.State -> T × spec.State}
{r assuming motive}
(f_valid : ∀ env_r s,
block_until s →
spec.validate (env_r + r) s → ∃ r' : spec.Reservation,
match f s with
| ⟨t, s'⟩ => spec.validate (env_r + r') s' ∧ motive t r'
)
: (atomic_blocking_rmr block_until f).valid r assuming motive :=by
rw [valid]
intro env_r s _ initial_valid
simp only [iterate_blocking_rmr, initial_valid, true_and]
exists r
exact ⟨initial_valid, valid_rmr f_valid⟩
end Mt.TaskM |
The Galveston County Daily News , founded in 1842 , is the city 's primary newspaper and the oldest continuously printed newspaper in Texas . It currently serves as the newspaper of record for the city and the Texas City Post serves as the newspaper of record for the County . Radio station <unk> , on air from 1947 @-@ 2010 , has previously served as a local media outlet . Television station KHOU signed on the air as KGUL @-@ TV on March 23 , 1953 . Originally licensed in Galveston , KGUL was the second television station to launch in the Houston area after <unk> @-@ TV . One of the original investors in the station was actor James Stewart , along with a small group of other Galveston investors . In June 1959 , KGUL changed its call sign to KHOU and moved their main office to Houston . The local hip hop name for Galveston is " G @-@ town . "
|
data ⊥ : Set where -- keep this on the first line!!
Type = Set -- This should not be gobbled up under the data declaration
-- Andreas, 2021-04-15, issue #1145
-- Allow stacking of layout keywords on the same line.
-- This files contains tests for the new layout rules
-- and also tests from debugging the original implementation.
--
-- Some original failures where influenced by whether
-- there were comments or black lines, so please don't
-- "clean up" this file.
postulate A : Set; B : Set
C : Set
-- First example
private postulate -- Bar
D : Set
_>>_ : Set → Set → Set
module _ where
private {- this block comment
removes the line breaks
between the layout keywords -} postulate
D1 : Set
E1 : Set
-- Second example
private module PM where -- stack
module PMM where
private module PMPM1 where
module PMPM2 where private -- stack
module PMPM2M where
module PMM2 where
-- Testing whether do-notation still works
test = do A; B
C
let-example =
let X = A
in B
do-example = do
Set
where
Y : Set
Y = A
F = Set
private
doey = do
Set
where postulate poey : Set
E = Set
|
/-
Copyright 2022 Google LLC
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
Authors: Moritz Firsching
-/
import tactic
/-!
# Pigeon-hole and double counting
## TODO
- statement
- 1. Numbers
- Claim
- Claim
- 2. Sequences
- Claim
- proof
- 3. Sums
- Claim
- Double Counting
- 4. Numbers again
- 5. Graphs
- Theorem
- proof
- Claim
- 6. Sperner's Lemma
- proof
- Proof of Brouwer's fixed point theorem (for $n = 2$)
-/
|
# Get dtm data ------------------------------------------------------------
if (file.exists("data/fal_school_dtm.tif")) {
fal_school = raster("data/fal_school_dtm.tif")
} else {
#Get lidar data
fal_school = get_from_xy(
xy = c(179011, 033245),
radius = 500,
resolution = 0.5,
model_type = 'DSM'
)
}
# Get bike route data -----------------------------------------------------
p = read_sf("data/full_track.gpx", layer = "tracks")
pp = read_sf("data/full_track.gpx", layer = "track_points") # track points
resting_area = sf::read_sf("data/Resting-Area/Resting-area.shp")
|
import combinatorics.simple_graph.coloring
--OUTPUT 1
theorem graph_2_colorable_iff_bipartite : ∀ G : graph, (∃ c : coloring G 2, @is_valid_coloring G c) ↔ is_bipartite G :=
begin
-- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue,
-- and no edge will have both endpoints colored the same color.
assume (G : graph),
have h1 : ∀ c : coloring G 2, (@is_valid_coloring G c), from
by {
assume (c : coloring G 2),
sorry,
},
-- Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.
have h2 : ∀ c : coloring G 2, (@is_valid_coloring G c) →
(∃ A : set (vertex G), ∃ B : set (vertex G),
(∀ v : vertex G, c v = red ↔ v ∈ A) ∧ (∀ v : vertex G, c v = blue ↔ v ∈ B)), from
by {
assume (c : coloring G 2) (h2 : @is_valid_coloring G c),
sorry,
},
-- $A$ and $B$ are subsets of vertics of $G$, by definition of valid coloring.
have h3 : ∀ c : coloring G 2, (@is_valid_coloring G c) →
(∃ A : set (vertex G), ∃ B : set (vertex G),
(∀ v : vertex G, c v = red ↔ v ∈ A) ∧ (∀ v : vertex G, c v = blue ↔ v ∈ B) ∧
A ⊆ vertex G ∧ B ⊆ vertex G), from
by {
assume (c : coloring G 2) (h3 : @is_valid_coloring G c),
sorry,
},
-- Since all vertices of $A$ are red, there are no edges within $A$,
-- and similarly for $B$.
have h4 : ∀ c : coloring G 2, (@is_valid_coloring G c) →
(∃ A : set (vertex G), ∃ B : set (vertex G),
(∀ v : vertex G, c v = red ↔ v ∈ A) ∧ (∀ v : vertex G, c v = blue ↔ v ∈ B) ∧
A ⊆ vertex G ∧ B ⊆ vertex G ∧ ∀ e : edge G, (source e ∈ A ∧ target e ∈ A) ∨ (source e ∈ B ∧ target e ∈ B)), from
by {
assume (c : coloring G 2) (h4 : @is_valid_coloring G c),
sorry,
},
-- This implies that every edge has one endpoint in $A$ and the other in $B$,
-- which means $G$ is bipartite.
have h5 : ∀ c : coloring G 2, (@is_valid_coloring G c) →
(∃ A : set (vertex G), ∃ B : set (vertex G),
(∀ v : vertex G, c v = red ↔ v ∈ A) ∧ (∀ v : vertex G, c v = blue ↔ v ∈ B) ∧
A ⊆ vertex G ∧ B ⊆ vertex G ∧ (∀ e : edge G, source e ∈ A ∨ source e ∈ B) ∧ (∀ e : edge G, target e ∈ A ∨ target e ∈ B)), from
by {
assume (c : coloring G 2) (h5 : @is_valid_coloring G c),
sorry,
},
-- Hence the existence of a valid coloring implies that the graph is bipartite.
have h6 : ∀ c : coloring G 2, (@is_valid_coloring G c) → (∃ A : set (vertex G), ∃ B : set (vertex G),
(∀ v : vertex G, c v = red ↔ v ∈ A) ∧ (∀ v : vertex G, c v = blue ↔ v ∈ B) ∧
A ⊆ vertex G ∧ B ⊆ vertex G ∧ A ⊥ B ∧ A ∪ B = vertex G),
from
by {
assume (c : coloring G 2) (h6 : @is_valid_coloring G c),
sorry,
},
-- Conversely,
have h7 : ∀ A B : set (vertex G), A ⊥ B → A ⊆ vertex G ∧ B ⊆ vertex G → A ∪ B = vertex G →
(∃ c : coloring G 2, @is_valid_coloring G c ∧
(∀ v : vertex G, c v = red ↔ v ∈ A) ∧ (∀ v : vertex G, c v = blue ↔ v ∈ B)), from
by {
assume (A B : set (vertex G)) (h7 : A ⊥ B) (h8 : A ⊆ vertex G ∧ B ⊆ vertex G) (h9 : A ∪ B = vertex G),
sorry,
},
-- Suppose $G$ is bipartite,
have h8 : ∀ A B : set (vertex G), A ⊥ B → A ⊆ vertex G ∧ B ⊆ vertex G → A ∪ B = vertex G →
is_bipartite G, from
by {
assume (A B : set (vertex G)) (h8 : A ⊥ B) (h9 : A ⊆ vertex G ∧ B ⊆ vertex G) (h10 : A ∪ B = vertex G),
sorry,
},
-- Hence the existence of a valid coloring is equivalent to the graph being bipartite.
sorry,
end
--OUTPUT 2
theorem two_colorable_bipartite {G : Type*} : two_colorable G ↔ bipartite G := sorry
/--`theorem`
Positive Real Numbers are Closed Under Multiplication
Let $x, y$ be positive real numbers. Then $x y$ is a positive real number.
`proof`
Let $x, y$ be positive real numbers, i.e. $\exists z \in \R : z > 0$. Now,
{{begin-eqn}}
{{eqn | l = z \times z
| r = z \times z + z \times z
| c = Real numbers can be multiplied
}}
{{eqn | r = z \times z
| c = Real numbers are commutative with multiplication
}}
{{eqn | r = z^{2}
| c = Real numbers exponentiation is multiplication
}}
{{eqn | r = x^{2}
| c = Real numbers are equal if the difference is zero
}}
{{eqn | r = y^{2}
| c = Real numbers are equal if the difference is zero
}}
{{eqn | r = (x y)^{2}
| c = Real numbers exponentiation is multiplication
}}
{{eqn | r = (x y) \times (x y)
| c = Real numbers exponentiation is multiplication
}}
{{eqn | r = x \times (x y)
| c = Real numbers are commutative with multiplication
}}
{{eqn | r = x \times y
| c = Real numbers are commutative with multiplication
}}
{{end-eqn}
Note that we've shown that $z > 0 \implies z^{2} > 0$.
{{qed}}
-/
theorem positive_real_multiply_closed : ∀ x y : ℝ, x > 0 → y > 0 → x*y > 0 :=
begin
-- Let x, y be positive real numbers
assume (x y : ℝ) (hx : x > 0) (hy : y > 0),
-- Then $x y$ is a positive real number
have h1 : x * y > 0, from sorry,
show x * y > 0, from h1,
end
--OUTPUT 3
theorem bipartite_colorable (G : Type*) [graph G] :
∀ red blue,
∀ v : G, v ∈ red ∨ v ∈ blue :=
begin
sorry,
end
--OUTPUT 4
theorem two_colorable_bipartite (G : Type*) : sorry := sorry
/-
jarrick
`theorem`
`proof`
-/
/--`theorem`
`proof`
-/
/--`theorem`
`proof`
-/
/--`theorem`
`proof`
-/
/--`theorem`
`proof`
-/
/--`theorem`
`proof`
-/
/--`theorem`
`proof`
-/
end
--OUTPUT 5
theorem bipartite_graph_two_colorable {G : Type*} [graph G] : (∃ c : G → ℕ, c valid) ↔ is_bipartite G :=
begin
have h1 : ∃ c : G → ℕ, c valid → is_bipartite G, from sorry,
have h2 : is_bipartite G → ∃ c : G → ℕ, c valid, from sorry,
show (∃ c : G → ℕ, c valid) ↔ is_bipartite G, from sorry,
end
--OUTPUT 6
theorem bipartite_graph_colorable {G : Type*} [graph G] : bipartite G ↔ 2-colorable G :=
begin
-- Let G be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color
have h1 : 2-colorable G → ∀ a b : G, ∃! v : G, (a, v) ∈ edges G ∨ (v, a) ∈ edges G, from sorry,
-- Let A denote the subset of vertices colored red, and let B denote the subset of vertices colored blue
have h2 : 2-colorable G → ∃ A B : set G, ∀ v : G, (v ∈ A ∧ ¬ v ∈ B) ∨ (v ∈ B ∧ ¬ v ∈ A), from sorry,
-- Since all vertices of A are red, there are no edges within A
have h3 : 2-colorable G → ∀ A B : set G, ∃! v : G, (v ∈ A ∧ ¬ v ∈ B) ∨ (v ∈ B ∧ ¬ v ∈ A) → ∀ a, b ∈ A, (a, b) ∉ edges G, from sorry,
-- similarly for B
have h4 : 2-colorable G → ∀ A B : set G, ∃! v : G, (v ∈ A ∧ ¬ v ∈ B) ∨ (v ∈ B ∧ ¬ v ∈ A) → ∀ a, b ∈ B, (a, b) ∉ edges G, from sorry,
-- This implies that every edge has one endpoint in A and the other in B
have h5 : 2-colorable G → ∀ A B : set G, ∃! v : G, (v ∈ A ∧ ¬ v ∈ B) ∨ (v ∈ B ∧ ¬ v ∈ A) → ∀ e : G × G, e ∈ edges G → e.1 ∈ A ∨ e.2 ∈ B, from sorry,
-- which means G is bipartite
have h6 : 2-colorable G → ∀ A B : set G, ∃! v : G, (v ∈ A ∧ ¬ v ∈ B) ∨ (v ∈ B ∧ ¬ v ∈ A) → ∀ e : G × G, e ∈ edges G → e.1 ∈ A ∨ e.2 ∈ B → bipartite G, from sorry,
-- Conversely, suppose G is bipartite, that is, we can partition the vertices into two subsets V1, V2 every edge has one endpoint in V1 and the other in V2
have h7 : bipartite G → ∃ A B : set G, ∀ e : G × G, e ∈ edges G → e.1 ∈ A ∨ e.2 ∈ B, from sorry,
-- Then coloring every vertex of V1 red and every vertex of V2 blue yields a valid coloring
have h8 : bipartite G → ∃ A B : set G, ∀ e : G × G, e ∈ edges G → e.1 ∈ A ∨ e.2 ∈ B → ∃! v : G, (v ∈ A ∧ ¬ v ∈ B) ∨ (v ∈ B ∧ ¬ v ∈ A), from sorry,
-- so G is 2-colorable
have h9 : bipartite G → ∃ A B : set G, ∀ e : G × G, e ∈ edges G → e.1 ∈ A ∨ e.2 ∈ B → ∃! v : G, (v ∈ A ∧ ¬ v ∈ B) ∨ (v ∈ B ∧ ¬ v ∈ A) → 2-colorable G, from sorry,
show bipartite G ↔ 2-colorable G, from by {
sorry,
}
end
--OUTPUT 7
theorem two_colorable_bipartite (G : Type*) [graph G] : is_two_colorable G ↔ is_bipartite G :=
begin
-- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
assume h1 : is_two_colorable G,
have h2 : ∃ A B : set G, ∀ a : G, (a ∈ A ∨ a ∈ B) ∧ (∀ v₁ : G, ∀ v₂ : G, ¬ (a = v₁ ∧ a = v₂)), from sorry,
have h3 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a = v₁ ∧ a = v₂), from sorry,
have h4 : ∀ a : G, ∀ v₁ v₂ : G, (a = v₁ ∧ a = v₂) → false, from sorry,
have h5 : ∀ a : G, ∀ v₁ v₂ : G, (a = v₁ ∧ a = v₂) → ⊥, from sorry,
have h6 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ A ∧ v₁ ∈ A ∧ v₂ ∈ A ∧ (v₁, v₂) ∈ E a), from sorry,
have h7 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ B ∧ v₁ ∈ B ∧ v₂ ∈ B ∧ (v₁, v₂) ∈ E a), from sorry,
have h8 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ A ∧ v₁ ∈ A ∧ v₂ ∈ B ∧ (v₁, v₂) ∈ E a), from sorry,
have h9 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ B ∧ v₁ ∈ B ∧ v₂ ∈ A ∧ (v₁, v₂) ∈ E a), from sorry,
show is_bipartite G, from sorry,
-- Conversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.
assume h1 : is_bipartite G,
have h2 : ∃ (A B : set G) (f : G → Prop ∧ Prop), ∀ a : G, (a ∈ A ∨ a ∈ B) ∧ (∀ v₁ v₂ : G, ¬ (a = v₁ ∧ a = v₂))
∧ (∀ v₁ v₂ : G, ¬ (a ∈ A ∧ v₁ ∈ A ∧ v₂ ∈ A ∧ (v₁, v₂) ∈ E a))
∧ (∀ v₁ v₂ : G, ¬ (a ∈ B ∧ v₁ ∈ B ∧ v₂ ∈ B ∧ (v₁, v₂) ∈ E a))
∧ (∀ v₁ v₂ : G, ¬ (a ∈ A ∧ v₁ ∈ A ∧ v₂ ∈ B ∧ (v₁, v₂) ∈ E a))
∧ (∀ v₁ v₂ : G, ¬ (a ∈ B ∧ v₁ ∈ B ∧ v₂ ∈ A ∧ (v₁, v₂) ∈ E a))
∧ ∀ a : G, f a = (a ∈ A, a ∈ B), from sorry,
have h3 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a = v₁ ∧ a = v₂), from sorry,
have h4 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ A ∧ v₁ ∈ A ∧ v₂ ∈ A ∧ (v₁, v₂) ∈ E a), from sorry,
have h5 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ B ∧ v₁ ∈ B ∧ v₂ ∈ B ∧ (v₁, v₂) ∈ E a), from sorry,
have h6 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ A ∧ v₁ ∈ A ∧ v₂ ∈ B ∧ (v₁, v₂) ∈ E a), from sorry,
have h7 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ B ∧ v₁ ∈ B ∧ v₂ ∈ A ∧ (v₁, v₂) ∈ E a), from sorry,
have h8 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ A ∧ v₁ ∈ B ∧ v₂ ∈ A ∧ (v₁, v₂) ∈ E a), from sorry,
have h9 : ∀ a : G, ∀ v₁ v₂ : G, ¬ (a ∈ B ∧ v₁ ∈ A ∧ v₂ ∈ B ∧ (v₁, v₂) ∈ E a), from sorry,
show is_two_colorable G, from by {
use A ∪ B,
use sorry,
use sorry,
use sorry,
use sorry,
use sorry,
use sorry,
},
end
--OUTPUT 8
theorem two_colorable_iff_bipartite (G : Type*) [pgraph G] : two_colorable G ↔ bipartite G :=
begin
-- $G$ is 2-colorable,
assume (h1 : two_colorable G),
-- We can color every vertex either red or blue,
have h2 : ∃ f : finset (vertex G) → finset (vertex G), ∀ (x : vertex G), x ∈ f x ∨ x ∈ f ((finset.univ : finset (vertex G))\(f x)), from sorry,
-- No edge will have both endpoints colored the same color.
have h3 : ∀ (x y : vertex G), x ≠ y → x ∈ f x → y ∈ f ((finset.univ : finset (vertex G))\(f x)), from sorry,
let f : finset (vertex G) → finset (vertex G) := fml,
let A : finset (vertex G) := finset.image f (finset.univ : finset (vertex G)),
let B : finset (vertex G) := finset.univ\A,
-- $A$ denote the subset of vertices colored red,
have h4 : ∀ (x : vertex G), x ∈ A ↔ x ∈ f x, from sorry,
-- $B$ denote the subset of vertices colored blue,
have h5 : ∀ (x : vertex G), x ∈ B ↔ x ∉ f x, from sorry,
-- Since all vertices of $A$ are red,
have h6 : ∀ (x : vertex G), x ∈ A → x ∉ B, from sorry,
-- There are no edges within $A$,
have h7 : ∀ (x y : vertex G), x ∈ A → y ∈ A → ¬ edge G x y, from sorry,
-- Similarly, there are no edges within $B$,
have h8 : ∀ (x y : vertex G), x ∈ B → y ∈ B → ¬ edge G x y, from sorry,
-- This implies that every edge has one endpoint in $A$ and the other in $B$,
-- which means $G$ is bipartite.
have h9 : bipartite G, from sorry,
show bipartite G, from sorry,
-- Conversely, suppose $G$ is bipartite,
assume (h1 : bipartite G),
-- That is, we can partition the vertices into two subsets $V_1, V_2$
have h2 : ∃ A B : finset (vertex G), is_independent_set G A ∧ is_independent_set G B ∧ ∀ (x : vertex G), x ∉ A ∨ x ∉ B, from sorry,
let (A B : finset (vertex G)) := fml,
assume (h3 : is_independent_set G A),
assume (h4 : is_independent_set G B),
assume (h5 : ∀ (x : vertex G), x ∉ A ∨ x ∉ B),
-- Every edge has one endpoint in $V_1$ and the other in $V_2$
have h6 : ∀ (x y : vertex G), edge G x y → x ∉ A → y ∉ A, from sorry,
have h7 : ∀ (x y : vertex G), edge G x y → x ∉ B → y ∉ B, from sorry,
-- Then coloring every vertex of $V_1$ red and every vertex of $V_2$ blue
have h8 : two_colorable G, from sorry,
show two_colorable G, from sorry,
end
--OUTPUT 9
theorem bipartite_is_two_colorable (G : Type*):
(∀ v : ℕ, v ∈ G ∨ v ∉ G) →
(∀ u v : ℕ, (u ≠ v) ∧ (v ∈ G) ∧ (u ∈ G) → u ↔ v) →
(∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → (∀ v : ℕ, v ∈ A ∪ B) ∧ ((v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B))) →
∃ (A B : ℕ → Prop), ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) ∧ ((v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B))
:= by {
-- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color.
assume (h1 : ∀ v : ℕ, v ∈ G ∨ v ∉ G),
have h2 : ∀ u v : ℕ, (u ≠ v) ∧ (v ∈ G) ∧ (u ∈ G) → u ↔ v, from sorry,
have h3 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → (∀ v : ℕ, v ∈ A ∪ B) ∧ ((v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B)), from sorry,
-- Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.
show ∃ (A B : ℕ → Prop), ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) ∧ ((v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B)), from by {
show ∃ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) ∧ ((v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B)), from sorry,
-- Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.
have h4 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → v ∈ A → w ∈ A → ¬ (v ↔ w), from sorry,
have h5 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → v ∈ B → w ∈ B → ¬ (v ↔ w), from sorry,
-- This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
have h6 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B) → v ↔ w, from sorry,
have h6 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B) → v ∈ G ∧ w ∈ G, from sorry,
have h7 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B) → v ↔ w, from sorry,
have h8 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B) → (∀ v w : ℕ, v ∈ A ∧ w ∈ B) ∧ (∀ v w : ℕ, v ∈ B ∧ w ∈ A), from sorry,
have h9 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B) → (∀ v w : ℕ, v ∈ A ∧ w ∈ B) ∧ (∀ v w : ℕ, v ∈ B ∧ w ∈ A), from sorry,
have h10 : ∀ A B : ℕ → Prop, ∀ v w : ℕ, (∀ u : ℕ, u ∈ G → (u ∈ A → ¬ u ∈ B) ∧ (u ∈ B → ¬ u ∈ A)) → (v ∈ A ∧ w ∈ B) ∨ (v ∈ B ∧ w ∈ A) ∨ (v ∈ A ∧ w ∈ A) ∨ (v ∈ B ∧ w ∈ B) → (∃ A B : ℕ →
end --Needs more than 2000 tokens!
--OUTPUT 10
theorem is_2_colorable_iff_bipartite (G : Type*) [graph G] : 2-colorable G ↔ bipartite G :=
begin
assume (h1 : colorable G 2),
-- Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue.
have h2 : ∃ red blue : G, (∀ vertex : G, vertex ∈ red ∨ vertex ∈ blue) ∧ (∀ {u v}, edge u v → u ∈ red ∧ v ∈ blue ∨ u ∈ blue ∧ v ∈ red), from sorry,
let h3 : ∃ red blue : G, (∀ vertex : G, vertex ∈ red ∨ vertex ∈ blue) ∧ (∀ {u v}, edge u v → u ∈ red ∧ v ∈ blue ∨ u ∈ blue ∧ v ∈ red) := h2,
let red blue := classical.some h3,
let h4 : (∀ vertex : G, vertex ∈ red ∨ vertex ∈ blue) ∧ (∀ {u v}, edge u v → u ∈ red ∧ v ∈ blue ∨ u ∈ blue ∧ v ∈ red) := classical.some_spec h3,
let h5 : ∀ vertex : G, vertex ∈ red ∨ vertex ∈ blue := h4.left,
let h6 : ∀ {u v}, edge u v → u ∈ red ∧ v ∈ blue ∨ u ∈ blue ∧ v ∈ red := h4.right,
-- Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$.
have h7 : ∀ {u v}, edge u v → u ∈ red ∧ v ∈ blue ∨ u ∈ blue ∧ v ∈ red, from
begin
assume (u v : G) (h7 : edge u v),
show u ∈ red ∧ v ∈ blue ∨ u ∈ blue ∧ v ∈ red, from sorry,
end,
have h8 : ∀ {u v}, u ∈ red ∧ v ∈ blue → edge u v, from
begin
assume (u v : G) (h8 : u ∈ red ∧ v ∈ blue),
show edge u v, from sorry,
end,
have h9 : ∀ {u v}, u ∈ blue ∧ v ∈ red → edge u v, from
begin
assume (u v : G) (h9 : u ∈ blue ∧ v ∈ red),
show edge u v, from sorry,
end,
have h10 : ∀ {u v}, u ∈ red ∧ v ∈ red → ¬ edge u v, from
begin
assume (u v : G) (h10 : u ∈ red ∧ v ∈ red),
have h11 : u ∉ red ∨ v ∉ red, from sorry,
have h12 : ¬ (u ∈ red ∧ v ∈ blue), from sorry,
show ¬ edge u v, from sorry,
end,
have h11 : ∀ {u v}, u ∈ blue ∧ v ∈ blue → ¬ edge u v, from
begin
assume (u v : G) (h11 : u ∈ blue ∧ v ∈ blue),
have h12 : u ∉ blue ∨ v ∉ blue, from sorry,
have h13 : ¬ (u ∈ blue ∧ v ∈ red), from sorry,
show ¬ edge u v, from sorry,
end,
-- This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
have h12 : bipartite G, from sorry,
have h13 : ∃! c : colorable G 2, bipartite G, from
begin
use h1,
show bipartite G, from sorry,
end,
have h14 : ∀ c : colorable G 2, bipartite G, from
begin
assume (c : colorable G 2),
show bipartite G, from sorry,
end,
show 2-colorable G ↔ bipartite G, from sorry,
end
/- FEW SHOT PROMPTS TO CODEX(START)
/--`theorem`
Power Set is Closed under Intersection
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then:
:$\forall A, B \in \powerset S: A \cap B \in \powerset S$
`proof`
Let $A, B \in \powerset S$.
Then by the definition of power set, $A \subseteq S$ and $B \subseteq S$.
From Intersection is Subset we have that $A \cap B \subseteq A$.
It follows from Subset Relation is Transitive that $A \cap B \subseteq S$.
Thus $A \cap B \in \powerset S$ and closure is proved.
{{qed}}
-/
theorem power_set_intersection_closed {α : Type*} (S : set α) : ∀ A B ∈ 𝒫 S, (A ∩ B) ∈ 𝒫 S :=
begin
-- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$
assume (A : set α) (hA : A ∈ 𝒫 S) (B : set α) (hB : B ∈ 𝒫 S),
-- Then $A ⊆ S$ and $B ⊆ S$, by power set definition
have h1 : (A ⊆ S) ∧ (B ⊆ S), from sorry,
-- Then $(A ∩ B) ⊆ A$, by intersection of set is a subset
have h2 : (A ∩ B) ⊆ A, from sorry,
-- Then $(A ∩ B) ⊆ S$, by subset relation is transitive
have h3 : (A ∩ B) ⊆ S, from sorry,
-- Hence $(A ∩ B) ∈ 𝒫 S$, by power set definition
show (A ∩ B) ∈ 𝒫 S, from sorry,
end
/--`theorem`
Square of Sum
:$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$
`proof`
Follows from the distribution of multiplication over addition:
{{begin-eqn}}
{{eqn | l = \left({x + y}\right)^2
| r = \left({x + y}\right) \cdot \left({x + y}\right)
}}
{{eqn | r = x \cdot \left({x + y}\right) + y \cdot \left({x + y}\right)
| c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x \cdot x + x \cdot y + y \cdot x + y \cdot y
| c = Real Multiplication Distributes over Addition
}}
{{eqn | r = x^2 + 2xy + y^2
| c =
}}
{{end-eqn}}
{{qed}}
-/
theorem square_of_sum (x y : ℝ) : (x + y)^2 = (x^2 + 2*x*y + y^2) :=
begin
-- expand the power
calc (x + y)^2 = (x+y)*(x+y) : by sorry
-- distributive property of multiplication over addition gives:
... = x*(x+y) + y*(x+y) : by sorry
-- applying the above property further gives:
... = x*x + x*y + y*x + y*y : by sorry
-- rearranging the terms using commutativity and adding gives:
... = x^2 + 2*x*y + y^2 : by sorry,
end
/--`theorem`
Identity of Group is Unique
Let $\struct {G, \circ}$ be a group. Then there is a unique identity element $e \in G$.
`proof`
From Group has Latin Square Property, there exists a unique $x \in G$ such that:
:$a x = b$
and there exists a unique $y \in G$ such that:
:$y a = b$
Setting $b = a$, this becomes:
There exists a unique $x \in G$ such that:
:$a x = a$
and there exists a unique $y \in G$ such that:
:$y a = a$
These $x$ and $y$ are both $e$, by definition of identity element.
{{qed}}
-/
theorem group_identity_unique {G : Type*} [group G] : ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a :=
begin
-- Group has Latin Square Property
have h1 : ∀ a b : G, ∃! x : G, a * x = b, from sorry,
have h2 : ∀ a b : G, ∃! y : G, y * a = b, from sorry,
-- Setting $b = a$, this becomes:
have h3 : ∀ a : G, ∃! x : G, a * x = a, from sorry,
have h4 : ∀ a : G, ∃! y : G, y * a = a, from sorry,
-- These $x$ and $y$ are both $(1 : G)$, by definition of identity element
have h5 : ∀ a : G, classical.some (h3 a) = (1 : G), from sorry,
have h6 : ∀ a : G, classical.some (h4 a) = (1 : G), from sorry,
show ∃! e : G, ∀ a : G, e * a = a ∧ a * e = a, from by {
use (1 : G),
have h7 : ∀ e : G, (∀ a : G, e * a = a ∧ a * e = a) → e = 1, from by {
assume (e : G) (h7 : ∀ a : G, e * a = a ∧ a * e = a),
have h8 : ∀ a : G, e = classical.some (h3 a), from sorry,
have h9 : ∀ a : G, e = classical.some (h4 a), from sorry,
show e = (1 : G), from sorry,
},
sorry,
}
end
/--`theorem`
Bipartite Graph is two colorable
Let $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.
`proof`
Let $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.
Conversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.
QED
-/
theorem
FEW SHOT PROMPTS TO CODEX(END)-/
|
C02W82DBHV2R:~ i847419$ coqtop
Welcome to Coq 8.8.0 (April 2018)
Coq < Require Import Classical.
Coq <
Coq < Theorem exp002: (forall I R : Prop, ((I /\ ~R) /\ ~I) -> R).
1 subgoal
============================
forall I R : Prop, (I /\ ~ R) /\ ~ I -> R
exp002 < intros.
1 subgoal
I, R : Prop
H : (I /\ ~ R) /\ ~ I
============================
R
exp002 < destruct H.
1 subgoal
I, R : Prop
H : I /\ ~ R
H0 : ~ I
============================
R
exp002 < destruct H.
1 subgoal
I, R : Prop
H : I
H1 : ~ R
H0 : ~ I
============================
R
exp002 < tauto.
No more subgoals.
exp002 < Qed.
exp002 is defined
Coq <
|
module Assume where
postulate
assume : ∀ {a} {A : Set a} → A |
theory Sat
imports IMP
begin
fun Sat :: "bexp \<Rightarrow> bool" where
"Sat (Bc b) = b" |
"Sat (Not b) = (\<exists>s. \<not> bval b s)" |
"Sat (And a b) = (\<exists>s. bval a s \<and> bval b s)" |
"Sat (Less a b) = (\<exists>s. aval a s < aval b s)"
theorem sat_equiv: "Sat a \<longleftrightarrow> (\<exists>s. bval a s)"
apply (induction a)
apply auto
done
theorem unsat_contradiction: "\<not> Sat (And a (Not a))"
by simp
theorem sat_taut_incl: "(\<forall>s. bval a s) \<Longrightarrow> Sat b \<Longrightarrow> Sat (And a b)"
apply (induction b arbitrary: a)
apply simp_all
done
theorem sat_split: "Sat (And a b) \<Longrightarrow> Sat a \<and> Sat b"
proof -
assume "Sat (And a b)"
hence "\<exists>s. bval a s \<and> bval b s" by simp
hence "(\<exists>s. bval a s) \<and> (\<exists>s. bval b s)" by auto
thus ?thesis using sat_equiv by simp
qed
end |
-- Copyright (c) 2017 Scott Morrison. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Scott Morrison
import tidy.tidy
open tactic
set_option trace.tidy true
@[reducible] def {u} auto_cast {α β : Sort u} {h : α = β} (a : α) := cast h a
@[simp] lemma {u} auto_cast_identity {α : Sort u} (p : α = α) (a : α) : @auto_cast α α p a = a :=
begin unfold auto_cast, unfold cast, end
notation `⟬` p `⟭` := @auto_cast _ _ (by obviously) p
|
import Lean
open Lean
open Lean.Meta
open Lean.Elab.Tactic
universe u
axiom elimEx (motive : Nat → Nat → Sort u) (x y : Nat)
(diag : (a : Nat) → motive a a)
(upper : (delta a : Nat) → motive a (a + delta.succ))
(lower : (delta a : Nat) → motive (a + delta.succ) a)
: motive y x
theorem ex1 (p q : Nat) : p ≤ q ∨ p > q := by
cases p, q using elimEx with
| diag => apply Or.inl; apply Nat.leRefl
| lower d => apply Or.inl; show p ≤ p + d.succ; admit
| upper d => apply Or.inr; show q + d.succ > q; admit
theorem ex2 (p q : Nat) : p ≤ q ∨ p > q := by
cases p, q using elimEx
case lower => admit
case upper => admit
case diag => apply Or.inl; apply Nat.leRefl
axiom Nat.parityElim (motive : Nat → Sort u)
(even : (n : Nat) → motive (2*n))
(odd : (n : Nat) → motive (2*n+1))
(n : Nat)
: motive n
theorem time2Eq (n : Nat) : 2*n = n + n := by
rw [Nat.mul_comm]
show (0 + n) + n = n+n
simp
theorem ex3 (n : Nat) : Exists (fun m => n = m + m ∨ n = m + m + 1) := by
cases n using Nat.parityElim with
| even i =>
apply Exists.intro i
apply Or.inl
rw [time2Eq]
| odd i =>
apply Exists.intro i
apply Or.inr
rw [time2Eq]
open Nat in
theorem ex3b (n : Nat) : Exists (fun m => n = m + m ∨ n = m + m + 1) := by
cases n using parityElim with
| even i =>
apply Exists.intro i
apply Or.inl
rw [time2Eq]
| odd i =>
apply Exists.intro i
apply Or.inr
rw [time2Eq]
def ex4 {α} (xs : List α) (h : xs = [] → False) : α := by
cases he:xs with
| nil => contradiction
| cons x _ => exact x
def ex5 {α} (xs : List α) (h : xs = [] → False) : α := by
cases he:xs using List.casesOn with
| nil => contradiction
| cons x _ => exact x
theorem ex6 {α} (f : List α → Bool) (h₁ : {xs : List α} → f xs = true → xs = []) (xs : List α) (h₂ : xs ≠ []) : f xs = false :=
match he:f xs with
| true => False.elim (h₂ (h₁ he))
| false => rfl
theorem ex7 {α} (f : List α → Bool) (h₁ : {xs : List α} → f xs = true → xs = []) (xs : List α) (h₂ : xs ≠ []) : f xs = false := by
cases he:f xs with
| true => exact False.elim (h₂ (h₁ he))
| false => rfl
theorem ex8 {α} (f : List α → Bool) (h₁ : {xs : List α} → f xs = true → xs = []) (xs : List α) (h₂ : xs ≠ []) : f xs = false := by
cases he:f xs using Bool.casesOn with
| true => exact False.elim (h₂ (h₁ he))
| false => rfl
theorem ex9 (xs : List α) (h : xs = [] → False) : Nonempty α := by
cases xs using List.rec with
| nil => contradiction
| cons x _ => apply Nonempty.intro; assumption
theorem modLt (x : Nat) {y : Nat} (h : y > 0) : x % y < y := by
induction x, y using Nat.mod.inductionOn with
| ind x y h₁ ih =>
rw [Nat.mod_eq_sub_mod h₁.2]
exact ih h
| base x y h₁ =>
match Iff.mp (Decidable.notAndIffOrNot ..) h₁ with
| Or.inl h₁ => contradiction
| Or.inr h₁ =>
have hgt := Nat.gtOfNotLe h₁
have heq := Nat.mod_eq_of_lt hgt
rw [← heq] at hgt
assumption
theorem ex11 {p q : Prop } (h : p ∨ q) : q ∨ p := by
induction h using Or.casesOn with
| inr h => ?myright
| inl h => ?myleft
case myleft => exact Or.inr h
case myright => exact Or.inl h
theorem ex12 {p q : Prop } (h : p ∨ q) : q ∨ p := by
cases h using Or.casesOn with
| inr h => ?myright
| inl h => ?myleft
case myleft => exact Or.inr h
case myright => exact Or.inl h
theorem ex13 (p q : Nat) : p ≤ q ∨ p > q := by
cases p, q using elimEx with
| diag => ?hdiag
| lower d => ?hlower
| upper d => ?hupper
case hdiag => apply Or.inl; apply Nat.leRefl
case hlower => apply Or.inl; show p ≤ p + d.succ; admit
case hupper => apply Or.inr; show q + d.succ > q; admit
theorem ex14 (p q : Nat) : p ≤ q ∨ p > q := by
cases p, q using elimEx with
| diag => ?hdiag
| lower d => _
| upper d => ?hupper
case hdiag => apply Or.inl; apply Nat.leRefl
case lower => apply Or.inl; show p ≤ p + d.succ; admit
case hupper => apply Or.inr; show q + d.succ > q; admit
theorem ex15 (p q : Nat) : p ≤ q ∨ p > q := by
cases p, q using elimEx with
| diag => ?hdiag
| lower d => _
| upper d => ?hupper
{ apply Or.inl; apply Nat.leRefl }
{ apply Or.inr; show q + d.succ > q; admit }
{ apply Or.inl; show p ≤ p + d.succ; admit }
theorem ex16 {p q : Prop} (h : p ∨ q) : q ∨ p := by
induction h
case inl h' => exact Or.inr h'
case inr h' => exact Or.inl h'
theorem ex17 (n : Nat) : 0 + n = n := by
induction n
case zero => rfl
case succ m ih =>
show Nat.succ (0 + m) = Nat.succ m
rw [ih]
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
! This file was ported from Lean 3 source module linear_algebra.matrix.invariant_basis_number
! leanprover-community/mathlib commit 843240b048bbb19942c581fd64caecbbe96337be
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.LinearAlgebra.Matrix.ToLin
import Mathbin.LinearAlgebra.InvariantBasisNumber
/-!
# Invertible matrices over a ring with invariant basis number are square.
-/
variable {n m : Type _} [Fintype n] [DecidableEq n] [Fintype m] [DecidableEq m]
variable {R : Type _} [Semiring R] [InvariantBasisNumber R]
open Matrix
theorem Matrix.square_of_invertible (M : Matrix n m R) (N : Matrix m n R) (h : M ⬝ N = 1)
(h' : N ⬝ M = 1) : Fintype.card n = Fintype.card m :=
card_eq_of_linearEquiv R (Matrix.toLinearEquivRight'OfInv h' h)
#align matrix.square_of_invertible Matrix.square_of_invertible
|
[STATEMENT]
lemma inv_rule_basic:
"\<lbrakk> init T \<subseteq> P; {P} (trans T) {> P} \<rbrakk>
\<Longrightarrow> reach T \<subseteq> P"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<lbrakk>init T \<subseteq> P; {P} TS.trans T {> P}\<rbrakk> \<Longrightarrow> reach T \<subseteq> P
[PROOF STEP]
by (safe, erule reach.induct, auto simp add: PO_hoare_def) |
module AsP where
f
: {A : Set}
→ A
→ A
→ A
f x@y z@w
= x
g
: {A : Set}
→ A
→ A
→ A
g x@y z'@w'
with y
... | _
= x
|
(* Title: HOL/Examples/Cantor.thy
Author: Makarius
*)
section \<open>Cantor's Theorem\<close>
theory Cantor
imports Main
begin
subsection \<open>Mathematical statement and proof\<close>
text \<open>
Cantor's Theorem states that there is no surjection from
a set to its powerset. The proof works by diagonalization. E.g.\ see
\<^item> \<^url>\<open>http://mathworld.wolfram.com/CantorDiagonalMethod.html\<close>
\<^item> \<^url>\<open>https://en.wikipedia.org/wiki/Cantor's_diagonal_argument\<close>
\<close>
theorem Cantor: "\<nexists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. A = f x"
proof
assume "\<exists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. A = f x"
then obtain f :: "'a \<Rightarrow> 'a set" where *: "\<forall>A. \<exists>x. A = f x" ..
let ?D = "{x. x \<notin> f x}"
from * obtain a where "?D = f a" by blast
moreover have "a \<in> ?D \<longleftrightarrow> a \<notin> f a" by blast
ultimately show False by blast
qed
subsection \<open>Automated proofs\<close>
text \<open>
These automated proofs are much shorter, but lack information why and how it
works.
\<close>
theorem "\<nexists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. f x = A"
by best
theorem "\<nexists>f :: 'a \<Rightarrow> 'a set. \<forall>A. \<exists>x. f x = A"
by force
subsection \<open>Elementary version in higher-order predicate logic\<close>
text \<open>
The subsequent formulation bypasses set notation of HOL; it uses elementary
\<open>\<lambda>\<close>-calculus and predicate logic, with standard introduction and elimination
rules. This also shows that the proof does not require classical reasoning.
\<close>
lemma iff_contradiction:
assumes *: "\<not> A \<longleftrightarrow> A"
shows False
proof (rule notE)
show "\<not> A"
proof
assume A
with * have "\<not> A" ..
from this and \<open>A\<close> show False ..
qed
with * show A ..
qed
theorem Cantor': "\<nexists>f :: 'a \<Rightarrow> 'a \<Rightarrow> bool. \<forall>A. \<exists>x. A = f x"
proof
assume "\<exists>f :: 'a \<Rightarrow> 'a \<Rightarrow> bool. \<forall>A. \<exists>x. A = f x"
then obtain f :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where *: "\<forall>A. \<exists>x. A = f x" ..
let ?D = "\<lambda>x. \<not> f x x"
from * have "\<exists>x. ?D = f x" ..
then obtain a where "?D = f a" ..
then have "?D a \<longleftrightarrow> f a a" by (rule arg_cong)
then have "\<not> f a a \<longleftrightarrow> f a a" .
then show False by (rule iff_contradiction)
qed
subsection \<open>Classic Isabelle/HOL example\<close>
text \<open>
The following treatment of Cantor's Theorem follows the classic example from
the early 1990s, e.g.\ see the file \<^verbatim>\<open>92/HOL/ex/set.ML\<close> in
Isabelle92 or \<^cite>\<open>\<open>\S18.7\<close> in "paulson-isa-book"\<close>. The old tactic scripts
synthesize key information of the proof by refinement of schematic goal
states. In contrast, the Isar proof needs to say explicitly what is proven.
\<^bigskip>
Cantor's Theorem states that every set has more subsets than it has
elements. It has become a favourite basic example in pure higher-order logic
since it is so easily expressed:
@{text [display]
\<open>\<forall>f::\<alpha> \<Rightarrow> \<alpha> \<Rightarrow> bool. \<exists>S::\<alpha> \<Rightarrow> bool. \<forall>x::\<alpha>. f x \<noteq> S\<close>}
Viewing types as sets, \<open>\<alpha> \<Rightarrow> bool\<close> represents the powerset of \<open>\<alpha>\<close>. This
version of the theorem states that for every function from \<open>\<alpha>\<close> to its
powerset, some subset is outside its range. The Isabelle/Isar proofs below
uses HOL's set theory, with the type \<open>\<alpha> set\<close> and the operator \<open>range :: (\<alpha> \<Rightarrow>
\<beta>) \<Rightarrow> \<beta> set\<close>.
\<close>
theorem "\<exists>S. S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
proof
let ?S = "{x. x \<notin> f x}"
show "?S \<notin> range f"
proof
assume "?S \<in> range f"
then obtain y where "?S = f y" ..
then show False
proof (rule equalityCE)
assume "y \<in> f y"
assume "y \<in> ?S"
then have "y \<notin> f y" ..
with \<open>y \<in> f y\<close> show ?thesis by contradiction
next
assume "y \<notin> ?S"
assume "y \<notin> f y"
then have "y \<in> ?S" ..
with \<open>y \<notin> ?S\<close> show ?thesis by contradiction
qed
qed
qed
text \<open>
How much creativity is required? As it happens, Isabelle can prove this
theorem automatically using best-first search. Depth-first search would
diverge, but best-first search successfully navigates through the large
search space. The context of Isabelle's classical prover contains rules for
the relevant constructs of HOL's set theory.
\<close>
theorem "\<exists>S. S \<notin> range (f :: 'a \<Rightarrow> 'a set)"
by best
end
|
-- import ..src.struct_tact
-- lemma test :
-- forall (o1 : option nat) (o2 : nat → option nat) n
-- o1 >>= o2 = some n ->
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import algebra.algebra.spectrum
import analysis.calculus.deriv
/-!
# The spectrum of elements in a complete normed algebra
This file contains the basic theory for the resolvent and spectrum of a Banach algebra.
## Main definitions
* `spectral_radius : ℝ≥0∞`: supremum of `∥k∥₊` for all `k ∈ spectrum 𝕜 a`
## Main statements
* `spectrum.is_open_resolvent_set`: the resolvent set is open.
* `spectrum.is_closed`: the spectrum is closed.
* `spectrum.subset_closed_ball_norm`: the spectrum is a subset of closed disk of radius
equal to the norm.
* `spectrum.is_compact`: the spectrum is compact.
* `spectrum.spectral_radius_le_nnnorm`: the spectral radius is bounded above by the norm.
* `spectrum.has_deriv_at_resolvent`: the resolvent function is differentiable on the resolvent set.
## TODO
* after we have Liouville's theorem, prove that the spectrum is nonempty when the
scalar field is ℂ.
* compute all derivatives of `resolvent a`.
-/
open_locale ennreal
/-- The *spectral radius* is the supremum of the `nnnorm` (`∥⬝∥₊`) of elements in the spectrum,
coerced into an element of `ℝ≥0∞`. Note that it is possible for `spectrum 𝕜 a = ∅`. In this
case, `spectral_radius a = 0`. It is also possible that `spectrum 𝕜 a` be unbounded (though
not for Banach algebras, see `spectrum.is_bounded`, below). In this case,
`spectral_radius a = ∞`. -/
noncomputable def spectral_radius (𝕜 : Type*) {A : Type*} [normed_field 𝕜] [ring A]
[algebra 𝕜 A] (a : A) : ℝ≥0∞ :=
⨆ k ∈ spectrum 𝕜 a, ∥k∥₊
namespace spectrum
section spectrum_compact
variables {𝕜 : Type*} {A : Type*}
variables [normed_field 𝕜] [normed_ring A] [normed_algebra 𝕜 A] [complete_space A]
local notation `σ` := spectrum 𝕜
local notation `ρ` := resolvent_set 𝕜
local notation `↑ₐ` := algebra_map 𝕜 A
lemma is_open_resolvent_set (a : A) : is_open (ρ a) :=
units.is_open.preimage ((algebra_map_isometry 𝕜 A).continuous.sub continuous_const)
lemma is_closed (a : A) : is_closed (σ a) :=
(is_open_resolvent_set a).is_closed_compl
lemma mem_resolvent_of_norm_lt {a : A} {k : 𝕜} (h : ∥a∥ < ∥k∥) :
k ∈ ρ a :=
begin
rw [resolvent_set, set.mem_set_of_eq, algebra.algebra_map_eq_smul_one],
have hk : k ≠ 0 := ne_zero_of_norm_pos (by linarith [norm_nonneg a]),
let ku := units.map (↑ₐ).to_monoid_hom (units.mk0 k hk),
have hku : ∥-a∥ < ∥(↑ku⁻¹:A)∥⁻¹ := by simpa [ku, algebra_map_isometry] using h,
simpa [ku, sub_eq_add_neg, algebra.algebra_map_eq_smul_one] using (ku.add (-a) hku).is_unit,
end
lemma norm_le_norm_of_mem {a : A} {k : 𝕜} (hk : k ∈ σ a) :
∥k∥ ≤ ∥a∥ :=
le_of_not_lt $ mt mem_resolvent_of_norm_lt hk
lemma subset_closed_ball_norm (a : A) :
σ a ⊆ metric.closed_ball (0 : 𝕜) (∥a∥) :=
λ k hk, by simp [norm_le_norm_of_mem hk]
lemma is_bounded (a : A) : metric.bounded (σ a) :=
(metric.bounded_iff_subset_ball 0).mpr ⟨∥a∥, subset_closed_ball_norm a⟩
theorem is_compact [proper_space 𝕜] (a : A) : is_compact (σ a) :=
metric.is_compact_of_is_closed_bounded (is_closed a) (is_bounded a)
theorem spectral_radius_le_nnnorm (a : A) :
spectral_radius 𝕜 a ≤ ∥a∥₊ :=
begin
suffices h : ∀ k ∈ σ a, (∥k∥₊ : ℝ≥0∞) ≤ ∥a∥₊,
{ exact bsupr_le h, },
{ by_cases ha : (σ a).nonempty,
{ intros _ hk,
exact_mod_cast norm_le_norm_of_mem hk },
{ rw set.not_nonempty_iff_eq_empty at ha,
simp [ha, set.ball_empty_iff] } }
end
end spectrum_compact
section resolvent_deriv
variables {𝕜 : Type*} {A : Type*}
variables [nondiscrete_normed_field 𝕜] [normed_ring A] [normed_algebra 𝕜 A] [complete_space A]
local notation `ρ` := resolvent_set 𝕜
local notation `↑ₐ` := algebra_map 𝕜 A
theorem has_deriv_at_resolvent {a : A} {k : 𝕜} (hk : k ∈ ρ a) :
has_deriv_at (resolvent a) (-(resolvent a k) ^ 2) k :=
begin
have H₁ : has_fderiv_at ring.inverse _ (↑ₐk - a) := has_fderiv_at_ring_inverse hk.unit,
have H₂ : has_deriv_at (λ k, ↑ₐk - a) 1 k,
{ simpa using (algebra.linear_map 𝕜 A).has_deriv_at.sub_const a },
simpa [resolvent, sq, hk.unit_spec, ← ring.inverse_unit hk.unit] using H₁.comp_has_deriv_at k H₂,
end
end resolvent_deriv
end spectrum
|
import algebra.group_power
import data.real.basic
import tactic.norm_num
-- The integers, rationals and reals are all different types in Lean
-- and the obvious inclusions between them are all denoted by ↑ .
lemma rational_half_not_an_integer : ¬ ∃ n : ℤ, (1/2 : ℚ) = ↑n :=
begin
-- proof by contradiction
rintros ⟨n,Hn⟩, -- n is an integer, Hn the proof that 1/2 = n
-- goal is "false"
have H := rat.coe_int_denom n, -- H says denominator of n is 1
rw ←Hn at H, -- H now says denominator of 1/2 is 1...
exact absurd H dec_trivial -- ...but denominator of 1/2 isn't 1.
end
lemma real_half_not_an_integer : ¬ (∃ n : ℤ, (1/2 : ℝ) = (n : ℝ) ) :=
begin
rintro ⟨n,Hn⟩, -- n is an integer, Hn the proof that it's 1/2
apply rational_half_not_an_integer,
existsi n,
-- now our hypothesis is that 1/2 = n as reals, and we want to
-- deduce 1/2 = n as rationals!
-- This is possible by some messing around with coercions
-- from integers to rationals to reals. I wish this were easier
-- for beginners in Lean...
rw ←@rat.cast_inj ℝ _ _,
rw rat.cast_coe_int,
rw ←Hn, --goal now is to prove that real 1/2 = rational 1/2
simp -- simplifier is good at that sort of thing
end
-- proof that the real numbers which are integers and whose squares are less than three
-- are precisely -1, 0 and 1
lemma square_lt_three_of_ge_two (n : ℕ) : ¬ (n + 2) * (n + 2) < 3 :=
begin
intro H,
suffices Hab : 4 < 3,
exact absurd Hab dec_trivial,
exact calc
4 = 2 * 2 : rfl
... ≤ (n + 2) * 2 : nat.mul_le_mul_right 2 (show 2 ≤ n+2, from dec_trivial)
... ≤ (n + 2) * (n + 2) : nat.mul_le_mul_left (n+2) (show 2 ≤ n+2, from dec_trivial)
... < 3 : H
end
lemma int_squared_lt_three {z : ℤ} : z ^ 2 < 3 → z = -1 ∨ z = 0 ∨ z = 1 :=
begin
cases z with n n,
{ rw pow_two,
show ↑n * ↑n < ↑3 → _,
rw [←int.coe_nat_mul,int.coe_nat_lt],
intro Hn,
cases n,
right,left,refl,
cases n,
right,right,refl,
cases square_lt_three_of_ge_two n Hn,
},
{ rw [pow_two,←int.nat_abs_mul_self],
show ↑((n+1)*(n+1)) < ↑3 → _,
rw int.coe_nat_lt,
intro Hn,
cases n,
left,trivial,
cases square_lt_three_of_ge_two n Hn,
}
end
theorem B_is_minus_one_zero_one (x : ℝ) : x ∈ { x : ℝ | x^2 < 3 ∧ ∃ y : ℤ, x = ↑y} ↔ x = -1 ∨ x = 0 ∨ x = 1 :=
begin
split,
{ intro H,
cases H.right with y Hy,
have Hleft := H.left,
rw [Hy,pow_two,←int.cast_mul] at Hleft,
have Htemp : (3 : ℝ) = (3 : ℤ),
refl,
rw Htemp at Hleft,
rw [int.cast_lt,←pow_two] at Hleft,
rw Hy,
cases int_squared_lt_three Hleft with h h,
left,rw h,refl,
cases h with h h,
right,left,rw h,refl,
right,right,rw h,refl
},
{ intro H,
cases H,
rw H,
split,norm_num,existsi (-1 : ℤ),refl,
cases H,
rw H,
split,norm_num,existsi (0 : ℤ),refl,
rw H,
split,norm_num,existsi (1 : ℤ),refl
}
end
|
lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im] |
! Ensure that loops not affiliated with acc compute regions cause an error.
subroutine test1
!$acc loop gang ! { dg-error "loop directive must be associated with an OpenACC compute region" }
DO i = 1,10
ENDDO
end subroutine test1
|
import topology.basic
import data.real.basic
import topology.metric_space.basic
import topology.uniform_space.uniform_convergence
import topology.uniform_space.basic
-- set_option pp.all true
open filter
theorem unif_cont
(F : ℕ → ℝ → ℝ) (hF : ∀ n, continuous (F n))
(f : ℝ → ℝ) (hFf : tendsto_uniformly F f at_top) :
continuous f :=
begin
-- set up fully "peeled" proof state:
rw metric.continuous_iff, -- metric def of continuity
intros x ε hε, -- introduce universally quantified variables into the proof state
-- solution to disallowed metavarible dependencies: skolimize and rcases first!
simp [metric.continuous_iff] at hF, -- metric definition of continuity
simp [classical.skolem] at hF,
-- sometimes with skolem behind other universal quantifiers, you need to help Lean a bit
-- (simp and rw don't work)
-- rw [classical.skolem] at hF, -- try yourself
-- simp [classical.skolem] at hF, -- try yourself
have hF := λ n b, classical.skolem.mp (hF n b),
have hF' := λ n, classical.skolem.mp (hF n),
simp [classical.skolem] at hF',
clear hF,
clear hF,
rcases hF' with ⟨δ, hF⟩,
have hδ := λ n x ε hε, and.elim_left (hF n x ε hε),
have hF := λ n x ε hε, and.elim_right (hF n x ε hε),
-- the same for the other hypothesis
rw metric.tendsto_uniformly_iff at hFf,
simp at hFf, -- get rid of `at_top`
simp [classical.skolem] at hFf,
rcases hFf with ⟨M, hM⟩,
apply exists.intro, -- delay choice, creates mvars
apply exists.intro, -- delay choice, creates mvars
rotate 1, -- postpone a goal by rotating the goal list
intros y hxy,
-- start the proof
apply lt_of_le_of_lt, -- transitivity
apply dist_triangle, -- triangle inequality
rotate 1,
apply lt_of_lt_of_le,
apply add_lt_add_left, -- bound a summand
rw dist_comm, -- symmetry
apply hM, -- apply the hypothesis
rotate 4,
apply le_trans,
apply add_le_of_add_le_right,
rotate 1,
apply dist_triangle,
rotate 9,
-- apply le_trans,
apply add_le_of_add_le_right,
rotate 1,
apply add_le_of_add_le_right,
rotate 1,
apply le_of_lt,
apply hM,
rotate 14,
apply add_le_of_add_le_left,
rotate 1,
have hF' := hF _, -- apply the hypothesis to an argument to be specified later
apply le_of_lt,
apply hF',
-- supply all the epsilons
rotate 2,
use ε / 3,
apply le_refl,
rotate 1,
use ε / 3,
rotate 2,
rotate 1,
-- use ε,
use δ (M (ε / 3) (by linarith)) x (ε / 3) (by linarith), -- now we can supply "the" delta belonging to "the" M
rotate 1,
apply le_refl,
apply le_of_eq,
apply add_eq_of_eq_sub,
refl,
apply le_of_eq,
apply eq_sub_of_add_eq,
ring_nf,
exact hxy,
apply le_refl,
apply hδ,
end
-- Toy example for this sort of behavior:
-- if we defer the choice of `y` at the very beginning,
-- the goal cannot be closed with `hx`
-- because it would assign `x` to the second goal which (syntactically) is not allowed.
-- But above we **need** to delay the choice of `δ` early so that we can continue to unpack and work with the goal.
theorem test (h : ∃ x, x = 0) : ∃ y, y = 0 :=
begin
-- apply exists.intro, -- uncomment
rcases h with ⟨x, hx⟩,
apply exists.intro, -- comment
exact hx,
end |
import analysis.inner_product_space.adjoint
variables {E 𝕜 : Type*}
[is_R_or_C 𝕜]
[inner_product_space 𝕜 E]
[finite_dimensional 𝕜 E]
namespace inner_product_space
local notation `⟪`x`, `y`⟫` := @inner 𝕜 E _ x y
lemma gram_self_adjoint (T : E →ₗ[𝕜] E): is_self_adjoint (T.adjoint * T) :=
begin
intros x y,
simp only [linear_map.mul_apply, linear_map.adjoint_inner_left, linear_map.adjoint_inner_right],
end
lemma gram_positive (T : E →ₗ[𝕜] E) :
∀ (x : E), is_R_or_C.re ⟪ (T.adjoint * T) x, x ⟫ ≥ 0 ∧ is_R_or_C.im ⟪ (T.adjoint * T) x, x⟫ = 0 :=
begin
intro x,
rw [linear_map.mul_apply, linear_map.adjoint_inner_left, inner_self_eq_norm_sq_to_K],
norm_cast,
split,
{apply sq_nonneg _},
{refl},
end
end inner_product_space |
/-
Copyright (c) 2022 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Selsam
-/
import data.nat.basic
import data.list.basic
import data.list.pairwise
import tactic.interval_cases
open tactic
/--
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each.
Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron.
At least one person sat between Karen and Darren. Who sat in the middle car?
-/
theorem amc8_2020_p6 (aaron darren karen maren sharon : ℕ) :
list.pairwise (≠) [aaron, darren, karen, maren, sharon]
→ list.all [aaron, darren, karen, maren, sharon] (λ n, n < 5)
→ maren = 0
→ aaron + 1 = sharon
→ darren > aaron
→ max darren karen > min darren karen + 1
→ aaron = 2 :=
begin
intros h_distinct h_bounds h_maren h_aaron_sharon h_darren_aaron h_darren_karen,
subst h_maren,
simp at h_distinct h_bounds,
cases_matching* [_ ∧ _],
interval_cases aaron ; try { contradiction } ; subst_vars ;
interval_cases darren ; try { contradiction } ; subst_vars ;
interval_cases karen ; try { contradiction } ; subst_vars ;
simp at h_darren_karen ; contradiction
end
|
theory EdmondsKarp_Algo
imports EdmondsKarp_Termination_Abstract FordFulkerson_Algo
begin
text \<open>
In this theory, we formalize an abstract version of
Edmonds-Karp algorithm, which we obtain by refining the
Ford-Fulkerson algorithm to always use shortest augmenting paths.
Then, we show that the algorithm always terminates within $O(VE)$ iterations.
\<close>
subsection \<open>Algorithm\<close>
context Network
begin
text \<open>First, we specify the refined procedure for finding augmenting paths\<close>
definition "find_shortest_augmenting_spec f \<equiv> assert (NFlow c s t f) \<then>
(select p. Graph.isShortestPath (residualGraph c f) s p t)"
text \<open>We show that our refined procedure is actually a refinement\<close>
thm SELECT_refine
lemma find_shortest_augmenting_refine[refine]:
"(f',f)\<in>Id \<Longrightarrow> find_shortest_augmenting_spec f' \<le> \<Down>(\<langle>Id\<rangle>option_rel) (find_augmenting_spec f)"
unfolding find_shortest_augmenting_spec_def find_augmenting_spec_def
apply (refine_vcg)
apply (auto
simp: NFlow.shortest_is_augmenting RELATESI
dest: NFlow.augmenting_path_imp_shortest)
done
text \<open>Next, we specify the Edmonds-Karp algorithm.
Our first specification still uses partial correctness,
termination will be proved afterwards. \<close>
definition "edka_partial \<equiv> do {
let f = (\<lambda>_. 0);
(f,_) \<leftarrow> while\<^bsup>fofu_invar\<^esup>
(\<lambda>(f,brk). \<not>brk)
(\<lambda>(f,_). do {
p \<leftarrow> find_shortest_augmenting_spec f;
case p of
None \<Rightarrow> return (f,True)
| Some p \<Rightarrow> do {
assert (p\<noteq>[]);
assert (NPreflow.isAugmentingPath c s t f p);
assert (Graph.isShortestPath (residualGraph c f) s p t);
let f = NFlow.augment_with_path c f p;
assert (NFlow c s t f);
return (f, False)
}
})
(f,False);
assert (NFlow c s t f);
return f
}"
lemma edka_partial_refine[refine]: "edka_partial \<le> \<Down>Id fofu"
unfolding edka_partial_def fofu_def
apply (refine_rcg bind_refine')
apply (refine_dref_type)
apply (vc_solve simp: find_shortest_augmenting_spec_def)
done
end \<comment> \<open>Network\<close>
subsubsection \<open>Total Correctness\<close>
context Network begin
text \<open>We specify the total correct version of Edmonds-Karp algorithm.\<close>
definition "edka \<equiv> do {
let f = (\<lambda>_. 0);
(f,_) \<leftarrow> while\<^sub>T\<^bsup>fofu_invar\<^esup>
(\<lambda>(f,brk). \<not>brk)
(\<lambda>(f,_). do {
p \<leftarrow> find_shortest_augmenting_spec f;
case p of
None \<Rightarrow> return (f,True)
| Some p \<Rightarrow> do {
assert (p\<noteq>[]);
assert (NPreflow.isAugmentingPath c s t f p);
assert (Graph.isShortestPath (residualGraph c f) s p t);
let f = NFlow.augment_with_path c f p;
assert (NFlow c s t f);
return (f, False)
}
})
(f,False);
assert (NFlow c s t f);
return f
}"
text \<open>Based on the measure function, it is easy to obtain a well-founded
relation that proves termination of the loop in the Edmonds-Karp algorithm:\<close>
definition "edka_wf_rel \<equiv> inv_image
(less_than_bool <*lex*> measure (\<lambda>cf. ek_analysis_defs.ekMeasure cf s t))
(\<lambda>(f,brk). (\<not>brk,residualGraph c f))"
text \<open>The following theorem states that the total correct
version of Edmonds-Karp algorithm refines the partial correct one.\<close>
theorem edka_refine[refine]: "edka \<le> \<Down>Id edka_partial"
unfolding edka_def edka_partial_def
apply (refine_rcg bind_refine'
WHILEIT_refine_WHILEI[where V=edka_wf_rel])
apply (refine_dref_type)
apply (simp; fail)
subgoal
txt \<open>Unfortunately, the verification condition for introducing
the variant requires a bit of manual massaging to be solved:\<close>
apply (simp)
apply (erule bind_sim_select_rule)
apply (auto split: option.split
simp: NFlow.augment_with_path_def
simp: assert_bind_spec_conv Let_def
simp: find_shortest_augmenting_spec_def
simp: edka_wf_rel_def NFlow.shortest_path_decr_ek_measure
; fail) []
done
txt \<open>The other VCs are straightforward\<close>
apply (vc_solve)
done
subsubsection \<open>Complexity Analysis\<close>
text \<open>For the complexity analysis, we additionally show that the measure
function is bounded by $O(VE)$. Note that our absolute bound is not as
precise as possible, but clearly $O(VE)$.\<close>
(* TODO: #edgesSp even bound by |E|, as either e or swap e lays on shortest path! *)
interpret ek: ek_analysis cf
by unfold_locales auto
have cardV_positive: "card V > 0" and cardE_positive: "card E > 0"
using card_0_eq[OF finite_V] V_not_empty apply blast
using card_0_eq[OF finite_E] E_not_empty apply blast
done
show ?thesis proof (cases "cf.connected s t")
case False hence "ek.ekMeasure = 0" by (auto simp: ek.ekMeasure_def)
with cardV_positive cardE_positive show ?thesis
by auto
next
case True
have "cf.min_dist s t > 0"
apply (rule ccontr)
apply (auto simp: Graph.min_dist_z_iff True s_not_t[symmetric])
done
have "cf = c"
unfolding residualGraph_def E_def
by auto
hence "ek.uE = E\<union>E\<inverse>" unfolding ek.uE_def by simp
from True have "ek.ekMeasure
= (card cf.V - cf.min_dist s t) * (card ek.uE + 1) + (card (ek.spEdges))"
unfolding ek.ekMeasure_def by simp
also from
mlex_bound[of "card cf.V - cf.min_dist s t" "card V",
OF _ ek.card_spEdges_less]
have "\<dots> < card V * (card ek.uE+1)"
using \<open>cf.min_dist s t > 0\<close> \<open>card V > 0\<close>
by (auto simp: resV_netV)
also have "card ek.uE \<le> 2*card E" unfolding \<open>ek.uE = E\<union>E\<inverse>\<close>
apply (rule order_trans)
apply (rule card_Un_le)
by auto
finally show ?thesis by (auto simp: algebra_simps)
qed
qed
text \<open>Finally, we present a version of the Edmonds-Karp algorithm
which is instrumented with a loop counter, and asserts that
there are less than $2|V||E|+|V| = O(|V||E|)$ iterations.
Note that we only count the non-breaking loop iterations.
\<close>
text \<open>The refinement is achieved by a refinement relation, coupling the
instrumented loop state with the uninstrumented one\<close>
definition "edkac_rel \<equiv> {((f,brk,itc), (f,brk)) | f brk itc.
itc + ek_analysis_defs.ekMeasure (residualGraph c f) s t
< 2 * card V * card E + card V
}"
definition "edka_complexity \<equiv> do {
let f = (\<lambda>_. 0);
(f,_,itc) \<leftarrow> while\<^sub>T
(\<lambda>(f,brk,_). \<not>brk)
(\<lambda>(f,_,itc). do {
p \<leftarrow> find_shortest_augmenting_spec f;
case p of
None \<Rightarrow> return (f,True,itc)
| Some p \<Rightarrow> do {
let f = NFlow.augment_with_path c f p;
return (f, False,itc + 1)
}
})
(f,False,0);
assert (itc < 2 * card V * card E + card V);
return f
}"
lemma edka_complexity_refine: "edka_complexity \<le> \<Down>Id edka"
proof -
have [refine_dref_RELATES]:
"RELATES edkac_rel"
by (auto simp: RELATES_def)
show ?thesis
unfolding edka_complexity_def edka_def
apply (refine_rcg)
apply (refine_dref_type)
apply (vc_solve simp: edkac_rel_def "NFlow.augment_with_path_def")
subgoal using ekMeasure_upper_bound by auto []
subgoal by (drule (1) NFlow.shortest_path_decr_ek_measure; auto)
done
qed
text \<open>We show that this algorithm never fails, and computes a maximum flow.\<close>
theorem "edka_complexity \<le> (spec f. isMaxFlow f)"
proof -
note edka_complexity_refine
also note edka_refine
also note edka_partial_refine
also note fofu_partial_correct
finally show ?thesis .
qed
end \<comment> \<open>Network\<close>
end \<comment> \<open>Theory\<close>
|
informal statement Suppose $T \in \mathcal{L}(V)$ is self-adjoint, $\lambda \in \mathbf{F}$, and $\epsilon>0$. Prove that if there exists $v \in V$ such that $\|v\|=1$ and $\|T v-\lambda v\|<\epsilon,$ then $T$ has an eigenvalue $\lambda^{\prime}$ such that $\left|\lambda-\lambda^{\prime}\right|<\epsilon$.formal statement theorem exercise_1_27 {n : ℕ} (hn : odd n) : 8 ∣ (n^2 - 1) := |
Require Import
Coq.Strings.String
Coq.Arith.Mult
Coq.Vectors.Vector.
Require Import
Fiat.Examples.DnsServer.SimplePacket
Fiat.Common.SumType
Fiat.Common.BoundedLookup
Fiat.Common.ilist
Fiat.Common.DecideableEnsembles
Fiat.Computation
Fiat.QueryStructure.Specification.Representation.Notations
Fiat.QueryStructure.Specification.Representation.Heading
Fiat.QueryStructure.Specification.Representation.Tuple
Fiat.BinEncoders.Env.BinLib.AlignedByteString
Fiat.BinEncoders.Env.BinLib.AlignWord
Fiat.BinEncoders.Env.Common.Specs
Fiat.BinEncoders.Env.Common.Compose
Fiat.BinEncoders.Env.Common.ComposeOpt
Fiat.BinEncoders.Env.Automation.Solver
Fiat.BinEncoders.Env.Lib2.WordOpt
Fiat.BinEncoders.Env.Lib2.NatOpt
Fiat.BinEncoders.Env.Lib2.StringOpt
Fiat.BinEncoders.Env.Lib2.EnumOpt
Fiat.BinEncoders.Env.Lib2.FixListOpt
Fiat.BinEncoders.Env.Lib2.SumTypeOpt
Fiat.BinEncoders.Env.Lib2.DomainNameOpt
Fiat.Common.IterateBoundedIndex
Fiat.Common.Tactics.CacheStringConstant.
Require Import
Bedrock.Word.
Section AlignedDecoders.
Context {cache : Cache}.
Context {cacheAddNat : CacheAdd cache nat}.
Lemma aligned_encode_char_eq
: forall (w : word 8) cd,
refine (encode_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) w cd)
(ret (build_aligned_ByteString (Vector.cons _ w _ (Vector.nil _)), addE cd 8)).
Proof.
intros; shatter_word w; simpl.
unfold encode_word_Spec; simpl.
compute.
intros.
computes_to_inv; subst.
match goal with
|- computes_to (ret ?c) ?v => replace c with v
end.
computes_to_econstructor.
f_equal.
eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _).
rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
erewrite eq_rect_Vector_cons; repeat f_equal.
instantiate (1 := eq_refl _); reflexivity.
Grab Existential Variables.
reflexivity.
Qed.
Lemma AlignedDecodeChar {C}
{numBytes}
: forall (v : Vector.t (word 8) (S numBytes))
(t : (word 8 * ByteString * CacheDecode) -> C)
(e : C)
cd,
Ifopt (decode_word
(transformerUnit := ByteString_QueueTransformerOpt) (sz := 8) (build_aligned_ByteString v) cd)
as w Then t w Else e
=
LetIn (Vector.nth v Fin.F1)
(fun w => t (w, build_aligned_ByteString (snd (Vector_split 1 _ v)), addD cd 8)).
Proof.
unfold LetIn; intros.
unfold decode_word, WordOpt.decode_word.
rewrite aligned_decode_char_eq; simpl.
f_equal.
pattern numBytes, v; apply Vector.caseS; simpl; intros.
reflexivity.
Qed.
Lemma AlignedEncodeChar {numBytes}
: forall (w : word 8) ce ce' (c : _ -> Comp _) (v : Vector.t _ numBytes),
refine (c (addE ce 8)) (ret (build_aligned_ByteString v, ce'))
-> refine (((encode_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) w)
ThenC c) ce)
(ret (build_aligned_ByteString (Vector.cons _ w _ v), ce')).
Proof.
unfold compose; intros.
unfold Bind2.
setoid_rewrite aligned_encode_char_eq; simplify with monad laws.
simpl; rewrite H; simplify with monad laws.
simpl.
rewrite <- build_aligned_ByteString_append.
reflexivity.
Qed.
Lemma AlignedDecode2Char {C}
{numBytes}
: forall (v : Vector.t (word 8) (S (S numBytes)))
(t : (word 16 * ByteString * CacheDecode) -> C)
(e : C)
cd,
Ifopt (decode_word
(transformerUnit := ByteString_QueueTransformerOpt) (sz := 16) (build_aligned_ByteString v) cd) as w
Then t w Else e=
Let n := Core.append_word (Vector.nth v (Fin.FS Fin.F1)) (Vector.nth v Fin.F1) in
t (n, build_aligned_ByteString (snd (Vector_split 2 _ v)), addD cd 16).
Proof.
unfold LetIn; intros.
unfold decode_word, WordOpt.decode_word.
match goal with
|- context[Ifopt ?Z as _ Then _ Else _] => replace Z with
(let (v', v'') := Vector_split 2 numBytes v in Some (VectorByteToWord v', build_aligned_ByteString v'')) by (symmetry; apply (@aligned_decode_char_eq' _ 1 v))
end.
unfold Vector_split, If_Opt_Then_Else, If_Opt_Then_Else.
f_equal.
rewrite !Vector_nth_tl, !Vector_nth_hd.
erewrite VectorByteToWord_cons.
rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
f_equal.
erewrite VectorByteToWord_cons.
rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
Grab Existential Variables.
omega.
omega.
Qed.
Variable addE_addE_plus :
forall (ce : CacheEncode) (n m : nat), addE (addE ce n) m = addE ce (n + m).
Lemma encode_words {n m}
: forall (w : word (n + m)) ce,
refine (encode_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) w ce)
((encode_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) (split1' _ _ w)
ThenC (encode_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) (split2' _ _ w)))
ce).
Proof.
induction n.
- unfold compose; simpl; intros.
unfold encode_word_Spec at 2; simpl.
autorewrite with monad laws.
simpl; rewrite addE_addE_plus.
pose proof transform_id_left as H'; simpl in H'; rewrite H'.
reflexivity.
- simpl; intros.
rewrite (word_split_SW w) at 1.
rewrite encode_SW_word.
unfold compose, Bind2.
rewrite (IHn (word_split_tl w) (addE ce 1)).
unfold compose, Bind2.
unfold encode_word_Spec; autorewrite with monad laws.
simpl.
rewrite encode_word_S.
pose proof transform_assoc as H'; simpl in H'.
rewrite !H'.
rewrite !addE_addE_plus; simpl.
f_equiv.
f_equiv.
f_equiv.
rewrite !word_split_hd_SW_word, !word_split_tl_SW_word.
fold plus.
clear;
generalize (split1' n m (word_split_tl w))
(ByteString_enqueue (word_split_hd w) ByteString_id).
induction w0; simpl in *.
+ intros; pose proof (transform_id_right b) as H; simpl in H; rewrite H; eauto.
+ intros.
rewrite <- (IHw0 (wtl w) b0).
pose proof enqueue_transform_opt as H'''; simpl in H'''.
rewrite <- H'''; eauto.
+ eauto.
Qed.
Lemma AlignedEncode2Char {numBytes}
: forall (w : word 16) ce ce' (c : _ -> Comp _) (v : Vector.t _ numBytes),
refine (c (addE ce 16)) (ret (build_aligned_ByteString v, ce'))
-> refine (((encode_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) w)
ThenC c) ce)
(ret (build_aligned_ByteString (Vector.cons
_ (split1' 8 8 w) _
(Vector.cons _ (split2' 8 8 w) _ v)), ce')).
Proof.
unfold compose, Bind2; intros.
intros; setoid_rewrite (@encode_words 8 8 w).
rewrite (@AlignedEncodeChar 1) by apply aligned_encode_char_eq.
simplify with monad laws.
unfold snd.
rewrite addE_addE_plus.
rewrite H.
simplify with monad laws.
unfold fst.
rewrite <- build_aligned_ByteString_append.
unfold append.
reflexivity.
Qed.
Lemma BindOpt_assoc {A A' A''} :
forall (a_opt : option A)
(f : A -> option A')
(g : A' -> option A''),
BindOpt (BindOpt a_opt f) g =
BindOpt a_opt (fun a => BindOpt (f a) g).
Proof.
destruct a_opt as [ ? | ]; simpl; intros; eauto.
Qed.
Corollary AlignedDecode2Nat {C}
{numBytes}
: forall (v : Vector.t (word 8) (S (S numBytes)))
(t : _ -> C)
e
cd,
Ifopt (decode_nat (transformerUnit := ByteString_QueueTransformerOpt) 16 (build_aligned_ByteString v) cd) as w
Then t w Else e
=
Let n := wordToNat (Core.append_word (Vector.nth v (Fin.FS Fin.F1)) (Vector.nth v Fin.F1)) in
t (n, build_aligned_ByteString (snd (Vector_split 2 _ v)), addD cd 16).
Proof.
unfold CacheDecode.
unfold decode_nat, DecodeBindOpt2; intros.
unfold BindOpt at 1.
rewrite AlignedDecode2Char.
reflexivity.
Qed.
Corollary AlignedEncode2Nat
{numBytes}
: forall (n : nat) ce ce' (c : _ -> Comp _) (v : Vector.t _ numBytes),
refine (c (addE ce 16)) (ret (build_aligned_ByteString v, ce'))
-> refine (((encode_nat_Spec 16 (transformerUnit := ByteString_QueueTransformerOpt) n)
ThenC c) ce)
(ret (build_aligned_ByteString (Vector.cons
_ (split1' 8 8 (natToWord 16 n)) _
(Vector.cons _ (split2' 8 8 (natToWord 16 n)) _ v)), ce')).
Proof.
unfold encode_nat_Spec; cbv beta; intros.
rewrite <- AlignedEncode2Char; eauto.
reflexivity.
Qed.
Lemma optimize_under_if_opt {A ResultT}
: forall (a_opt : option A) (t t' : A -> ResultT) (e e' : ResultT),
(forall a, t a = t' a) -> e = e' ->
Ifopt a_opt as a Then t a Else e = Ifopt a_opt as a Then t' a Else e'.
Proof.
intros; subst; eauto.
destruct a_opt; eauto.
Qed.
Lemma rewrite_under_LetIn
{A B}
: forall (a : A) (k k' : A -> B),
(forall a, k a = k' a) -> LetIn a k = LetIn a k'.
Proof.
intros; unfold LetIn; eauto.
Qed.
Fixpoint Guarded_Vector_split
(sz n : nat)
{struct sz}
: Vector.t (word 8) n
-> Vector.t (word 8) (sz + (n - sz)) :=
match sz, n return
Vector.t _ n
-> Vector.t (word 8) (sz + (n - sz))
with
| 0, _ => fun v => (eq_rect _ (Vector.t _) v _ (minus_n_O n))
| S n', 0 =>
fun v =>
Vector.cons _ (wzero _) _ (Guarded_Vector_split n' _ v)
| S n', S sz' =>
fun v =>
Vector.cons _ (Vector.hd v) _ (Guarded_Vector_split n' _ (Vector.tl v))
end .
Lemma le_B_Guarded_Vector_split
{n}
(b : Vector.t _ n)
(m : nat)
: {b' : ByteString | le_B b' (build_aligned_ByteString b)}.
eexists (build_aligned_ByteString
(snd (Vector_split _ _ (Guarded_Vector_split m n b)))).
abstract (unfold build_aligned_ByteString, le_B; simpl;
unfold length_ByteString; simpl; omega).
Defined.
Lemma build_aligned_ByteString_eq_split
: forall m n b H0,
(m <= n)%nat
-> build_aligned_ByteString b =
(build_aligned_ByteString (eq_rect (m + (n - m)) (t (word 8)) (Guarded_Vector_split m n b) n H0)).
Proof.
intros.
intros; eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := eq_refl _).
simpl.
revert n b H0 H; induction m; simpl.
intros ? ?; generalize (minus_n_O n).
intros; rewrite <- Equality.transport_pp.
apply Eqdep_dec.eq_rect_eq_dec; auto with arith.
intros.
inversion H; subst.
- revert b H0 IHm; clear.
intro; pattern m, b; apply Vector.caseS; simpl; intros.
assert ((n + (n - n)) = n) by omega.
rewrite eq_rect_Vector_cons with (H' := H).
f_equal.
erewrite <- IHm; eauto.
- revert b H0 IHm H1; clear.
intro; pattern m0, b; apply Vector.caseS; simpl; intros.
assert ((m + (n - m)) = n) by omega.
erewrite eq_rect_Vector_cons with (H' := H).
f_equal.
erewrite <- IHm; eauto.
omega.
Qed.
Lemma ByteAlign_Decode_w_Measure_le {A}
: forall (dec_a : ByteString -> CacheDecode -> option (A * ByteString * CacheDecode))
(n m : nat)
(dec_a' : Vector.t (word 8) (m + (n - m)) -> A)
(cd : CacheDecode)
(f : CacheDecode -> CacheDecode)
(b : Vector.t (word 8) n)
decode_a_le
(dec_fail : ~ (m <= n)%nat
-> forall b cd, dec_a (build_aligned_ByteString (numBytes := n) b) cd = None),
(forall b cd, dec_a (build_aligned_ByteString b) cd =
Some (dec_a' b, build_aligned_ByteString (snd (Vector_split m (n - m) b)), f cd))
-> Decode_w_Measure_le dec_a (build_aligned_ByteString b) cd decode_a_le =
match Compare_dec.le_dec m n with
| left e => (Let a := dec_a' (Guarded_Vector_split m n b) in
Some (a, le_B_Guarded_Vector_split b m, f cd))
| right _ => None
end.
Proof.
intros.
destruct (Compare_dec.le_dec m n).
assert (m + (n - m) = n) by omega.
assert (forall b, Decode_w_Measure_le dec_a (build_aligned_ByteString b) cd decode_a_le
= Decode_w_Measure_le dec_a (build_aligned_ByteString ( eq_rect _ _ (Guarded_Vector_split m n b) _ H0)) cd decode_a_le).
{ revert l; clear; intros.
destruct (Decode_w_Measure_le dec_a (build_aligned_ByteString b) cd decode_a_le)
as [ [ [? [? ?] ] ?] | ] eqn: ?.
apply Decode_w_Measure_le_eq' in Heqo.
simpl in Heqo.
destruct (Decode_w_Measure_le dec_a
(build_aligned_ByteString (eq_rect (m + (n - m)) (t (word 8)) (Guarded_Vector_split m n b) n H0)) cd decode_a_le) as [ [ [? [? ?] ] ?] | ] eqn: ?.
apply Decode_w_Measure_le_eq' in Heqo0.
simpl in *.
rewrite <- build_aligned_ByteString_eq_split in Heqo0 by eauto.
rewrite Heqo0 in Heqo.
injection Heqo; intros.
rewrite H, H2;
repeat f_equal.
revert l0 l1. rewrite H1; intros; f_equal.
f_equal; apply Core.le_uniqueness_proof.
apply ByteString_id.
eapply Decode_w_Measure_le_eq'' in Heqo0.
rewrite <- build_aligned_ByteString_eq_split in Heqo0 by eauto.
rewrite Heqo0 in Heqo.
discriminate.
apply ByteString_id.
erewrite build_aligned_ByteString_eq_split in Heqo by eauto.
rewrite Heqo; reflexivity.
}
rewrite H1.
match goal with
|- ?a = _ => destruct a as [ [ [? ?] ? ] | ] eqn: ?
end.
eapply Decode_w_Measure_le_eq' in Heqo.
assert (dec_a (build_aligned_ByteString (Guarded_Vector_split m n b)) cd
= Some (a, projT1 s, c)).
{ destruct s; simpl in *.
rewrite <- Heqo.
unfold build_aligned_ByteString; repeat f_equal; simpl.
eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := sym_eq H0).
clear H1.
destruct H0; reflexivity.
}
rewrite H in H2; injection H2; intros.
rewrite H3, H5; unfold LetIn; simpl.
repeat f_equal.
destruct s; simpl in *.
unfold le_B_Guarded_Vector_split; simpl.
clear H1; revert l0.
rewrite <- H4; intros.
f_equal; apply Core.le_uniqueness_proof.
apply ByteString_id.
apply Decode_w_Measure_le_eq'' in Heqo.
pose proof (H (Guarded_Vector_split m n b) cd).
assert (Some (dec_a' (Guarded_Vector_split m n b),
build_aligned_ByteString (snd (Vector_split m (n - m) (Guarded_Vector_split m n b))),
f cd) = None).
{ rewrite <- Heqo.
rewrite <- H.
repeat f_equal.
eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := sym_eq H0).
clear H1.
destruct H0; reflexivity.
}
discriminate.
eapply dec_fail in n0; simpl.
eapply Specs.Decode_w_Measure_le_eq' in n0.
apply n0.
Qed.
Lemma If_Opt_Then_Else_BindOpt {A B C}
: forall (a_opt : option A)
(t : A -> option B)
(e : option B)
(k : _ -> option C),
BindOpt (Ifopt a_opt as a Then t a Else e) k
= Ifopt a_opt as a Then (BindOpt (t a) k) Else (BindOpt e k).
Proof.
destruct a_opt; simpl; intros; reflexivity.
Qed.
Lemma lt_B_Guarded_Vector_split
{n}
(b : Vector.t _ n)
(m : nat)
(m_OK : lt 0 m)
(_ : ~ lt n m)
: {b' : ByteString | lt_B b' (build_aligned_ByteString b)}.
eexists (build_aligned_ByteString
(snd (Vector_split _ _ (Guarded_Vector_split m n b)))).
abstract (unfold build_aligned_ByteString, lt_B; simpl;
unfold length_ByteString; simpl; omega).
Defined.
Fixpoint BytesToString {sz}
(b : Vector.t (word 8) sz)
: string :=
match b with
| Vector.nil => EmptyString
| Vector.cons a _ b' => String (Ascii.ascii_of_N (wordToN a)) (BytesToString b')
end.
Variable addD_addD_plus :
forall (cd : CacheDecode) (n m : nat), addD (addD cd n) m = addD cd (n + m).
Lemma decode_string_aligned_ByteString
{sz sz'}
: forall (b : t (word 8) (sz + sz'))
(cd : CacheDecode),
FixStringOpt.decode_string sz (build_aligned_ByteString b) cd =
Some (BytesToString (fst (Vector_split sz sz' b)),
build_aligned_ByteString (snd (Vector_split sz sz' b)),
addD cd (8 * sz)).
Proof.
induction sz; intros.
- simpl; repeat f_equal.
- simpl.
assert (forall A n b, exists a b', b = Vector.cons A a n b')
by (clear; intros; pattern n, b; apply caseS; eauto).
destruct (H _ _ b) as [? [? ?] ]; subst; simpl.
unfold AsciiOpt.decode_ascii, DecodeBindOpt2.
rewrite BindOpt_assoc; simpl.
etransitivity.
unfold BindOpt at 1.
rewrite AlignedDecodeChar; simpl.
rewrite IHsz. higher_order_reflexivity.
simpl.
destruct (Vector_split sz sz' x0); simpl.
unfold LetIn.
rewrite addD_addD_plus;
repeat f_equal.
omega.
Qed.
Fixpoint StringToBytes
(s : string)
: Vector.t (word 8) (String.length s) :=
match s return Vector.t (word 8) (String.length s) with
| EmptyString => Vector.nil _
| String a s' => Vector.cons _ (NToWord 8 (Ascii.N_of_ascii a)) _ (StringToBytes s')
end.
Lemma encode_string_ByteString
: forall (s : string)
(ce : CacheEncode),
refine (FixStringOpt.encode_string_Spec s ce)
(ret (build_aligned_ByteString (StringToBytes s), addE ce (8 * String.length s))).
Proof.
induction s; intros; simpl.
- f_equiv. f_equal.
eapply ByteString_f_equal.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := eq_refl _); reflexivity.
- unfold AsciiOpt.encode_ascii_Spec.
unfold Bind2; setoid_rewrite aligned_encode_char_eq.
simplify with monad laws.
rewrite IHs.
simplify with monad laws.
simpl.
rewrite <- build_aligned_ByteString_append; simpl.
rewrite !addE_addE_plus.
f_equiv; f_equiv; f_equiv; omega.
Qed.
Lemma encode_string_aligned_ByteString {numBytes}
: forall (s : string)
(ce : CacheEncode)
(ce' : CacheEncode)
(c : _ -> Comp _) (v : Vector.t _ numBytes),
refine (c (addE ce (8 * String.length s))) (ret (build_aligned_ByteString v, ce'))
-> refine (((FixStringOpt.encode_string_Spec s) ThenC c) ce)
(ret (build_aligned_ByteString (append (StringToBytes s) v), ce')).
Proof.
unfold compose, Bind2; autorewrite with monad laws; intros.
rewrite encode_string_ByteString.
simplify with monad laws.
unfold snd at 1.
rewrite H.
simplify with monad laws.
simpl.
rewrite <- build_aligned_ByteString_append; simpl.
reflexivity.
Qed.
Lemma decode_string_aligned_ByteString_overflow
{sz}
: forall {sz'}
(b : t (word 8) sz')
(cd : CacheDecode),
lt sz' sz
-> FixStringOpt.decode_string sz (build_aligned_ByteString b) cd = None.
Proof.
induction sz; intros.
- omega.
- simpl.
assert (forall A n b, exists a b', b = Vector.cons A a n b')
by (clear; intros; pattern n, b; apply caseS; eauto).
destruct sz'.
+ reflexivity.
+ destruct (H0 _ _ b) as [? [? ?] ]; subst; simpl.
unfold AsciiOpt.decode_ascii, DecodeBindOpt2.
rewrite BindOpt_assoc; simpl.
etransitivity.
unfold BindOpt.
rewrite AlignedDecodeChar; simpl.
rewrite IHsz. higher_order_reflexivity.
omega.
reflexivity.
Qed.
Lemma ByteAlign_Decode_w_Measure_lt {A}
: forall (dec_a : nat -> ByteString -> CacheDecode -> option (A * ByteString * CacheDecode))
(n m : nat)
(dec_a' : forall m n, Vector.t (word 8) (m + n) -> A)
(cd : CacheDecode)
(f : nat -> CacheDecode -> CacheDecode)
(b : Vector.t (word 8) n)
(m_OK : lt 0 m)
decode_a_le
(dec_fail : (lt n m)%nat
-> forall b cd, dec_a m (build_aligned_ByteString (numBytes := n) b) cd = None),
(forall n m b cd, dec_a m (build_aligned_ByteString b) cd =
Some (dec_a' _ _ b, build_aligned_ByteString (snd (Vector_split m n b)), f m cd))
-> Decode_w_Measure_lt (dec_a m) (build_aligned_ByteString b) cd decode_a_le =
match Compare_dec.lt_dec n m with
| left _ => None
| right n' => (Let a := dec_a' _ _ (Guarded_Vector_split m n b) in
Some (a, lt_B_Guarded_Vector_split b m m_OK n' , f m cd))
end.
Proof.
intros.
destruct (Compare_dec.lt_dec m n);
destruct (Compare_dec.lt_dec n m); try omega.
- assert (m + (n - m) = n) by omega.
assert (forall b, Decode_w_Measure_lt (dec_a m) (build_aligned_ByteString b) cd decode_a_le
= Decode_w_Measure_lt (dec_a m)(build_aligned_ByteString ( eq_rect _ _ (Guarded_Vector_split m n b) _ H0)) cd decode_a_le).
{ revert l; clear; intros.
destruct (Decode_w_Measure_lt (dec_a m) (build_aligned_ByteString b) cd decode_a_le)
as [ [ [? [? ?] ] ?] | ] eqn: ?.
apply Decode_w_Measure_lt_eq' in Heqo.
simpl in Heqo.
destruct (Decode_w_Measure_lt (dec_a m)
(build_aligned_ByteString (eq_rect _ (t (word 8)) (Guarded_Vector_split m n b) n H0)) cd decode_a_le) as [ [ [? [? ?] ] ?] | ] eqn: ?.
apply Decode_w_Measure_lt_eq' in Heqo0.
unfold proj1_sig in Heqo0.
rewrite <- build_aligned_ByteString_eq_split in Heqo0.
rewrite Heqo0 in Heqo.
injection Heqo; intros.
rewrite H, H2;
repeat f_equal.
revert l1 l0. rewrite H1; intros; f_equal.
f_equal; apply Core.le_uniqueness_proof.
omega.
apply ByteString_id.
eapply Decode_w_Measure_lt_eq'' in Heqo0.
rewrite <- build_aligned_ByteString_eq_split in Heqo0 by omega.
rewrite Heqo0 in Heqo.
discriminate.
apply ByteString_id.
erewrite (build_aligned_ByteString_eq_split m n) in Heqo by omega.
rewrite Heqo; reflexivity.
}
rewrite H1.
match goal with
|- ?a = _ => destruct a as [ [ [? ?] ? ] | ] eqn: ?
end.
eapply Decode_w_Measure_lt_eq' in Heqo.
assert (dec_a m (build_aligned_ByteString (Guarded_Vector_split m n b)) cd
= Some (a, projT1 s, c)).
{ destruct s; simpl in *.
rewrite <- Heqo.
unfold build_aligned_ByteString; repeat f_equal; simpl.
eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := sym_eq H0).
clear H1.
destruct H0; reflexivity.
}
rewrite H in H2; injection H2; intros.
rewrite H3, H5; unfold LetIn; simpl.
repeat f_equal.
destruct s; simpl in *.
unfold lt_B_Guarded_Vector_split; simpl.
clear H1; revert l0.
rewrite <- H4; intros.
f_equal. apply Core.le_uniqueness_proof.
apply ByteString_id.
apply Decode_w_Measure_lt_eq'' in Heqo.
pose proof (H _ _ (Guarded_Vector_split m n b) cd).
assert (Some (dec_a' _ _ (Guarded_Vector_split m n b),
build_aligned_ByteString (snd (Vector_split m (n - m) (Guarded_Vector_split m n b))),
f m cd) = None).
{ rewrite <- Heqo.
rewrite <- H.
repeat f_equal.
eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := sym_eq H0).
clear H1.
destruct H0; reflexivity.
}
discriminate.
- eapply dec_fail in l; simpl.
eapply Specs.Decode_w_Measure_lt_eq' in l.
apply l.
- assert (m = n) by omega; subst.
assert (n + (n - n) = n) by omega.
assert (forall b, Decode_w_Measure_lt (dec_a n) (build_aligned_ByteString b) cd decode_a_le
= Decode_w_Measure_lt (dec_a n)(build_aligned_ByteString ( eq_rect _ _ (Guarded_Vector_split n n b) _ H0)) cd decode_a_le).
{ clear; intros.
destruct (Decode_w_Measure_lt (dec_a n) (build_aligned_ByteString b) cd decode_a_le)
as [ [ [? [? ?] ] ?] | ] eqn: ?.
apply Decode_w_Measure_lt_eq' in Heqo.
simpl in Heqo.
destruct (Decode_w_Measure_lt (dec_a n)
(build_aligned_ByteString (eq_rect _ (t (word 8)) (Guarded_Vector_split n n b) n H0)) cd decode_a_le) as [ [ [? [? ?] ] ?] | ] eqn: ?.
apply Decode_w_Measure_lt_eq' in Heqo0.
unfold proj1_sig in Heqo0.
rewrite <- build_aligned_ByteString_eq_split in Heqo0.
rewrite Heqo0 in Heqo.
injection Heqo; intros.
rewrite H, H2;
repeat f_equal.
revert l l0. rewrite H1; intros; f_equal.
f_equal; apply Core.le_uniqueness_proof.
omega.
apply ByteString_id.
eapply Decode_w_Measure_lt_eq'' in Heqo0.
rewrite <- build_aligned_ByteString_eq_split in Heqo0 by omega.
rewrite Heqo0 in Heqo.
discriminate.
apply ByteString_id.
erewrite (build_aligned_ByteString_eq_split n n) in Heqo by omega.
rewrite Heqo; reflexivity.
}
rewrite H1.
match goal with
|- ?a = _ => destruct a as [ [ [? ?] ? ] | ] eqn: ?
end.
eapply Decode_w_Measure_lt_eq' in Heqo.
assert (dec_a n (build_aligned_ByteString (Guarded_Vector_split n n b)) cd
= Some (a, projT1 s, c)).
{ destruct s; simpl in *.
rewrite <- Heqo.
unfold build_aligned_ByteString; repeat f_equal; simpl.
eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := sym_eq H0).
clear H1.
destruct H0; reflexivity.
}
rewrite H in H2; injection H2; intros.
rewrite H3, H5; unfold LetIn; simpl.
repeat f_equal.
destruct s; simpl in *.
unfold lt_B_Guarded_Vector_split; simpl.
clear H1; revert l.
rewrite <- H4; intros.
f_equal. apply Core.le_uniqueness_proof.
apply ByteString_id.
apply Decode_w_Measure_lt_eq'' in Heqo.
pose proof (H _ _ (Guarded_Vector_split n n b) cd).
assert (Some (dec_a' _ _ (Guarded_Vector_split n n b),
build_aligned_ByteString (snd (Vector_split n (n - n) (Guarded_Vector_split n n b))),
f n cd) = None).
{ rewrite <- Heqo.
rewrite <- H.
repeat f_equal.
eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := sym_eq H0).
clear H1.
destruct H0; reflexivity.
}
discriminate.
Qed.
Lemma optimize_under_match {A B} {P}
: forall (a a' : A) (f : {P a a'} + {~P a a'}) (t t' : _ -> B)
(e e' : _ -> B),
(forall (a a' : A) (a_eq : _), t a_eq = t' a_eq)
-> (forall (a a' : A) (a_neq : _), e a_neq = e' a_neq)
-> match f with
| left e => t e
| right n => e n
end =
match f with
| left e => t' e
| right n => e' n
end.
Proof.
destruct f; simpl; intros; eauto.
Qed.
Lemma optimize_Fix {A}
: forall
(body : forall x : ByteString,
(forall y : ByteString,
lt_B y x -> (fun _ : ByteString => CacheDecode -> option (A * ByteString * CacheDecode)) y) ->
(fun _ : ByteString => CacheDecode -> option (A * ByteString * CacheDecode)) x)
(body' : forall x : nat,
(forall y : nat,
(lt y x)%nat ->
(fun m : nat =>
t (word 8) m -> CacheDecode ->
option (A * {n : _ & Vector.t _ n} * CacheDecode)) y) ->
t (word 8) x -> CacheDecode -> option (A * {n : _ & Vector.t _ n} * CacheDecode) )
n (b : Vector.t _ n) (cd : CacheDecode)
(body_Proper :
forall (x0 : ByteString)
(f g : forall y : ByteString, lt_B y x0 -> CacheDecode -> option (A * ByteString * CacheDecode)),
(forall (y : ByteString) (p : lt_B y x0), f y p = g y p) -> body x0 f = body x0 g)
(body'_Proper :
forall (x0 : nat)
(f
g : forall y : nat,
(lt y x0)%nat -> t (word 8) y -> CacheDecode -> option (A * {n0 : nat & t Core.char n0} * CacheDecode)),
(forall (y : nat) (p : (lt y x0)%nat), f y p = g y p) -> body' x0 f = body' x0 g)
,
(forall n (b : Vector.t (word 8) n)
(rec : forall x : ByteString,
lt_B x (build_aligned_ByteString b) -> CacheDecode -> option (A * ByteString * CacheDecode))
(rec' : forall x : nat,
(lt x n)%nat -> t Core.char x -> CacheDecode ->
option (A * {n : _ & Vector.t _ n} * CacheDecode))
cd,
(forall m cd b a b' cd' b_lt b_lt' ,
rec' m b_lt' b cd = Some (a, b', cd')
-> rec (build_aligned_ByteString b) b_lt cd = Some (a, build_aligned_ByteString (projT2 b'), cd'))
-> (forall m cd b b_lt b_lt' ,
rec' m b_lt' b cd = None
-> rec (build_aligned_ByteString b) b_lt cd = None)
-> body (build_aligned_ByteString b) rec cd
= match (body' n rec' b cd) with
| Some (a, b', cd') => Some (a, build_aligned_ByteString (projT2 b'), cd')
| None => None
end)
-> Fix well_founded_lt_b (fun _ : ByteString => CacheDecode -> option (A * ByteString * CacheDecode)) body (build_aligned_ByteString b) cd =
match Fix Wf_nat.lt_wf (fun m : nat => Vector.t (word 8) m -> CacheDecode -> option (A * { n : _ & Vector.t _ n} * CacheDecode)) body' n b cd with
| Some (a, b', cd') => Some (a, build_aligned_ByteString (projT2 b'), cd')
| None => None
end.
Proof.
intros.
revert cd b; pattern n.
eapply (well_founded_ind Wf_nat.lt_wf); intros.
rewrite Init.Wf.Fix_eq, Init.Wf.Fix_eq.
apply H; intros.
erewrite H0, H1; eauto.
rewrite H0, H1; eauto.
eauto.
eauto.
Qed.
Lemma lift_match_if_ByteAlign
{T1}
{T2 T3 T4 A : T1 -> Type}
{B B' C}
: forall (b : bool)
(t1 : T1)
(t e : option (A t1 * B * C))
(b' : forall t1, T2 t1 -> T3 t1 -> T4 t1 -> bool)
(t' e' : forall t1, T2 t1 -> T3 t1 -> T4 t1 -> option (A t1 * B' * C))
(f : B' -> B)
(t2 : T2 t1)
(t3 : T3 t1)
(t4 : T4 t1),
(b = b' t1 t2 t3 t4)
-> (t = match t' t1 t2 t3 t4 with
| Some (a, b', c) => Some (a, f b', c)
| None => None
end)
-> (e = match e' t1 t2 t3 t4 with
| Some (a, b', c) => Some (a, f b', c)
| None => None
end)
-> (if b then t else e) =
match (fun t1 t2 t3 t4 => if b' t1 t2 t3 t4 then t' t1 t2 t3 t4 else e' t1 t2 t3 t4) t1 t2 t3 t4 with
| Some (a, b', c) => Some (a, f b', c)
| None => None
end.
Proof.
intros; destruct b; eauto; rewrite <- H; simpl; eauto.
Qed.
Lemma lift_match_if_sumbool_ByteAlign
{T1}
{T3 : T1 -> Type}
{P : forall t1 (t3 : T3 t1), Prop}
{T2 T4 A : T1 -> Type}
{B B' C}
: forall (t1 : T1)
(t3 : T3 t1)
(b : forall t1 t3, {P t1 t3} + {~P t1 t3})
(t : _ -> option (A t1 * B * C))
(e : _ -> option (A t1 * B * C))
(b' : forall t1 t3, T2 t1 -> T4 t1 -> {P t1 t3} + {~P t1 t3})
(t' : forall t1 t3, T2 t1 -> T4 t1 -> _ -> option (A t1 * B' * C))
(e' : forall t1 t3, T2 t1 -> T4 t1 -> _ -> option (A t1 * B' * C))
(f : B' -> B)
(t2 : T2 t1)
(t4 : T4 t1),
(b t1 t3 = b' t1 t3 t2 t4)
-> (forall e'',
t e'' = match t' t1 t3 t2 t4 e'' with
| Some (a, b', c) => Some (a, f b', c)
| None => None
end)
-> (forall e'',
e e'' = match e' t1 t3 t2 t4 e'' with
| Some (a, b', c) => Some (a, f b', c)
| None => None
end)
-> (match b t1 t3 with
left e'' => t e''
| right e'' => e e''
end) =
match (fun t1 t2 t3 t4 =>
match b' t1 t3 t2 t4 with
| left e'' => t' t1 t3 t2 t4 e''
| right e'' => e' t1 t3 t2 t4 e''
end) t1 t2 t3 t4 with
| Some (a, b', c) => Some (a, f b', c)
| None => None
end.
Proof.
intros; destruct b; eauto; rewrite <- H; simpl; eauto.
Qed.
Lemma SW_word_append :
forall b sz (w : word sz) sz' (w' : word sz'),
SW_word b (Core.append_word w w')
= eq_rect _ word (Core.append_word w (SW_word b w')) _ (sym_eq (plus_n_Sm _ _)).
Proof.
induction w; simpl; intros.
- apply Eqdep_dec.eq_rect_eq_dec; auto with arith.
- erewrite <- !WS_eq_rect_eq.
rewrite IHw; reflexivity.
Qed.
Lemma decode_word_plus':
forall (n m : nat) (v : ByteString),
decode_word' (n + m) v =
(`(w, v') <- decode_word' n v;
`(w', v'') <- decode_word' m v';
Some (eq_rect _ _ (Core.append_word w' w) _ (plus_comm _ _), v'')).
Proof.
induction n.
- simpl; intros.
destruct (decode_word' m v) as [ [? ?] | ]; simpl; repeat f_equal.
revert w; clear.
induction w; simpl; eauto.
rewrite IHw at 1.
rewrite Core.succ_eq_rect; f_equal.
apply Eqdep_dec.UIP_dec; auto with arith.
- simpl; intros.
simpl; rewrite !DecodeBindOpt_assoc;
destruct (ByteString_dequeue v) as [ [? ?] | ]; try reflexivity.
simpl; rewrite !DecodeBindOpt_assoc.
rewrite IHn.
simpl; rewrite !DecodeBindOpt_assoc.
destruct (decode_word' n b0) as [ [? ?] | ]; try reflexivity.
simpl; rewrite !DecodeBindOpt_assoc.
destruct (decode_word' m b1) as [ [? ?] | ]; try reflexivity.
simpl; f_equal; f_equal; clear.
revert b n w; induction w0; simpl; intros.
+ apply SW_word_eq_rect_eq.
+ erewrite !SW_word_eq_rect_eq; simpl.
erewrite <- !WS_eq_rect_eq.
f_equal.
rewrite SW_word_append.
rewrite <- Equality.transport_pp.
f_equal.
Grab Existential Variables.
omega.
omega.
Qed.
Lemma decode_word_aligned_ByteString_overflow
{sz'}
: forall (b : t (word 8) sz')
{sz : nat}
(cd : CacheDecode),
lt sz' sz
-> decode_word (sz := 8 * sz) (build_aligned_ByteString b) cd = None.
Proof.
induction b; intros.
- unfold build_aligned_ByteString; simpl.
inversion H; subst; reflexivity.
- destruct sz; try omega.
apply lt_S_n in H.
pose proof (IHb _ cd H).
unfold decode_word, WordOpt.decode_word.
rewrite <- mult_n_Sm, plus_comm.
rewrite decode_word_plus'.
rewrite (@aligned_decode_char_eq' _ 0).
simpl.
unfold build_aligned_ByteString, decode_word in *.
simpl in H0.
destruct (decode_word' (sz + (sz + (sz + (sz + (sz + (sz + (sz + (sz + 0))))))))
{|
padding := 0;
front := WO;
paddingOK := build_aligned_ByteString_subproof n b;
numBytes := n;
byteString := b |}) as [ [? ?] | ]; simpl in *; try congruence.
Qed.
Lemma build_aligned_ByteString_eq_split'
: forall n sz v,
(n <= sz)%nat
->
build_aligned_ByteString v
= build_aligned_ByteString (Guarded_Vector_split n sz v).
Proof.
intros; eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := (le_plus_minus_r _ _ H)).
generalize (le_plus_minus_r n sz H); clear.
revert sz v; induction n; simpl; intros.
unfold Guarded_Vector_split.
rewrite <- Equality.transport_pp.
generalize (eq_trans (minus_n_O sz) e); clear;
intro.
apply Eqdep_dec.eq_rect_eq_dec; auto with arith.
destruct v; simpl in *.
omega.
unfold Guarded_Vector_split; fold Guarded_Vector_split;
simpl.
erewrite eq_rect_Vector_cons; eauto.
f_equal.
apply IHn.
Grab Existential Variables.
omega.
Qed.
Lemma optimize_Guarded_Decode {sz} {C} n
: forall (a_opt : ByteString -> option C)
(a_opt' : ByteString -> option C) v,
(~ (n <= sz)%nat
-> a_opt (build_aligned_ByteString v) = None)
-> (le n sz -> a_opt (build_aligned_ByteString (Guarded_Vector_split n sz v))
= a_opt'
(build_aligned_ByteString (Guarded_Vector_split n sz v)))
-> a_opt (build_aligned_ByteString v) =
If NPeano.leb n sz Then
a_opt' (build_aligned_ByteString (Guarded_Vector_split n sz v))
Else None.
Proof.
intros; destruct (NPeano.leb n sz) eqn: ?.
- apply NPeano.leb_le in Heqb.
rewrite <- H0.
simpl; rewrite <- build_aligned_ByteString_eq_split'; eauto.
eauto.
- rewrite H; simpl; eauto.
intro.
rewrite <- NPeano.leb_le in H1; congruence.
Qed.
Lemma AlignedDecode4Char {C}
{numBytes}
: forall (v : Vector.t (word 8) (S (S (S (S numBytes)))))
(t : _ -> C)
(e : C)
cd,
Ifopt (decode_word
(transformerUnit := ByteString_QueueTransformerOpt) (sz := 32) (build_aligned_ByteString v) cd) as w
Then t w Else e =
Let n := Core.append_word (Vector.nth v (Fin.FS (Fin.FS (Fin.FS Fin.F1))))
(Core.append_word (Vector.nth v (Fin.FS (Fin.FS Fin.F1)))
(Core.append_word (Vector.nth v (Fin.FS Fin.F1)) (Vector.nth v Fin.F1))) in
t (n, build_aligned_ByteString (snd (Vector_split 4 _ v)), addD cd 32).
Proof.
unfold LetIn; intros.
unfold decode_word, WordOpt.decode_word.
match goal with
|- context[Ifopt ?Z as _ Then _ Else _] => replace Z with
(let (v', v'') := Vector_split 4 numBytes v in Some (VectorByteToWord v', build_aligned_ByteString v'')) by (symmetry; apply (@aligned_decode_char_eq' _ 3 v))
end.
Local Transparent Vector_split.
unfold Vector_split, If_Opt_Then_Else, If_Opt_Then_Else.
f_equal.
rewrite !Vector_nth_tl, !Vector_nth_hd.
erewrite VectorByteToWord_cons.
rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
f_equal.
erewrite VectorByteToWord_cons.
rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
erewrite VectorByteToWord_cons.
rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
erewrite VectorByteToWord_cons.
rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
Grab Existential Variables.
omega.
omega.
omega.
omega.
Qed.
Lemma split2_split2
: forall n m o (w : word (n + (m + o))),
split2' m o (split2' n (m + o) w) =
split2' (n + m) o (eq_rect _ _ w _ (plus_assoc _ _ _)).
Proof.
induction n; simpl; intros.
- rewrite <- Eqdep_dec.eq_rect_eq_dec; auto with arith.
- rewrite IHn.
f_equal.
pose proof (shatter_word_S w); destruct_ex; subst.
clear.
rewrite <- WS_eq_rect_eq with (H := plus_assoc n m o).
revert m o x0 x; induction n; simpl; intros.
+ rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
+ erewrite <- WS_eq_rect_eq; fold plus; pose proof (shatter_word_S x0);
destruct_ex; subst; f_equal.
rewrite IHn; f_equal.
erewrite <- WS_eq_rect_eq; reflexivity.
Qed.
Lemma AlignedEncode32Char {numBytes}
: forall (w : word 32) ce ce' (c : _ -> Comp _) (v : Vector.t _ numBytes),
refine (c (addE ce 32)) (ret (build_aligned_ByteString v, ce'))
-> refine (((encode_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) w)
ThenC c) ce)
(ret (build_aligned_ByteString
(Vector.cons
_ (split1' 8 24 w) _
(Vector.cons
_
(split1' 8 16 (split2' 8 24 w)) _
(Vector.cons
_
(split1' 8 8 (split2' 16 16 w)) _
(Vector.cons
_
(split2' 24 8 w) _ v)))), ce')).
Proof.
unfold compose, Bind2; intros.
intros; setoid_rewrite (@encode_words 8 24 w).
rewrite (@AlignedEncodeChar 3).
simplify with monad laws.
unfold snd.
rewrite H.
simplify with monad laws.
unfold fst.
unfold transform.
unfold ByteStringQueueTransformer.
rewrite <- build_aligned_ByteString_append.
instantiate (1 := Vector.cons _ _ _ (Vector.cons _ _ _ (Vector.cons _ _ _ (Vector.nil _)))).
unfold append.
reflexivity.
setoid_rewrite (@encode_words 8 16 _).
rewrite (@AlignedEncodeChar 2).
reflexivity.
setoid_rewrite (@encode_words 8 8).
rewrite (@AlignedEncodeChar 1) by apply aligned_encode_char_eq.
rewrite !addE_addE_plus; simpl plus.
rewrite !split2_split2.
simpl plus.
rewrite <- !Eqdep_dec.eq_rect_eq_dec; auto with arith.
reflexivity.
Qed.
Fixpoint align_decode_list {A}
(A_decode_align : forall n,
Vector.t (word 8) n
-> CacheDecode
-> option (A * {n : _ & Vector.t _ n}
* CacheDecode))
(n : nat)
{sz}
(v : Vector.t (word 8) sz)
(cd : CacheDecode)
: option (list A * {n : _ & Vector.t _ n} * CacheDecode) :=
match n with
| 0 => Some (@nil _, existT _ _ v, cd)
| S s' => `(x, b1, e1) <- A_decode_align sz v cd;
`(xs, b2, e2) <- align_decode_list A_decode_align s' (projT2 b1) e1;
Some ((x :: xs)%list, b2, e2)
end.
Lemma optimize_align_decode_list
{A}
(A_decode :
ByteString
-> CacheDecode
-> option (A * ByteString * CacheDecode))
(A_decode_align : forall n,
Vector.t (word 8) n
-> CacheDecode
-> option (A * {n : _ & Vector.t _ n}
* CacheDecode))
(A_decode_OK :
forall n (v : Vector.t _ n) cd,
A_decode (build_aligned_ByteString v) cd =
Ifopt A_decode_align n v cd as a Then
Some (fst (fst a), build_aligned_ByteString (projT2 (snd (fst a))), snd a)
Else
None)
: forall (n : nat)
{sz}
(v : Vector.t (word 8) sz)
(cd : CacheDecode),
decode_list A_decode n (build_aligned_ByteString v) cd =
Ifopt align_decode_list A_decode_align n v cd as a Then
Some (fst (fst a), build_aligned_ByteString (projT2 (snd (fst a))), snd a)
Else
None.
Proof.
induction n; simpl; intros; eauto.
rewrite A_decode_OK.
rewrite (If_Opt_Then_Else_DecodeBindOpt).
destruct (A_decode_align sz v cd) as [ [ [? [? ?] ] ?] | ]; simpl; eauto.
rewrite IHn.
rewrite (If_Opt_Then_Else_DecodeBindOpt).
destruct (align_decode_list A_decode_align n t c)
as [ [ [? [? ?] ] ?] | ]; simpl; eauto.
Qed.
Fixpoint align_encode_list {A}
(A_encode_align :
A
-> CacheEncode
-> {n : _ & Vector.t (word 8) n} * CacheEncode)
(As : list A)
(ce : CacheEncode)
: {n : _ & Vector.t (word 8) n} * CacheEncode :=
match As with
| nil => (existT _ _ (Vector.nil _), ce)
| a :: As' =>
let (b, ce') := A_encode_align a ce in
let (b', ce'') := align_encode_list A_encode_align As' ce' in
(existT _ _ (append (projT2 b) (projT2 b')), ce'')
end.
Lemma optimize_align_encode_list
{A}
(A_encode_Spec : A -> CacheEncode -> Comp (ByteString * CacheEncode))
(A_encode_align :
A
-> CacheEncode
-> {n : _ & Vector.t (word 8) n} * CacheEncode)
(A_encode_OK :
forall a ce,
refine (A_encode_Spec a ce)
(ret (let (v', ce') := A_encode_align a ce in
(build_aligned_ByteString (projT2 v'), ce'))))
: forall (As : list A)
(ce : CacheEncode),
refine (encode_list_Spec A_encode_Spec As ce)
(let (v', ce') := (align_encode_list A_encode_align As ce) in
ret (build_aligned_ByteString (projT2 v'), ce')).
Proof.
induction As; simpl; intros; simpl.
- simpl.
repeat f_equiv.
eapply ByteString_f_equal.
instantiate (1 := eq_refl _); reflexivity.
instantiate (1 := eq_refl _); reflexivity.
- unfold Bind2.
rewrite A_encode_OK; simplify with monad laws.
rewrite IHAs.
destruct (A_encode_align a ce); simpl.
destruct (align_encode_list A_encode_align As c);
simplify with monad laws.
simpl.
rewrite <- build_aligned_ByteString_append.
reflexivity.
Qed.
Lemma LetIn_If_Opt_Then_Else {A B C}
: forall (a : A)
(k : A -> option B)
(t : B -> C)
(e : C),
(Ifopt LetIn a k as b Then t b Else e)
= LetIn a (fun a => Ifopt k a as b Then t b Else e).
Proof.
reflexivity.
Qed.
Lemma decode_unused_word_plus':
forall (n m : nat) (v : ByteString),
decode_unused_word' (n + m) v =
(`(w, v') <- decode_unused_word' n v;
`(w', v'') <- decode_unused_word' m v';
Some ((), v'')).
Proof.
induction n.
- simpl; intros.
destruct (decode_unused_word' m v) as [ [? ?] | ]; simpl; repeat f_equal.
destruct u; eauto.
- simpl; intros.
unfold decode_unused_word' in *; simpl.
destruct (ByteString_dequeue v) as [ [? ?] | ]; try reflexivity.
simpl.
pose proof (IHn m b0).
destruct (transformer_dequeue_word (n + m) b0) as [ [? ?] | ];
simpl in *; try congruence.
simpl in *.
destruct (transformer_dequeue_word n b0) as [ [? ?] | ];
simpl in *; try congruence.
destruct (transformer_dequeue_word n b0) as [ [? ?] | ];
simpl in *; try congruence.
Qed.
Lemma aligned_decode_unused_char_eq
{numBytes}
: forall (v : Vector.t _ (S numBytes)),
WordOpt.decode_unused_word' (transformerUnit := ByteString_QueueTransformerOpt) 8 (build_aligned_ByteString v)
= Some ((), build_aligned_ByteString (Vector.tl v)).
Proof.
unfold decode_unused_word'; simpl; intros.
etransitivity.
apply f_equal with (f := fun z => If_Opt_Then_Else z _ _ ).
eapply DecodeBindOpt_under_bind; intros; set_evars; rewrite !DecodeBindOpt_assoc.
repeat (unfold H; apply DecodeBindOpt_under_bind; intros; set_evars; rewrite !DecodeBindOpt_assoc).
unfold H5; higher_order_reflexivity.
simpl.
pattern numBytes, v; eapply Vector.caseS; intros; simpl; clear v numBytes.
replace (build_aligned_ByteString t) with (ByteString_enqueue_ByteString ByteString_id (build_aligned_ByteString t)).
unfold Core.char in h.
shatter_word h.
pose proof (@dequeue_transform_opt _ _ _ ByteString_QueueTransformerOpt).
rewrite build_aligned_ByteString_cons; simpl.
simpl in H7.
erewrite H7 with (t := x6)
(b' := {| front := WS x (WS x0 (WS x1 (WS x2 (WS x3 (WS x4 (WS x5 WO))))));
byteString := Vector.nil _ |}); simpl.
erewrite H7 with (t := x5)
(b' := {| front := WS x (WS x0 (WS x1 (WS x2 (WS x3 (WS x4 WO)))));
byteString := Vector.nil _ |}); simpl.
erewrite H7 with (t := x4)
(b' := {| front := WS x (WS x0 (WS x1 (WS x2 (WS x3 WO))));
byteString := Vector.nil _ |}); simpl.
erewrite H7 with (t := x3)
(b' := {| front := WS x (WS x0 (WS x1 (WS x2 WO)));
byteString := Vector.nil _ |}); simpl.
erewrite H7 with (t := x2)
(b' := {| front := WS x (WS x0 (WS x1 WO));
byteString := Vector.nil _ |}); simpl.
erewrite H7 with (t := x1)
(b' := {| front := WS x (WS x0 WO);
byteString := Vector.nil _ |}); simpl.
erewrite H7 with (t := x0)
(b' := {| front := WS x WO;
byteString := Vector.nil _ |}); simpl.
erewrite H7 with (t := x)
(b' := {| front := WO;
byteString := Vector.nil _ |}); simpl.
reflexivity.
unfold dequeue_opt.
simpl.
compute; repeat f_equal; apply Core.le_uniqueness_proof.
compute; repeat f_equal; apply Core.le_uniqueness_proof.
compute; repeat f_equal; apply Core.le_uniqueness_proof.
compute; repeat f_equal; apply Core.le_uniqueness_proof.
compute; repeat f_equal; apply Core.le_uniqueness_proof.
compute; repeat f_equal; apply Core.le_uniqueness_proof.
compute; repeat f_equal; apply Core.le_uniqueness_proof.
unfold build_aligned_ByteString.
unfold ByteString_dequeue; simpl.
repeat f_equal; apply Core.le_uniqueness_proof.
apply (@transform_id_left _ ByteStringQueueTransformer).
Qed.
Lemma decode_unused_word_aligned_ByteString_overflow
: forall {sz'}
(b : t (word 8) sz')
{sz}
(cd : CacheDecode),
lt sz' sz
-> decode_unused_word (8 * sz) (build_aligned_ByteString b) cd = None.
Proof.
induction b; intros.
- unfold build_aligned_ByteString; simpl.
inversion H; subst; reflexivity.
- destruct sz; try omega.
apply lt_S_n in H.
pose proof (IHb _ cd H).
unfold decode_unused_word, WordOpt.decode_word.
rewrite <- mult_n_Sm, plus_comm.
rewrite decode_unused_word_plus'.
rewrite (@aligned_decode_unused_char_eq ).
simpl.
unfold decode_unused_word in H0.
simpl in H0.
destruct (decode_unused_word' (sz + (sz + (sz + (sz + (sz + (sz + (sz + (sz + 0))))))))
(build_aligned_ByteString b));
simpl in *; try congruence.
Qed.
Lemma AlignedDecodeUnusedChar {C}
{numBytes}
: forall (v : Vector.t (word 8) (S numBytes))
(t : (() * ByteString * CacheDecode) -> C)
(e : C)
cd,
Ifopt (decode_unused_word
(transformerUnit := ByteString_QueueTransformerOpt) 8 (build_aligned_ByteString v) cd)
as w Then t w Else e
=
(t ((), build_aligned_ByteString (snd (Vector_split 1 _ v)), addD cd 8)).
Proof.
unfold LetIn; intros.
unfold decode_unused_word, WordOpt.decode_word.
rewrite aligned_decode_unused_char_eq; simpl.
f_equal.
Qed.
Lemma AlignedDecodeUnusedChars {C}
{numBytes numBytes'}
: forall (v : Vector.t (word 8) (numBytes' + numBytes))
(k : _ -> option C)
cd,
BindOpt (decode_unused_word
(transformerUnit := ByteString_QueueTransformerOpt) (8 * numBytes') (build_aligned_ByteString v) cd) k =
k ((), build_aligned_ByteString (snd (Vector_split numBytes' _ v)), addD cd (8 * numBytes')).
Proof.
induction numBytes'.
- Local Transparent Vector_split.
simpl; intros; unfold Vector_split; simpl.
reflexivity.
- simpl.
replace (S
(numBytes' +
S
(numBytes' +
S
(numBytes' +
S
(numBytes' +
S (numBytes' + S (numBytes' + S (numBytes' + S (numBytes' + 0))))))))) with (8 + 8 * numBytes') by omega.
unfold decode_unused_word; intros.
rewrite decode_unused_word_plus'.
rewrite (@aligned_decode_unused_char_eq ).
simpl BindOpt.
pose proof (IHnumBytes' (Vector.tl v) k (addD cd 8)).
simpl in H.
unfold decode_unused_word in H.
simpl in *.
fold plus in *. unfold Core.char in *.
revert H;
repeat match goal with
|- context [decode_unused_word' ?z ?b] =>
destruct (decode_unused_word' z b) as [ [? ?] | ] eqn: ?; simpl in *
end; intros.
destruct u.
rewrite addD_addD_plus in H; simpl in H; rewrite H.
destruct ((Vector_split numBytes' numBytes (Vector.tl v))); simpl.
reflexivity.
rewrite addD_addD_plus in H; simpl in H; rewrite H.
destruct ((Vector_split numBytes' numBytes (Vector.tl v))); simpl.
reflexivity.
Qed.
Lemma aligned_encode_unused_char_eq
: forall cd,
refine (encode_unused_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) 8 cd)
(ret (build_aligned_ByteString (Vector.cons _ (wzero 8) _ (Vector.nil _)), addE cd 8)).
Proof.
unfold encode_unused_word_Spec, encode_unused_word_Spec'; simpl.
intros; refine pick val (wzero 8); eauto; simplify with monad laws.
compute; intros.
computes_to_inv; subst.
match goal with
|- computes_to (ret ?c) ?v => replace c with v
end.
computes_to_econstructor.
f_equal.
eapply ByteString_f_equal; simpl.
instantiate (1 := eq_refl _).
rewrite <- !Eqdep_dec.eq_rect_eq_dec; eauto using Peano_dec.eq_nat_dec.
erewrite eq_rect_Vector_cons; repeat f_equal.
instantiate (1 := eq_refl _); reflexivity.
Grab Existential Variables.
reflexivity.
Qed.
Lemma AlignedEncodeUnusedChar {numBytes}
: forall ce ce' (c : _ -> Comp _) (v : Vector.t _ numBytes),
refine (c (addE ce 8)) (ret (build_aligned_ByteString v, ce'))
-> refine (((encode_unused_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) 8)
ThenC c) ce)
(ret (build_aligned_ByteString (Vector.cons _ (wzero 8) _ v), ce')).
Proof.
unfold compose, Bind2; simpl; intros.
rewrite aligned_encode_unused_char_eq.
simplify with monad laws.
simpl snd; rewrite H; simplify with monad laws.
simpl.
rewrite <- build_aligned_ByteString_append.
reflexivity.
Qed.
Lemma AlignedEncode2UnusedChar {numBytes}
: forall ce ce' (c : _ -> Comp _) (v : Vector.t _ numBytes),
refine (c (addE ce 16)) (ret (build_aligned_ByteString v, ce'))
-> refine (((encode_unused_word_Spec (transformerUnit := ByteString_QueueTransformerOpt) 16)
ThenC c) ce)
(ret (build_aligned_ByteString (Vector.cons _ (wzero 8) _ (Vector.cons _ (wzero 8) _ v)), ce')).
Proof.
unfold compose, Bind2; intros.
rewrite <- (AlignedEncode2Char (wzero 16)); eauto.
unfold encode_unused_word_Spec, encode_word_Spec, compose, Bind2.
simpl.
unfold encode_unused_word_Spec'; simpl.
intros; refine pick val (wzero 16); eauto; simpl.
simplify with monad laws.
rewrite refineEquiv_bind_unit; simpl.
reflexivity.
Qed.
Definition align_decode_sumtype
{m : nat}
{types : t Type m}
(decoders :
ilist (B := fun T =>
forall n,
Vector.t (word 8) n
-> CacheDecode
-> option (T * {n : _ & Vector.t (word 8) n} * CacheDecode)) types)
(idx : Fin.t m)
{n : nat}
(v : Vector.t (word 8) n)
(cd : CacheDecode)
:= `(a, b', cd') <- ith (decoders) idx n v cd;
Some (inj_SumType types idx a, b', cd').
Lemma align_decode_sumtype_OK'
{m : nat}
{types : t Type m}
(align_decoders :
ilist (B := fun T =>
forall n,
Vector.t (word 8) n
-> CacheDecode
-> option (T * {n : _ & Vector.t (word 8) n} * CacheDecode)) types)
(decoders : ilist (B := fun T => ByteString -> CacheDecode -> option (T * ByteString * CacheDecode)) types)
(decoders_OK : forall n v cd idx',
ith decoders idx' (build_aligned_ByteString v) cd
= Ifopt ith align_decoders idx' n v cd as a Then
Some (fst (fst a), build_aligned_ByteString (projT2 (snd (fst a))), snd a)
Else
None)
: forall
(idx : Fin.t m)
{n : nat}
(v : Vector.t (word 8) n)
(cd : CacheDecode),
decode_SumType types decoders idx (build_aligned_ByteString v) cd
=
Ifopt align_decode_sumtype align_decoders idx
v cd as a Then
Some (fst (fst a), build_aligned_ByteString (projT2 (snd (fst a))), snd a)
Else
None.
Proof.
intros.
unfold decode_SumType, align_decode_sumtype.
rewrite decoders_OK.
destruct (ith align_decoders idx n v cd) as [ [ [? ?] ?] | ];
reflexivity.
Qed.
Corollary align_decode_sumtype_OK
{m : nat}
{types : t Type m}
(align_decoders :
ilist (B := fun T =>
forall n,
Vector.t (word 8) n
-> CacheDecode
-> option (T * {n : _ & Vector.t (word 8) n} * CacheDecode)) types)
(decoders : ilist (B := fun T => ByteString -> CacheDecode -> option (T * ByteString * CacheDecode)) types)
(decoders_OK : forall n v cd,
Iterate_Ensemble_BoundedIndex
(fun idx' => ith decoders idx' (build_aligned_ByteString v) cd
= Ifopt ith align_decoders idx' n v cd as a Then
Some (fst (fst a), build_aligned_ByteString (projT2 (snd (fst a))), snd a)
Else
None))
: forall
(idx : Fin.t m)
{n : nat}
(v : Vector.t (word 8) n)
(cd : CacheDecode),
decode_SumType types decoders idx (build_aligned_ByteString v) cd
=
Ifopt align_decode_sumtype align_decoders idx
v cd as a Then
Some (fst (fst a), build_aligned_ByteString (projT2 (snd (fst a))), snd a)
Else
None.
Proof.
intros; eapply align_decode_sumtype_OK'; intros.
pose proof (decoders_OK n0 v0 cd0).
eapply Iterate_Ensemble_BoundedIndex_equiv in H.
apply H.
Qed.
Definition align_encode_sumtype
{m : nat}
{types : t Type m}
(encoders :
ilist (B := (fun T : Type => T -> @CacheEncode cache -> ({n : _ & Vector.t (word 8) n} * (CacheEncode)))) types)
(st : SumType types)
(ce : CacheEncode)
:= ith (encoders) (SumType_index types st) (SumType_proj types st) ce.
Lemma align_encode_sumtype_OK'
{m : nat}
{types : t Type m}
(align_encoders :
ilist (B := (fun T : Type => T -> @CacheEncode cache -> ({n : _ & Vector.t (word 8) n} * (CacheEncode)))) types)
(encoders :
ilist (B := (fun T : Type => T -> @CacheEncode cache -> Comp (ByteString * (CacheEncode)))) types)
(encoders_OK : forall idx t (ce : CacheEncode),
refine (ith encoders idx t ce)
(ret (build_aligned_ByteString (projT2 (fst (ith align_encoders idx t ce))),
snd (ith align_encoders idx t ce))))
: forall (st : SumType types)
(ce : CacheEncode),
refine (encode_SumType_Spec types encoders st ce)
(ret (build_aligned_ByteString (projT2 (fst (align_encode_sumtype align_encoders st ce))),
(snd (align_encode_sumtype align_encoders st ce)))).
Proof.
intros; unfold encode_SumType_Spec, align_encode_sumtype.
rewrite encoders_OK; reflexivity.
Qed.
Corollary align_encode_sumtype_OK
{m : nat}
{types : t Type m}
(align_encoders :
ilist (B := (fun T : Type => T -> @CacheEncode cache -> ({n : _ & Vector.t (word 8) n} * (CacheEncode)))) types)
(encoders :
ilist (B := (fun T : Type => T -> @CacheEncode cache -> Comp (ByteString * (CacheEncode)))) types)
(encoders_OK : Iterate_Ensemble_BoundedIndex (fun idx => forall t (ce : CacheEncode),
refine (ith encoders idx t ce)
(ret (build_aligned_ByteString (projT2 (fst (ith align_encoders idx t ce))),
snd (ith align_encoders idx t ce)))))
: forall (st : SumType types)
(ce : CacheEncode),
refine (encode_SumType_Spec types encoders st ce)
(ret (build_aligned_ByteString (projT2 (fst (align_encode_sumtype align_encoders st ce))),
(snd (align_encode_sumtype align_encoders st ce)))).
Proof.
intros; eapply align_encode_sumtype_OK'; intros.
eapply Iterate_Ensemble_BoundedIndex_equiv in encoders_OK.
apply encoders_OK.
Qed.
Lemma nth_Vector_split {A}
: forall {sz} n v idx,
Vector.nth (snd (Vector_split (A := A) n sz v)) idx
= Vector.nth v (Fin.R n idx).
Proof.
induction n; simpl; intros; eauto.
assert (forall A n b, exists a b', b = Vector.cons A a n b')
by (clear; intros; pattern n, b; apply caseS; eauto).
pose proof (H _ _ v); destruct_ex; subst.
simpl.
destruct (Vector_split n sz x0) as [? ?] eqn: ?.
rewrite <- IHn.
rewrite Heqp; reflexivity.
Qed.
Lemma eq_rect_Vector_tl {A}
: forall n (v : Vector.t A (S n)) m H H',
Vector.tl (eq_rect (S n) (t A) v (S m) H)
= eq_rect _ (Vector.t A) (Vector.tl v) _ H'.
Proof.
intros n v; pattern n, v; apply Vector.caseS; simpl; intros.
erewrite eq_rect_Vector_cons; simpl; eauto.
Qed.
Lemma Vector_split_merge {A}
: forall sz m n (v : Vector.t A _),
snd (Vector_split m _ (snd (Vector_split n (m + sz) v))) =
snd (Vector_split (n + m) _ (eq_rect _ _ v _ (plus_assoc _ _ _))).
Proof.
induction m; intros; simpl.
- induction n; simpl.
+ simpl in *.
apply Eqdep_dec.eq_rect_eq_dec; auto with arith.
+ simpl in v.
assert (forall A n b, exists a b', b = Vector.cons A a n b')
by (clear; intros; pattern n, b; apply caseS; eauto).
pose proof (H _ _ v); destruct_ex; subst.
simpl.
pose proof (IHn x0).
destruct (Vector_split n sz x0) eqn: ?.
simpl in *.
rewrite H0.
erewrite eq_rect_Vector_cons with (H' := (plus_assoc n 0 sz)); eauto; simpl.
destruct (Vector_split (n + 0) sz (eq_rect (n + sz) (Vector.t A) x0 (n + 0 + sz) (plus_assoc n 0 sz))); reflexivity.
- assert (n + (S m + sz) = S n + (m + sz)) by omega.
fold plus in *; unfold Core.char in *.
replace (Vector.tl (snd (Vector_split n (S (m + sz)) v)))
with ((snd (Vector_split n (m + sz) (Vector.tl (eq_rect _ _ v _ H))))).
+ pose proof (IHm n ((Vector.tl (eq_rect (n + (S m + sz)) (t A) v (S n + (m + sz)) H)))).
destruct (Vector_split m sz (snd (Vector_split n (m + sz) (Vector.tl (eq_rect (n + (S m + sz)) (t A) v (S n + (m + sz)) H))))) eqn: ?; simpl in *.
fold plus in *; rewrite Heqp.
simpl; rewrite H0.
clear.
assert ( S (n + (m + sz)) = S (n + m + sz)) by omega.
rewrite <- eq_rect_Vector_tl with (H1 := H0).
rewrite <- Equality.transport_pp; simpl; clear.
generalize (eq_trans H H0);
generalize (PeanoNat.Nat.add_assoc n (S m) sz); clear H H0.
revert sz m v; induction n; simpl.
* intros.
rewrite <- !Eqdep_dec.eq_rect_eq_dec; auto with arith.
destruct (Vector_split m sz (Vector.tl v)) eqn: ?.
simpl in *; fold plus in *; rewrite Heqp; reflexivity.
* intros.
assert (n + S (m + sz) = S (n + m + sz)) by omega.
assert (n + S (m + sz) = n + S m + sz) by omega.
erewrite eq_rect_Vector_tl with (H' := H).
erewrite eq_rect_Vector_tl with (H' := H0).
pose proof (IHn _ _ (Vector.tl v) H0 H).
destruct ((Vector_split (n + m) sz (Vector.tl (eq_rect (n + S (m + sz)) (t A) (Vector.tl v) (S (n + m + sz)) H)))) eqn: ?.
simpl in *; fold plus in *; rewrite Heqp, H1; simpl.
destruct (Vector_split (n + S m) sz (eq_rect (n + S (m + sz)) (Vector.t A) (Vector.tl v) (n + S m + sz) H0)) eqn: ?.
reflexivity.
+ clear.
revert H v.
assert (forall q (v : t A (n + (S q))) H,
snd (Vector_split n q (Vector.tl (eq_rect (n + (S q)) (t A) v (S n + (q)) H))) =
Vector.tl (snd (Vector_split n (S (q)) v))).
{ induction n; simpl; intros.
rewrite <- Eqdep_dec.eq_rect_eq_dec; auto with arith.
assert (n + S q = S (n + q)) by omega.
rewrite eq_rect_Vector_tl with (H' := H0).
pose proof (IHn q (Vector.tl v) H0).
destruct ((Vector_split n q (Vector.tl (eq_rect (n + S q) (t A) (Vector.tl v) (S n + q) H0))))
eqn: ?.
fold plus in *; simpl in *; rewrite Heqp; simpl.
rewrite H1.
destruct (Vector_split n (S q) (Vector.tl v)); reflexivity.
}
intros; rewrite H; reflexivity.
Qed.
Lemma zeta_to_fst {A B C}
: forall (ab : A * B) (k : A -> B -> C),
(let (a, b) := ab in (k a b)) =
k (fst ab) (snd ab).
Proof.
destruct ab; reflexivity.
Qed.
Lemma zeta_inside_ret {A B C}
: forall (ab : A * B) (k : A -> B -> C),
refine (let (a, b) := ab in ret (k a b))
(ret (let (a, b) := ab in k a b)).
Proof.
destruct ab; reflexivity.
Qed.
Ltac rewrite_DecodeOpt2_fmap :=
set_refine_evar;
progress rewrite ?BindOpt_map, ?DecodeOpt2_fmap_if,
?DecodeOpt2_fmap_if_bool;
subst_refine_evar.
Lemma Ifopt_Ifopt {A A' B}
: forall (a_opt : option A)
(t : A -> option A')
(e : option A')
(t' : A' -> B)
(e' : B),
Ifopt (Ifopt a_opt as a Then t a Else e) as a' Then t' a' Else e' =
Ifopt a_opt as a Then (Ifopt (t a) as a' Then t' a' Else e') Else (Ifopt e as a' Then t' a' Else e').
Proof.
destruct a_opt; simpl; reflexivity.
Qed.
Corollary AlignedDecodeNat {C}
{numBytes}
: forall (v : Vector.t (word 8) (S numBytes))
(t : _ -> C)
e
cd,
Ifopt (decode_nat (transformerUnit := ByteString_QueueTransformerOpt) 8 (build_aligned_ByteString v) cd) as w
Then t w Else e
=
Let n := wordToNat (Vector.nth v Fin.F1) in
t (n, build_aligned_ByteString (snd (Vector_split 1 _ v)), addD cd 8).
Proof.
unfold CacheDecode.
unfold decode_nat, DecodeBindOpt2; intros.
unfold BindOpt at 1.
rewrite AlignedDecodeChar.
reflexivity.
Qed.
Lemma optimize_Guarded_Decode' {sz} {C} n
: forall (a_opt : ByteString -> C)
(a_opt' : ByteString -> C) v c,
(~ (n <= sz)%nat
-> a_opt (build_aligned_ByteString v) = c)
-> (le n sz -> a_opt (build_aligned_ByteString (Guarded_Vector_split n sz v))
= a_opt'
(build_aligned_ByteString (Guarded_Vector_split n sz v)))
-> a_opt (build_aligned_ByteString v) =
If NPeano.leb n sz Then
a_opt' (build_aligned_ByteString (Guarded_Vector_split n sz v))
Else c.
Proof.
intros; destruct (NPeano.leb n sz) eqn: ?.
- apply NPeano.leb_le in Heqb.
rewrite <- H0.
simpl; rewrite <- build_aligned_ByteString_eq_split'; eauto.
eauto.
- rewrite H; simpl; eauto.
intro.
rewrite <- NPeano.leb_le in H1; congruence.
Qed.
End AlignedDecoders.
|
import assign
import general_assign
def get_pure_literals (f : formula) : list literal :=
list.filter (λ l, is_pure_literal l f) f.join
#eval get_pure_literals example_unsat_formula
#eval get_pure_literals example_sat_formula
#eval get_pure_literals example_complex_formula
def assign_pure_literals (f : formula) : formula :=
assign_all f (get_pure_literals f)
#eval assign_pure_literals example_unsat_formula
#eval assign_pure_literals example_sat_formula
#eval assign_pure_literals example_complex_formula
lemma pure_literal_impl_correct (f : formula) (pures : list literal)
: ↥(pures.all (λ l, is_pure_literal l f)) →
(sat f ↔ sat (assign_all f pures)) :=
begin
intro h,
rw assign_all,
induction' pures;
simp,
have h_f : ↥(pures.all (λ l, is_pure_literal l (assign_lit hd f))) :=
begin
rw list.all_iff_forall,
rw list.all_iff_forall at h,
intros a h_in,
have h := h a,
simp [h_in] at h,
rw is_pure_literal,
have sub : (assign_lit hd f).join ⊆ f.join := by apply assign_subset,
have final : l_not a ∉ list.join (assign_lit hd f) := begin
intro h_next,
have in_f : l_not a ∈ f.join := begin
apply sub,
exact h_next,
end,
rw is_pure_literal' at h,
simp [in_f] at h,
contradiction,
end,
simp at final,
simp,
apply final,
end,
have ih := ih (assign_lit hd f) h_f,
rw ←ih,
apply general_assign,
apply or.inr,
rw list.all_iff_forall at h,
apply h,
simp,
end
lemma pure_literal_correct (f : formula) :
sat f ↔ sat (assign_pure_literals f) := begin
apply pure_literal_impl_correct,
rw list.all_iff_forall,
rw get_pure_literals,
intro a,
apply list.of_mem_filter,
end
lemma pure_literals_leq_size (f : formula)
: formula_size (assign_pure_literals f) ≤ formula_size f :=
by apply assign_all_leq_size
|
import Complexes.Structures.SerreGraph
class CombinatorialTwoComplex (V : Sort _) extends SerreGraph V where
relator : {v : V} → Loop v → Sort _
inv : {v w : V} → (e : v ⟶ w) → relator (.cons (op e) $ .cons e .nil) -- the trivial relations are satisfied
flip : {v : V} → {l : Loop v} → relator l → relator l.inv
flip_inv : {v : V} → {l : Loop v} → (r : relator l) → (Eq.subst l.inverse_inv $ flip (flip r)) = r
inductive NullHomotopy {V : Sort _} [CombinatorialTwoComplex V] : {v : V} → (ℓ : Loop v) → Sort _
| nil : {v : V} → NullHomotopy (.nil v)
| relator : {v : V} → {l : Loop v} → CombinatorialTwoComplex.relator l → NullHomotopy l
| concat : {v : V} → {l l' : Loop v} → NullHomotopy l → NullHomotopy l' → NullHomotopy (.append l l')
| delete : {v : V} → {l l' : Loop v} → NullHomotopy l → NullHomotopy (.append l l') → NullHomotopy l'
| rotate : {v : V} → {l : Loop v} → NullHomotopy l → NullHomotopy l.rotate
| rotate' : {v : V} → {l : Loop v} → NullHomotopy l → NullHomotopy l.rotate'
inductive Path.Homotopy {V : Sort _} [CombinatorialTwoComplex V] : {v w : V} → (p q : Path v w) → Prop
| rel : {v w : V} → {p q : Path v w} → NullHomotopy (.append p q.inverse) → Homotopy p q
namespace NullHomotopy
variable {V : Sort _} [C : CombinatorialTwoComplex V] {u v w : V} (l l' : Loop v)
def subst : {u v : V} → (h : u = v) → (l : Loop u) → (l' : Loop v) → (l = (congrArg Loop h) ▸ l') → NullHomotopy l → NullHomotopy l'
| _, _, rfl, _, _, rfl => id
def swap {u v : V} : {p : Path u v} → {q : Path v u} → NullHomotopy (.append p q) → NullHomotopy (.append q p)
| .nil, _ => by rw [Path.append_nil]; exact id
| .cons _ _, _ => by
dsimp [Path.append]
intro r
let r' := NullHomotopy.rotate r
dsimp [Loop.next, Path.first, Loop.rotate] at r'
rw [Path.append_cons]
apply swap
rw [Path.append_snoc]
exact r'
def delete' {v : V} {l l' : Loop v} (r : NullHomotopy l) (r' : NullHomotopy (.append l' l)) : NullHomotopy l' :=
NullHomotopy.delete r $ swap r'
def contract {u v : V} {p : Path u v} {l : Loop v} (rel : NullHomotopy l) {q : Path v u} : NullHomotopy (.append p (.append l q)) → NullHomotopy (.append p q) := by
intro r
let r' := swap r
rw [Path.append_assoc] at r'
let r'' := delete rel r'
exact swap r''
def splice {u v : V} {p : Path u v} {l : Loop v} (rel : NullHomotopy l) {q : Path v u} : NullHomotopy (.append p q) → NullHomotopy (.append p (.append l q)) := by
intro r
apply swap
rw [Path.append_assoc]
apply concat rel
apply swap
exact r
def trivial {u v : V} : (p : Path u v) → NullHomotopy (.append p p.inverse)
| .nil => .nil
| .cons e p' => by
rename_i x
dsimp [Path.append, Path.inverse]
rw [Path.append_snoc]
rw [← Path.snoc_cons]
let erel : Loop x := .cons (SerreGraph.op e) (.cons e .nil)
let l : Loop x := .append erel (.append p' p'.inverse)
show NullHomotopy l.rotate
apply NullHomotopy.rotate
apply NullHomotopy.concat
· apply NullHomotopy.relator
apply CombinatorialTwoComplex.inv
· apply trivial
def inv {v : V} {l : Loop v} : NullHomotopy l → NullHomotopy l.inv
| .nil => .nil
| .relator d => .relator $ CombinatorialTwoComplex.flip d
| .concat r r' => by
rw [Loop.inv, Path.inverse_append]
apply NullHomotopy.concat
· exact inv r'
· exact inv r
| .delete r r' => by
let r'' := inv r'
rw [Loop.inv, Path.inverse_append] at r''
exact NullHomotopy.delete' (inv r) r''
| .rotate r => by
apply subst (Loop.prev_inv _ _)
· apply Loop.rotate'_inv
· exact rotate' <| inv r
| .rotate' r => by
apply subst (Loop.next_inv _ _)
· apply Loop.rotate_inv
· exact rotate <| inv r
end NullHomotopy
namespace Path.Homotopy
variable {V : Sort _} [C : CombinatorialTwoComplex V] {u v : V} (p q r : Path u v)
theorem refl : (p : Path u v) → Path.Homotopy p p := (.rel $ NullHomotopy.trivial ·)
theorem symm : Path.Homotopy p q → Path.Homotopy q p
| .rel h => by
let h' := h.inv
rw [Loop.inv, Path.inverse_append, Path.inverse_inv] at h'
exact .rel h'
theorem trans : Path.Homotopy p q → Path.Homotopy q r → Path.Homotopy p r
|.rel h, .rel h' => by
let H := NullHomotopy.concat h h'
rw [Path.append_assoc p _ _, ← Path.append_assoc _ q _] at H
let H' := NullHomotopy.contract (.swap $ .trivial _) H
exact .rel H'
instance equivalence (u v : V) : Equivalence (@Path.Homotopy V C u v) where
refl := refl
symm := symm _ _
trans := trans _ _ _
instance setoid (u v : V) : Setoid (Path u v) where
r := Path.Homotopy
iseqv := equivalence u v
theorem inv_cancel_left (p : Path u v) : Path.Homotopy (.append (.inverse p) p) .nil :=
.rel $ by
simp [inverse]
apply NullHomotopy.swap
apply NullHomotopy.trivial
theorem inv_cancel_right (p : Path u v) : Path.Homotopy (.append p (.inverse p)) .nil :=
.rel $ by
simp [inverse]
apply NullHomotopy.trivial
theorem mul_sound {u v w : V} {p q : Path u v} {r s : Path v w} :
Path.Homotopy p q → Path.Homotopy r s →
Path.Homotopy (.append p r) (.append q s)
| .rel a, .rel b => .rel $ by
rw [inverse_append, append_assoc, ← append_assoc _ _ (inverse q)]
exact NullHomotopy.splice b a
theorem inv_sound {u v : V} {p q : Path u v} :
Path.Homotopy p q →
Path.Homotopy p.inverse q.inverse
| .rel r =>.rel $ by
rw [← inverse_append]
apply NullHomotopy.inv
apply NullHomotopy.swap
exact r
end Path.Homotopy
|
/-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f
-/
import mathzoo.imports.miniF2F
open_locale nat rat real big_operators topological_space
theorem numbertheory_4x3m7y3neq2003
(x y : ℤ) :
4 * x^3 - 7 * y^3 ≠ 2003 :=
begin
intro hneq,
apply_fun (coe : ℤ → zmod 7) at hneq,
push_cast at hneq,
have : (2003 : zmod 7) = (1 : zmod 7),
dec_trivial,
rw this at hneq,
have : (7 : zmod 7) = (0 : zmod 7),
dec_trivial,
rw this at hneq,
rw zero_mul at hneq,
rw sub_zero at hneq,
have main : ∀ (x : zmod 7), x^3 ∈ [(0 : zmod 7), 1, -1],
dec_trivial,
rcases main x with h' | h' | h' | h,
iterate 3 {
rw h' at hneq,
revert hneq,
dec_trivial,
},
exact h,
end |
[STATEMENT]
lemma a_calc_i_0: "$a^{m}_n = n" if "m \<noteq> 0" "i = 0" for n m :: nat
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. $a^{m}_n = real n
[PROOF STEP]
unfolding ann_def v_pres_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<Sum>k<n * m. (1 / (1 + i)) .^ (real (k + 1) / real m) / real m) = real n
[PROOF STEP]
using that
[PROOF STATE]
proof (prove)
using this:
m \<noteq> 0
i = 0
goal (1 subgoal):
1. (\<Sum>k<n * m. (1 / (1 + i)) .^ (real (k + 1) / real m) / real m) = real n
[PROOF STEP]
by simp |
From Coq Require Import ssreflect.
From stdpp Require Import base gmap.
From iris.proofmode Require Import tactics.
From aneris.prelude Require Import gset_map.
From aneris.aneris_lang.lib Require Import list_proof.
From aneris.aneris_lang.lib.serialization Require Import serialization_proof.
From aneris.aneris_lang Require Import aneris_lifting proofmode.
From aneris.examples.crdt.spec Require Import crdt_base crdt_time crdt_events crdt_denot crdt_resources.
From aneris.examples.crdt.oplib Require Import oplib_code.
From aneris.examples.crdt.oplib.spec Require Import model spec.
From aneris.examples.crdt.oplib.examples.two_p_set Require Import two_p_set_code.
Section tpsCrdt.
Context `{!Log_Time} `{!EqDecision vl} `{!Countable vl}.
Definition tpsOp : Type := vl + vl.
Definition tpsSt : Type := gset vl * gset vl.
Definition update_state (op : tpsOp) (st : tpsSt) : tpsSt :=
match op with
| inl a => ({[a]} ∪ st.1, st.2)
| inr a => (st.1, {[a]} ∪ st.2)
end.
Definition tps_denot (s : gset (Event tpsOp)) (state : tpsSt) : Prop :=
set_fold (λ ev st, update_state (EV_Op ev) st) (∅, ∅) s = state.
Global Instance tps_denot_fun : Rel2__Fun tps_denot.
Proof. constructor; intros ? ? ? <- <-; done. Qed.
Global Instance tps_denot_instance : CrdtDenot tpsOp tpsSt := {
crdt_denot := tps_denot;
}.
End tpsCrdt.
Global Arguments tpsOp _ : clear implicits.
Global Arguments tpsSt _ {_ _}.
Section OpTps.
Context `{!Log_Time}
`{!EqDecision vl} `{!Countable vl}.
Definition op_tps_effect (st : tpsSt vl) (ev : Event (tpsOp vl)) (st' : tpsSt vl) : Prop :=
st' = update_state (EV_Op ev) st.
Lemma op_tps_effect_fun st : Rel2__Fun (op_tps_effect st).
Proof. constructor; intros ??? -> ->; done. Qed.
Instance op_tps_effect_coh : OpCrdtEffectCoh op_tps_effect.
Proof.
intros s ev st st' Hst Hevs Hmax Hext.
rewrite /op_tps_effect /crdt_denot /= /tps_denot.
rewrite set_fold_disj_union_strong; [| |set_solver]; last first.
{ intros [[] ] [[] ]; rewrite /update_state /=; intros; f_equal; set_solver. }
rewrite Hst set_fold_singleton; split; done.
Qed.
Definition op_tps_init_st : tpsSt vl := (∅, ∅).
Lemma op_tps_init_st_coh : ⟦ (∅ : gset (Event (tpsOp vl))) ⟧ ⇝ op_tps_init_st.
Proof. done. Qed.
Global Instance op_tps_model_instance : OpCrdtModel (tpsOp vl) (tpsSt vl) := {
op_crdtM_effect := op_tps_effect;
op_crdtM_effect_fun := op_tps_effect_fun;
op_crdtM_effect_coh := op_tps_effect_coh;
op_crdtM_init_st := op_tps_init_st;
op_crdtM_init_st_coh := op_tps_init_st_coh
}.
End OpTps.
From aneris.aneris_lang.lib Require Import set_code set_proof.
From aneris.aneris_lang.lib Require Import inject.
From aneris.examples.crdt.oplib.proof Require Import time.
Section tps_proof.
Context `{!EqDecision vl} `{!Countable vl}
`{!Inject vl val} `{!∀ (a : vl), Serializable vl_serialization $a}.
Context `{!anerisG M Σ}.
Context `{!CRDT_Params} `{!OpLib_Res (tpsOp vl)}.
Definition tps_OpLib_Op_Coh := λ (op : tpsOp vl) (v : val), v = $op.
Lemma tps_OpLib_Op_Coh_Inj (o1 o2 : tpsOp vl) (v : val) :
tps_OpLib_Op_Coh o1 v → tps_OpLib_Op_Coh o2 v → o1 = o2.
Proof. intros Ho1 Ho2; apply (inj (@inject _ _ Inject_sum)); rewrite -Ho1 -Ho2; done. Qed.
Lemma tps_OpLib_Coh_Ser (op : tpsOp vl) (v : val) :
tps_OpLib_Op_Coh op v → Serializable (sum_serialization vl_serialization vl_serialization) v.
Proof. intros Heq. rewrite Heq; destruct op; apply _. Qed.
Definition tps_OpLib_State_Coh :=
λ (st : tpsSt vl) v, ∃ v1 v2, v = PairV v1 v2 ∧ is_set st.1 v1 ∧ is_set st.2 v2.
Global Instance tps_OpLib_Params : OpLib_Params (tpsOp vl) (tpsSt vl) :=
{|
OpLib_Serialization := (sum_serialization vl_serialization vl_serialization);
OpLib_State_Coh := tps_OpLib_State_Coh;
OpLib_Op_Coh := tps_OpLib_Op_Coh;
OpLib_Op_Coh_Inj := tps_OpLib_Op_Coh_Inj;
OpLib_Coh_Ser := tps_OpLib_Coh_Ser
|}.
Lemma tps_init_st_fn_spec : ⊢ init_st_fn_spec init_st.
Proof.
iIntros (addr).
iIntros "!#" (Φ) "_ HΦ".
rewrite /init_st.
wp_pures.
wp_apply wp_set_empty; first done.
iIntros (v Hv).
wp_apply wp_set_empty; first done.
iIntros (w Hw).
wp_pures.
iApply "HΦ".
iPureIntro; eexists _, _; split_and!; simpl; done.
Qed.
Lemma tps_effect_spec : ⊢ effect_spec effect.
Proof.
iIntros (addr ev st s log_ev log_st).
iIntros "!#" (Φ) "(%Hev & %Hst & %Hs & %Hevs) HΦ".
rewrite /effect.
destruct log_ev as [log_ev orig vc].
destruct Hev as (evpl&evvc&evorig& ?&Hopcoh&?&?).
destruct Hevs as (Hnin & Hmax & Hext).
destruct log_st as [log_st1 log_st2].
destruct Hst as (st1&st2&?&?&?).
simplify_eq/=.
rewrite Hopcoh /=.
wp_pures.
destruct log_ev; wp_pures.
- wp_apply wp_set_add; first by iPureIntro.
iIntros (w Hw).
wp_pures.
iApply "HΦ".
iExists _; iSplit; last by eauto.
simpl; iPureIntro; eexists _, _; split_and!; done.
- wp_apply wp_set_add; first by iPureIntro.
iIntros (w Hw).
wp_pures.
iApply "HΦ".
iExists _; iSplit; last by eauto.
simpl; iPureIntro; eexists _, _; split_and!; done.
Qed.
Lemma tps_crdt_fun_spec : ⊢ crdt_fun_spec tps_crdt.
Proof.
iIntros (addr).
iIntros "!#" (Φ) "_ HΦ".
rewrite /tps_crdt.
wp_pures.
iApply "HΦ".
iExists _, _; iSplit; first done.
iSplit.
- iApply tps_init_st_fn_spec; done.
- iApply tps_effect_spec; done.
Qed.
Lemma tps_init_spec :
init_spec
(oplib_init
(s_ser (s_serializer (sum_serialization vl_serialization vl_serialization)))
(s_deser (s_serializer (sum_serialization vl_serialization vl_serialization)))) -∗
init_spec_for_specific_crdt
(tps_init (s_ser (s_serializer vl_serialization)) (s_deser (s_serializer vl_serialization))).
Proof.
iIntros "#Hinit" (repId addr addrs_val).
iIntros (Φ) "!# (%Haddrs & %Hrepid & Hprotos & Hskt & Hfr & Htoken) HΦ".
rewrite /tps_init.
wp_pures.
wp_apply ("Hinit" with "[$Hprotos $Htoken $Hskt $Hfr]").
{ do 2 (iSplit; first done). iApply tps_crdt_fun_spec; done. }
iIntros (get update) "(HLS & #Hget & #Hupdate)".
wp_pures.
iApply "HΦ"; eauto.
Qed.
End tps_proof.
|
{-# OPTIONS --without-K --safe --no-sized-types --no-guardedness
--no-subtyping #-}
module Agda.Builtin.Char.Properties where
open import Agda.Builtin.Char
open import Agda.Builtin.Equality
primitive
primCharToNatInjective : ∀ a b → primCharToNat a ≡ primCharToNat b → a ≡ b
|
(** * Mapping predicates over [StringLike] things *)
Require Import Coq.Numbers.Natural.Peano.NPeano.
Require Import Coq.omega.Omega.
Require Import Fiat.Parsers.StringLike.Core.
Require Import Fiat.Parsers.StringLike.Properties.
Require Import Fiat.Common.
Set Implicit Arguments.
Local Open Scope string_like_scope.
Section forall_chars.
Context {Char} {HSLM : StringLikeMin Char} {HSL : StringLike Char} {HSLP : StringLikeProperties Char}.
Definition forall_chars (str : String) (P : Char -> Prop)
:= forall n ch,
take 1 (drop n str) ~= [ ch ]
-> P ch.
Lemma forall_chars_from_get (str : String) (P : Char -> Prop)
: (forall n ch, get n str = Some ch -> P ch) -> forall_chars str P.
Proof.
unfold forall_chars.
intros H n ch Heq.
apply (H n); clear H.
rewrite get_drop, <- get_0; assumption.
Qed.
Global Instance forall_chars_Proper
: Proper (beq ==> pointwise_relation _ impl ==> impl) forall_chars.
Proof.
unfold pointwise_relation, respectful, forall_chars, impl.
intros ?? H' ?? H'' H''' ?? H.
rewrite <- H' in H.
eauto using is_char_Proper.
Qed.
Global Instance forall_chars_Proper_flip
: Proper (beq ==> pointwise_relation _ (flip impl) ==> flip impl) forall_chars.
Proof.
unfold pointwise_relation, respectful, forall_chars, flip, impl.
intros ?? H' ?? H'' H''' ?? H.
rewrite H' in H.
eauto using is_char_Proper.
Qed.
Global Instance forall_chars_Proper_iff
: Proper (beq ==> pointwise_relation _ iff ==> iff) forall_chars.
Proof.
unfold pointwise_relation, respectful.
repeat intro; split;
apply forall_chars_Proper; try assumption; repeat intro;
match goal with
| [ H : _ |- _ ] => apply H; assumption
end.
Qed.
Global Instance forall_chars_Proper_eq
: Proper (beq ==> pointwise_relation _ eq ==> impl) forall_chars.
Proof.
unfold pointwise_relation, respectful, forall_chars, impl.
intros ?? H' ?? H'' H''' ?? H.
rewrite <- H' in H.
rewrite <- H''; eauto.
Qed.
Global Instance forall_chars_Proper_eq_flip
: Proper (beq ==> pointwise_relation _ eq ==> flip impl) forall_chars.
Proof.
unfold pointwise_relation, respectful, forall_chars, flip, impl.
intros ?? H' ?? H'' H''' ?? H.
rewrite H' in H.
rewrite H''; eauto.
Qed.
Global Instance forall_chars_Proper_eq_iff
: Proper (beq ==> pointwise_relation _ eq ==> iff) forall_chars.
Proof.
unfold pointwise_relation, respectful.
repeat intro; split;
apply forall_chars_Proper; try assumption; repeat intro;
match goal with
| [ H : _ |- _ ] => apply H; assumption
| [ H : _ |- _ ] => rewrite H; assumption
| [ H : _ |- _ ] => rewrite <- H; assumption
end.
Qed.
Lemma forall_chars_nil (str : String) P
: length str = 0 -> forall_chars str P.
Proof.
intros H n ch H'.
apply length_singleton in H'.
rewrite take_length, drop_length, H in H'.
simpl in H'; omega.
Qed.
Lemma helper
(P : nat -> nat -> Type)
n
(H0 : forall n0, P (min 1 (n - n0)) n0)
(H1 : forall n0, P 1 (n0 + n))
{n0}
: P 1 n0.
Proof.
destruct (Compare_dec.le_dec n n0) as [H'|H'].
{ specialize (H1 (n0 - n)).
rewrite Nat.sub_add in H1 by assumption; assumption. }
{ apply Compare_dec.not_le in H'.
specialize (H0 n0).
destruct (n - n0) as [|[|]] eqn:?; simpl in *; trivial; omega. }
Defined.
Lemma forall_chars__split (str : String) P n
: forall_chars str P
<-> (forall_chars (take n str) P /\ forall_chars (drop n str) P).
Proof.
unfold forall_chars; repeat (split || intro);
repeat match goal with
| [ H : _ |- _ ] => setoid_rewrite drop_length in H
| [ H : _ |- _ ] => setoid_rewrite take_length in H
| [ H : _ |- _ ] => setoid_rewrite drop_take in H
| [ H : _ |- _ ] => setoid_rewrite take_take in H
| [ H : _ |- _ ] => setoid_rewrite drop_drop in H
| [ H : _ /\ _ |- _ ] => destruct H
| [ H : context[min 1 ?x] |- _ ] => destruct x eqn:?; simpl in H
| [ H : is_true (take 0 _ ~= [ _ ]) |- _ ] => exfalso; apply length_singleton in H
| _ => omega
| _ => progress simpl in *; omega
| _ => solve [ eauto ]
| _ => solve [ eapply (@helper (fun a b => take a (drop b str) ~= [ ch ] -> P ch)); eauto ]
end.
Qed.
Lemma forall_chars_singleton (str : String) P ch
: str ~= [ ch ] -> (P ch <-> forall_chars str P).
Proof.
intro H.
pose proof (length_singleton _ _ H).
unfold forall_chars.
split; intro H'; repeat intro.
{ match goal with
| [ n : nat |- _ ] => destruct n; [ | exfalso ]
end;
repeat match goal with
| _ => intro
| _ => omega
| [ H : _ |- _ ] => rewrite drop_0 in H
| [ H : _, H' : _ |- _ ] => rewrite (singleton_take H') in H
| [ H : _ |- False ] => apply length_singleton in H
| [ H : _ |- _ ] => rewrite take_length in H
| [ H : _ |- _ ] => rewrite drop_length in H
| [ H : ?x = 1, H' : context[?x] |- _ ] => rewrite H in H'
| _ => erewrite singleton_unique; eassumption
| [ H : context[min] |- _ ] => revert H; apply Min.min_case_strong
end. }
{ match goal with
| [ H : _ |- _ ] => apply (H 0)
end.
rewrite drop_0.
rewrite take_long; trivial; omega. }
Qed.
Lemma forall_chars_singleton_length (str : String) P (H : length str = 1)
: forall_chars str P <-> (forall ch, str ~= [ ch ] -> P ch).
Proof.
split.
{ intro H''.
intros ch' H'''.
apply (forall_chars_singleton _ _ _ H'''); assumption. }
{ destruct (singleton_exists _ H) as [ch H'].
intro H''.
apply (forall_chars_singleton _ P ch H'); eauto. }
Qed.
Lemma forall_chars_False (str : String) P
: (forall ch, ~P ch) -> forall_chars str P -> length str = 0.
Proof.
intros H' H.
case_eq (length str); trivial.
specialize (H (length str - 1)).
pose proof (singleton_exists (drop (length str - 1) str)) as H''.
rewrite drop_length in H''.
intros n H'''.
rewrite H''' in *.
rewrite sub_plus in H'' by omega.
rewrite Minus.minus_diag in *.
specialize (H'' eq_refl).
destruct H'' as [ch H''].
exfalso; eapply H', H.
rewrite take_long; try eassumption.
apply length_singleton in H''; omega.
Qed.
Global Opaque forall_chars.
Lemma forall_chars__split_forall (str : String) P
: forall_chars str P
<-> (forall n, forall_chars (take n str) P /\ forall_chars (drop n str) P).
Proof.
split.
{ intros H n.
rewrite <- forall_chars__split; assumption. }
{ intro H.
specialize (H 0).
rewrite forall_chars__split; eassumption. }
Qed.
Lemma forall_chars_take (str : String) P
: forall_chars str P <-> (forall n, forall_chars (take (S n) str) P).
Proof.
split.
{ intros H n.
revert H; rewrite forall_chars__split; intro H; destruct H; eassumption. }
{ intro H.
specialize (H (length str)).
rewrite take_long in H by omega; assumption. }
Qed.
Lemma forall_chars_drop (str : String) P
: forall_chars str P <-> (forall n, forall_chars (drop n str) P).
Proof.
split.
{ intros H n.
revert H; rewrite forall_chars__split; intro H; destruct H; eassumption. }
{ intro H.
specialize (H 0).
rewrite drop_0 in H; assumption. }
Qed.
Lemma forall_chars_get (str : String) P
: forall_chars str P <-> (forall n ch, get n str = Some ch -> P ch).
Proof.
split.
{ intros H n ch Heq.
rewrite get_drop in Heq.
apply get_0 in Heq.
apply forall_chars__split with (n := n) in H.
destruct H as [_ H].
apply forall_chars__split with (n := 1) in H.
destruct H as [H _].
rewrite forall_chars_singleton_length in H.
{ auto. }
{ apply length_singleton in Heq; assumption. } }
{ apply forall_chars_from_get. }
Qed.
Definition forall_chars__char_in (str : String) (ls : list Char)
:= forall_chars str (fun ch => List.In ch ls).
Definition forall_chars__impl__forall_chars__char_in {str ls} {P : _ -> Prop}
(H : forall ch, P ch -> List.In ch ls)
: impl (forall_chars str P) (forall_chars__char_in str ls).
Proof.
unfold forall_chars__char_in.
apply forall_chars_Proper; trivial; reflexivity.
Qed.
Definition forall_chars__char_in__impl__forall_chars {str ls} {P : _ -> Prop}
(H : forall ch, List.In ch ls -> P ch)
: impl (forall_chars__char_in str ls) (forall_chars str P).
Proof.
unfold forall_chars__char_in.
apply forall_chars_Proper; trivial; reflexivity.
Qed.
Global Instance forall_chars__char_in__Proper
: Proper (beq ==> eq ==> impl) forall_chars__char_in.
Proof.
unfold pointwise_relation, respectful, forall_chars__char_in, impl.
repeat intro; subst.
match goal with
| [ H : _ |- _ ] => rewrite <- H; assumption
end.
Qed.
Global Instance forall_chars__char_in__Proper_iff
: Proper (beq ==> eq ==> iff) forall_chars__char_in.
Proof.
unfold pointwise_relation, respectful, forall_chars__char_in, impl.
repeat intro; subst.
match goal with
| [ H : _ |- _ ] => rewrite <- H; reflexivity
end.
Qed.
Lemma forall_chars__char_in__split n (str : String) ls
: forall_chars__char_in str ls
<-> (forall_chars__char_in (take n str) ls /\ forall_chars__char_in (drop n str) ls).
Proof.
unfold forall_chars__char_in; apply forall_chars__split.
Qed.
Lemma forall_chars__char_in__app_or_iff (str : String) (ls1 ls2 : list Char)
: forall_chars__char_in str (ls1 ++ ls2)
<-> (forall_chars str (fun ch => List.In ch ls1 \/ List.In ch ls2)).
Proof.
unfold forall_chars__char_in; split; repeat intro.
{ eapply forall_chars_Proper; [ .. | eassumption ]; [ reflexivity | ].
intro; hnf.
apply List.in_app_or. }
{ eapply forall_chars_Proper; [ .. | eassumption ]; [ reflexivity | ].
intro; hnf.
apply List.in_or_app. }
Qed.
Lemma forall_chars__char_in__or_app (str : String) (ls1 ls2 : list Char)
: forall_chars__char_in str ls1 \/ forall_chars__char_in str ls2 -> forall_chars__char_in str (ls1 ++ ls2).
Proof.
unfold forall_chars__char_in.
intros [?|?]; repeat intro;
(eapply forall_chars_Proper; [ .. | eassumption ]; [ reflexivity | ]; intros ??);
apply List.in_or_app; eauto.
Qed.
Lemma forall_chars__char_in_nil (str : String)
: forall_chars__char_in str nil <-> length str = 0.
Proof.
unfold forall_chars__char_in.
split.
{ eapply forall_chars_False; simpl; eauto. }
{ apply forall_chars_nil. }
Qed.
Lemma forall_chars__char_in_empty (str : String) (H : length str = 0) ls
: forall_chars__char_in str ls.
Proof.
unfold forall_chars__char_in.
apply forall_chars_nil; assumption.
Qed.
Lemma forall_chars__char_in_singleton_str (str : String) ls ch (H : str ~= [ ch ])
: forall_chars__char_in str ls <-> List.In ch ls.
Proof.
unfold forall_chars__char_in.
rewrite <- forall_chars_singleton; try eassumption; reflexivity.
Qed.
Global Opaque forall_chars__char_in.
End forall_chars.
|
lemma nat.not_self_lt_lt_succ_self {a b : nat}
: a < b → b < a + 1 → false :=
begin
intros h1 h2,
cases nat.le.dest h1 with w h_1,
cases nat.le.dest h2 with w_1 h_2,
rw ← h_1 at h_2,
rw ← nat.succ_add at h_2,
apply nat.lt_irrefl (nat.succ a),
apply nat.lt_of_lt_of_le,
apply lt_add_of_pos_right (nat.succ a),
refine (_ : nat.succ (w + w_1) > 0),
apply nat.succ_pos,
have : ∀ x y z, nat.succ x + y + z = x + nat.succ (y + z),
{ intros, rw ← nat.add_one, rw ← nat.add_one, ac_refl },
rw this at h_2, rw h_2
end
def sub_lt_if_ge : ∀ n m k : ℕ, n ≤ k → k < n + m → k - n < m :=
begin
intros n m k hnk h,
rw ← nat.add_sub_of_le hnk at h,
apply lt_of_add_lt_add_left h
end
def ge_if_not_lt {n k : ℕ} (h : ¬ (n < k)) : n ≥ k :=
by { cases nat.lt_or_ge n k, exfalso, exact h h_1, exact h_1 }
lemma pos_of_prod_pos_l {n m : ℕ} : 0 < n * m → 0 < n :=
begin
intro h, apply decidable.by_contradiction, intro h',
have : n = 0,
{ cases nat.lt_trichotomy n 0,
{ exfalso, apply nat.not_lt_zero _ h_1 },
{ cases h_1, assumption, trivial } },
rw [this, zero_mul] at h, apply nat.lt_irrefl _ h,
end
lemma mod_of_add_multiple (n m k : ℕ) : (n + m * k) % m = n % m :=
begin
induction k, simp,
{ have : m ≤ m * nat.succ k_n,
{ rw nat.mul_succ, apply nat.le_add_left },
rw nat.mod_eq_sub_mod (nat.le_trans this (nat.le_add_left _ _)),
rw nat.add_sub_assoc this,
rw (_ : m * nat.succ k_n - m = m * nat.succ k_n - m * 1),
rw ← nat.mul_sub_left_distrib, rw ← nat.add_one,
rw nat.add_sub_cancel, assumption, simp }
end
lemma nat.mul_two : ∀ n, n * 2 = n + n :=
by simp [(*), nat.mul]
def binomial_coefficient : nat → nat → nat
| _ 0 := 1
| 0 _ := 0
| (n+1) (k+1) := binomial_coefficient n (k+1) + binomial_coefficient n k
lemma binomial_coefficient_lt
: ∀ n k, n < k → binomial_coefficient n k = 0 :=
begin
intro, induction n with n ih; intros k h,
{ cases k, cases h, refl },
cases k, cases h,
simp [binomial_coefficient],
rw [ih, ih], transitivity,
apply nat.lt_succ_self, assumption,
apply nat.lt_of_succ_lt_succ h
end
lemma binomial_coefficient_self
: ∀ n, binomial_coefficient n n = 1 :=
begin
intro, induction n with n ih, refl,
simp [binomial_coefficient], rw ih,
rw binomial_coefficient_lt, apply nat.lt_succ_self
end
lemma binomial_coefficient_zero
: ∀ n, binomial_coefficient n 0 = 1 :=
by intro; cases n; refl
theorem nat.even_odd_induction (P : nat → Sort _)
(bc₀ : P 0) (bc₁ : P 1) (rec : ∀ n, P n → P n.succ → P n.succ.succ) : ∀ n, P n
| 0 := bc₀
| 1 := bc₁
| (n+2) := rec n (nat.even_odd_induction n) (nat.even_odd_induction n.succ) |
using Distributions, Interpolations
function simulate(
model::CartpoleFriction,
policy,
K,
z_nom, u_nom,
Q, R,
T_sim, Δt,
z0, w;
_norm = 2,
ul = -Inf * ones(length(u_nom[1])),
uu = Inf * ones(length(u_nom[1])),
friction = false,
μ = 0.1)
T = length(K) + 1
times = [(t-1) * Δt for t = 1:T-1]
tf = Δt * T
t_sim = range(0, stop = tf, length = T_sim)
t_ctrl = range(0, stop = tf, length = T)
dt_sim = tf / (T_sim - 1)
p = 1:policy.output
A_state = hcat(z_nom...)
A_ctrl = hcat(u_nom...)
z_rollout = [z0]
u_rollout = []
J = 0.0
Jx = 0.0
Ju = 0.0
for tt = 1:T_sim-1
t = t_sim[tt]
k = searchsortedlast(times, t)
z_cubic = zeros(model.n)
for i = 1:model.n
interp_cubic = CubicSplineInterpolation(t_ctrl, A_state[i,:])
z_cubic[i] = interp_cubic(t)
end
z = z_rollout[end] + dt_sim * w[:,tt]
u = eval_policy(policy, K[k], z, z_cubic, u_nom[k])
# clip controls
u = max.(u, ul[p])
u = min.(u, uu[p])
if friction
_u = [u[1] - μ * sign(z_cubic[3]) * model.g * (model.mp + model.mc);
0.0;
0.0]
else
_u = u[1]
end
push!(z_rollout, rk3(model, z, _u, zeros(model.d), dt_sim))
push!(u_rollout, u)
if _norm == 2
J += (z_rollout[end] - z_cubic)' * Q[k + 1] * (z_rollout[end] - z_cubic)
J += (u_rollout[end] - u_nom[k][p])' * R[k][p,
p] * (u_rollout[end] - u_nom[k][p])
Jx += (z_rollout[end] - z_cubic)' * Q[k + 1] * (z_rollout[end] - z_cubic)
Ju += (u_rollout[end] - u_nom[k][p])' * R[k][p,
p] * (u_rollout[end] - u_nom[k][p])
else
J += norm(sqrt(Q[k + 1]) * (z_rollout[end] - z_cubic), _norm)
J += norm(sqrt(R[k][p, p]) * (u - u_nom[k][p]), _norm)
Jx += norm(sqrt(Q[k + 1]) * (z_rollout[end] - z_cubic), _norm)
Ju += norm(sqrt(R[k][p, p]) * (u - u_nom[k][p]), _norm)
end
end
return z_rollout, u_rollout, J / (T_sim - 1), Jx / (T_sim - 1), Ju / (T_sim - 1)
nothing
end
# Nominal trajectories
X̄_nominal, Ū_nominal = unpack(Z̄_nominal, prob_nominal)
X̄_friction, Ū_friction = unpack(Z̄_friction, prob_friction)
X̄_dpo, Ū_dpo = unpack(Z[prob_dpo.prob.idx.nom], prob_dpo.prob.prob.nom)
# Policies
K_nominal = tvlqr(model, X̄_nominal, [Ū_nominal[t][1:1] for t = 1:T-1],
Q, [R[t][1:1, 1:1] for t = 1:T-1], h)
K_friction = tvlqr(model, X̄_friction, [Ū_friction[t][1:1] for t = 1:T-1],
Q, [R[t][1:1, 1:1] for t = 1:T-1], h)
θ = [reshape(Z[prob_dpo.prob.idx.policy[prob_dpo.prob.idx.θ[t]]],
1, model.n) for t = 1:T-1]
# Simulation
T_sim = 10T
Δt = h
dt_sim = tf / (T_sim - 1)
W = Distributions.MvNormal(zeros(model.n), Diagonal(1.0e-5 * ones(model.n)))
w = rand(W, T_sim)
W0 = Distributions.MvNormal(zeros(model.n), Diagonal(1.0e-5 * ones(model.n)))
w0 = rand(W0, 1)
model_sim = model_friction
μ_sim = 0.1
t_sim = range(0, stop = tf, length = T_sim)
z_lqr, u_lqr, J_lqr, Jx_lqr, Ju_lqr = simulate(model_sim,
policy,
K_nominal,
X̄_nominal, Ū_nominal,
Q, R,
T_sim, h,
x1 + vec(w0),
w,
ul = ul_friction, uu = uu_friction,
friction = true,
μ = μ0)
z_lqr_fr, u_lqr_fr, J_lqr_fr, Jx_lqr_fr, Ju_lqr_fr = simulate(model_sim,
policy,
K_friction,
X̄_friction, Ū_friction,
Q, R,
T_sim, h,
x1 + vec(w0),
w,
ul = ul_friction, uu = uu_friction,
friction = true,
μ = μ0)
z_dpo, u_dpo, J_dpo, Jx_dpo, Ju_dpo = simulate(model_sim,
policy,
θ,
X̄_dpo, Ū_dpo,
Q, R,
T_sim, h,
x1 + vec(w0),
w,
ul = ul_friction, uu = uu_friction,
friction = true,
μ = μ0)
# objective value
J_lqr
J_lqr_fr
J_dpo
# state tracking
Jx_lqr
Jx_lqr_fr
Jx_dpo
# control tracking
Ju_lqr
Ju_lqr_fr
Ju_dpo
|
module maryjohn2 where
postulate Person : Set
postulate john : Person
postulate mary : Person
postulate barbara : Person
postulate IsStudent : Person -> Set
postulate maryIsStudent : IsStudent mary
postulate implication : IsStudent mary -> IsStudent john
Lemma1 : Set
Lemma1 = IsStudent john
proof-lemma1 : Lemma1
proof-lemma1 = implication maryIsStudent
Lemma2 : Set
Lemma2 = IsStudent john -> IsStudent barbara
proof-lemma2 : Lemma2
proof-lemma2 = \(x : IsStudent john) -> _
|
module Categories.Functor.CartesianClosed where
open import Categories.Category using (Category; _[_∘_]; _[_,_])
open import Categories.Category.Cartesian using (Cartesian)
open import Categories.Category.Cartesian.Bundle using (CartesianCategory)
open import Categories.Category.CartesianClosed using (CartesianClosed)
open import Categories.Category.CartesianClosed.Bundle using (CartesianClosedCategory)
open import Categories.Functor using (Functor)
open import Categories.Functor.Cartesian using (IsCartesianF)
import Categories.Morphism as Morphism
open import Level
private
variable
o ℓ e o′ ℓ′ e′ : Level
record IsCartesianClosedF
(C : CartesianClosedCategory o ℓ e)
(D : CartesianClosedCategory o′ ℓ′ e′)
(F : Functor (CartesianClosedCategory.U C) (CartesianClosedCategory.U D))
: Set (levelOfTerm C ⊔ levelOfTerm D) where
open Morphism (CartesianClosedCategory.U D) using (IsIso; Iso)
open Functor F
private
CU = CartesianClosedCategory.U C
DU = CartesianClosedCategory.U D
CC₁ = CartesianClosedCategory.cartesianClosed C
CC₂ = CartesianClosedCategory.cartesianClosed D
C₁ = CartesianClosed.cartesian CC₁
C₂ = CartesianClosed.cartesian CC₂
C′ = record { U = CU ; cartesian = C₁ }
D′ = record { U = DU ; cartesian = C₂ }
_^C_ = CartesianClosed._^_ CC₁
_^D_ = CartesianClosed._^_ CC₂
field
isCartesianF : IsCartesianF C′ D′ F
module isCartesianF = IsCartesianF isCartesianF
conv : forall {A B : Category.Obj CU} -> DU [ F₀ (B ^C A) , F₀ B ^D F₀ A ]
conv {A} {B} =
CartesianClosed.λg CC₂
(DU [ F₁ (CartesianClosed.eval′ CC₁) ∘ IsCartesianF.×-iso.to isCartesianF (B ^C A) A ])
field
F-preserve-^ : forall {A B : Category.Obj CU} -> IsIso (conv {A} {B})
module F-preserve-^ {A B : Category.Obj CU} = IsIso (F-preserve-^ {A} {B})
-- Note that F is called strict if `conv` ≈ id.
record CartesianClosedF
(C : CartesianClosedCategory o ℓ e)
(D : CartesianClosedCategory o′ ℓ′ e′)
: Set (levelOfTerm C ⊔ levelOfTerm D) where
field
F : Functor (CartesianClosedCategory.U C) (CartesianClosedCategory.U D)
isCartesianClosed : IsCartesianClosedF C D F
open Functor F public
open IsCartesianClosedF isCartesianClosed public
|
lemma continuous_on_of_real_o_iff [simp]: "continuous_on S (\<lambda>x. complex_of_real (g x)) = continuous_on S g" |
lemma succ_le_succ (a b : mynat) (h : a ≤ b) : succ a ≤ succ b :=
begin
cases h with c hc,
rw hc,
use c,
rw succ_add,
refl,
end |
! RUN: bbc -emit-fir %s -o - | FileCheck %s
! CHECK-LABEL: anint_test
subroutine anint_test(a, b)
real :: a, b
! CHECK: fir.call @llvm.round.f32
b = anint(a)
end subroutine
|
/-
Copyright (c) 2021 Huub Vromen. All rights reserved.
Author: Huub Vromen
-/
/- outline of Lewis' theory of common knowledge (formalisation according to
the proposal of Jaap van der Does) -/
-- type of individuals in the population P
variable {indiv : Type}
variables {i j: indiv}
-- states of affairs and propositions are of the same type
variables {p ψ χ : Prop}
-- Reason-to-believe is a two-place relation between individuals and propositions
constant R : indiv → Prop → Prop
-- Indication is a three-place relation between individuals and two propositions
constant Ind : indiv → Prop → Prop → Prop
/-- The following axioms represent that state of affairs α is a basis for common
knowledge of proposition φ -/
constants {α φ: Prop}
axiom CK1 : R i α
axiom CK2 : Ind i α (R j α)
axiom CK3 : Ind i α φ
/-- `Indicative modus ponens represents the meaning of indication.
This axiom corresponds to axiom A1 in Cubitt and Sugden's account. -/
axiom I_MP : Ind i ψ χ → R i ψ → R i χ
/-- `I-introspection` represents that all individuals in P share inductive
standards and background knowledge. This axiom is a variant of axiom C4
in Cubitt and Sugden's account -/
axiom I_introspection : Ind i α ψ → Ind i α (Ind j α ψ)
/-- `LR` (Lewis' Rule) is the assumption that the population allows for introspection
with regard to using I-MP. This axiom is a variant of axiom A6 in
Cubitt and Sugden's account -/
axiom LR : (Ind i α (R j α) ∧ Ind i α (Ind j α ψ) → Ind i α (R j ψ))
/-- Now we can prove Lewis' theorem. First, we inductively define G (`generated
by φ), the property of being a proposition `p` for which we have to prove
that `R i p` holds. -/
inductive G : Prop → Prop
| base : G φ
| step (p : Prop) {i : indiv} : G p → G (R i p)
theorem Lewis (p : Prop) (hG : @G indiv p) : ∀i : indiv, R i p :=
begin
intro i,
have h1 : Ind i α p :=
begin
induction hG with u j hu ih,
{ exact CK3 },
{ have h2 : Ind i α (Ind j α u) := I_introspection ih,
have h3 : Ind i α (R j u) := LR (and.intro CK2 h2),
assumption }
end,
exact I_MP h1 CK1
end
/- Both I-MP and LR can be derived as lemmas in the theory of reasoning with
reasons. -/
|
{-
--{-# OPTIONS --allow-unsolved-metas #-}
--{-# OPTIONS -v 100 #-}
{- Yellow highlighting should be accompanied by an error message -}
{- By default, command-line agda reports "unsolved metas" and gives a source-code location, but no information about the nature of the unsolved metas. (Emacs agda gives only yellow-highlighting.). With Without increasing verbosity (via "-v 100") -}
postulate
yellow-highlighting-but-no-error-message : Set _
{- --allow-unsolved-metas -}
{-
Running from the command-line, agda reports that there are unsolved metas but doesn't say anything about what they are. Is there a way to increase verbosity about the unsolved metas?
-}
postulate
error-message-but-no-link-to-source-code-location : _ _
-}
postulate
Σ : (A : Set) (B : A → Set) → Set
X : Set
confusing-message--not-empty-type-of-sizes : Σ _ λ x → X
confusing-message--not-empty-type-of-sizes = {!!}
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import algebra.module.pi
import algebra.ordered_pi
import algebra.module.prod
import algebra.ordered_field
/-!
# Ordered modules
In this file we define
* `ordered_module R M` : an ordered additive commutative monoid `M` is an `ordered_module`
over an `ordered_semiring` `R` if the scalar product respects the order relation on the
monoid and on the ring. There is a correspondence between this structure and convex cones,
which is proven in `analysis/convex/cone.lean`.
## Implementation notes
* We choose to define `ordered_module` as a `Prop`-valued mixin, so that it can be
used for both modules and algebras
(the axioms for an "ordered algebra" are exactly that the algebra is ordered as a module).
* To get ordered modules and ordered vector spaces, it suffices to replace the
`order_add_comm_monoid` and the `ordered_semiring` as desired.
## References
* https://en.wikipedia.org/wiki/Ordered_module
## Tags
ordered module, ordered module, ordered vector space
-/
/--
An ordered module is an ordered additive commutative monoid
with a partial order in which the scalar multiplication is compatible with the order.
-/
@[protect_proj]
class ordered_module (R M : Type*)
[ordered_semiring R] [ordered_add_comm_monoid M] [module R M] : Prop :=
(smul_lt_smul_of_pos : ∀ {a b : M}, ∀ {c : R}, a < b → 0 < c → c • a < c • b)
(lt_of_smul_lt_smul_of_pos : ∀ {a b : M}, ∀ {c : R}, c • a < c • b → 0 < c → a < b)
section ordered_module
variables {R M : Type*}
[ordered_semiring R] [ordered_add_comm_monoid M] [module R M] [ordered_module R M]
{a b : M} {c : R}
lemma smul_lt_smul_of_pos : a < b → 0 < c → c • a < c • b := ordered_module.smul_lt_smul_of_pos
lemma smul_le_smul_of_nonneg (h₁ : a ≤ b) (h₂ : 0 ≤ c) : c • a ≤ c • b :=
begin
by_cases H₁ : c = 0,
{ simp [H₁, zero_smul] },
{ by_cases H₂ : a = b,
{ rw H₂ },
{ exact le_of_lt
(smul_lt_smul_of_pos (lt_of_le_of_ne h₁ H₂) (lt_of_le_of_ne h₂ (ne.symm H₁))), } }
end
lemma eq_of_smul_eq_smul_of_pos_of_le (h₁ : c • a = c • b) (hc : 0 < c) (hle : a ≤ b) :
a = b :=
hle.lt_or_eq.resolve_left $ λ hlt, (smul_lt_smul_of_pos hlt hc).ne h₁
lemma lt_of_smul_lt_smul_of_nonneg (h : c • a < c • b) (hc : 0 ≤ c) : a < b :=
hc.eq_or_lt.elim (λ hc, false.elim $ lt_irrefl (0:M) $ by rwa [← hc, zero_smul, zero_smul] at h)
(ordered_module.lt_of_smul_lt_smul_of_pos h)
lemma smul_lt_smul_iff_of_pos (hc : 0 < c) : c • a < c • b ↔ a < b :=
⟨λ h, lt_of_smul_lt_smul_of_nonneg h hc.le, λ h, smul_lt_smul_of_pos h hc⟩
lemma smul_pos_iff_of_pos (hc : 0 < c) : 0 < c • a ↔ 0 < a :=
calc 0 < c • a ↔ c • 0 < c • a : by rw smul_zero
... ↔ 0 < a : smul_lt_smul_iff_of_pos hc
end ordered_module
/-- If `R` is a linear ordered semifield, then it suffices to verify only the first axiom of
`ordered_module`. Moreover, it suffices to verify that `a < b` and `0 < c` imply
`c • a ≤ c • b`. We have no semifields in `mathlib`, so we use the assumption `∀ c ≠ 0, is_unit c`
instead. -/
lemma ordered_module.mk'' {R M : Type*} [linear_ordered_semiring R] [ordered_add_comm_monoid M]
[module R M] (hR : ∀ {c : R}, c ≠ 0 → is_unit c)
(hlt : ∀ ⦃a b : M⦄ ⦃c : R⦄, a < b → 0 < c → c • a ≤ c • b) :
ordered_module R M :=
begin
have hlt' : ∀ ⦃a b : M⦄ ⦃c : R⦄, a < b → 0 < c → c • a < c • b,
{ refine λ a b c hab hc, (hlt hab hc).lt_of_ne _,
rw [ne.def, (hR hc.ne').smul_left_cancel],
exact hab.ne },
refine { smul_lt_smul_of_pos := hlt', .. },
intros a b c h hc,
rcases (hR hc.ne') with ⟨c, rfl⟩,
rw [← c.inv_smul_smul a, ← c.inv_smul_smul b],
refine hlt' h (pos_of_mul_pos_left _ hc.le),
simp only [c.mul_inv, zero_lt_one]
end
/-- If `R` is a linear ordered field, then it suffices to verify only the first axiom of
`ordered_module`. -/
lemma ordered_module.mk' {k M : Type*} [linear_ordered_field k] [ordered_add_comm_monoid M]
[module k M] (hlt : ∀ ⦃a b : M⦄ ⦃c : k⦄, a < b → 0 < c → c • a ≤ c • b) :
ordered_module k M :=
ordered_module.mk'' (λ c hc, is_unit.mk0 _ hc) hlt
instance linear_ordered_semiring.to_ordered_module {R : Type*} [linear_ordered_semiring R] :
ordered_module R R :=
{ smul_lt_smul_of_pos := ordered_semiring.mul_lt_mul_of_pos_left,
lt_of_smul_lt_smul_of_pos := λ _ _ _ h hc, lt_of_mul_lt_mul_left h hc.le }
section field
variables {k M N : Type*} [linear_ordered_field k]
[ordered_add_comm_group M] [module k M] [ordered_module k M]
[ordered_add_comm_group N] [module k N] [ordered_module k N]
{a b : M} {c : k}
lemma smul_le_smul_iff_of_pos (hc : 0 < c) : c • a ≤ c • b ↔ a ≤ b :=
⟨λ h, inv_smul_smul' hc.ne' a ▸ inv_smul_smul' hc.ne' b ▸
smul_le_smul_of_nonneg h (inv_nonneg.2 hc.le),
λ h, smul_le_smul_of_nonneg h hc.le⟩
lemma smul_le_smul_iff_of_neg (hc : c < 0) : c • a ≤ c • b ↔ b ≤ a :=
begin
rw [← neg_neg c, neg_smul, neg_smul (-c), neg_le_neg_iff, smul_le_smul_iff_of_pos (neg_pos.2 hc)],
apply_instance,
end
lemma smul_lt_iff_of_pos (hc : 0 < c) : c • a < b ↔ a < c⁻¹ • b :=
calc c • a < b ↔ c • a < c • c⁻¹ • b : by rw [smul_inv_smul' hc.ne']
... ↔ a < c⁻¹ • b : smul_lt_smul_iff_of_pos hc
lemma smul_le_iff_of_pos (hc : 0 < c) : c • a ≤ b ↔ a ≤ c⁻¹ • b :=
calc c • a ≤ b ↔ c • a ≤ c • c⁻¹ • b : by rw [smul_inv_smul' hc.ne']
... ↔ a ≤ c⁻¹ • b : smul_le_smul_iff_of_pos hc
lemma le_smul_iff_of_pos (hc : 0 < c) : a ≤ c • b ↔ c⁻¹ • a ≤ b :=
calc a ≤ c • b ↔ c • c⁻¹ • a ≤ c • b : by rw [smul_inv_smul' hc.ne']
... ↔ c⁻¹ • a ≤ b : smul_le_smul_iff_of_pos hc
instance prod.ordered_module : ordered_module k (M × N) :=
ordered_module.mk' $ λ v u c h hc,
⟨smul_le_smul_of_nonneg h.1.1 hc.le, smul_le_smul_of_nonneg h.1.2 hc.le⟩
instance pi.ordered_module {ι : Type*} {M : ι → Type*} [Π i, ordered_add_comm_group (M i)]
[Π i, module k (M i)] [∀ i, ordered_module k (M i)] :
ordered_module k (Π i : ι, M i) :=
begin
refine (ordered_module.mk' $ λ v u c h hc i, _),
change c • v i ≤ c • u i,
exact smul_le_smul_of_nonneg (h.le i) hc.le,
end
-- Sometimes Lean fails to apply the dependent version to non-dependent functions,
-- so we define another instance
instance pi.ordered_module' {ι : Type*} {M : Type*} [ordered_add_comm_group M]
[module k M] [ordered_module k M] :
ordered_module k (ι → M) :=
pi.ordered_module
end field
section order_dual
variables {R M : Type*}
instance [semiring R] [ordered_add_comm_monoid M] [module R M] : has_scalar R (order_dual M) :=
{ smul := @has_scalar.smul R M _ }
instance [semiring R] [ordered_add_comm_monoid M] [module R M] : mul_action R (order_dual M) :=
{ one_smul := @mul_action.one_smul R M _ _,
mul_smul := @mul_action.mul_smul R M _ _ }
instance [semiring R] [ordered_add_comm_monoid M] [module R M] :
distrib_mul_action R (order_dual M) :=
{ smul_add := @distrib_mul_action.smul_add R M _ _ _,
smul_zero := @distrib_mul_action.smul_zero R M _ _ _ }
instance [semiring R] [ordered_add_comm_monoid M] [module R M] : module R (order_dual M) :=
{ add_smul := @module.add_smul R M _ _ _,
zero_smul := @module.zero_smul R M _ _ _ }
instance [ordered_semiring R] [ordered_add_comm_monoid M] [module R M]
[ordered_module R M] :
ordered_module R (order_dual M) :=
{ smul_lt_smul_of_pos := λ a b, @ordered_module.smul_lt_smul_of_pos R M _ _ _ _ b a,
lt_of_smul_lt_smul_of_pos := λ a b,
@ordered_module.lt_of_smul_lt_smul_of_pos R M _ _ _ _ b a }
end order_dual
|
From Undecidability.L Require Import L HaltMuRec_to_HaltL.
From Undecidability.MuRec Require Import MuRec.
Theorem MuRec_computable_to_L_computable {k} (R : Vector.t nat k -> nat -> Prop) :
MuRec_computable R -> L_computable R.
Proof.
eapply HaltMuRec_to_HaltL.computable_MuRec_to_L.
Qed.
|
(*
File: AllLambda.thy
Time-stamp: <2016-01-06T17:58:00Z>
Author: JRF
Web: http://jrf.cocolog-nifty.com/software/2016/01/post.html
Logic Image: ZF (of Isabelle2015)
*)
theory AllLambda
imports dB dBeta SndLAlpha SndLBeta Length FNumber FinBndEx
begin
lemma LAlpha3_iff_LAlpha2:
assumes infv: "Infinite(LVariable)"
shows "LAlpha3(M, N) <-> <M, N>: LAlpha2"
apply (rule iff_trans [OF LAlpha3_iff_LAlpha
infv [THEN LAlpha2_iff_LAlpha, THEN iff_sym]])
done
end
|
Formal statement is: lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] Informal statement is: $\mathbb{R} \cong \mathbb{C}$ |
From SegmentQueue.lib.blocking_pool
Require Export outer_storage_interfaces.
From iris.program_logic Require Import atomic.
From iris.heap_lang Require Export proofmode notation lang.
Definition option_to_value (v: option val): val :=
match v with
None => NONEV
| Some v => SOMEV v
end.
Theorem option_to_value_inj:
FinFun.Injective option_to_value.
Proof.
intros a b. rewrite /option_to_value.
case: a; case: b=> //=.
move=> a b. case. by intros ->.
Qed.
Inductive outerStorageState :=
OuterStorageState {
failuresInStorage: nat;
valuesInStorage: gmultiset val
}.
Definition insertionAddsValue (state: outerStorageState) (v: val) :=
match state with
OuterStorageState fs vs => OuterStorageState fs ({[v]} ⊎ vs)
end.
Definition insertionRemovesFailure (state: outerStorageState) :=
match state with
OuterStorageState fs vs => match fs with
0 => None
| S fs => Some (OuterStorageState fs vs)
end
end.
Definition retrievalAddsFailure (state: outerStorageState) :=
match state with
OuterStorageState fs vs => OuterStorageState (S fs) vs
end.
Locate multiplicity.
Definition retrievalRemovesValue (state: outerStorageState) (value: val) :=
match state with
OuterStorageState fs vs =>
match multiplicity value vs with
0 => None
| _ => Some (OuterStorageState fs (vs ∖ {[value]}))
end
end.
Record outerStorageSpec Σ `{!heapG Σ} (impl: outerStorageInterface) :=
OuterStorageSpec {
is_outer_storage:
namespace -> gname -> val -> iProp Σ;
is_outer_storage_persistent N γ s:
Persistent (is_outer_storage N γ s);
outer_storage_contents:
gname -> outerStorageState -> iProp Σ;
tryInsert_spec N γ s (value : val):
is_outer_storage N γ s -∗
<<< ∀ state, ▷ outer_storage_contents γ state >>>
tryInsert impl s value @ ⊤∖↑N
<<< ∃ (b: bool),
if b then
outer_storage_contents γ (insertionAddsValue state value)
else match insertionRemovesFailure state with
None => False
| Some state' => outer_storage_contents γ state'
end, RET #b >>>;
tryRetrieve_spec N γ s:
is_outer_storage N γ s -∗
<<< ∀ state, ▷ outer_storage_contents γ state >>>
tryRetrieve impl s @ ⊤ ∖ ↑N
<<< ∃ (v : option val),
match v with
| Some v' => match retrievalRemovesValue state v' with
| None => False
| Some state' => outer_storage_contents γ state'
end
| None => outer_storage_contents γ (retrievalAddsFailure state)
end, RET option_to_value v >>>;
newStorage_spec N:
{{{ True }}}
newStorage impl #()
{{{ γ v, RET v; is_outer_storage N γ v
∗ outer_storage_contents γ
{| failuresInStorage := 0; valuesInStorage := ∅ |} }}}
}.
Existing Instances is_outer_storage_persistent.
|
{-# OPTIONS -WShadowingInTelescope #-}
bad : Set → Set → Set
bad = λ x x → x
|
import Mathlib.analysis.normed_space.basic
import Mathlib.topology.local_homeomorph
namespace Mathlib
open filter
open set
namespace asymptotics
-- TODO: Type* syntax
variable {α : Type} {β : Type} {E : Type} {F : Type} {G : Type}
{E' : Type} {F' : Type} {G' : Type} {R : Type} {R' : Type}
variable [has_norm E] [has_norm F] [has_norm G] [normed_group E'] [normed_group F']
[normed_group G'] [normed_ring R] [normed_ring R']
{c c' : ℝ} {f : α → E} {g : α → F} {k : α → G} {f' : α → E'} {g' : α → F'} {k' : α → G'}
{l l' : filter α}
section defs
def is_O_with (c : ℝ) (f : α → E) (g : α → F) (l : filter α) : Prop :=
filter.eventually (fun (x : α) => norm (f x) ≤ c * norm (g x)) l
theorem is_O_with_iff {c : real} {f : α → E} {g : α → F} {l : filter α} :
is_O_with c f g l ↔ filter.eventually (fun (x : α) => norm (f x) ≤ c * norm (g x)) l :=
iff.rfl
theorem is_O_with.of_bound {c : real} {f : α → E} {g : α → F} {l : filter α} :
filter.eventually (λ (x : α) => norm (f x) ≤ c * norm (g x)) l → is_O_with c f g l :=
λ (h : filter.eventually (fun (x : α) => norm (f x) ≤ c * norm (g x)) l) => h
def is_O (f : α → E) (g : α → F) (l : filter α) : Prop :=
∃ c : ℝ, is_O_with c f g l
theorem is_O_iff_is_O_with {f : α → E} {g : α → F} {l : filter α} :
is_O f g l ↔ ∃ c : ℝ, is_O_with c f g l :=
iff.rfl
theorem is_O_iff {f : α → E} {g : α → F} {l : filter α} :
is_O f g l ↔ ∃ c : ℝ, filter.eventually (λ x => norm (f x) ≤ c * norm (g x)) l :=
iff.rfl
theorem is_O.of_bound (c : ℝ) {f : α → E} {g : α → F} {l : filter α}
(h : filter.eventually (λ x => norm (f x) ≤ c * norm (g x)) l) : is_O f g l :=
⟨c, h⟩
def is_o (f : α → E) (g : α → F) (l : filter α) : Prop :=
∀ (c : ℝ), 0 < c → is_O_with c f g l
theorem is_o_iff_forall_is_O_with {f : α → E} {g : α → F} {l : filter α} :
is_o f g l ↔ ∀ (c : ℝ), 0 < c → is_O_with c f g l :=
iff.rfl
theorem is_o_iff {f : α → E} {g : α → F} {l : filter α} :
is_o f g l ↔ ∀ (c : ℝ), 0 < c → filter.eventually (λ x => norm (f x) ≤ c * norm (g x)) l
:= iff.rfl
theorem is_o.def {f : α → E} {g : α → F} {l : filter α} (h : is_o f g l) {c : ℝ} (hc : 0 < c) :
filter.eventually (λ x => norm (f x) ≤ c * norm (g x)) l :=
h _ hc
theorem is_o.def' {f : α → E} {g : α → F} {l : filter α} (h : is_o f g l) {c : ℝ} (hc : 0 < c) :
is_O_with c f g l :=
h _ hc
end defs
theorem is_O_with.is_O (h : is_O_with c f g l) : is_O f g l := ⟨c, h⟩
theorem is_o.is_O_with (hgf : is_o f g l) : is_O_with 1 f g l := hgf _ zero_lt_one
theorem is_o.is_O (hgf : is_o f g l) : is_O f g l := hgf.is_O_with.is_O
end asymptotics
end Mathlib
|
def f : IO Nat := do
IO.println "hello"
IO.getStdin
return 10
def f1 : ExceptT String (StateT Nat Id) Nat := do
modify (· + 1)
get
def f2 (x : Nat) : ExceptT String (StateT Nat Id) Nat := do
modify (· + x)
get
def g1 : ExceptT String (StateT Nat Id) Unit := do
let x : String ← f1
return ()
def g2 : ExceptT String (StateT Nat Id) Unit := do
let x : String ← f2 10
return ()
def g3 : ExceptT String (StateT Nat Id) String := do
let x ← f2
f1
example : Nat := Id.run do
let mut n : Nat := 0
(n, _) := (false, false)
n
|
#ifndef _GP_H_
#define _GP_H_
#include "armadillo"
using namespace arma;
#include "myTypes.h"
#include "KernelFunction.h"
#include "MeanFunction.h"
#include <gsl/gsl_math.h>
#include <gsl/gsl_multimin.h>
double f_eval_mean(const gsl_vector *x, void *param);
void df_eval_mean(const gsl_vector *x, void *param, gsl_vector *g);
void fdf_eval_mean(const gsl_vector *x, void *param, double *f, gsl_vector *g);
double f_eval_kernel(const gsl_vector *x, void *param);
void df_eval_kernel(const gsl_vector *x, void *param, gsl_vector *g);
void fdf_eval_kernel(const gsl_vector *x, void *param, double *f, gsl_vector *g);
double f_eval_noise(const gsl_vector *x, void *param);
void df_eval_noise(const gsl_vector *x, void *param, gsl_vector *g);
void fdf_eval_noise(const gsl_vector *x, void *param, double *f, gsl_vector *g);
class GP {
public:
GP(REAL s2_n, KernelFunction *kernel, MeanFunction *mean);
void SetTraining(const Mat<REAL>& X, const Row<REAL> &y);
void AddTraining(const Mat<REAL>& X, const Row<REAL> &y);
void Predict(const Mat<REAL> &Xs, Row<REAL> &mu);
void Predict(const Mat<REAL> &Xs, Row<REAL> &mu, Row<REAL> &var);
void Predict(const Col<REAL> &Xs, REAL &mu);
void Predict(const Col<REAL> &Xs, REAL &mu, REAL &var);
void PredictGradient(const Col<REAL> &Xs, Col<REAL> &grad);
void PredictGradient(const Col<REAL> &Xs, Col<REAL> &grad, Col<REAL> &vargrad);
void ComputeAlpha();
void ComputeChol();
void ComputeW();
REAL ComputeLikelihood();
void SetKernelFuncParams(const Col<REAL>& param);
void SetMeanFuncParams(const Col<REAL>& param);
void SetNoise(const REAL &s2_n);
void GradLikelihoodMeanParams(Col<REAL> &grad);
void GradLikelihoodKernelParams(Col<REAL> &grad);
void GradLikelihoodNoise(Col<REAL> &grad);
void HessianLikelihoodMeanParams(Mat<REAL> &hessian);
void OptimizeNoiseParam(REAL &noise_param, int max_iterations=10);
void OptimizeMeanParam(Col<REAL> &mean_param, int max_iterations=10);
void OptimizeKernelParam(Col<REAL> &kernel_param, int max_iterations=10);
inline Mat<REAL> &GetTrainingData() { return X; }
inline KernelFunction *GetKernelFunction() { return this->kernel; }
inline MeanFunction *GetMeanFunction() { return this->mean; }
protected:
KernelFunction *kernel;
MeanFunction *mean;
Mat<REAL> K;
Mat<REAL> W;
Mat<REAL> X;
Mat<REAL> L;
Row<REAL> y;
Row<REAL> meanvals;
Col<REAL> alpha;
REAL s2_n;
REAL loglikelihood;
bool need_to_compute_alpha;
bool need_to_compute_chol;
bool need_to_compute_w;
void MatrixMap(Mat<REAL> &matrix, const Mat<REAL> &a, const Mat<REAL> &b);
};
#endif
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
! This file was ported from Lean 3 source module field_theory.finite.basic
! leanprover-community/mathlib commit 2196ab363eb097c008d4497125e0dde23fb36db2
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.FieldTheory.Separable
import Mathbin.FieldTheory.SplittingField
import Mathbin.RingTheory.IntegralDomain
import Mathbin.Tactic.ApplyFun
/-!
# Finite fields
This file contains basic results about finite fields.
Throughout most of this file, `K` denotes a finite field
and `q` is notation for the cardinality of `K`.
See `ring_theory.integral_domain` for the fact that the unit group of a finite field is a
cyclic group, as well as the fact that every finite integral domain is a field
(`fintype.field_of_domain`).
## Main results
1. `fintype.card_units`: The unit group of a finite field is has cardinality `q - 1`.
2. `sum_pow_units`: The sum of `x^i`, where `x` ranges over the units of `K`, is
- `q-1` if `q-1 ∣ i`
- `0` otherwise
3. `finite_field.card`: The cardinality `q` is a power of the characteristic of `K`.
See `card'` for a variant.
## Notation
Throughout most of this file, `K` denotes a finite field
and `q` is notation for the cardinality of `K`.
## Implementation notes
While `fintype Kˣ` can be inferred from `fintype K` in the presence of `decidable_eq K`,
in this file we take the `fintype Kˣ` argument directly to reduce the chance of typeclass
diamonds, as `fintype` carries data.
-/
variable {K : Type _} {R : Type _}
-- mathport name: exprq
local notation "q" => Fintype.card K
open Finset Function
open BigOperators Polynomial
namespace FiniteField
section Polynomial
variable [CommRing R] [IsDomain R]
open Polynomial
/-- The cardinality of a field is at most `n` times the cardinality of the image of a degree `n`
polynomial -/
theorem card_image_polynomial_eval [DecidableEq R] [Fintype R] {p : R[X]} (hp : 0 < p.degree) :
Fintype.card R ≤ natDegree p * (univ.image fun x => eval x p).card :=
Finset.card_le_mul_card_image _ _ fun a _ =>
calc
_ = (p - C a).roots.toFinset.card :=
congr_arg card (by simp [Finset.ext_iff, mem_roots_sub_C hp])
_ ≤ (p - C a).roots.card := (Multiset.toFinset_card_le _)
_ ≤ _ := card_roots_sub_C' hp
#align finite_field.card_image_polynomial_eval FiniteField.card_image_polynomial_eval
/-- If `f` and `g` are quadratic polynomials, then the `f.eval a + g.eval b = 0` has a solution. -/
theorem exists_root_sum_quadratic [Fintype R] {f g : R[X]} (hf2 : degree f = 2) (hg2 : degree g = 2)
(hR : Fintype.card R % 2 = 1) : ∃ a b, f.eval a + g.eval b = 0 :=
letI := Classical.decEq R
suffices ¬Disjoint (univ.image fun x : R => eval x f) (univ.image fun x : R => eval x (-g))
by
simp only [disjoint_left, mem_image] at this
push_neg at this
rcases this with ⟨x, ⟨a, _, ha⟩, ⟨b, _, hb⟩⟩
exact ⟨a, b, by rw [ha, ← hb, eval_neg, neg_add_self]⟩
fun hd : Disjoint _ _ =>
lt_irrefl (2 * ((univ.image fun x : R => eval x f) ∪ univ.image fun x : R => eval x (-g)).card) <|
calc
2 * ((univ.image fun x : R => eval x f) ∪ univ.image fun x : R => eval x (-g)).card ≤
2 * Fintype.card R :=
Nat.mul_le_mul_left _ (Finset.card_le_univ _)
_ = Fintype.card R + Fintype.card R := (two_mul _)
_ <
nat_degree f * (univ.image fun x : R => eval x f).card +
nat_degree (-g) * (univ.image fun x : R => eval x (-g)).card :=
(add_lt_add_of_lt_of_le
(lt_of_le_of_ne (card_image_polynomial_eval (by rw [hf2] <;> exact by decide))
(mt (congr_arg (· % 2)) (by simp [nat_degree_eq_of_degree_eq_some hf2, hR])))
(card_image_polynomial_eval (by rw [degree_neg, hg2] <;> exact by decide)))
_ = 2 * ((univ.image fun x : R => eval x f) ∪ univ.image fun x : R => eval x (-g)).card := by
rw [card_disjoint_union hd] <;>
simp [nat_degree_eq_of_degree_eq_some hf2, nat_degree_eq_of_degree_eq_some hg2, bit0,
mul_add]
#align finite_field.exists_root_sum_quadratic FiniteField.exists_root_sum_quadratic
end Polynomial
theorem prod_univ_units_id_eq_neg_one [CommRing K] [IsDomain K] [Fintype Kˣ] :
(∏ x : Kˣ, x) = (-1 : Kˣ) := by
classical
have : (∏ x in (@univ Kˣ _).eraseₓ (-1), x) = 1 :=
prod_involution (fun x _ => x⁻¹) (by simp)
(fun a => by simp (config := { contextual := true }) [Units.inv_eq_self_iff])
(fun a => by simp [@inv_eq_iff_eq_inv _ _ a]) (by simp)
rw [← insert_erase (mem_univ (-1 : Kˣ)), prod_insert (not_mem_erase _ _), this, mul_one]
#align finite_field.prod_univ_units_id_eq_neg_one FiniteField.prod_univ_units_id_eq_neg_one
section
variable [GroupWithZero K] [Fintype K]
theorem pow_card_sub_one_eq_one (a : K) (ha : a ≠ 0) : a ^ (q - 1) = 1 :=
calc
a ^ (Fintype.card K - 1) = (Units.mk0 a ha ^ (Fintype.card K - 1) : Kˣ) := by
rw [Units.val_pow_eq_pow_val, Units.val_mk0]
_ = 1 := by
classical
rw [← Fintype.card_units, pow_card_eq_one]
rfl
#align finite_field.pow_card_sub_one_eq_one FiniteField.pow_card_sub_one_eq_one
theorem pow_card (a : K) : a ^ q = a :=
by
have hp : 0 < Fintype.card K := lt_trans zero_lt_one Fintype.one_lt_card
by_cases h : a = 0;
· rw [h]
apply zero_pow hp
rw [← Nat.succ_pred_eq_of_pos hp, pow_succ, Nat.pred_eq_sub_one, pow_card_sub_one_eq_one a h,
mul_one]
#align finite_field.pow_card FiniteField.pow_card
theorem pow_card_pow (n : ℕ) (a : K) : a ^ q ^ n = a :=
by
induction' n with n ih
· simp
· simp [pow_succ, pow_mul, ih, pow_card]
#align finite_field.pow_card_pow FiniteField.pow_card_pow
end
variable (K) [Field K] [Fintype K]
theorem card (p : ℕ) [CharP K p] : ∃ n : ℕ+, Nat.Prime p ∧ q = p ^ (n : ℕ) :=
by
haveI hp : Fact p.prime := ⟨CharP.char_is_prime K p⟩
letI : Module (ZMod p) K := { (ZMod.castHom dvd_rfl K : ZMod p →+* _).toModule with }
obtain ⟨n, h⟩ := VectorSpace.card_fintype (ZMod p) K
rw [ZMod.card] at h
refine' ⟨⟨n, _⟩, hp.1, h⟩
apply Or.resolve_left (Nat.eq_zero_or_pos n)
rintro rfl
rw [pow_zero] at h
have : (0 : K) = 1 := by apply fintype.card_le_one_iff.mp (le_of_eq h)
exact absurd this zero_ne_one
#align finite_field.card FiniteField.card
-- this statement doesn't use `q` because we want `K` to be an explicit parameter
theorem card' : ∃ (p : ℕ)(n : ℕ+), Nat.Prime p ∧ Fintype.card K = p ^ (n : ℕ) :=
let ⟨p, hc⟩ := CharP.exists K
⟨p, @FiniteField.card K _ _ p hc⟩
#align finite_field.card' FiniteField.card'
@[simp]
theorem cast_card_eq_zero : (q : K) = 0 :=
by
rcases CharP.exists K with ⟨p, _char_p⟩; skip
rcases card K p with ⟨n, hp, hn⟩
simp only [CharP.cast_eq_zero_iff K p, hn]
conv =>
congr
rw [← pow_one p]
exact pow_dvd_pow _ n.2
#align finite_field.cast_card_eq_zero FiniteField.cast_card_eq_zero
theorem forall_pow_eq_one_iff (i : ℕ) : (∀ x : Kˣ, x ^ i = 1) ↔ q - 1 ∣ i := by
classical
obtain ⟨x, hx⟩ := IsCyclic.exists_generator Kˣ
rw [← Fintype.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hx,
orderOf_dvd_iff_pow_eq_one]
constructor
· intro h
apply h
· intro h y
simp_rw [← mem_powers_iff_mem_zpowers] at hx
rcases hx y with ⟨j, rfl⟩
rw [← pow_mul, mul_comm, pow_mul, h, one_pow]
#align finite_field.forall_pow_eq_one_iff FiniteField.forall_pow_eq_one_iff
/-- The sum of `x ^ i` as `x` ranges over the units of a finite field of cardinality `q`
is equal to `0` unless `(q - 1) ∣ i`, in which case the sum is `q - 1`. -/
theorem sum_pow_units [Fintype Kˣ] (i : ℕ) :
(∑ x : Kˣ, (x ^ i : K)) = if q - 1 ∣ i then -1 else 0 :=
by
let φ : Kˣ →* K :=
{ toFun := fun x => x ^ i
map_one' := by rw [Units.val_one, one_pow]
map_mul' := by
intros
rw [Units.val_mul, mul_pow] }
have : Decidable (φ = 1) := by classical infer_instance
calc
(∑ x : Kˣ, φ x) = if φ = 1 then Fintype.card Kˣ else 0 := sum_hom_units φ
_ = if q - 1 ∣ i then -1 else 0 := _
suffices q - 1 ∣ i ↔ φ = 1 by
simp only [this]
split_ifs with h h
swap
rfl
rw [Fintype.card_units, Nat.cast_sub, cast_card_eq_zero, Nat.cast_one, zero_sub]
show 1 ≤ q
exact fintype.card_pos_iff.mpr ⟨0⟩
rw [← forall_pow_eq_one_iff, MonoidHom.ext_iff]
apply forall_congr'
intro x
rw [Units.ext_iff, Units.val_pow_eq_pow_val, Units.val_one, MonoidHom.one_apply]
rfl
#align finite_field.sum_pow_units FiniteField.sum_pow_units
/-- The sum of `x ^ i` as `x` ranges over a finite field of cardinality `q`
is equal to `0` if `i < q - 1`. -/
theorem sum_pow_lt_card_sub_one (i : ℕ) (h : i < q - 1) : (∑ x : K, x ^ i) = 0 :=
by
by_cases hi : i = 0
· simp only [hi, nsmul_one, sum_const, pow_zero, card_univ, cast_card_eq_zero]
classical
have hiq : ¬q - 1 ∣ i := by
contrapose! h
exact Nat.le_of_dvd (Nat.pos_of_ne_zero hi) h
let φ : Kˣ ↪ K := ⟨coe, Units.ext⟩
have : univ.map φ = univ \ {0} := by
ext x
simp only [true_and_iff, embedding.coe_fn_mk, mem_sdiff, Units.exists_iff_ne_zero, mem_univ,
mem_map, exists_prop_of_true, mem_singleton]
calc
(∑ x : K, x ^ i) = ∑ x in univ \ {(0 : K)}, x ^ i := by
rw [← sum_sdiff ({0} : Finset K).subset_univ, sum_singleton,
zero_pow (Nat.pos_of_ne_zero hi), add_zero]
_ = ∑ x : Kˣ, x ^ i := by
rw [← this, univ.sum_map φ]
rfl
_ = 0 := by
rw [sum_pow_units K i, if_neg]
exact hiq
#align finite_field.sum_pow_lt_card_sub_one FiniteField.sum_pow_lt_card_sub_one
section IsSplittingField
open Polynomial
section
variable (K' : Type _) [Field K'] {p n : ℕ}
theorem x_pow_card_sub_x_natDegree_eq (hp : 1 < p) : (X ^ p - X : K'[X]).natDegree = p :=
by
have h1 : (X : K'[X]).degree < (X ^ p : K'[X]).degree :=
by
rw [degree_X_pow, degree_X]
exact_mod_cast hp
rw [nat_degree_eq_of_degree_eq (degree_sub_eq_left_of_degree_lt h1), nat_degree_X_pow]
#align finite_field.X_pow_card_sub_X_nat_degree_eq FiniteField.x_pow_card_sub_x_natDegree_eq
theorem x_pow_card_pow_sub_x_natDegree_eq (hn : n ≠ 0) (hp : 1 < p) :
(X ^ p ^ n - X : K'[X]).natDegree = p ^ n :=
x_pow_card_sub_x_natDegree_eq K' <| Nat.one_lt_pow _ _ (Nat.pos_of_ne_zero hn) hp
#align finite_field.X_pow_card_pow_sub_X_nat_degree_eq FiniteField.x_pow_card_pow_sub_x_natDegree_eq
theorem x_pow_card_sub_x_ne_zero (hp : 1 < p) : (X ^ p - X : K'[X]) ≠ 0 :=
ne_zero_of_natDegree_gt <|
calc
1 < _ := hp
_ = _ := (x_pow_card_sub_x_natDegree_eq K' hp).symm
#align finite_field.X_pow_card_sub_X_ne_zero FiniteField.x_pow_card_sub_x_ne_zero
theorem x_pow_card_pow_sub_x_ne_zero (hn : n ≠ 0) (hp : 1 < p) : (X ^ p ^ n - X : K'[X]) ≠ 0 :=
x_pow_card_sub_x_ne_zero K' <| Nat.one_lt_pow _ _ (Nat.pos_of_ne_zero hn) hp
#align finite_field.X_pow_card_pow_sub_X_ne_zero FiniteField.x_pow_card_pow_sub_x_ne_zero
end
variable (p : ℕ) [Fact p.Prime] [Algebra (ZMod p) K]
theorem roots_x_pow_card_sub_x : roots (X ^ q - X : K[X]) = Finset.univ.val := by
classical
have aux : (X ^ q - X : K[X]) ≠ 0 := X_pow_card_sub_X_ne_zero K Fintype.one_lt_card
have : (roots (X ^ q - X : K[X])).toFinset = Finset.univ :=
by
rw [eq_univ_iff_forall]
intro x
rw [Multiset.mem_toFinset, mem_roots aux, is_root.def, eval_sub, eval_pow, eval_X,
sub_eq_zero, pow_card]
rw [← this, Multiset.toFinset_val, eq_comm, Multiset.dedup_eq_self]
apply nodup_roots
rw [separable_def]
convert is_coprime_one_right.neg_right using 1
·
rw [derivative_sub, derivative_X, derivative_X_pow, CharP.cast_card_eq_zero K, C_0,
MulZeroClass.zero_mul, zero_sub]
#align finite_field.roots_X_pow_card_sub_X FiniteField.roots_x_pow_card_sub_x
instance (F : Type _) [Field F] [Algebra F K] : IsSplittingField F K (X ^ q - X)
where
Splits :=
by
have h : (X ^ q - X : K[X]).natDegree = q :=
X_pow_card_sub_X_nat_degree_eq K Fintype.one_lt_card
rw [← splits_id_iff_splits, splits_iff_card_roots, Polynomial.map_sub, Polynomial.map_pow,
map_X, h, roots_X_pow_card_sub_X K, ← Finset.card_def, Finset.card_univ]
adjoin_roots := by
classical
trans Algebra.adjoin F ((roots (X ^ q - X : K[X])).toFinset : Set K)
· simp only [Polynomial.map_pow, map_X, Polynomial.map_sub]
· rw [roots_X_pow_card_sub_X, val_to_finset, coe_univ, Algebra.adjoin_univ]
end IsSplittingField
variable {K}
theorem frobenius_pow {p : ℕ} [Fact p.Prime] [CharP K p] {n : ℕ} (hcard : q = p ^ n) :
frobenius K p ^ n = 1 := by
ext; conv_rhs => rw [RingHom.one_def, RingHom.id_apply, ← pow_card x, hcard]; clear hcard
induction n; · simp
rw [pow_succ, pow_succ', pow_mul, RingHom.mul_def, RingHom.comp_apply, frobenius_def, n_ih]
#align finite_field.frobenius_pow FiniteField.frobenius_pow
open Polynomial
theorem expand_card (f : K[X]) : expand K q f = f ^ q :=
by
cases' CharP.exists K with p hp
letI := hp
rcases FiniteField.card K p with ⟨⟨n, npos⟩, ⟨hp, hn⟩⟩
haveI : Fact p.prime := ⟨hp⟩
dsimp at hn
rw [hn, ← map_expand_pow_char, frobenius_pow hn, RingHom.one_def, map_id]
#align finite_field.expand_card FiniteField.expand_card
end FiniteField
namespace ZMod
open FiniteField Polynomial
theorem sq_add_sq (p : ℕ) [hp : Fact p.Prime] (x : ZMod p) : ∃ a b : ZMod p, a ^ 2 + b ^ 2 = x :=
by
cases' hp.1.eq_two_or_odd with hp2 hp_odd
· subst p
change Fin 2 at x
fin_cases x
· use 0
simp
· use 0, 1
simp
let f : (ZMod p)[X] := X ^ 2
let g : (ZMod p)[X] := X ^ 2 - C x
obtain ⟨a, b, hab⟩ : ∃ a b, f.eval a + g.eval b = 0 :=
@exists_root_sum_quadratic _ _ _ _ f g (degree_X_pow 2) (degree_X_pow_sub_C (by decide) _)
(by rw [ZMod.card, hp_odd])
refine' ⟨a, b, _⟩
rw [← sub_eq_zero]
simpa only [eval_C, eval_X, eval_pow, eval_sub, ← add_sub_assoc] using hab
#align zmod.sq_add_sq ZMod.sq_add_sq
end ZMod
namespace CharP
theorem sq_add_sq (R : Type _) [CommRing R] [IsDomain R] (p : ℕ) [NeZero p] [CharP R p] (x : ℤ) :
∃ a b : ℕ, (a ^ 2 + b ^ 2 : R) = x :=
by
haveI := char_is_prime_of_pos R p
obtain ⟨a, b, hab⟩ := ZMod.sq_add_sq p x
refine' ⟨a.val, b.val, _⟩
simpa using congr_arg (ZMod.castHom dvd_rfl R) hab
#align char_p.sq_add_sq CharP.sq_add_sq
end CharP
open Nat
open ZMod
/-- The **Fermat-Euler totient theorem**. `nat.modeq.pow_totient` is an alternative statement
of the same theorem. -/
@[simp]
theorem ZMod.pow_totient {n : ℕ} (x : (ZMod n)ˣ) : x ^ φ n = 1 :=
by
cases n
· rw [Nat.totient_zero, pow_zero]
· rw [← card_units_eq_totient, pow_card_eq_one]
#align zmod.pow_totient ZMod.pow_totient
/-- The **Fermat-Euler totient theorem**. `zmod.pow_totient` is an alternative statement
of the same theorem. -/
theorem Nat.ModEq.pow_totient {x n : ℕ} (h : Nat.coprime x n) : x ^ φ n ≡ 1 [MOD n] :=
by
rw [← ZMod.eq_iff_modEq_nat]
let x' : Units (ZMod n) := ZMod.unitOfCoprime _ h
have := ZMod.pow_totient x'
apply_fun (coe : Units (ZMod n) → ZMod n) at this
simpa only [-ZMod.pow_totient, Nat.succ_eq_add_one, Nat.cast_pow, Units.val_one, Nat.cast_one,
coe_unit_of_coprime, Units.val_pow_eq_pow_val]
#align nat.modeq.pow_totient Nat.ModEq.pow_totient
section
variable {V : Type _} [Fintype K] [DivisionRing K] [AddCommGroup V] [Module K V]
-- should this go in a namespace?
-- finite_dimensional would be natural,
-- but we don't assume it...
theorem card_eq_pow_finrank [Fintype V] : Fintype.card V = q ^ FiniteDimensional.finrank K V :=
by
let b := IsNoetherian.finsetBasis K V
rw [Module.card_fintype b, ← FiniteDimensional.finrank_eq_card_basis b]
#align card_eq_pow_finrank card_eq_pow_finrank
end
open FiniteField
namespace ZMod
/-- A variation on Fermat's little theorem. See `zmod.pow_card_sub_one_eq_one` -/
@[simp]
theorem pow_card {p : ℕ} [Fact p.Prime] (x : ZMod p) : x ^ p = x :=
by
have h := FiniteField.pow_card x
rwa [ZMod.card p] at h
#align zmod.pow_card ZMod.pow_card
@[simp]
theorem pow_card_pow {n p : ℕ} [Fact p.Prime] (x : ZMod p) : x ^ p ^ n = x :=
by
induction' n with n ih
· simp
· simp [pow_succ, pow_mul, ih, pow_card]
#align zmod.pow_card_pow ZMod.pow_card_pow
@[simp]
theorem frobenius_zMod (p : ℕ) [Fact p.Prime] : frobenius (ZMod p) p = RingHom.id _ :=
by
ext a
rw [frobenius_def, ZMod.pow_card, RingHom.id_apply]
#align zmod.frobenius_zmod ZMod.frobenius_zMod
@[simp]
theorem card_units (p : ℕ) [Fact p.Prime] : Fintype.card (ZMod p)ˣ = p - 1 := by
rw [Fintype.card_units, card]
#align zmod.card_units ZMod.card_units
/-- **Fermat's Little Theorem**: for every unit `a` of `zmod p`, we have `a ^ (p - 1) = 1`. -/
theorem units_pow_card_sub_one_eq_one (p : ℕ) [Fact p.Prime] (a : (ZMod p)ˣ) : a ^ (p - 1) = 1 := by
rw [← card_units p, pow_card_eq_one]
#align zmod.units_pow_card_sub_one_eq_one ZMod.units_pow_card_sub_one_eq_one
/-- **Fermat's Little Theorem**: for all nonzero `a : zmod p`, we have `a ^ (p - 1) = 1`. -/
theorem pow_card_sub_one_eq_one {p : ℕ} [Fact p.Prime] {a : ZMod p} (ha : a ≠ 0) :
a ^ (p - 1) = 1 := by
have h := pow_card_sub_one_eq_one a ha
rwa [ZMod.card p] at h
#align zmod.pow_card_sub_one_eq_one ZMod.pow_card_sub_one_eq_one
theorem orderOf_units_dvd_card_sub_one {p : ℕ} [Fact p.Prime] (u : (ZMod p)ˣ) : orderOf u ∣ p - 1 :=
orderOf_dvd_of_pow_eq_one <| units_pow_card_sub_one_eq_one _ _
#align zmod.order_of_units_dvd_card_sub_one ZMod.orderOf_units_dvd_card_sub_one
theorem orderOf_dvd_card_sub_one {p : ℕ} [Fact p.Prime] {a : ZMod p} (ha : a ≠ 0) :
orderOf a ∣ p - 1 :=
orderOf_dvd_of_pow_eq_one <| pow_card_sub_one_eq_one ha
#align zmod.order_of_dvd_card_sub_one ZMod.orderOf_dvd_card_sub_one
open Polynomial
theorem expand_card {p : ℕ} [Fact p.Prime] (f : Polynomial (ZMod p)) :
expand (ZMod p) p f = f ^ p :=
by
have h := FiniteField.expand_card f
rwa [ZMod.card p] at h
#align zmod.expand_card ZMod.expand_card
end ZMod
/-- **Fermat's Little Theorem**: for all `a : ℤ` coprime to `p`, we have
`a ^ (p - 1) ≡ 1 [ZMOD p]`. -/
theorem Int.ModEq.pow_card_sub_one_eq_one {p : ℕ} (hp : Nat.Prime p) {n : ℤ} (hpn : IsCoprime n p) :
n ^ (p - 1) ≡ 1 [ZMOD p] := by
haveI : Fact p.prime := ⟨hp⟩
have : ¬(n : ZMod p) = 0 :=
by
rw [CharP.int_cast_eq_zero_iff _ p, ← (nat.prime_iff_prime_int.mp hp).coprime_iff_not_dvd]
· exact hpn.symm
exact ZMod.charP p
simpa [← ZMod.int_cast_eq_int_cast_iff] using ZMod.pow_card_sub_one_eq_one this
#align int.modeq.pow_card_sub_one_eq_one Int.ModEq.pow_card_sub_one_eq_one
section
namespace FiniteField
variable {F : Type _} [Field F]
section Finite
variable [Finite F]
/-- In a finite field of characteristic `2`, all elements are squares. -/
theorem isSquare_of_char_two (hF : ringChar F = 2) (a : F) : IsSquare a :=
haveI hF' : CharP F 2 := ringChar.of_eq hF
isSquare_of_charTwo' a
#align finite_field.is_square_of_char_two FiniteField.isSquare_of_char_two
/-- In a finite field of odd characteristic, not every element is a square. -/
theorem exists_nonsquare (hF : ringChar F ≠ 2) : ∃ a : F, ¬IsSquare a :=
by
-- Idea: the squaring map on `F` is not injective, hence not surjective
let sq : F → F := fun x => x ^ 2
have h : ¬injective sq :=
by
simp only [injective, not_forall, exists_prop]
refine' ⟨-1, 1, _, Ring.neg_one_ne_one_of_char_ne_two hF⟩
simp only [sq, one_pow, neg_one_sq]
rw [Finite.injective_iff_surjective] at h
-- sq not surjective
simp_rw [IsSquare, ← pow_two, @eq_comm _ _ (_ ^ 2)]
push_neg at h⊢
exact h
#align finite_field.exists_nonsquare FiniteField.exists_nonsquare
end Finite
variable [Fintype F]
/-- The finite field `F` has even cardinality iff it has characteristic `2`. -/
theorem even_card_iff_char_two : ringChar F = 2 ↔ Fintype.card F % 2 = 0 :=
by
rcases FiniteField.card F (ringChar F) with ⟨n, hp, h⟩
rw [h, Nat.pow_mod]
constructor
· intro hF
rw [hF]
simp only [Nat.bit0_mod_two, zero_pow', Ne.def, PNat.ne_zero, not_false_iff, Nat.zero_mod]
· rw [← Nat.even_iff, Nat.even_pow]
rintro ⟨hev, hnz⟩
rw [Nat.even_iff, Nat.mod_mod] at hev
exact (Nat.Prime.eq_two_or_odd hp).resolve_right (ne_of_eq_of_ne hev zero_ne_one)
#align finite_field.even_card_iff_char_two FiniteField.even_card_iff_char_two
theorem even_card_of_char_two (hF : ringChar F = 2) : Fintype.card F % 2 = 0 :=
even_card_iff_char_two.mp hF
#align finite_field.even_card_of_char_two FiniteField.even_card_of_char_two
theorem odd_card_of_char_ne_two (hF : ringChar F ≠ 2) : Fintype.card F % 2 = 1 :=
Nat.mod_two_ne_zero.mp (mt even_card_iff_char_two.mpr hF)
#align finite_field.odd_card_of_char_ne_two FiniteField.odd_card_of_char_ne_two
/-- If `F` has odd characteristic, then for nonzero `a : F`, we have that `a ^ (#F / 2) = ±1`. -/
theorem pow_dichotomy (hF : ringChar F ≠ 2) {a : F} (ha : a ≠ 0) :
a ^ (Fintype.card F / 2) = 1 ∨ a ^ (Fintype.card F / 2) = -1 :=
by
have h₁ := FiniteField.pow_card_sub_one_eq_one a ha
rw [← Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF), mul_comm, pow_mul,
pow_two] at h₁
exact mul_self_eq_one_iff.mp h₁
#align finite_field.pow_dichotomy FiniteField.pow_dichotomy
/-- A unit `a` of a finite field `F` of odd characteristic is a square
if and only if `a ^ (#F / 2) = 1`. -/
theorem unit_isSquare_iff (hF : ringChar F ≠ 2) (a : Fˣ) :
IsSquare a ↔ a ^ (Fintype.card F / 2) = 1 := by
classical
obtain ⟨g, hg⟩ := IsCyclic.exists_generator Fˣ
obtain ⟨n, hn⟩ : a ∈ Submonoid.powers g :=
by
rw [mem_powers_iff_mem_zpowers]
apply hg
have hodd := Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF)
constructor
· rintro ⟨y, rfl⟩
rw [← pow_two, ← pow_mul, hodd]
apply_fun @coe Fˣ F _ using Units.ext
· push_cast
exact FiniteField.pow_card_sub_one_eq_one (y : F) (Units.ne_zero y)
· subst a
intro h
have key : 2 * (Fintype.card F / 2) ∣ n * (Fintype.card F / 2) :=
by
rw [← pow_mul] at h
rw [hodd, ← Fintype.card_units, ← orderOf_eq_card_of_forall_mem_zpowers hg]
apply orderOf_dvd_of_pow_eq_one h
have : 0 < Fintype.card F / 2 := Nat.div_pos Fintype.one_lt_card (by norm_num)
obtain ⟨m, rfl⟩ := Nat.dvd_of_mul_dvd_mul_right this key
refine' ⟨g ^ m, _⟩
rw [mul_comm, pow_mul, pow_two]
#align finite_field.unit_is_square_iff FiniteField.unit_isSquare_iff
/-- A non-zero `a : F` is a square if and only if `a ^ (#F / 2) = 1`. -/
theorem isSquare_iff (hF : ringChar F ≠ 2) {a : F} (ha : a ≠ 0) :
IsSquare a ↔ a ^ (Fintype.card F / 2) = 1 :=
by
apply
(iff_congr _ (by simp [Units.ext_iff])).mp (FiniteField.unit_isSquare_iff hF (Units.mk0 a ha))
simp only [IsSquare, Units.ext_iff, Units.val_mk0, Units.val_mul]
constructor
· rintro ⟨y, hy⟩
exact ⟨y, hy⟩
· rintro ⟨y, rfl⟩
have hy : y ≠ 0 := by
rintro rfl
simpa [zero_pow] using ha
refine' ⟨Units.mk0 y hy, _⟩
simp
#align finite_field.is_square_iff FiniteField.isSquare_iff
end FiniteField
end
|
"""
ESS(chain, k)
Calculate the Effective Sample Size.
# Arguments
- `chain`: matrix of draws from the target distribution
- `k`: (integer) number of lags
# Output
- `ESS`: vector of effective sample sizes
"""
function ESS(chain::Matrix{T} where T<:AbstractFloat, k = 10::Integer)
M, J = size(chain)
@assert M > k "The chain of β must be longer than k."
ρ = sum(pacf(chain, 1:k); dims = 1)
τ = 1 .+ 2 .* sum(ρ; dims = 1)
ESS = round.(Int, vec(M ./ τ))
return ESS
end
"""
probability(x, chain)
Calculate probabilities based on the probit link.ß
# Arguments
- `x`: matrix of inputs
- `chain`: matrix of draws from the target distribution
# Output
- `p`: matrix of probabilities
"""
function probability(x::Matrix{T} where T<:Real, chain::Matrix{T} where T<:AbstractFloat)
z = x * chain'
p = cdf(Normal(0, 1), z)
return p
end
probability(x::Vector{T} where T<:Real, chain::Matrix{T} where T<:AbstractFloat) = probability(repeat(x, 1, 1), chain) |
import tactic
open list
variable {α : Type}
@[simp]
lemma drop_eq_nil_of_le
{l : list α}
{k : ℕ}
(h : length l ≤ k)
: drop k l = [] :=
by simpa [←length_eq_zero] using nat.sub_eq_zero_of_le h
theorem drop_nil :
∀ n, drop n [] = ([] : list α) :=
λ _, list.drop_eq_nil_of_le (nat.zero_le _)
|
lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}" |
_ : @ω Set → Set₁
_ = λ (@0 _) → Set
|
[STATEMENT]
lemma lemma_2_reachability_s:
fixes PROB :: "'a problem" and as s
assumes "finite PROB" "(s \<in> valid_states PROB)" "(as \<in> valid_plans PROB)"
"(length as > card (reachable_s PROB s) - 1)"
shows "(\<exists>as1 as2 as3.
(as1 @ as2 @ as3 = as) \<and> (exec_plan s (as1 @ as2) = exec_plan s as1) \<and> \<not>(as2 = []))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
{
[PROOF STATE]
proof (state)
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
have "Suc (length as) > card (reachable_s PROB s)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. card (reachable_s PROB s) < Suc (length as)
[PROOF STEP]
using assms(4)
[PROOF STATE]
proof (prove)
using this:
card (reachable_s PROB s) - 1 < length as
goal (1 subgoal):
1. card (reachable_s PROB s) < Suc (length as)
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
card (reachable_s PROB s) < Suc (length as)
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
card (reachable_s PROB s) < Suc (length as)
[PROOF STEP]
have "card (state_set (state_list s as)) > card (reachable_s PROB s)"
[PROOF STATE]
proof (prove)
using this:
card (reachable_s PROB s) < Suc (length as)
goal (1 subgoal):
1. card (reachable_s PROB s) < card (state_set (state_list s as))
[PROOF STEP]
using card_state_set
[PROOF STATE]
proof (prove)
using this:
card (reachable_s PROB s) < Suc (length as)
Suc (length ?as) = card (state_set (state_list ?s ?as))
goal (1 subgoal):
1. card (reachable_s PROB s) < card (state_set (state_list s as))
[PROOF STEP]
by metis
[PROOF STATE]
proof (state)
this:
card (reachable_s PROB s) < card (state_set (state_list s as))
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
card (reachable_s PROB s) < card (state_set (state_list s as))
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
note 1 = this
[PROOF STATE]
proof (state)
this:
card (reachable_s PROB s) < card (state_set (state_list s as))
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
{
[PROOF STATE]
proof (state)
this:
card (reachable_s PROB s) < card (state_set (state_list s as))
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
have "finite (reachable_s PROB s)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. finite (reachable_s PROB s)
[PROOF STEP]
using assms(1, 2) reachable_s_finite_thm
[PROOF STATE]
proof (prove)
using this:
finite PROB
s \<in> valid_states PROB
\<lbrakk>finite ?PROB; ?s \<in> valid_states ?PROB\<rbrakk> \<Longrightarrow> finite (reachable_s ?PROB ?s)
goal (1 subgoal):
1. finite (reachable_s PROB s)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
finite (reachable_s PROB s)
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
finite (reachable_s PROB s)
[PROOF STEP]
have "\<not>(inj last (state_set (state_list s as)) (reachable_s PROB s))"
[PROOF STATE]
proof (prove)
using this:
finite (reachable_s PROB s)
goal (1 subgoal):
1. \<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)
[PROOF STEP]
using assms(4) 1 lemma_2_reachability_s_i
[PROOF STATE]
proof (prove)
using this:
finite (reachable_s PROB s)
card (reachable_s PROB s) - 1 < length as
card (reachable_s PROB s) < card (state_set (state_list s as))
\<lbrakk>finite ?t; card ?t < card ?s\<rbrakk> \<Longrightarrow> \<not> FactoredSystem.inj ?f ?s ?t
goal (1 subgoal):
1. \<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
}
[PROOF STATE]
proof (state)
this:
\<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
note 2 = this
[PROOF STATE]
proof (state)
this:
\<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
obtain slist_1 slist_2 where 3:
"slist_1 \<in> state_set (state_list s as)" "slist_2 \<in> state_set (state_list s as)"
"(last slist_1 = last slist_2)" "length slist_1 \<noteq> length slist_2"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (\<And>slist_1 slist_2. \<lbrakk>slist_1 \<in> state_set (state_list s as); slist_2 \<in> state_set (state_list s as); last slist_1 = last slist_2; length slist_1 \<noteq> length slist_2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
using assms(2, 3) 2 not_eq_last_diff_paths_reachability_s
[PROOF STATE]
proof (prove)
using this:
s \<in> valid_states PROB
as \<in> valid_plans PROB
\<not> FactoredSystem.inj last (state_set (state_list s as)) (reachable_s PROB s)
\<lbrakk>?s \<in> valid_states ?PROB; ?as \<in> valid_plans ?PROB; \<not> FactoredSystem.inj last (state_set (state_list ?s ?as)) (reachable_s ?PROB ?s)\<rbrakk> \<Longrightarrow> \<exists>slist_1 slist_2. slist_1 \<in> state_set (state_list ?s ?as) \<and> slist_2 \<in> state_set (state_list ?s ?as) \<and> last slist_1 = last slist_2 \<and> length slist_1 \<noteq> length slist_2
goal (1 subgoal):
1. (\<And>slist_1 slist_2. \<lbrakk>slist_1 \<in> state_set (state_list s as); slist_2 \<in> state_set (state_list s as); last slist_1 = last slist_2; length slist_1 \<noteq> length slist_2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
slist_1 \<in> state_set (state_list s as)
slist_2 \<in> state_set (state_list s as)
last slist_1 = last slist_2
length slist_1 \<noteq> length slist_2
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
slist_1 \<in> state_set (state_list s as)
slist_2 \<in> state_set (state_list s as)
last slist_1 = last slist_2
length slist_1 \<noteq> length slist_2
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
slist_1 \<in> state_set (state_list s as)
slist_2 \<in> state_set (state_list s as)
last slist_1 = last slist_2
length slist_1 \<noteq> length slist_2
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
using assms
[PROOF STATE]
proof (prove)
using this:
slist_1 \<in> state_set (state_list s as)
slist_2 \<in> state_set (state_list s as)
last slist_1 = last slist_2
length slist_1 \<noteq> length slist_2
finite PROB
s \<in> valid_states PROB
as \<in> valid_plans PROB
card (reachable_s PROB s) - 1 < length as
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
proof(cases as)
[PROOF STATE]
proof (state)
goal (2 subgoals):
1. \<lbrakk>slist_1 \<in> state_set (state_list s as); slist_2 \<in> state_set (state_list s as); last slist_1 = last slist_2; length slist_1 \<noteq> length slist_2; finite PROB; s \<in> valid_states PROB; as \<in> valid_plans PROB; card (reachable_s PROB s) - 1 < length as; as = []\<rbrakk> \<Longrightarrow> \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
2. \<And>a list. \<lbrakk>slist_1 \<in> state_set (state_list s as); slist_2 \<in> state_set (state_list s as); last slist_1 = last slist_2; length slist_1 \<noteq> length slist_2; finite PROB; s \<in> valid_states PROB; as \<in> valid_plans PROB; card (reachable_s PROB s) - 1 < length as; as = a # list\<rbrakk> \<Longrightarrow> \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
case (Cons a list)
[PROOF STATE]
proof (state)
this:
as = a # list
goal (2 subgoals):
1. \<lbrakk>slist_1 \<in> state_set (state_list s as); slist_2 \<in> state_set (state_list s as); last slist_1 = last slist_2; length slist_1 \<noteq> length slist_2; finite PROB; s \<in> valid_states PROB; as \<in> valid_plans PROB; card (reachable_s PROB s) - 1 < length as; as = []\<rbrakk> \<Longrightarrow> \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
2. \<And>a list. \<lbrakk>slist_1 \<in> state_set (state_list s as); slist_2 \<in> state_set (state_list s as); last slist_1 = last slist_2; length slist_1 \<noteq> length slist_2; finite PROB; s \<in> valid_states PROB; as \<in> valid_plans PROB; card (reachable_s PROB s) - 1 < length as; as = a # list\<rbrakk> \<Longrightarrow> \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
as = a # list
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
as = a # list
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
using assms(2, 3) 3 eq_last_state_imp_append_nempty_as state_set_thm list.distinct(1)
[PROOF STATE]
proof (prove)
using this:
as = a # list
s \<in> valid_states PROB
as \<in> valid_plans PROB
slist_1 \<in> state_set (state_list s as)
slist_2 \<in> state_set (state_list s as)
last slist_1 = last slist_2
length slist_1 \<noteq> length slist_2
\<lbrakk>?as \<noteq> []; ?s \<in> valid_states ?PROB; ?as \<in> valid_plans ?PROB; ?slist_1.0 \<noteq> []; ?slist_2.0 \<noteq> []; ?slist_1.0 \<in> state_set (state_list ?s ?as); ?slist_2.0 \<in> state_set (state_list ?s ?as); length ?slist_1.0 \<noteq> length ?slist_2.0; last ?slist_1.0 = last ?slist_2.0\<rbrakk> \<Longrightarrow> \<exists>as1 as2 as3. as1 @ as2 @ as3 = ?as \<and> exec_plan ?s (as1 @ as2) = exec_plan ?s as1 \<and> as2 \<noteq> []
(?s1.0 \<in> state_set ?s2.0) = (prefix ?s1.0 ?s2.0 \<and> ?s1.0 \<noteq> [])
[] \<noteq> ?x21.0 # ?x22.0
goal (1 subgoal):
1. \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
by metis
[PROOF STATE]
proof (state)
this:
\<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
goal (1 subgoal):
1. \<lbrakk>slist_1 \<in> state_set (state_list s as); slist_2 \<in> state_set (state_list s as); last slist_1 = last slist_2; length slist_1 \<noteq> length slist_2; finite PROB; s \<in> valid_states PROB; as \<in> valid_plans PROB; card (reachable_s PROB s) - 1 < length as; as = []\<rbrakk> \<Longrightarrow> \<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
[PROOF STEP]
qed force
[PROOF STATE]
proof (state)
this:
\<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []
goal:
No subgoals!
[PROOF STEP]
qed |
open import Relation.Binary.Core
module InsertSort.Impl1.Correctness.Order {A : Set}
(_≤_ : A → A → Set)
(tot≤ : Total _≤_) where
open import Data.List
open import Data.Sum
open import InsertSort.Impl1 _≤_ tot≤
open import List.Sorted _≤_
lemma-insert-sorted : {xs : List A}(x : A) → Sorted xs → Sorted (insert x xs)
lemma-insert-sorted {xs = .[]} x nils = singls x
lemma-insert-sorted {xs = .([ y ])} x (singls y)
with tot≤ x y
... | inj₁ x≤y = conss x≤y (singls y)
... | inj₂ y≤x = conss y≤x (singls x)
lemma-insert-sorted x (conss {y} {z} {ys} y≤z szys)
with tot≤ x y
... | inj₁ x≤y = conss x≤y (conss y≤z szys)
... | inj₂ y≤x
with tot≤ x z | lemma-insert-sorted x szys
... | inj₁ x≤z | _ = conss y≤x (conss x≤z szys)
... | inj₂ z≤x | h = conss y≤z h
theorem-insertSort-sorted : (xs : List A) → Sorted (insertSort xs)
theorem-insertSort-sorted [] = nils
theorem-insertSort-sorted (x ∷ xs) = lemma-insert-sorted x (theorem-insertSort-sorted xs)
|
import Aesop
example : True := by
aesop (rule_sets [Nonexistent])
|
Formal statement is: lemma reflect_poly_0 [simp]: "reflect_poly 0 = 0" Informal statement is: The reflection of the polynomial $0$ is $0$. |
theorem Nat.zeroDiv {n : Nat} : 0 / n = 0 := by
rw Nat.divDef
have key : ¬ (0 < n ∧ n ≤ 0) by
intro h
have h' := Nat.ltOfLtOfLe h.1 h.2
exact Nat.notLtZero 0 h'
rw ifNeg key
theorem Nat.div_le_div_of_mul_aux (k n m : Nat) : n / k ≤ (n + m) / k := by {
induction m with
| zero => { exact Nat.leRefl _ }
| succ m ih => { exact sorry }
}
theorem Nat.div_le_div_of_mul {k n m : Nat} (h : n ≤ m) : n / k ≤ m / k := by {
have h' := Nat.div_le_div_of_mul_aux k n (m - n);
exact sorry
}
theorem Nat.div_le_of_le_mul {k n m : Nat} (h : k ≤ n * m) : k / m ≤ n := by {
induction k with
| zero => { rw Nat.zeroDiv; exact Nat.zeroLe _ }
| succ k ih => {
have hk : k ≤ n * m := Nat.leTrans (Nat.leSucc _) h;
have ih' := ih (Nat.leTrans (Nat.leSucc _) h);
exact sorry
}
}
theorem Nat.div_lt_of_lt_mul {k n m : Nat} (h : k < n * m) : k / m < n := by {
exact sorry
} |
theorem le_trans (a b c : mynat) (hab : a ≤ b) (hbc : b ≤ c) : a ≤ c :=
begin
cases hab,
cases hbc,
use (hab_w + hbc_w),
rw ← add_assoc,
rw hbc_h,
rw hab_h,
refl,
end
|
import data.fintype data.finset algebra.group_power
open fintype finset
/- 6. Say A and B are finite sets, with sizes a and b respectively. Prove that the set B^A of functions A → B has size b^a. What about the case a = b = 0?
-/
#check finset.card
variables {A B C : Type} [fintype A] [decidable_eq A] [fintype B]
variables {a b : ℕ }
theorem Q1006 (hp: card A = a) (hq : card B = b) : card (A → B) = b^a :=
begin
simp,
rw hp,
rw hq,
end |
[GOAL]
⊢ MeasurablePow ℝ≥0∞ ℝ
[PROOFSTEP]
refine' ⟨ENNReal.measurable_of_measurable_nnreal_prod _ _⟩
[GOAL]
case refine'_1
⊢ Measurable fun p => (↑p.fst, p.snd).fst ^ (↑p.fst, p.snd).snd
[PROOFSTEP]
simp_rw [ENNReal.coe_rpow_def]
[GOAL]
case refine'_1
⊢ Measurable fun p => if p.fst = 0 ∧ p.snd < 0 then ⊤ else ↑(p.fst ^ p.snd)
[PROOFSTEP]
refine' Measurable.ite _ measurable_const (measurable_fst.pow measurable_snd).coe_nnreal_ennreal
[GOAL]
case refine'_1
⊢ MeasurableSet {a | a.fst = 0 ∧ a.snd < 0}
[PROOFSTEP]
exact MeasurableSet.inter (measurable_fst (measurableSet_singleton 0)) (measurable_snd measurableSet_Iio)
[GOAL]
case refine'_2
⊢ Measurable fun x => (⊤, x).fst ^ (⊤, x).snd
[PROOFSTEP]
simp_rw [ENNReal.top_rpow_def]
[GOAL]
case refine'_2
⊢ Measurable fun x => if 0 < x then ⊤ else if x = 0 then 1 else 0
[PROOFSTEP]
refine' Measurable.ite measurableSet_Ioi measurable_const _
[GOAL]
case refine'_2
⊢ Measurable fun x => if x = 0 then 1 else 0
[PROOFSTEP]
exact Measurable.ite (measurableSet_singleton 0) measurable_const measurable_const
|
/-
Copyright © 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Yury Kudryashov
! This file was ported from Lean 3 source module geometry.manifold.diffeomorph
! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Geometry.Manifold.ContMdiffMap
/-!
# Diffeomorphisms
This file implements diffeomorphisms.
## Definitions
* `diffeomorph I I' M M' n`: `n`-times continuously differentiable diffeomorphism between
`M` and `M'` with respect to I and I'; we do not introduce a separate definition for the case
`n = ∞`; we use notation instead.
* `diffeomorph.to_homeomorph`: reinterpret a diffeomorphism as a homeomorphism.
* `continuous_linear_equiv.to_diffeomorph`: reinterpret a continuous equivalence as
a diffeomorphism.
* `model_with_corners.trans_diffeomorph`: compose a given `model_with_corners` with a diffeomorphism
between the old and the new target spaces. Useful, e.g, to turn any finite dimensional manifold
into a manifold modelled on a Euclidean space.
* `diffeomorph.to_trans_diffeomorph`: the identity diffeomorphism between `M` with model `I` and `M`
with model `I.trans_diffeomorph e`.
## Notations
* `M ≃ₘ^n⟮I, I'⟯ M'` := `diffeomorph I J M N n`
* `M ≃ₘ⟮I, I'⟯ M'` := `diffeomorph I J M N ⊤`
* `E ≃ₘ^n[𝕜] E'` := `E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E'`
* `E ≃ₘ[𝕜] E'` := `E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, E')⟯ E'`
## Implementation notes
This notion of diffeomorphism is needed although there is already a notion of structomorphism
because structomorphisms do not allow the model spaces `H` and `H'` of the two manifolds to be
different, i.e. for a structomorphism one has to impose `H = H'` which is often not the case in
practice.
## Keywords
diffeomorphism, manifold
-/
open Manifold Topology
open Function Set
variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {E' : Type _} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {F : Type _}
[NormedAddCommGroup F] [NormedSpace 𝕜 F] {H : Type _} [TopologicalSpace H] {H' : Type _}
[TopologicalSpace H'] {G : Type _} [TopologicalSpace G] {G' : Type _} [TopologicalSpace G']
{I : ModelWithCorners 𝕜 E H} {I' : ModelWithCorners 𝕜 E' H'} {J : ModelWithCorners 𝕜 F G}
{J' : ModelWithCorners 𝕜 F G'}
variable {M : Type _} [TopologicalSpace M] [ChartedSpace H M] {M' : Type _} [TopologicalSpace M']
[ChartedSpace H' M'] {N : Type _} [TopologicalSpace N] [ChartedSpace G N] {N' : Type _}
[TopologicalSpace N'] [ChartedSpace G' N'] {n : ℕ∞}
section Defs
variable (I I' M M' n)
/--
`n`-times continuously differentiable diffeomorphism between `M` and `M'` with respect to I and I'
-/
@[protect_proj, nolint has_nonempty_instance]
structure Diffeomorph extends M ≃ M' where
contMdiff_toFun : ContMdiff I I' n to_equiv
contMdiff_inv_fun : ContMdiff I' I n to_equiv.symm
#align diffeomorph Diffeomorph
end Defs
-- mathport name: diffeomorph
scoped[Manifold] notation M " ≃ₘ^" n:1000 "⟮" I ", " J "⟯ " N => Diffeomorph I J M N n
-- mathport name: diffeomorph.top
scoped[Manifold] notation M " ≃ₘ⟮" I ", " J "⟯ " N => Diffeomorph I J M N ⊤
-- mathport name: diffeomorph.self
scoped[Manifold]
notation E " ≃ₘ^" n:1000 "[" 𝕜 "] " E' =>
Diffeomorph (modelWithCornersSelf 𝕜 E) (modelWithCornersSelf 𝕜 E') E E' n
-- mathport name: diffeomorph.self.top
scoped[Manifold]
notation E " ≃ₘ[" 𝕜 "] " E' =>
Diffeomorph (modelWithCornersSelf 𝕜 E) (modelWithCornersSelf 𝕜 E') E E' ⊤
namespace Diffeomorph
instance : CoeFun (M ≃ₘ^n⟮I, I'⟯ M') fun _ => M → M' :=
⟨fun e => e.toEquiv⟩
instance : Coe (M ≃ₘ^n⟮I, I'⟯ M') C^n⟮I, M; I', M'⟯ :=
⟨fun Φ => ⟨Φ, Φ.contMdiff_toFun⟩⟩
@[continuity]
protected theorem continuous (h : M ≃ₘ^n⟮I, I'⟯ M') : Continuous h :=
h.contMdiff_toFun.Continuous
#align diffeomorph.continuous Diffeomorph.continuous
protected theorem contMdiff (h : M ≃ₘ^n⟮I, I'⟯ M') : ContMdiff I I' n h :=
h.contMdiff_toFun
#align diffeomorph.cont_mdiff Diffeomorph.contMdiff
protected theorem contMdiffAt (h : M ≃ₘ^n⟮I, I'⟯ M') {x} : ContMdiffAt I I' n h x :=
h.ContMdiff.ContMdiffAt
#align diffeomorph.cont_mdiff_at Diffeomorph.contMdiffAt
protected theorem contMdiffWithinAt (h : M ≃ₘ^n⟮I, I'⟯ M') {s x} : ContMdiffWithinAt I I' n h s x :=
h.ContMdiffAt.ContMdiffWithinAt
#align diffeomorph.cont_mdiff_within_at Diffeomorph.contMdiffWithinAt
protected theorem contDiff (h : E ≃ₘ^n[𝕜] E') : ContDiff 𝕜 n h :=
h.ContMdiff.ContDiff
#align diffeomorph.cont_diff Diffeomorph.contDiff
protected theorem smooth (h : M ≃ₘ⟮I, I'⟯ M') : Smooth I I' h :=
h.contMdiff_toFun
#align diffeomorph.smooth Diffeomorph.smooth
protected theorem mdifferentiable (h : M ≃ₘ^n⟮I, I'⟯ M') (hn : 1 ≤ n) : Mdifferentiable I I' h :=
h.ContMdiff.Mdifferentiable hn
#align diffeomorph.mdifferentiable Diffeomorph.mdifferentiable
protected theorem mdifferentiableOn (h : M ≃ₘ^n⟮I, I'⟯ M') (s : Set M) (hn : 1 ≤ n) :
MdifferentiableOn I I' h s :=
(h.Mdifferentiable hn).MdifferentiableOn
#align diffeomorph.mdifferentiable_on Diffeomorph.mdifferentiableOn
@[simp]
theorem coe_toEquiv (h : M ≃ₘ^n⟮I, I'⟯ M') : ⇑h.toEquiv = h :=
rfl
#align diffeomorph.coe_to_equiv Diffeomorph.coe_toEquiv
@[simp, norm_cast]
theorem coe_coe (h : M ≃ₘ^n⟮I, I'⟯ M') : ⇑(h : C^n⟮I, M; I', M'⟯) = h :=
rfl
#align diffeomorph.coe_coe Diffeomorph.coe_coe
theorem toEquiv_injective : Injective (Diffeomorph.toEquiv : (M ≃ₘ^n⟮I, I'⟯ M') → M ≃ M')
| ⟨e, _, _⟩, ⟨e', _, _⟩, rfl => rfl
#align diffeomorph.to_equiv_injective Diffeomorph.toEquiv_injective
@[simp]
theorem toEquiv_inj {h h' : M ≃ₘ^n⟮I, I'⟯ M'} : h.toEquiv = h'.toEquiv ↔ h = h' :=
toEquiv_injective.eq_iff
#align diffeomorph.to_equiv_inj Diffeomorph.toEquiv_inj
/-- Coercion to function `λ h : M ≃ₘ^n⟮I, I'⟯ M', (h : M → M')` is injective. -/
theorem coeFn_injective : Injective fun (h : M ≃ₘ^n⟮I, I'⟯ M') (x : M) => h x :=
Equiv.coe_fn_injective.comp toEquiv_injective
#align diffeomorph.coe_fn_injective Diffeomorph.coeFn_injective
@[ext]
theorem ext {h h' : M ≃ₘ^n⟮I, I'⟯ M'} (Heq : ∀ x, h x = h' x) : h = h' :=
coeFn_injective <| funext Heq
#align diffeomorph.ext Diffeomorph.ext
section
variable (M I n)
/-- Identity map as a diffeomorphism. -/
protected def refl : M ≃ₘ^n⟮I, I⟯ M
where
contMdiff_toFun := contMdiff_id
contMdiff_inv_fun := contMdiff_id
toEquiv := Equiv.refl M
#align diffeomorph.refl Diffeomorph.refl
@[simp]
theorem refl_toEquiv : (Diffeomorph.refl I M n).toEquiv = Equiv.refl _ :=
rfl
#align diffeomorph.refl_to_equiv Diffeomorph.refl_toEquiv
@[simp]
theorem coe_refl : ⇑(Diffeomorph.refl I M n) = id :=
rfl
#align diffeomorph.coe_refl Diffeomorph.coe_refl
end
/-- Composition of two diffeomorphisms. -/
protected def trans (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : M' ≃ₘ^n⟮I', J⟯ N) : M ≃ₘ^n⟮I, J⟯ N
where
contMdiff_toFun := h₂.contMdiff_toFun.comp h₁.contMdiff_toFun
contMdiff_inv_fun := h₁.contMdiff_inv_fun.comp h₂.contMdiff_inv_fun
toEquiv := h₁.toEquiv.trans h₂.toEquiv
#align diffeomorph.trans Diffeomorph.trans
@[simp]
theorem trans_refl (h : M ≃ₘ^n⟮I, I'⟯ M') : h.trans (Diffeomorph.refl I' M' n) = h :=
ext fun _ => rfl
#align diffeomorph.trans_refl Diffeomorph.trans_refl
@[simp]
theorem refl_trans (h : M ≃ₘ^n⟮I, I'⟯ M') : (Diffeomorph.refl I M n).trans h = h :=
ext fun _ => rfl
#align diffeomorph.refl_trans Diffeomorph.refl_trans
@[simp]
theorem coe_trans (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : M' ≃ₘ^n⟮I', J⟯ N) : ⇑(h₁.trans h₂) = h₂ ∘ h₁ :=
rfl
#align diffeomorph.coe_trans Diffeomorph.coe_trans
/-- Inverse of a diffeomorphism. -/
protected def symm (h : M ≃ₘ^n⟮I, J⟯ N) : N ≃ₘ^n⟮J, I⟯ M
where
contMdiff_toFun := h.contMdiff_inv_fun
contMdiff_inv_fun := h.contMdiff_toFun
toEquiv := h.toEquiv.symm
#align diffeomorph.symm Diffeomorph.symm
@[simp]
theorem apply_symm_apply (h : M ≃ₘ^n⟮I, J⟯ N) (x : N) : h (h.symm x) = x :=
h.toEquiv.apply_symm_apply x
#align diffeomorph.apply_symm_apply Diffeomorph.apply_symm_apply
@[simp]
theorem symm_apply_apply (h : M ≃ₘ^n⟮I, J⟯ N) (x : M) : h.symm (h x) = x :=
h.toEquiv.symm_apply_apply x
#align diffeomorph.symm_apply_apply Diffeomorph.symm_apply_apply
@[simp]
theorem symm_refl : (Diffeomorph.refl I M n).symm = Diffeomorph.refl I M n :=
ext fun _ => rfl
#align diffeomorph.symm_refl Diffeomorph.symm_refl
@[simp]
theorem self_trans_symm (h : M ≃ₘ^n⟮I, J⟯ N) : h.trans h.symm = Diffeomorph.refl I M n :=
ext h.symm_apply_apply
#align diffeomorph.self_trans_symm Diffeomorph.self_trans_symm
@[simp]
theorem symm_trans_self (h : M ≃ₘ^n⟮I, J⟯ N) : h.symm.trans h = Diffeomorph.refl J N n :=
ext h.apply_symm_apply
#align diffeomorph.symm_trans_self Diffeomorph.symm_trans_self
@[simp]
theorem symm_trans' (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : M' ≃ₘ^n⟮I', J⟯ N) :
(h₁.trans h₂).symm = h₂.symm.trans h₁.symm :=
rfl
#align diffeomorph.symm_trans' Diffeomorph.symm_trans'
@[simp]
theorem symm_toEquiv (h : M ≃ₘ^n⟮I, J⟯ N) : h.symm.toEquiv = h.toEquiv.symm :=
rfl
#align diffeomorph.symm_to_equiv Diffeomorph.symm_toEquiv
@[simp, mfld_simps]
theorem toEquiv_coe_symm (h : M ≃ₘ^n⟮I, J⟯ N) : ⇑h.toEquiv.symm = h.symm :=
rfl
#align diffeomorph.to_equiv_coe_symm Diffeomorph.toEquiv_coe_symm
theorem image_eq_preimage (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set M) : h '' s = h.symm ⁻¹' s :=
h.toEquiv.image_eq_preimage s
#align diffeomorph.image_eq_preimage Diffeomorph.image_eq_preimage
theorem symm_image_eq_preimage (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set N) : h.symm '' s = h ⁻¹' s :=
h.symm.image_eq_preimage s
#align diffeomorph.symm_image_eq_preimage Diffeomorph.symm_image_eq_preimage
@[simp, mfld_simps]
theorem range_comp {α} (h : M ≃ₘ^n⟮I, J⟯ N) (f : α → M) : range (h ∘ f) = h.symm ⁻¹' range f := by
rw [range_comp, image_eq_preimage]
#align diffeomorph.range_comp Diffeomorph.range_comp
@[simp]
theorem image_symm_image (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set N) : h '' (h.symm '' s) = s :=
h.toEquiv.image_symm_image s
#align diffeomorph.image_symm_image Diffeomorph.image_symm_image
@[simp]
theorem symm_image_image (h : M ≃ₘ^n⟮I, J⟯ N) (s : Set M) : h.symm '' (h '' s) = s :=
h.toEquiv.symm_image_image s
#align diffeomorph.symm_image_image Diffeomorph.symm_image_image
/-- A diffeomorphism is a homeomorphism. -/
def toHomeomorph (h : M ≃ₘ^n⟮I, J⟯ N) : M ≃ₜ N :=
⟨h.toEquiv, h.Continuous, h.symm.Continuous⟩
#align diffeomorph.to_homeomorph Diffeomorph.toHomeomorph
@[simp]
theorem toHomeomorph_toEquiv (h : M ≃ₘ^n⟮I, J⟯ N) : h.toHomeomorph.toEquiv = h.toEquiv :=
rfl
#align diffeomorph.to_homeomorph_to_equiv Diffeomorph.toHomeomorph_toEquiv
@[simp]
theorem symm_toHomeomorph (h : M ≃ₘ^n⟮I, J⟯ N) : h.symm.toHomeomorph = h.toHomeomorph.symm :=
rfl
#align diffeomorph.symm_to_homeomorph Diffeomorph.symm_toHomeomorph
@[simp]
theorem coe_toHomeomorph (h : M ≃ₘ^n⟮I, J⟯ N) : ⇑h.toHomeomorph = h :=
rfl
#align diffeomorph.coe_to_homeomorph Diffeomorph.coe_toHomeomorph
@[simp]
theorem coe_toHomeomorph_symm (h : M ≃ₘ^n⟮I, J⟯ N) : ⇑h.toHomeomorph.symm = h.symm :=
rfl
#align diffeomorph.coe_to_homeomorph_symm Diffeomorph.coe_toHomeomorph_symm
@[simp]
theorem contMdiffWithinAt_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {s x}
(hm : m ≤ n) :
ContMdiffWithinAt I I' m (f ∘ h) s x ↔ ContMdiffWithinAt J I' m f (h.symm ⁻¹' s) (h x) :=
by
constructor
· intro Hfh
rw [← h.symm_apply_apply x] at Hfh
simpa only [(· ∘ ·), h.apply_symm_apply] using
Hfh.comp (h x) (h.symm.cont_mdiff_within_at.of_le hm) (maps_to_preimage _ _)
· rw [← h.image_eq_preimage]
exact fun hf => hf.comp x (h.cont_mdiff_within_at.of_le hm) (maps_to_image _ _)
#align diffeomorph.cont_mdiff_within_at_comp_diffeomorph_iff Diffeomorph.contMdiffWithinAt_comp_diffeomorph_iff
@[simp]
theorem contMdiffOn_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {s} (hm : m ≤ n) :
ContMdiffOn I I' m (f ∘ h) s ↔ ContMdiffOn J I' m f (h.symm ⁻¹' s) :=
h.toEquiv.forall_congr' fun x => by
simp only [hm, coe_to_equiv, symm_apply_apply, cont_mdiff_within_at_comp_diffeomorph_iff,
mem_preimage]
#align diffeomorph.cont_mdiff_on_comp_diffeomorph_iff Diffeomorph.contMdiffOn_comp_diffeomorph_iff
@[simp]
theorem contMdiffAt_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} {x} (hm : m ≤ n) :
ContMdiffAt I I' m (f ∘ h) x ↔ ContMdiffAt J I' m f (h x) :=
h.contMdiffWithinAt_comp_diffeomorph_iff hm
#align diffeomorph.cont_mdiff_at_comp_diffeomorph_iff Diffeomorph.contMdiffAt_comp_diffeomorph_iff
@[simp]
theorem contMdiff_comp_diffeomorph_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : N → M'} (hm : m ≤ n) :
ContMdiff I I' m (f ∘ h) ↔ ContMdiff J I' m f :=
h.toEquiv.forall_congr' fun x => h.contMdiffAt_comp_diffeomorph_iff hm
#align diffeomorph.cont_mdiff_comp_diffeomorph_iff Diffeomorph.contMdiff_comp_diffeomorph_iff
@[simp]
theorem contMdiffWithinAt_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n)
{s x} : ContMdiffWithinAt I' J m (h ∘ f) s x ↔ ContMdiffWithinAt I' I m f s x :=
⟨fun Hhf => by
simpa only [(· ∘ ·), h.symm_apply_apply] using
(h.symm.cont_mdiff_at.of_le hm).comp_contMdiffWithinAt _ Hhf,
fun Hf => (h.ContMdiffAt.of_le hm).comp_contMdiffWithinAt _ Hf⟩
#align diffeomorph.cont_mdiff_within_at_diffeomorph_comp_iff Diffeomorph.contMdiffWithinAt_diffeomorph_comp_iff
@[simp]
theorem contMdiffAt_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) {x} :
ContMdiffAt I' J m (h ∘ f) x ↔ ContMdiffAt I' I m f x :=
h.contMdiffWithinAt_diffeomorph_comp_iff hm
#align diffeomorph.cont_mdiff_at_diffeomorph_comp_iff Diffeomorph.contMdiffAt_diffeomorph_comp_iff
@[simp]
theorem contMdiffOn_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) {s} :
ContMdiffOn I' J m (h ∘ f) s ↔ ContMdiffOn I' I m f s :=
forall₂_congr fun x hx => h.contMdiffWithinAt_diffeomorph_comp_iff hm
#align diffeomorph.cont_mdiff_on_diffeomorph_comp_iff Diffeomorph.contMdiffOn_diffeomorph_comp_iff
@[simp]
theorem contMdiff_diffeomorph_comp_iff {m} (h : M ≃ₘ^n⟮I, J⟯ N) {f : M' → M} (hm : m ≤ n) :
ContMdiff I' J m (h ∘ f) ↔ ContMdiff I' I m f :=
forall_congr' fun x => h.contMdiffWithinAt_diffeomorph_comp_iff hm
#align diffeomorph.cont_mdiff_diffeomorph_comp_iff Diffeomorph.contMdiff_diffeomorph_comp_iff
theorem toLocalHomeomorph_mdifferentiable (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) :
h.toHomeomorph.toLocalHomeomorph.Mdifferentiable I J :=
⟨h.MdifferentiableOn _ hn, h.symm.MdifferentiableOn _ hn⟩
#align diffeomorph.to_local_homeomorph_mdifferentiable Diffeomorph.toLocalHomeomorph_mdifferentiable
section Constructions
/-- Product of two diffeomorphisms. -/
def prodCongr (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : N ≃ₘ^n⟮J, J'⟯ N') :
(M × N) ≃ₘ^n⟮I.Prod J, I'.Prod J'⟯ M' × N'
where
contMdiff_toFun := (h₁.ContMdiff.comp contMdiff_fst).prod_mk (h₂.ContMdiff.comp contMdiff_snd)
contMdiff_inv_fun :=
(h₁.symm.ContMdiff.comp contMdiff_fst).prod_mk (h₂.symm.ContMdiff.comp contMdiff_snd)
toEquiv := h₁.toEquiv.prodCongr h₂.toEquiv
#align diffeomorph.prod_congr Diffeomorph.prodCongr
@[simp]
theorem prodCongr_symm (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : N ≃ₘ^n⟮J, J'⟯ N') :
(h₁.prodCongr h₂).symm = h₁.symm.prodCongr h₂.symm :=
rfl
#align diffeomorph.prod_congr_symm Diffeomorph.prodCongr_symm
@[simp]
theorem coe_prodCongr (h₁ : M ≃ₘ^n⟮I, I'⟯ M') (h₂ : N ≃ₘ^n⟮J, J'⟯ N') :
⇑(h₁.prodCongr h₂) = Prod.map h₁ h₂ :=
rfl
#align diffeomorph.coe_prod_congr Diffeomorph.coe_prodCongr
section
variable (I J J' M N N' n)
/-- `M × N` is diffeomorphic to `N × M`. -/
def prodComm : (M × N) ≃ₘ^n⟮I.Prod J, J.Prod I⟯ N × M
where
contMdiff_toFun := contMdiff_snd.prod_mk contMdiff_fst
contMdiff_inv_fun := contMdiff_snd.prod_mk contMdiff_fst
toEquiv := Equiv.prodComm M N
#align diffeomorph.prod_comm Diffeomorph.prodComm
@[simp]
theorem prodComm_symm : (prodComm I J M N n).symm = prodComm J I N M n :=
rfl
#align diffeomorph.prod_comm_symm Diffeomorph.prodComm_symm
@[simp]
theorem coe_prodComm : ⇑(prodComm I J M N n) = Prod.swap :=
rfl
#align diffeomorph.coe_prod_comm Diffeomorph.coe_prodComm
/-- `(M × N) × N'` is diffeomorphic to `M × (N × N')`. -/
def prodAssoc : ((M × N) × N') ≃ₘ^n⟮(I.Prod J).Prod J', I.Prod (J.Prod J')⟯ M × N × N'
where
contMdiff_toFun :=
(contMdiff_fst.comp contMdiff_fst).prod_mk
((contMdiff_snd.comp contMdiff_fst).prod_mk contMdiff_snd)
contMdiff_inv_fun :=
(contMdiff_fst.prod_mk (contMdiff_fst.comp contMdiff_snd)).prod_mk
(contMdiff_snd.comp contMdiff_snd)
toEquiv := Equiv.prodAssoc M N N'
#align diffeomorph.prod_assoc Diffeomorph.prodAssoc
end
end Constructions
variable [SmoothManifoldWithCorners I M] [SmoothManifoldWithCorners J N]
theorem uniqueMdiffOn_image_aux (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set M}
(hs : UniqueMdiffOn I s) : UniqueMdiffOn J (h '' s) :=
by
convert hs.unique_mdiff_on_preimage (h.to_local_homeomorph_mdifferentiable hn)
simp [h.image_eq_preimage]
#align diffeomorph.unique_mdiff_on_image_aux Diffeomorph.uniqueMdiffOn_image_aux
@[simp]
theorem uniqueMdiffOn_image (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set M} :
UniqueMdiffOn J (h '' s) ↔ UniqueMdiffOn I s :=
⟨fun hs => h.symm_image_image s ▸ h.symm.uniqueMdiffOn_image_aux hn hs,
h.uniqueMdiffOn_image_aux hn⟩
#align diffeomorph.unique_mdiff_on_image Diffeomorph.uniqueMdiffOn_image
@[simp]
theorem uniqueMdiffOn_preimage (h : M ≃ₘ^n⟮I, J⟯ N) (hn : 1 ≤ n) {s : Set N} :
UniqueMdiffOn I (h ⁻¹' s) ↔ UniqueMdiffOn J s :=
h.symm_image_eq_preimage s ▸ h.symm.uniqueMdiffOn_image hn
#align diffeomorph.unique_mdiff_on_preimage Diffeomorph.uniqueMdiffOn_preimage
@[simp]
theorem uniqueDiffOn_image (h : E ≃ₘ^n[𝕜] F) (hn : 1 ≤ n) {s : Set E} :
UniqueDiffOn 𝕜 (h '' s) ↔ UniqueDiffOn 𝕜 s := by
simp only [← uniqueMdiffOn_iff_uniqueDiffOn, unique_mdiff_on_image, hn]
#align diffeomorph.unique_diff_on_image Diffeomorph.uniqueDiffOn_image
@[simp]
theorem uniqueDiffOn_preimage (h : E ≃ₘ^n[𝕜] F) (hn : 1 ≤ n) {s : Set F} :
UniqueDiffOn 𝕜 (h ⁻¹' s) ↔ UniqueDiffOn 𝕜 s :=
h.symm_image_eq_preimage s ▸ h.symm.uniqueDiffOn_image hn
#align diffeomorph.unique_diff_on_preimage Diffeomorph.uniqueDiffOn_preimage
end Diffeomorph
namespace ContinuousLinearEquiv
variable (e : E ≃L[𝕜] E')
/-- A continuous linear equivalence between normed spaces is a diffeomorphism. -/
def toDiffeomorph : E ≃ₘ[𝕜] E'
where
contMdiff_toFun := e.ContDiff.ContMdiff
contMdiff_inv_fun := e.symm.ContDiff.ContMdiff
toEquiv := e.toLinearEquiv.toEquiv
#align continuous_linear_equiv.to_diffeomorph ContinuousLinearEquiv.toDiffeomorph
@[simp]
theorem coe_toDiffeomorph : ⇑e.toDiffeomorph = e :=
rfl
#align continuous_linear_equiv.coe_to_diffeomorph ContinuousLinearEquiv.coe_toDiffeomorph
@[simp]
theorem symm_toDiffeomorph : e.symm.toDiffeomorph = e.toDiffeomorph.symm :=
rfl
#align continuous_linear_equiv.symm_to_diffeomorph ContinuousLinearEquiv.symm_toDiffeomorph
@[simp]
theorem coe_toDiffeomorph_symm : ⇑e.toDiffeomorph.symm = e.symm :=
rfl
#align continuous_linear_equiv.coe_to_diffeomorph_symm ContinuousLinearEquiv.coe_toDiffeomorph_symm
end ContinuousLinearEquiv
namespace ModelWithCorners
variable (I) (e : E ≃ₘ[𝕜] E')
/-- Apply a diffeomorphism (e.g., a continuous linear equivalence) to the model vector space. -/
def transDiffeomorph (I : ModelWithCorners 𝕜 E H) (e : E ≃ₘ[𝕜] E') : ModelWithCorners 𝕜 E' H
where
toLocalEquiv := I.toLocalEquiv.trans e.toEquiv.toLocalEquiv
source_eq := by simp
unique_diff' := by simp [range_comp e, I.unique_diff]
continuous_toFun := e.Continuous.comp I.Continuous
continuous_invFun := I.continuous_symm.comp e.symm.Continuous
#align model_with_corners.trans_diffeomorph ModelWithCorners.transDiffeomorph
@[simp, mfld_simps]
theorem coe_transDiffeomorph : ⇑(I.transDiffeomorph e) = e ∘ I :=
rfl
#align model_with_corners.coe_trans_diffeomorph ModelWithCorners.coe_transDiffeomorph
@[simp, mfld_simps]
theorem coe_transDiffeomorph_symm : ⇑(I.transDiffeomorph e).symm = I.symm ∘ e.symm :=
rfl
#align model_with_corners.coe_trans_diffeomorph_symm ModelWithCorners.coe_transDiffeomorph_symm
theorem transDiffeomorph_range : range (I.transDiffeomorph e) = e '' range I :=
range_comp e I
#align model_with_corners.trans_diffeomorph_range ModelWithCorners.transDiffeomorph_range
theorem coe_extChartAt_transDiffeomorph (x : M) :
⇑(extChartAt (I.transDiffeomorph e) x) = e ∘ extChartAt I x :=
rfl
#align model_with_corners.coe_ext_chart_at_trans_diffeomorph ModelWithCorners.coe_extChartAt_transDiffeomorph
theorem coe_extChartAt_transDiffeomorph_symm (x : M) :
⇑(extChartAt (I.transDiffeomorph e) x).symm = (extChartAt I x).symm ∘ e.symm :=
rfl
#align model_with_corners.coe_ext_chart_at_trans_diffeomorph_symm ModelWithCorners.coe_extChartAt_transDiffeomorph_symm
theorem extChartAt_transDiffeomorph_target (x : M) :
(extChartAt (I.transDiffeomorph e) x).target = e.symm ⁻¹' (extChartAt I x).target := by
simp only [range_comp e, e.image_eq_preimage, preimage_preimage, mfld_simps]
#align model_with_corners.ext_chart_at_trans_diffeomorph_target ModelWithCorners.extChartAt_transDiffeomorph_target
end ModelWithCorners
namespace Diffeomorph
variable (e : E ≃ₘ[𝕜] F)
instance smoothManifoldWithCorners_transDiffeomorph [SmoothManifoldWithCorners I M] :
SmoothManifoldWithCorners (I.transDiffeomorph e) M :=
by
refine' smoothManifoldWithCorners_of_contDiffOn _ _ fun e₁ e₂ h₁ h₂ => _
refine'
e.cont_diff.comp_cont_diff_on
(((contDiffGroupoid ⊤ I).compatible h₁ h₂).1.comp e.symm.cont_diff.cont_diff_on _)
mfld_set_tac
#align diffeomorph.smooth_manifold_with_corners_trans_diffeomorph Diffeomorph.smoothManifoldWithCorners_transDiffeomorph
variable (I M)
/-- The identity diffeomorphism between a manifold with model `I` and the same manifold
with model `I.trans_diffeomorph e`. -/
def toTransDiffeomorph (e : E ≃ₘ[𝕜] F) : M ≃ₘ⟮I, I.transDiffeomorph e⟯ M
where
toEquiv := Equiv.refl M
contMdiff_toFun x :=
by
refine' contMdiffWithinAt_iff'.2 ⟨continuousWithinAt_id, _⟩
refine' e.cont_diff.cont_diff_within_at.congr' (fun y hy => _) _
·
simp only [Equiv.coe_refl, id, (· ∘ ·), I.coe_ext_chart_at_trans_diffeomorph,
(extChartAt I x).right_inv hy.1]
exact
⟨(extChartAt I x).map_source (mem_ext_chart_source I x), trivial, by simp only [mfld_simps]⟩
contMdiff_inv_fun x :=
by
refine' contMdiffWithinAt_iff'.2 ⟨continuousWithinAt_id, _⟩
refine' e.symm.cont_diff.cont_diff_within_at.congr' (fun y hy => _) _
· simp only [mem_inter_iff, I.ext_chart_at_trans_diffeomorph_target] at hy
simp only [Equiv.coe_refl, Equiv.refl_symm, id, (· ∘ ·),
I.coe_ext_chart_at_trans_diffeomorph_symm, (extChartAt I x).right_inv hy.1]
exact
⟨(extChartAt _ x).map_source (mem_ext_chart_source _ x), trivial, by
simp only [e.symm_apply_apply, Equiv.refl_symm, Equiv.coe_refl, mfld_simps]⟩
#align diffeomorph.to_trans_diffeomorph Diffeomorph.toTransDiffeomorph
variable {I M}
@[simp]
theorem contMdiffWithinAt_transDiffeomorph_right {f : M' → M} {x s} :
ContMdiffWithinAt I' (I.transDiffeomorph e) n f s x ↔ ContMdiffWithinAt I' I n f s x :=
(toTransDiffeomorph I M e).contMdiffWithinAt_diffeomorph_comp_iff le_top
#align diffeomorph.cont_mdiff_within_at_trans_diffeomorph_right Diffeomorph.contMdiffWithinAt_transDiffeomorph_right
@[simp]
theorem contMdiffAt_transDiffeomorph_right {f : M' → M} {x} :
ContMdiffAt I' (I.transDiffeomorph e) n f x ↔ ContMdiffAt I' I n f x :=
(toTransDiffeomorph I M e).contMdiffAt_diffeomorph_comp_iff le_top
#align diffeomorph.cont_mdiff_at_trans_diffeomorph_right Diffeomorph.contMdiffAt_transDiffeomorph_right
@[simp]
theorem contMdiffOn_transDiffeomorph_right {f : M' → M} {s} :
ContMdiffOn I' (I.transDiffeomorph e) n f s ↔ ContMdiffOn I' I n f s :=
(toTransDiffeomorph I M e).contMdiffOn_diffeomorph_comp_iff le_top
#align diffeomorph.cont_mdiff_on_trans_diffeomorph_right Diffeomorph.contMdiffOn_transDiffeomorph_right
@[simp]
theorem contMdiff_transDiffeomorph_right {f : M' → M} :
ContMdiff I' (I.transDiffeomorph e) n f ↔ ContMdiff I' I n f :=
(toTransDiffeomorph I M e).contMdiff_diffeomorph_comp_iff le_top
#align diffeomorph.cont_mdiff_trans_diffeomorph_right Diffeomorph.contMdiff_transDiffeomorph_right
@[simp]
theorem smooth_transDiffeomorph_right {f : M' → M} :
Smooth I' (I.transDiffeomorph e) f ↔ Smooth I' I f :=
contMdiff_transDiffeomorph_right e
#align diffeomorph.smooth_trans_diffeomorph_right Diffeomorph.smooth_transDiffeomorph_right
@[simp]
theorem contMdiffWithinAt_transDiffeomorph_left {f : M → M'} {x s} :
ContMdiffWithinAt (I.transDiffeomorph e) I' n f s x ↔ ContMdiffWithinAt I I' n f s x :=
((toTransDiffeomorph I M e).contMdiffWithinAt_comp_diffeomorph_iff le_top).symm
#align diffeomorph.cont_mdiff_within_at_trans_diffeomorph_left Diffeomorph.contMdiffWithinAt_transDiffeomorph_left
@[simp]
theorem contMdiffAt_transDiffeomorph_left {f : M → M'} {x} :
ContMdiffAt (I.transDiffeomorph e) I' n f x ↔ ContMdiffAt I I' n f x :=
((toTransDiffeomorph I M e).contMdiffAt_comp_diffeomorph_iff le_top).symm
#align diffeomorph.cont_mdiff_at_trans_diffeomorph_left Diffeomorph.contMdiffAt_transDiffeomorph_left
@[simp]
theorem contMdiffOn_transDiffeomorph_left {f : M → M'} {s} :
ContMdiffOn (I.transDiffeomorph e) I' n f s ↔ ContMdiffOn I I' n f s :=
((toTransDiffeomorph I M e).contMdiffOn_comp_diffeomorph_iff le_top).symm
#align diffeomorph.cont_mdiff_on_trans_diffeomorph_left Diffeomorph.contMdiffOn_transDiffeomorph_left
@[simp]
theorem contMdiff_transDiffeomorph_left {f : M → M'} :
ContMdiff (I.transDiffeomorph e) I' n f ↔ ContMdiff I I' n f :=
((toTransDiffeomorph I M e).contMdiff_comp_diffeomorph_iff le_top).symm
#align diffeomorph.cont_mdiff_trans_diffeomorph_left Diffeomorph.contMdiff_transDiffeomorph_left
@[simp]
theorem smooth_transDiffeomorph_left {f : M → M'} :
Smooth (I.transDiffeomorph e) I' f ↔ Smooth I I' f :=
e.contMdiff_transDiffeomorph_left
#align diffeomorph.smooth_trans_diffeomorph_left Diffeomorph.smooth_transDiffeomorph_left
end Diffeomorph
|
-- Desigualdad de naturales (2)
-- ============================
import tactic
import data.pnat.basic
lemma pnat.add_sub_cancel
(a b : ℕ+)
: a + b - b = a :=
pnat.eq
begin
rw [pnat.sub_coe, if_pos],
{ apply nat.add_sub_cancel },
{ exact lt_add_of_pos_left b.1 a.pos },
end
lemma eq_with_2
(x z: ℕ+)
(h2: ¬x + 2 * z < 2 * z)
: x + 2 * z - 2 * z = x :=
pnat.add_sub_cancel _ _
|
(* File: Syntax.v *)
(* Author: Peter Urbak <[email protected]> *)
(* Version: 2012-05-02 *)
(*
This script shows how to:
1. Define a simple arithmetic language.
*)
(* -*- The Source Syntax. -*- *)
Inductive arithmetic_expression : Type :=
| Lit : nat -> arithmetic_expression
| Plus : arithmetic_expression -> arithmetic_expression ->
arithmetic_expression
| Minus : arithmetic_expression -> arithmetic_expression ->
arithmetic_expression
| Times : arithmetic_expression -> arithmetic_expression ->
arithmetic_expression
| Divide : arithmetic_expression -> arithmetic_expression ->
arithmetic_expression.
(* end-of-Syntax.v *) |
lemma cCons_Cons_eq [simp]: "x ## y # ys = x # y # ys" |
(* ***********************************************************************)
(* *)
(* Synchronously executed Interpreted Time Petri Nets (SITPNs) *)
(* *)
(* *)
(* Copyright Université de Montpellier, contributor(s): Vincent *)
(* Iampietro, David Andreu, David Delahaye (May 2020) *)
(* *)
(* This software is governed by the CeCILL-C license under French law *)
(* and abiding by the rules of distribution of free software. You can *)
(* use, modify and/ or redistribute the software under the terms of *)
(* the CeCILL-C license as circulated by CEA, CNRS and INRIA at the *)
(* following URL "http://www.cecill.info". The fact that you are *)
(* presently reading this means that you have had knowledge of the *)
(* CeCILL-C license and that you accept its terms. *)
(* *)
(* ***********************************************************************)
(** * Global type definitions. *)
Require Import Coqlib.
Require Import String.
Require Import Ascii.
Require Import HexString.
(** Type definitions used in all part of the Hilecop-Cert project. *)
(** Defines the set of strictly positive natural numbers. *)
Definition natstar := { n : nat | n > 0 }.
(** Casts a natstar into a nat. *)
Definition natstar_to_nat (ns : natstar) := proj1_sig ns.
Coercion natstar_to_nat : natstar >-> nat.
(** Defines some natstar. *)
Definition onens := exist _ 1 (gt_Sn_O 0).
Definition twons := exist _ 2 (gt_Sn_O 1).
Definition threens := exist _ 3 (gt_Sn_O 2).
(** Defines the type of relation that are a strict order
over a type A.
*)
Inductive IsStrictOrderBRel {A} (brel : A -> A -> bool) : Prop :=
MkStrictOrderB {
brel_irrefl : forall a, brel a a = false;
brel_trans : forall a b c, brel a b = true -> brel b c = true -> brel a c = true;
(* Irreflexivity and transitivity entail anti-symmetry. *)
(* brel_antisym : forall a b, brel a b = true -> brel b a = false; *)
}.
(** States that two elements of type A are comparable through
the boolean relation [brel]. *)
Definition AreComparableWithBRel {A} (x y : A) (brel : A -> A -> bool) : Prop :=
brel x y <> false \/ brel y x <> false.
(** States that [brel] is a strict total order over a type A, that is:
- [brel] is a strict order over type A.
- all elements of A that are different are comparable with [brel].
*)
Definition IsStrictTotalOrderBRel {A}
(eqA : A -> A -> Prop)
(decEqA : forall x y, {eqA x y} + {~eqA x y})
(brel : A -> A -> bool) :=
IsStrictOrderBRel brel /\ forall x y, ~eqA x y -> AreComparableWithBRel x y brel.
(** Defines the type of Petri net arcs. *)
Inductive ArcT : Type := basic | test | inhibitor.
(** Implements the equality between two arc_t values. *)
Definition arct_eqb (a a' : ArcT) : bool :=
match a, a' with
| basic, basic => true
| test, test => true
| inhibitor, inhibitor => true
| _, _ => false
end.
(** Cast from ArcT to nat. *)
Definition ArcT_in_nat (a : ArcT) :=
match a with
| basic => 0
| test => 1
| inhibitor => 2
end.
Coercion ArcT_in_nat : ArcT >-> nat.
(** Defines the type of Petri net transitions. *)
Inductive TransitionT : Type := not_temporal | temporal_a_b |
temporal_a_a | temporal_a_inf.
(** Cast from TransitionT to nat. *)
Definition TransitionT_in_nat (t : TransitionT) :=
match t with
| not_temporal => 0
| temporal_a_b => 1
| temporal_a_a => 2
| temporal_a_inf => 3
end.
Coercion TransitionT_in_nat : TransitionT >-> nat.
(** Implements the equality between two transition_t values. *)
Definition transt_eqb (t t' : TransitionT) : bool :=
match t, t' with
| not_temporal, not_temporal => true
| temporal_a_b, temporal_a_b => true
| temporal_a_a, temporal_a_a => true
| temporal_a_inf, temporal_a_inf => true
| _, _ => false
end.
(** Defines an option type able to return error messages. *)
Inductive optionE (A : Type) : Type :=
| Success : A -> optionE A
| Err : string -> optionE A.
Arguments Success {A} a.
Arguments Err {A}.
(** Converts a nat into its hexadecimal string representation. Useful
to display variable values in error messages. *)
Notation "$$ n" := (of_nat n) (at level 0, only parsing).
|
import Smt
theorem trans (p q r : Prop) : p = q → q = r → p = r := by
smt
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.PrePort
import Mathlib.Lean3Lib.init.default
import Mathlib.data.padics.padic_integers
import Mathlib.topology.metric_space.cau_seq_filter
import Mathlib.analysis.specific_limits
import Mathlib.data.polynomial.identities
import Mathlib.topology.algebra.polynomial
import Mathlib.PostPort
namespace Mathlib
/-!
# Hensel's lemma on ℤ_p
This file proves Hensel's lemma on ℤ_p, roughly following Keith Conrad's writeup:
<http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf>
Hensel's lemma gives a simple condition for the existence of a root of a polynomial.
The proof and motivation are described in the paper
[R. Y. Lewis, *A formal proof of Hensel's lemma over the p-adic integers*][lewis2019].
## References
* <http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf>
* [R. Y. Lewis, *A formal proof of Hensel's lemma over the p-adic integers*][lewis2019]
* <https://en.wikipedia.org/wiki/Hensel%27s_lemma>
## Tags
p-adic, p adic, padic, p-adic integer
-/
-- We begin with some general lemmas that are used below in the computation.
theorem padic_polynomial_dist {p : ℕ} [fact (nat.prime p)] (F : polynomial (padic_int p)) (x : padic_int p) (y : padic_int p) : norm (polynomial.eval x F - polynomial.eval y F) ≤ norm (x - y) := sorry
theorem limit_zero_of_norm_tendsto_zero {p : ℕ} [fact (nat.prime p)] {ncs : cau_seq (padic_int p) norm} {F : polynomial (padic_int p)} (hnorm : filter.tendsto (fun (i : ℕ) => norm (polynomial.eval (coe_fn ncs i) F)) filter.at_top (nhds 0)) : polynomial.eval (cau_seq.lim ncs) F = 0 :=
tendsto_nhds_unique (comp_tendsto_lim ncs) (tendsto_zero_of_norm_tendsto_zero hnorm)
/-- `T` is an auxiliary value that is used to control the behavior of the polynomial `F`. -/
/-- We will construct a sequence of elements of ℤ_p satisfying successive values of `ih`. -/
/-- Given `z : ℤ_[p]` satisfying `ih n z`, construct `z' : ℤ_[p]` satisfying `ih (n+1) z'`. We need
the hypothesis `ih n z`, since otherwise `z'` is not necessarily an integer. -/
-- why doesn't "noncomputable theory" stick here?
theorem hensels_lemma {p : ℕ} [fact (nat.prime p)] {F : polynomial (padic_int p)} {a : padic_int p} (hnorm : norm (polynomial.eval a F) < norm (polynomial.eval a (coe_fn polynomial.derivative F)) ^ bit0 1) : ∃ (z : padic_int p),
polynomial.eval z F = 0 ∧
norm (z - a) < norm (polynomial.eval a (coe_fn polynomial.derivative F)) ∧
norm (polynomial.eval z (coe_fn polynomial.derivative F)) =
norm (polynomial.eval a (coe_fn polynomial.derivative F)) ∧
∀ (z' : padic_int p),
polynomial.eval z' F = 0 → norm (z' - a) < norm (polynomial.eval a (coe_fn polynomial.derivative F)) → z' = z := sorry
|
-- | Premisa |
lemma Example_1 (A B : Prop) : A ∧ (A → B) → B :=
begin
-- Complete
end
-- | Premisas |
lemma Example_2 (A B C D : Prop) : (A ∨ B) → (A → C) → (B → D) → C ∨ D :=
begin
-- Complete
end
-- | Premisa |
lemma Example_3 (A B : Prop) : ¬ (A ∧ B) → (A → ¬ B) :=
begin
-- Complete
end |
import ..lectures.love05_inductive_predicates_demo
/-! # LoVe Demo 7: Metaprogramming
Note for Brown FPV students: we are not following this demo directly!
But I'm posting it because some definitions in here are used in the exercises.
--Rob
Users can extend Lean with custom tactics and tools. This kind of
programming—programming the prover—is called metaprogramming.
Lean's metaprogramming framework uses mostly the same notions and syntax as
Lean's input language itself. Abstract syntax trees __reflect__ internal data
structures, e.g., for expressions (terms). The prover's C++ internals are
exposed through Lean interfaces, which we can use for
* accessing the current context and goal;
* unifying expressions;
* querying and modifying the environment;
* setting attributes.
Most of Lean's predefined tactics are implemented in Lean (and not in C++).
Example applications:
* proof goal transformations;
* heuristic proof search;
* decision procedures;
* definition generators;
* advisor tools;
* exporters;
* ad hoc automation.
Advantages of Lean's metaprogramming framework:
* Users do not need to learn another programming language to write
metaprograms; they can work with the same constructs and notation used to
define ordinary objects in the prover's library.
* Everything in that library is available for metaprogramming purposes.
* Metaprograms can be written and debugged in the same interactive environment,
encouraging a style where formal libraries and supporting automation are
developed at the same time. -/
set_option pp.beta true
set_option pp.generalized_field_notation false
namespace LoVe
/-! ## Tactics and Tactic Combinators
When programming our own tactics, we often need to repeat some actions on
several goals, or to recover if a tactic fails. Tactic combinators help in such
case.
`repeat` applies its argument repeatedly on all (sub…sub)goals until it cannot
be applied any further. -/
lemma repeat_example :
even 4 ∧ even 7 ∧ even 3 ∧ even 0 :=
begin
repeat { apply and.intro },
repeat { apply even.add_two },
repeat { sorry }
end
/-! The "orelse" combinator `<|>` tries its first argument and applies its
second argument in case of failure. -/
lemma repeat_orelse_example :
even 4 ∧ even 7 ∧ even 3 ∧ even 0 :=
begin
repeat { apply and.intro },
repeat {
apply even.add_two
<|> apply even.zero },
repeat { sorry }
end
/-! `iterate` works repeatedly on the first goal until it fails; then it
stops. -/
lemma iterate_orelse_example :
even 4 ∧ even 7 ∧ even 3 ∧ even 0 :=
begin
repeat { apply and.intro },
iterate {
apply even.add_two
<|> apply even.zero },
repeat { sorry }
end
/-! `all_goals` applies its argument exactly once to each goal. It succeeds only
if the argument succeeds on **all** goals. -/
lemma all_goals_example :
even 4 ∧ even 7 ∧ even 3 ∧ even 0 :=
begin
repeat { apply and.intro },
all_goals { apply even.add_two }, -- fails
repeat { sorry }
end
/-! `try` transforms its argument into a tactic that never fails. -/
lemma all_goals_try_example :
even 4 ∧ even 7 ∧ even 3 ∧ even 0 :=
begin
repeat { apply and.intro },
all_goals { try { apply even.add_two } },
repeat { sorry }
end
/-! `any_goals` applies its argument exactly once to each goal. It succeeds
if the argument succeeds on **any** goal. -/
lemma any_goals_example :
even 4 ∧ even 7 ∧ even 3 ∧ even 0 :=
begin
repeat { apply and.intro },
any_goals { apply even.add_two },
repeat { sorry }
end
/-! `solve1` transforms its argument into an all-or-nothing tactic. If the
argument does not prove the goal, `solve1` fails. -/
lemma any_goals_solve1_repeat_orelse_example :
even 4 ∧ even 7 ∧ even 3 ∧ even 0 :=
begin
repeat { apply and.intro },
any_goals { solve1 { repeat {
apply even.add_two
<|> apply even.zero } } },
repeat { sorry }
end
/-! The combinators `repeat`, `iterate`, `all_goals`, and `any_goals` can easily
lead to infinite looping: -/
/-
lemma repeat_not_example :
¬ even 1 :=
begin
repeat { apply not.intro },
sorry
end
-/
/-! Let us start with the actual metaprogramming, by coding a custom tactic. The
tactic embodies the behavior we hardcoded in the `solve1` example above: -/
meta def intro_and_even : tactic unit :=
do
tactic.repeat (tactic.applyc ``and.intro),
tactic.any_goals (tactic.solve1 (tactic.repeat
(tactic.applyc ``even.add_two
<|> tactic.applyc ``even.zero))),
pure ()
/-! The `meta` keyword makes it possible for the function to call other
metafunctions. The `do` keyword enters a monad, and the `<|>` operator is the
"orelse" operator of alternative monads. At the end, we return `()`, of type
`unit`, to ensure the metaprogram has the desired type.
Any executable Lean definition can be used as a metaprogram. In addition, we can
put `meta` in front of a definition to indicate that is a metadefinition. Such
definitions need not terminate but cannot be used in non-`meta` contexts.
Let us apply our custom tactic: -/
lemma any_goals_solve1_repeat_orelse_example₂ :
even 4 ∧ even 7 ∧ even 3 ∧ even 0 :=
begin
intro_and_even,
repeat { sorry }
end
/-! ## The Metaprogramming Monad
Tactics have access to
* the list of **goals** as metavariables (each metavariables has a type and a
local context (hypothesis); they can optionally be instantiated);
* the **elaborator** (to elaborate expressions and compute their type);
* the **environment**, containing all declarations and inductive types;
* the **attributes** (e.g., the list of `@[simp]` rules).
The tactic monad is an alternative monad, with `fail` and `<|>`. Tactics can
also produce trace messages. -/
lemma even_14 :
even 14 :=
by do
tactic.trace "Proving evenness …",
intro_and_even
meta def hello_then_intro_and_even : tactic unit :=
do
tactic.trace "Proving evenness …",
intro_and_even
lemma even_16 :
even 16 :=
by hello_then_intro_and_even
run_cmd tactic.trace "Hello, Metaworld!"
meta def trace_goals : tactic unit :=
do
tactic.trace "local context:",
ctx ← tactic.local_context,
tactic.trace ctx,
tactic.trace "target:",
P ← tactic.target,
tactic.trace P,
tactic.trace "all missing proofs:",
Hs ← tactic.get_goals,
tactic.trace Hs,
τs ← list.mmap tactic.infer_type Hs,
tactic.trace τs
lemma even_18_and_even_20 (α : Type) (a : α) :
even 18 ∧ even 20 :=
by do
tactic.applyc ``and.intro,
trace_goals,
intro_and_even
lemma triv_imp (a : Prop) (h : a) :
a :=
by do
h ← tactic.get_local `h,
tactic.trace "h:",
tactic.trace h,
tactic.trace "raw h:",
tactic.trace (expr.to_raw_fmt h),
tactic.trace "type of h:",
τ ← tactic.infer_type h,
tactic.trace τ,
tactic.trace "type of type of h:",
υ ← tactic.infer_type τ,
tactic.trace υ,
tactic.apply h
meta def exact_list : list expr → tactic unit
| [] := tactic.fail "no matching expression found"
| (h :: hs) :=
do {
tactic.trace "trying",
tactic.trace h,
tactic.exact h }
<|> exact_list hs
meta def hypothesis : tactic unit :=
do
hs ← tactic.local_context,
exact_list hs
lemma app_of_app {α : Type} {p : α → Prop} {a : α}
(h : p a) :
p a :=
by hypothesis
/-! ## Names, Expressions, Declarations, and Environments
The metaprogramming framework is articulated around five main types:
* `tactic` manages the proof state, the global context, and more;
* `name` represents a structured name (e.g., `x`, `even.add_two`);
* `expr` represents an expression (a term) as an abstract syntax tree;
* `declaration` represents a constant declaration, a definition, an axiom, or a
lemma;
* `environment` stores all the declarations and notations that make up the
global context. -/
#print expr
#check expr tt -- elaborated expressions
#check expr ff -- unelaborated expressions (pre-expressions)
#print name
#check (expr.const `ℕ [] : expr)
#check expr.sort level.zero -- Sort 0, i.e., Prop
#check expr.sort (level.succ level.zero)
-- Sort 1, i.e., Type
#check expr.var 0 -- bound variable with De Bruijn index 0
#check (expr.local_const `uniq_name `pp_name binder_info.default
`(ℕ) : expr)
#check (expr.mvar `uniq_name `pp_name `(ℕ) : expr)
#check (expr.pi `pp_name binder_info.default `(ℕ)
(expr.sort level.zero) : expr)
#check (expr.lam `pp_name binder_info.default `(ℕ)
(expr.var 0) : expr)
#check expr.elet
#check expr.macro
/-! We can create literal expressions conveniently using backticks and
parentheses:
* Expressions with a single backtick must be fully elaborated.
* Expressions with two backticks are __pre-expressions__: They may contain some
holes to be filled in later, based on some context.
* Expressions with three backticks are pre-expressions without name checking. -/
run_cmd do
let e : expr := `(list.map (λn : ℕ, n + 1) [1, 2, 3]),
tactic.trace e
run_cmd do
let e : expr := `(list.map _ [1, 2, 3]), -- fails
tactic.trace e
run_cmd do
let e₁ : pexpr := ``(list.map (λn, n + 1) [1, 2, 3]),
let e₂ : pexpr := ``(list.map _ [1, 2, 3]),
tactic.trace e₁,
tactic.trace e₂
run_cmd do
let e : pexpr := ```(seattle.washington),
tactic.trace e
/-! We can also create literal names with backticks:
* Names with a single backtick, `n, are not checked for existence.
* Names with two backticks, ``n, are resolved and checked. -/
run_cmd tactic.trace `and.intro
run_cmd tactic.trace `intro_and_even
run_cmd tactic.trace `seattle.washington
run_cmd tactic.trace ``and.intro
run_cmd tactic.trace ``intro_and_even
run_cmd tactic.trace ``seattle.washington -- fails
/-! __Antiquotations__ embed an existing expression in a larger expression. They
are announced by the prefix `%%` followed by a name from the current context.
Antiquotations are available with one, two, and three backticks: -/
run_cmd do
let x : expr := `(2 : ℕ),
let e : expr := `(%%x + 1),
tactic.trace e
run_cmd do
let x : expr := `(@id ℕ),
let e : pexpr := ``(list.map %%x),
tactic.trace e
run_cmd do
let x : expr := `(@id ℕ),
let e : pexpr := ```(a _ %%x),
tactic.trace e
lemma one_add_two_eq_three :
1 + 2 = 3 :=
by do
`(%%a + %%b = %%c) ← tactic.target,
tactic.trace a,
tactic.trace b,
tactic.trace c,
`(@eq %%α %%l %%r) ← tactic.target,
tactic.trace α,
tactic.trace l,
tactic.trace r,
tactic.exact `(refl _ : 3 = 3)
#print declaration
/-! The `environment` type is presented as an abstract type, equipped with some
operations to query and modify it. The `environment.fold` metafunction iterates
over all declarations making up the environment. -/
run_cmd do
env ← tactic.get_env,
tactic.trace (environment.fold env 0 (λdecl n, n + 1))
/-! ## First Example: A Conjuction-Destructing Tactic
We define a `destruct_and` tactic that automates the elimination of `∧` in
premises, automating proofs such as these: -/
lemma abcd_a (a b c d : Prop) (h : a ∧ (b ∧ c) ∧ d) :
a :=
and.elim_left h
lemma abcd_b (a b c d : Prop) (h : a ∧ (b ∧ c) ∧ d) :
b :=
and.elim_left (and.elim_left (and.elim_right h))
lemma abcd_bc (a b c d : Prop) (h : a ∧ (b ∧ c) ∧ d) :
b ∧ c :=
and.elim_left (and.elim_right h)
/-! Our tactic relies on a helper metafunction, which takes as argument the
hypothesis `h` to use as an expression rather than as a name: -/
meta def destruct_and_helper : expr → tactic unit
| h :=
do
t ← tactic.infer_type h,
match t with
| `(%%a ∧ %%b) :=
tactic.exact h
<|>
do {
ha ← tactic.to_expr ``(and.elim_left %%h),
destruct_and_helper ha }
<|>
do {
hb ← tactic.to_expr ``(and.elim_right %%h),
destruct_and_helper hb }
| _ := tactic.exact h
end
meta def destruct_and (nam : name) : tactic unit :=
do
h ← tactic.get_local nam,
destruct_and_helper h
/-! Let us check that our tactic works: -/
lemma abc_a (a b c : Prop) (h : a ∧ b ∧ c) :
a :=
by destruct_and `h
lemma abc_b (a b c : Prop) (h : a ∧ b ∧ c) :
b :=
by destruct_and `h
lemma abc_bc (a b c : Prop) (h : a ∧ b ∧ c) :
b ∧ c :=
by destruct_and `h
lemma abc_ac (a b c : Prop) (h : a ∧ b ∧ c) :
a ∧ c :=
by destruct_and `h -- fails
/-! ## Second Example: A Provability Advisor
Next, we implement a `prove_direct` tool that traverses all lemmas in the
database and checks whether one of them can be used to prove the current goal. A
similar tactic is available in `mathlib` under the name `library_search`. -/
meta def is_theorem : declaration → bool
| (declaration.defn _ _ _ _ _ _) := ff
| (declaration.thm _ _ _ _) := tt
| (declaration.cnst _ _ _ _) := ff
| (declaration.ax _ _ _) := tt
meta def get_all_theorems : tactic (list name) :=
do
env ← tactic.get_env,
pure (environment.fold env [] (λdecl nams,
if is_theorem decl then declaration.to_name decl :: nams
else nams))
meta def prove_with_name (nam : name) : tactic unit :=
do
tactic.applyc nam
({ md := tactic.transparency.reducible, unify := ff }
: tactic.apply_cfg),
tactic.all_goals tactic.assumption,
pure ()
meta def prove_direct : tactic unit :=
do
nams ← get_all_theorems,
list.mfirst (λnam,
do
prove_with_name nam,
tactic.trace ("directly proved by " ++ to_string nam))
nams
lemma nat.eq_symm (x y : ℕ) (h : x = y) :
y = x :=
by prove_direct
lemma nat.eq_symm₂ (x y : ℕ) (h : x = y) :
y = x :=
by library_search
lemma list.reverse_twice (xs : list ℕ) :
list.reverse (list.reverse xs) = xs :=
by prove_direct
lemma list.reverse_twice_symm (xs : list ℕ) :
xs = list.reverse (list.reverse xs) :=
by prove_direct -- fails
/-! As a small refinement, we propose a version of `prove_direct` that also
looks for equalities stated in symmetric form. -/
meta def prove_direct_symm : tactic unit :=
prove_direct
<|>
do {
tactic.applyc `eq.symm,
prove_direct }
lemma list.reverse_twice₂ (xs : list ℕ) :
list.reverse (list.reverse xs) = xs :=
by prove_direct_symm
lemma list.reverse_twice_symm₂ (xs : list ℕ) :
xs = list.reverse (list.reverse xs) :=
by prove_direct_symm
/-! ## A Look at Two Predefined Tactics
Quite a few of Lean's predefined tactics are implemented as metaprograms and
not in C++. We can find these definitions by clicking the name of a construct
in Visual Studio Code while holding the control or command key. -/
#check tactic.intro
#check tactic.assumption
end LoVe
|
\section{Russian Government Links to and Contacts with The Trump Campaign}
\markboth{Russian Government Links to and Contacts with The Trump Campaign}{Russian Government Links to and Contacts with The Trump Campaign}
The Office identified multiple contacts---``links,'' in the words of the Appointment Order---between Trump Campaign officials and individuals with ties to the Russian government.
The Office investigated whether those contacts constituted a third avenue of attempted Russian interference with or influence on the 2016 presidential election.
In particular, the investigation examined whether these contacts involved or resulted in coordination or a conspiracy with the Trump Campaign and Russia, including with respect to Russia providing assistance to the Campaign in exchange for any sort of favorable treatment in the future.
Based on the available information, the investigation did not establish such coordination.
This Section describes the principal links between the Trump Campaign and individuals with ties to the Russian government, including some contacts with Campaign officials or associates that have been publicly reported to involve Russian contacts.
Each subsection begins with an overview of the Russian contact at issue and then describes in detail the relevant facts, which are generally presented in chronological order, beginning with the early months of the Campaign and extending through the post-election, transition period.
\subsection{Campaign Period (September 2015--November~8, 2016)}
Russian-government-connected individuals and media entities began showing interest in Trump's campaign in the months after he announced his candidacy in June 2015.% 288
\footnote{For example, on August~18, 2015, on behalf of the editor-in-chief of the internet newspaper \textit{Vzglyad}, Georgi Asatryan emailed campaign press secretary Hope Hicks asking for a phone or in-person candidate interview.
8/18/15 Email, Asatryan to Hicks.
One day earlier, the publication's founder (and former Russian parliamentarian) Konstantin Rykov had registered two Russian websites---\UseVerb{Trump2016ru} and \UseVerb{DonaldTrump2016ru}.
No interview took place.}
Because Trump's status as a public figure at the time was attributable in large part to his prior business and entertainment dealings, this Office investigated whether a business contact with Russia-linked individuals and entities during the campaign period---the Trump Tower Moscow project, \textit{see} \hyperlink{subsubsection.1.4.1.1}{Volume~I, Section~IV.A.1}, \textit{infra}---led to or involved coordination of election assistance.
Outreach from individuals with ties to Russia continued in the spring and summer of 2016, when Trump was moving toward---and eventually becoming---the Republican nominee for President.
As set forth below, the Office also evaluated a series of links during this period: outreach to two of Trump's then-recently named foreign policy advisors, including a representation that Russia had ``dirt'' on Clinton in the form of thousands of emails (\hyperlink{subsubsection.1.4.1.2}{Volume~I, Sections~IV.A.2} \& \hyperlink{subsubsection.1.4.1.3}{IV.A.3});
dealings with a D.C.-based think tank that specializes in Russia and has connections with its government (\hyperlink{subsubsection.1.4.1.4}{Volume~I, Section~IV.A.4});
a meeting at Trump Tower between the Campaign and a Russian lawyer promising dirt on candidate Clinton that was ``part of Russia and its government's support for [Trump]'' (\hyperlink{subsubsection.1.4.1.5}{Volume~I, Section~IV.A.5});
events at the Republican National Convention (\hyperlink{subsubsection.1.4.1.6}{Volume~I, Section~IV.A.6});
post-Convention contacts between Trump Campaign officials and Russia's ambassador to the United States (\hyperlink{subsubsection.1.4.1.7}{Volume~I, Section~IV.A.7});
and contacts through campaign chairman Paul Manafort, who had previously worked for a Russian oligarch and a pro-Russian political party in Ukraine (\hyperlink{subsubsection.1.4.1.8}{Volume~I, Section~IV.A.8}).
\subsubsection{Trump Tower Moscow Project}
The Trump Organization has pursued and completed projects outside the United States as part of its real estate portfolio.
Some projects have involved the acquisition and ownership (through subsidiary corporate structures) of property.
In other cases, the Trump Organization has executed licensing deals with real estate developers and management companies, often local to the country where the project was located.% 289
\footnote{\textit{See, e.g., Interview of Donald J Trump, Jr, Senate Judiciary Committee}, 115th Cong.~151--52 (Sept.~7, 2017) (discussing licensing deals of specific projects).}
Between at least 2013 and~2016, the Trump Organization explored a similar licensing deal in Russia involving the construction of a Trump-branded property in Moscow.
The project, commonly referred to as a ``Trump Tower Moscow'' or ``Trump Moscow'' project, anticipated a combination of commercial, hotel, and residential properties all within the same building.
Between 2013 and June 2016, several employees of the Trump Organization, including then-president of the organization Donald J. Trump, pursued a Moscow deal with several Russian counterparties.
From the fall of 2015 until the middle of 2016, Michael Cohen spearheaded the Trump Organization's pursuit of a Trump Tower Moscow project, including by reporting on the project's status to candidate Trump and other executives in the Trump Organization.% 290
\footnote{As noted in \hyperlink{subsubsection.1.3.4.1}{Volume~I, Section~III.D.1}, \textit{supra}, in November 2018, Cohen pleaded guilty to making false statements to Congress concerning, among other things, the duration of the Trump Tower Moscow project.
\textit{See} Information \P~7(a), \textit{United States~v.\ Michael Cohen}, 1:18-cr-850 (S.D.N.Y. Nov.~29, 2018), Doc.~2 (``\textit{Cohen} Information'').}
\paragraph{Trump Tower Moscow Venture with the Crocus Group~(2013--2014)}
The Trump Organization and the Crocus Group, a Russian real estate conglomerate owned and controlled by Aras Agalarov, began discussing a Russia-based real estate project shortly after the conclusion of the 2013 Miss Universe pageant in Moscow.% 291
\footnote{\textit{See Interview of Donald J Trump, Jr, Senate Judiciary Committee}, 115th Cong.~13 (Sept.~7, 2017) (``Following the pageant the Trump Organization and Mr.~Agalarov's company, Crocus Group, began preliminarily discussion [sic] potential real estate projects in Moscow.'').
As has been widely reported, the Miss Universe pageant---which Trump co-owned at the time---was held at the Agalarov-owned Crocus City Hall in Moscow in November
2013. Both groups were involved in organizing the pageant, and Aras Agalarov's son Emin was a musical performer at the event, which Trump attended.}
Donald J. Trump~Jr.\ served as the primary negotiator on behalf of the Trump Organization; Emin Agalarov (son of Aras Agalarov) and Irakli ``Ike'' Kaveladze represented the Crocus Group during negotiations,% 292
\footnote{Kaveladze 11/16/17 302, at~2, 4--6; \blackout{Grand Jury} OSC-KAV\_00385 (12/6/13 Email, Trump~Jr.\ to Kaveladze \& E.~Agalarov).}
with the occasional assistance of Robert Goldstone.% 293
\footnote{\blackout{Grand Jury}}
In December 2013, Kaveladze and Trump~Jr.\ negotiated and signed preliminary terms of an agreement for the Trump Tower Moscow project.% 294
\footnote{\blackout{Grand Jury}}
On December~23, 2013, after discussions with Donald J. Trump, the Trump Organization agreed to accept an arrangement whereby the organization received a flat 3.5\%~commission on all sales, with no licensing fees or incentives.% 295
\footnote{OSC-KAV\_00452 (12/23/13 Email, Trump~Jr.\ to Kaveladze \& E.~Agalarov).}
The parties negotiated a letter of intent during January and February 2014.% 296
\footnote{\textit{See, e.g.}, OSC-KAV\_01158 (Letter agreement signed by Trump~Jr.\ \& E.~Agalarov); OSC-KAV\_01147 (1/20/14 Email, Kaveladze to Trump~Jr.\ et~al.).}
From January 2014 through November 2014, the Trump Organization and Crocus Group discussed development plans for the Moscow project.
Some time before January~24, 2014, the Crocus Group sent the Trump Organization a proposal for a 800-unit, 194-meter building to be constructed at an Agalarov-owned site in Moscow called ``Crocus City,'' which had also been the site of the Miss Universe pageant.% 297
\footnote{\textit{See, e.g.}, OSC-KAV\_00972 (10/14/14 Email, McGee to Khoo et~al.) (email from Crocus Group contractor about specifications); OSC-KAV\_00540 (1/24/14 Email, McGee to Trump~Jr.\ et~al.).}
In February 2014, Ivanka Trump met with Emin Agalarov and toured the Crocus City site during a visit to Moscow.% 298
\footnote{\textit{See} OSC-KA V 00631 (2/5/14 Email, E.~Agalarov to Ivanka Trump~Jr.\ \& Kaveladze); Goldstone Facebook post, 2/4/14 (8:01~a.m) \blackout{Investigative Technique}}
From March 2014 through July 2014, the groups discussed ``design standards'' and other architectural elements.% 299
\footnote{\textit{See, e.g.}, OSC-KAV\_00791 (6/3/14 Email, Kaveladze to Trump~Jr.\ et~al.[)]; OSC-KAV\_00799 (6/10/14 Email, Trump~Jr.\ to Kaveladze et~al.); OSC-KAV\_00817 (6/16/14 Email, Trump~Jr.\ to Kaveladze et~al.).}
For example, in July 2014, members of the Trump Organization sent Crocus Group counterparties questions about the ``demographics of these prospective buyers'' in the Crocus City area, the development of neighboring parcels in Crocus City, and concepts for redesigning portions of the building.% 300
\footnote{OSC-KAV\_00870 (7/17/14 Email, Khoo to McGee et~al.).}
In August 2014, the Trump Organization requested specifications for a competing Marriott-branded tower being built in Crocus City.% 301
\footnote{OSC-KAV\_00855 (8/4/14 Email, Khoo to McGee et~al.).}
Beginning in September 2014, the Trump Organization stopped responding in a timely fashion to correspondence and proposals from the Crocus Group.% 302
\footnote{OSC-KAV\_00903 (9/29/14 Email, Tropea to McGee \& Kaveladze (noting last response was on August~26, 2014)); OSC-KAV\_00906 (9/29/14 Email, Kaveladze to Tropea \& McGee (suggesting silence ``proves my fear that those guys are bailing out of the project'')); OSC-KAV\_00972 (10/14/14 Email, McGee to Khoo et~al.) (email from Crocus Group contractor about development specifications)[].}
Communications between the two groups continued through November 2014 with decreasing frequency; what appears to be the last communication is dated November~24, 2014.% 303
\footnote{OSC-KAV\_01140 (11/24/14 Email, Khoo to McGee et~al.).}
The project appears not to have developed past the planning stage, and no construction occurred.
\paragraph{Communications with I.C.~Expert Investment Company and Giorgi Rtskhiladze (Summer and Fall 2015)}
In the late summer of 2015, the Trump Organization received a new inquiry about pursuing a Trump Tower project in Moscow.
In approximately September 2015, Felix Sater, a New York-based real estate advisor, contacted Michael Cohen, then-executive vice president of the Trump Organization and special counsel to Donald J. Trump.% 304
\footnote{Sater provided information to our Office in two 2017 interviews conducted under a proffer agreement.
\blackout{Grand Jury}}
Sater had previously worked with the Trump Organization and advised it on a number of domestic and international projects.
Sater had explored the possibility of a Trump Tower project in Moscow while working with the Trump Organization and therefore knew of the organization's general interest in completing a deal there.% 305
\footnote{\blackout{Grand Jury}}
Sater had also served as an informal agent of the Trump Organization in Moscow previously and had accompanied Ivanka Trump and Donald Trump~Jr.\ to Moscow in the mid-2000s.% 306
\footnote{Sater 9/19/17 302, at~1--2, 5.}
Sater contacted Cohen on behalf of I.C.~Expert Investment Company (I.C.~Expert), a Russian real-estate development corporation controlled by Andrei Vladimirovich Rozov.% 307
\footnote{Sater 9/19/17 302, at~3.}
Sater had known Rozov since approximately 2007 and, in 2014, had served as an agent on behalf of Rozov during Rozov's purchase of a building in New York City.% 308
\footnote{Rozov 1/25/18 302, at~1.}
Sater later contacted Rozov and proposed that I.C.~Expert pursue a Trump Tower Moscow project in which I.C.~Expert would license the name and brand from the Trump Organization but construct the building on its own.
Sater worked on the deal with Rozov and another employee of I.C.~Expert.% 309
\footnote{Rozov 1/25/18 302, at~1;
\textit{see also} 11/2/15 Email, Cohen to Rozov et~al.\ (sending letter of intent).}
Cohen was the only Trump Organization representative to negotiate directly with I.C.~Expert or its agents.
In approximately September 2015, Cohen obtained approval to negotiate with I.C.~Expert from candidate Trump, who was then president of the Trump Organization.
Cohen provided updates directly to Trump about the project throughout 2015 and into 2016, assuring him the project was continuing.% 310
\footnote{Cohen 9/12/18 302, at~1--2, 4--6.}
Cohen also discussed the Trump Moscow project with Ivanka Trump as to design elements (such as possible architects to use for the project% 311
\footnote{Cohen 9/12/18 302, at~5.}%
) and Donald J. Trump~Jr.\ (about his experience in Moscow and possible involvement in the project% 312
\footnote{Cohen 9/12/18 302, at~4--5.}%
) during the fall of 2015.
Also during the fall of 2015, Cohen communicated about the Trump Moscow proposal with Giorgi Rtskhiladze, a business executive who previously had been involved in a development deal with the Trump Organization in Batumi, Georgia.% 313
\footnote{Rtskhiladze was a U.S.-based executive of the Georgian company Silk Road Group.
In approximately 2011, Silk Road Group and the Trump Organization entered into a licensing agreement to build a Trump-branded property in Batumi, Georgia.
Rtskhiladze was also involved in discussions for a Trump-branded project in Astana, Kazakhstan.
The Office twice interviewed Rtskhiladze, \blackout{Grand Jury}}
Cohen stated that he spoke to Rtskhiladze in part because Rtskhiladze had pursued business ventures in Moscow, including a licensing deal with the Agalarov-owned Crocus Group.% 314
\footnote{Cohen 9/12/18 302, at~12;
\textit{see also} Rtskhiladze 5/10/18 302, at~1.}
On September~22, 2015, Cohen forwarded a preliminary design study for the Trump Moscow project to Rtskhiladze, adding ``I look forward to your reply about this spectacular project in Moscow.''
Rtskhiladze forwarded Cohen's email to an associate and wrote, ``[i]f we could organize the meeting in New York at the highest level of the Russian Government and Mr.~Trump this project would definitely receive the worldwide attention.''% 315
\footnote{9/22/15 Email, Rtskhiladze to Nizharadze.}
On September~24, 2015, Rtskhiladze sent Cohen an attachment that he described as a proposed ``[l]etter to the Mayor of Moscow from Trump org,'' explaining that ``[w]e need to send this letter to the Mayor of Moscow (second guy in Russia) he is aware of the potential project and will pledge his support.''% 316
\footnote{9/24/15 Email, Rtskhiladze to Cohen.}
In a second email to Cohen sent the same day, Rtskhiladze provided a translation of the letter, which described the Trump Moscow project as a ``symbol of stronger economic, business and cultural relationships between New York and Moscow and therefore United States and the Russian Federation.''% 317
\footnote{9/24/15 Email, Rtskhiladze to Cohen.}
On September~27, 2015, Rtskhiladze sent another email to Cohen, proposing that the Trump Organization partner on the Trump Moscow project with ``Global Development Group LLC,'' which he described as being controlled by Michail Posikhin, a Russian architect, and Simon Nizharadze.% 318
\footnote{9/27/15 Email, Rtskhiladze to Cohen.}
Cohen told the Office that he ultimately declined the proposal and instead continued to work with I.C.~Expert, the company represented by Felix Sater.% 319
\footnote{Cohen 9/12/18 302, at~12.}
\paragraph{Letter of Intent and Contacts to Russian Government (October 2015--January 2016)}
\subparagraph{Trump Signs the Letter of Intent on behalf of the Trump Organization}
Between approximately October~13, 2015 and November~2, 2015, the Trump Organization (through its subsidiary Trump Acquisition, LLC) and I.C.~Expert completed a letter of intent (LOI) for a Trump Moscow property.
The LOI, signed by Trump for the Trump Organization and Rozov on behalf of I.C.~Expert, was ``intended to facilitate further discussions'' in order to ``attempt to enter into a mutually acceptable agreement'' related to the Trump-branded project in Moscow.% 320
\footnote{11/2/15 Email, Cohen to Rozov et~al.\ (attachment) (hereinafter ``LOI'');
\textit{see also} 10/13/15 Email, Sater to Cohen \& Davis (attaching proposed letter of intent).}
The LOI contemplated a development with residential, hotel, commercial, and office components, and called for ``[a]pproximately 250 first class, luxury residential condominiums,'' as well as ``[o]ne first class, luxury hotel consisting of approximately 15 floors and containing not fewer than 150 hotel rooms.''% 321
\footnote{LOI, p.~2.}
For the residential and commercial portions of the project, the Trump Organization would receive between 1\% and 5\% of all condominium sales,% 322
\footnote{The LOI called for the Trump Organization to receive 5\% of all gross sales up to \$100~million;
4\% of all gross sales from \$100~million to \$250~million;
3\% of all gross sales from \$250~million to \$500~million;
2\% of all gross sales from \$500~million to \$1~billion;
and 1\% of all gross sales over \$1~billion.
LOI, Schedule 2.}
plus 3\% of all rental and other revenue.% 323
\footnote{LOI, Schedule 2.}
For the project's hotel portion, the Trump Organization would receive a base fee of 3\% of gross operating revenues for the first five years and 4\% thereafter, plus a separate incentive fee of 20\% of operating profit.% 324
\footnote{LOI, Schedule 1.}
Under the LOI, the Trump Organization also would receive a \$4 million ``up-front fee'' prior to groundbreaking.% 325
\footnote{LOI, Schedule 2.}
Under these terms, the Trump Organization stood to earn substantial sums over the lifetime of the project, without assuming significant liabilities or financing commitments.% 326
\footnote{Cohen 9/12/18 302, at~3.}
On November~3, 2015, the day after the Trump Organization transmitted the LOI, Sater emailed Cohen suggesting that the Trump Moscow project could be used to increase candidate Trump's chances at being elected, writing:
\begin{quote}
Buddy our boy can become President of the USA and we can engineer it.
I will get all of Putins team to buy in on this, I will manage this process\dots.
Michael, Putin gets on stage with Donald for a ribbon cutting for Trump Moscow, and Donald owns the republican nomination.
And possibly beats Hillary and our boy is in\dots.
We will manage this process better than anyone.
You and I will get Donald and Vladimir on a stage together very shortly.
That the game changer.% 327
\footnote{11/3/15 Email, Sater to Cohen (12:14~p.m.).}
\end{quote}
Later that day, Sater followed up:
\begin{quote}
Donald doesn't stare down, he negotiates and understands the economic issues and Putin only wants to deal with a pragmatic leader, and a successful business man is a good candidate for someone who knows how to negotiate.
``Business, politics, whatever it all is the same for someone who knows how to deal''
I think I can get Putin to say that at the Trump Moscow press conference.
If he says it we own this election.
America's most difficult adversary agreeing that Donald is a good guy to negotiate\dots.
We can own this election.
Michael my next steps are very sensitive with Putin's very very close people, we can pull this off.
Michael lets go.
2 boys from Brooklyn getting a USA president elected.
This is good really good.% 328
\footnote{11/3/15 Email, Sater to Cohen (12:40~p.m.).}
\end{quote}
According to Cohen, he did not consider the political import of the Trump Moscow project to the 2016 U.S. presidential election at the time.
Cohen also did not recall candidate Trump or anyone affiliated with the Trump Campaign discussing the political implications of the Trump Moscow project with him.
However, Cohen recalled conversations with Trump in which the candidate suggested that his campaign would be a significant ``infomercial'' for Trump-branded properties.% 329
\footnote{Cohen 9/12/18 302, at~3--4; Cohen 8/7/18 302, at~15.}
\subparagraph{Post-LOI Contacts with Individuals in Russia}
Given the size of the Trump Moscow project, Sater and Cohen believed the project required approval (whether express or implicit) from the Russian national government, including from the Presidential Administration of Russia.% 330
\footnote{\blackout{Grand Jury}}
Sater stated that he therefore began to contact the Presidential Administration through another Russian business contact.% 331
\footnote{Sater 12/15/17 302, at~3--4.}
In early negotiations with the Trump Organization, Sater had alluded to the need for government approval and his attempts to set up meetings with Russian officials.
On October~12, 2015, for example, Sater wrote to Cohen that ``all we need is Putin on board and we are golden,'' and that a ``meeting with Putin and top deputy is tentatively set for the 14th [of October].''% 332
\footnote{10/12/15 Email, Sater to Cohen (8:07~a.m.).}
\blackout{Grand Jury} this meeting was being coordinated by associates in Russia and that he had no direct interaction with the Russian government.% 333
\footnote{\blackout{Grand Jury}}
Approximately a month later, after the LOI had been signed, Lana Erchova emailed Ivanka Trump on behalf of Erchova's then-husband Dmitry Klokov, to offer Klokov's assistance to the Trump Campaign.% 334
\footnote{Ivanka Trump received an email from a woman who identified herself as ``Lana E. Alexander,'' which said in part, ``If you ask anyone who knows Russian to google my husband Dmitry Klokov, you'll see who he is close to and that he has done Putin's political campaigns.''
11/16/15 Email, Erchova to I.~Trump.}
Klokov was at that time Director of External Communications for PJSC Federal Grid Company of Unified Energy System, a large Russian electricity transmission company, and had been previously employed as an aide and press secretary to Russia's energy minister.
Ivanka Trump forwarded the email to Cohen.% 335
\footnote{11/16/15 Email, I.~Trump to Cohen.}
He told the Office that, after receiving this inquiry, he had conducted an internet search for Klokov's name and concluded (incorrectly) that Klokov was a former Olympic weightlifter.% 336
\footnote{Cohen 8/7/18 302, at~17.
During his interviews with the Office, Cohen still appeared to believe that the Klokov he spoke with was that Olympian.
The investigation, however, established that the email address used to communicate with Cohen belongs to a different Dmitry Klokov, as described above.}
Between November~18 and~19, 2015, Klokov and Cohen had at least one telephone call and exchanged several emails.
Describing himself in emails to Cohen as a ``trusted person'' who could offer the Campaign ``political synergy'' and ``synergy on a government level,'' Klokov recommended that Cohen travel to Russia to speak with him and an unidentified intermediary.
Klokov said that those conversations could facilitate a later meeting in Russia between the candidate and an individual Klokov described as ``our person of interest.''% 337
\footnote{11/18/15 Email, Klokov to Cohen (6:51~a.m.).}
In an email to the Office, Erchova later identified the ``person of interest'' as Russian President Vladimir Putin.% 338
\footnote{In July 2018, the Office received an unsolicited email purporting to be from Erchova, in which she wrote that ``[a]t the end of 2015 and beginning of 2016 I was asked by my ex-husband to contact Ivanka Trump \dots\ and offer cooperation to Trump's team on behalf of the Russian officials.''
7/27/18 Email, Erchova to Special Counsel's Office.
The email claimed that the officials wanted to offer candidate Trump ``land in Crimea among other things and unofficial meeting with Putin.''
\textit{Id.}
In order to vet the email's claims, the Office responded requesting more details.
The Office did not receive any reply.}
In the telephone call and follow-on emails with Klokov, Cohen discussed his desire to use a near-term trip to Russia to do site surveys and talk over the Trump Moscow project with local developers.
Cohen registered his willingness also to meet with Klokov and the unidentified intermediary, but was emphatic that all meetings in Russia involving him or candidate Trump---including a possible meeting between candidate Trump and Putin---would need to be ``in conjunction with the development and an official visit'' with the Trump Organization receiving a formal invitation to visit.% 339
\footnote{11/18/15 Email, Cohen to Klokov (7:15~a.m.).}
(Klokov had written previously that ``the visit [by candidate Trump to Russia] has to be informal.'')% 340
\footnote{11/18/15 Email, Klokov to Cohen (6:51~a.m.).}
Klokov had also previously recommended to Cohen that he separate their negotiations over a possible meeting between Trump and ``the person of interest'' from any existing business track.% 341
\footnote{11/18/15 Email, Klokov to Cohen (6:51~a.m.) (``I would suggest separating your negotiations and our proposal to meet.
I assure you, after the meeting level of projects and their capacity can be completely different, having the most important support.'').}
Re-emphasizing that his outreach was not done on behalf of any business, Klokov added in second email to Cohen that, if publicized well, such a meeting could have ``phenomenal'' impact ``in a business dimension'' and that the ``person of interest['s]'' ``most important support'' could have significant ramifications for the ``level of projects and their capacity.''
Klokov concluded by telling Cohen that there was ``no bigger warranty in any project than [the] consent of the person of interest.''% 342
\footnote{11/19/15 Email, Klokov to Cohen (7:40~a.m.).}
Cohen rejected the proposal, saying that ``[c]urrently our LOI developer is in talks with VP's Chief of Staff and arranging a formal invite for the two to meet.''% 343
\footnote{11/19/15 Email, Cohen to Klokov (12:56~p.m.).}
This email appears to be their final exchange, and the investigation did not identify evidence that Cohen brought Klokov's initial offer of assistance to the Campaign's attention or that anyone associated with the Trump Organization or the Campaign dealt with Klokov at a later date.
Cohen explained that he did not pursue the proposed meeting because he was already working on the Moscow Project with Sater, who Cohen understood to have his own connections to the Russian government.% 344
\footnote{Cohen 9/18/18 302, at~12.}
By late December 2015, however, Cohen was complaining that Sater had not been able to use those connections to set up the promised meeting with Russian government officials.
Cohen told Sater that he was ``setting up the meeting myself.''% 345
\footnote{FS00004 (12/30/15 Text Message, Cohen to Sater (6:17~p.m.)).}
On January~11, 2016, Cohen emailed the office of Dmitry Peskov, the Russian government's press secretary, indicating that he desired contact with Sergei Ivanov, Putin's chief of staff.
Cohen erroneously used the email address ``\UseVerb{PrpeskovaATprpressgofru}'' instead of ``\UseVerb{PrpeskovaATprpressgovru},'' so the email apparently did not go through.% 346
\footnote{1/11/16 Email, Cohen to \UseVerb{prpeskovaATprpressgofru} (9:12~a.m.).}
On January~14, 2016, Cohen emailed a different address (\UseVerb{infoATprpressgovru}) with the following message:
\begin{quote}
Dear Mr.~Peskov,
Over the past few months, I have been working with a company based in Russia regarding the development of a Trump Tower-Moscow project in Moscow City.
Without getting into lengthy specifics, the communication between our two sides has stalled.
As this project is too important, I am hereby requesting your assistance.
I respectfully request someone, preferably you; contact me so that I might discuss the specifics as well as arranging meetings with the appropriate individuals.
I thank you in advance for your assistance and look forward to hearing from you soon.% 347
\footnote{1/14/16 Email, Cohen to \UseVerb{infoATprpressgovru} (9:21~a.m.).}
\end{quote}
Two days later, Cohen sent an email to \UseVerb{PrpeskovaATprpressgovru}, repeating his request to speak with Sergei Ivanov.% 348
\footnote{1/16/16 Email, Cohen to \UseVerb{prpeskovaATprpressgovru} (10:28~a.m.).}
Cohen testified to Congress, and initially told the Office, that he did not recall receiving a response to this email inquiry and that he decided to terminate any further work on the Trump Moscow project as of January 2016.
Cohen later admitted that these statements were false.
In fact, Cohen had received (and recalled receiving) a response to his inquiry, and he continued to work on and update candidate Trump on the project through as late as June 2016.% 349
\footnote{\textit{Cohen} Information \P\P~4,~7.
Cohen's interactions with President Trump and the President's lawyers when preparing his congressional testimony are discussed further in \hyperref[chap:volume-2]{Volume~II}\null.
\textit{See} \hyperlink{subsubsection.2.2.11.3}{Vol.~II, Section~II.K.3}, \textit{infra}.}
On January~20, 2016, Cohen received an email from Elena Poliakova, Peskov's personal assistant.
Writing from her personal email account, Poliakova stated that she had been trying to reach Cohen and asked that he call her on the personal number that she provided.% 350
\footnote{1/20/16 Email, Poliakova to Cohen (5:57~a.m.)
(``Mr.~Cohen[,] I can't get through to both your phones. Pls, call me.'').}
Shortly after receiving Poliakova's email, Cohen called and spoke to her for 20~minutes.% 351
\footnote{Telephone records show a 20-minute call on January~20, 2016 between Cohen and the number Poliakova provided in her email.
Call Records of Michael Cohen \blackout{Grand Jury}
After the call, Cohen saved Poliakova's contact information in his Trump Organization Outlook contact list.
1/20/16 Cohen Microsoft Outlook Entry (6:22~a.m.).}
Cohen described to Poliakova his position at the Trump Organization and outlined the proposed Trump Moscow project, including information about the Russian counterparty with which the Trump Organization had partnered.
Cohen requested assistance in moving the project forward, both in securing land to build the project and with financing.
According to Cohen, Poliakova asked detailed questions and took notes, stating that she would need to follow up with others in Russia.% 352
\footnote{Cohen 9/12/18 302, at~2--3.}
Cohen could not recall any direct follow-up from Poliakova or from any other representative of the Russian government, nor did the Office identify any evidence of direct follow-up.
However, the day after Cohen's call with Poliakova, Sater texted Cohen, asking him to ``[c]all me when you have a few minutes to chat \dots\ It's about Putin they called today.''% 353
\footnote{FS00011 (1/21/16 Text Messages, Sater to Cohen).}
Sater then sent a draft invitation for Cohen to visit Moscow to discuss the Trump Moscow project,% 354
\footnote{The invitation purported to be from Genbank, a Russian bank that was, according to Sater, working at the behest of a larger bank, VTB, and would consider providing financing.
FS00008 (12/31/15 Text Messages, Sater \& Cohen).
Additional information about Genbank can be found \textit{infra}.}
along with a note to ``[t]ell me if the letter is good as amended by me or make whatever changes you want and send it back to me.''% 355
\footnote{FSO0011 (1/21/16 Text Message, Sater to Cohen (7:44~p.m.));
1/21/16 Email, Sater to Cohen (6:49~p.m.).}
After a further round of edits, on January~25, 2016, Sater sent Cohen an invitation---signed by Andrey Ryabinskiy of the company MHJ---to travel to ``Moscow for a working visit'' about the ``prospects of development and the construction business in Russia,'' ``the various land plots available suited for construction of this enormous Tower,'' and ``the opportunity to co-ordinate a follow up visit to Moscow by Mr.~Donald Trump.''% 356
\footnote{1/25/16 Email, Sater to Cohen (12:01~p.m.) (attachment).}
According to Cohen, he elected not to travel at the time because of concerns about the lack of concrete proposals about land plots that could be considered as options for the project.% 357
\footnote{Cohen 9/12/18 302, at~6--7.}
\paragraph{Discussions about Russia Travel by Michael Cohen or Candidate Trump (December 2015--June 2016)}
\subparagraph{Sater's Overtures to Cohen to Travel to Russia}
The late January communication was neither the first nor the last time that Cohen contemplated visiting Russia in pursuit of the Trump Moscow project.
Beginning in late 2015, Sater repeatedly tried to arrange for Cohen and candidate Trump, as representatives of the Trump Organization, to travel to Russia to meet with Russian government officials and possible financing partners.
In December 2015, Sater sent Cohen a number of emails about logistics for traveling to Russia for meetings.% 358
\footnote{\textit{See, e.g.}, 12/1/15 Email, Sater to Cohen (12:41~p.m.)
(``Please scan and send me a copy of your passport for the Russian Ministry of Foreign Affairs.'').}
On December~19, 2015, Sater wrote:
\begin{quote}
Please call me I have Evgeney [Dvoskin] on the other line.% 359
\footnote{Toll records show that Sater was speaking to Evgeny Dvoskin.
Call Records of Felix Sater
\blackout{Grand Jury}
Dvoskin is an executive of Genbank, a large bank with lending focused in Crimea, Ukraine.
At the time that Sater provided this financing letter to Cohen, Genbank was subject to U.S. government sanctions,
\textit{see Russia/Ukraine-related Sanctions and Identifications}, Office of Foreign Assets Control (Dec.~22, 2015), \textit{available at} \url{https://www.treasury.gov/resource-center/sanctions/OFACEnforcement/Pages/20151222.aspx}.
Dvoskin, who had been deported from the United States in 2000 for criminal activity, was under indictment in the United States for stock fraud under the aliases Eugene Slusker and Gene Shustar.
\textit{See United States~v.\ Rizzo, et~al.}, 2:03-cr-63 (E.D.N.Y. Feb.~6, 2003).}
He needs a copy of your and Donald's passports they need a scan of every page of the passports.
Invitations \& Visas will be issued this week by VTB Bank to discuss financing for Trump Tower Moscow. Politically neither Putins office nor Ministry of Foreign Affairs cannot issue invite, so they are inviting commercially/business.
VTB is Russia's 2 biggest bank and VTB Bank CEO Andrey Kostin, will be at all meetings with Putin so that it is a business meeting not political.
We will be invited to Russian consulate this week to receive invite \& have visa issued.% 360
\footnote{12/19/15 Email, Sater to Cohen (10:50~a.m.);
FS00002 (12/19/15 Text Messages, Sater to Cohen, (10:53~a.m.)[)].}
\end{quote}
In response, Cohen texted Sater an image of his own passport.% 361
\footnote{FS00004 (12/19/15 Text Message, Cohen to Sater);
ERT\_0198-256 (12/19/15 Text Messages, Cohen \& Sater).}
Cohen told the Office that at one point he requested a copy of candidate Trump's passport from Rhona Graff, Trump's executive assistant at the Trump Organization, and that Graff later brought Trump's passport to Cohen's office.% 362
\footnote{Cohen 9/12/18 302, at~5.}
The investigation did not, however, establish that the passport was forwarded to Sater.% 363
\footnote{On December~21, 2015, Sater sent Cohen a text message that read, ``They need a copy of DJT passport,'' to which Cohen responded, ``After I return from Moscow with you with a date for him.''
FS00004 (12/21/15 Text Messages, Cohen \& Sater).}
Into the spring of 2016, Sater and Cohen continued to discuss a trip to Moscow in connection with the Trump Moscow project.
On April~20, 2016, Sater wrote Cohen, ``[t]he People wanted to know when you are coming?''% 364
\footnote{FS00014 (4/20/16 Text Message, Sater to Cohen (9:06~p.m.)).}
On May~4, 2016, Sater followed up:
\begin{quote}
I had a chat with Moscow.
ASSUMING the trip does happen the question is before or after the convention.
I said I believe, but don't know for sure, that it's probably after the convention.
Obviously the pre-meeting trip (you only) can happen anytime you want but the 2 big guys where [sic] the question.
I said I would confirm and revert\dots.
Let me know about If I was right by saying I believe after Cleveland and also when you want to speak to them and possibly fly over.% 365
\footnote{FS00015 (5/4/16 Text Message, Sater to Cohen (7:38~p.m.)).}
\end{quote}
Cohen responded, ``My trip before Cleveland.
Trump once he becomes the nominee after the convention.''% 366
\footnote{FS00015 (5/4/16 Text Message, Cohen to Sater (8:03~p.m.)).}
The day after this exchange, Sater tied Cohen's travel to Russia to the St.~Petersburg International Economic Forum (``Forum''), an annual event attended by prominent Russian politicians and businessmen.
Sater told the Office that he was informed by a business associate that Peskov wanted to invite Cohen to the Forum.% 367
\footnote{Sater 12/15/17 302, at~4.}
On May~5, 2016, Sater wrote to Cohen:
\begin{quote}
Peskov would like to invite you as his guest to the St.~Petersburg Forum which is Russia's Davos it's June~16--19.
He wants to meet there with you and possibly introduce you to either Putin or Medvedev, as they are not sure if 1 or both will be there.
This is perfect.
The entire business class of Russia will be there as well.
He said anything you want to discuss including dates and subjects are on the table to discuss[.]% 368
\footnote{FS00016 (5/5/16 Text Messages, Sater to Cohen (6:26 \& 6:27~a.m.)).}
\end{quote}
The following day, Sater asked Cohen to confirm those dates would work for him to travel; Cohen wrote back, ``[w]orks for me.''% 369
\footnote{FS00016 (5/6/16 Text Messages, Cohen \& Sater).}
On June~9, 2016, Sater sent Cohen a notice that he (Sater) was completing the badges for the Forum, adding, ``Putin is there on the 17th very strong chance you will meet him as well.''% 370
\footnote{FS00018 (6/9/16 Text Messages, Sater \& Cohen).}
On June~13, 2016, Sater forwarded Cohen an invitation to the Forum signed by the Director of the Roscongress Foundation, the Russian entity organizing the Forum.% 371
\footnote{6/13/16 Email, Sater to Cohen (2:10~p.m.).}
Sater also sent Cohen a Russian visa application and asked him to send two passport photos.% 372
\footnote{FS00018 (6/13/16 Text Message, Sater to Cohen (2:20~p.m.));
6/13/16 Email, Sater to Cohen.}
According to Cohen, the invitation gave no indication that Peskov had been involved in inviting him.
Cohen was concerned that Russian officials were not actually involved or were not interested in meeting with him (as Sater had alleged), and so he decided not to go to the Forum.% 373
\footnote{Cohen 9/12/18 302, at~6--8.}
On June~14, 2016, Cohen met Sater in the lobby of the Trump Tower in New York and informed him that he would not be traveling at that time.% 374
\footnote{FS00019 (6/14/16 Text Messages, Cohen \& Sater (12:06 and 2:50~p.m.)).}
\subparagraph{Candidate Trump's Opportunities to Travel to Russia}
The investigation identified evidence that, during the period the Trump Moscow project was under consideration, the possibility of candidate Trump visiting Russia arose in two contexts.
First, in interviews with the Office, Cohen stated that he discussed the subject of traveling to Russia with Trump twice: once in late 2015; and again in spring 2016.% 375
\footnote{Cohen 9/12/18 302, at~2.}
According to Cohen, Trump indicated a willingness to travel if it would assist the project significantly.
On one occasion, Trump told Cohen to speak with then-campaign manager Corey Lewandowski to coordinate the candidate's schedule.
Cohen recalled that he spoke with Lewandowski, who suggested that they speak again when Cohen had actual dates to evaluate.
Cohen indicated, however, that he knew that travel prior to the Republican National Convention would be impossible given the candidate's preexisting commitments to the Campaign.% 376
\footnote{Cohen 9/12/18 302, at~7.}
Second, like Cohen, Trump received and turned down an invitation to the St.~Petersburg International Economic Forum.
In late December 2015, Mira Duma---a contact of Ivanka Trump's from the fashion industry---first passed along invitations for Ivanka Trump and candidate Trump from Sergei Prikhodko, a Deputy Prime Minister of the Russian Federation.% 377
\footnote{12/21/15 Email, Mira to Ivanka Trump (6:57~a.m.) (attachments);
TRUMPORG\_16\_000057 (1/7/16 Email, I.~Trump to Graff (9:18~a.m.)).}
On January~14, 2016, Rhona Graff sent an email to Duma stating that Trump was ``honored to be asked to participate in the highly prestigious'' Forum event, but that he would ``have to decline'' the invitation given his ``very grueling and full travel schedule'' as a presidential candidate.% 378
\footnote{1/14/16 Email, Graff to Mira.}
Graff asked Duma whether she recommended that Graff ``send a formal note to the Deputy Prime Minister'' declining his invitation; Duma replied that a formal note would be ``great.''% 379
\footnote{1/15/16 Email, Mira to Graff.}
It does not appear that Graff prepared that note immediately.
According to written answers from President Trump,% 380
\footnote{As explained in \hyperref[chap:volume-2]{Volume~II} and \hyperlink{section.3.3}{Appendix~C}, on September~17, 2018, the Office sent written questions to the President's counsel.
On November~20, 2018, the President provided written answers to those questions through counsel.}
Graff received an email from Deputy Prime Minister Prikhodko on March~17, 2016, again inviting Trump to participate in the 2016 Forum in St.~Petersburg.% 381
\footnote{Written Responses of Donald J. Trump (Nov.~20, 2018), at~17 (Response to Question~IV, Part~(e)) (``[D]ocuments show that Ms.~Graff prepared for my signature a brief response declining the invitation.'').}
Two weeks later, on March~31, 2016, Graff prepared for Trump's signature a two-paragraph letter declining the invitation.% 382
\footnote{Written Responses of Donald J. Trump (Nov.~20, 2018), at~17 (Response to Question~IV, Part~(e));
\textit{see also} TRUMPORG\_16\_000134 (unsigned letter dated March~31, 2016).}
The letter stated that Trump's ``schedule has become extremely demanding'' because of the presidential campaign, that he ``already ha[d] several commitments in the United States'' for the time of the Forum, but that he otherwise ``would have gladly given every consideration to attending such an important event.''% 383
\footnote{TRUMPORG\_16\_000134 (unsigned letter).}
Graff forwarded the letter to another executive assistant at the Trump Organization with instructions to print the document on letterhead for Trump to sign.% 384
\footnote{TRUMPORG\_16000133 (3/31/16 Email, Graff to Macchia).}
At approximately the same time that the letter was being prepared, Robert Foresman---a New York-based investment banker---began reaching out to Graff to secure an in-person meeting with candidate Trump.
According to Foresman, he had been asked by Anton Kobyakov, a Russian presidential aide involved with the Roscongress Foundation, to see if Trump could speak at the Forum.% 385
\footnote{Foresman 10/17/18 302, at~3--4.}
Foresman first emailed Graff on March~31, 2016, following a phone introduction brokered through Trump business associate Mark Burnett (who produced the television show \textit{The Apprentice}).
In his email, Foresman referenced his long-standing personal and professional expertise in Russia and Ukraine, his work setting up an early ``private channel'' between Vladimir Putin and former U.S. President George W. Bush, and an ``approach'' he had received from ``senior Kremlin officials'' about the candidate.
Foresman asked Graff for a meeting with the candidate, Corey Lewandowski, or ``another relevant person'' to discuss this and other ``concrete things'' Foresman felt uncomfortable discussing over ``unsecure email.''% 386
\footnote{\textit{See} TRUMPORG\_16\_00136 (3/31/16 Email, Foresman to Graff);
\textit{see also} Foresman 10/17/18 302, at~3--4.}
On April~4, 2016, Graff forwarded Foresman's meeting request to Jessica Macchia, another executive assistant to Trump.% 387
\footnote{\textit{See} TRUMPORG\_16\_00136 (4/4/16 Email, Graff to Macchia).}
With no response forthcoming, Foresman twice sent reminders to Graff---first on April~26 and again on April~30, 2016.% 388
\footnote{\textit{See} TRUMPORG\_16\_00137 (4/26/16 Email, Foresman to Graff);
TRUMPORG\_16\_00141 (4/30/16 Email, Foresman to Graff).}
Graff sent an apology to Foresman and forwarded his April~26 email (as well as his initial March 2016 email) to Lewandowski.% 389
\footnote{\textit{See} TRUMPORG\_16\_00139 (4/27/16 Email, Graff to Foresman);
TRUMPORG\_16\_00137 (4/27/16 Email, Graff to Lewandowski).}
On May~2, 2016, Graff forwarded Foresman's April~30 email---which suggested an alternative meeting with Donald Trump~Jr.\ or Eric Trump so that Foresman could convey to them information that ``should be conveyed to [the candidate] personally or [to] someone [the candidate] absolutely trusts''---to policy advisor Stephen Miller.% 390
\footnote{TRUMPORG\_16\_00142 (5/2/16 Email, Graff to S.~Miller);
\textit{see also} TRUMPORG\_16\_00143 (5/2/16 Email, Graff to S.~Miller) (forwarding March 2016 email from Foresman).}
No communications or other evidence obtained by the Office indicate that the Trump Campaign learned that Foresman was reaching out to invite the candidate to the Forum or that the Campaign otherwise followed up with Foresman until after the election, when he interacted with the Transition Team as he pursued a possible position in the incoming Administration.% 391
\footnote{Foresman's contacts during the transition period are discussed further in \hyperlink{subsubsection.1.4.2.3}{Volume~I, Section~IV.B.3}, \textit{infra}.}
When interviewed by the Office, Foresman denied that the specific ``approach'' from ``senior Kremlin officials'' noted in his March~31, 2016 email was anything other than Kobyakov's invitation to Roscongress.
According to Foresman, the ``concrete things'' he referenced in the same email were a combination of the invitation itself, Foresman's personal perspectives on the invitation and Russia policy in general, and details of a Ukraine plan supported by a U.S. think tank (EastWest Institute).
Foresman told the Office that Kobyakov had extended similar invitations through him to another Republican presidential candidate and one other politician.
Foresman also said that Kobyakov had asked Foresman to invite Trump to speak after that other presidential candidate withdrew from the race and the other politician's participation did not work out.% 392
\footnote{Foresman 10/17/18 302, at~4.}
Finally, Foresman claimed to have no plans to establish a back channel involving Trump, stating the reference to his involvement in the Bush--Putin back channel was meant to burnish his credentials to the Campaign.
Foresman commented that he had not recognized any of the experts announced as Trump's foreign policy team in March 2016, and wanted to secure an in-person meeting with the candidate to share his professional background and policy views, including that Trump should decline Kobyakov's invitation to speak at the Forum.% 393
\footnote{Foresman 10/17/18 302, at~8--9.}
\subsubsection{George Papadopoulos}
George Papadopoulos was a foreign policy advisor to the Trump Campaign from March 2016 to early October 2016.% 394
\footnote{Papadopoulos met with our Office for debriefings on several occasions in the summer and fall of 2017, after he was arrested and charged in a sealed criminal complaint with making false statements in a January 2017 FBI interview about, \textit{inter alia}, the timing, extent, and nature of his interactions and communications with Joseph Mifsud and two Russian nationals: Olga Polonskaya and Ivan Timofeev.
Papadopoulos later pleaded guilty, pursuant to a plea agreement, to an information charging him with making false statements to the FBI, in violation of 18~U.S.C. \S~1001(a).}
In late April 2016, Papadopoulos was told by London-based professor Joseph Mifsud, immediately after Mifsud's return from a trip to Moscow, that the Russian government had obtained ``dirt'' on candidate Clinton in the form of thousands of emails.
One week later, on May~6, 2016, Papadopoulos suggested to a representative of a foreign government that the Trump Campaign had received indications from the Russian government that it could assist the Campaign through the anonymous release of information that would be damaging to candidate Clinton.
Papadopoulos shared information about Russian ``dirt'' with people outside of the Campaign, and the Office investigated whether he also provided it to a Campaign official.
Papadopoulos and the Campaign officials with whom he interacted told the Office that they did not recall that Papadopoulos passed them the information.
Throughout the relevant period of time and for several months thereafter, Papadopoulos worked with Mifsud and two Russian nationals to arrange a meeting between the Campaign and the Russian government.
That meeting never came to pass.
\paragraph{Origins of Campaign Work}
In March 2016, Papadopoulos became a foreign policy advisor to the Trump Campaign.% 395
\footnote{\textit{A Transcript of Donald Trump's Meeting with the Washington Post Editorial Board}, Washington Post (Mar.~21, 2016).}
As early as the summer of 2015, he had sought a role as a policy advisor to the Campaign but, in a September~30, 2015 email, he was told that the Campaign was not hiring policy advisors.% 396
\footnote{7/15/15 LinkedIn Message, Papadopoulos to Lewandowski (6:57~a.m.); 9/30/15 Email, Glassner to Papadopoulos (7:42:21~a.m.).}
In late 2015, Papadopoulos obtained a paid position on the campaign of Republican presidential candidate Ben Carson.% 397
\footnote{Papadopoulos 8/10/17 302, at~2.}
Although Carson remained in the presidential race until early March 2016, Papadopoulos had stopped actively working for his campaign by early February 2016.% 398
\footnote{Papadopoulos 8/10/17 302, at~2; 2/4/16 Email, Papadopoulos to Idris.}
At that time, Papadopoulos reached out to a contact at the London Centre of International Law Practice (LCILP), which billed itself as a ``unique institution \dots\ comprising high-level professional international law practitioners, dedicated to the advancement of global legal knowledge and the practice of international law.''% 399
\footnote{London Centre of International Law Practice, at \url{https://www.lcilp.org/} (via \url{web.archive.org}).}
Papadopoulos said that he had finished his role with the Carson campaign and asked if LCILP was hiring.% 400
\footnote{2/4/16 Email, Papadopoulos to Idris.}
In early February, Papadopoulos agreed to join LCILP and arrived in London to begin work.% 401
\footnote{2/5/16 Email, Idris to Papadopoulos (6:11:25~p.m.);
2/6/16 Email, Idris to Papadopoulos (5:34:15~p.m.).}
As he was taking his position at LCILP, Papadopoulos contacted Trump campaign manager Corey Lewandowski via LinkedIn and emailed campaign official Michael Glassner about his interest in joining the Trump Campaign.% 402
\footnote{2/4/16 LinkedIn Message, Papadopoulos to Lewandowski (1:28~p.m.);
2/4/16 Email, Papadopoulos to Glassner (2:10:36~p.m.).}
On March~2, 2016, Papadopoulos sent Glassner another message reiterating his interest.% 403
\footnote{3/2/16 Email, Papadopoulos to Glassner (11:17:23~a.m.).}
Glassner passed along word of Papadopoulos's interest to another campaign official, Joy Lutes, who notified Papadopoulos by email that she had been told by Glassner to introduce Papadopoulos to Sam Clovis, the Trump Campaign's national co-chair and chief policy advisor.% 404
\footnote{3/2/16 Email, Lutes to Papadopoulos (10:08:15~p.m.).}
At the time of Papadopoulos's March~2 email, the media was criticizing the Trump Campaign for lack of experienced foreign policy or national security advisors within its ranks.% 405
\footnote{Clovis 10/3/17 302 (1 of 2), at~4.}
To address that issue, senior Campaign officials asked Clovis to put a foreign policy team together on short notice.% 406
\footnote{Clovis 10/3/17 302 (1 of 2), at~4.}
After receiving Papadopoulos's name from Lutes, Clovis performed a Google search on Papadopoulos, learned that he had worked at the Hudson Institute, and believed that he had credibility on energy issues.% 407
\footnote{\blackout{Grand Jury}; 3/3/16 Email, Lutes to Clovis \& Papadopoulos (6:05:47~p.m.).}
On March~3, 2016, Clovis arranged to speak with Papadopoulos by phone to discuss Papadopoulos joining the Campaign as a foreign policy advisor, and on March~6, 2016, the two spoke.% 408
\footnote{3/6/16 Email, Papadopoulos to Clovis (4:24:21~p.m.).}
Papadopoulos recalled that Russia was mentioned as a topic, and he understood from the conversation that Russia would be an important aspect of the Campaign's foreign policy.% 409
\footnote{Statement of Offense \P~4, \textit{United States~v. George Papadopoulos}, 1:17-cr-182 (D.D.C. Oct.~5,
2017), Doc.~19 (``\textit{Papadopoulos} Statement of Offense'').}
At the end of the conversation, Clovis offered Papadopoulos a role as a foreign policy advisor to the Campaign, and Papadopoulos accepted the offer.% 410
\footnote{Papadopoulos 8/10/17 302, at~2.}
\paragraph{Initial Russia-Related Contacts}
Approximately a week after signing on as a foreign policy advisor, Papadopoulos traveled to Rome, Italy, as part of his duties with LCILP.% 411
\footnote{Papadopoulos 8/10/17 302, at~2--3; \textit{Papadopoulos} Statement of Offense \P~5.}
The purpose of the trip was to meet officials affiliated with Link Campus University, a for-profit institution headed by a former Italian government official.% 412
\footnote{Papadopoulos 8/10/17 302, at~2--3;
Stephanie Kirchgaessner et~al., \textit{Joseph Mifsud: more questions than answers about mystery professor linked to Russia}, The Guardian (Oct.~31, 2017) (``Link Campus University \dots\ is headed by a former Italian interior minister named Vincenzo Scotti.'').}
During the visit, Papadopoulos was introduced to Joseph Mifsud.
Mifsud is a Maltese national who worked as a professor at the London Academy of Diplomacy in London, England.% 413
\footnote{\textit{Papadopoulos} Statement of Offense \P~5.}
Although Mifsud worked out of London and was also affiliated with LCILP, the encounter in Rome was the first time that Papadopoulos met him.% 414
\footnote{Papadopoulos 8/10/17 302, at~3.}
Mifsud maintained various Russian contacts while living in London, as described further below.
Among his contacts was \blackout{Investigative Technique},% 415
\footnote{\textit{See, e.g.} \blackout{Investigative Technique}.
\blackout{Harm to Ongoing Matters}}
a one-time employee of the IRA, the entity that carried out the Russian social media campaign (\textit{see} \hyperlink{section.1.2}{Volume~I, Section~II}, \textit{supra}).
In January and February 2016, Mifsud and \blackout{Investigative Technique} discussed \blackout{Investigative Technique} possibly meeting in Russia.
The investigation did not identify evidence of them meeting.
Later, in the spring of 2016, \blackout{Investigative Technique} was also in contact \blackout{Investigative Technique} that was linked to an employee of the Russian Ministry of Defense, and that account had overlapping contacts with a group of Russian military-controlled Facebook accounts that included accounts used to promote the DCLeaks releases in the course of the GRU's hack-and-release operations (\textit{see} \hyperlink{subsubsection.1.3.2.1}{Volume~I, Section~III.B.1}, \textit{supra}).
According to Papadopoulos, Mifsud at first seemed uninterested in Papadopoulos when they met in Rome.% 416
\footnote{\textit{Papadopoulos} Statement of Offense \P~5.}
After Papadopoulos informed Mifsud about his role in the Trump Campaign, however, Mifsud appeared to take greater interest in Papadopoulos.% 417
\footnote{\textit{Papadopoulos} Statement of Offense \P~5.}
The two discussed Mifsud's European and Russian contacts and had a general discussion about Russia; Mifsud also offered to introduce Papadopoulos to European leaders and others with contacts to the Russian government.% 418
\footnote{Papadopoulos 8/10/17 302, at~3; Papadopoulos 8/11/17 302, at~2.}
Papadopoulos told the Office that Mifsud's claim of substantial connections with Russian government officials interested Papadopoulos, who thought that such connections could increase his importance as a policy advisor to the Trump Campaign.% 419
\footnote{\textit{Papadopoulos} Statement of Offense \P~5.}
On March~17, 2016, Papadopoulos returned to London.% 420
\footnote{Papadopoulos 8/10/17 302, at~2.}
Four days later, candidate Trump publicly named him as a member of the foreign policy and national security advisory team chaired by Senator Jeff Sessions, describing Papadopoulos as ``an oil and energy consultant'' and an ``[e]xcellent guy.''% 421
\footnote{Phillip Rucker \& Robert Costa, \textit{Trump Questions Need for NATO, Outlines Noninterventionist Foreign Policy}, Washington Post (Mar.~21, 2016).}
On March~24, 2016, Papadopoulos met with Mifsud in London.% 422
\footnote{Papadopoulos 8/10/17 302, at~3; 3/24/16 Text Messages, Mifsud \& Papadopoulos.}
Mifsud was accompanied by a Russian female named Olga Polonskaya.
Mifsud introduced Polonskaya as a former student of his who had connections to Vladimir Putin.% 423
\footnote{Papadopoulos 8/10/17 302, at~3.}
Papadopoulos understood at the time that Polonskaya may have been Putin's niece but later learned that this was not true.% 424
\footnote{Papadopoulos 8/10/17 302, at~3;
Papadopoulos 2/10/17 302, at~2--3;
Papadopoulos Internet Search History (3/24/16) (revealing late-morning and early-afternoon searches on March~24, 2016 for ``putin's niece,'' ``olga putin,'' and ``russian president niece olga,'' among other terms).}
During the meeting, Polonskaya offered to help Papadopoulos establish contacts in Russia and stated that the Russian ambassador in London was a friend of hers.% 425
\footnote{Papadopoulos 8/10/17 302, at~3.}
Based on this interaction, Papadopoulos expected Mifsud and Polonskaya to introduce him to the Russian ambassador in London, but that did not occur.% 426
\footnote{\textit{Papadopoulos} Statement of Offense \P~8 n.1.}
Following his meeting with Mifsud, Papadopoulos sent an email to members of the Trump Campaign's foreign policy advisory team. The subject line of the message was ``Meeting with Russian leadership -- including Putin.''% 427
\footnote{3/24/16 Email, Papadopoulos to Page et~al. (8:48:21~a.m.).}
The message stated in pertinent part:
\begin{quote}
I just finished a very productive lunch with a good friend of mine, Joseph Mifsud, the director of the London Academy of Diplomacy -- who introduced me to both Putin's niece and the Russian Ambassador in London -- who also acts as the Deputy Foreign Minister.% 428
\footnote{Papadopoulos's statements to the Campaign were false.
As noted above, the woman he met was not Putin's niece, he had not met the Russian Ambassador in London, and the Ambassador did not also serve as Russia's Deputy Foreign Minister.}
The topic of the lunch was to arrange a meeting between us and the Russian leadership to discuss U.S.--Russia ties under President Trump.
They are keen to host us in a ``neutral'' city, or directly in Moscow.
They said the leadership, including Putin, is ready to meet with us and Mr.~Trump should there be interest.
Waiting for everyone's thoughts on moving forward with this very important issue.% 429
\footnote{3/24/16 Email, Papadopoulos to Page et~al.\ (8:48:21~a.m.).}
\end{quote}
Papadopoulos's message came at a time when Clovis perceived a shift in the Campaign's approach toward Russia---from one of engaging with Russia through the NATO framework and taking a strong stance on Russian aggression in Ukraine, \blackout{Grand Jury}% 430
\footnote{\blackout{Grand Jury}}
Clovis's response to Papadopoulos, however, did not reflect that shift.
Replying to Papadopoulos and the other members of the foreign policy advisory team copied on the initial email, Clovis wrote:
\begin{quote}
This is most informative.
Let me work it through the campaign.
No commitments until we see how this plays out.
My thought is that we probably should not go forward with any meetings with the Russians until we have had occasion to sit with our NATO allies, especially France, Germany and Great Britain.
We need to reassure our allies that we are not going to advance anything with Russia until we have everyone on the same page.
More thoughts later today.
Great work.% 431
\footnote{3/24/16 Email, Clovis to Papadopoulos et~al.\ (8:55:04~a.m.).}
\end{quote}
\paragraph{March~31 Foreign Policy Team Meeting}
The Campaign held a meeting of the foreign policy advisory team with Senator Sessions and candidate Trump approximately one week later, on March~31, 2016, in Washington, D.C.% 432
\footnote{Papadopoulos 8/10/17 302, at~4; Papadopoulos 8/11/17 302, at~3.}
The meeting---which was intended to generate press coverage for the Campaign% 433
\footnote{Sessions 1/17/18 302, at~16--17.}%
---took place at the Trump International Hotel.% 434
\footnote{Papadopoulos 8/10/17 302, at~4.}
Papadopoulos flew to Washington for the event.
At the meeting, Senator Sessions sat at one end of an oval table, while Trump sat at the other.
As reflected in the photograph below (which was posted to Trump's Instagram account), Papadopoulos sat between the two, two seats to Sessions's left[.]
\begin{figure}[t]
\vspace{-20pt}
\begin{center}
\includegraphics[width=5in]{images/p-86-foreign-policy-team.png}%
\end{center}
\vspace{-20pt}
\caption*{March~31, 2016 Meeting of Foreign Policy Team, with Papadopoulos (Fourth from Right of Candidate Trump}
\vspace{-10pt}
\label{fig:foreign-policy-team}
\end{figure}
During the meeting, each of the newly announced foreign policy advisors introduced themselves and briefly described their areas of experience or expertise.% 435
\footnote{Papadopoulos 8/10/17 302, at~4.}
Papadopoulos spoke about his previous work in the energy sector and then brought up a potential meeting with Russian officials.% 436
\footnote{Papadopoulos 8/10/17 302, at~4.}
Specifically, Papadopoulos told the group that he had learned through his contacts in London that Putin wanted to meet with candidate Trump and that these connections could help arrange that meeting.% 437
\footnote{\textit{Papadopoulos} Statement of Offense \P~9; \textit{see} Gordon 8/29/17 302, at~14; Carafano 9/12/17 302, at~2; Hoskins 9/14/17 302, at~1.}
Trump and Sessions both reacted to Papadopoulos's statement. Papadopoulos and Campaign advisor J.D.~Gordon---who told investigators in an interview that he had a ``crystal clear'' recollection of the meeting---have stated that Trump was interested in and receptive to the idea of a meeting with Putin.% 438
\footnote{Papadopoulos 8/10/17 302, at~4--5; Gordon 9/7/17 302, at~4--5.}
Papadopoulos understood Sessions to be similarly supportive of his efforts to arrange a meeting.% 439
\footnote{Papadopoulos 8/10/17 302, at~5; Papadopoulos 8/11/17 302, at~3.}
Gordon and two other attendees, however, recall that Sessions generally opposed the proposal, though they differ in their accounts of the concerns he voiced or the strength of the opposition he expressed.% 440
\footnote{Sessions 1/17/18 302, at~17; Gordon 9/7/17 302, at~5; Hoskins 9/14/17 302, at~1; Carafano 9/12/17 302, at~2.}
\paragraph{George Papadopoulos Learns That Russia Has ``Dirt'' in the Form of Clinton Emails}
Whatever Sessions's precise words at the March~31 meeting, Papadopoulos did not understand Sessions or anyone else in the Trump Campaign to have directed that he refrain from making further efforts to arrange a meeting between the Campaign and the Russian government.
To the contrary, Papadopoulos told the Office that he understood the Campaign to be supportive of his efforts to arrange such a meeting.% 441
\footnote{Papadopoulos 8/10/17 302, at~4--5; Papadopoulos 8/11/17 302, at~3; Papadopoulos 9/20/17 302, at~2.}
Accordingly, when he returned to London, Papadopoulos resumed those efforts.% 442
\footnote{\textit{Papadopoulos} Statement of Offense \P~10.}
Throughout April 2016, Papadopoulos continued to correspond with, meet with, and seek Russia contacts through Mifsud and, at times, Polonskaya.% 443
\footnote{\textit{Papadopoulos} Statement of Offense \P\P~10--15.}
For example, within a week of her initial March~24 meeting with him, Polonskaya attempted to send Papadopoulos a text message---which email exchanges show to have been drafted or edited by Mifsud---addressing Papadopoulos's ``wish to engage with the Russian Federation.''% 444
\footnote{3/29/16 Emails, Mifsud to Polonskaya (3:39~a.m. and 5:36~a.m.).}
When Papadopoulos learned from Mifsud that Polonskaya had tried to message him, he sent her an email seeking another meeting.% 445
\footnote{4/10/16 Email, Papadopoulos to Polonskaya (2:45:59~p.m.).}
Polonskaya responded the next day that she was ``back in St.~Petersburg'' but ``would be very pleased to support [Papadopoulos's] initiatives between our two countries'' and ``to meet [him] again.''% 446
\footnote{4/11/16 Email, Polonskaya to Papadopoulos (3:11:24~a.m.).}
Papadopoulos stated in reply that he thought ``a good step'' would be to introduce him to ``the Russian Ambassador in London,'' and that he would like to talk to the ambassador, ``or anyone else you recommend, about a potential foreign policy trip to Russia.''% 447
\footnote{4/11/16 Email, Papadopoulos to Polonskaya (9:21:56~a.m.).}
Mifsud, who had been copied on the email exchanges, replied on the morning of April~11, 2016.
He wrote, ``This is already been agreed.
I am flying to Moscow on the 18th for a Valdai meeting, plus other meetings at the Duma.
We will talk tomorrow.''% 448
\footnote{4/11/16 Email, Mifsud to Papadopoulos (11:43:53).}
The two bodies referenced by Mifsud are part of or associated with the Russian government: the Duma is a Russian legislative assembly,% 449
\footnote{\textit{Papadopoulos} Statement of Offense \P~10(c).}
while ``Valdai'' refers to the Valdai Discussion Club, a Moscow-based group that ``is close to Russia's foreign-policy establishment.''% 450
\footnote{Anton Troianovski, \textit{Putin Ally Warns of Arms Race as Russia Considers Response to U.S. Nuclear Stance}, Washington Post (Feb.~10, 2018).}
Papadopoulos thanked Mifsud and said that he would see him ``tomorrow.''% 451
\footnote{4/11/16 Email, Papadopoulos to Mifsud (11:51:53~a.m.).}
For her part, Polonskaya responded that she had ``already alerted my personal links to our conversation and your request,'' that ``we are all very excited [about] the possibility of a good relationship with Mr.~Trump,'' and that ``[t]he Russian Federation would love to welcome him once his candidature would be officially announced.''% 452
\footnote{4/12/16 Email, Polonskaya to Papadopoulos (4:47:06~a.m.).}
Papadopoulos's and Mifsud's mentions of seeing each other ``tomorrow'' referenced a meeting that the two had scheduled for the next morning, April~12, 2016, at the Andaz Hotel in London.
Papadopoulos acknowledged the meeting during interviews with the Office,% 453
\footnote{Papadopoulos 9/19/17 302, at~7.}
and records from Papadopoulos's UK cellphone and his internet-search history all indicate that the meeting took place.% 454
\footnote{4/12/16 Email, Mifsud to Papadopoulos (5:44:39~a.m.) (forwarding Libya-related document); 4/12/16 Email, Mifsud to Papadopoulos \& Obaid (10:28:20~a.m.); Papadopoulos Internet Search History (Apr.~11, 2016 10:56:49~p.m.) (search for ``andaz hotel liverpool street''); 4/12/16 Text Messages, Mifsud \& Papadopoulos.}
Following the meeting, Mifsud traveled as planned to Moscow.% 455
\footnote{\textit{See, e.g.}, 4/18/16 Email, Mifsud to Papadopoulos (8:04:54~a.m.).}
On April~18, 2016, while in Russia, Mifsud introduced Papadopoulos over email to Ivan Timofeev, a member of the Russian International Affairs Council (RIAC).% 456
\footnote{Papadopoulos 8/10/17 302, at~5.}
Mifsud had described Timofeev as having connections with the Russian Ministry of Foreign Affairs (MFA),% 457
\footnote{\textit{Papadopoulos} Statement of Offense \P~11.}
the executive entity in Russia responsible for Russian foreign relations.% 458
\footnote{During the campaign period, Papadopoulos connected over LinkedIn with several MFA-affiliated individuals in addition to Timofeev.
On April~25, 2016, he connected with Dmitry Andreyko, publicly identified as a First Secretary at the Russian Embassy in Ireland.
In July 2016, he connected with Yuriy Melnik, the spokesperson for the Russian Embassy in Washington and with Alexey Krasilnikov, publicly identified as a counselor with the MFA\null.
And on September~16, 2016, he connected with Sergei Nalobin, also identified as an MFA official.
\textit{See} Papadopoulos LinkedIn Connections \blackout{Investigative Technique}}
Over the next several weeks, Papadopoulos and Timofeev had multiple conversations over Skype and email about setting ``the groundwork'' for a ``potential'' meeting between the Campaign and Russian government officials.% 459
\footnote{\textit{Papadopoulos} Statement of Offense \P~11.}
Papadopoulos told the Office that, on one Skype call, he believed that his conversation with Timofeev was being monitored or supervised by an unknown third party, because Timofeev spoke in an official manner and Papadopoulos heard odd noises on the line.% 460
\footnote{Papadopoulos 8/10/17 302, at~5; Papadopoulos 9/19/17 302, at~10.}
Timofeev also told Papadopoulos in an April~25, 2016 email that he had just spoken ``to Igor Ivanov[,] the President of RIAC and former Foreign Minister of Russia,'' and conveyed Ivanov's advice about how best to arrange a ``Moscow visit.''% 461
\footnote{4/25/16 Email, Timofeev to Papadopoulos (8:16:35~a.m.).}
After a stop in Rome, Mifsud returned to England on April~25, 2016.% 462
\footnote{4/22/16 Email, Mifsud to Papadopoulos (12:41:01~a.m.).}
The next day, Papadopoulos met Mifsud for breakfast at the Andaz Hotel (the same location as their last meeting).% 463
\footnote{\textit{Papadopoulos} Statement of Offense \P~14;
4/25/16 Text Messages, Mifsud \& Papadopoulos.}
During that meeting, Mifsud told Papadopoulos that he had met with high-level Russian government officials during his recent trip to Moscow.
Mifsud also said that, on the trip, he learned that the Russians had obtained ``dirt'' on candidate Hillary Clinton.
As Papadopoulos later stated to the FBI, Mifsud said that the ``dirt'' was in the form of ``emails of Clinton,'' and that they ``have thousands of emails.''% 464
\footnote{\textit{Papadopoulos} Statement of Offense \P~14.}
On May~6, 2016, 10 days after that meeting with Mifsud, Papadopoulos suggested to a representative of a foreign government that the Trump Campaign had received indications from the Russian government that it could assist the Campaign through the anonymous release of information that would be damaging to Hillary Clinton.% 465
\footnote{\label{fnFourSixFive}This information is contained in the FBI case-opening document and related materials.
\sout{The information is law enforcement sensitive (LES) and must be treated accordingly in any external dissemination.}
The foreign government conveyed this information to the U.S. government on July~26, 2016, a few days after WikiLeaks's release of Clinton-related emails.
The FBI opened its investigation of potential coordination between Russia and the Trump Campaign a few days later based on the information.}
\paragraph{Russia-Related Communications With The Campaign}
While he was discussing with his foreign contacts a potential meeting of campaign officials with Russian government officials, Papadopoulos kept campaign officials apprised of his efforts.
On April~25, 2016, the day before Mifsud told Papadopoulos about the emails, Papadopoulos wrote to senior policy advisor Stephen Miller that ``[t]he Russian government has an open invitation by Putin for Mr.~Trump to meet him when he is ready,'' and that ``[t]he advantage of being in London is that these governments tend to speak a bit more openly in `neutral' cities.''% 466
\footnote{4/25/16 Email, Papadopoulos to S.~Miller (8:12:44~p.m.).}
On April~27, 2016, after his meeting with Mifsud, Papadopoulos wrote a second message to Miller stating that ``some interesting messages [were] coming in from Moscow about a trip when the time is right.''% 467
\footnote{4/27/16 Email, Papadopoulos to S.~Miller (6:55:58~p.m.).}
The same day, Papadopoulos sent a similar email to campaign manager Corey Lewandowski, telling Lewandowski that Papadopoulos had ``been receiving a lot of calls over the last month about Putin wanting to host [Trump] and the team when the time is right.''% 468
\footnote{4/27/16 Email, Papadopoulos to Lewandowski (7:15:14~p.m.).}
Papadopoulos's Russia-related communications with Campaign officials continued throughout the spring and summer of 2016.
On May~4, 2016, he forwarded to Lewandowski an email from Timofeev raising the possibility of a meeting in Moscow, asking Lewandowski whether that was ``something we want to move forward with.''% 469
\footnote{5/4/16 Email, Papadopoulos to Lewandowski (8:14:49~a.m.).}
The next day, Papadopoulos forwarded the same Timofeev email to Sam Clovis, adding to the top of the email ``Russia update.''% 470
\footnote{5/5/16 Email, Papadopoulos to Clovis (7:15:21~p.m.).}
He included the same email in a May~21, 2016 message to senior Campaign official Paul Manafort, under the subject line ``Request from Russia to meet Mr.~Trump,'' stating that ``Russia has been eager to meet Mr.~Trump for quite sometime and have been reaching out to me to discuss.''% 471
\footnote{5/21/16 Email, Papadopoulos to Manafort (2:30:14~p.m.).}
Manafort forwarded the message to another Campaign official, without including Papadopoulos, and stated: ``Let[']s discuss.
We need someone to communicate that [Trump] is not doing these trips.
It should be someone low level in the Campaign so as not to send any signal.''% 472
\footnote{\textit{Papadopoulos} Statement of Offense \P~19 n.2.}
On June~1, 2016, Papadopoulos replied to an earlier email chain with Lewandowski about a Russia visit, asking if Lewandowski ``want[ed] to have a call about this topic'' and whether ``we were following up with it.''% 473
\footnote{6/1/16 Email, Papadopoulos to Lewandowski (3:08:18~p.m.).}
After Lewandowski told Papadopoulos to ``connect with'' Clovis because he was ``running point,'' Papadopoulos emailed Clovis that ``the Russian MFA'' was asking him ``if Mr.~Trump is interested in visiting Russia at some point.''% 474
\footnote{6/1/16 Email, Lewandowski to Papadopoulos (3:20:03~p.m.);
6/1/16 Email, Papadopoulos to Clovis (3:29:14~p.m.).}
Papadopoulos wrote in an email that he ``[w]anted to pass this info along to you for you to decide what's best to do with it and what message I should send (or to ignore).''% 475
\footnote{6/1/16 Email, Papadopoulos to Clovis (3:29:14~p.m.).
Papadopoulos's email coincided in time with another message to Clovis suggesting a Trump--Putin meeting.
First, on May~15, 2016, David Klein---a distant relative of then-Trump Organization lawyer Jason Greenblatt---emailed Clovis about a potential Campaign meeting with Berel Lazar, the Chief Rabbi of Russia.
The email stated that Klein had contacted Lazar in February about a possible Trump--Putin meeting and that Lazar was ``a very close confidante of Putin.''
DJTFP00011547 (5/15/16 Email, Klein to Clovis (5:45:24~p.m.)).
The investigation did not find evidence that Clovis responded to Klein's email or that any further contacts of significance came out of Klein's subsequent meeting with Greenblatt and Rabbi Lazar at Trump Tower.
Klein 8/30/18 302, at~2.}
After several email and Skype exchanges with Timofeev,% 476
\footnote{\textit{Papadopoulos} Statement of Offense \P~21(a).}
Papadopoulos sent one more email to Lewandowski on June~19, 2016, Lewandowski's last day as campaign manager.% 477
\footnote{\blackout{Grand Jury}}
The email stated that ``[t]he Russian ministry of foreign affairs'' had contacted him and asked whether, if Mr.~Trump could not travel to Russia, a campaign representative such as Papadopoulos could attend meetings.% 478
\footnote{6/19/16 Email, Papadopoulos to Lewandowski (1:11:11~p.m.).}
Papadopoulos told Lewandowski that he was ``willing to make the trip off the record if it's in the interest of Mr.~Trump and the campaign to meet specific people.''% 479
\footnote{6/19/16 Email, Papadopoulos to Lewandowski (1:11:11~p.m.).}
Following Lewandowski's departure from the Campaign, Papadopoulos communicated with Clovis and Walid Phares, another member of the foreign policy advisory team, about an off-the-record meeting between the Campaign and Russian government officials or with Papadopoulos's other Russia connections, Mifsud and Timofeev.% 480
\footnote{\textit{Papadopoulos} Statement of Offense \P~21;
7/14/16 Email, Papadopoulos to Timofeev (11:57:24~p.m.);
7/15/16 Email, Papadopoulos to Mifsud;
7/27/16 Email, Papadopoulos to Mifsud (2:14:18~p.m.).}
Papadopoulos also interacted directly with Clovis and Phares in connection with the summit of the Transatlantic Parliamentary Group on Counterterrorism (TAG), a group for which Phares was co-secretary general.% 481
\footnote{Papadopoulos 9/19/17 302, at~16--17;
\textit{9th TAG Summit in Washington DC}, Transatlantic Parliament Group on Counter Terrorism.}
On July~16, 2016, Papadopoulos attended the TAG summit in Washington, D.C., where he sat next to Clovis (as reflected in the photograph below).% 482
\footnote{\textit{9th TAG Summit in Washington DC}, Transatlantic Parliament Group on Counter Terrorism.}
\begin{figure}[t]
\vspace{-20pt}
\begin{center}
\includegraphics[width=5in]{images/p-91-papadopolous-clovis.png}%
\end{center}
\vspace{-20pt}
\caption*{George Papadopoulos (far right) and Sam Clovis (second from right)}
\vspace{-10pt}
\label{fig:papadopolous-clovis}
\end{figure}
Although Clovis claimed to have no recollection of attending the TAG summit,% 483
\footnote{\blackout{Grand Jury}}
Papadopoulos remembered discussing Russia and a foreign policy trip with Clovis and Phares during the event.% 484
\footnote{Papadopoulos 9/19/17 302, at~16--17.}
Papadopoulos's recollection is consistent with emails sent before and after the TAG summit.
The pre-summit messages included a July~11, 2016 email in which Phares suggested meeting Papadopoulos the day after the summit to chat,% 485
\footnote{7/11/16 Email, Phares to Papadopoulos.}
and a July~12 message in the same chain in which Phares advised Papadopoulos that other summit attendees ``are very nervous about Russia. So be aware.''% 486
\footnote{7/12/16 Email, Phares to Papadopoulos (14:52:29).}
Ten days after the summit, Papadopoulos sent an email to Mifsud listing Phares and Clovis as other ``participants'' in a potential meeting at the London Academy of Diplomacy.% 487
\footnote{7/27/16 Email, Papadopoulos to Mifsud (14:14:18).}
Finally, Papadopoulos's recollection is also consistent with handwritten notes from a journal that he kept at the time.% 488
\footnote{Papadopoulos 9/20/17 302, at~3.}
Those notes, which are reprinted in part below, appear to refer to potential September 2016 meetings in London with representatives of the ``office of Putin,'' and suggest that Phares, Clovis, and Papadopoulos (``Walid/Sam me'') would attend without the official backing of the Campaign (``no official letter/no message from Trump'').% 489
\footnote{Papadopoulos declined to assist in deciphering his notes, telling investigators that he could not read his own handwriting from the journal.
Papadopoulos 9/19/17 302, at~21.
The notes, however, appear to read as listed in the column to the left of the image above.}
\begin{wrapfigure}{tr}{3.1in}
\vspace{-20pt}
\begin{center}
\includegraphics[width=3in]{images/p-92-papadopolous-notes.png}%
\end{center}
\vspace{-20pt}
\caption*{}
\vspace{-10pt}
\label{fig:papadopolous-notes}
\end{wrapfigure}
\begin{quote}
September:
Have an exploratory meeting \sout{to} or lose.
In September -- if allowed they will blast Mr.~Trump.
We want the meeting in London/England
Walid/Sam me
No official letter/no message from Trump
They are talking to us.
-It is a lot of risk.
-Office of Putin.
-Explore: we are a campaign.
off Israel! EGYPT
Willingness to meet the FM sp with Walid/Sam
-FM coming
-Useful to have a session with him.
\end{quote}
Later communications indicate that Clovis determined that he (Clovis) could not travel.
On August~15, 2016, Papadopoulos emailed Clovis that he had received requests from multiple foreign governments, ``even Russia[],'' for ``closed door workshops/consultations abroad,'' and asked whether there was still interest for Clovis, Phares, and Papadopoulos ``to go on that trip.''% 490
\footnote{8/15/16 Email, Papadopoulos to Clovis (11:59:07~a.m.).}
Clovis copied Phares on his response, which said that he could not ``travel before the election'' but that he ``would encourage [Papadopoulos] and Walid to make the trips, if it is feasible.''% 491
\footnote{8/15/16 Email, Clovis to Papadopoulos (12:01:45~p.m.).}
Papadopoulos was dismissed from the Trump Campaign in early October 2016, after an interview he gave to the Russian news agency Interfax generated adverse publicity.% 492
\footnote{\textit{George Papadopoulos: Sanctions Have Done Little More Than to Turn Russia Towards China}, Interfax (Sept.~30, 2016).}
\paragraph{Trump Campaign Knowledge of ``Dirt''}
Papadopoulos admitted telling at least one individual outside of the Campaign---specifically, the then-Greek foreign minister---about Russia's obtaining Clinton-related emails.% 493
\footnote{Papadopoulos 9/19/17 302, at~14--15;
Def.~Sent.~Mem., United States~v.\ George Papadopoulos, 1:17-cr-182 (D.D.C. Aug.~31, 2018), Doc.~45.}
In addition, a different foreign government informed the FBI that, 10 days after meeting with Mifsud in late April 2016, Papadopoulos suggested that the Trump Campaign had received indications from the Russian government that it could assist the Campaign through the anonymous release of information that would be damaging to Hillary Clinton.% 494
\footnote{\textit{See} footnote \ref{fnFourSixFive} of \hyperlink{paragraph.1.4.1.2.4}{Volume~I, Section~IV.A.2.d}, \textit{supra}.}
(This conversation occurred after the GRU spearphished Clinton Campaign chairman John Podesta and stole his emails, and the GRU hacked into the DCCC and DNC, \textit{see} \hyperlink{subsection.1.3.1}{Volume~I, Sections~III.A} \&~\hyperlink{subsection.1.3.2}{III.B}, \textit{supra}.)
Such disclosures raised questions about whether Papadopoulos informed any Trump Campaign official about the emails.
When interviewed, Papadopoulos and the Campaign officials who interacted with him told the Office that they could not recall Papadopoulos's sharing the information that Russia had obtained ``dirt'' on candidate Clinton in the form of emails or that Russia could assist the Campaign through the anonymous release of information about Clinton.
Papadopoulos stated that he could not clearly recall having told anyone on the Campaign anyone on the Campaign and wavered about whether he accurately remembered an incident in which Clovis had been upset after hearing Papadopoulos tell Clovis that Papadopoulos thought ``they have her emails.''% 495
\footnote{Papadopoulos 8/10/17 302, at~5;
Papadopoulos 8/11/17 302, at~5;
Papadopoulos 9/20/17 302, at~2.}
The Campaign officials who interacted or corresponded with Papadopoulos have similarly stated, with varying degrees of certainty, that he did not tell them.
Senior policy advisor Stephen Miller, for example, did not remember hearing anything from Papadopoulos or Clovis about Russia having emails of or dirt on candidate Clinton.% 496
\footnote{S. Miller 12/14/17 302, at~10.}
Clovis stated that he did not recall anyone, including Papadopoulos, having given him non-public information that a foreign government might be in possession of material damaging to Hillary Clinton.% 497
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 498
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 499
\footnote{\blackout{Grand Jury}}
No documentary evidence, and nothing in the email accounts or other communications facilities reviewed by the Office, shows that Papadopoulos shared this information with the Campaign.
\paragraph{Additional George Papadopoulos Contact}
The Office investigated another Russia-related contact with Papadopoulos.
The Office was not fully able to explore the contact because the individual at issue---Sergei Millian---remained out of the country since the inception of our investigation and declined to meet with members of the Office despite our repeated efforts to obtain an interview.
Papadopoulos first connected with Millian via LinkedIn on July~15, 2016, shortly after Papadopoulos had attended the TAG Summit with Clovis.% 500
\footnote{7/15/16 LinkedIn Message, Millian to Papadopoulos.}
Millian, an American citizen who is a native of Belarus, introduced himself ``as president of [the] New York-based Russian American Chamber of Commerce,'' and claimed that through that position he had ``insider knowledge and direct access to the top hierarchy in Russian politics.''% 501
\footnote{7/15/16 LinkedIn Message, Millian to Papadopoulos.}
Papadopoulos asked Timofeev whether he had heard of Millian.% 502
\footnote{7/22/16 Facebook Message, Papadopoulos to Timofeev (7:40:23~p.m.);
7/26/16 Facebook Message, Papadopoulos to Timofeev (3:08:57~p.m.).}
Although Timofeev said no,% 503
\footnote{7/23/16 Facebook Message, Timofeev to Papadopoulos (4:31:37~a.m.);
7/26/16 Facebook Message, Timofeev to Papadopoulos (3:37:16~p.m.).}
Papadopoulos met Millian in New York City.% 504
\footnote{7/16/16 Text Messages, Papadopoulos \& Millian (7:55:43~p.m.).}
The meetings took place on July~30 and August~1, 2016.% 505
\footnote{7/30/16 Text Messages, Papadopoulos \& Millian (5:38 \& 6:05~p.m.);
7/31/16 Text Messages, Millian \& Papadopoulos (3:48 \& 4:18~p.m.);
8/1/16 Text Message, Millian to Papadopoulos (8:19~p.m.).}
Afterwards, Millian invited Papadopoulos to attend---and potentially speak at---two international energy conferences, including one that was to be held in Moscow in September 2016.% 506
\footnote{8/2/16 Text Messages, Millian \& Papadopoulos (3:04 \& 3:05~p.m.);
8/3/16 Facebook Messages, Papadopoulos \& Millian (4:07:37~a.m. \& 1:11:58~p.m.).}
Papadopoulos ultimately did not attend either conference.
On July~31, 2016, following his first in-person meeting with Millian, Papadopoulos emailed Trump Campaign official Bo Denysyk to say that he had been contacted ``by some leaders of Russian-American voters here in the US about their interest in voting for Mr.~Trump,'' and to ask whether he should ``put you in touch with their group (US--Russia chamber of commerce).''% 507
\footnote{7/31/16 Email, Papadopoulos to Denysyk (12:29:59~p.m.).}
Denysyk thanked Papadopoulos ``for taking the initiative,'' but asked him to ``hold off with outreach to Russian-Americans'' because ``too many articles'' had already portrayed the Campaign, then-campaign chairman Paul Manafort, and candidate Trump as ``being pro-Russian.''% 508
\footnote{7/31/16 Email, Denysyk to Papadopoulos (21:54:52).}
On August~23, 2016, Millian sent a Facebook message to Papadopoulos promising that he would ``share with you a disruptive technology that might be instrumental in your political work for the campaign.''% 509
\footnote{8/23/16 Facebook Message, Millian to Papadopoulos (2:55:36~a.m.).}
Papadopoulos claimed to have no recollection of this matter.% 510
\footnote{Papadopoulos 9/20/17 302, at~2.}
On November~9, 2016, shortly after the election, Papadopoulos arranged to meet Millian in Chicago to discuss business opportunities, including potential work with Russian ``billionaires who are not under sanctions.''% 511
\footnote{11/10/16 Facebook Message, Millian to Papadopoulos (9:35:05~p.m.).}
The meeting took place on November~14, 2016, at the Trump Hotel and Tower in Chicago.% 512
\footnote{11/14/16 Facebook Message, Millian to Papadopoulos (1:32:11~a.m.).}
According to Papadopoulos, the two men discussed partnering on business deals, but Papadopoulos perceived that Millian's attitude toward him changed when Papadopoulos stated that he was only pursuing private-sector opportunities and was not interested in a job in the Administration.% 513
\footnote{Papadopoulos 9/19/17 302, at~19.}
The two remained in contact, however, and had extended online discussions about possible business opportunities in Russia.% 514
\footnote{\textit{E.g.}, 11/29/16 Facebook Messages, Papadopoulos \& Millian (5:09--5:11~p.m.);
12/7/16 Facebook Message, Millian to Papadopoulos (5:10:54~p.m.).}
The two also arranged to meet at a Washington, D.C. bar when both attended Trump's inauguration in late January 2017.% 515
\footnote{1/20/17 Facebook Messages, Papadopoulos \& Millian (4:37--4:39~a.m.).}
\subsubsection{Carter Page}
Carter Page worked for the rump Campaign from January 2016 to September 2016.
He was formally and publicly announced as a foreign policy advisor by the candidate in March 2016.% 516
\footnote{Page was interviewed by the FBI during five meetings in March 2017, before the Special Counsel's appointment.
\blackout{Grand Jury}}
Page had lived and worked in Russia, and he had been approached by Russian intelligence officers several years before he volunteered for the Trump Campaign.
During his time with the Campaign, Page advocated pro-Russia foreign policy positions and traveled to Moscow in his personal capacity.
Russian intelligence officials had formed relationships with Page in 2008 and~2013 and Russian officials may have focused on Page in 2016 because of his affiliation with the Campaign.
However, the investigation did not establish that Page coordinated with the Russian government in its efforts to interfere with the 2016 presidential election.
\paragraph{Background}
Before he began working for the Campaign in January 2016, Page had substantial prior experience studying Russian policy issues and living and working in Moscow.
From 2004 to 2007, Page was the deputy branch manager of Merrill Lynch's Moscow office.% 517
\footnote{\textit{Testimony of Carter Page, Hearing Before the U.S. House of Representatives, Permanent Select Committee on Intelligence,} 115th Cong.~40 (Nov.~2, 2017) (exhibit).}
There, he worked on transactions involving the Russian energy company Gazprom and came to know Gazprom's deputy chief financial officer, Sergey Yatsenko.% 518
\footnote{Page 3/30/17 302, at~10.}
In 2008, Page founded Global Energy Capital LLC (GEC), an investment management and advisory firm focused on the energy sector in emerging markets.% 519
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 520
\footnote{\blackout{Grand Jury}}
The company otherwise had no sources of income, and Page was forced to draw down his life savings to support himself and pursue his business venture.% 521
\footnote{\blackout{Grand Jury}}
Page asked Yatsenko to work with him at GEC as a senior advisor on a contingency basis,
\blackout{Grand Jury}% 522
\footnote{Page 3/30/17 302, at~10; \blackout{Grand Jury}}
In 2008, Page met Alexander Bulatov, a Russian government official who worked at the Russian Consulate in New York.% 523
\footnote{\blackout{Grand Jury}}
Page later learned that Bulatov was a Russian intelligence officer,
\blackout{Grand Jury}% 524
\footnote{\blackout{Grand Jury}}
In 2013, Victor Podobnyy, another Russian intelligence officer working covertly in the United States under diplomatic cover, formed a relationship with Page.% 525
\footnote{\blackout{Grand Jury}; Complaint \P\P~22, 24, 32, \textit{United States~v.\ Buryakov}, 1:15-mj-215 (S.D.N.Y. Jan.~23, 2015), Doc.~1 ``\textit{Buryakov} Complaint'').}
Podobnyy met Page at an energy symposium in New York City and began exchanging emails with him.% 526
\footnote{\textit{Buryakov} Complaint \P~34.}
Podobnyy and Page also met in person on multiple occasions, during which Page offered his outlook on the future of the energy industry and provided documents to Podobnyy about the energy business.% 527
\footnote{\textit{Buryakov} Complaint \P~34.}
In a recorded conversation on April~8, 2013, Podobnyy told another intelligence officer that Page was interested in business opportunities in Russia.% 528
\footnote{\textit{Buryakov} Complaint \P~32.}
In Podobnyy's words, Page ``got hooked on Gazprom thinking that if they have a project, he could \dots\ rise up. Maybe he can\dots. [I]t's obvious that he wants to earn lots of money.''% 529
\footnote{\textit{Buryakov} Complaint.}
Podobnyy said that he had led Page on by ``feed[ing] him empty promises'' that Podobnyy would use his Russian business connections to help Page.% 530
\footnote{\textit{Buryakov} Complaint.}
Podobnyy told the other intelligence officer that his method of recruiting foreign sources was to promise them favors and then discard them once he obtained relevant information from them.% 531
\footnote{\textit{Buryakov} Complaint.}
In 2015, Podobnyy and two other Russian intelligence officers were charged with conspiracy to act as an unregistered agent of a foreign government.% 532
\footnote{\textit{See Buryakov} Complaint;
\textit{see also} Indictment, \textit{United States~v.\ Buryakov}, 1:15-cr-73 (S.D.N.Y. Feb.~9, 2015), Doc.~10;
\blackout{Grand Jury}}
The criminal complaint detailed Podobnyy's interactions with and conversations about Page, who was identified only as ``\hbox{Male-1}.''% 533
\footnote{\textit{Buryakov} Complaint \P\P~32--34; \blackout{Grand Jury}}
Based on the criminal complaint's description of the interactions, Page was aware that he was the individual described as ``\hbox{Male-1}.''% 534
\footnote{\blackout{Grand Jury}}
Page later spoke with a Russian government official at the United Nations General Assembly and identified himself so that the official would understand he was ``\hbox{Male-1}'' from the Podobnyy complaint. % 535
\footnote{Page 3/16/17 302, at~4; \blackout{Grand Jury}}
Page told the official that he ``didn't do anything''
\blackout{Grand Jury}% 536
\footnote{Page 3/16/17 302, at~4; \blackout{Grand Jury}}
In interviews with the FBI before the Office's opening, Page acknowledged that he understood that the individuals he had associated with were members of the Russian intelligence services, but he stated that he had only provided immaterial non-public information to them and that he did not view this relationship as a backchannel.% 537
\footnote{Page 3/30/17 302, at~6; Page 3/31/17 302, at~1.}
Page told investigating agents that ``the more immaterial non-public information I give them, the better for this country.''% 538
\footnote{Page 3/31/17 302, at~1.}
\paragraph{Origins of and Early Campaign Work}
In January 2016, Page began volunteering on an informal, unpaid basis for the Trump Campaign after Ed Cox, a state Republican Party official, introduced Page to Trump Campaign officials.% 539
\footnote{Page 3/16/17 302, at~1; \blackout{Grand Jury}}
Page told the Office that his goal in working on the Campaign was to help candidate Trump improve relations with Russia.% 540
\footnote{Page 3/10/17 302, at~2.}
To that end, Page emailed Campaign officials offering his thoughts on U.S.--Russia relations, prepared talking points and briefing memos on Russia, and proposed that candidate Trump meet with President Vladimir Putin in Moscow.% 541
\footnote{\textit{See, e.g.}, 1/30/16 Email, Page to Glassner et~al.;
3/17/16 Email, Page to Clovis (attaching a ``President's Daily Brief\thinspace'' prepared by Page that discussed the ``severe degradation of U.S.--Russia relations following Washington's meddling'' in Ukraine); \blackout{Grand Jury}}
In communications with Campaign officials, Page also repeatedly touted his high-level contacts in Russia and his ability to forge connections between candidate Trump and senior Russian governmental officials.
For example, on January~30, 2016, Page sent an email to senior Campaign officials stating that he had ``spent the past week in Europe and ha[d] been in discussions with some individuals with close ties to the Kremlin'' who recognized that Trump could have a ``game-changing effect \dots\ in bringing the end of the new Cold War.''% 542
\footnote{1/30/16 Email, Page to Glassner et~al.}
The email stated that ``[t]hrough [his] discussions with these high level contacts,'' Page believed that ``a direct meeting in Moscow between Mr[.] Trump and Putin could be arranged.''% 543
\footnote{1/30/16 Email, Page to Glassner et~al.}
Page closed the email by criticizing U.S. sanctions on Russia.% 544
\footnote{1/30/16 Email, Page to Glassner et~al.}
\blackout{Grand Jury}% 545
\footnote{\blackout{Grand Jury}}
On March~21, 2016, candidate Trump formally and publicly identified Page as a member of his foreign policy team to advise on Russia and the energy sector.% 546
\footnote{\textit{A Transcript of Donald Trump's Meeting with the Washington Post Editorial Board}, Washington Post (Mar.~21, 2016); \blackout{Grand Jury}}
Over the next several months, Page continued providing policy-related work product to Campaign officials.
For example, in April 2016, Page provided feedback on an outline for a foreign policy speech that the candidate gave at the Mayflower Hotel,% 547
\footnote{\blackout{Grand Jury}}
\textit{see} \hyperlink{subsubsection.1.4.1.4}{Volume~I, Section~IV.A.4}, \textit{infra}.
In May~2016, Page prepared an outline of an energy policy speech for the Campaign and then traveled to Bismarck, North Dakota, to watch the candidate deliver the speech.% 548
\footnote{\blackout{Grand Jury}}
Chief policy advisor Sam Clovis expressed appreciation for Page's work and praised his work to other Campaign officials.% 549
\footnote{\textit{See, e.g.}, 3/28/16 Email, Clovis to Lewandowski et~al.\
(forwarding notes prepared by Page and stating, ``I wanted to let you know the type of work some of our advisors are capable of.'').}
\paragraph{Carter Page's July 2016 Trip To Moscow}
Page's affiliation with the Trump Campaign took on a higher profile and drew the attention of Russian officials after the candidate named him a foreign policy advisor.
As a result, in late April 2016, Page was invited to give a speech at the July 2016 commencement ceremony at the New Economic School (NES) in Moscow.% 550
\footnote{Page 3/16/17 302, at~2--3; Page 3/10/17 302, at~3.}
The NES commencement ceremony generally featured high-profile speakers; for example, President Barack Obama delivered a commencement address at the school in 2009.% 551
\footnote{S. Weber 7/28/17 302, at~3.}
NES officials told the Office that the interest in inviting Page to speak at NES was based entirely on his status as a Trump Campaign advisor who served as the candidate's Russia expert.% 552
\footnote{Y. Weber 6/1/17 302, at~4--5;
S. Weber 7/28/17 302, at~3.}
Andrej Krickovic, an associate of Page's and assistant professor at the Higher School of Economics in Russia, recommended that NES rector Shlomo Weber invite Page to give the commencement address based on his connection to the Trump Campaign.% 553
\footnote{\textit{See} Y[.]~Weber 6/1/17 302, at~4;
S. Weber 7/28/17 302, at~3.}
Denis Klimentov, an employee of NES, said that when Russians learned of Page's involvement in the Trump Campaign in March 2016, the excitement was palpable.% 554
\footnote{De[.]~Klimentov 6/9/17 302, at~2.}
Weber recalled that in summer 2016 there was substantial interest in the Trump Campaign in Moscow, and he felt that bringing a member of the Campaign to the school would be beneficial.% 555
\footnote{S. Weber 7/28/17 302, at~3.}
Page was eager to accept the invitation to speak at NES, and he sought approval from Trump Campaign officials to make the trip to Russia.% 556
\footnote{\textit{See} 5/16/16 Email, Page to Phares et~al.\
(referring to submission of a ``campaign advisor request form'').}
On May~16, 2016, while that request was still under consideration, Page emailed Clovis, J.D.~Gordon, and Walid Phares and suggested that candidate Trump take his place speaking at the commencement ceremony in Moscow.% 557
\footnote{\blackout{Grand Jury}; 5/16/16 Email, Page to Phares et~al.}
On June~19, 2016, Page followed up again to request approval to speak at the NES event and to reiterate that NES ``would love to have Mr.~Trump speak at this annual celebration'' in Page's place.% 558
\footnote{6/19/16 Email, Page to Gordon et~al.}
Campaign manager Corey Lewandowski responded the same day, saying, ``If you want to do this, it would be out side [sic] of your role with the DJT for President campaign. I am certain Mr.~Trump will not be able to attend.% 559
\footnote{6/19/16 Email, Lewandowski to Page et~al.}
In early July 2016, Page traveled to Russia for the NES events.
On July~5, 2016, Denis Klimentov, copying his brother, Dmitri Klimentov,% 560
\footnote{Dmitri Klimentov is a New York-based public relations consultant.}
emailed Maria Zakharova, the Director of the Russian Ministry of Foreign Affairs' Information and Press Department, about Page's visit and his connection to the Trump Campaign.% 561
\footnote{7/5/16 Email, Klimentov to Zakharova (translated).}
Denis Klimentov said in the email that he wanted to draw the Russian government's attention to Page's visit in Moscow.% 562
\footnote{7/5/16 Email, Klimentov to Zakharova (translated).}
His message to Zakharova continued: ``Page is Trump's adviser on foreign policy.
He is a known businessman; he used to work in Russia\dots.
If you have any questions, I will be happy to help contact him.''% 563
\footnote{7/5/16 Email, Klimentov to Zakharova (translated).}
Dmitri Klimentov then contacted Russian Press Secretary Dmitry Peskov about Page's visit to see if Peskov wanted to introduce Page to any Russian government officials.% 564
\footnote{De.~Klimentov 11/27/18 302, at~1--2.}
The following day, Peskov responded to what appears to have been the same Denis Klimentov--Zakharova email thread.
Peskov wrote, ``I have read about [Page].
Specialists say that he is far from being the main one.
So I better not initiate a meeting in the Kremlin.''% 565
\footnote{7/6/16 Email, Peskov to Klimentov (translated).}
On July~7, 2016, Page delivered the first of his two speeches in Moscow at NES.% 566
\footnote{Page 3/10/17 302, at~3.}
In the speech, Page criticized the U.S. government's foreign policy toward Russia, stating that ``Washington and other Western capitals have impeded potential progress through their often hypocritical focus on ideas such as democratization, inequality, corruption and regime change.''% 567
\footnote{\textit{See} Carter W. Page, \textit{The Lecture of Trump's Advisor Carter Page in Moscow}, YouTube Channel Katehon Think Tank, Posted July~7, 2016,
\textit{available at} \url{https://www.youtube.com/watch?time\_continue=28\&v=1CYF29saA9w}.
Page also provided the FBI with a copy of his speech and slides from the speech.
\textit{See} Carter Page, ``The Evolution of the World Economy: Trends and Potential,'' Speech at N[ew] Economic S[chool] (July~7, 2016).}
On July~8, 2016, Page delivered a speech during the NES commencement.% 568
\footnote{Page 3/10/17 302, at~3.}
After Page delivered his commencement address, Russian Deputy Prime Minister and NES board member Arkady Dvorkovich spoke at the ceremony and stated that the sanctions the United States had imposed on Russia had hurt the NES\null.% 569
\footnote{Page 3/16/17 302, at~3.}
Page and Dvorkovich shook hands at the commencement ceremony, and Weber recalled that Dvorkovich made statements to Page about working together in the future.% 570
\footnote{S. Weber 7/28/17 302, at~4.}
\blackout{Grand Jury}% 571
\footnote{\blackout{Grand Jury}}
Page said that, during his time in Moscow, he met with friends and associates he knew from when he lived in Russia, including Andrey Baranov, a former Gazprom employee who had become the head of investor relations at Rosneft, a Russian energy company.% 572
\footnote{Page 3/10/17 302, at~3;
Page 3/30/17 302, at~3;
Page 3/31/17 302, at~2.}
Page stated that he and Baranov talked about ``immaterial non-public'' information.% 573
\footnote{Page 3/30/17 302, at~3.}
Page believed he and Baranov discussed Rosneft president Igor Sechin, and he thought Baranov might have mentioned the possibility of a sale of a stake in Rosneft in passing.% 574
\footnote{Page 3/30/17 302, at~9. \blackout{Grand Jury}}
Page recalled mentioning his involvement in the Trump Campaign with Baranov, although he did not remember details of the conversation.% 575
\footnote{\blackout{Grand Jury} Page 3/30/17 302, at~3.}
Page also met with individuals from Tatneft, a Russian energy company, to discuss possible business deals, including having Page work as a consultant.% 576
\footnote{Page 3/10/17 302, at~3;
Page 3/30/17 302, at~7;
Page 3/31/17 302, at~2.}
On July~8, 2016, while he was in Moscow, Page emailed several Campaign officials and stated he would send ``a readout soon regarding some incredible insights and outreach I've received from a few Russian legislators and senior members of the Presidential Administration here.''% 577
\footnote{\blackout{Grand Jury} 7/8/16 Email, Page to Dahl \& Gordon.}
On July~9, 2016, Page emailed Clovis, writing in pertinent part:
\begin{quote}
Russian Deputy Prime minister and NES board member Arkady Dvorkovich also spoke before the event.
In a private conversation, Dvorkovich expressed strong support for Mr.~Trump and a desire to work together toward devising better solutions in response to the vast range of current international problems.
Based on feedback from a diverse array of other sources close to the Presidential Administration, it was readily apparent that this sentiment is widely held at all levels of government.% 578
\footnote{\blackout{Grand Jury} 7/9/16 Email, Page to Clovis.}
\end{quote}
Despite these representations to the Campaign,
\blackout{Grand Jury}% 579
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 580
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 581
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 582
\footnote{\blackout{Grand Jury}}
The Office was unable to obtain additional evidence or testimony about who Page may have met or communicated with in Moscow; thus, Page's activities in Russia---as described in his emails with the Campaign---were not fully explained.
\paragraph{Later Campaign Work and Removal from the Campaign}
In July 2016, after returning from Russia, Page traveled to the Republican National Convention in Cleveland.% 583
\footnote{Page 3/10/17 302, at~4;
Page 3/16/17 302, at~3.}
While there, Page met Russian Ambassador to the United States Sergey Kislyak; that interaction is described in \hyperlink{paragraph.1.4.1.6.1}{Volume~I, Section~IV.A.6.a}, \textit{infra}.% 584
\footnote{Page 3/10/17 302, at~4;
Page 3/16/17 302, at~3.}
Page later emailed Campaign officials with feedback he said he received from ambassadors he had met at the Convention, and he wrote that Ambassador Kislyak was very worried about candidate Clinton's world views.% 585
\footnote{\blackout{Grand Jury}; 7/23/16 Email, Page to Clovis;
7/25/16 Email, Page to Gordon \& Schmitz.}
\blackout{Grand Jury}% 586
\footnote{\blackout{Grand Jury}}
Following the Convention, Page's trip to Moscow and his advocacy for pro-Russia foreign policy drew the media's attention and began to generate substantial press coverage.
The Campaign responded by distancing itself from Page, describing him as an ``informal foreign policy advisor'' who did ``not speak for Mr.~Trump or the campaign.''% 587
\footnote{\textit{See, e.g.}, Steven Mufson \& Tom Hamburger, \textit{Trump Advisor's Public Comments, Ties to Moscow Stir Unease in Both Parties}, Washington Post (Aug.~5, 2016).}
On September~23, 2016, Yahoo!\ News reported that U.S. intelligence officials were investigating whether Page had opened private communications with senior Russian officials to discuss U.S. sanctions policy under a possible Trump Administration.% 588
\footnote{Michael Isikoff, \textit{U.S. Intel Officials Probe Ties Between Trump Adviser and Kremlin}, Yahoo!\ News (Sept.~23, 2016).}
A Campaign spokesman told Yahoo!\ News that Page had ``no role'' in the Campaign and that the Campaign was ``not aware of any of his activities, past or present.''% 589
\footnote{Michael Isikoff, \textit{U.S. Intel Officials Probe Ties Between Trump Adviser and Kremlin}, Yahoo!\ News (Sept.~23, 2016);
\textit{see also} 9/25/16 Email, Hicks to Conway \& Bannon (instructing that inquiries about Page should be answered with ``[h]e was announced as an informal adviser in March.
Since then he has had no role or official contact with the campaign.
We have no knowledge of activities past or present and he now officially has been removed from all lists etc.'').}
On September~24, 2016, Page was formally removed from the Campaign.% 590
\footnote{Page 3/16/17 302, at~2;
\textit{see, e.g.}, 9/23/16 Email, J.~Miller to Bannon \& S.~Miller (discussing plans to remove Page from the campaign).}
Although Page had been removed from the Campaign, after the election he sought a position in the Trump Administration.% 591
\footnote{\blackout{Grand Jury}, ``Transition Online Form,'' 11/14/16 \blackout{Grand Jury}}
On November~14, 2016, he submitted an application to the Transition Team that inflated his credentials and experiences, stating that in his capacity as a Trump Campaign foreign policy advisor he had met with ``top world leaders'' and ``effectively responded to diplomatic outreach efforts from senior government officials in Asia, Europe, the Middle East, Africa, [and] the Americas.''% 592
\footnote{\blackout{Grand Jury} ``Transition Online Form,'' 11/14/16 \blackout{Grand Jury}}
Page received no response from the Transition Team.
When Page took a personal trip to Moscow in December 2016, he met again with at least one Russian government official.
That interaction and a discussion of the December trip are set forth in \hyperlink{subsubsection.1.4.2.6}{Volume~I, Section~IV.B.6}, \textit{infra}.
\subsubsection{Dimitri Simes and the Center for the National Interest}
Members of the Trump Campaign interacted on several occasions with the Center for the National Interest (CNI), principally through its President and Chief Executive Officer, Dimitri Simes.
CNI is a think tank with expertise in and connections to the Russian government.
Simes was born in the former Soviet Union and immigrated to the United States in the 1970s.
In April 2016, candidate Trump delivered his first speech on foreign policy and national security at an event hosted by the \textit{National Interest}, a publication affiliated with CNI\null.
Then-Senator Jeff Sessions and Russian Ambassador Kislyak both attended the event and, as a result, it gained some attention in relation to Sessions's confirmation hearings to become Attorney General.
Sessions had various other contacts with CNI during the campaign period on foreign-policy matters, including Russia.
Jared Kushner also interacted with Simes about Russian issues during the campaign.
The investigation did not identify evidence that the Campaign passed or received any messages to or from the Russian government through CNI or Simes.
\paragraph{CNI and Dimitri Simes Connect with the Trump Campaign}
CNI is a Washington-based non-profit organization that grew out of a center founded by former President Richard Nixon.% 593
\footnote{Simes 3/8/18 302, at~1--2.}
CNI describes itself ``as a voice for strategic realism in U.S. foreign policy,'' and publishes a bi-monthly foreign policy magazine, the \textit{National Interest}.% 594
\footnote{\textit{About the Center}, CNI, \textit{available at} \url{https://cftni.org/about/}.}
CNI is overseen by a board of directors and an advisory council that is largely honorary and whose members at the relevant time included Sessions, who served as an advisor to candidate Trump on national security and foreign policy issues.% 595
\footnote{\textit{Advisory Counsel}, CNI, \textit{available at} \url{https://web.archive.org/web/20161030025331/http://cftni.org/about/advisory-council/};
Simes 3/8/18 302, at~3--4;
Saunders 2/15/18 302, at~4;
Sessions 1/17/18 302, at~16.}
Dimitri Simes is president and CEO of CNI and the publisher and CEO of the \textit{National Interest}.% 596
\footnote{Simes 3/8/18 302, at~2.}
Simes was born in the former Soviet Union, emigrated to the United States in the early 1970s, and joined CNI's predecessor after working at the Carnegie Endowment for International Peace.% 597
\footnote{ Simes 3/8/18 302, at~1--2;
Simes 3/27/18 302, at~19.}
Simes personally has many contacts with current and former Russian government officials,% 598
\footnote{Simes 3/27/18 302, at~10--15.}
as does CNI collectively.
As CNI stated when seeking a grant from the Carnegie Corporation in 2015, CNI has ``unparalleled access to Russian officials and politicians among Washington think tanks,''% 599
\footnote{C00011656 (\textit{Rethinking U.S.--Russia Relations}, CNI (Apr.~18, 2015)).}
in part because CNI has arranged for U.S. delegations to visit Russia and for Russian delegations to visit the United States as part of so-called ``Track~II'' diplomatic efforts.% 600
\footnote{Simes 3/8/18 302, at~5;
Saunders 2/15/18 302, at~29--30;
Zakheim 1/25/18 302, at~3.}
On March~14, 2016, CNI board member Richard Plepler organized a luncheon for CNI and its honorary chairman, Henry Kissinger, at the Time Warner Building in New York.% 601
\footnote{Simes 3/8/18 302, at~6;
CO0006784 (3/11/16 Email, Gilbride to Saunders (3:43:12~p.m.)[)];
\textit{cf.}~Zakheim 1/25/18 302, at~1 (Kissinger was CNI's ``Honorary Chairman of the Board'');
Boyd 1/24/18 302, at~2;
P. Sanders 2/15/18 302, at~5.}
The idea behind the event was to generate interest in CNI's work and recruit new board members for CNI\null.% 602
\footnote{Simes 3/8/18 302, at~5--6; Simes 3/27/18 302, at~2.}
Along with Simes, attendees at the event included Jared Kushner, son-in-law of candidate Trump.% 603
\footnote{Simes 3/8/18 302, at~6; Kushner 4/11/18 302, at~2.}
Kushner told the Office that the event came at a time when the Trump Campaign was having trouble securing support from experienced foreign policy professionals and that, as a result, he decided to seek Simes's assistance during the March~14 event.% 604
\footnote{Kushner 4/11/18 302, at~2.}
Simes and Kushner spoke again on a March~24, 2016 telephone call,% 605
\footnote{Simes 3/8/18 302, at~6--7.}
three days after Trump had publicly named the team of foreign policy advisors that had been put together on short notice.% 606
\footnote{\blackout{Grand Jury} \textit{see} \hyperlink{subsubsection.1.4.1.2}{Volume~I, Section~IV.A.2}, \textit{supra}.}
On March~31, 2016, Simes and Kushner had an in-person, one-on-one meeting in Kushner's New York office.% 607
\footnote{Simes 3/8/18 302, at~7--9.}
During that meeting, Simes told Kushner that the best way to handle foreign-policy issues for the Trump Campaign would be to organize an advisory group of experts to meet with candidate Trump and develop a foreign policy approach that was consistent with Trump's voice.% 608
\footnote{Simes 3/8/18 302, at~7--8.}
Simes believed that Kushner was receptive to that suggestion.% 609
\footnote{Simes 3/8/18 302, at~8;
\textit{see also} Boyd 1/24/18 302, at~2.}
Simes also had contact with other individuals associated with the Trump Campaign regarding the Campaign's foreign policy positions.
For example, on June~17, 2016, Simes sent J.D.~Gordon an email with a ``memo to Senator Sessions that we discussed at our recent meeting'' and asked Gordon to both read it and share it with Sessions.
The memorandum proposed building a ``small and carefully selected group of experts'' to assist Sessions with the Campaign, operating under the assumption ``that Hillary Clinton is very vulnerable on national security and foreign policy issues.''
The memorandum outlined key issues for the Campaign, including a ``new beginning with Russia.''% 610
\footnote{C00008187 (6/17/16 Email, Simes to Gordon (3:35:45~p.m.)).}
\paragraph{National Interest Hosts a Foreign Policy Speech at the Mayflower Hotel}
During both their March~24 phone call and their March~31 in-person meeting, Simes and Kushner discussed the possibility of CNI hosting a foreign policy speech by candidate Trump.% 611
\footnote{Simes 3/8/18 302, at~7.}
Following those conversations, Simes agreed that he and others associated with CNI would provide behind-the-scenes input on the substance of the foreign-policy speech and that CNI officials would coordinate the logistics of the speech with Sessions and his staff, including Sessions's chief of staff, Rick Dearborn.% 612
\footnote{Simes 3/8/18 302, at~8--11;
C00008923 (4/6/16 Email, Simes to Burt (2:22:28~p.m.));
Burt 2/9/18 302, at~7.}
In mid-April~2016, Kushner put Simes in contact with senior policy advisor Stephen Miller and forwarded to Simes an outline of the foreign-policy speech that Miller had prepared.% 613
\footnote{C00008551 (4/17/16 Email, Kushner to Simes (2:44:25~p.m.));
C00006759 (4/14/16 Email Kushner to Simes \& S.~Miller (12:30~p.m.)).}
Simes sent back to the Campaign bullet points with ideas for the speech that he had drafted with CNI Executive Director Paul Saunders and board member Richard Burt.% 614
\footnote{Burt 2/9/18 302, at~7;
Saunders 2/15/18 302, at~7--8.}
Simes received subsequent draft outlines from Miller, and he and Saunders spoke to Miller by phone about substantive changes to the speech.% 615
\footnote{Simes 3/8/18 302, at~13;
Saunders 2/15/18 302, at~7--8.}
It is not clear, however, whether CNI officials received an actual draft of the speech for comment; while Saunders recalled having received an actual draft, Simes did not, and the emails that CNI produced to this Office do not contain such a draft.% 616
\footnote{Simes 3/8/18 302, at~13;
Saunders 2/15/18 302, at~7--8.}
After board members expressed concern to Simes that CNI's hosting the speech could be perceived as an endorsement of a particular candidate, CNI decided to have its publication, the \textit{National Interest}, serve as the host and to have the event at the National Press Club.% 617
\footnote{Saunders 2/15/18 302, at~8;
Simes 3/8/18 302, at~12;
CO00003834--43 (4/22/16 Email, Simes to Boyd et~al.\ (8:47~a.m.)).}
Kushner later requested that the event be moved to the Mayflower Hotel, which was another venue that Simes had mentioned during initial discussions with the Campaign, in order to address concerns about security and capacity.% 618
\footnote{Simes 3/8/18 302, at~12, 18;
Saunders 2/15/18 302, at~11.}
On April~25, 2016, Saunders booked event rooms at the Mayflower to host both the speech and a VIP reception that was to be held beforehand.% 619
\footnote{Saunders 2/15/18 302, at~11--12;
CO0006651--57 (Mayflower Group Sales Agreement).}
Saunders understood that the reception---at which invitees would have the chance to meet candidate Trump---would be a small event.% 620
\footnote{Saunders 2/15/18 302, at~12--13.}
Saunders decided who would attend by looking at the list of CNI's invitees to the speech itself and then choosing a subset for the reception.% 621
\footnote{Saunders 2/15/18 302, at~12.}
CNI's invitees to the reception included Sessions and~Kislyak.% 622
\footnote{C00002575 (Attendee List);
C00008536 (4/25/16 Email, Simes to Kushner (4:53:45~p.m.)).}
The week before the speech Simes had informed Kislyak that he would be invited to the speech, and that he would have the opportunity to meet Trump.% 623
\footnote{Simes 3/8/18 302, at~19--20.}
When the pre-speech reception began on April~27, a receiving line was quickly organized so that attendees could meet Trump.% 624
\footnote{Simes 3/8/18 302, at~21.}
Sessions first stood next to Trump to introduce him to the members of Congress who were in attendance.% 625
\footnote{Simes 3/8/18 302, at~21.}
After those members had been introduced, Simes stood next to Trump and introduced him to the CNI invitees in attendance, including Kislyak.% 626
\footnote{Simes 3/8/18 302, at~21.}
Simes perceived the introduction to be positive and friendly, but thought it clear that Kislyak and Trump had just met for the first time.% 627
\footnote{Simes 3/8/18 302, at~21.}
Kislyak also met Kushner during the pre-speech reception.
The two shook hands and chatted for a minute or two, during which Kushner recalled Kislyak saying, ``we like what your candidate is saying \dots\ it's refreshing.''% 628
\footnote{Kushner 4/11/18 302, at~4.}
Several public reports state that, in addition to speaking to Kushner at the pre-speech reception, Kislyak also met or conversed with Sessions at that time.% 629
\footnote{\textit{See, e.g.}, Ken Dilanian, \textit{Did Trump, Kushner, Sessions Have an Undisclosed Meeting With Russian?}, NBC News (June~1, 2016);
Julia Ioffe, \textit{Why Did Jeff Sessions Really Meet With Sergey Kislyak}, The Atlantic (June~13, 2017).}
Sessions stated to investigators, however, that he did not remember any such conversation.% 630
\footnote{Sessions 1/17/18 302, at~22.}
Nor did anyone else affiliated with CNI or the \textit{National Interest} specifically recall a conversation or meeting between Sessions and Kislyak at the pre-speech reception.% 631
\footnote{Simes 3/8/18 302, at~21;
Saunders 2/15/18 302, at~14, 21;
Boyd 1/24/18 302, at~3--4;
Heilbrunn 2/1/18 302, at~6;
\textit{Statement Regarding President Trump's April~27, 2016 Foreign Policy Speech at the Center for the National Interest}, CNI (Mar.~8, 2017).}
It appears that, if a conversation occurred at the pre-speech reception, it was a brief one conducted in public view, similar to the exchange between Kushner and~Kislyak.
The Office found no evidence that Kislyak conversed with either Trump or Sessions after the speech, or would have had the opportunity to do so.
Simes, for example, did not recall seeing Kislyak at the post-speech luncheon,% 632
\footnote{Simes 3/8/18 302, at~22;
Heilbrunn 2/1/18 302, at~7.}
and the only witness who accounted for Sessions's whereabouts stated that Sessions may have spoken to the press after the event but then departed for Capitol Hill.% 633
\footnote{Luff 1/30/18 302, at~4.}
Saunders recalled, based in part on a food-related request he received from a Campaign staff member, that Trump left the hotel a few minutes after the speech to go to the airport.% 634
\footnote{Saunders 2/15/18 302, at~15.}
\paragraph{Jeff Sessions's Post-Speech Interactions with CNI}
In the wake of Sessions's confirmation hearings as Attorney General, questions arose about whether Sessions's campaign-period interactions with CNI apart from the Mayflower speech included any additional meetings with Ambassador Kislyak or involved Russian-related matters.
With respect to Kislyak contacts, on May~23, 2016, Sessions attended CNI's Distinguished Service Award dinner at the Four Seasons Hotel in Washington, D.C.% 635
\footnote{Sessions 1/17/18 302, at~22;
Saunders 2/15/18 302, at~17.}
Sessions attended a pre-dinner reception and was seated at one of two head tables for the event.% 636
\footnote{Saunders 2/15/18 302, at~17;
C00004779--80 (5/23/16 Email, Cantelmo to Saunders \& Hagberg (9:30:12~a.m.)[)];
C00004362 (5/23/16 Email, Bauman to Cantelmo et~al.\ (2:02:32~a.m.)[)].}
A seating chart prepared by Saunders indicates that Sessions was scheduled to be seated next to Kislyak, who appears to have responded to the invitation by indicating he would attend the event.% 637
\footnote{C00004362 (5/23/16 Email Bauman to Cantelmo et~al.\ (2:02:32~a.m.)[)].}
Sessions, however, did not remember seeing, speaking with, or sitting next to Kislyak at the dinner.% 638
\footnote{Sessions 1/17/18 302, at~22.}
Although CNI board member Charles Boyd said he may have seen Kislyak at the dinner,% 639
\footnote{Boyd 1/24/18 302, at~4.}
Simes, Saunders, and Jacob Heilbrunn---editor of the \textit{National Interest}---all had no recollection of seeing Kislyak at the May~23 event.% 640
\footnote{Simes 3/8/18 302, at~23;
Saunders 2/15/18 302, at~18;
Heilbrunn 2/1/18 302, at~7.}
Kislyak also does not appear in any of the photos from the event that the Office obtained.
In the summer of 2016, CNI organized at least two dinners in Washington, D.C. for Sessions to meet with experienced foreign policy professionals.% 641
\footnote{Simes 3/8/18 302, at~31;
Saunders 2/15/18 302, at~19;
Burt 2/9/18 302, at~9--10;
Khalilzad 1/9/18 302, at~5.}
The dinners included CNI-affiliated individuals, such as Richard Burt and Zalmay Khalilzad, a former U.S. ambassador to Afghanistan and Iraq and the person who had introduced Trump before the April~27, 2016 foreign-policy speech.% 642
\footnote{Burt 2/9/18 302, at~9--10;
Khalilzad 1/9/18 302, at~1--2, 5.}
Khalilzad also met with Sessions one-on-one separately from the dinners.% 643
\footnote{Khalilzad 1/9/18 302, at~5--6.}
At the dinners and in the meetings, the participants addressed U.S. relations with Russia, including how U.S. relations with NATO and European countries affected U.S. policy toward Russia.% 644
\footnote{Simes 3/8/18 302, at~31;
Burt 2/9/18 302, at~9--10;
Khalilzad 1/9/18 302, at~5.}
But the discussions were not exclusively focused on Russia.% 645
\footnote{Saunders 2/15/18 302, at~20.}
Khalilzad, for example, recalled discussing ``nation-building'' and violent extremism with Sessions.% 646
\footnote{Khalilzad 1/9/18 302, at~6.}
In addition, Sessions asked Saunders (of CNI) to draft two memoranda not specific to Russia: one on Hillary Clinton's foreign policy shortcomings and another on Egypt.% 647
\footnote{Saunders 2/15/18 302, at~19--20.}
\paragraph{Jared Kushner's Continuing Contacts with Simes}
Between the April 2016 speech at the Mayflower Hotel and the presidential election, Jared Kushner had periodic contacts with Simes.% 648
\footnote{Simes 3/8/18 302, at~27.}
Those contacts consisted of both in-person meetings and phone conversations, which concerned how to address issues relating to Russia in the Campaign and how to move forward with the advisory group of foreign policy experts that Simes had proposed.% 649
\footnote{Simes 3/8/18 302, at~27.}
Simes recalled that he, not Kushner, initiated all conversations about Russia, and that Kushner never asked him to set up back-channel conversations with Russians.% 650
\footnote{Simes 3/8/18 302, at~27.}
According to Simes, after the Mayflower speech in late April, Simes raised the issue of Russian contacts with Kushner, advised that it was bad optics for the Campaign to develop hidden Russian contacts, and told Kushner both that the Campaign should not highlight Russia as an issue and should handle any contacts with Russians with care.% 651
\footnote{Simes 3/8/18 302, at~27.
During this period of time, the Campaign received a request for a high level Campaign official to meet with an officer at a Russian state-owned bank ``to discuss an offer [that officer] claims to be carrying from President Putin to meet with'' candidate Trump.
NOSC00005653 (5/17/16 Email, Dearborn to Kushner (8:12~a.m.)).
Copying Manafort and Gates, Kushner responded, ``Pass on this.
A lot of people come claiming to carry messages.
Very few are able to verify.
For now I think we decline such meetings.
Most likely these people go back home and claim they have special access to gain importance for themselves.
Be careful.''
NOSC00005653 (5/17/16 Email, Kushner to Dearborn).}
Kushner generally provided a similar account of his interactions with Simes.% 652
\footnote{Kushner 4/11/18 302, at~11--13.}
Among the Kushner--Simes meetings was one held on August~17, 2016, at Simes's request, in Kushner's New York office.
The meeting was to address foreign policy advice that CNI was providing and how to respond to the Clinton Campaign's Russia-related attacks on candidate Trump.% 653
\footnote{Simes 3/8/18 302, at~29--30;
Simes 3/27/18 302, at~6;
Kushner 4/11/18 302, at~12;
C00007269 (8/10/16 Meeting Invitation, Vargas to Simes et~al.);
DJTFP00023484 (8/11/16 Email, Hagan to Manafort (5:57:15~p.m.)).}
In advance of the meeting, Simes sent Kushner a ``Russia Policy Memo'' laying out ``what Mr.~Trump may want to say about Russia.''% 654
\footnote{C00007981--84 (8/9/16 Email, Simes to Kushner (6:09:21~p.m.)).
The memorandum recommended ``downplaying Russia as a U.S. foreign policy priority at this time'' and suggested that ``some tend to exaggerate Putin's flaws.''
The memorandum also recommended approaching general Russian related questions in the framework of ``how to work with Russia to advance important U.S. national interests'' and that a Trump Administration ``not go abroad in search of monsters to destroy.''
The memorandum did not discuss sanctions but did address how to handle Ukraine-related questions, including questions about Russia's invasion and annexation of Crimea.}
In a cover email transmitting that memo and a phone call to set up the meeting, Simes mentioned ``a well-documented story of highly questionable connections between Bill Clinton'' and the Russian government, ``parts of [which]'' (according to Simes) had even been ``discussed with the CIA and the FBI in the late 1990s and shared with the [Independent Counsel] at the end of the Clinton presidency.''% 655
\footnote{C00007981 (8/9/16 Email, Simes to Kushner (6:09:21~p.m.)).}
Kushner forwarded the email to senior Trump Campaign officials Stephen Miller, Paul Manafort, and Rick Gates, with the note ``suggestion only.''% 656
\footnote{DJTFP00023459 (8/10/16 Email, Kushner to S.~Miller et~al.\ (11:30:13~a.m.)).}
Manafort subsequently forwarded the email to his assistant and scheduled a meeting with Simes.% 657
\footnote{DJTFP00023484 (8/11/16 Email, Hagan to Manafort (5:57:15~p.m.)).}
(Manafort was on the verge of leaving the Campaign by the time of the scheduled meeting with Simes, and Simes ended up meeting only with Kushner[.])[]
During the August~17 meeting, Simes provided Kushner the Clinton-related information that he had promised.% 658
\footnote{Simes 3/8/18 302, at~29--30;
Simes 3/27/18 302, at~6;
Kushner 4/11/18 302, at~12.}
Simes told Kushner that,
\blackout{Personal Privacy}% 659
\footnote{Simes 3/8/18 302, at~30;
Simes 3/27/18 302, at~6.}
Simes claimed that he had received this information from former CIA and Reagan White House official Fritz Ermarth, who claimed to have learned it from U.S. intelligence sources, not from Russians.% 660
\footnote{Simes 3/8/18 302, at~30.}
Simes perceived that Kushner did not find the information to be of interest or use to the Campaign because it was, in Simes's words, ``old news.''% 661
\footnote{Simes 3/8/18 302, at~30;
Simes 3/27/18 302, at~6.}
When interviewed by the Office, Kushner stated that he believed that there was little chance of something new being revealed about the Clintons given their long career as public figures, and that he never received from Simes information that could be ``operationalized'' for the Trump Campaign.% 662
\footnote{Kushner 4/11/18 302, at~12.}
Despite Kushner's reaction, Simes believed that he provided the same information at a small group meeting of foreign policy experts that CNI organized for Sessions.% 663
\footnote{Simes 3/8/18 302, at~30.}
\subsubsection{June~9, 2016 Meeting at Trump Tower}
On June~9, 2016, senior representatives of the Trump Campaign met in Trump Tower with a Russian attorney expecting to receive derogatory information about Hillary Clinton from the Russian government.
The meeting was proposed to Donald Trump~Jr.\ in an email from Robert Goldstone, at the request of his then-client Emin Agalarov, the son of Russian real-estate developer Aras Agalarov.
Goldstone relayed to Trump~Jr.\ that the ``Crown prosecutor of Russia \dots\ offered to provide the Trump Campaign with some official documents and information that would incriminate Hillary and her dealings with Russia'' as ``part of Russia and its government's support for Mr.~Trump.''
Trump~Jr.\ immediately responded that ``if it's what you say I love it,'' and arranged the meeting through a series of emails and telephone calls.
Trump~Jr.\ invited campaign chairman Paul Manafort and senior advisor Jared Kushner to attend the meeting, and both attended.
Members of the Campaign discussed the meeting before it occurred, and Michael Cohen recalled that Trump~Jr.\ may have told candidate Trump about an upcoming meeting to receive adverse information about Clinton, without linking the meeting to Russia.
According to written answers submitted by President Trump, he has no recollection of learning of the meeting at the time, and the Office found no documentary evidence showing that he was made aware of the meeting---or its Russian connection---before it occurred.
The Russian attorney who spoke at the meeting, Natalia Veselnitskaya, had previously worked for the Russian government and maintained a relationship with that government throughout this period of time.
She claimed that funds derived from illegal activities in Russia were provided to Hillary Clinton and other Democrats.
Trump~Jr.\ requested evidence to support those claims, but Veselnitskaya did not provide such information.
She and her associates then turned to a critique of the origins of the Magnitsky Act, a 2012 statute that imposed financial and travel sanctions on Russian officials and that resulted in a retaliatory ban on adoptions of Russian children.
Trump~Jr.\ suggested that the issue could be revisited when and if candidate Trump was elected.
After the election, Veselnitskaya made additional efforts to follow up on the meeting, but the Trump Transition Team did not engage.
\paragraph{Setting Up the June~9 Meeting}
\subparagraph{Outreach to Donald Trump~Jr}
Aras Agalarov is a Russian real-estate developer with ties to Putin and other members of the Russian government, including Russia's Prosecutor General, Yuri Chaika.% 664
\footnote{\blackout{Grand Jury} Goldstone 2/8/18 302, at~4.}
Aras Agalarov is the president of the Crocus Group, a Russian enterprise that holds substantial Russian government construction contracts and that---as discussed above, \hyperlink{subsubsection.1.4.1.1}{Volume~I, Section~IV.A.1}, \textit{supra}---worked with Trump in connection with the 2013 Miss Universe pageant in Moscow and a potential Trump Moscow real-estate project.% 665
\footnote{\blackout{Grand Jury} Kaveladze 11/16/17 302, at~3;
Shugart 9/25/17 302, at~2--3;
\blackout{Grand Jury}}
The relationship continued over time, as the parties pursued the Trump Moscow project in 2013--2014 and exchanged gifts and letters in 2016.% 666
\footnote{\blackout{Grand Jury} Goldstone 2/8/18 302, at~10;
\blackout{Grand Jury} Kaveladze 11/16/17 302, at~5--6;
4/25/16 Email, Graff to Goldstone.}
For example, in April 2016, Trump responded to a letter from Aras Agalarov with a handwritten note.% 667
\footnote{RG000033--34 (4/25/16 Email, Graff to Goldstone (attachment)[)].}
Aras Agalarov expressed interest in Trump's campaign, passed on ``congratulations'' for winning in the primary and---according to one email drafted by Goldstone---an ``offer'' of his ``support and that of many of his important Russian friends and colleagues[,] especially with reference to U.S./Russian relations.''% 668
\footnote{DJTJROO008 (2/29/16 Email, Goldstone to Trump~Jr.\ et~al.);
\blackout{Grand Jury}}
On June~3, 2016, Emin Agalarov called Goldstone, Emin's then-publicist.% 669
\footnote{Call Records of Robert Goldstone \blackout{Grand Jury} Goldstone 2/8/18 302, at~6.}
Goldstone is a music and events promoter who represented Emin Agalarov from approximately late 2012 until late 2016.% 670
\footnote{Goldstone 2/8/18 302, at~1--2; \blackout{Grand Jury} Beniaminov 1/6/18 302, at~3.}
While representing Emin Agalarov, Goldstone facilitated the ongoing contact between the Trumps and the Agalarovs---including an invitation that Trump sent to Putin to attend the 2013 Miss Universe Pageant in Moscow.% 671
\footnote{Goldstone 2/8/18 302, at~1--5; \blackout{Grand Jury}
DJTJRO0008 (2/29/19 Email, Goldstone to Trump~Jr.);
Beniaminov 1/6/18 302, at~3;
Shugart 9/25/17 302, at~2;
TRUMPORG\_18\_001325 (6/21/13 Email, Goldstone to Graff);
TRUMPORG\_18\_001013 (6/24/13 Email, Goldstone to Graff);
TRUMPORG\_18\_001014 (6/24/13 Email, Graff to Shugart);
TRUMPORG\_18\_001018 (6/26/13 Email, Graff to Goldstone);
TRUMPORG\_18\_001022 (6/27/13 Email, Graff to L.~Kelly);
TRUMPORG\_18\_001333 (9/12/13 Email, Goldstone to Graff, Shugart);
MU000004289 (7/27/13 Email, Goldstone to Graff, Shugart).}
\blackout{Grand Jury}% 672
\footnote{\blackout{Grand Jury} \textit{see} Goldstone 2/8/18 302, at~6--7.}
Goldstone understood
\blackout{Grand Jury}
a Russian political connection, and Emin Agalarov indicated that the attorney was a prosecutor.% 673
\footnote{\blackout{Grand Jury}}
Goldstone recalled that the information that might interest the Trumps involved Hillary Clinton
\blackout{Grand Jury}% 674
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 675
\footnote{\blackout{Grand Jury}}
The \blackout{Grand Jury} mentioned by Emin Agalarov was Natalia Veselnitskaya.% 676
\footnote{In December 2018, a grand jury in the Southern District of New York returned an indictment charging Veselnitskaya with obstructing the Prevezon litigation discussed in the text above.
\textit{See} Indictment, \textit{United States~v.\ Natalia Vladimirovna Veselnitskaya}, No.~18-cr-904 (S.D.N.Y.).
The indictment alleges, among other things, that Veselnitskaya lied to the district court about her relationship to the Russian Prosecutor General's Office and her involvement in responding to a U.S. document request sent to the Russian government.}
From approximately 1998 until 2001, Veselnitskaya worked as a prosecutor for the Central Administrative District of the Russian Prosecutor's Office,% 677
\footnote{Veselnitskaya 11/20/17 Statement to the Senate Committee on the Judiciary, at~2;
\blackout{Grand Jury}
}
and she continued to perform government-related work and maintain ties to the Russian government following her departure.% 678
\footnote{Testimony of Natalia Veselnitskaya Before the Senate Committee on Judiciary (Nov.~20, 2017) at~33;
Keir Simmons \& Rachel Elbaum, \textit{Russian Lawyer Veselnitskaya Says She Didn't Give Trump~Jr.\ Info on Clinton}, NBC News (July~11, 2017);
Maria Tsvetkova \& Jack Stubbs, \textit{Moscow Lawyer Who Met Trump~Jr.\ Had Russian Spy Agency As Client}, Reuters (July~21, 2017);
Andrew E. Kramer \& Sharon LaFraniere, \textit{Lawyer Who Was Said to Have Dirt on Clinton Had Closer Ties to Kremlin than She Let On}, New York Times (Apr.~27, 2018).}
She lobbied and testified about the Magnitsky Act, which imposed financial sanctions and travel restrictions on Russian officials and which was named for a Russian tax specialist who exposed a fraud and later died in a Russian prison.% 679
\footnote{\textit{See} Pub.\ L.~No.~112-208 \S\S~402, 404(a)(1), 126 Stat.~1502, 1502--1506.
Sergei Magnitsky was a Russian tax specialist who worked for William Browder, a former investment fund manager in Russia.
Browder hired Magnitsky to investigate tax fraud by Russian officials, and Magnitsky was charged with helping Browder embezzle money.
After Magnitsky died in a Russian prison, Browder lobbied Congress to pass the Magnitsky Act.
\textit{See, e.g}, Andrew E. Kramer, \textit{Turning Tables in Magnitsky Case, Russia Accuses Nemesis of Murder}, New York Times (Oct.~22, 2017);
Testimony of Natalia Veselnitskaya Before the Senate Committee on Judiciary (Nov.~20, 2017), Exhibits at~1--4;
Rosie Gray, \textit{Bill Browder's Testimony to the Senate Judiciary Committee}, The Atlantic (July~25, 2017).}
Putin called the statute ``a purely political, unfriendly act,'' and Russia responded by barring a list of current and former U.S. officials from entering Russia and by halting the adoption of Russian children by U.S. citizens.% 680
\footnote{Rilen Barry, \textit{Russia Bars 18 Americans After Sanctions by US}, New York Times (Apr.~13, 2013);
Tom Porter, \textit{Supporters of the Magnitsky Act Claim They've Been Targets of Russian Assassination and Kidnapping Bids}, Newsweek (July~16, 2017).}
Veselnitskaya performed legal work for Denis Katsyv,% 681
\footnote{Testimony of Natalia Veselnitskaya Before the Senate Committee on Judiciary (Nov.~20, 2017), at~21.}
the son of Russian businessman Peter Katsyv, and for his company Prevezon Holdings Ltd., which was a defendant in a civil-forfeiture action alleging the laundering of proceeds from the fraud exposed by Magnitsky.% 682
\footnote{\textit{See} Veselnitskaya Decl., \textit{United States~v.\ Prevezon Holdings, Ltd.}, No.~13-cv-6326 (S.D.N.Y.);
\textit{see Prevezon Holdings}, Second Amended Complaint;
\textit{Prevezon Holdings}, Mem.\ and Order;
\textit{Prevezon Holdings}, Deposition of Oleg Lurie.}
She also appears to have been involved in an April 2016 approach to a U.S. congressional delegation in Moscow offering ``confidential information'' from ``the Prosecutor General of Russia'' about ``interactions between certain political forces in our two countries.''% 683
\footnote{\textit{See} Gribbin 8/31/17 302, at~1--2 \& 1A (undated one-page document given to congressional delegation).
The Russian Prosecutor General is an official with broad national responsibilities in the Russian legal system.
\textit{See Federal Law on the Prosecutor's Office of the Russian Federation} (1992, amended 2004).}
Shortly after his June~3 call with Emin Agalarov, Goldstone emailed Trump~Jr.% 684
\footnote{RG000061 (6/3/16 Email, Goldstone to Trump~Jr.);
DJTJRO0446 (6/3/16 Email, Goldstone to Donald Trump~Jr.);
\UseVerb{DJTJ} 07/11/17 (11:00) Tweet.}
The email stated:
\begin{quote}
Good morning
Emin just called and asked me to contact you with something very interesting.
The Crown prosecutor of Russia met with his father Aras this morning and in their meeting offered to provide the Trump campaign with some official documents and information that would incriminate Hillary and her dealings with Russia and would be very useful to your father.
This is obviously very high level and sensitive information but is part of Russia and its government's support for Mr.~Trump -- helped along by Aras and Emin.
What do you think is the best way to handle this information and would you be able to speak to Emin about it directly?
I can also send this info to your father via Rhona, but it is ultra sensitive so wanted to send to you first.
Best
Rob Goldstone
\end{quote}
Within minutes of this email, Trump~Jr.\ responded, emailing back: ``Thanks Rob I appreciate that.
I am on the road at the moment but perhaps I just speak to Emin first.
Seems we have some time and if it's what you say I love it especially later in the summer.
Could we do a call first thing next week when I am back?''% 685
\footnote{DJTJROO446 (6/3/16 Email, Trump~Jr.\ to Goldstone);
\UseVerb{DJTJ} 07/11/17 (11:00) Tweet;
RG000061 (6/3/16 Email, Trump~Jr.\ to Goldstone).}
Goldstone conveyed Trump~Jr.'s interest to Emin Agalarov, emailing that Trump~Jr.\ ``wants to speak personally on the issue.''% 686
\footnote{\blackout{Grand Jury} RG000062 (6/3/16 Email, Goldstone \& Trump~Jr.).}
On June~6, 2016, Emin Agalarov asked Goldstone if there was ``[a]ny news,'' and Goldstone explained that Trump~Jr.\ was likely still traveling for the ``final elections \dots\ where [T]rump will be `crowned' the official nominee.''% 687
\footnote{RG000063 (6/6/16 Email, A.~Agalarov to Goldstone);
RG000064 (6/6/16 Email, Goldstone to A.~Agalarov).}
On the same day, Goldstone again emailed Trump~Jr.\ and asked when Trump~Jr.\ was ``free to talk with Emin about this Hillary info.''% 688
\footnote{RG000065 (6/6/16 Email, Goldstone to Trump~Jr.);
DJTJR00446 (6/6/16 Email, Goldstone to Trump~Jr.).}
Trump~Jr.\ asked if they could ``speak now,'' and Goldstone arranged a call between Trump~Jr.\ and Emin Agalarov.% 689
\footnote{DJTJRO0445 (6/6/16 Email, Goldstone and Trump~Jr.);
RG000065--67 (6/6/16 Email, Goldstone and Trump~Jr.);
\blackout{Grand Jury}}
On June~6 and June~7, Trump~Jr.\ and Emin Agalarov had multiple brief calls.% 690
\footnote{DJTJRO0499 (Call Records of Donald Trump~Jr.\ \blackout{Grand Jury});
Call Records of Donald Trump~Jr.\ \blackout{Grand Jury}.}
Also on June~6, 2016, Aras Agalarov called Ike Kaveladze and asked him to attend a meeting in New York with the Trump Organization.% 691
\footnote{Kaveladze 11/16/17 302, at~6; \blackout{Grand Jury}}
Kaveladze is a Georgia-born, naturalized U.S. citizen who worked in the United States for the Crocus Group and reported to Aras Agalarov.% 692
\footnote{Kaveladze 11/16/17 302, at~1--2;
\blackout{Grand Jury} Beniaminov 1/6/18 302, at~2--3;
\blackout{Grand Jury}}
Kaveladze told the Office that, in a second phone call on June~6, 2016, Aras Agalarov asked Kaveladze if he knew anything about the Magnitsky Act, and Aras sent him a short synopsis for the meeting and Veselnitskaya's business card.
According to Kaveladze, Aras Agalarov said the purpose of the meeting was to discuss the Magnitsky Act, and he asked Kaveladze to translate.% 693
\footnote{Kaveladze 11/16/17 302, at~6.}
\subparagraph{Awareness of the Meeting Within the Campaign}
On June~7, Goldstone emailed Trump~Jr.\ and said that ``Emin asked that I schedule a meeting with you and [t]he Russian government attorney who is flying over from Moscow.''% 694
\footnote{DJTJROO467 (6/7/16 Email, Goldstone to Trump~Jr.);
\UseVerb{DJTJ} 07/11/17 (11:00) Tweet;
RG000068 (6/7/16 Email, Goldstone to Trump~Jr.);
\blackout{Grand Jury}}
Trump~Jr.\ replied that Manafort (identified as the ``campaign boss''), Jared Kushner, and Trump~Jr.\ would likely attend.% 695
\footnote{DJTJROO469 (6/7/16 Email, Trump~Jr.\ to Goldstone);
\UseVerb{DJTJ} 07/11/17 (11:00) Tweet;
RG000071 (6/7/16 Email, Trump~Jr.\ to Goldstone);
OSC-KAV\_00048 (6/7/16 Email, Goldstone to Kaveladze);
\blackout{Grand Jury}}
Goldstone was surprised to learn that Trump~Jr., Manafort, and Kushner would attend.% 696
\footnote{Goldstone 2/8/18 302, at~7;
\blackout{Grand Jury}}
Kaveladze \blackout{Grand Jury}
``puzzled'' by the list of attendees and that he checked with one of Emin Agalarov's assistants, Roman Beniaminov, who said that the purpose of the meeting was for Veselnitskaya to convey ``negative information on Hillary Clinton.''% 697
\footnote{\blackout{Grand Jury} \textit{see} Kaveladze 11/16/17 302, at~7;
OSC-KAV\_00048 (6/7/16 Email, Goldstone to Kaveladze).}
Beniaminov, however, stated that he did not recall having known or said that.% 698
\footnote{Beniaminov 1/6/18 302, at~3.}
Early on June~8, 2016 Kushner emailed his assistant, asking her to discuss a 3:00~p.m. meeting the following day with Trump~Jr.% 699
\footnote{NOSC0000007--08 (6/8/18 Email, Kushner to Vargas).}
Later that day, Trump~Jr.\ forwarded the entirety of his email correspondence regarding the meeting with Goldstone to Manafort and Kushner, under the subject line ``FW: Russia -- Clinton---private and confidential,'' adding a note that the ``[m]eeting got moved to 4 tomorrow at my offices.''% 700
\footnote{NOSC00000039--42 (6/8/16 Email, Trump~Jr.\ to Kushner \& Manafort);
DJTJR00485 (6/8/16 Email, Trump~Jr.\ to Kushner \& Manafort).}
Kushner then sent his assistant a second email, informing her that the ``[m]eeting with don jr is 4pm now.''% 701
\footnote{NOSC0000004 (6/8/16 Email, Kushner to Vargas).}
Manafort responded, ``See you then.~P.''% 702
\footnote{6/8/16 Email, Manafort to Trump~Jr.}
Rick Gates, who was the deputy campaign chairman, stated during interviews with the Office that in the days before June~9, 2016 Trump~Jr.\ announced at a regular morning meeting of senior campaign staff and Trump family members that he had a lead on negative information about the Clinton Foundation.% 703
\footnote{Gates 1/30/18 302, at~7;
Gates 3/1/18 302, at~3--4.
Although the March~1 302 refers to ``June~19,'' that is likely a typographical error;
external emails indicate that a meeting with those participants occurred on June~6.
\textit{See} NOSC00023603 (6/6/16 Email, Gates to Trump~Jr.\ et~al.).}
Gates believed that Trump~Jr.\ said the information was coming from a group in Kyrgyzstan and that he was introduced to the group by a friend.% 704
\footnote{Gates 1/30/18 302, at~7.
Aras Agalarov is originally from Azerbaijan, and public reporting indicates that his company, the Crocus Group, has done substantial work in Kyrgyzstan.
\textit{See} Neil MacFarquhar, \textit{A Russian Developer Helps Out the Kremlin on Occasion. Was He a Conduit to Trump?}, New York Times (July~16, 2017).}
Gates recalled that the meeting was attended by Trump~Jr., Eric Trump, Paul Manafort, Hope Hicks, and, joining late, Ivanka Trump and Jared Kushner.
According to Gates, Manafort warned the group that the meeting likely would not yield vital information and they should be careful.% 705
\footnote{Gates 3/1/18 302, at~3--4.}
Hicks denied any knowledge of the June~9 meeting before 2017,% 706
\footnote{Hicks 12/7/17 302, at~6.}
and Kushner did not recall if the planned June~9 meeting came up at all earlier that week.% 707
\footnote{Kushner 4/11/18 302, at~8.}
Michael Cohen recalled being in Donald J. Trump's office on June~6 or~7 when Trump~Jr.\ told his father that a meeting to obtain adverse information about Clinton was going forward.% 708
\footnote{Cohen 8/7/18 302, at~4--6.}
Cohen did not recall Trump~Jr.\ stating that the meeting was connected to Russia.% 709
\footnote{Cohen 8/7/18 302, at~4--5.}
From the tenor of the conversation, Cohen believed that Trump~Jr.\ had previously discussed the meeting with his father, although Cohen was not involved in any such conversation.% 710
\footnote{Cohen 9/12/18 302, at~15--16.}
In an interview with the Senate Judiciary Committee, however, Trump~Jr.\ stated that he did not inform his father about the emails or the upcoming meeting.% 711
\footnote{\textit{Interview of: Donald J. Trump, Jr., Senate Judiciary Committee}, 115th Cong.~28--29, 84, 94--95
(Sept.~7, 2017).
The Senate Judiciary Committee interview was not under oath, but Trump~Jr.\ was advised that it is a violation of 18~U.S.C. \S~1001 to make materially false statements in a congressional investigation.
\textit{Id.}~at~10--11.}
Similarly, neither Manafort nor Kushner recalled anyone informing candidate Trump of the meeting, including Trump~Jr.% 712
\footnote{Manafort 9/11/18 302, at~3--4;
Kushner 4/11/18 302, at~10.}
President Trump has stated to this Office, in written answers to questions, that he has ``no recollection of learning at the time'' that his son, Manafort, or ``Kushner was considering participating in a meeting in June 2016 concerning potentially negative information about Hillary Clinton.''% 713
\footnote{Written Responses of Donald J. Trump (Nov.~20, 2018), at~8 (Response to Question~I, Parts~(a)--(c)).
We considered whether one sequence of events suggested that candidate Trump had contemporaneous knowledge of the June~9 meeting.
On June~7, 2016 Trump announced his intention to give ``a major speech'' ``probably Monday of next week''---which would have been June~13---about ``all of the things that have taken place with the Clintons.''
\textit{See, e.g.}, Phillip Bump, \textit{What we know about the Trump Tower meeting}, Washington Post (Aug.~7, 2018).
Following the June~9 meeting, Trump changed the subject of his planned speech to national security.
But the Office did not find evidence that the original idea for the speech was connected to the anticipated June~9 meeting or that the change of topic was attributable to the failure of that meeting to produce concrete evidence about Clinton.
Other events, such as the Pulse nightclub shooting on June~12, could well have caused the change.
The President's written answers to our questions state that the speech's focus was altered ``[i]n light of\thinspace'' the Pulse nightclub shooting.
\textit{See} Written Responses, \textit{supra}.
As for the original topic of the June~13 speech, Trump has said that ``he expected to give a speech referencing the publicly available, negative information about the Clintons,'' and that the draft of the speech prepared by Campaign staff ``was based on publicly available material, including, in particular, information from the book \textit{Clinton Cash} by Peter Schweizer.''
Written Responses, \textit{supra}.
In a later June~22 speech, Trump did speak extensively about allegations that Clinton was corrupt, drawing from the \textit{Clinton Cash} book.
\textit{See Full Transcript: Donald Trump NYC Speech on Stakes of the Election}, \UseVerb{politicocom} (June~22, 2016).}
\paragraph{The Events of June~9, 2016}
\subparagraph{Arrangements for the Meeting}
Veselnitskaya was in New York on June~9, 2016, for appellate proceedings in the \textit{Prevezon} civil forfeiture litigation.% 714
\footnote{Testimony of Natalia Veselnitskaya Before the Senate Committee on Judiciary (Nov.~20, 2017) at~41, 42;
Alison Frankel, \textit{How Did Russian Lawyer Veselnitskaya Get into U.S. for Trump Tower Meeting?} Reuters, (Nov.~6, 2017);
Michael Kranish et~al., \textit{Russian Lawyer who Met with Trump~Jr.\ Has Long History Fighting Sanctions}, Washington Post (July~11, 2017);
\textit{see} OSC-KAV\_00113 (6/8/16 Email, Goldstone to Kaveladze);
RG000073 (6/8/16 Email, Goldstone to Trump~Jr.);
Lieberman 12/13/17 302, at~5;
\textit{see also Prevezon Holdings} Order (Oct.~17, 2016).}
That day, Veselnitskaya called Rinat Akhmetshin, a Soviet-born U.S. lobbyist,
\blackout{Grand Jury}
and when she learned that he was in New York, invited him to lunch.% 715
\footnote{\blackout{Grand Jury}}
Akhmetshin told the Office that he had worked on issues relating to the Magnitsky Act and had worked on the \textit{Prevezon} litigation.% 716
\footnote{Akhmetshin 11/14/17 302, at~4--6; \blackout{Grand Jury}}
Kaveladze and Anatoli Samochornov, a Russian-born translator who had assisted Veselnitska with Magnitsky-related lobbying and the \textit{Prevezon} case, also attended the lunch.% 717
\footnote{Kaveladze 11/16/17 302, at~7;
\blackout{Grand Jury};
Samochornov 7/13/17 302, at~2, 4;
\blackout{Grand Jury}}
\blackout{Grand Jury} Veselnitskaya said she was meeting \blackout{Grand Jury} and asked Akhmetshin what she should tell him.% 718
\footnote{\blackout{Grand Jury}}
According to several participants in the lunch, Veselnitskaya showed Akhmetshin a document alleging financial misconduct by Bill Browder and the Ziff brothers (Americans with business in Russia), and those individuals subsequently making political donations to the DNC\null.% 719
\footnote{\blackout{Grand Jury};
Kaveladze 11/16/17 302, at~7;
\blackout{Grand Jury};
Samochornov did not recall the planned subject matter of the Trump Tower meeting coming up at lunch.
\blackout{Grand Jury} Samochornov 7/12/17 302, at~4.
In her later Senate statement and interactions with the press, Veselnitskaya produced what she claimed were the talking points that she brought to the June~9 meeting.}
\blackout{Grand Jury}% 720
\footnote{\blackout{Grand Jury}}
The group then went to Trump Tower for the meeting.% 721
\footnote{\textit{E.g.}, Samochornov 7/12/17 302, at~4.}
\subparagraph{Conduct of the Meeting}
Trump~Jr., Manafort, and Kushner participated on the Trump side, while Kaveladze, Samochomov, Akhmetshin, and Goldstone attended with Veselnitskaya.% 722
\footnote{\textit{E.g.}, Samochornov 7/12/17 302, at~4.}
The Office spoke to every participant except Veselnitskaya and Trump~Jr., the latter of whom declined to be voluntarily interviewed by the Office
\blackout{Grand Jury}
The meeting lasted approximately 20~minutes.% 723
\footnote{\textit{E.g.}, Samochornov 7/12/17 302, at~4; Goldstone 2/8/18 302, at~9.}
\blackout{Grand Jury}% 724
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury} Goldstone recalled that Trump~Jr.\ invited Veselnitskaya to begin but did not say anything about the subject of the meeting.% 725
\footnote{\blackout{Grand Jury}}
Participants agreed that Veselnitskaya stated that the Ziff brothers had broken Russian laws and had donated their profits to the DNC or the Clinton Campaign.% 726
\footnote{\blackout{Grand Jury}}
She asserted that the Ziff brothers had engaged in tax evasion and money laundering in both the United States and Russia,% 727
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 728
\footnote{\blackout{Grand Jury}}
According to Akhmetshin, Trump~Jr.\ asked follow-up questions about how the alleged payments could be tied specifically to the Clinton Campaign, but Veselnitskaya indicated that she could not trace the money once it entered the United States.% 729
\footnote{\blackout{Grand Jury}; Akhmetshin 11/14/17 302, at~12.}
Kaveladze similarly recalled that Trump~Jr.\ asked what they have on Clinton, and Kushner became aggravated and asked ``[w]hat are we doing here?''% 730
\footnote{Kaveladze 11/16/17 302, at~8;
\blackout{Grand Jury}}
Akhmetshin then spoke about U.S. sanctions imposed under the Magnitsky Act and Russia's response prohibiting U.S. adoption of Russian children.% 731
\footnote{Samochornov 7/13/17 302, at~3;
\blackout{Grand Jury}}
Several participants recalled that Trump~Jr.\ commented that Trump is a private citizen, and there was nothing they could do at that time.% 732
\footnote{\textit{E.g.}, Akhmetshin 11/14/17 302, at~12--13;
\blackout{Grand Jury}}
Trump~Jr.\ also said that they could revisit the issue if and when they were in government.% 733
\footnote{Akhmetshin 11/14/17 302, at~12--13;
\blackout{Grand Jury}
Samochornov 7/13/17 302, at~3.
Trump~Jr.\ confirmed this in a statement he made in July 2017 after news of the June 2016 meeting broke.
\textit{Interview of: Donald J. Trump, Jr., Senate Judiciary Committee U.S. Senate Washington DC}, 115th Cong.~57 (Sept.~7, 2017).}
Notes that Manafort took on his phone reflect the general flow of the conversation, although not all of its details.% 734
\footnote{Manafort's notes state:
\begin{quote}
Bill Browder \\
Offshore -- Cyprus \\
133m shares \\
Companies \\
Not invest -- loan \\
Value in Cyprus as inter \\
Illici \\
Active sponsors of RNC \\
Browder hired Joanna Glover \\
Tied into Cheney \\
Russian adoption by American families
\end{quote}
PJM-SJC-00000001--02 (Notes Produced to Senate Judiciary Committee).}
At some point in the meeting, Kushner sent an iMessage to Manafort stating ``waste of time,'' followed immediately by two separate emails to assistants at Kushner Companies with requests that they call him to give him an excuse to leave.% 735
\footnote{NOSC00003992 (6/9/16 Text Message, Kushner to Manafort);
Kushner 4/11/18 302, at~9;
Vargas 4/4/18 302, at~7;
NOSC00000044 (6/9/16 Email, Kushner to Vargas);
NOSC00000045 (6/9/16 Email, Kushner to Cain).}
Samochornov recalled that Kushner departed the meeting before it concluded; Veselnitskaya recalled the same when interviewed by the press in July 2017.% 736
\footnote{Samochornov 7/12/17 302, at~4;
\blackout{Grand Jury};
Kushner 4/11/18 302, at~9--10;
\textit{see also Interview of: Donald J. Trump, Jr., Senate Judiciary Committee}, 115th Cong.~48--49 (Sept.~7, 2017).}
Veselnitskaya's press interviews and written statements to Congress differ materially from other accounts.
In a July 2017 press interview, Veselnitskaya claimed that she has no connection to the Russian government and had not referred to any derogatory information concerning the Clinton Campaign when she met with Trump Campaign officials.% 737
\footnote{\textit{Russian Lawyer Veselnitskaya Says She Didn't Give Trump~Jr.\ Info on Clinton}, NBC News (July~11, 2017).}
Veselnitskaya's November 2017 written submission to the Senate Judiciary Committee stated that the purpose of the June~9 meeting was not to connect with ``the Trump Campaign'' but rather to have ``a private meeting with Donald Trump~Jr.---a friend of my good acquaintance's son on the matter of assisting me or my colleagues in informing the Congress members as to the criminal nature of manipulation and interference with the legislative activities of the US Congress.''% 738
\footnote{\textit{Testimony of Natalia Veselnitskaya before the United States Senate Committee on the Judiciary}, 115th Cong.~10 (Nov.~20, 2017).}
In other words, Veselnitskaya claimed her focus was on Congress and not the Campaign.
No witness, however, recalled any reference to Congress during the meeting.
Veselnitskaya also maintained that she ``attended the meeting as a lawyer of Denis Katsyv,'' the previously mentioned owner of Prevezon Holdings, but she did not ``introduce [her]self in this capacity.''% 739
\footnote{\textit{Testimony of Natalia Veselnitskaya before the United States Senate Committee on the Judiciary}, 115th Cong.~10 (Nov.~20, 2017).}
In a July 2017 television interview, Trump~Jr.\ stated that while he had no way to gauge the reliability, credibility, or accuracy of what Goldstone had stated was the purpose of the meeting, if ``someone has information on our opponent \dots\ maybe this is something.
I should hear them out.''% 740
\footnote{Sean Hannity, \textit{Transcript---Donald Trump~Jr}, Fox News (July~11, 2017).}
Trump~Jr.\ further stated in September 2017 congressional testimony that he thought he should ``listen to what Rob and his colleagues had to say.''% 741
\footnote{\textit{Interview of: Donald J. Trump, Jr, Senate Judiciary Committee}, 115th Cong.~16 (Sept.~7, 2017).}
Depending on what, if any, information was provided, Trump~Jr.\ stated he could then ``consult with counsel to make an informed decision as to whether to give it any further consideration.''% 742
\footnote{\textit{Interview of: Donald J. Trump, Jr, Senate Judiciary Committee}, 115th Cong.~16--17 (Sept.~7, 2017).}
After the June~9 meeting, Goldstone apologized to Trump~Jr.% 743
\footnote{Kaveladze 11/16/17 302, at~8; \blackout{Grand Jury}; Goldstone 2/8/18 302, at~9; \blackout{Grand Jury}}
According to Goldstone, he told Trump~Jr.
\blackout{Grand Jury}% 744
\footnote{\blackout{Grand Jury}}
and told Emin Agalarov in a phone call that the meeting was about adoption
\blackout{Grand Jury}% 745
\footnote{\blackout{Grand Jury};
The week after the June~9 meeting, a cybersecurity firm and the DNC announced the Russian hack of the DNC\null.
\textit{See} \hyperlink{subsubsection.1.3.2.2}{Volume~I, Section~III.B.2}, \textit{supra}.
\blackout{Grand Jury}
(and one text message shows) that, shortly after the DNC announcement, Goldstone made comments connecting the DNC hacking announcement to the June~9 meeting.
\blackout{Grand Jury}; OSC-KAV\_00029 (6/14/16 Email, Goldstone to E.~Agalarov \& Kaveladze (10:09~a.m.)).
The investigation did not identify evidence connecting the events of June~9 to the GRU's hack-and-dump operation.
OSC-KAV\_00029--30 (6/14/16 Email, Goldstone to E.~Agalarov).}
\blackout{Grand Jury}% 746
\footnote{\blackout{Grand Jury}}
Aras Agalarov asked Kaveladze to report in after the meeting, but before Kaveladze could call, Aras Agalarov called him.% 747
\footnote{Kaveladze 11/16/17 302, at~8; Call Records of Ike Kaveladze \blackout{Grand Jury}}
With Veselnitskaya next to him, Kaveladze reported that the meeting had gone well, but he later told Aras Agalarov that the meeting about the Magnitsky Act had been a waste of time because it was not with lawyers and they were ``preaching to the wrong crowd.''% 748
\footnote{Kaveladze 11/16/17 302, at~8; Call Records of Ike Kaveladze \blackout{Grand Jury}.
On June~14, 2016 Kaveladze's teenage daughter emailed asking how the June~9 meeting had gone, and Kaveladze responded, ``meeting was boring.
The Russians did not have any bad info on Hilary [sic].''
OSC-KAV\_00257 (6/14/16 Email, I.~Kaveladze to A.~Kaveladze; \blackout{Grand Jury}).}
\paragraph{Post-June~9 Events}
Veselnitskaya and Aras Agalarov made at least two unsuccessful attempts after the election to meet with Trump representatives to convey similar information about Browder and the Magnitsky Act.% 749
\footnote{Goldstone 2/8/18 302, at~11; \blackout{Grand Jury}}
On November~23, 2016, Kaveladze emailed Goldstone about setting up another meeting ``with T people'' and sent a document bearing allegations similar to those conveyed on June~9.% 750
\footnote{OSC-KAV\_00138 1/23/16 Email, Goldstone to Kaveladze); \blackout{Grand Jury}}
Kaveladze followed up with Goldstone, stating that ``Mr.~A,'' which Goldstone understood to mean Aras Agalarov, called to ask about the meeting.% 751
\footnote{RG000196 (11/26--29/16 Text Messages, Goldstone \& Kaveladze); \blackout{Grand Jury}}
Goldstone emailed the document to Rhona Graff, saying that ``Aras Agalarov has asked me to pass on this document in the hope it can be passed on to the appropriate team.
If needed, a lawyer representing the case is in New York currently and happy to meet with any member of his transition team.''% 752
\footnote{Goldstone 2/8/18 302, at~11; \blackout{Grand Jury}; 57700118 (11/28/16 Email, Goldstone to Graff).}
According to Goldstone, around January 2017, Kaveladze contacted him again to set up another meeting, but Goldstone did not make the request.% 753
\footnote{\blackout{Grand Jury}}
The investigation did not identify evidence of the transition team following up.
Participants in the June~9, 2016 meeting began receiving inquiries from attorneys representing the Trump Organization starting in approximately June 2017.% 754
\footnote{\blackout{Grand Jury}}
On approximately June~2, 2017, Goldstone spoke with Alan Garten, general counsel of the Trump Organization, about his participation in the June~9 meeting.% 755
\footnote{\blackout{Grand Jury}}
The same day, Goldstone emailed Veselnitskaya's name to Garten, identifying her as the ``woman who was the attorney who spoke at the meeting from Moscow.''% 756
\footnote{RG000256 (6/2/17 Email, Goldstone to Garten).}
Later in June 2017, Goldstone participated in a lengthier call with Garten and Alan Futerfas, outside counsel for the Trump Organization (and, subsequently, personal counsel for Trump~Jr.).% 757
\footnote{\blackout{Grand Jury}}
On June~27, 2017, Goldstone emailed Emin Agalarov with the subject ``Trump attorneys'' and stated that he was ``interviewed by attorneys'' about the June~9 meeting who were ``concerned because it links Don Jr.\ to officials from Russia---which he has always denied meeting.''% 758
\footnote{RG000092 (6/27/17 Email, Goldstone to E.~Agalarov).}
Goldstone stressed that he ``did say at the time this was an awful idea and a terrible meeting.''% 759
\footnote{RG000092 (6/27/17 Email, Goldstone to E.~Agalarov). \blackout{Grand Jury}}
Emin Agalarov sent a screenshot of the message to Kaveladze.% 760
\footnote{OSC-KAV\_01190 (6/27/17 Text Message, E.~Agalarov to Kaveladze).}
The June~9 meeting became public in July 2017.
In a July~9, 2017 text message to Emin Agalarov, Goldstone wrote ``I made sure I kept you and your father out of [t]his story,''% 761
\footnote{RG000286--87 (7/9/17 Text Messages, E.~Agalarov \& Goldstone); \blackout{Grand Jury}}
and ``[i]f contacted I can do a dance and keep you out of it.''% 762
\footnote{\blackout{Investigative Technique}}
Goldstone added, ``FBI now investigating,'' and ``I hope this favor was worth for your dad---it could blow up.''% 763
\footnote{\blackout{Investigative Technique} \blackout{Grand Jury}}
On July~12, 2017 Emin Agalarov complained to Kaveladze that his father, Aras, ``never listens'' to him and that their relationship with ``mr T has been thrown down the drain.''% 764
\footnote{OSC-KAV\_01197 (7/11--12/17 Text Messages, Kaveladze \& E.~Agalarov); \blackout{Grand Jury}}
The next month, Goldstone commented to Emin Agalarov about the volume of publicity the June~9 meeting had generated, stating that his ``reputation [was] basically destroyed by this dumb meeting which your father insisted on even though Ike and Me told him would be bad news and not to do.''% 765
\footnote{\blackout{Investigative Technique}}
Goldstone added, ``I am not able to respond out of courtesy to you and your father.
So am painted as some mysterious link to Putin.''% 766
\footnote{\blackout{Investigative Technique}}
After public reporting on the June~9 meeting began, representatives from the Trump Organization again reached out to participants.
On July~10, 2017, Futerfas sent Goldstone an email with a proposed statement for Goldstone to issue, which read:
\begin{quote}
As the person who arranged the meeting, I can definitively state that the statements I have read by Donald Trump~Jr.\ are 100\% accurate.
The meeting was a complete waste of time and Don was never told Ms.~Veselnitskaya's name prior to the meeting.
Ms.~Veselnitskaya mostly talked about the Magnitsky Act and Russian adoption laws and the meeting lasted 20 to 30~minutes at most.
There was never any follow up and nothing ever came of the meeting.% 767
\footnote{7/10/17 Email, Goldstone to Futerfas \& Garten.}
\end{quote}
\blackout{Grand Jury}
the statement drafted by Trump Organization representatives was
\blackout{Grand Jury}% 768
\footnote{\blackout{Grand Jury}}
He proposed a different statement, asserting that he had been asked ``by [his] client in Moscow---Emin Agalarov---to facilitate a meeting between a Russian attorney (Natalia Veselnitzkaya [sic]) and Donald Trump~Jr.
The lawyer had apparently stated that she had some information regarding funding to the DNC from Russia, which she believed Mr.~Trump~Jr.\ might find interesting.''% 769
\footnote{7/10/17 Email, Goldstone to Futerfas \& Garten.}
Goldstone never released either statement.% 770
\footnote{\blackout{Grand Jury}}
On the Russian end, there were also communications about what participants should say about the June~9 meeting.
Specifically, the organization that hired Samochornov---an anti-Magnitsky Act group controlled by Veselnitskaya and the owner of Prevezon---offered to pay \$90,000 of Samochornov's legal fees.% 771
\footnote{Samochornov 7/13/17 302, at~1; \blackout{Grand Jury}}
At Veselnitskaya's request, the organization sent Samochornov a transcript of a Veselnitskaya press interview, and Samochornov understood that the organization would pay his legal fees only if he made statements consistent with Veselnitskaya's.% 772
\footnote{\blackout{Grand Jury} Samochornov 7/13/17 302, at~1.}
Samochornov declined, telling the Office that he did not want to perjure himself.% 773
\footnote{Samochornov 7/13/17 302, at~1.}
The individual who conveyed Veselnitskaya's request to Samochornov stated that he did not expressly condition payment on following Veselnitskaya's answers but, in hindsight, recognized that by sending the transcript, Samochornov could have interpreted the offer of assistance to be conditioned on his not contradicting Veselnitskaya's account.% 774
\footnote{\blackout{Grand Jury}}
\hyperlink{subsection.2.2.7}{Volume~II, Section~II.G}, \textit{infra}, discusses interactions between President Trump, Trump~Jr., and others in June and July 2017 regarding the June~9 meeting.
\subsubsection{Events at the Republican National Convention}
Trump Campaign officials met with Russian Ambassador Sergey Kislyak during the week of the Republican National Convention.
The evidence indicates that those interactions were brief and non-substantive.
During platform committee meetings immediately before the Convention, J.D.~Gordon, a senior Campaign advisor on policy and national security, diluted a proposed amendment to the Republican Party platform expressing support for providing ``lethal'' assistance to Ukraine in response to Russian aggression.
Gordon requested that platform committee personnel revise the proposed amendment to state that only ``appropriate'' assistance be provided to Ukraine.
The original sponsor of the ``lethal'' assistance amendment stated that Gordon told her (the sponsor) that he was on the phone with candidate Trump in connection with his request to dilute the language.
Gordon denied making that statement to the sponsor, although he acknowledged it was possible he mentioned having previously spoken to the candidate about the subject matter.
The investigation did not establish that Gordon spoke to or was directed by the candidate to make that proposal.
Gordon said that he sought the change because he believed the proposed language was inconsistent with Trump's position on Ukraine.
\paragraph{Ambassador Kislyak's Encounters with Senator Sessions and J.D.~Gordon the Week of the RNC}
In July 2016, Senator Sessions and Gordon spoke at the Global Partners in Diplomacy event, a conference co-sponsored by the State Department and the Heritage Foundation held in Cleveland, Ohio the same week as the Republican National Convention (RNC or ``Convention'').% 775
\footnote{Gordon 8/29/17 302, at~9;
Sessions 1/17/18 302, at~22;
Allan Smith, \textit{We Now Know More About why Jeff Sessions and a Russian Ambassador Crossed Paths at the Republican Convention}, Business Insider (Mar.~2, 2017).}
Approximately 80 foreign ambassadors to the United States, including Kislyak, were invited to the conference.% 776
\footnote{Gordon 8/29/17 302, at~9;
Laura DeMarco, \textit{Global Cleveland and Sen.~Bob Corker Welcome International Republican National Convention Guests}, Cleveland Plain Dealer (July~20, 2016).
}
On July~20, 2016, Gordon and Sessions delivered their speeches at the conference.% 777
\footnote{Gordon 8/29/17 302, at~9;
Sessions 1/17/18 302, at~22.}
In his speech, Gordon stated in pertinent part that the United States should have better relations with Russia.% 778
\footnote{Gordon 8/29/17 302, at~9.}
During Sessions's speech, he took questions from the audience, one of which may have been asked by~Kislyak.% 779
\footnote{Sessions 1/17/18 302, at~22;
Luff 1/30/18 302, at~3.}
When the speeches concluded, several ambassadors lined up to greet the speakers.% 780
\footnote{Gordon 8/29/17 302, at~9;
Luff 1/30/18 302, at~3.}
Gordon shook hands with Kislyak and reiterated that he had meant what he said in the speech about improving U.S.--Russia relations.% 781
\footnote{Gordon 8/29/17 302, at~9.}
Sessions separately spoke with between six and 12 ambassadors, including Kislyak.% 782
\footnote{Sessions 1/17/18 302, at~22; Luff 1/30/18 302, at~3;
\textit{see also} \hyperlink{paragraph.1.4.1.4.2}{Volume~I, Section~IV.A.4.b}, \textit{supra} (explaining that Sessions and Kislyak may have met three months before this encounter during a reception held on April~26, 2016, at the Mayflower Hotel).}
Although Sessions stated during interviews with the Office that he had no specific recollection of what he discussed with Kislyak, he believed that the two spoke for only a few minutes and that they would have exchanged pleasantries and said some things about U.S.--Russia relations.% 783
\footnote{Sessions 1/17/18 302, at~22.}
Later that evening, Gordon attended a reception as part of the conference.% 784
\footnote{Gordon 8/29/17 302, at~9--10.}
Gordon ran into Kislyak as the two prepared plates of food, and they decided to sit at the same table to eat.% 785
\footnote{Gordon 8/29/17 302, at~9--10.}
They were joined at that table by the ambassadors from Azerbaijan and Kazakhstan, and by Trump Campaign advisor Carter Page.% 786
\footnote{Gordon 8/29/17 302, at~10;
\textit{see also} \hyperlink{paragraph.1.4.1.3.4}{Volume~I, Section~IV.A.3.d}, \textit{supra} (explaining that Page acknowledged meeting Kislyak at this event).
}
As they ate, Gordon and Kislyak talked for what Gordon estimated to have been three to five minutes, during which Gordon again mentioned that he meant what he said in his speech about improving U.S.--Russia relations.% 787
\footnote{Gordon 8/29/17 302, at~9--10.}
\paragraph{Change to Republican Party Platform}
In preparation for the 2016 Convention, foreign policy advisors to the Trump Campaign, working with the Republican National Committee, reviewed the 2012 Convention's foreign policy platform to identify divergence between the earlier platform and candidate Trump's positions.% 788
\footnote{Gordon 8/29/17 302, at~9--10.}
The Campaign team discussed toning down language from the 2012 platform that identified Russia as the country's number one threat, given the candidate's belief that there needed to be better U.S. relations with Russia.% 789
\footnote{Gordon 8/29/17 302, at~9--10.}
The RNC Platform Committee sent the 2016 draft platform to the National Security and Defense Platform Subcommittee on July~10, 2016, the evening before its first meeting to propose amendments.% 790
\footnote{Gordon 8/29/17 302, at~10;
Hoff 5/26/17 302, at~1--2.}
Although only delegates could participate in formal discussions and vote on the platform, the Trump Campaign could request changes, and members of the Trump Campaign attended committee meetings.% 791
\footnote{Hoff 5/26/17 302, at~1;
Gordon 9/7/17 302, at~10.}
John Mashburn, the Campaign's policy director, helped oversee the Campaign's involvement in the platform committee meetings.% 792
\footnote{Mashburn 6/25/18 302, at~4;
Manafort 9/20/18 302, at~7--8.}
He told the Office that he directed Campaign staff at the Convention, including J.D.~Gordon, to take a hands-off approach and only to challenge platform planks if they directly contradicted Trump's wishes.% 793
\footnote{Mashburn 6/25/18 302, at~4;
Gordon 8/29/17 302, at~10.}
On July~11, 2016, delegate Diana Denman submitted a proposed platform amendment that included provision of armed support for Ukraine.% 794
\footnote{DENMAN 000001--02, DENMAN 000012, DENMAN 000021--22;
Denman 12/4/17 302, at~1;
Denman 6/7/17 302, at~2.}
The amendment described Russia's ``ongoing military aggression'' in Ukraine and announced ``support'' for ``maintaining (and, if warranted, increasing) sanctions against Russia until Ukraine's sovereignty and territorial integrity are fully restored'' and for ``providing lethal defensive weapons to Ukraine's armed forces and greater coordination with NATO on defense planning.''% 795
\footnote{DENMAN 000001--02, DENMAN 000012, DENMAN 000021--22.}
Gordon reviewed the proposed platform changes, including Denman's.% 796
\footnote{Gordon 8/29/17 302, at~10--11.}
Gordon stated that he flagged this amendment because of Trump's stated position on Ukraine, which Gordon personally heard the candidate say at the March~31 foreign policy meeting---namely, that the Europeans should take primary responsibility for any assistance to Ukraine, that there should be improved U.S.--Russia relations, and that he did not want to start World War~III over that region.% 797
\footnote{Gordon 8/29/17 302, at~11;
Gordon 9/7/17 302, at~11;
Gordon 2/14/19 302, at~1--2, 5--6.}
Gordon told the Office that Trump's statements on the campaign trail following the March meeting underscored those positions to the point where Gordon felt obliged to object to the proposed platform change and seek its dilution.% 798
\footnote{Gordon 2/14/19 302, at~5--6.}
On July~11, 2016, at a meeting of the National Security and Defense Platform Subcommittee, Denman offered her amendment.% 799
\footnote{Denman 6/7/17 302, at~2;
\textit{see} DENMAN 000014.}
Gordon and another Campaign staffer, Matt Miller, approached a committee co-chair and asked him to table the amendment to permit further discussion.% 800
\footnote{Denman 6/7/17 302, at~2;
Denman 12/4/17 302, at~2;
Gordon 9/7/17 302, at~11--12;
\textit{see} Hoff 5/26/17 302, at~2.}
Gordon's concern with the amendment was the language about providing ``lethal defensive weapons to Ukraine.''% 801
\footnote{Denman 6/7/17 302, at~3.}
Miller did not have any independent basis to believe that this language contradicted Trump's views and relied on Gordon's recollection of the candidate's views.% 802
\footnote{M. Miller 10/25/17 302, at~3.}
According to Denman, she spoke with Gordon and Matt Miller, and they told her that they had to clear the language and that Gordon was ``talking to New York.''% 803
\footnote{Denman 12/4/17 302, at~2;
Denman 6/7/17 302, at~2.}
Denman told others that she was asked by the two Trump Campaign staffers to strike ``lethal defens[iv]e weapons'' from the proposal but that she refused.% 804
\footnote{Hoff 5/26/17 302, at~2.}
Denman recalled Gordon saying that he was on the phone with candidate Trump, but she was skeptical whether that was true.% 805
\footnote{Denman 6/7/17 302, at~2--3, 3--4;
Denman 12/4/17 302, at~2.}
Gordon denied having told Denman that he was on the phone with Trump, although he acknowledged it was possible that he mentioned having previously spoken to the candidate about the subject matter.% 806
\footnote{Gordon 2/14/19 302, at~7.}
Gordon's phone records reveal a call to Sessions's office in Washington that afternoon, but do not include calls directly to a number associated with Trump.% 807
\footnote{Call Records of J.D.~Gordon \blackout{Grand Jury}.
Gordon stated to the Office that his calls with Sessions were unrelated to the platform change. Gordon 2/14/19 302, at~7.}
And according to the President's written answers to the Office's questions, he does not recall being involved in the change in language of the platform amendment.% 808
\footnote{Written Responses of Donald J. Trump (Nov.~20, 2018), at~17 (Response to Question~IV, Part~(f)).}
Gordon stated that he tried to reach Rick Dearborn, a senior foreign policy advisor, and Mashburn, the Campaign policy director.
Gordon stated that he connected with both of them (he could not recall if by phone or in person) and apprised them of the language he took issue with in the proposed amendment.
Gordon recalled no objection by either Dearborn or Mashburn and that all three Campaign advisors supported the alternative formulation (``appropriate assistance'').% 809
\footnote{Gordon 2/14/19 302, at~6--7;
Gordon 9/7/17 302, at~11--12;
\textit{see} Gordon 8/29/17 302, at~11.}
Dearborn recalled Gordon warning them about the amendment, but not weighing in because Gordon was more familiar with the Campaign's foreign policy stance.% 810
\footnote{Dearborn 11/28/17 302, at~7--8.}
Mashburn stated that Gordon reached him, and he told Gordon that Trump had not taken a stance on the issue and that the Campaign should not intervene.% 811
\footnote{Mashburn 6/25/18 302, at~4.}
When the amendment came up again in the committee's proceedings, the subcommittee changed the amendment by striking the ``lethal defens[iv]e weapons'' language and replacing it with ``appropriate assistance.''% 812
\footnote{Hoff 5/26/17 302, at~2--3;
\textit{see} Denman 12/4/17 302, at~2--3;
Gordon 8/29/17 302, at~11.}
Gordon stated that he and the subcommittee co-chair ultimately agreed to replace the language about armed assistance with ``appropriate assistance.''% 813
\footnote{Gordon 8/29/17 302, at~11;
Gordon 9/7/17 302, at~12.}
The subcommittee accordingly approved Denman's amendment but with the term ``appropriate assistance.''% 814
\footnote{Hoff 5/26/17 302, at~2--3.}
Gordon stated that, to his recollection, this was the only change sought by the Campaign.% 815
\footnote{Gordon 2/14/19 302, at~6.}
Sam Clovis, the Campaign's national co-chair and chief policy advisor, stated he was surprised by the change and did not believe it was in line with Trump's stance.% 816
\footnote{Clovis 10/3/17 302, at~10--11.}
Mashburn stated that when he saw the word ``appropriate assistance,'' he believed that Gordon had violated Mashburn's directive not to intervene.% 817
\footnote{Mashburn 6/25/18 302, at~4.}
\subsubsection{Post-Convention Contacts with Kislyak}
Ambassador Kislyak continued his efforts to interact with Campaign officials with responsibility for the foreign-policy portfolio---among them Sessions and Gordon---in the weeks after the Convention.
The Office did not identify evidence in those interactions of coordination between the Campaign and the Russian government.
\paragraph{Ambassador Kislyak Invites J.D.~Gordon to Breakfast at the Ambassador's Residence}
On August~3, 2016, an official from the Embassy of the Russian Federation in the United States wrote to Gordon ``[o]n behalf of\thinspace'' Ambassador Kislyak inviting Gordon ``to have breakfast/tea with the Ambassador at his residence'' in Washington, D.C. the following week.% 818
\footnote{DJTFP00004828 (8/3/16 Email, Pchelyakov [\UseVerb{russianembassyorg}] to Gordon).}
Gordon responded five days later to decline the invitation.
He wrote, ``[t]hese days are not optimal for us, as we are busily knocking down a constant stream of false media stories while also preparing for the first debate with HRC\null.
Hope to take a raincheck for another time when things quiet down a bit.
Please pass along my regards to the Ambassador.''% 819
\footnote{DJTFP00004953 (8/8/16 Email, Gordon to \UseVerb{russianembassyorg}).}
The investigation did not identify evidence that Gordon made any other arrangements to meet (or met) with Kislyak after this email.
\paragraph{Senator Sessions's September 2016 Meeting with Ambassador Kislyak}
Also in August 2016, a representative of the Russian Embassy contacted Sessions's Senate office about setting up a meeting with~Kislyak.% 820
\footnote{Luff 1/30/18 302, at~5.}
At the time, Sessions was a member of the Senate Foreign Relations Committee and would meet with foreign officials in that capacity.% 821
\footnote{Sessions 1/17/18 302, at~23--24;
Luff 1/30/18 302, at~5.}
But Sessions's staff reported, and Sessions himself acknowledged, that meeting requests from ambassadors increased substantially in 2016, as Sessions assumed a prominent role in the Trump Campaign and his name was mentioned for potential cabinet-level positions in a future Trump Administration.% 822
\footnote{Sessions 1/17/18 302, at~23--24;
Luff 1/30/18 302, at~5;
Landrum 2/27/18 302, at~3--5.}
On September~8, 2016, Sessions met with Kislyak in his Senate office.% 823
\footnote{Sessions 1/17/18 302, at~23.}
Sessions said that he believed he was doing the Campaign a service by meeting with foreign ambassadors, including Kislyak.% 824
\footnote{Sessions 1/17/18 302, at~23.}
He was accompanied in the meeting by at least two of his Senate staff: Sandra Luff, his legislative director; and Pete Landrum, who handled military affairs.% 825
\footnote{Sessions 1/17/18 302, at~23;
Luff 1/30/18 302, at~5--6;
Landrum 2/27/18 302, at~4--5 (stating he could not remember if election was discussed).}
The meeting lasted less than 30~minutes.% 826
\footnote{Luff 1/30/18 302, at~6;
Landrum 2/27/18 302, at~5.}
Sessions voiced concerns about Russia's sale of a missile-defense system to Iran, Russian planes buzzing U.S. military assets in the Middle East, and Russian aggression in emerging democracies such as Ukraine and Moldova.% 827
\footnote{Luff 1/30/18 302, at~6;
Landrum 2/27/18 302, at~4--5.}
Kislyak offered explanations on these issues and complained about NATO land forces in former Soviet-bloc countries that border Russia.% 828
\footnote{Luff 1/30/18 302, at~6;
Landrum 2/27/18 302, at~4--5.}
Landrum recalled that Kislyak referred to the presidential campaign as ``an interesting campaign,''% 829
\footnote{Landrum 2/27/18 302, at~5.}
and Sessions also recalled Kislyak saying that the Russian government was receptive to the overtures Trump had laid out during his campaign.% 830
\footnote{Sessions 1/17/18 302, at~23.
Sessions also noted that ambassadors came to him for information about Trump and hoped he would pass along information to Trump.
Sessions 1/17/18 302, at~23--24.}
None of the attendees, though, remembered any discussion of Russian election interference or any request that Sessions convey information from the Russian government to the Trump Campaign.% 831
\footnote{Sessions 1/17/18 302, at~23;
Luff 1/30/18 302, at~6;
Landrum 2/27/18 302, at~5.}
During the meeting, Kislyak invited Sessions to further discuss U.S.--Russia relations with him over a meal at the ambassador's residence.% 832
\footnote{Luff 1/30/18 302, at~5;
Landrum 2/27/18 302, at~4.}
Sessions was non-committal when Kislyak extended the invitation.
After the meeting ended, Luff advised Sessions against accepting the one-on-one meeting with Kislyak, whom she assessed to be an ``old school KGB guy.''% 833
\footnote{Luff 1/30/18 302, at~5.}
Neither Luff nor Landrum recalled that Sessions followed up on the invitation or made any further effort to dine or meet with Kislyak before the November 2016 election.% 834
\footnote{Luff 1/30/18 302, at~6;
Landrum 2/27/18 302, at~4--5.}
Sessions and Landrum recalled that, after the election, some efforts were made to arrange a meeting between Sessions and~Kislyak.% 835
\footnote{Sessions 1/17/18 302, at~23.}
According to Sessions, the request came through CNI and would have involved a meeting between Sessions and Kislyak, two other ambassadors, and the Governor of Alabama.% 836
\footnote{Sessions 1/17/18 302, at~23.}
Sessions, however, was in New York on the day of the anticipated meeting and was unable to attend.% 837
\footnote{Sessions 1/17/18 302, at~23.}
The investigation did not identify evidence that the two men met at any point after their September~8 meeting.
\subsubsection{Paul Manafort}
Paul Manafort served on the Trump Campaign, including a period as campaign chairman, from March to August 2016.% 838
\footnote{On August~21, 2018, Manafort was convicted in the Eastern District of Virginia on eight tax, Foreign Bank Account Registration (FBAR), and bank fraud charges. On September~14, 2018, Manafort pleaded guilty in the District of Columbia to (1)~conspiracy to defraud the United States and conspiracy to commit offenses against the United States (money laundering, tax fraud, FBAR, Foreign Agents Registration Act (FARA), and FARA false statements), and (2)~conspiracy to obstruct justice (witness tampering).
Manafort also admitted criminal conduct with which he had been charged in the Eastern District of Virginia, but as to which the jury hung.
The conduct at issue in both cases involved Manafort's work in Ukraine and the money he earned for that work, as well as crimes after the Ukraine work ended. On March~7, 2019, Manafort was sentenced to 47 months of imprisonment in the Virginia prosecution.
On March~13, the district court in D.C. sentenced Manafort to a total term of 73 months: 60 months on the Count 1 conspiracy (with 30 of those months to run concurrent to the Virginia sentence), and 13 months on the Count 1 conspiracy, to be served consecutive to the other two sentences.
The two sentences resulted in a total term of 90 months.
}
Manafort had connections to Russia through his prior work for Russian oligarch Oleg Deripaska and later through his work for a pro-Russian regime in Ukraine.
Manafort stayed in touch with these contacts during the campaign period through Konstantin Kilimnik, a longtime Manafort employee who previously ran Manafort's office in Kiev and who the FBI assesses to have ties to Russian intelligence.
Manafort instructed Rick Gates, his deputy on the Campaign and a longtime employee,% 839
\footnote{As noted in \hyperlink{paragraph.1.3.4.1.2}{Volume~I, Section~III.D.1.b}, \textit{supra}, Gates pleaded guilty to two criminal charges in the District of Columbia, including making a false statement to the FBI, pursuant to a plea agreement.
He has provided information and in-court testimony that the Office has deemed to be reliable. \textit{See also} Transcript at~16, \textit{United States~v.\ Paul J. Manafort, Jr.}, 1:17-cr-201 (D.D.C. Feb.~13, 2019), Doc.~514 (``\textit{Manafort} 2/13/19 Transcript'') (court's explanation of reasons to credit Gates's statements in one instance).}
to provide Kilimnik with updates on the Trump Campaign---including internal polling data, although Manafort claims not to recall that specific instruction.
Manafort expected Kilimnik to share that information with others in Ukraine and with Deripaska.
Gates periodically sent such polling data to Kilimnik during the campaign.
Manafort also twice met Kilimnik in the United States during the campaign period and conveyed campaign information.
The second meeting took place on August~2, 2016, in New York City.
Kilimnik requested the meeting to deliver in person a message from former Ukrainian President Viktor Yanukovych, who was then living in Russia.
The message was about a peace plan for Ukraine that Manafort has since acknowledged was a ``backdoor'' means for Russia to control eastern Ukraine.
Several months later, after the presidential election, Kilimnik wrote an email to Manafort expressing the view---which Manafort later said he shared---that the plan's success would require U.S. support to succeed: ``all that is required to start the process is a very minor `wink' (or slight push) from [Donald Trump].''% 840
\footnote{The email was drafted in Kilimnik's DMP email account (in English) \blackout{Investigative Technique}}
The email also stated that if Manafort were designated as the U.S. representative and started the process, Yanukovych would ensure his reception in Russia ``at the very top level.''
Manafort communicated with Kilimnik about peace plans for Ukraine on at least four occasions after their first discussion of the topic on August~2, December 2016 (the Kilimnik email described above); January 2017; February 2017; and again in the spring of 2018.
The Office reviewed numerous Manafort email and text communications, and asked President Trump about the plan in written questions.% 841
\footnote{According to the President's written answers, he does not remember Manafort communicating to him any particular positions that Ukraine or Russia would want the United States to support.
Written Responses of Donald J. Trump (Nov.~20, 2018), at~16--17 (Response to Question~IV, Part~(d)).}
The investigation did not uncover evidence of Manafort's passing along information about Ukrainian peace plans to the candidate or anyone else in the Campaign or the Administration.
The Office was not, however, able to gain access to all of Manafort's electronic communications (in some instances, messages were sent using encryption applications).
And while Manafort denied that he spoke to members of the Trump Campaign or the new Administration about the peace plan, he lied to the Office and the grand jury about the peace plan and his meetings with Kilimnik, and his unreliability on this subject was among the reasons that the district judge found that he breached his cooperation agreement.% 842
\footnote{Manafort made several false statements during debriefings. Based on that conduct, the Office determined that Manafort had breached his plea agreement and could not be a cooperating witness. The judge presiding in Manafort's D.C. criminal case found by a preponderance of the evidence that Manafort intentionally made multiple false statements to the FBI, the Office, and the grand jury concerning his interactions and communications with Kilimnik (and concerning two other issues). Although the report refers at times to Manafort's statements, it does so only when those statements are sufficiently corroborated to be trustworthy, to identify issues on which Manafort's untruthful responses may themselves be of evidentiary value, or to provide Manafort's explanations for certain events, even when we were unable to determine whether that explanation was credible.}
The Office could not reliably determine Manafort's purpose in sharing internal polling data with Kilimnik during the campaign period.
Manafort
\blackout{Grand Jury}
did not see a downside to sharing campaign information, and told Gates that his role in the Campaign would be ``good for business'' and potentially a way to be made whole for work he previously completed in the Ukraine.
As to Deripaska, Manafort claimed that by sharing campaign information with him, Deripaska might see value in their relationship and resolve a ``disagreement''---a reference to one or more outstanding lawsuits.
Because of questions about Manafort's credibility and our limited ability to gather evidence on what happened to the polling data after it was sent to Kilimnik, the Office could not assess what Kilimnik (or others he may have given it to) did with it.
The Office did not identify evidence of a connection between Manafort's sharing polling data and Russia's interference in the election, which had already been reported by U.S. media outlets at the time of the August~2 meeting.
The investigation did not establish that Manafort otherwise coordinated with the Russian government on its election-interference efforts.
\paragraph{Paul Manafort's Ties to Russia and Ukraine}
Manafort's Russian contacts during the campaign and transition periods stem from his consulting work for Deripaska from approximately 2005 to 2009 and his separate political consulting work in Ukraine from 2005 to 2015, including through his company DMP International LLC (DMI).
Kilimnik worked for Manafort in Kiev during this entire period and continued to communicate with Manafort through at least June 2018.
Kilimnik, who speaks and writes Ukrainian and Russian, facilitated many of Manafort's communications with Deripaska and Ukrainian oligarchs.
\subparagraph{Oleg Deripaska Consulting Work}
In approximately 2005, Manafort began working for Deripaska, a Russian oligarch who has a global empire involving aluminum and power companies and who is closely aligned with Vladimir Putin.% 843
\footnote{Dinchuk et~al., \textit{Russian Tycoon Deripaska in Putin Delegation to China}, Reuters (June~8, 2018).}
A memorandum describing work that Manafort performed for Deripaska in 2005 regarding the post-Soviet republics referenced the need to brief the Kremlin and the benefits that the work could confer on ``the Putin Government.''% 844
\footnote{6/23/05 Memo, Manafort \& Davis to Deripaska \& Rothchild.}
Gates described the work Manafort did for Deripaska as ``political risk insurance,'' and explained that Deripaska used Manafort to install friendly political officials in countries where Deripaska had business interests.% 845
\footnote{Gates 2/2/18 302, at~7.}
Manafort's company earned tens of millions of dollars from its work for Deripaska and was loaned millions of dollars by Deripaska as well.% 846
\footnote{Manafort 9/20/18 302, at~2--5;
Manafort Income by Year, 2005--2015;
Manafort Loans from Wire Transfers, 2005--2015.}
In 2007, Deripaska invested through another entity in Pericles Emerging Market Partners L.P. (``Pericles''), an investment fund created by Manafort and former Manafort business partner Richard Davis.
The Pericles fund was established to pursue investments in Eastern Europe.% 847
\footnote{Gates 3/12/18 302, at~5.}
Deripaska was the sole investor.% 848
\footnote{Manafort 12/16/15 Dep., at~157:8--11.}
Gates stated in interviews with the Office that the venture led to a deterioration of the relationship between Manafort and Deripaska.% 849
\footnote{Gates 2/2/18 302, at~9.}
In particular, when the fund failed, litigation between Manafort and Deripaska ensued.
Gates stated that, by 2009, Manafort's business relationship with Deripaska had ``dried up.''% 850
\footnote{Gates 2/2/18 302, at~6.}
According to Gates, various interactions with Deripaska and his intermediaries over the past few years have involved trying to resolve the legal dispute.% 851
\footnote{Gates 2/2/18 302, at~9--10.}
As described below, in 2016, Manafort, Gates, Kilimnik, and others engaged in efforts to revive the Deripaska relationship and resolve the litigation.
\subparagraph{Political Consulting Work}
Through Deripaska, Manafort was introduced to Rinat Akhmetov, a Ukrainian oligarch who hired Manafort as a political consultant.% 852
\footnote{Manafort 7/30/14 302, at~1;
Manafort 9/20/18 302, at~2.}
In 2005, Akhmetov hired Manafort to engage in political work supporting the Party of Regions,% 853
\footnote{Manafort 9/11/18 302, at~5--6.}
a political party in Ukraine that was generally understood to align with Russia.
Manafort assisted the Party of Regions in regaining power, and its candidate, Viktor Yanukovych, won the presidency in 2010.
Manafort became a close and trusted political advisor to Yanukovych during his time as President of Ukraine.
Yanukovych served in that role until 2014, when he fled to Russia amidst popular protests.% 854
\footnote{Gates 3/16/18 302, at~1;
Davis 2/8/18 302, at~9;
Devine 7/6/18 302, at~2--3.}
\subparagraph{Konstantin Kilimnik}
Kilimnik is a Russian national who has lived in both Russia and Ukraine and was a longtime Manafort employee.% 855
\footnote{Patten 5/22/18 302, at~5;
Gates 1/29/18 302, at~18--19;
10/28/97 Kilimnik Visa Record, U.S. Department of State.}
Kilimnik had direct and close access to Yanukovych and his senior entourage, and he facilitated communications between Manafort and his clients, including Yanukovych and multiple Ukrainian oligarchs.% 856
\footnote{Gates 1/29/18 302, at~18--19;
Patten 5/22/18 302, at~8;
Gates 1/31/18 302, at~4--5;
Gates 1/30/18 302, at~2;
Gates 2/2/18 302, at~11.}
Kilimnik also maintained a relationship with Deripaska's deputy, Viktor Boyarkin,% 857
\footnote{Gates 1/29/18 302, at~18;
Patten 5/22/18 302, at~8.}
a Russian national who previously served in the defense attaché office of the Russian Embassy to the United States.% 858
\footnote{Boyarkin Visa Record, U.S. Department of State.}
Manafort told the Office that he did not believe Kilimnik was working as a Russian ``spy.''% 859
\footnote{Manafort 9/11/18 302, at~5.}
The FBI, however, assesses that Kilimnik has ties to Russian intelligence.% 860
\footnote{The Office has noted Kilimnik's assessed ties to Russian intelligence in public court filings.
\textit{E.g.}, Gov't Opp.\ to Mot.\ to Modify, \textit{United States~v.\ Paul J. Manafort, Jr.}, 1:17-cr-201 (D.D.C. Dec.~4, 2017), Doc.~73, at~2 (``\textit{Manafort} (D.D.C.) Gov't Opp.\ to Mot.\ to Modify'').}
Several pieces of the Office's evidence---including witness interviews and emails obtained through court-authorized search warrants---support that assessment:
\begin{itemize}
\item Kilimnik was born on April~27, 1970, in Dnipropetrovsk Oblast, then of the Soviet Union, and attended the Military Institute of the Ministry of Defense from 1987 until 1992.% 861
\footnote{12/17/16 Kilimnik Visa Record, U.S., Department of State.}
Sam Patten, a business partner to Kilimnik,% 862
\footnote{In August 2018, Patten pleaded guilty pursuant to a plea agreement to violating the Foreign Agents Registration Act, and admitted in his Statement of Offense that he also misled and withheld documents from the Senate Select Committee on Intelligence in the course of its investigation of Russian election interference.
Plea Agreement, \textit{United States~v.\ W.~Samuel Patten}, 1:18-cr-260 (D.D.C. Aug.~31, 2018), Doc.~6;
Statement of Offense, \textit{United States~v.\ W.~Samuel Patten}, 1:18-cr-260 (D.D.C. Aug.~31, 2018), Doc.~7.}
stated that Kilimnik told him that he was a translator in the Russian army for seven years and that he later worked in the Russian armament industry selling arms and military equipment.% 863
\footnote{Patten 5/22/18 302, at~5--6.}
\item U.S. government visa records reveal that Kilimnik obtained a visa to travel to the United States with a Russian diplomatic passport in 1997.% 864
\footnote{10/28/97 Kilimnik Visa Record, U.S. Department of State.}
\item Kilimnik worked for the International Republican Institute's (IRI) Moscow office, where he did translation work and general office management from 1998 to 2005.% 865
\footnote{Nix 3/30/18 302, at~1--2.}
While another official recalled the incident differently,% 866
\footnote{Nix 3/30/18 302, at~2.}
one former associate of Kilimnik's at TRI told the FBI that Kilimnik was fired from his post because his links to Russian intelligence were too strong.
The same individual stated that it was well known at IRI that Kilimnik had links to the Russian government.% 867
\footnote{Lenzi 1/30/18 302, at~2.}
\item Jonathan Hawker, a British national who was a public relations consultant at FTI Consulting, worked with DMI on a public relations campaign for Yanukovych.
After Hawker's work for DMI ended, Kilimnik contacted Hawker about working for a Russian government entity on a public-relations project that would promote, in Western and Ukrainian media, Russia's position on its 2014 invasion of Crimea.% 868
\footnote{Hawker 1/9/18 302, at~13;
3/18/14 Email, Hawker \& Tulukbaev.}
\item Gates suspected that Kilimnik was a ``spy,'' a view that he shared with Manafort, Hawker, and Alexander van der Zwaan,% 869
\footnote{van der Zwaan pleaded guilty in the U.S. District Court for the District of Columbia to making false statements to the Special Counsel's Office.
Plea Agreement, \textit{United States~v.\ Alex van der Zwaan}, 1:18-cr-31 (D.D.C. Feb.~20, 2018), Doc.~8.}
an attorney who had worked with DMI on a report for the Ukrainian Ministry of Foreign Affairs.% 870
\footnote{Hawker 6/9/18 302, at~4;
van der Zwaan 11/3/17 302, at~22.
Manafort said in an interview that Gates had joked with Kilimnik about Kilimnik's going to meet with his KGB handler.
Manafort 10/16/18 302, at~7.}
\end{itemize}
\blackout{Investigative Technique}
\paragraph{Contacts during Paul Manafort's Time with the Trump Campaign}
\subparagraph{Paul Manafort Joins the Campaign}
Manafort served on the Trump Campaign from late March to August~19, 2016.
On March~29, 2016, the Campaign announced that Manafort would serve as the Campaign's ``Convention Manager.''% 871
\footnote{\textit{Press Release---Donald J. Trump Announces Campaign Convention Manager Paul J. Manafort}, The American Presidency Project -- U.C. Santa Barbara (Mar.~29, 2016).}
On May~19, 2016, Manafort was promoted to campaign chairman and chief strategist, and Gates, who had been assisting Manafort on the Campaign, was appointed deputy campaign chairman.% 872
\footnote{Gates 1/29/18 302, at~8; Meghan Keneally, \textit{Timeline of Manafort's role in the Trump Campaign}, ABC News (Oct.~20, 2017).}
Thomas Barrack and Roger Stone both recommended Manafort to candidate Trump.% 873
\footnote{Gates 1/29/18 302, at~7--8;
Manafort 9/11/18 302, at~1--2;
Barrack 12/12/17 302, at~3.}
In early 2016, at Manafort's request, Barrack suggested to Trump that Manafort join the Campaign to manage the Republican Convention.% 874
\footnote{Barrack 12/12/17 302, at~3;
Gates 1/29/18 302, at~7--8.}
Stone had worked with Manafort from approximately 1980 until the mid-1990s through various consulting and lobbying firms.
Manafort met Trump in 1982 when Trump hired the Black, Manafort, Stone and Kelly lobbying firm.% 875
\footnote{Manafort 10/16/18 302, at~6.}
Over the years, Manafort saw Trump at political and social events in New York City and at Stone's wedding, and Trump requested VIP status at the 1988 and~1996 Republican conventions worked by Manafort.% 876
\footnote{Manafort 10/16/18 302, at~6.}
According to Gates, in March 2016, Manafort traveled to Trump's Mar-a-Lago estate in Florida to meet with Trump.
Trump hired him at that time.% 877
\footnote{Gates 2/2/18 302, at~10.}
Manafort agreed to work on the Campaign without pay.
Manafort had no meaningful income at this point in time, but resuscitating his domestic political campaign career could be financially beneficial in the future.
Gates reported that Manafort intended, if Trump won the Presidency, to remain outside the Administration and monetize his relationship with the Administration.% 878
\footnote{Gates 1/30/18 302, at~4.}
\subparagraph{Paul Manafort's Campaign-Period Contacts}
Immediately upon joining the Campaign, Manafort directed Gates to prepare for his review separate memoranda addressed to Deripaska, Akhmetov, Serhiy Lyovochkin, and Boris Kolesnikov,% 879
\footnote{Gates 2/2/18 302, at~11.}
the last three being Ukrainian oligarchs who were senior Opposition Bloc officials.% 880
\footnote{\textit{See} Sharon La Franiere, Manafort's Trial Isn't About Russia, but It Will Be in the Air, New York Times (July~30, 2018);
Tierney Sneed, \textit{Prosecutors Believe Manafort Made \$60 Million Consulting in Ukraine}, Talking Points Memo (July~30, 2018);
Mykola Vorobiov, \textit{How Pro-Russian Forces Will Take Revenge on Ukraine}, Atlantic Council (Sept.~23, 2018);
Sergii Leshchenko, \textit{Ukraine's Oligarchs Are Still Calling the Shots}, Foreign Policy (Aug.~14, 2014);
Interfax-Ukraine, \textit{Kolesnikov: Inevitability of Punishment Needed for Real Fight Against Smuggling in Ukraine}, Kyiv Post (June~23, 2018);
Igor Kossov, \textit{Kyiv Hotel Industry Makes Room for New Entrants}, Kyiv Post (Mar.~7, 2019);
Markian Kuzmowycz, \textit{How the Kremlin Can Win Ukraine's Elections}, Atlantic Council (Nov.~19, 2018).
The Opposition Bloc is a Ukraine political party that largely reconstituted the Party of Regions.
}
The memoranda described Manafort's appointment to the Trump Campaign and indicated his willingness to consult on Ukrainian politics in the future.
On March~30, 2016, Gates emailed the memoranda and a press release announcing Manafort's appointment to Kilimnik for translation and dissemination.% 881
\footnote{3/30/16 Email, Gates to Kilimnik.}
Manafort later followed up with Kilimnik to ensure his messages had been delivered, emailing on April~11, 2016 to ask whether Kilimnik had shown ``our friends'' the media coverage of his new role.% 882
\footnote{4/11/16 Email, Manafort \& Kilimnik.}
Kilimnik replied, ``Absolutely.
Every article.''
Manafort further asked: ``How do we use to get whole.
Has Ovd [Oleg Vladimirovich Deripaska] operation seen?''
Kilimnik wrote back the same day, ``Yes, I have been sending everything to Victor [Boyarkin, Deripaska's deputy], who has been forwarding the coverage directly to OVD.''% 883
\footnote{4/11/16 Email, Manafort \& Kilimnik.}
Gates reported that Manafort said that being hired on the Campaign would be ``good for business'' and increase the likelihood that Manafort would be paid the approximately \$2 million he was owed for previous political consulting work in Ukraine.% 884
\footnote{Gates 2/2/18 302, at~10.}
Gates also explained to the Office that Manafort thought his role on the Campaign could help ``confirm'' that Deripaska had dropped the Pericles lawsuit, and that Gates believed Manafort sent polling data to Deripaska (as discussed further below) so that Deripaska would not move forward with his lawsuit against Manafort.% 885
\footnote{Gates 2/2/18 302, at~11;
Gates 9/27/18 302 (serial 740), at~2.}
Gates further stated that Deripaska wanted a visa to the United States, that Deripaska could believe that having Manafort in a position inside the Campaign or Administration might be helpful to Deripaska, and that Manafort's relationship with Trump could help Deripaska in other ways as well.% 886
\footnote{Gates 2/2/18 302, at~12.}
Gates stated, however, that Manafort never told him anything specific about what, if anything, Manafort might be offering Deripaska.% 887
\footnote{Gates 2/2/18 302, at~12.}
Gates also reported that Manafort instructed him in April 2016 or early May~2016 to send Kilimnik Campaign internal polling data and other updates so that Kilimnik, in turn, could share it with Ukrainian oligarchs.% 888
\footnote{Gates 1/31/18 302, at~17;
Gates 9/27/18 302 (serial 740), at~2.
In a later interview with the Office, Gates stated that Manafort directed him to send polling data to Kilimnik after a May~7, 2016 meeting between Manafort and Kilimnik in New York, discussed in \hyperlink{subparagraph.1.4.1.8.2.3}{Volume~I, Section~IV.A.8.b.iii}, \textit{infra}.
Gates 11/7/18 302, at~3.}
Gates understood that the information would also be shared with Deripaska,
\blackout{Grand Jury}.% 889
\footnote{Gates 9/27/18 302, Part~II, at~2; \blackout{Grand Jury}}
Gates reported to the Office that he did not know why Manafort wanted him to send polling information, but Gates thought it was a way to showcase Manafort's work, and Manafort wanted to open doors to jobs after the Trump Campaign ended.% 890
\footnote{Gates 2/12/18 302, at~10;
Gates 1/31/18 302, at~17.}
Gates said that Manafort's instruction included sending internal polling data prepared for the Trump Campaign by pollster Tony Fabrizio.% 891
\footnote{Gates 9/27/18 302 (serial 740), at~2;
Gates 2/7/18 302, at~15.}
Fabrizio had worked with Manafort for years and was brought into the Campaign by Manafort.
Gates stated that, in accordance with Manafort's instruction, he periodically sent Kilimnik polling data via WhatsApp; Gates then deleted the communications on a daily basis.% 892
\footnote{Gates 1/31/18 302, at~17.}
Gates further told the Office that, after Manafort left the Campaign in mid-August, Gates sent Kilimnik polling data less frequently and that the data he sent was more publicly available information and less internal data.% 893
\footnote{Gates 2/12/18 302, at~11--12.
According to Gates, his access to internal polling data was more limited because Fabrizio was himself distanced from the Campaign at that point.}
Gates's account about polling data is consistent
\blackout{Grand Jury}% 894
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}
with multiple emails that Kilimnik sent to U.S. associates and press contacts between late July and mid-August of~2016.
Those emails referenced ``internal polling,'' described the status of the Trump Campaign and Manafort's role in it, and assessed Trump's prospects for victory.% 895
\footnote{8/18/16 Email, Kilimnik to Dirkse;
8/18/16 Email, Kilimnik to Schultz;
8/18/16 Email, Kilimnik to Marson;
7/27/16 Email, Kilimnik to Ash;
8/18/16 Email, Kilimnik to Ash;
8/18/16 Email, Kilimnik to Jackson;
8/18/16 Email, Kilimnik to Mendoza-Wilson;
8/19/16 Email, Kilimnik to Patten.}
Manafort did not acknowledge instructing Gates to send Kilimnik internal data,
\blackout{Grand Jury}% 896
\footnote{\blackout{Grand Jury}}
The Office also obtained contemporaneous emails that shed light on the purpose of the communications with Deripaska and that are consistent with Gates's account.
For example, in response to a July~7, 2016, email from a Ukrainian reporter about Manafort's failed Deripaska-backed investment, Manafort asked Kilimnik whether there had been any movement on ``this issue with our friend.''% 897
\footnote{7/7/16 Email, Manafort to Kilimnik.}
Gates stated that ``our friend'' likely referred to Deripaska,% 898
\footnote{Gates 2/2/18 302, at~13.}
and Manafort told the Office that the ``issue'' (and ``our biggest interest,'' as stated below) was a solution to the Deripaska--Pericles issue.% 899
\footnote{Manafort 9/11/18 302, at~6.}
Kilimnik replied:
\begin{quote}
I am carefully optimistic on the question of our biggest interest.
Our friend [Boyarkin] said there is lately significantly more attention to the campaign in his boss' [Deripaska's] mind, and he will be most likely looking for ways to reach out to you pretty soon, understanding all the time sensitivity.
I am more than sure that it will be resolved and we will get back to the original relationship with V.'s boss [Deripaska].% 900
\footnote{7/8/16 Email, Kilimnik to Manafort.}
\end{quote}
Eight minutes later, Manafort replied that Kilimnik should tell Boyarkin's ``boss,'' a reference to Deripaska, ``that if he needs private briefings we can accommodate.''% 901
\footnote{7/8/16 Email, Kilimnik to Manafort;
Gates 2/2/18 302, at~13.}
Manafort has alleged to the Office that he was willing to brief Deripaska only on public campaign matters and gave an example: why Trump selected Mike Pence as the Vice-Presidential running mate.% 902
\footnote{Manafort 9/11/18 302, at~6.}
Manafort said he never gave Deripaska a briefing.% 903
\footnote{Manafort 9/11/18 302, at~6.}
Manafort noted that if Trump won, Deripaska would want to use Manafort to advance whatever interests Deripaska had in the United States and elsewhere.% 904
\footnote{Manafort 9/11/18 302, at~6.}
\subparagraph{Paul Manafort's Two Campaign-Period Meetings with Konstantin Kilimnik in the United States}
Manafort twice met with Kilimnik in person during the campaign period---once in May and again in August 2016.
The first meeting took place on May~7, 2016, in New York City.% 905
\footnote{\blackout{Investigative Technique}}
In the days leading to the meeting, Kilimnik had been working to gather information about the political situation in Ukraine.
That included information gleaned from a trip that former Party of Regions official Yuriy Boyko had recently taken to Moscow---a trip that likely included meetings between Boyko and high-ranking Russian officials.% 906
\footnote{4/26/16 Email, Kilimnik to Purcell, at~2;
Gates 2/2/18 302, at~12;
Patten 5/22/18 302, at~6--7;
Gates 11/7/18 302, at~3.}
Kilimnik then traveled to Washington, D.C. on or about May~5, 2016; while in Washington, Kilimnik had pre-arranged meetings with State Department employees.% 907
\footnote{5/7/16 Email, Kilimnik to Charap \& Kimmage;
5/7/16 Email, Kansanof to Kilimnik.}
Late on the evening of May~6, Gates arranged for Kilimnik to take a 3:00~a.m.\ train to meet Manafort in New York for breakfast on May~7.% 908
\footnote{5/6/16 Email, Manafort to Gates;
5/6/16 Email, Gates to Kilimnik.}
According to Manafort, during the meeting, he and Kilimnik talked about events in Ukraine, and Manafort briefed Kilimnik on the Trump Campaign, expecting Kilimnik to pass the information back to individuals in Ukraine and elsewhere.% 909
\footnote{Manafort 10/11/18 302, at~1.}
Manafort stated that Opposition Bloc members recognized Manafort's position on the Campaign was an opportunity, but Kilimnik did not ask for anything.% 910
\footnote{Manafort 10/11/18 302, at~1.}
Kilimnik spoke about a plan of Boyko to boost election participation in the eastern zone of Ukraine, which was the base for the Opposition Bloc.% 911
\footnote{Manafort 10/11/18 302, at~1.}
Kilimnik returned to Washington, D.C. right after the meeting with Manafort.
Manafort met with Kilimnik a second time at the Grand Havana Club in New York City on the evening of August~2, 2016.
The events leading to the meeting are as follows.
On July~28, 2016, Kilimnik flew from Kiev to Moscow.% 912
\footnote{7/25/16 Email, Kilimnik to \UseVerb{katrinyana} (2:17:34~a.m.).}
The next day, Kilimnik wrote to Manafort requesting that they meet, using coded language about a conversation he had that day.% 913
\footnote{7/99/16 Email, Kilimnik to Manafort (10:51~a.m.).}
In an email with a subject line ``Black Caviar,'' Kilimnik wrote:
\begin{quote}
I met today with the guy who gave you your biggest black caviar jar several years ago.
We spent about 5 hours talking about his story, and I have several important messages from him to you.
He asked me to go and brief you on our conversation.
I said I have to run it by you first, but in principle I am prepared to do it\dots.
It has to do about the future of his country, and is quite interesting.% 914
\footnote{7/29/16 Email, Kilimnik to Manafort (10:51~a.m.).}
\end{quote}
Manafort identified ``the guy who gave you your biggest black caviar jar'' as Yanukovych.
He explained that, in 2010, he and Yanukovych had lunch to celebrate the recent presidential election.
Yanukovych gave Manafort a large jar of black caviar that was worth approximately \$30,000 to \$40,000.% 915
\footnote{Manafort 9/12/18 302, at~3.}
Manafort's identification of Yanukovych as ``the guy who gave you your biggest black caviar jar'' is consistent with Kilimnik being in Moscow---where Yanukovych resided---when Kilimnik wrote ``I met today with a guy,'' and with a December 2016 email in which Kilimnik referred to Yanukovych as ``BG,''
\blackout{Grand Jury}% 916
\footnote{7/29/16 Email, Manafort to Kilimnik; \blackout{Investigative Technique}; \blackout{Grand Jury}}
Manafort replied to Kilimnik's July~29 email, ``Tuesday [August~2] is best \dots\ Tues or weds in NYC.''% 917
\footnote{7/29/16 Email, Manafort to Kilimnik.}
Three days later, on July~31, 2016, Kilimnik flew back to Kiev from Moscow, and on that same day, wrote to Manafort that he needed ``about 2 hours'' for their meeting ``because it is a long caviar story to tell.''% 918
\footnote{7/31/16 Email, Manafort to Kilimnik.}
Kilimnik wrote that he would arrive at JFK on August~2 at~7:30~p.m., and he and Manafort agreed to a late dinner that night.% 919
\footnote{7/31/16 Email, Manafort to Kilimnik.}
Documentary evidence---including flight, phone, and hotel records, and the timing of text messages exchanged% 920
\footnote{Kilimnik 8/2/16 CBP Record; Call Records of Konstantin Kilimnik \blackout{Grand Jury}; Call Records of Rick Gates \blackout{Grand Jury}; 8/2--3/16, Kilimnik Park Lane Hotel Receipt.}%
---confirms the dinner took place as planned on August~2.% 921
\footnote{Deripaska's private plane also flew to Teterboro Airport in New Jersey on the evening of August~2, 2016. According to Customs and Border Protection records, the only passengers on the plane were Deripaska's wife, daughter, mother, and father-in-law, and separate records obtained by our Office confirm that Kilimnik flew on a commercial flight to New York.}
As to the contents of the meeting itself, the accounts of Manafort and Gates---who arrived late to the dinner---differ in certain respects.
But their versions of events, when assessed alongside available documentary evidence and what Kilimnik told business associate Sam Patten, indicate that at least three principal topics were discussed.
First, Manafort and Kilimnik discussed a plan to resolve the ongoing political problems in Ukraine by creating an autonomous republic in its more industrialized eastern region of Donbas,% 922
\footnote{The Luhansk and Donetsk People's Republics, which are located in the Donbas region of Ukraine, declared themselves independent in response to the popular unrest in 2014 that removed President Yanukovych from power.
Pro-Russian Ukrainian militia forces, with backing from the Russian military, have occupied the region since 2014.
Under the Yanukovych-backed plan, Russia would assist in withdrawing the military, and Donbas would become an autonomous region within Ukraine with its own prime minister.
The plan emphasized that Yanukovych would be an ideal candidate to bring peace to the region as prime minister of the republic, and facilitate the reintegration of the region into Ukraine with the support of the U.S. and Russian presidents.
As noted above, according to \blackout{Grand Jury} the written documentation describing the plan, for the plan to work, both U.S. and Russian support were necessary.
\blackout{Grand Jury} 2/21/18 Email, Manafort, Ward, \& Fabrizio, at~3--5.}
and having Yanukovych, the Ukrainian President ousted in 2014, elected to head that republic.% 923
\footnote{Manafort 9/11/18 302, at~4; \blackout{Grand Jury}}
That plan, Manafort later acknowledged, constituted a ``backdoor'' means for Russia to control eastern Ukraine.% 924
\footnote{\blackout{Grand Jury}}
Manafort initially said that, if he had not cut off the discussion, Kilimnik would have asked Manafort in the August~2 meeting to convince Trump to come out in favor of the peace plan, and Yanukovych would have expected Manafort to use his connections in Europe and Ukraine to support the plan.% 925
\footnote{Manafort 9/11/18 302, at~4.}
Manafort also initially told the Office that he had said to Kilimnik that the plan was crazy, that the discussion ended, and that he did not recall Kilimnik asking Manafort to reconsider the plan after their August~2 meeting.% 926
\footnote{Manafort 9/11/18 302, at~4.}
Manafort said
\blackout{Grand Jury}
that he reacted negatively to Yanukovych sending---years later---an ``urgent'' request when Yanukovych needed him.% 927
\footnote{\blackout{Grand Jury} Manafort 9/11/18 302, at~5;
Manafort 9/12/18 302, at~4.}
When confronted with an email written by Kilimnik on or about December~8, 2016, however, Manafort acknowledged Kilimnik raised the peace plan again in that email.% 928
\footnote{Manafort 9/12/18 302, at~4; \blackout{Investigative Technique}}
Manafort ultimately acknowledged Kilimnik also raised the peace plan in January and February 2017 meetings with Manafort
\blackout{Grand Jury}% 929
\footnote{\blackout{Grand Jury} Documentary evidence confirms the peace-plan discussions in 2018.
2/19/18 Email, Fabrizio to Ward (forwarding email from Manafort);
2/21/18 Email, Manafort to Ward \& Fabrizio.}
Second, Manafort briefed Kilimnik on the state of the Trump Campaign and Manafort's plan to win the election.% 930
\footnote{Manafort 9/11/18 302, at~5.}
That briefing encompassed the Campaign's messaging and its internal polling data.
According to Gates, it also included discussion of ``battleground'' states, which Manafort identified as Michigan, Wisconsin, Pennsylvania, and Minnesota.% 931
\footnote{Gates 1/30/18 302, at~3, 5.}
Manafort did not refer explicitly to ``battle ground'' states in his telling of the August~2 discussion,
\blackout{Grand Jury}% 932
\footnote{\blackout{Grand Jury}}
Third, according to Gates and what Kilimnik told Patten, Manafort and Kilimnik discussed two sets of financial disputes related to Manafort's previous work in the region.
Those consisted of the unresolved Deripaska lawsuit and the funds that the Opposition Bloc owed to Manafort for his political consulting work and how Manafort might be able to obtain payment.% 933
\footnote{Gates 1/30/18 302, at~2--4;
Patten 5/22/18 302, at~7.}
After the meeting, Gates and Manafort both stated that they left separately from Kilimnik because they knew the media was tracking Manafort and wanted to avoid media reporting on his connections to Kilimnik.% 934
\footnote{Gates 1/30/18 302, at~5;
Manafort 9/11/18 302, at~5.}
\paragraph{Post-Resignation Activities}
Manafort resigned from the Trump Campaign in mid-August~2016, approximately two weeks after his second meeting with Kilimnik, amidst negative media reporting about his political consulting work for the pro-Russian Party of Regions in Ukraine.
Despite his resignation, Manafort continued to offer advice to various Campaign officials through the November election.
Manafort told Gates that he still spoke with Kushner, Bannon, and candidate Trump,% 935
\footnote{Gates 2/12/18 302, at~12.}
and some of those post-resignation contacts are documented in emails.
For example, on October~21, 2016, Manafort sent Kushner an email and attached a strategy memorandum proposing that the Campaign make the case against Clinton ``as the failed and corrupt champion of the establishment'' and that ``Wikileaks provides the Trump campaign the ability to make the case in a very credible way---by using the words of Clinton, its campaign officials and DNC members.''% 936
\footnote{NOSC00021517--20 (10/21/16 Email, Manafort to Kushner).}
Later, in a November~5, 2016 email to Kushner entitled ``Securing the Victory,'' Manafort stated that he was ``really feeling good about our prospects on Tuesday and focusing on preserving the victory,'' and that he was concerned the Clinton Campaign would respond to a loss by ``mov[ing] immediately to discredit the [Trump] victory and claim voter fraud and cyber-fraud, including the claim that the Russians have hacked into the voting machines and tampered with the results.''% 937
\footnote{NOSC00021573--75 (11/5/16 Email, Manafort to Kushner).}
Trump was elected President on November~8, 2016.
Manafort told the Office that, in the wake of Trump's victory, he was not interested in an Administration job.
Manafort instead preferred to stay on the ``outside,'' and monetize his campaign position to generate business given his familiarity and relationship with Trump and the incoming Administration.% 938
\footnote{Manafort 9/12/18 302, at~1, 4--5;
Gates 1/30/18 302, at~4.}
Manafort appeared to follow that plan, as he traveled to the Middle East, Cuba, South Korea, Japan, and China and was paid to explain what a Trump presidency would entail.% 939
\footnote{Manafort 9/12/18 302, at~1.}
Manafort's activities in early 2017 included meetings relating to Ukraine and Russia.
The first meeting, which took place in Madrid, Spain in January 2017, was with Georgiy Oganov.
Oganov, who had previously worked at the Russian Embassy in the United States, was a senior executive at a Deripaska company and was believed to report directly to Deripaska.% 940
\footnote{Kalashnikova 5/17/18 302, at~4;
Gary Lee, Soviet Embassy's Identity Crisis, Washington Post (Dec.~20, 1991);
Georgy S. Oganov Executive Profile \& Biography, Bloomberg (Mar.~12, 2019).}
Manafort initially denied attending the meeting.
When he later acknowledged it, he claimed that the meeting had been arranged by his lawyers and concerned only the Pericles lawsuit.% 941
\footnote{Manafort 9/11/18 302, at~7.}
Other evidence, however, provides reason to doubt Manafort's statement that the sole topic of the meeting was the Pericles lawsuit.
In particular, text messages to Manafort from a number associated with Kilimnik suggest that Kilimnik and Boyarkin---not Manafort's counsel---had arranged the meeting between Manafort and Oganov.% 942
\footnote{Text Message, Manafort \& Kilimnik.}
Kilimnik's message states that the meeting was supposed to be ``not about money or Pericles'' but instead ``about recreating [the] old friendship''---ostensibly between Manafort and Deripaska---``and talking about global politics.''% 943
\footnote{Text Message, Manafort \& Kilimnik;
Manafort 9/12/18 302, at~5.}
Manafort also replied by text that he ``need[s] this finished before Jan.~20,''% 944
\footnote{Text Message, Manafort \& Kilimnik.}
which appears to be a reference to resolving Pericles before the inauguration.
On January~15, 2017, three days after his return from Madrid, Manafort emailed K.T. McFarland, who was at that time designated to be Deputy National Security Advisor and was formally appointed to that position on January~20, 2017.% 945
\footnote{1/15/17 Email, Manafort, McFarland, \& Flynn.}
Manafort's January~15 email to McFarland stated: ``I have some important information I want to share that I picked up on my travels over the last month.''% 946
\footnote{1/15/17 Email, Manafort, McFarland, \& Flynn.}
Manafort told the Office that the email referred to an issue regarding Cuba, not Russia or Ukraine, and Manafort had traveled to Cuba in the past month.% 947
\footnote{Manafort 9/11/18 302, at~7.}
Either way, McFarland---who was advised by Flynn not to respond to the Manafort inquiry---appears not to have responded to Manafort.% 948
\footnote{1/15/17 Email, Manafort, McFarland, \& Flynn;
McFarland 12/22/17 302, at~18--19.}
Manafort told the Office that around the time of the Presidential Inauguration in January, he met with Kilimnik and Ukrainian oligarch Serhiy Lyovochkin at the Westin Hotel in Alexandria, Virginia.% 949
\footnote{\blackout{Grand Jury} Manafort 9/11/18 302, at~7;
Manafort 9/21/18 302, at~3;
1/19/17 \& 1/22/17 Kilimnik CBP Records, Jan.~19 and~22, 2017;
2016--17 Text Messages, Kilimnik \& Patten, at~1--2.}
During this meeting, Kilimnik again discussed the Yanukovych peace plan that he had broached at the August~2 meeting and in a detailed December~8, 2016 message found in Kilimnik's DMP email account.% 950
\footnote{\blackout{Investigative Technique}}
In that December~8 email, which Manafort acknowledged having read,% 951
\footnote{Manafort 9/11/18 302, at~6;
\blackout{Grand Jury}}
Kilimnik wrote, ``[a]ll that is required to start the process is a very minor `wink' (or slight push) from DT''---an apparent reference to President-elect Trump---``and a decision to authorize you to be a `special representative' and manage this process.''
Kilimnik assured Manafort, with that authority, he ``could start the process and within 10 days visit Russia [Yanukovych] guarantees your reception at the very top level,'' and that ``DT could have peace in Ukraine basically within a few months after inauguration.''% 952
\footnote{\blackout{Investigative Technique}}
As noted above,
\blackout{Grand Jury}
and statements to the Office, Manafort sought to qualify his engagement on and support for the plan.
\blackout{Grand Jury}% 953
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 954
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 955
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}
On February~26, 2017, Manafort met Kilimnik in Madrid, where Kilimnik had flown from Moscow.% 956
\footnote{9/21/17 Email, Zatynaiko to Kilimnik.}
In his first two interviews with the Office, Manafort denied meeting with Kilimnik on his Madrid trip and then---after being confronted with documentary evidence that Kilimnik was in Madrid at the same time as him---recognized that he met him in Madrid.
Manafort said that Kilimnik had updated him on a criminal investigation into so-called ``black ledger'' payments to Manafort that was being conducted by Ukraine's National Anti-Corruption Bureau.% 957
\footnote{Manafort 9/13/18 302, at~1.}
\blackout{Grand Jury}% 958
\footnote{\blackout{Grand Jury} In resolving whether Manafort breached his cooperation plea agreement by lying to the Office, the district court found that Manafort lied about, among other things, his contacts with Kilimnik regarding the peace plan, including the meeting in Madrid.}
Manafort remained in contact with Kilimnik throughout 2017 and into the spring of 2018.
Those contacts included matters pertaining to the criminal charges brought by the Office,% 959
\footnote{\textit{Manafort} (D.D.C.) Gov't Opp.\ to Mot.\ to Modify, at~2;
Superseding Indictment \P\P~48--51, \textit{United States~v.\ Paul J. Manafort}, Jr., 1:17-cr-201 (D.D.C. June~8, 2018), Doc.~318.}
and the Ukraine peace plan.
In early 2018, Manafort retained his longtime polling firm to craft a draft poll in Ukraine, sent the pollsters a three-page primer on the plan sent by Kilimnik, and worked with Kilimnik to formulate the polling questions.% 960
\footnote{9/12/18 Email, Fabrizio to Manafort \& Ward;
2/16/18 Email, Fabrizio to Manafort;
2/19/18 Email, Fabrizio to Ward;
2/21/18 Email, Manafort to Ward \& Fabrizio.}
The primer sent to the pollsters specifically called for the United States and President Trump to support the Autonomous Republic of Donbas with Yanukovych as Prime Minister,% 961
\footnote{9/21/18 Email, Manafort to Ward \& Fabrizio (7:16:49~a.m.) (attachment).}
and a series of questions in the draft poll asked for opinions on Yanukovych's role in resolving the conflict in Donbas.% 962
\footnote{3/9/18 Email, Ward to Manafort \& Fabrizio (attachment).}
(The poll was not solely about Donbas; it also sought participants' views on leaders apart from Yanukovych as they pertained to the 2019 Ukraine presidential election.)
The Office has not uncovered evidence that Manafort brought the Ukraine peace plan to the attention of the Trump Campaign or the Trump Administration.
Kilimnik continued his efforts to promote the peace plan to the Executive Branch (\textit{e.g.}, U.S. Department of State) into the summer of 2018.% 963
\footnote{\blackout{Investigative Technique}}
\subsection{Post-Election and Transition-Period Contacts}
Trump was elected President on November~8, 2016.
Beginning immediately after the election, individuals connected to the Russian government started contacting officials on the Trump Campaign and Transition Team through multiple channels---sometimes through Russian Ambassador Kislyak and at other times through individuals who sought reliable contacts through U.S. persons not formally tied to the Campaign or Transition Team.
The most senior levels of the Russian government encouraged these efforts.
The investigation did not establish that these efforts reflected or constituted coordination between the Trump Campaign and Russia in its election-interference activities.
\subsubsection{Immediate Post-Election Activity}
As soon as news broke that Trump had been elected President, Russian government officials and prominent Russian businessmen began trying to make inroads into the new Administration.
They appeared not to have preexisting contacts and struggled to connect with senior officials around the President-Elect.
As explained below, those efforts entailed both official contact through the Russian Embassy in the United States and outreaches---sanctioned at high levels of the Russian government---through business rather than political contacts.
\paragraph{Outreach from the Russian Government}
At approximately 3~a.m.\ on election night, Trump Campaign press secretary Hope Hicks received a telephone call on her personal cell phone from a person who sounded foreign but was calling from a number with a DC area code.% 964
\footnote{Hicks 12/8/17 302, at~3.}
Although Hicks had a hard time understanding the person, she could make out the words ``Putin call.''% 965
\footnote{Hicks 12/8/17 302, at~3.}
Hicks told the caller to send her an email.% 966
\footnote{Hicks 12/8/17 302, at~3.}
The following morning, on November~9, 2016, Sergey Kuznetsov, an official at the Russian Embassy to the United States, emailed Hicks from his Gmail address with the subject line, ``Message from Putin.''% 967
\footnote{NOSC00044381 (11/9/16 Email, Kuznetsov to Hicks (5:27~a.m.)).}
Attached to the email was a message from Putin, in both English and Russian, which Kuznetsov asked Hicks to convey to the President-Elect.% 968
\footnote{NOSC00044381--82 (11/9/16 Email, Kuznetsov to Hicks (5:27~a.m.)).}
In the message, Putin offered his congratulations to Trump for his electoral victory, stating he ``look[ed] forward to working with [Trump] on leading Russian--American relations out of crisis.''% 969
\footnote{NOSC00044382 (11/9/16 Letter from Putin to President-Elect Trump (Nov.~9, 2016) (translation)).}
Hicks forwarded the email to Kushner, asking, ``Can you look into this?
Don't want to get duped but don't want to blow off Putin!''% 970
\footnote{NOSC00044381 (11/9/16 Email, Hicks to Kushner (10:26~a.m.)).}
Kushner stated in Congressional testimony that he believed that it would be possible to verify the authenticity of the forwarded email through the Russian Ambassador, whom Kushner had previously met in April 2016.% 971
\footnote{Statement of Jared C. Kushner to Congressional Committees, at~4 (Jul.~24, 2017).}
Unable to recall the Russian Ambassador's name, Kushner emailed Dimitri Simes of CNI, whom he had consulted previously about Russia, \textit{see} \hyperlink{subsubsection.1.4.1.4}{Volume~I, Section~IV.A.4}, \textit{supra}, and asked, ``What is the name of Russian ambassador?''% 972
\footnote{NOSC00000058 (11/9/16 Email, Kushner to Simes (10:28~a.m.));
Statement of Jared Kushner to Congressional Committees, at~4 (Jul.~24, 2017).}
Kushner forwarded Simes's response---which identified Kislyak by name---to Hicks.% 973
\footnote{NOSC00000058 (11/9/16 Email, Kushner to Hicks (11:05:44~a.m.)).}
After checking with Kushner to see what he had learned, Hicks conveyed Putin's letter to transition officials.% 974
\footnote{Hicks 12/8/17 302, at~3--4.}
Five days later, on November~14, 2016, Trump and Putin spoke by phone in the presence of Transition Team members, including incoming National Security Advisor Michael Flynn.% 975
\footnote{Flynn 11/16/17 302, at~8--10;
\textit{see} Doug G. Ware, \textit{Trump, Russia's Putin Talk about Syria, Icy Relations in Phone Call}, UPI (Nov.~14, 2016).}
\paragraph{High-Level Encouragement of Contacts through Alternative Channels}
As Russian officials in the United States reached out to the President-Elect and his team, a number of Russian individuals working in the private sector began their own efforts to make contact.
Petr Aven, a Russian national who heads Alfa-Bank, Russia's largest commercial bank, described to the Office interactions with Putin during this time period that might account for the flurry of Russian activity.% 976
\footnote{Aven provided information to the Office in an interview and through an attorney proffer, \blackout{Grand Jury}}
Aven told the Office that he is one of approximately 50 wealthy Russian businessmen who regularly meet with Putin in the Kremlin; these 50 men are often referred to as ``oligarchs.''% 977
\footnote{Aven 8/2/18 302, at~7.}
Aven told the Office that he met on a quarterly basis with Putin, including in the fourth quarter (Q4) of 2016, shortly after the U.S. presidential election.% 978
\footnote{\blackout{Grand Jury}}
Aven said that he took these meetings seriously and understood that any suggestions or critiques that Putin made during these meetings were implicit directives, and that there would be consequences for Aven if he did not follow through.% 979
\footnote{Aven 8/2/18 302, at~2--3.}
As was typical, the 2016 Q4 meeting with Putin was preceded by a preparatory meeting with Putin's chief of staff, Anton Vaino.% 980
\footnote{\blackout{Grand Jury} and interview with the Office, Aven referred to the high-ranking Russian government officials using numbers (\textit{e.g.}, Official~1, Official~2).
Aven separately confirmed through an attorney proffer that Official~1 was Putin and Official~2 was Putin's chief of staff, Vaino.
\textit{See} Affidavit of Ryan Junck (Aug.~2, 2018) (hard copy on file).}
According to Aven, at his Q4 2016 one-on-one meeting with Putin,% 981
\footnote{At the time of his Q4 2016 meeting with Putin, Aven was generally aware of the press coverage about Russian interference in the U.S. election.
According to Aven, he did not discuss that topic with Putin at any point, and Putin did not mention the rationale behind the threat of new sanctions.
Aven 8/2/18 302, at~5--7.}
Putin raised the prospect that the United States would impose additional sanctions on Russian interests, including sanctions against Aven and/or Alfa-Bank.% 982
\footnote{\blackout{Grand Jury}}
Putin suggested that Aven needed to take steps to protect himself and Alfa-Bank.% 983
\footnote{\blackout{Grand Jury}}
Aven also testified that Putin spoke of the difficulty faced by the Russian government in getting in touch with the incoming Trump Administration.% 984
\footnote{\blackout{Grand Jury}}
According to Aven, Putin indicated that he did not know with whom formally to speak and generally did not know the people around the President-Elect.% 985
\footnote{\blackout{Grand Jury}}
Aven
\blackout{Grand Jury}
told Putin he would take steps to protect himself and the Alfa-Bank shareholders from potential sanctions, and one of those steps would be to try to reach out to the incoming Administration to establish a line of communication.% 986
\footnote{\blackout{Grand Jury}}
Aven described Putin responding with skepticism about Aven's prospect for success.% 987
\footnote{\blackout{Grand Jury} Aven 8/2/18 302, at~6.}
According to Aven, although Putin did not expressly direct him to reach out to the Trump Transition Team, Aven understood that Putin expected him to try to respond to the concerns he had raised.% 988
\footnote{Aven 8/2/18 302, at~4--8; \blackout{Grand Jury}}
Aven's efforts are described in \hyperlink{subsubsection.1.4.2.5}{Volume~I, Section~IV.B.5}, \textit{infra}.
\subsubsection{Kirill Dmitriev's Transition-Era Outreach to the Incoming Administration}
Aven's description of his interactions with Putin is consistent with the behavior of Kirill Dmitriev, a Russian national who heads Russia's sovereign wealth fund and is closely connected to Putin.
Dmitriev undertook efforts to meet members of the incoming Trump Administration in the months after the election.
Dmitriev asked a close business associate who worked for the United Arab Emirates (UAE) royal court, George Nader, to introduce him to Trump transition officials, and Nader eventually arranged a meeting in the Seychelles between Dmitriev and Erik Prince, a Trump Campaign supporter and an associate of Steve Bannon.% 989
\footnote{Nader provided information to the Office in multiple interviews, all but one of which were conducted under a proffer agreement \blackout{Grand Jury}.
The investigators also interviewed Prince under a proffer agreement.
Bannon was interviewed by the Office, \blackout{Grand Jury} under a proffer agreement.}
In addition, the UAE national security advisor introduced Dmitriev to a hedge fund manager and friend of Jared Kushner, Rick Gerson, in late November 2016.
In December 2016 and January 2017, Dmitriev and Gerson worked on a proposal for reconciliation between the United States and Russia, which Dmitriev implied he cleared through Putin.
Gerson provided that proposal to Kushner before the inauguration, and Kushner later gave copies to Bannon and Secretary of State Rex Tillerson.
\paragraph{Background}
Dmitriev is a Russian national who was appointed CEO of Russia's sovereign wealth fund, the Russian Direct Investment Fund (RDIF), when it was founded in 2011.% 990
\footnote{Kirill Dmitriev Biography, Russian Direct Investment Fund, \textit{available at} \url{https://rdif.ru/Eng\_person\_dmitriev\_kirill/}.
\textit{See also} Overview, Russian Direct Investment Fund, \textit{available at} \url{https://rdif.ru/Eng\_About/}.}
Dmitriev reported directly to Putin and frequently referred to Putin as his ``boss.''% 991
\footnote{Gerson 6/15/18 302, at~1.
\textit{See also, e.g.}, 12/14/16 Text Message, Dmitriev to Gerson;
1/9/17 Text Message, Dmitriev to Gerson.}
RDIF has co-invested in various projects with UAE sovereign wealth funds.% 992
\footnote{\blackout{Grand Jury}}
Dmitriev regularly interacted with Nader, a senior advisor to UAE Crown Prince Mohammed bin Zayed (Crown Prince Mohammed), in connection with RDIF's dealings with the UAE\null.% 993
\footnote{Nader 1/22/18 302, at~1--2;
Nader 1/23/18 302, at~2--3;
5/3/16 Email, Nader to Phares;
\blackout{Grand Jury}}
Putin wanted Dmitriev to be in charge of both the financial and the political relationship between Russia and the Gulf states, in part because Dmitriev had been educated in the West and spoke English fluently.% 994
\footnote{Nader 1/22/18 302, at~1--2.}
Nader considered Dmitriev to be Putin's interlocutor in the Gulf region, and would relay Dmitriev's views directly to Crown Prince Mohammed.% 995
\footnote{Nader 1/22/18 302, at~3.}
Nader developed contacts with both U.S. presidential campaigns during the 2016 election, and kept Dmitriev abreast of his efforts to do so.% 996
\footnote{Nader 1/22/18 302, at~3; \blackout{Grand Jury}}
According to Nader, Dmitriev said that his and the government of Russia's preference was for candidate Trump to win and asked Nader to assist him in meeting members of the Trump Campaign.% 997
\footnote{Nader 1/22/18 302, at~3; \blackout{Grand Jury}}
\blackout{Grand Jury}% 998
\footnote{\blackout{Grand Jury}}
Nader did not introduce Dmitriev to anyone associated with the Trump Campaign before the election.% 999
\footnote{Nader 1/22/18 302, at~3.}
\blackout{Grand Jury}% 1000
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1001
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1002
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1003
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1004
\footnote{\blackout{Grand Jury}}
Erik Prince is a businessman who had relationships with various individuals associated with the Trump Campaign, including Steve Bannon, Donald Trump~Jr., and Roger Stone.% 1005
\footnote{Prince 4/4/18 302, at~1--5;
Bannon 2/14/18 302, at~21.}
Prince did not have a formal role in the Campaign, although he offered to host a fundraiser for Trump and sent unsolicited policy papers on issues such as foreign policy, trade, and Russian election interference to Bannon.% 1006
\footnote{Prince 4/4/18 302, at~1, 3--4;
Prince 5/3/18 302, at~2;
Bannon 2/14/18 302, at~19--20;
10/18/16 Email, Prince to Bannon.}
After the election, Prince frequently visited transition offices at Trump Tower, primarily to meet with Bannon but on occasion to meet Michael Flynn and others.% 1007
\footnote{Flynn 11/20/17 302, at~6;
Flynn 1/11/18 302, at~5;
Flynn 1/24/18 302, at~5--6;
Flynn 5/1/18 302, at~11;
Prince 4/4/18 302, at~5, 8;
Bannon 2/14/18 302, at~20--21;
11/12/16 Email, Prince to Corallo.}
Prince and Bannon would discuss, \textit{inter alia}, foreign policy issues and Prince's recommendations regarding who should be appointed to fill key national security positions.% 1008
\footnote{Prince 4/4/18 302, at~5;
Bannon 2/14/18 302, at~21.}
Although Prince was not formally affiliated with the transition, Nader
\blackout{Grand Jury}
received assurances
\blackout{Grand Jury}
that the incoming Administration considered Prince a trusted associate.% 1009
\footnote{\blackout{Grand Jury}}
\paragraph{Kirill Dmitriev's Post-Election Contacts With the Incoming Administration}
Soon after midnight on election night, Dmitriev messaged
\blackout{Investigative Technique}
who was traveling to New York to attend the 2016 World Chess Championship.
\blackout{Investigative Technique}
Dmitry Peskov, the Russian Federation's press secretary, who was also attending the World Chess Championship.% 1010
\footnote{\blackout{Investigative Technique} Nader 1/22/18 302, at~5--6; \blackout{Grand Jury}}
\blackout{Investigative Technique}% 1011
\footnote{\blackout{Investigative Technique}}
\blackout{Investigative Technique}% 1012
\footnote{\blackout{Investigative Technique}}
\blackout{Investigative Technique}% 1013
\footnote{\blackout{Investigative Technique}}
At approximately 2:40~a.m.\ on November~9, 2016, news reports stated that candidate Clinton had called President-Elect Trump to concede.
At
\blackout{Investigative Technique}% 1014
\footnote{\blackout{Investigative Technique}}
\blackout{Investigative Technique}
wrote to Dmitriev, ``Putin has won.''% 1015
\footnote{\blackout{Investigative Technique}}
Later that morning, Dmitriev contacted Nader, who was in New York, to request a meeting with the ``key people'' in the incoming Administration as soon as possible in light of the ``[g]reat results.''% 1016
\footnote{11/9/16 Text Message, Dmitriev to Nader (9:34~a.m.);
Nader 1/22/18 302, at~4.}
He asked Nader to convey to the incoming Administration that ``we want to start rebuilding the relationship in whatever is a comfortable pace for them.
We understand all of the sensitivities and are not in a rush.''% 1017
\footnote{11/9/16 Text Message, Dmitriev to Nader (11:58~p.m.).}
Dmitriev and Nader had previously discussed Nader introducing him to the contacts Nader had made within the Trump Campaign.% 1018
\footnote{Nader 1/22/18 302, at~3.}
Dmitriev also told Nader that he would ask Putin for permission to travel to the United States, where he would be able to speak to media outlets about the positive impact of Trump's election and the need for reconciliation between the United States and Russia.% 1019
\footnote{11/9/16 Text Message, Dmitriev to Nader (10:06~a.m.);
11/9/16 Text Message, Dmitriev to Nader (10:10~a.m.);
\blackout{Grand Jury}}
Later that day, Dmitriev flew to New York, where Peskov was separately traveling to attend the chess tournament.% 1020
\footnote{11/9/16 Text Message, Dmitriev to Nader (10:08~a.m.);
11/9/16 Text Message, Dmitriev to Nader (3:40~p.m.);
Nader 1/22/18 302, at~5.}
Dmitriev invited Nader to the opening of the tournament and noted that, if there was ``a chance to see anyone key from Trump camp,'' he ``would love to start building for the future.''% 1021
\footnote{11/9/16 Text Message, Dmitriev to Nader (7:10~p.m.).}
Dmitriev also asked Nader to invite Kushner to the event so that he (Dmitriev) could meet him.% 1022
\footnote{11/10/16 Text Message, Dmitriev to Nader (5:20~a.m.).}
Nader did not pass along Dmitriev's invitation to anyone connected with the incoming Administration.% 1023
\footnote{Nader 1/22/18 302, at~5--6.}
Although one World Chess Federation official recalled hearing from an attendee that President-Elect Trump had stopped by the tournament, the investigation did not establish that Trump or any Campaign or Transition Team official attended the event.% 1024
\footnote{Marinello 5/31/18 302, at~2--3;
Nader 1/22/18 302, at~5--6.}
And the President's written answers denied that he had.% 1025
\footnote{Written Responses of Donald J. Trump (Nov.~20, 2018), at~17--18 (Response to Question~V, Part~(a).}
Nader stated that Dmitriev continued to press him to set up a meeting with transition officials, and was particularly focused on Kushner and Trump~Jr.% 1026
\footnote{Nader 1/22/18 302, at~6;
\blackout{Grand Jury}}
Dmitriev told Nader that Putin would be very grateful to Nader and that a meeting would make history.% 1027
\footnote{Nader 1/22/18 302, at~6;
\blackout{Grand Jury}}
\blackout{Grand Jury}% 1028
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1029
\footnote{\blackout{Grand Jury}}
According to Nader, Dmitriev was very anxious to connect with the incoming Administration and told Nader that he would try other routes to do so besides Nader himself.% 1030
\footnote{Nader 1/22/18 302, at~6.}
Nader did not ultimately introduce Dmitriev to anyone associated with the incoming Administration during Dmitriev's post-election trip to New York.% 1031
\footnote{Nader 1/22/18 302, at~5--7.}
In early December 2016, Dmitriev again broached the topic of meeting incoming Administration officials with Nader in January or February.% 1032
\footnote{12/8/16 Text Messages, Dmitriev to Nader (12:10:31~a.m.);
Nader 1/22/18 302, at~11.}
Dmitriev sent Nader a list of publicly available quotes of Dmitriev speaking positively about Donald Trump ``in case they [were] helpful.''% 1033
\footnote{12/8/16 Text Message, Dmitriev to Nader (12:10:31~a.m.);
12/8/16 Text Message, Dmitriev to Nader (12:10:57~a.m.).}
\paragraph{Erik Prince and Kirill Dmitriev Meet in the Seychelles}
\subparagraph{George Nader and Erik Prince Arrange Seychelles Meeting with Dmitriev}
Nader traveled to New York in early January 2017 and had lunchtime and dinner meetings with Erik Prince on January~3, 2017.% 1034
\footnote{Prince 4/4/18 302, at~8.}
Nader and Prince discussed Dmitriev.% 1035
\footnote{Prince 5/3/18 302, at~3; \blackout{Grand Jury}}
Nader informed Prince that the Russians were looking to build a link with the incoming Trump Administration.% 1036
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}
he told Prince that Dmitriev had been pushing Nader to introduce him to someone from the incoming Administration
\blackout{Grand Jury}.% 1037
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}
Nader suggested, in light of Prince's relationship with Transition Team officials, that Prince and Dmitriev meet to discuss issues of mutual concern.% 1038
\footnote{\blackout{Grand Jury}}
Prince told Nader that he needed to think further about it and to check with Transition Team officials.% 1039
\footnote{\blackout{Grand Jury}}
After his dinner with Prince, Nader sent Prince a link to a Wikipedia entry about Dmitriev, and sent Dmitriev a message stating that he had just met ``with some key people within the family and inner circle''---a reference to Prince---and that he had spoken at length and positively about Dmitriev.% 1040
\footnote{1/4/17 Text Message, Nader to Prince;
1/4/17 Text Messages, Nader to Dmitriev (5:24~a.m.--5:26~a.m.);
Nader 1/22/18 302, at~8--9
\blackout{Grand Jury}}
Nader told Dmitriev that the people he met had asked for Dmitriev's bio, and Dmitriev replied that he would update and send it.% 1041
\footnote{1/4/17 Text Messages, Nader \& Dmitriev (7:24:27~a.m.).}
Nader later received from Dmitriev two files concerning Dmitriev: one was a two-page biography, and the other was a list of Dmitriev's positive quotes about Donald Trump.% 1042
\footnote{1/4/17 Text Messages, Dmitriev to Nader (7:25--7:29~a.m.).}
The next morning, Nader forwarded the message and attachments Dmitriev had sent him to Prince.% 1043
\footnote{1/4/17 Text Messages, Nader to Prince.}
Nader wrote to Prince that these documents were the versions ``to be used with some additional details for them'' (with ``them'' referring to members of the incoming Administration).% 1044
\footnote{1/4/17 Text Messages, Nader to Prince; \blackout{Grand Jury}}
Prince opened the attachments at Trump Tower within an hour of receiving them.% 1045
\footnote{Prince 5/3/18 302, at~1--3.}
Prince stated that, while he was at Trump Tower that day, he spoke with Kellyanne Conway, Wilbur Ross, Steve Mnuchin, and others while waiting to see Bannon.% 1046
\footnote{Prince 5/3/18 302, at~2--3.}
Cell-site location data for Prince's mobile phone indicates that Prince remained at Trump Tower for approximately three hours.% 1047
\footnote{Cell-site location data for Prince's mobile phone \blackout{Investigative Technique}}
Prince said that he could not recall whether, during those three hours, he met with Bannon and discussed Dmitriev with him.% 1048
\footnote{Prince 5/3/18 302, at~3.}
\blackout{Grand Jury}.% 1049
\footnote{\blackout{Grand Jury}}
Prince booked a ticket to the Seychelles on January~7, 2017.% 1050
\footnote{1/5/17 Email, Kasbo to Prince.}
The following day, Nader wrote to Dmitriev that he had a ``pleasant surprise'' for him, namely that he had arranged for Dmitriev to meet ``a Special Guest'' from ``the New Team,'' referring to Prince.% 1051
\footnote{1/8/17 Text Messages, Nader to Dmitriev (6:05--6:10~p.m.).}
Nader asked Dmitriev if he could come to the Seychelles for the meeting on January~12, 2017, and Dmitriev agreed.% 1052
\footnote{1/8/17 Text Messages, Nader \& Dmitriev (6:10--7:27~p.m.).}
The following day, Dmitriev sought assurance from Nader that the Seychelles meeting would be worthwhile.% 1053
\footnote{1/9/17 Text Message, Dmitriev to Nader.}
\blackout{Grand Jury}
Dmitriev was not enthusiastic about the idea of meeting with Prince, and that Nader assured him that Prince wielded influence with the incoming Administration.% 1054
\footnote{\blackout{Grand Jury}}
Nader wrote to Dmitriev, ``This guy [Prince] is designated by Steve [Bannon] to meet you!
I know him and he is very very well connected and trusted by the New Team.
His sister is now a Minister of Education.''% 1055
\footnote{1/9/17 Text Message, Nader to Dmitriev (2:12:59~p.m.);
Nader 1/19/18 302, at~13;
\blackout{Grand Jury}}
According to Nader, Prince had led him to believe that Bannon was aware of Prince's upcoming meeting with Dmitriev, and Prince acknowledged that it was fair for Nader to think that Prince would pass information on to the Transition Team.% 1056
\footnote{Nader 1/19/18 302, at~13; \blackout{Grand Jury} Prince 5/3/18 302, at~3.}
Bannon, however, told the Office that Prince did not tell him in advance about his meeting with Dmitriev.% 1057
\footnote{Bannon 2/14/18 302, at~25--26.}
\subparagraph{The Seychelles Meetings}
Dmitriev arrived with his wife in the Seychelles on January~11, 2017, and checked into the Four Seasons Resort where Crown Prince Mohammed and Nader were staying.% 1058
\footnote{1/10/17 Text Messages, Dmitriev \& Nader (2:05:54--3:30:25~p.m.);
1/11/17 Text Messages, Dmitriev \& Nader (2:16:16--5:17:59~p.m.).}
Prince arrived that same day.% 1059
\footnote{1/7/17 Email, Kasbo to Prince.}
Prince and Dmitriev met for the first time that afternoon in Nader's villa, with Nader present.% 1060
\footnote{1/11/17 Text Messages, Nader \& Dmitriev (5:18:24--5:37:14~p.m.);
\blackout{Grand Jury}}
The initial meeting lasted approximately 30--45~minutes.% 1061
\footnote{Prince 5/3/18 302, at~4; \blackout{Grand Jury}}
\blackout{Grand Jury}% 1062
\footnote{\blackout{Grand Jury}}
Prince described the eight years of the Obama Administration in negative terms, and stated that he was looking forward to a new era of cooperation and conflict resolution.% 1063
\footnote{\blackout{Grand Jury}}
According to Prince, he told Dmitriev that Bannon was effective if not conventional, and that Prince provided policy papers to Bannon.% 1064
\footnote{Prince 5/3/18 302, at~4.}
\blackout{Grand Jury}% 1065
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1066
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1067
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1068
\footnote{\blackout{Grand Jury}}
The topic of Russian interference in the 2016 election did not come up.% 1069
\footnote{Prince 5/3/18 302, at~4--5.}
\blackout{Grand Jury}% 1070
\footnote{\blackout{Grand Jury}}
Prince added that he would inform Bannon about his meeting with Dmitriev, and that if there was interest in continuing the discussion, Bannon or someone else on the Transition Team would do so.% 1071
\footnote{Prince 5/3/18 302, at~4; \blackout{Grand Jury}}
\blackout{Grand Jury}% 1072
\footnote{\blackout{Grand Jury}}
Afterwards, Prince returned to his room, where he learned that a Russian aircraft carrier had sailed to Libya, which led him to call Nader and ask him to set up another meeting with Dmitriev.% 1073
\footnote{Prince 4/4/18 302, at~10;
Prince 5/3/18 302, at~4;
\blackout{Grand Jury}}
According to Nader, Prince called and said he had checked with his associates back home and needed to convey to Dmitriev that Libya was ``off the table.''% 1074
\footnote{Nader 1/22/18 302, at~14;
\blackout{Grand Jury}}
Nader wrote to Dmitriev that Prince had ``received an urgent message that he needs to convey to you immediately,'' and arranged for himself, Dmitriev, and Prince to meet at a restaurant on the Four Seasons property.% 1075
\footnote{\blackout{Grand Jury} 1/11/17 Text Messages, Dmitriev \& Nader (9:13:54--10:24:25~p.m.).}
At the second meeting, Prince told Dmitriev that the United States could not accept any Russian involvement in Libya, because it would make the situation there much worse.% 1076
\footnote{\blackout{Grand Jury}
Prince, however, denied that and recalled that he was making these remarks to Dmitriev not in an official capacity for the transition but based on his experience as a former naval officer. Prince 5/3/18 302, at~4.}
\blackout{Grand Jury}% 1077
\footnote{\blackout{Grand Jury}}
After the brief second meeting concluded, Nader and Dmitriev discussed what had transpired.% 1078
\footnote{Nader 1/22/18 302, at~15;
\blackout{Grand Jury}}
Dmitriev told Nader that he was disappointed in his meetings with Prince for two reasons: first, he believed the Russians needed to be communicating with someone who had more authority within the incoming Administration than Prince had.% 1079
\footnote{Nader 1/22/18 302, at~9, 15;
\blackout{Grand Jury}}
Second, he had hoped to have a discussion of greater substance, such as outlining a strategic roadmap for both countries to follow.% 1080
\footnote{Nader 1/22/18 302, at~15;
\blackout{Grand Jury}}
Dmitriev told Nader that
\blackout{Grand Jury}
Prince's comments
\blackout{Grand Jury}
were insulting
\blackout{Grand Jury}% 1081
\footnote{\blackout{Grand Jury} Nader 1/22/18 302, at~15.}
Hours after the second meeting, Prince sent two text messages to Bannon from the Seychelles.% 1082
\footnote{Call Records of Erik Prince \blackout{Grand Jury}}
As described further below, investigators were unable to obtain the content of these or other messages between Prince and Bannon, and the investigation also did not identify evidence of any further communication between Prince and Dmitriev after their meetings in the Seychelles.
\subparagraph{Erik Prince's Meeting with Steve Bannon after the Seychelles Trip}
After the Seychelles meetings, Prince told Nader that he would inform Bannon about his discussion with Dmitriev and would convey that someone within the Russian power structure was interested in seeking better relations with the incoming Administration.% 1083
\footnote{Prince 4/4/18 302, at~10;
Prince 5/3/18 302, at~4;
\blackout{Grand Jury}}
On January~12, 2017, Prince contacted Bannon's personal assistant to set up a meeting for the following week.% 1084
\footnote{1/12/17 Text Messages, Prince to Preate.}
Several days later, Prince messaged her again asking about Bannon's schedule.% 1085
\footnote{1/15/17 Text Message, Prince to Preate.}
Prince said that he met Bannon at Bannon's home after returning to the United States in mid-January and briefed him about several topics, including his meeting with Dmitriev.% 1086
\footnote{Prince 4/4/18 302, at~11;
Prince 5/3/18 302, at~5.}
Prince told the Office that he explained to Bannon that Dmitriev was the head of a Russian sovereign wealth fund and was interested in improving relations between the United States and Russia.% 1087
\footnote{Prince 4/4/18 302, at~11;
Prince 5/3/18 302, at~5.}
Prince had on his cellphone a screenshot of Dmitriev's Wikipedia page dated January~16, 2017, and Prince told the Office that he likely showed that image to Bannon.% 1088
\footnote{Prince 5/3/18 302, at~5;
1/16/17 Image on Prince Phone (on file with the Office).}
Prince also believed he provided Bannon with Dmitriev's contact information.% 1089
\footnote{Prince 5/3/18 302, at~5.}
According to Prince, Bannon instructed Prince not to follow up with Dmitriev, and Prince had the impression that the issue was not a priority for Bannon.% 1090
\footnote{Prince 5/3/18 302, at~5.}
Prince related that Bannon did not appear angry, just relatively uninterested.% 1091
\footnote{Prince 5/3/18 302, at~5.}
Bannon, by contrast, told the Office that he never discussed with Prince anything regarding Dmitriev, RDIF, or any meetings with Russian individuals or people associated with Putin.% 1092
\footnote{Bannon 10/26/18 302, at~10--11.}
Bannon also stated that had Prince mentioned such a meeting, Bannon would have remembered it, and Bannon would have objected to such a meeting having taken place.% 1093
\footnote{Bannon 10/26/18 302, at~10--11.}
The conflicting accounts provided by Bannon and Prince could not be independently clarified by reviewing their communications, because neither one was able to produce any of the messages they exchanged in the time period surrounding the Seychelles meeting.
Prince's phone contained no text messages prior to March 2017, though provider records indicate that he and Bannon exchanged dozens of messages.% 1094
\footnote{Call Records of Erik Prince \blackout{Grand Jury}}
Prince denied deleting any messages but claimed he did not know why there were no messages on his device before March 2017.% 1095
\footnote{Prince 4/4/18 302, at~6.}
Bannon's devices similarly contained no messages in the relevant time period, and Bannon also stated he did not know why messages did not appear on his device.% 1096
\footnote{Bannon 10/26/18 302, at~11;
Bannon 2/14/18 302, at~36.}
Bannon told the Office that, during both the months before and after the Seychelles meeting, he regularly used his personal Blackberry and personal email for work-related communications (including those with Prince), and he took no steps to preserve these work communications.% 1097
\footnote{Bannon 10/26/18 302, at~11.}
\paragraph{Kirill Dmitriev's Post-Election Contact with Rick Gerson Regarding U.S.--Russia Relations}
Dmitriev's contacts during the transition period were not limited to those facilitated by Nader.
In approximately late November 2016, the UAE national security advisor introduced Dmitriev to Rick Gerson, a friend of Jared Kushner who runs a hedge fund in New York.% 1098
\footnote{Gerson 6/5/18 302, at~1, 3;
11/26/16 Text Message, Dmitriev to Gerson;
1/25/17 Text Message, Dmitriev to Nader.}
Gerson stated he had no formal role in the transition and had no involvement in the Trump Campaign other than occasional casual discussions about the Campaign with Kushner.% 1099
\footnote{Gerson 6/5/18 302, at~1.}
After the election, Gerson assisted the transition by arranging meetings for transition officials with former UK prime minister Tony Blair and a UAE delegation led by Crown Prince Mohammed.% 1100
\footnote{Gerson 6/5/18 302, at~1--2;
Kushner 4/11/18 302, at~21.}
When Dmitriev and Gerson met, they principally discussed potential joint ventures between Gerson's hedge fund and RDIF\null.% 1101
\footnote{Gerson 6/5/18 302, at~3--4;
\textit{see, e.g.}, 12/2/16 Text Messages, Dmitriev \& Gerson;
12/14/16 Text Messages, Dmitriev \& Gerson;
1/3/17 Text Message, Gerson to Dmitriev;
12/2/16 Email, Tolokonnikov to Gerson.}
Dmitriev was interested in improved economic cooperation between the United States and Russia and asked Gerson who he should meet with in the incoming Administration who would be helpful towards this goal.% 1102
\footnote{Gerson 6/5/18 302, at~3;
12/14/16 Text Message, Dmitriev to Gerson.}
Gerson replied that he would try to figure out the best way to arrange appropriate introductions, but noted that confidentiality would be required because of the sensitivity of holding such meetings before the new Administration took power, and before Cabinet nominees had been confirmed by the Senate.% 1103
\footnote{12/14/16 Text Message, Gerson to Dmitriev.}
Gerson said he would ask Kushner and Michael Flynn who the ``key person or people'' were on the topics of reconciliation with Russia, joint security concerns, and economic matters.% 1104
\footnote{12/14/16 Text Message, Gerson to Dmitriev.}
Dmitriev told Gerson that he had been tasked by Putin to develop and execute a reconciliation plan between the United States and Russia.
He noted in a text message to Gerson that if Russia was ``approached with respect and willingness to understand our position, we can have Major Breakthroughs quickly.''% 1105
\footnote{12/14/16 Text Messages, Dmitriev \& Gerson;
Gerson 6/15/18 302, at~1.}
Gerson and Dmitriev exchanged ideas in December 2016 about what such a reconciliation plan would include.% 1106
\footnote{12/14/16 Text Messages, Dmitriev \& Gerson.}
Gerson told the Office that the Transition Team had not asked him to engage in these discussions with Dmitriev, and that he did so on his own initiative and as a private citizen.% 1107
\footnote{Gerson 6/15/18 302, at~1.}
On January~9, 2017, the same day he asked Nader whether meeting Prince would be worthwhile, Dmitriev sent his biography to Gerson and asked him if he could ``share it with Jared (or somebody else very senior in the team)---so that they know that we are focused from our side on improving the relationship and my boss asked me to play a key role in that.''% 1108
\footnote{1/9/17 Text Messages, Dmitriev to Gerson;
1/9/17 Text Message, Dmitriev to Nader.}
Dmitriev also asked Gerson if he knew Prince, and if Prince was somebody important or worth spending time with.% 1109
\footnote{Gerson 6/5/18 302, at~4.}
After his trip to the Seychelles, Dmitriev told Gerson that Bannon had asked Prince to meet with Dmitriev and that the two had had a positive meeting.% 1110
\footnote{1/18/17 Text Messages, Dmitriev \& Gerson.}
On January~16, 2017, Dmitriev consolidated the ideas for U.S.--Russia reconciliation that he and Gerson had been discussing into a two-page document that listed five main points:
(1)~jointly fighting terrorism;
(2)~jointly engaging in anti-weapons of mass destruction efforts;
(3)~developing ``win-win'' economic and investment initiatives;
(4)~maintaining an honest, open, and continual dialogue regarding issues of disagreement; and
(5)~ensuring proper communication and trust by ``key people'' from each country.% 1111
\footnote{1/16/17 Text Messages, Dmitriev \& Gerson.}
On January~18, 2017, Gerson gave a copy of the document to Kushner.% 1112
\footnote{Gerson 6/5/18 302, at~3;
Gerson 6/15/18 302, at~2.}
Kushner had not heard of Dmitriev at that time.% 1113
\footnote{Gerson 6/5/18 302, at~3.}
Gerson explained that Dmitriev was the head of RDIF, and Gerson may have alluded to Dmitriev's being well connected.% 1114
\footnote{Gerson 6/5/18 302, at~3;
Gerson 6/15/18 302, at~1--2;
Kushner 4/11/18 302, at~22.}
Kushner placed the document in a file and said he would get it to the right people.% 1115
\footnote{Gerson 6/5/18 302, at~3.}
Kushner ultimately gave one copy of the document to Bannon and another to Rex Tillerson; according to Kushner, neither of them followed up with Kushner about it.% 1116
\footnote{Kushner 4/11/18 302, at~32.}
On January~19, 2017, Dmitriev sent Nader a copy of the two-page document, telling him that this was ``a view from our side that I discussed in my meeting on the islands and with you and with our friends.
Please share with them---we believe this is a good foundation to start from.''% 1117
\footnote{1/19/17 Text Message, Dmitriev to Nader (11:11:56~a.m.).}
Gerson informed Dmitriev that he had given the document to Kushner soon after delivering it.% 1118
\footnote{1/18/17 Text Message, Gerson to Dmitriev;
Gerson 6/15/18 302, at~2.}
On January~26, 2017, Dmitriev wrote to Gerson that his ``boss''---an apparent reference to Putin---was asking if there had been any feedback on the proposal.% 1119
\footnote{1/26/17 Text Message, Dmitriev to Gerson.}
Dmitriev said, ``[w]e do not want to rush things and move at a comfortable speed.
At the same time, my boss asked me to try to have the key US meetings in the next two weeks if possible.''% 1120
\footnote{1/26/17 Text Message, Dmitriev to Gerson.}
He informed Gerson that Putin and President Trump would speak by phone that Saturday, and noted that that information was ``very confidential.''% 1121
\footnote{1/96/17 Text Message, Dmitriev to Gerson.}
The same day, Dmitriev wrote to Nader that he had seen his ``boss'' again yesterday who had ``emphasized that this is a great priority for us and that we need to build this communication channel to avoid bureaucracy.''% 1122
\footnote{1/26/17 Text Message, Dmitriev to Nader (10:04:41~p.m.).}
On January~28, 2017, Dmitriev texted Nader that he wanted ``to see if I can confirm to my boss that your friends may use some of the ideas from the 2 pager I sent you in the telephone call that will happen at~12 EST,''% 1123
\footnote{1/28/17 Text Message, Dmitriev to Nader (11:05:39~a.m.).}
an apparent reference to the call scheduled between President Trump and Putin.
Nader replied, ``Definitely paper was so submitted to Team by Rick and me.
They took it seriously!''% 1124
\footnote{1/28/17 Text Message, Nader to Dmitriev (11:11:33~a.m.).}
After the call between President Trump and Putin occurred, Dmitriev wrote to Nader that ``the call went very well.
My boss wants me to continue making some public statements that us [sic] Russia cooperation is good and important.''% 1125
\footnote{1/29/17 Text Message, Dmitriev to Nader (11:06:35~a.m.).}
Gerson also wrote to Dmitriev to say that the call had gone well, and Dmitriev replied that the document they had drafted together ``played an important role.''% 1126
\footnote{1/28/17 Text Message, Gerson to Dmitriev;
1/29/17 Text Message, Dmitriev to Gerson.}
Gerson and Dmitriev appeared to stop communicating with one another in approximately March 2017, when the investment deal they had been working on together showed no signs of progressing.% 1127
\footnote{Gerson 6/15/18 302, at~4;
3/21/17 Text Message, Gerson to Dmitriev.}
\subsubsection{Ambassador Kislyak's Meeting with Jared Kushner and Michael Flynn in Trump Tower Following the Election}
On November~16, 2016, Catherine Vargas, an executive assistant to Kushner, received a request for a meeting with Russian Ambassador Sergey Kislyak.% 1128
\footnote{\textit{Statement of Jared C. Kushner to Congressional Committees} (``Kushner Stmt.''), at~6 (7/24/17) (written statement by Kushner to the Senate Judiciary Committee).}
That same day, Vargas sent Kushner an email with the subject, ``MISSED CALL: Russian Ambassador to the US, Sergey Ivanovich Kislyak\dots.''% 1129
\footnote{NOSC00004356 (11/16/16 Email, Vargas to Kushner (6:44~p.m.)).}
The text of the email read, ``RE: setting up a time to meet w/you on 12/1. LMK how to proceed.''
Kushner responded in relevant part, ``I think I do this one -- confirm with Dimitri [Simes of CNI] that this is the right guy.''% 1130
\footnote{NOSC00004356 (11/16/16 Email, Kushner to Vargas (9:54~p.m.)).}
After reaching out to a colleague of Simes at CNI, Vargas reported back to Kushner that Kislyak was ``the best go-to guy for routine matters in the US,'' while Yuri Ushakov, a Russian foreign policy advisor, was the contact for ``more direct/substantial matters.''% 1131
\footnote{11/17/16 Email, Brown to Simes (10:41~a.m.);
Brown 10/13/17 302, at~4;
11/17/16 Email, Vargas to Kushner (12:31:18).}
Bob Foresman, the UBS investment bank executive who had previously tried to transmit to candidate Trump an invitation to speak at an economic forum in Russia, \textit{see} \hyperlink{subparagraph.1.4.1.1.4.2}{Volume~I, Section~IV.A.1.d.ii}, \textit{supra}, may have provided similar information to the Transition Team.
According to Foresman, at the end of an early December 2016 meeting with incoming National Security Advisor Michael Flynn and his designated deputy (K.T. McFarland) in New York, Flynn asked Foresman for his thoughts on~Kislyak.
Foresman had not met Kislyak but told Flynn that, while Kislyak was an important person, Kislyak did not have a direct line to Putin.% 1132
\footnote{Foresman 10/17/18 302, at~17.}
Foresman subsequently traveled to Moscow, inquired of a source he believed to be close to Putin, and heard back from that source that Ushakov would be the official channel for the incoming U.S. national security advisor.% 1133
\footnote{Foresman 10/17/18 302, at~17--18.}
Foresman acknowledged that Flynn had not asked him to undertake that inquiry in Russia but told the Office that he nonetheless felt obligated to report the information back to Flynn, and that he worked to get a face-to-face meeting with Flynn in January 2017 so that he could do so.% 1134
\footnote{Foresman 10/17/18 302, at~18.}
Email correspondence suggests that the meeting ultimately went forward,% 1135
\footnote{RMF-SCO-00000015 (1/5/17 Email, Foresman to Atencio \& Flaherty);
RMF-SCO-00000015 (1/5/17 Email, Flaherty to Foresman \& Atencio).}
but Flynn has no recollection of it or of the earlier December meeting.% 1136
\footnote{9/26/18 Attorney Proffer from Covington \& Burling LLP (reflected in email on file with the Office).}
(The investigation did not identify evidence of Flynn or Kushner meeting with Ushakov after being given his name.% 1137
\footnote{Vargas 4/4/18 302, at~5.}%
)
In the meantime, although he had already formed the impression that Kislyak was not necessarily the right point of contact,% 1138
\footnote{Kushner 11/1/17 302, at~4.}
Kushner went forward with the meeting that Kislyak had requested on November~16. It took place at Trump Tower on November~30, 2016.% 1139
\footnote{AKIN\_GUMP\_BERKOWITZ\_0000016--019 (11/29/16 Email, Vargas to Kuznetsov).}
At Kushner's invitation, Flynn also attended; Bannon was invited but did not attend.% 1140
\footnote{Flynn 1/11/18 302, at~2;
NOS00004240 (Calendar Invite, Vargas to Kushner \& Flynn).}
During the meeting, which lasted approximately 30~minutes, Kushner expressed a desire on the part of the incoming Administration to start afresh with U.S.--Russian relations.% 1141
\footnote{Kushner Stmt. at~6.}
Kushner also asked Kislyak to identify the best person (whether Kislyak or someone else) with whom to direct future discussions---someone who had contact with Putin and the ability to speak for him.% 1142
\footnote{Kushner Stmt.\ at~6;
Kushner 4/11/18 302, at~18.}
The three men also discussed U.S. policy toward Syria, and Kislyak floated the idea of having Russian generals brief the Transition Team on the topic using a secure communications line.% 1143
\footnote{Kushner Stmt.\ at~7;
Kushner 4/11/18 302, at~18;
Flynn 1/11/18 302, at~2.}
After Flynn explained that there was no secure line in the Transition Team offices, Kushner asked Kislyak if they could communicate using secure facilities at the Russian Embassy.% 1144
\footnote{Kushner 4/11/18 302, at~18.}
Kislyak quickly rejected that idea.% 1145
\footnote{Kushner 4/11/18 302, at~18.}
\subsubsection{Jared Kushner's Meeting with Sergey Gorkov}
On December~6, 2016, the Russian Embassy reached out to Kushner's assistant to set up a second meeting between Kislyak and Kushner.% 1146
\footnote{Kushner Stmt.\ at~7;
NOSC00000123 (12/6/16 Email, Vargas to Kushner (12:11:40~p.m.)).}
Kushner declined several proposed meeting dates, but Kushner's assistant indicated that Kislyak was very insistent about securing a second meeting.% 1147
\footnote{Kushner 4/11/18 302, at~19;
NOSC00000130 (12/12/16 Email, Kushner to Vargas (10:41~p.m.)).}
Kushner told the Office that he did not want to take another meeting because he had already decided Kislyak was not the right channel for him to communicate with Russia, so he arranged to have one of his assistants, Avi Berkowitz, meet with Kislyak in his stead.% 1148
\footnote{Kushner 4/11/18 302, at~19;
Kushner Stmt.\ at~7;
DJTFP\_SCO\_01442290 (12/6/16 Email, Berkowitz to \blackout{Personal Privacy}}
Although embassy official Sergey Kuznetsov wrote to Berkowitz that Kislyak thought it ``important'' to ``continue the conversation with Mr.~Kushner in person,''% 1149
\footnote{DJTFP\_SCO\_01442290 (12/7/16 Email \blackout{Personal Privacy} to Berkowitz (12:31:39~p.m.)).}
Kislyak nonetheless agreed to meet instead with Berkowitz once it became apparent that Kushner was unlikely to take a meeting.
Berkowitz met with Kislyak on December~12, 2016, at Trump Tower.% 1150
\footnote{Berkowitz 1/12/18 302, at~7;
AKIN\_GUMP\_BERKOWITZ\_000001--04 (12/12/16 Text Messages, Berkowitz \& 202-701-8532).}
The meeting lasted only a few minutes, during which Kislyak indicated that he wanted Kushner to meet someone who had a direct line to Putin: Sergey Gorkov, the head of the Russian-government-owned bank Vnesheconombank (VEB).
Kushner agreed to meet with Gorkov.% 1151
\footnote{Kushner 4/11/18 302, at~19;
NOSC00000130--135 (12/12/16 Email, Kushner to Berkowitz).}
The one-on-one meeting took place the next day, December~13, 2016, at the Colony Capital building in Manhattan, where Kushner had previously scheduled meetings.% 1152
\footnote{Kushner 4/11/18 302, at~19;
NOSC00000130--135 (12/12/16 Email, Kushner to Berkowitz).}
VEB was (and is) the subject of Department of Treasury economic sanctions imposed in response to Russia's annexation of Crimea.% 1153
\footnote{\textit{Announcement of Treasury Sanctions on Entities Within the Financial Services and Energy Sectors of Russia, Against Arms or Related Materiel Entities, and those Undermining Ukraine's Sovereignty}, United States Department of the Treasury (Jul.~16, 2014).}
Kushner did not, however, recall any discussion during his meeting with Gorkov about the sanctions against VEB or sanctions more generally.% 1154
\footnote{Kushner 4/11/18 302, at~20.}
Kushner stated in an interview that he did not engage in any preparation for the meeting and that no one on the Transition Team even did a Google search for Gorkov's name.% 1155
\footnote{Kushner 4/11/18 302, at~19.
Berkowitz, by contrast, stated to the Office that he had googled Gorkov's name and told Kushner that Gorkov appeared to be a banker.
Berkowitz 1/12/18 302, at~8.}
At the start of the meeting, Gorkov presented Kushner with two gifts: a painting and a bag of soil from the town in Belarus where Kushner's family originated.% 1156
\footnote{Kushner 4/11/18 302, at~19--20.}
The accounts from Kushner and Gorkov differ as to whether the meeting was diplomatic or business in nature.
Kushner told the Office that the meeting was diplomatic, with Gorkov expressing disappointment with U.S.--Russia relations under President Obama and hopes for improved relations with the incoming Administration.% 1157
\footnote{Kushner Stmt.\ at~8.}
According to Kushner, although Gorkov told Kushner a little bit about his bank and made some statements about the Russian economy, the two did not discuss Kushner's companies or private business dealings of any kind.% 1158
\footnote{Kushner Stmt.\ at~8.}
(At the time of the meeting, Kushner Companies had a debt obligation coming due on the building it owned at 666~Fifth Avenue, and there had been public reporting both about efforts to secure lending on the property and possible conflicts of interest for Kushner arising out of his company's borrowing from foreign lenders.% 1159
\footnote{\textit{See, e.g.}, Peter Grant, \textit{Donald Trump Son-in-Law Jared Kushner Could Face His Own Conflict-of-Interest Questions}, Wall Street Journal (Nov.~29, 2016).}%
)
In contrast, in a 2017 public statement, VEB suggested Gorkov met with Kushner in Kushner's capacity as CEO of Kushner Companies for the purpose of discussing business, rather than as part of a diplomatic effort.
In particular, VEB characterized Gorkov's meeting with Kushner as part of a series of ``roadshow meetings'' with ``representatives of major US banks and business circles,'' which included ``negotiations'' and discussion of the ``most promising business lines and sectors.''% 1160
\footnote{Patrick Reevell \& Matthew Mosk, \textit{Russian Banker Sergey Gorkov Brushes off Questions About Meeting with Jared Kushner}, ABC News (June~1, 2017).}
Foresman, the investment bank executive mentioned in \hyperlink{subsubsection.1.4.1.1}{Volume~I, Sections~IV.A.1} and~\hyperlink{subsubsection.1.4.2.3}{IV.B.3}, \textit{supra}, told the Office that he met with Gorkov and VEB deputy chairman Nikolay Tsekhomsky in Moscow just before Gorkov left for New York to meet Kushner.% 1161
\footnote{Foresman 10/17/18 302, at~14--15.}
According to Foresman, Gorkov and Tsekhomsky told him that they were traveling to New York to discuss post-election issues with U.S. financial institutions, that their trip was sanctioned by Putin, and that they would be reporting back to Putin upon their return.% 1162
\footnote{Foresman 10/17/18 302, at~15--16.}
The investigation did not resolve the apparent conflict in the accounts of Kushner and Gorkov or determine whether the meeting was diplomatic in nature (as Kushner stated), focused on business (as VEB's public statement indicated), or whether it involved some combination of those matters or other matters.
Regardless, the investigation did not identify evidence that Kushner and Gorkov engaged in any substantive follow-up after the meeting.
Rather, a few days after the meeting, Gorkov's assistant texted Kushner's assistant, ``Hi, please inform your side that the information about the meeting had a very positive response!''% 1163
\footnote{AKIN\_GUMP\_BERKOWITZ\_0000011 (12/19/16 Text Message, Ivanchenko to Berkowitz (9:56~a.m.)).}
Over the following weeks, the two assistants exchanged a handful of additional cordial texts.% 1164
\footnote{AKIN\_GUMP\_BERKOWITZ\_0000011--15 (12/19/16--2/16/17 Text Messages, Ivanchenko \& Berkowitz).}
On February~8, 2017, Gorkov's assistant texted Kushner's assistant (Berkowitz) to try to set up another meeting, and followed up by text at least twice in the days that followed.% 1165
\footnote{AKIN\_GUMP\_BERKOWITZ\_0000015 (2/8/17 Text Message, Ivanchenko to Berkowitz (10:41~a.m.)).}
According to Berkowitz, he did not respond to the meeting request in light of the press coverage regarding the Russia investigation, and did not tell Kushner about the meeting request.% 1166
\footnote{Berkowitz 3/22/18 302, at~4--5.}
\subsubsection{Petr Aven's Outreach Efforts to the Transition Team}
In December 2016, weeks after the one-on-one meeting with Putin described in \hyperlink{paragraph.1.4.2.1.2}{Volume~I, Section~IV.B.1.b}, \textit{supra}, Petr Aven attended what he described as a separate ``all-hands'' oligarch meeting between Putin and Russia's most prominent businessmen.% 1167
\footnote{Aven 8/2/18 302, at~7; \blackout{Grand Jury}}
As in Aven's one-on-one meeting, a main topic of discussion at the oligarch meeting in December 2016 was the prospect of forthcoming U.S. economic sanctions.% 1168
\footnote{\blackout{Grand Jury}}
After the December 2016 all-hands meeting, Aven tried to establish a connection to the Trump team.
Aven instructed Richard Burt to make contact with the incoming Trump Administration.
Burt was on the board of directors for LetterOne (L1), another company headed by Aven, and had done work for Alfa-Bank.% 1169
\footnote{\blackout{Grand Jury} Aven 8/2/18 302, at~6.}
Burt had previously served as U.S. ambassador to Germany and Assistant Secretary of State for European and Canadian Affairs, and one of his primary roles with Alfa-Bank and L1 was to facilitate introductions to business contacts in the United States and other Western countries.% 1170
\footnote{\blackout{Grand Jury} Aven 8/2/18 302, at~6;
Burt 2/9/18 302, at~2.}
While at a L1 board meeting held in Luxembourg in late December 2016, Aven pulled Burt aside and told him that he had spoken to someone high in the Russian government who expressed interest in establishing a communications channel between the Kremlin and the Trump Transition Team.% 1171
\footnote{Burt 2/9/18 302, at~2; \blackout{Grand Jury}}
Aven asked for Burt's help in contacting members of the Transition Team.% 1172
\footnote{\blackout{Grand Jury}}
Although Burt had been responsible for helping Aven build connections in the past, Burt viewed Aven's request as unusual and outside the normal realm of his dealings with Aven.% 1173
\footnote{Burt 2/9/18 302, at~4.}
Burt, who is a member of the board of CNI (discussed at \hyperlink{subsubsection.1.4.1.4}{Volume~I, Section~IV.A.4}, \textit{supra}),% 1174
\footnote{Burt 2/9/18 302, at~5.}
decided to approach CNI president Dimitri Simes for help facilitating Aven's request, recalling that Simes had some relationship with Kushner.% 1175
\footnote{Burt 2/9/18 302, at~3.}
At the time, Simes was lobbying the Trump Transition Team, on Burt's behalf, to appoint Burt U.S. ambassador to Russia.% 1176
\footnote{Burt 2/9/18 302, at~3.}
Burt contacted Simes by telephone and asked if he could arrange a meeting with Kushner to discuss setting up a high-level communications channel between Putin and the incoming Administration.% 1177
\footnote{Burt 2/9/18 302, at~3;
Simes 3/27/18 302, at~4.}
Simes told the Office that he declined and stated to Burt that setting up such a channel was not a good idea in light of the media attention surrounding Russian influence in the U.S. presidential election.% 1178
\footnote{Burt 2/9/18 302, at~3;
Simes 3/27/18 302, at~4.}
According to Simes, he understood that Burt was seeking a secret channel, and Simes did not want CNI to be seen as an intermediary between the Russian government and the incoming Administration.% 1179
\footnote{Simes 3/27/18 302, at~5.}
Based on what Simes had read in the media, he stated that he already had concerns that Trump's business connections could be exploited by Russia, and Simes said that he did not want CNI to have any involvement or apparent involvement in facilitating any connection.% 1180
\footnote{Simes 3/27/18 302, at~5.}
In an email dated December~22, 2016, Burt recounted for Aven his conversation with Simes:
\begin{quote}
Through a trusted third party, I have reached out to the very influential person I mentioned in Luxembourg concerning Project~A\null.
There is an interest and an understanding for the need to establish such a channel.
But the individual emphasized that at this moment, with so much intense interest in the Congress and the media over the question of cyber-hacking (and who ordered what), Project~A was too explosive to discuss.
The individual agreed to discuss it again after the New Year.
I trust the individual's instincts on this.
If this is unclear or you would like to discuss, don't hesitate to call.% 1181
\footnote{12/22/16 Email, Burt to Aven (7:23~p.m.).}
\end{quote}
According to Burt, the ``very influential person'' referenced in his email was Simes, and the reference to a ``trusted third party'' was a fabrication, as no such third party existed.
``Project~A'' was a term that Burt created for Aven's effort to help establish a communications channel between Russia and the Trump team, which he used in light of the sensitivities surrounding what Aven was requesting, especially in light of the recent attention to Russia's influence in the U.S. presidential election.% 1182
\footnote{Burt 2/9/18 302, at~3.}
According to Burt, his report that there was ``interest'' in a communications channel reflected Simes's views, not necessarily those of the Transition Team, and in any event, Burt acknowledged that he added some ``hype'' to that sentence to make it sound like there was more interest from the Transition Team than may have actually existed.% 1183
\footnote{Burt 2/9/18 302, at~3--4.}
Aven replied to Burt's email on the same day, saying ``Thank you. All clear.''% 1184
\footnote{12/22/16 Email, Aven to Burt (4:58:22~p.m.).}
According to Aven, this statement indicated that he did not want the outreach to continue.% 1185
\footnote{Aven 8/2/18 302, at~7.}
Burt spoke to Aven some time thereafter about his attempt to make contact with the Trump team, explaining that the current environment made it impossible,
\blackout{Grand Jury}% 1186
\footnote{\blackout{Grand Jury}}
Burt did not recall discussing Aven's request with Simes again, nor did he recall speaking to anyone else about the request.% 1187
\footnote{Burt 2/9/18 302, at~3--4.}
In the first quarter of 2017, Aven met again with Putin and other Russian officials.% 1188
\footnote{\blackout{Grand Jury}}
At that meeting, Putin asked about Aven's attempt to build relations with the Trump Administration and Aven recounted his lack of success.% 1189
\footnote{\blackout{Grand Jury} Aven 8/2/18 302, at~7.}
\blackout{Grand Jury}% 1190
\footnote{\blackout{Grand Jury}}
Putin continued to inquire about Aven's efforts to connect to the Trump Administration in several subsequent quarterly meetings.% 1191
\footnote{\blackout{Grand Jury}}
Aven also told Putin's chief of staff that he had been subpoenaed by the~FBI\null.% 1192
\footnote{Aven 8/2/18 302, at~8.}
As part of that conversation, he reported that he had been asked by the FBI about whether he had worked to create a back channel between the Russian government and the Trump Administration.% 1193
\footnote{Aven 8/2/18 302, at~8; \blackout{Grand Jury}}
According to Aven, the official showed no emotion in response to this report and did not appear to care.% 1194
\footnote{Aven 8/2/18 302, at~8; \blackout{Grand Jury}}
\subsubsection{Carter Page Contact with Deputy Prime Minister Arkady Dvorkovich}
In December 2016, more than two months after he was removed from the Trump Campaign, former Campaign foreign policy advisor Carter Page again visited Moscow in an attempt to pursue business opportunities.% 1195
\footnote{Page 3/10/17 302, at~4;
Page 3/16/17 302, at~3;
\blackout{Grand Jury} Among other meetings, Page contacted Andrey Baranov, head of investor relations at Rosneft, and they discussed the sale of Rosneft and meetings Baranov had attended with Rosneft CEO Igor Sechin.
\blackout{Grand Jury}}
\blackout{Grand Jury}% 1196
\footnote{\blackout{Grand Jury}}
According to Konstantin Kilimnik, Paul Manafort's associate, Page also gave some individuals in Russia the impression that he had maintained his connections to President-Elect Trump.
In a December~8, 2016 email intended for Manafort, Kilimnik wrote, ``Carter Page is in Moscow today, sending messages he is authorized to talk to Russia on behalf of DT on a range of issues of mutual interest, including Ukraine.''% 1197
\footnote{\blackout{Investigative Technique}}
On December~9, 2016, Page went to dinner with NES employees Shlomo Weber and Andrej Krickovic.% 1198
\footnote{Page 3/16/17 302, at~3;
Page 3/30/17 302, at~8.}
Weber had contacted Dvorkovich to let him know that Page was in town and to invite him to stop by the dinner if he wished to do so, and Dvorkovich came to the restaurant for a few minutes to meet with Page.% 1199
\footnote{Weber 7/28/17 302, at~4;
Page 3/16/17 302, at~3;
\blackout{Grand Jury}}
Dvorkovich congratulated Page on Trump's election and expressed interest in starting a dialogue between the United States and Russia.% 1200
\footnote{Page 3/16/17 302, at~3;
\blackout{Grand Jury}}
Dvorkovich asked Page if he could facilitate connecting Dvorkovich with individuals involved in the transition to be in a discussion of future cooperation.% 1201
\footnote{Page 3/16/17 302, at~3;
\blackout{Grand Jury}}
\blackout{Grand Jury}% 1202
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1203
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}
Dvorkovich separately discussed working together in the future by forming an academic partnership.% 1204
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1205
\footnote{\blackout{Grand Jury}}
\blackout{Grand Jury}% 1206
\footnote{\blackout{Grand Jury}}
\subsubsection{Contacts With and Through Michael T. Flynn}
Incoming National Security Advisor Michael Flynn was the Transition Team's primary conduit for communications with the Russian Ambassador and dealt with Russia on two sensitive matters during the transition period: a United Nations Security Council vote and the Russian government's reaction to the United States's imposition of sanctions for Russian interference in the 2016 election.% 1207
\footnote{As discussed further in \hyperlink{subsubsection.1.5.3.4}{Volume~I, Section~V.C.4}, \textit{infra}, Flynn pleaded guilty to making false statements to the FBI, in violation of 18 U.S.C. \S~1001, about these communications with Ambassador Kislyak.
Plea Agreement, \textit{United States~v.\ Michael T. Flynn}, No.~1:17-cr-232 (D.D.C. Dec.~1, 2017), Doc.~3.
Flynn's plea agreement required that he cooperate with this Office, and the statements from Flynn in this report reflect his cooperation over the course of multiple debriefings in 2017 and~2018.}
Despite Kushner's conclusion that Kislyak did not wield influence inside the Russian government, the Transition Team turned to Flynn's relationship with Kislyak on both issues.
As to the sanctions, Flynn spoke by phone to K.T. McFarland, his incoming deputy, to prepare for his call to Kislyak; McFarland was with the President-Elect and other senior members of the Transition Team at Mar-a-Lago at the time.
Although transition officials at Mar-a-Lago had some concern about possible Russian reactions to the sanctions, the investigation did not identify evidence that the President-Elect asked Flynn to make any request to~Kislyak.
Flynn asked Kislyak not to escalate the situation in response to U.S. sanctions imposed on December~29, 2016, and Kislyak later reported to Flynn that Russia acceded to that request.
\paragraph{United Nations Vote on Israeli Settlements}
On December~21, 2016, Egypt submitted a resolution to the United Nations Security Council calling on Israel to cease settlement activities in Palestinian territory.% 1208
\footnote{Karen DeYoung, \textit{How the U.S. Came to Abstain on a U.N. Resolution Condemning Israeli Settlements}, Washington Post (Dec.~28, 2016).}
The Security Council, which includes Russia, was scheduled to vote on the resolution the following day.% 1209
\footnote{Karen DeYoung, \textit{How the U.S. Came to Abstain on a U.N. Resolution Condemning Israeli Settlements}, Washington Post (Dec.~28, 2016).}
There was speculation in the media that the Obama Administration would not oppose the resolution.% 1210
\footnote{Michelle Nichols \& Lesley Wroughton, \textit{U.S. Intended to Allow Passage of U.N. Draft Critical of Israel}, Reuters (Dec.~21, 2016).}
According to Flynn, the Transition Team regarded the vote as a significant issue and wanted to support Israel by opposing the resolution.% 1211
\footnote{Flynn 11/16/17 302, at~12;
Flynn 11/17/17 302, at~2.}
On December~22, 2016, multiple members of the Transition Team, as well as President-Elect Trump, communicated with foreign government officials to determine their views on the resolution and to rally support to delay the vote or defeat the resolution.% 1212
\footnote{Flynn 11/16/17 302, at~12--14;
Flynn 11/17/17 302, at~2.}
Kushner led the effort for the Transition Team; Flynn was responsible for the Russian government.% 1213
\footnote{Flynn 11/16/17 302, at~12--14;
Flynn 11/17/17 302, at~2;
Kushner 11/1/17 302, at~3;
12/22/16 Email, Kushner to Flynn;
12/22/16 Email, McFarland to \blackout{Personal Privacy} et~al.}
Minutes after an early morning phone call with Kushner on December~22, Flynn called Kislyak.% 1214
\footnote{Flynn 11/16/17 302, at~13;
Call Records of Michael T. Flynn \blackout{Grand Jury}}
According to Flynn, he informed Kislyak about the vote and the Transition Team's opposition to the resolution, and requested that Russia vote against or delay the resolution.% 1215
\footnote{Statement of Offense \P~3(d), \textit{United States~v.\ Michael T. Flynn}, No.~1:17-cr-232 (D.D.C. Dec.~1, 2017), Doc.~4 (``\textit{Flynn} Statement of Offense'');
Flynn 11/16/17 302, at~12--13.}
Later that day, President-Elect Trump spoke with Egyptian President Abdel Fattah al-Sisi about the vote.% 1216
\footnote{Flynn 11/17/17 302, at~2;
Flynn 11/16/17 302, at~13.}
Ultimately, Egypt postponed the vote.% 1217
\footnote{\textit{U.N. Vote on Israeli Settlement Postponed, ``Potentially Indefinitely''}, Reuters (Dec.~22, 2016).}
On December~23, 2016, Malaysia, New Zealand, Senegal, and Venezuela resubmitted the resolution.% 1218
\footnote{Somini Sengupta \& Rick Gladstone, \textit{Rebuffing Israel, U.S. Allows Censure Over Settlements}, New York Times (Dec.~23, 2016).}
Throughout the day, members of the Transition Team continued to talk with foreign leaders about the resolution, with Flynn continuing to lead the outreach with the Russian government through Kislyak.% 1219
\footnote{Flynn 11/16/17 302, at~12--14;
Kushner 11/1/17 302, at~3;
12/23/16 Email, Flynn to Kushner et~al.}
When Flynn again spoke with Kislyak, Kislyak informed Flynn that if the resolution came to a vote, Russia would not vote against it.% 1220
\footnote{\textit{Flynn} Statement of Offense \P~3(g).}
The resolution later passed 14--0, with the United States abstaining.% 1221
\footnote{\textit{Israel's Settlements Have No Legal Validity, Constitute Flagrant Violation of International Law, Security Council Reaffirms}, 7853rd Meeting (PM), United Nations Security Council (Dec.~23, 2016).}
\paragraph{U.S. Sanctions Against Russia}
Flynn was also the Transition Team member who spoke with the Russian government when the Obama Administration imposed sanctions and other measures against Russia in response to Russia's interference in the 2016 presidential election.
On December~28, 2016, then-President Obama signed Executive Order 13757, which took effect at 12:01~a.m. the following day and imposed sanctions on nine Russian individuals and entities.% 1222
\footnote{\textit{Taking Additional Steps to Address the National Emergency With Respect to Significant Malicious Cyber-Enabled Activities}, The White House, Office of the Press Secretary (Dec.~29, 2016).}
On December~29, 2016, the Obama Administration also expelled 35 Russian government officials and closed two Russian government-owned compounds in the United States.% 1223
\footnote{\textit{Statement by the President on Actions in Response to Russian Malicious Cyber Activity and Harassment}, The White House, Office of the Press Secretary (Dec.~29, 2016).}
During the rollout of the sanctions, President-Elect Trump and multiple Transition Team senior officials, including McFarland, Steve Bannon, and Reince Priebus, were staying at the Mar-a-Lago club in Palm Beach, Florida.
Flynn was on vacation in the Dominican Republic,% 1224
\footnote{Flynn 11/16/17 302, at~14;
McFarland 12/22/17 302, at~3--8;
Bannon 2/12/18 302, at~5.}
but was in daily contact with McFarland.% 1225
\footnote{Flynn 11/17/17 302, at~5;
Flynn 1/19/18 302, at~1;
McFarland 11/22/17 302, at~3--9.}
The Transition Team and President-Elect Trump were concerned that these sanctions would harm the United States's relationship with Russia.% 1226
\footnote{Flynn 11/17/17 302, at~3.}
Although the details and timing of sanctions were unknown on December~28, 2016, the media began reporting that retaliatory measures from the Obama Administration against Russia were forthcoming.% 1227
\footnote{Christine Wang, \textit{US to announce new sanctions against Russia in response to election hacking}, CNBC (Dec.~28, 2016).}
When asked about imposing sanctions on Russia for its alleged interference in the 2016 presidential election, President-Elect Trump told the media, ``I think we ought to get on with our lives.''% 1228
\footnote{John Wagner, \textit{Trump on alleged election interference by Russia: ``Get on with our lives''}, Washington Post (Dec.~29, 2016).}
Russia initiated the outreach to the Transition Team.
On the evening of December~28, 2016, Kislyak texted Flynn, ``can you kindly call me back at your convenience.''% 1229
\footnote{SF000006 (12/28/16 Text Message, Kislyak to Flynn).}
Flynn did not respond to the text message that evening.
Someone from the Russian Embassy also called Flynn the next morning, at 10:38~a.m., but they did not talk.% 1230
\footnote{Call Records of Michael T. Flynn \blackout{Grand Jury}}
The sanctions were announced publicly on December~29, 2016.% 1231
\footnote{Flynn 11/17/17 302, at~2--3;
McFarland 12/22/17 302, at~4--5.}
At 1:53~p.m.\ that day, McFarland began exchanging emails with multiple Transition Team members and advisors about the impact the sanctions would have on the incoming Administration.% 1232
\footnote{12/29/16 Email, McFarland to O'Brien et~al.;
12/29/16 Email, McFarland to Flynn et~al.}
At 2:07~p.m., a Transition Team member texted Flynn a link to a New York Times article about the sanctions.% 1233
\footnote{SF000001 (12/29/16 Text Message, Flaherty to Flynn).}
At 2:29~p.m., McFarland called Flynn, but they did not talk.% 1234
\footnote{Call Records of K.T. McFarland \blackout{Grand Jury}}
Shortly thereafter, McFarland and Bannon discussed the sanctions.% 1235
\footnote{McFarland 12/22/17 302, at~5--6.}
According to McFarland, Bannon remarked that the sanctions would hurt their ability to have good relations with Russia, and that Russian escalation would make things more difficult.% 1236
\footnote{McFarland 12/22/17 302, at~5--6.}
McFarland believed she told Bannon that Flynn was scheduled to talk to Kislyak later that night.% 1237
\footnote{McFarland 12/22/17 302, at~6.}
McFarland also believed she may have discussed the sanctions with Priebus, and likewise told him that Flynn was scheduled to talk to Kislyak that night.% 1238
\footnote{McFarland 12/22/17 302, at~6.}
At 3:14~p.m., Flynn texted a Transition Team member who was assisting McFarland, ``Time for a call???''% 1239
\footnote{SF000001 (12/29/16 Text Message, Flynn to Flaherty).}
The Transition Team member responded that McFarland was on the phone with Tom Bossert, a Transition Team senior official, to which Flynn responded, ``Tit for tat w Russia not good.
Russian AMBO reaching out to me today.''% 1240
\footnote{SF000001 (12/29/16 Text Message, Flynn to Flaherty).}
Flynn recalled that he chose not to communicate with Kislyak about the sanctions until he had heard from the team at Mar-a-Lago.% 1241
\footnote{Flynn 11/20/17 302, at~3.}
He first spoke with Michael Ledeen,% 1242
\footnote{Michael Ledeen is married to Barbara Ledeen, the Senate staffer whose 2016 efforts to locate Hillary Clinton's missing emails are described in \hyperlink{subsubsection.1.3.4.2}{Volume~I, Section~III.D.2}, \textit{supra}.}
a Transition Team member who advised on foreign policy and national security matters, for 20~minutes.% 1243
\footnote{Flynn 11/17/17 302, at~3;
Call Records of Michael Ledeen \blackout{Grand Jury}}
Flynn then spoke with McFarland for almost 20~minutes to discuss what, if anything, to communicate to Kislyak about the sanctions.% 1244
\footnote{Flynn 11/17/17 302, at~3--4;
\textit{Flynn} Statement of Offense \P~3(c);
Call Records of K.T. McFarland \blackout{Grand Jury};
Call Records of Michael T. Flynn \blackout{Grand Jury}.
}
On that call, McFarland and Flynn discussed the sanctions, including their potential impact on the incoming Trump Administration's foreign policy goals.% 1245
\footnote{Flynn 11/17/17 302, at~3--4.}
McFarland and Flynn also discussed that Transition Team members in Mar-a-Lago did not want Russia to escalate the situation.% 1246
\footnote{Flynn 11/17/17 302, at~3--4;
\textit{Flynn} Statement of Offense \P~3(c);
McFarland 12/22/17 302, at~6--7.}
They both understood that Flynn would relay a message to Kislyak in hopes of making sure the situation would not get out of hand.% 1247
\footnote{Flynn 11/17/17 302, at~4;
McFarland 12/22/17 302, at~6--7.}
Immediately after speaking with McFarland, Flynn called and spoke with~Kislyak.% 1248
\footnote{\textit{Flynn} Statement of Offense \P~3(d).}
Flynn discussed multiple topics with Kislyak, including the sanctions, scheduling a video teleconference between President-Elect Trump and Putin, an upcoming terrorism conference, and Russia's views about the Middle East.% 1249
\footnote{Flynn 11/17/17 302, at~3--4;
\textit{Flynn} Statement of Offense \P~3(c);
12/30/16 Email, Flynn to McFarland.}
With respect to the sanctions, Flynn requested that Russia not escalate the situation, not get into a ``tit for tat,'' and only respond to the sanctions in a reciprocal manner.% 1250
\footnote{Flynn 11/17/17 302, at~1;
\textit{Flynn} Statement of Offense \P~3(d).}
Multiple Transition Team members were aware that Flynn was speaking with Kislyak that day.
In addition to her conversations with Bannon and Reince Priebus, at 4:43~p.m., McFarland sent an email to Transition Team members about the sanctions, informing the group that ``Gen [F]lynn is talking to russian ambassador this evening.''% 1251
\footnote{12/29/16 Email, McFarland to Flynn et~al.}
Less than an hour later, McFarland briefed President-Elect Trump.
Bannon, Priebus, Sean Spicer, and other Transition Team members were present.% 1252
\footnote{12/29/16 Email, Westerhout to Flaherty;
McFarland 12/22/17 302, at~7.}
During the briefing, President-Elect Trump asked McFarland if the Russians did ``it,'' meaning the intrusions intended to influence the presidential election.% 1253
\footnote{McFarland 12/22/17 302, at~7.}
McFarland said yes, and President-Elect Trump expressed doubt that it was the Russians.% 1254
\footnote{McFarland 12/22/17 302, at~7.}
McFarland also discussed potential Russian responses to the sanctions, and said Russia's response would be an indicator of what the Russians wanted going forward.% 1255
\footnote{McFarland 12/22/17 302, at~7.}
President-Elect Trump opined that the sanctions provided him with leverage to use with the Russians.% 1256
\footnote{McFarland 12/22/17 302, at~7.}
McFarland recalled that at the end of the meeting, someone may have mentioned to President-Elect Trump that Flynn was speaking to the Russian ambassador that evening.% 1257
\footnote{McFarland 12/22/17 302, at~7.}
After the briefing, Flynn and McFarland spoke over the phone.% 1258
\footnote{McFarland 12/22/17 302, at~7.}
Flynn reported on the substance of his call with Kislyak, including their discussion of the sanctions.% 1259
\footnote{Flynn 11/17/17 302, at~4;
\textit{Flynn} Statement of Offense \P~3(e).}
According to McFarland, Flynn mentioned that the Russian response to the sanctions was not going to be escalatory because they wanted a good relationship with the incoming Administration.% 1260
\footnote{McFarland 12/22/17 302, at~7.}
McFarland also gave Flynn a summary of her recent briefing with President-Elect Trump.% 1261
\footnote{McFarland 12/22/17 302, at~7.}
The next day, December~30, 2016, Russian Foreign Minister Sergey Lavrov remarked that Russia would respond in kind to the sanctions.% 1262
\footnote{\textit{Comment by Foreign Minister Sergey Lavrov on recent US sanctions and the expulsion of Russian diplomats, Moscow, December~20, 2016}, The Ministry of Foreign Affairs of the Russian Federation (Dec.~30, 2016 (5:32~a.m.)).}
Putin superseded that comment two hours later, releasing a statement that Russia would not take retaliatory measures in response to the sanctions at that time.% 1263
\footnote{\textit{Statement of the President of the Russian Federation}, Kremlin, Office of the President (Dec.~30, 2016 (7:15~a.m,)).}
Hours later President-Elect Trump tweeted, ``Great move on delay (by V.~Putin).''% 1264
\footnote{\UseVerb{DJT} 12/30/16 (11:41~a.m.) Tweet.}
Shortly thereafter, Flynn sent a text message to McFarland summarizing his call with Kislyak from the day before, which she emailed to Kushner, Bannon, Priebus, and other Transition Team members.% 1265
\footnote{12/30/16 Email, Flynn to McFarland;
12/30/16 Email, McFarland to Kushner et~al.}
The text message and email did not include sanctions as one of the topics discussed with~Kislyak.% 1266
\footnote{12/30/16 Email, McFarland to Kushner et~al.}
Flynn told the Office that he did not document his discussion of sanctions because it could be perceived as getting in the way of the Obama Administration's foreign policy.% 1267
\footnote{Flynn 11/17/17 302, at~4.}
On December~31, 2016, Kislyak called Flynn and told him the request had been received at the highest levels and that Russia had chosen not to retaliate to the sanctions in response to the request.% 1268
\footnote{Call Records of Michael T. Flynn \blackout{Grand Jury};
Flynn 11/17/17 302, at~1;
Flynn 1/19/17 302, at~3;
\textit{Flynn} Statement of Offense \P~3(g).}
Two hours later, Flynn spoke with McFarland and relayed his conversation with~Kislyak.% 1269
\footnote{Call Records of Michael T. Flynn \blackout{Grand Jury};
Flynn 11/17/17 302, at~5;
Flynn 1/19/17 302, at~3;
McFarland 12/22/17 302, at~10.}
According to McFarland, Flynn remarked that the Russians wanted a better relationship and that the relationship was back on track.% 1270
\footnote{McFarland 12/22/17 302, at~10.}
Flynn also told McFarland that he believed his phone call had made a difference.% 1271
\footnote{McFarland 12/22/17 302, at~10.}
McFarland recalled congratulating Flynn in response.% 1272
\footnote{McFarland 12/22/17 302, at~10.}
Flynn spoke with other Transition Team members that day, but does not recall whether they discussed the sanctions.% 1273
\footnote{Flynn 11/17/17 302, at~5--6.}
Flynn recalled discussing the sanctions with Bannon the next day and that Bannon appeared to know about Flynn's conversation with~Kislyak.% 1274
\footnote{Flynn 11/21/17 302, at~1;
Flynn 11/20/17 302, at~3;
Flynn 1/19/17 302, at~5;
\textit{Flynn} Statement of Offense \P~3(h).}
Bannon, for his part, recalled meeting with Flynn that day, but said that he did not remember discussing sanctions with him.% 1275
\footnote{Bannon 2/12/18 302, at~9.}
Additional information about Flynn's sanctions-related discussions with Kislyak, and the handling of those discussions by the Transition Team and the Trump Administration, is provided in \hyperref[chap:volume-2]{Volume~II} of this report.
\hr
In sum, the investigation established multiple links between Trump Campaign officials and individuals tied to the Russian government.
Those links included Russian offers of assistance to the Campaign.
In some instances, the Campaign was receptive to the offer, while in other instances the Campaign officials shied away.
Ultimately, the investigation did not establish that the Campaign coordinated or conspired with the Russian government in its election-interference activities.
|
From Coq Require Import String.
From MatchingLogic Require Import
Logic
DerivedOperators_Syntax
ProofSystem
ProofMode.MLPM
.
Import
MatchingLogic.Logic.Notations
MatchingLogic.DerivedOperators_Syntax.Notations
.
Open Scope string_scope.
Open Scope list_scope.
Open Scope ml_scope.
(** Low-level proof, using only the proof system *)
Lemma ex3_low : forall {Σ : Signature} (Γ : Theory) (A B : Pattern) x,
well_formed A = true ->
well_formed B = true ->
x ∉ free_evars A ->
x ∉ free_evars B ->
Γ ⊢ (ex, A^{{evar:x ↦ 0}} and B^{{evar:x↦0}}) ---> ex, A^{{evar:x↦0}}.
Proof.
intros Σ Γ A B x WFA WFB Hx1 Hx2.
(* inline proof of ex_quan_monotone *)
assert (forall A B, well_formed A -> well_formed B -> x ∉ free_evars A -> x ∉ free_evars B ->
Γ ⊢ A ---> B -> Γ ⊢ (ex, A^{{evar:x↦0}}) ---> ex, B^{{evar:x↦0}}) as H. {
intros. apply Ex_gen. 1-2: shelve.
eapply syllogism_meta. 1-3: shelve. exact H3.
rewrite <- (evar_open_evar_quantify x 0 B0) at 1. 2: shelve.
unfold evar_open. fold (instantiate (ex, B0^{{evar:x↦0}}) (patt_free_evar x)).
epose proof (Ex_quan Γ (B0^{{evar:x↦0}}) x _) as H4.
eapply useGenericReasoning in H4. exact H4. shelve.
}
apply (H (A and B)). 1-4: shelve.
epose proof (pf_conj_elim_l Γ A B _ _).
eapply useGenericReasoning in H0. exact H0. shelve.
Unshelve.
(* 9 well_formed goals *)
3-7,9-10,13-14: try wf_auto2.
(* 3 variable membership goals *)
4,5: cbn; solve_free_evars 1. 2: { apply evar_quantify_no_occurrence. }
(* 3 ProofInfoLe goals *)
1-3: try_solve_pile.
Defined.
From Coq Require Import ssreflect ssrfun ssrbool.
Open Scope string_scope.
Open Scope list_scope.
Open Scope ml_scope.
(** Proof using only FOL proof mode *)
Lemma ex3_fol_pm : forall {Σ : Signature} (Γ : Theory) (A B : Pattern) x,
well_formed A = true ->
well_formed B = true ->
x ∉ free_evars A ->
x ∉ free_evars B ->
Γ ⊢ (ex, A^{{evar:x ↦ 0}} and B^{{evar:x↦0}}) ---> ex, A^{{evar:x↦0}}.
Proof.
intros Σ Γ A B x WFA WFB Hx1 Hx2.
mlIntro "H". mlDestructEx "H" as x.
mlSimpl. mlExists x. fromMLGoal.
epose proof (pf_conj_elim_l Γ A _ _ _) as H0.
eapply useGenericReasoning in H0. exact H0. shelve.
Unshelve.
(* 2 well_formedness *)
1-2: wf_auto2.
(* 1 ProofInfoLe *)
try_solve_pile.
Defined.
(** Proof using proof mode *)
Lemma ex3_pm : forall {Σ : Signature} (Γ : Theory) (A B : Pattern) x,
well_formed A = true ->
well_formed B = true ->
x ∉ free_evars A ->
x ∉ free_evars B ->
Γ ⊢ (ex, A^{{evar:x ↦ 0}} and B^{{evar:x↦0}}) ---> ex, A^{{evar:x↦0}}.
Proof.
intros Σ Γ A B x WFA WFB Hx1 Hx2.
mlIntro "H". mlDestructEx "H" as x.
mlSimpl. mlDestructAnd "H" as "H0" "H1".
mlExists x. mlAssumption.
Defined.
Section compute.
From MatchingLogic.Theories Require Import Definedness_Syntax
Definedness_Semantics
Sorts_Syntax
Sorts_Semantics
Definedness_ProofSystem.
From stdpp Require Import base fin_sets sets propset finite.
Inductive Symbols :=
| sym_import_definedness (d : Definedness_Syntax.Symbols)
| Zero | Succ (* constructors for Nats *)
| TT | FF
| even
.
Instance Symbols_eqdec : EqDecision Symbols.
Proof. solve_decision. Defined.
#[local]
Program Instance Symbols_fin : Finite Symbols :=
{|
enum := [Zero; Succ; TT ; FF; even;
sym_import_definedness Definedness_Syntax.definedness] ;
|}.
Next Obligation.
repeat constructor; set_solver.
Qed.
Next Obligation.
destruct x; try set_solver.
destruct d; set_solver.
Qed.
Instance signature : Signature :=
{| variables := StringMLVariables ;
ml_symbols := {|
symbols := Symbols ;
|}
|}.
Instance definedness_syntax : Definedness_Syntax.Syntax :=
{|
Definedness_Syntax.inj := sym_import_definedness;
|}.
Open Scope string_scope.
Let X0 := patt_free_evar "X0".
Let X := patt_free_evar "X".
Let sym_even := patt_sym even.
Let sym_succ := patt_sym Succ.
Let sym_zero := patt_sym Zero.
Let sym_tt := patt_sym TT.
Let sym_ff := patt_sym FF.
(* axioms *)
Definition defined : Pattern := Definedness_Syntax.axiom AxDefinedness.
Definition A : Pattern :=
sym_zero.
Definition B : Pattern :=
patt_app sym_succ sym_zero.
Definition proof3_low := ex3_low ∅ A B "X" ltac:(wf_auto2) ltac:(wf_auto2) ltac:(set_solver) ltac:(set_solver).
Definition proof3_fol_pm := ex3_fol_pm ∅ A B "X" ltac:(wf_auto2) ltac:(wf_auto2) ltac:(set_solver) ltac:(set_solver).
Definition proof3_pm := ex3_pm ∅ A B "X" ltac:(wf_auto2) ltac:(wf_auto2) ltac:(set_solver) ltac:(set_solver).
End compute.
|
lemma open_lessThan [continuous_intros, simp]: "open {..< a}" |
import data.real.basic
def converges_to (s : ℕ → ℝ) (a : ℝ) :=
∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε
example : (λ x y : ℝ, (x + y)^2) = (λ x y : ℝ, x^2 + 2*x*y + y^2) :=
begin
ext x y,
ring,
end
example (a b : ℝ) : abs a = abs (a - b + b) :=
begin
congr,
ring,
end
example {a : ℝ} (h : 1 < a) : a < a * a :=
begin
convert (mul_lt_mul_right _).2 h,
rw one_mul,
linarith,
end
theorem converges_to_const (a : ℝ) : converges_to (λ x : ℕ, a) a :=
begin
intros ε epos,
use 0,
intros n nge,
dsimp,
rw [sub_self, abs_zero],
apply epos,
end
theorem converges_to_add {s t : ℕ → ℝ} {a b : ℝ}
(cs : converges_to s a) (ct : converges_to t b):
converges_to (λ n, s n + t n) (a + b) :=
begin
intros ε epos,
dsimp,
have e2pos : 0 < ε / 2 := by linarith,
cases cs (ε / 2) e2pos with Ns fs,
cases ct (ε / 2) e2pos with Nt ft,
let mN := max Ns Nt,
use mN,
intros n nge,
have : |s n - a| < ε / 2 := fs n (le_of_max_le_left nge),
have : |t n - b| < ε / 2 := ft n (le_of_max_le_right nge),
apply lt_of_le_of_lt,
show |s n + t n - (a + b)| ≤ |s n - a| + |t n - b|,
calc
|s n + t n - (a + b)| = |(s n - a) + (t n - b)| : by congr; ring
... ≤ |s n - a| + |t n - b| : by apply abs_add,
show |s n - a| + |t n - b| < ε,
have : ε = ε / 2 + ε / 2 := by norm_num,
rw this,
apply add_lt_add; assumption,
end
theorem converges_to_mul_const {s : ℕ → ℝ} {a : ℝ}
(c : ℝ) (cs : converges_to s a) :
converges_to (λ n, c * s n) (c * a) :=
begin
by_cases h : c = 0,
{ convert converges_to_const 0,
ext,
rw [h, zero_mul],
rw [h, zero_mul],
},
{ have acpos : |c| > 0,
from abs_pos.mpr h,
intros ε epos,
have : ε / |c| > 0,
from div_pos epos acpos,
dsimp,
cases cs (ε / |c|) this with N fn,
use N,
intros n nge,
have hec : |s n - a| < ε / |c| := fn n nge,
have hmul : |c * s n - c * a| = |c| * |s n - a|,
calc
|c * s n - c * a| = |c * (s n - a)| : by congr; ring
... = |c| * |s n - a| : by apply abs_mul,
have : ε = ε * |c| / |c|,
calc
ε = ε * 1 : (mul_one ε).symm
... = ε * (|c| / |c|) : by rw div_self (ne_of_gt acpos)
... = ε * |c| / |c| : by ring,
rw this,
calc
|c * s n - c * a| = |c| * |s n - a| : by assumption
... < |c| * (ε / |c|) : by apply (mul_lt_mul_left acpos).mpr hec
... = ε * |c| / |c| : by ring,
}
end
theorem exists_abs_le_of_converges_to {s : ℕ → ℝ} {a : ℝ}
(cs : converges_to s a) :
∃ N b, ∀ n, N ≤ n → abs (s n) < b :=
begin
cases cs 1 zero_lt_one with N h,
use [N, |a| + 1],
intros n nge,
have h := h n nge,
apply lt_of_le_of_lt,
show |s n| ≤ |a| + |s n - a|,
calc
|s n| = |a + (s n - a)| : by congr; ring_nf
... ≤ |a| + |s n - a| : by apply abs_add,
show |a| + |s n - a| < |a| + 1,
apply add_lt_add_of_le_of_lt,
apply le_refl,
exact h,
end
lemma aux {s t : ℕ → ℝ} {a : ℝ}
(cs : converges_to s a) (ct : converges_to t 0) :
converges_to (λ n, s n * t n) 0 :=
begin
intros ε epos, dsimp,
rcases exists_abs_le_of_converges_to cs with ⟨N, b, h⟩,
have bpos : 0 < b,
from lt_of_le_of_lt (abs_nonneg _) (h N (le_refl _)),
have pos : ε / b > 0,
from div_pos epos bpos,
cases ct (ε / b) pos with N' h',
let mN := max N N',
use mN,
intros n nge,
have : |s n * t n - 0| = |s n * t n|,
by congr; apply sub_zero,
rw this,
have ht : |t n - 0| = |t n|,
by congr; apply sub_zero,
rw abs_mul,
have : ε = b * (ε / b),
calc
ε = ε * 1 : (mul_one ε).symm
... = ε * (b / b) : by rw div_self (ne_of_gt bpos)
... = b * (ε / b) : by ring,
rw this,
apply mul_lt_mul',
show |s n| ≤ b,
apply le_of_lt,
apply h n,
from le_trans (le_max_left N N') nge,
show |t n| < ε / b,
rw ht.symm,
apply h' n,
from le_trans (le_max_right N N') nge,
show 0 ≤ |t n|,
from abs_nonneg (t n),
show 0 < b,
from bpos,
end
theorem converges_to_mul {s t : ℕ → ℝ} {a b : ℝ}
(cs : converges_to s a) (ct : converges_to t b):
converges_to (λ n, s n * t n) (a * b) :=
begin
have h₁ : converges_to (λ n, s n * (t n - b)) 0,
apply aux cs,
convert converges_to_add ct (converges_to_const (-b)),
ring,
convert converges_to_add h₁ (converges_to_mul_const b cs),
ext, ring,
ring,
end
theorem converges_to_unique {s : ℕ → ℝ} {a b : ℝ}
(sa : converges_to s a) (sb : converges_to s b) :
a = b :=
begin
by_contra abne,
have : |a - b| > 0,
apply abs_pos.mpr,
intro h',
apply abne,
linarith,
exact has_add.to_covariant_class_left ℝ,
let ε := |a - b| / 2,
have epos : ε > 0,
change |a - b| / 2 > 0,
linarith,
cases sa ε epos with Na hNa,
cases sb ε epos with Nb hNb,
let N := max Na Nb,
have ha : | 2 * (s N - a)| < |a - b|,
have : |s N - a| < |a - b| / 2,
apply hNa N,
apply le_max_left,
have : |2| * |s N - a| < |2| * (|a - b| / 2),
apply mul_lt_mul',
apply le_refl,
from this,
apply abs_nonneg,
norm_num,
rw abs_mul,
have : |2| * (|a - b| / 2) = |a - b|,
rw mul_div,
rw mul_comm,
rw ←mul_div,
norm_num,
linarith,
have hb : | 2 * (s N - b)| < |a - b|,
have : |s N - b| < |a - b| / 2,
apply hNb N,
apply le_max_right,
have : |2| * |s N - b| < |2| * (|a - b| / 2),
apply mul_lt_mul',
apply le_refl,
from this,
apply abs_nonneg,
norm_num,
rw abs_mul,
have : |2| * (|a - b| / 2) = |a - b|,
rw mul_div,
rw mul_comm,
rw ←mul_div,
norm_num,
linarith,
have ha : |s N - a| < |a - b| / 2,
apply hNa N,
apply le_max_left,
have hb : |s N - b| < |a - b| / 2,
apply hNb N,
apply le_max_right,
have ab : |a - b| = |a - b|/2 + |a - b|/2,
norm_num,
have : |a - b| = | -(s N - a) + (s N - b)|,
congr,
linarith,
have : |a - b| < |a - b|,
apply lt_of_le_of_lt,
show |a - b| ≤ | -(s N - a) + (s N - b)|,
rw this,
show | -(s N - a) + (s N - b)| < |a - b|,
apply lt_of_le_of_lt,
apply abs_add,
rw ab,
apply add_lt_add,
rw abs_neg,
from ha,
from hb,
from lt_irrefl _ this,
end
variables {α : Type*} [linear_order α]
def converges_to' (s : α → ℝ) (a : ℝ) :=
∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε |
State Before: a b : ℤ
n✝ : ℕ
n : ℤ
⊢ 0 ≤ 2 State After: no goals Tactic: decide State Before: a b : ℤ
n✝ : ℕ
n : ℤ
⊢ abs 2 ≠ 0 State After: no goals Tactic: decide State Before: a b : ℤ
n✝ : ℕ
n : ℤ
h : n % 2 < 2
⊢ 2 ≠ 0 State After: no goals Tactic: decide State Before: a b : ℤ
n✝ : ℕ
n : ℤ
h : n % 2 < 2
h₁✝ : 0 ≤ n % 2
k : ℕ
h₁ : ↑(k + 2) < 2
x✝ : 0 ≤ ↑(k + 2)
⊢ 2 ≤ ↑(k + 2) State After: a b : ℤ
n✝ : ℕ
n : ℤ
h : n % 2 < 2
h₁✝ : 0 ≤ n % 2
k : ℕ
h₁ : ↑(k + 2) < 2
x✝ : 0 ≤ ↑(k + 2)
⊢ 2 ≤ ↑k + ↑2 Tactic: rw [Nat.cast_add] State Before: a b : ℤ
n✝ : ℕ
n : ℤ
h : n % 2 < 2
h₁✝ : 0 ≤ n % 2
k : ℕ
h₁ : ↑(k + 2) < 2
x✝ : 0 ≤ ↑(k + 2)
⊢ 2 ≤ ↑k + ↑2 State After: no goals Tactic: exact (le_add_iff_nonneg_left 2).2 (NonNeg.mk k) State Before: a✝ b : ℤ
n✝ : ℕ
n : ℤ
h : n % 2 < 2
h₁✝ : 0 ≤ n % 2
a : ℕ
x✝ : -[a+1] < 2
h₁ : 0 ≤ -[a+1]
⊢ -[a+1] = 0 ∨ -[a+1] = 1 State After: no goals Tactic: cases h₁ |
** Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
** See https://llvm.org/LICENSE.txt for license information.
** SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
* VMS IMPLICIT NONE statement.
program bb10
implicit none
integer f
call check( f(-1), 7, 1)
end
function f(a)
implicitnone
integer f, a
f = iabs(-6) + JIABS(a)
end
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
! This file was ported from Lean 3 source module analysis.specific_limits.floor_pow
! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathbin.Analysis.SpecificLimits.Basic
import Mathbin.Analysis.SpecialFunctions.Pow
/-!
# Results on discretized exponentials
We state several auxiliary results pertaining to sequences of the form `⌊c^n⌋₊`.
* `tendsto_div_of_monotone_of_tendsto_div_floor_pow`: If a monotone sequence `u` is such that
`u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all `c > 1`, then `u n / n` tends to `l`.
* `sum_div_nat_floor_pow_sq_le_div_sq`: The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable
to `1/j^2`, up to a multiplicative constant.
-/
open Filter Finset
open Topology BigOperators
/-- If a monotone sequence `u` is such that `u n / n` tends to a limit `l` along subsequences with
exponential growth rate arbitrarily close to `1`, then `u n / n` tends to `l`. -/
theorem tendsto_div_of_monotone_of_exists_subseq_tendsto_div (u : ℕ → ℝ) (l : ℝ)
(hmono : Monotone u)
(hlim :
∀ a : ℝ,
1 < a →
∃ c : ℕ → ℕ,
(∀ᶠ n in atTop, (c (n + 1) : ℝ) ≤ a * c n) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / c n) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) :=
by
/- To check the result up to some `ε > 0`, we use a sequence `c` for which the ratio
`c (N+1) / c N` is bounded by `1 + ε`. Sandwiching a given `n` between two consecutive values of
`c`, say `c N` and `c (N+1)`, one can then bound `u n / n` from above by `u (c N) / c (N - 1)`
and from below by `u (c (N - 1)) / c N` (using that `u` is monotone), which are both comparable
to the limit `l` up to `1 + ε`.
We give a version of this proof by clearing out denominators first, to avoid discussing the sign
of different quantities. -/
have lnonneg : 0 ≤ l :=
by
rcases hlim 2 one_lt_two with ⟨c, cgrowth, ctop, clim⟩
have : tendsto (fun n => u 0 / c n) at_top (𝓝 0) :=
tendsto_const_nhds.div_at_top (tendsto_nat_cast_atTop_iff.2 Ctop)
apply le_of_tendsto_of_tendsto' this clim fun n => _
simp_rw [div_eq_inv_mul]
exact mul_le_mul_of_nonneg_left (hmono (zero_le _)) (inv_nonneg.2 (Nat.cast_nonneg _))
have A : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in at_top, u n - n * l ≤ ε * (1 + ε + l) * n :=
by
intro ε εpos
rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
have L : ∀ᶠ n in at_top, u (c n) - c n * l ≤ ε * c n :=
by
rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isOCat_one_iff ℝ, Asymptotics.isOCat_iff] at
clim
filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)]with n hn cnpos'
have cnpos : 0 < c n := cnpos'
calc
u (c n) - c n * l = (u (c n) / c n - l) * c n := by
simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, field_simps]
_ ≤ ε * c n := by
refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
exact le_trans (le_abs_self _) hn
obtain ⟨a, ha⟩ :
∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
eventually_at_top.1 (cgrowth.and L)
let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
filter_upwards [Ici_mem_at_top M]with n hn
have exN : ∃ N, n < c N :=
by
rcases(tendsto_at_top.1 Ctop (n + 1)).exists with ⟨N, hN⟩
exact ⟨N, by linarith only [hN]⟩
let N := Nat.find exN
have ncN : n < c N := Nat.find_spec exN
have aN : a + 1 ≤ N := by
by_contra' h
have cNM : c N ≤ M := by
apply le_max'
apply mem_image_of_mem
exact mem_range.2 h
exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
have cNn : c (N - 1) ≤ n :=
by
have : N - 1 < N := Nat.pred_lt Npos.ne'
simpa only [not_lt] using Nat.find_min exN this
have IcN : (c N : ℝ) ≤ (1 + ε) * c (N - 1) :=
by
have A : a ≤ N - 1 := by linarith only [aN, Npos]
have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
have := (ha _ A).1
rwa [B] at this
calc
u n - n * l ≤ u (c N) - c (N - 1) * l :=
by
apply sub_le_sub (hmono ncN.le)
apply mul_le_mul_of_nonneg_right (Nat.cast_le.2 cNn) lnonneg
_ = u (c N) - c N * l + (c N - c (N - 1)) * l := by ring
_ ≤ ε * c N + ε * c (N - 1) * l := by
apply add_le_add
· apply (ha _ _).2
exact le_trans (by simp only [le_add_iff_nonneg_right, zero_le']) aN
· apply mul_le_mul_of_nonneg_right _ lnonneg
linarith only [IcN]
_ ≤ ε * ((1 + ε) * c (N - 1)) + ε * c (N - 1) * l :=
(add_le_add (mul_le_mul_of_nonneg_left IcN εpos.le) le_rfl)
_ = ε * (1 + ε + l) * c (N - 1) := by ring
_ ≤ ε * (1 + ε + l) * n :=
by
refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
apply mul_nonneg εpos.le
linarith only [εpos, lnonneg]
have B : ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in at_top, (n : ℝ) * l - u n ≤ ε * (1 + l) * n :=
by
intro ε εpos
rcases hlim (1 + ε) ((lt_add_iff_pos_right _).2 εpos) with ⟨c, cgrowth, ctop, clim⟩
have L : ∀ᶠ n : ℕ in at_top, (c n : ℝ) * l - u (c n) ≤ ε * c n :=
by
rw [← tendsto_sub_nhds_zero_iff, ← Asymptotics.isOCat_one_iff ℝ, Asymptotics.isOCat_iff] at
clim
filter_upwards [clim εpos, Ctop (Ioi_mem_at_top 0)]with n hn cnpos'
have cnpos : 0 < c n := cnpos'
calc
(c n : ℝ) * l - u (c n) = -(u (c n) / c n - l) * c n := by
simp only [cnpos.ne', Ne.def, Nat.cast_eq_zero, not_false_iff, neg_sub, field_simps]
_ ≤ ε * c n := by
refine' mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _)
simp only [mul_one, Real.norm_eq_abs, abs_one] at hn
exact le_trans (neg_le_abs_self _) hn
obtain ⟨a, ha⟩ :
∃ a : ℕ,
∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ (c b : ℝ) * l - u (c b) ≤ ε * c b :=
eventually_at_top.1 (cgrowth.and L)
let M := ((Finset.range (a + 1)).image fun i => c i).max' (by simp)
filter_upwards [Ici_mem_at_top M]with n hn
have exN : ∃ N, n < c N :=
by
rcases(tendsto_at_top.1 Ctop (n + 1)).exists with ⟨N, hN⟩
exact ⟨N, by linarith only [hN]⟩
let N := Nat.find exN
have ncN : n < c N := Nat.find_spec exN
have aN : a + 1 ≤ N := by
by_contra' h
have cNM : c N ≤ M := by
apply le_max'
apply mem_image_of_mem
exact mem_range.2 h
exact lt_irrefl _ ((cNM.trans hn).trans_lt ncN)
have Npos : 0 < N := lt_of_lt_of_le Nat.succ_pos' aN
have aN' : a ≤ N - 1 := by linarith only [aN, Npos]
have cNn : c (N - 1) ≤ n :=
by
have : N - 1 < N := Nat.pred_lt Npos.ne'
simpa only [not_lt] using Nat.find_min exN this
calc
(n : ℝ) * l - u n ≤ c N * l - u (c (N - 1)) :=
by
refine' add_le_add (mul_le_mul_of_nonneg_right (Nat.cast_le.2 ncN.le) lnonneg) _
exact neg_le_neg (hmono cNn)
_ ≤ (1 + ε) * c (N - 1) * l - u (c (N - 1)) :=
by
refine' add_le_add (mul_le_mul_of_nonneg_right _ lnonneg) le_rfl
have B : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
have := (ha _ aN').1
rwa [B] at this
_ = c (N - 1) * l - u (c (N - 1)) + ε * c (N - 1) * l := by ring
_ ≤ ε * c (N - 1) + ε * c (N - 1) * l := (add_le_add (ha _ aN').2 le_rfl)
_ = ε * (1 + l) * c (N - 1) := by ring
_ ≤ ε * (1 + l) * n :=
by
refine' mul_le_mul_of_nonneg_left (Nat.cast_le.2 cNn) _
exact mul_nonneg εpos.le (add_nonneg zero_le_one lnonneg)
refine' tendsto_order.2 ⟨fun d hd => _, fun d hd => _⟩
· obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, d + ε * (1 + l) < l ∧ 0 < ε :=
by
have L : tendsto (fun ε => d + ε * (1 + l)) (𝓝[>] 0) (𝓝 (d + 0 * (1 + l))) :=
by
apply tendsto.mono_left _ nhdsWithin_le_nhds
exact tendsto_const_nhds.add (tendsto_id.mul tendsto_const_nhds)
simp only [MulZeroClass.zero_mul, add_zero] at L
exact (((tendsto_order.1 L).2 l hd).And self_mem_nhdsWithin).exists
filter_upwards [B ε εpos, Ioi_mem_at_top 0]with n hn npos
simp_rw [div_eq_inv_mul]
calc
d < n⁻¹ * n * (l - ε * (1 + l)) :=
by
rw [inv_mul_cancel, one_mul]
· linarith only [hε]
· exact Nat.cast_ne_zero.2 (ne_of_gt npos)
_ = n⁻¹ * (n * l - ε * (1 + l) * n) := by ring
_ ≤ n⁻¹ * u n :=
by
refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
linarith only [hn]
· obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, l + ε * (1 + ε + l) < d ∧ 0 < ε :=
by
have L : tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l))) :=
by
apply tendsto.mono_left _ nhdsWithin_le_nhds
exact
tendsto_const_nhds.add
(tendsto_id.mul ((tendsto_const_nhds.add tendsto_id).add tendsto_const_nhds))
simp only [MulZeroClass.zero_mul, add_zero] at L
exact (((tendsto_order.1 L).2 d hd).And self_mem_nhdsWithin).exists
filter_upwards [A ε εpos, Ioi_mem_at_top 0]with n hn npos
simp_rw [div_eq_inv_mul]
calc
(n : ℝ)⁻¹ * u n ≤ (n : ℝ)⁻¹ * (n * l + ε * (1 + ε + l) * n) :=
by
refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (Nat.cast_nonneg _))
linarith only [hn]
_ = (n : ℝ)⁻¹ * n * (l + ε * (1 + ε + l)) := by ring
_ < d := by
rwa [inv_mul_cancel, one_mul]
exact Nat.cast_ne_zero.2 (ne_of_gt npos)
#align tendsto_div_of_monotone_of_exists_subseq_tendsto_div tendsto_div_of_monotone_of_exists_subseq_tendsto_div
/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic tactic.field_simp.ne_zero -/
/-- If a monotone sequence `u` is such that `u ⌊c^n⌋₊ / ⌊c^n⌋₊` converges to a limit `l` for all
`c > 1`, then `u n / n` tends to `l`. It is even enough to have the assumption for a sequence of
`c`s converging to `1`. -/
theorem tendsto_div_of_monotone_of_tendsto_div_floor_pow (u : ℕ → ℝ) (l : ℝ) (hmono : Monotone u)
(c : ℕ → ℝ) (cone : ∀ k, 1 < c k) (clim : Tendsto c atTop (𝓝 1))
(hc : ∀ k, Tendsto (fun n : ℕ => u ⌊c k ^ n⌋₊ / ⌊c k ^ n⌋₊) atTop (𝓝 l)) :
Tendsto (fun n => u n / n) atTop (𝓝 l) :=
by
apply tendsto_div_of_monotone_of_exists_subseq_tendsto_div u l hmono
intro a ha
obtain ⟨k, hk⟩ : ∃ k, c k < a := ((tendsto_order.1 clim).2 a ha).exists
refine'
⟨fun n => ⌊c k ^ n⌋₊, _,
tendsto_nat_floor_at_top.comp (tendsto_pow_atTop_atTop_of_one_lt (cone k)), hc k⟩
have H : ∀ n : ℕ, (0 : ℝ) < ⌊c k ^ n⌋₊ := by
intro n
refine' zero_lt_one.trans_le _
simp only [Nat.one_le_cast, Nat.one_le_floor_iff, one_le_pow_of_one_le (cone k).le n]
have A :
tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / c k ^ (n + 1) * c k / (⌊c k ^ n⌋₊ / c k ^ n))
at_top (𝓝 (1 * c k / 1)) :=
by
refine' tendsto.div (tendsto.mul _ tendsto_const_nhds) _ one_ne_zero
· refine' tendsto_nat_floor_div_at_top.comp _
exact (tendsto_pow_atTop_atTop_of_one_lt (cone k)).comp (tendsto_add_at_top_nat 1)
· refine' tendsto_nat_floor_div_at_top.comp _
exact tendsto_pow_atTop_atTop_of_one_lt (cone k)
have B : tendsto (fun n : ℕ => (⌊c k ^ (n + 1)⌋₊ : ℝ) / ⌊c k ^ n⌋₊) at_top (𝓝 (c k)) :=
by
simp only [one_mul, div_one] at A
convert A
ext1 n
simp (disch :=
run_tac
tactic.field_simp.ne_zero) only [(zero_lt_one.trans (cone k)).ne',
Ne.def, not_false_iff, (H n).ne', field_simps]
ring
filter_upwards [(tendsto_order.1 B).2 a hk]with n hn
exact (div_le_iff (H n)).1 hn.le
#align tendsto_div_of_monotone_of_tendsto_div_floor_pow tendsto_div_of_monotone_of_tendsto_div_floor_pow
/-- The sum of `1/(c^i)^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
constant. -/
theorem sum_div_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
(∑ i in (range N).filterₓ fun i => j < c ^ i, 1 / (c ^ i) ^ 2) ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 :=
by
have cpos : 0 < c := zero_lt_one.trans hc
have A : 0 < c⁻¹ ^ 2 := sq_pos_of_pos (inv_pos.2 cpos)
have B : c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ ≤ c ^ 3 * (c - 1)⁻¹ :=
by
rw [← div_eq_mul_inv, ← div_eq_mul_inv, div_le_div_iff _ (sub_pos.2 hc)]
swap
· exact sub_pos.2 (pow_lt_one (inv_nonneg.2 cpos.le) (inv_lt_one hc) two_ne_zero)
have : c ^ 3 = c ^ 2 * c := by ring
simp only [mul_sub, this, mul_one, inv_pow, sub_le_sub_iff_left]
rw [mul_assoc, mul_comm c, ← mul_assoc, mul_inv_cancel (sq_pos_of_pos cpos).ne', one_mul]
simpa using pow_le_pow hc.le one_le_two
calc
(∑ i in (range N).filterₓ fun i => j < c ^ i, 1 / (c ^ i) ^ 2) ≤
∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, 1 / (c ^ i) ^ 2 :=
by
refine'
sum_le_sum_of_subset_of_nonneg _ fun i hi hident => div_nonneg zero_le_one (sq_nonneg _)
intro i hi
simp only [mem_filter, mem_range] at hi
simp only [hi.1, mem_Ico, and_true_iff]
apply Nat.floor_le_of_le
apply le_of_lt
rw [div_lt_iff (Real.log_pos hc), ← Real.log_pow]
exact Real.log_lt_log hj hi.2
_ = ∑ i in Ico ⌊Real.log j / Real.log c⌋₊ N, (c⁻¹ ^ 2) ^ i :=
by
congr 1 with i
simp [← pow_mul, mul_comm]
_ ≤ (c⁻¹ ^ 2) ^ ⌊Real.log j / Real.log c⌋₊ / (1 - c⁻¹ ^ 2) :=
by
apply geom_sum_Ico_le_of_lt_one (sq_nonneg _)
rw [sq_lt_one_iff (inv_nonneg.2 (zero_le_one.trans hc.le))]
exact inv_lt_one hc
_ ≤ (c⁻¹ ^ 2) ^ (Real.log j / Real.log c - 1) / (1 - c⁻¹ ^ 2) :=
by
apply div_le_div _ _ _ le_rfl
· apply Real.rpow_nonneg_of_nonneg (sq_nonneg _)
· rw [← Real.rpow_nat_cast]
apply Real.rpow_le_rpow_of_exponent_ge A
· exact pow_le_one _ (inv_nonneg.2 (zero_le_one.trans hc.le)) (inv_le_one hc.le)
· exact (Nat.sub_one_lt_floor _).le
· simpa only [inv_pow, sub_pos] using inv_lt_one (one_lt_pow hc two_ne_zero)
_ = c ^ 2 * (1 - c⁻¹ ^ 2)⁻¹ / j ^ 2 :=
by
have I : (c⁻¹ ^ 2) ^ (Real.log j / Real.log c) = 1 / j ^ 2 :=
by
apply Real.log_injOn_pos (Real.rpow_pos_of_pos A _)
· rw [one_div]
exact inv_pos.2 (sq_pos_of_pos hj)
rw [Real.log_rpow A]
simp only [one_div, Real.log_inv, Real.log_pow, Nat.cast_bit0, Nat.cast_one, mul_neg,
neg_inj]
field_simp [(Real.log_pos hc).ne']
ring
rw [Real.rpow_sub A, I]
have : c ^ 2 - 1 ≠ 0 := (sub_pos.2 (one_lt_pow hc two_ne_zero)).ne'
field_simp [hj.ne', (zero_lt_one.trans hc).ne']
ring
_ ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2 :=
by
apply div_le_div _ B (sq_pos_of_pos hj) le_rfl
exact mul_nonneg (pow_nonneg cpos.le _) (inv_nonneg.2 (sub_pos.2 hc).le)
#align sum_div_pow_sq_le_div_sq sum_div_pow_sq_le_div_sq
theorem mul_pow_le_nat_floor_pow {c : ℝ} (hc : 1 < c) (i : ℕ) : (1 - c⁻¹) * c ^ i ≤ ⌊c ^ i⌋₊ :=
by
have cpos : 0 < c := zero_lt_one.trans hc
rcases Nat.eq_zero_or_pos i with (rfl | hi)
· simp only [pow_zero, Nat.floor_one, Nat.cast_one, mul_one, sub_le_self_iff, inv_nonneg, cpos.le]
have hident : 1 ≤ i := hi
calc
(1 - c⁻¹) * c ^ i = c ^ i - c ^ i * c⁻¹ := by ring
_ ≤ c ^ i - 1 := by
simpa only [← div_eq_mul_inv, sub_le_sub_iff_left, one_le_div cpos, pow_one] using
pow_le_pow hc.le hident
_ ≤ ⌊c ^ i⌋₊ := (Nat.sub_one_lt_floor _).le
#align mul_pow_le_nat_floor_pow mul_pow_le_nat_floor_pow
/-- The sum of `1/⌊c^i⌋₊^2` above a threshold `j` is comparable to `1/j^2`, up to a multiplicative
constant. -/
theorem sum_div_nat_floor_pow_sq_le_div_sq (N : ℕ) {j : ℝ} (hj : 0 < j) {c : ℝ} (hc : 1 < c) :
(∑ i in (range N).filterₓ fun i => j < ⌊c ^ i⌋₊, (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2) ≤
c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2 :=
by
have cpos : 0 < c := zero_lt_one.trans hc
have A : 0 < 1 - c⁻¹ := sub_pos.2 (inv_lt_one hc)
calc
(∑ i in (range N).filterₓ fun i => j < ⌊c ^ i⌋₊, (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2) ≤
∑ i in (range N).filterₓ fun i => j < c ^ i, (1 : ℝ) / ⌊c ^ i⌋₊ ^ 2 :=
by
apply sum_le_sum_of_subset_of_nonneg
· intro i hi
simp only [mem_filter, mem_range] at hi
simpa only [hi.1, mem_filter, mem_range, true_and_iff] using
hi.2.trans_le (Nat.floor_le (pow_nonneg cpos.le _))
· intro i hi hident
exact div_nonneg zero_le_one (sq_nonneg _)
_ ≤ ∑ i in (range N).filterₓ fun i => j < c ^ i, (1 - c⁻¹)⁻¹ ^ 2 * (1 / (c ^ i) ^ 2) :=
by
apply sum_le_sum fun i hi => _
rw [mul_div_assoc', mul_one, div_le_div_iff]; rotate_left
· apply sq_pos_of_pos
refine' zero_lt_one.trans_le _
simp only [Nat.le_floor, one_le_pow_of_one_le, hc.le, Nat.one_le_cast, Nat.cast_one]
· exact sq_pos_of_pos (pow_pos cpos _)
rw [one_mul, ← mul_pow]
apply pow_le_pow_of_le_left (pow_nonneg cpos.le _)
rw [← div_eq_inv_mul, le_div_iff A, mul_comm]
exact mul_pow_le_nat_floor_pow hc i
_ ≤ (1 - c⁻¹)⁻¹ ^ 2 * (c ^ 3 * (c - 1)⁻¹) / j ^ 2 :=
by
rw [← mul_sum, ← mul_div_assoc']
refine' mul_le_mul_of_nonneg_left _ (sq_nonneg _)
exact sum_div_pow_sq_le_div_sq N hj hc
_ = c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2 := by
congr 1
field_simp [cpos.ne', (sub_pos.2 hc).ne']
ring
#align sum_div_nat_floor_pow_sq_le_div_sq sum_div_nat_floor_pow_sq_le_div_sq
|
[STATEMENT]
lemma divmod_abs_code [code]:
"divmod_abs (Pos k) (Pos l) = divmod k l"
"divmod_abs (Neg k) (Neg l) = divmod k l"
"divmod_abs (Neg k) (Pos l) = divmod k l"
"divmod_abs (Pos k) (Neg l) = divmod k l"
"divmod_abs j 0 = (0, \<bar>j\<bar>)"
"divmod_abs 0 j = (0, 0)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
1. (divmod_abs (Pos k) (Pos l) = divmod k l &&& divmod_abs (Neg k) (Neg l) = divmod k l &&& divmod_abs (Neg k) (Pos l) = divmod k l) &&& divmod_abs (Pos k) (Neg l) = divmod k l &&& divmod_abs j 0 = (0, \<bar>j\<bar>) &&& divmod_abs 0 j = (0, 0)
[PROOF STEP]
by (simp_all add: prod_eq_iff) |
(* -*- mode: coq; coq-prog-args: ("-emacs" "-top" "min_bug_univpoly") -*- *)
(* File reduced by coq-bug-finder from original input, then from 2673 lines to 1040 lines, then from 1056 lines to 1040 lines, then from 1055 lines to 696 lines, then from 704 lines to 696 lines *)
(* coqc version 8.6.1 (August 2017) compiled on Aug 8 2017 15:55:58 with OCaml 4.02.3
coqtop version blackbox:/home/janno/.opam/ra-gps/build/coq.8.6.dev,master (6dbffe1db990966321ce47fede1840252dc67688) *)
(* commenting this makes it work *)
Set Universe Polymorphism.
Unset Universe Minimization ToSet.
Module Export AdmitTactic.
Module Import LocalFalse.
Inductive False := .
End LocalFalse.
Axiom proof_admitted : False.
Tactic Notation "admit" := abstract case proof_admitted.
End AdmitTactic.
Require Coq.Strings.String.
Module Import LocalFalse.
Inductive False := .
End LocalFalse.
Inductive False := .
Set Implicit Arguments.
Inductive eq (A:Type) (x:A) : A -> Prop :=
eq_refl : x = x :>A
where "x = y :> A" := (@eq A x y) : type_scope.
Notation "x = y" := (x = y :>_) : type_scope.
Notation "x <> y :> T" := (~ x = y :>T) : type_scope.
Notation "x <> y" := (x <> y :>_) : type_scope.
Arguments eq_refl {A x} , [A] x.
Arguments eq_rect [A] x P _ y _ : rename.
Section Logic_lemmas.
Section equality.
Variables A B : Type.
Variable f : A -> B.
Variables x y z : A.
Theorem eq_sym : x = y -> y = x.
admit.
Defined.
Theorem eq_trans : x = y -> y = z -> x = z.
admit.
Defined.
Theorem f_equal : x = y -> f x = f y.
admit.
Defined.
Theorem not_eq_sym : x <> y -> y <> x.
admit.
Defined.
End equality.
Definition eq_ind_r :
forall (A:Type) (x:A) (P:A -> Prop), P x -> forall y:A, y = x -> P y.
admit.
Defined.
Definition eq_rec_r :
forall (A:Type) (x:A) (P:A -> Set), P x -> forall y:A, y = x -> P y.
admit.
Defined.
Definition eq_rect_r :
forall (A:Type) (x:A) (P:A -> Type), P x -> forall y:A, y = x -> P y.
admit.
Defined.
End Logic_lemmas.
Notation "'rew' H 'in' H'" := (eq_rect _ _ H' _ H)
(at level 10, H' at level 10,
format "'[' 'rew' H in '/' H' ']'").
Notation "'rew' [ P ] H 'in' H'" := (eq_rect _ P H' _ H)
(at level 10, H' at level 10,
format "'[' 'rew' [ P ] '/ ' H in '/' H' ']'").
Notation "'rew' <- H 'in' H'" := (eq_rect_r _ H' H)
(at level 10, H' at level 10,
format "'[' 'rew' <- H in '/' H' ']'").
Lemma rew_opp_r : forall A (P:A->Type) (x y:A) (H:x=y) (a:P y), rew H in rew <- H in a = a.
admit.
Defined.
Lemma rew_opp_l : forall A (P:A->Type) (x y:A) (H:x=y) (a:P x), rew <- H in rew H in a = a.
admit.
Defined.
Theorem f_equal2 :
forall (A1 A2 B:Type) (f:A1 -> A2 -> B) (x1 y1:A1)
(x2 y2:A2), x1 = y1 -> x2 = y2 -> f x1 x2 = f y1 y2.
admit.
Defined.
Theorem f_equal3 :
forall (A1 A2 A3 B:Type) (f:A1 -> A2 -> A3 -> B) (x1 y1:A1)
(x2 y2:A2) (x3 y3:A3),
x1 = y1 -> x2 = y2 -> x3 = y3 -> f x1 x2 x3 = f y1 y2 y3.
admit.
Defined.
Theorem f_equal4 :
forall (A1 A2 A3 A4 B:Type) (f:A1 -> A2 -> A3 -> A4 -> B)
(x1 y1:A1) (x2 y2:A2) (x3 y3:A3) (x4 y4:A4),
x1 = y1 -> x2 = y2 -> x3 = y3 -> x4 = y4 -> f x1 x2 x3 x4 = f y1 y2 y3 y4.
admit.
Defined.
Theorem f_equal5 :
forall (A1 A2 A3 A4 A5 B:Type) (f:A1 -> A2 -> A3 -> A4 -> A5 -> B)
(x1 y1:A1) (x2 y2:A2) (x3 y3:A3) (x4 y4:A4) (x5 y5:A5),
x1 = y1 ->
x2 = y2 ->
x3 = y3 -> x4 = y4 -> x5 = y5 -> f x1 x2 x3 x4 x5 = f y1 y2 y3 y4 y5.
admit.
Defined.
Theorem f_equal_compose : forall A B C (a b:A) (f:A->B) (g:B->C) (e:a=b),
f_equal g (f_equal f e) = f_equal (fun a => g (f a)) e.
admit.
Defined.
Theorem eq_trans_refl_l : forall A (x y:A) (e:x=y), eq_trans eq_refl e = e.
admit.
Defined.
Theorem eq_trans_refl_r : forall A (x y:A) (e:x=y), eq_trans e eq_refl = e.
admit.
Defined.
Theorem eq_sym_involutive : forall A (x y:A) (e:x=y), eq_sym (eq_sym e) = e.
admit.
Defined.
Theorem eq_trans_sym_inv_l : forall A (x y:A) (e:x=y), eq_trans (eq_sym e) e = eq_refl.
admit.
Defined.
Theorem eq_trans_sym_inv_r : forall A (x y:A) (e:x=y), eq_trans e (eq_sym e) = eq_refl.
admit.
Defined.
Theorem eq_trans_assoc : forall A (x y z t:A) (e:x=y) (e':y=z) (e'':z=t),
eq_trans e (eq_trans e' e'') = eq_trans (eq_trans e e') e''.
admit.
Defined.
Theorem eq_id_comm_l : forall A (f:A->A) (Hf:forall a, a = f a), forall a, f_equal f (Hf a) = Hf (f a).
admit.
Defined.
Theorem eq_id_comm_r : forall A (f:A->A) (Hf:forall a, f a = a), forall a, f_equal f (Hf a) = Hf (f a).
admit.
Defined.
Lemma eq_refl_map_distr : forall A B x (f:A->B), f_equal f (eq_refl x) = eq_refl (f x).
admit.
Defined.
Lemma eq_trans_map_distr : forall A B x y z (f:A->B) (e:x=y) (e':y=z), f_equal f (eq_trans e e') = eq_trans (f_equal f e) (f_equal f e').
admit.
Defined.
Lemma eq_sym_map_distr : forall A B (x y:A) (f:A->B) (e:x=y), eq_sym (f_equal f e) = f_equal f (eq_sym e).
admit.
Defined.
Lemma eq_trans_sym_distr : forall A (x y z:A) (e:x=y) (e':y=z), eq_sym (eq_trans e e') = eq_trans (eq_sym e') (eq_sym e).
admit.
Defined.
Lemma eq_trans_rew_distr : forall A (P:A -> Type) (x y z:A) (e:x=y) (e':y=z) (k:P x),
rew (eq_trans e e') in k = rew e' in rew e in k.
admit.
Defined.
Lemma rew_const : forall A P (x y:A) (e:x=y) (k:P),
rew [fun _ => P] e in k = k.
admit.
Defined.
Inductive option (A:Type) : Type :=
| Some : A -> option A
| None : option A.
Arguments None {A}.
Definition option_map (A B:Type) (f:A->B) (o : option A) : option B :=
match o with
| Some a => @Some B (f a)
| None => @None B
end.
Inductive sum (A B:Type) : Type :=
| inl : A -> sum A B
| inr : B -> sum A B.
Inductive prod (A B:Type) : Type :=
pair : A -> B -> prod A B.
Notation "x * y" := (prod x y) : type_scope.
Notation "( x , y , .. , z )" := (pair .. (pair x y) .. z) : core_scope.
Inductive list (A : Type) : Type :=
| nil : list A
| cons : A -> list A -> list A.
Arguments nil {A}.
Infix "::" := cons (at level 60, right associativity) : list_scope.
Local Open Scope list_scope.
Definition length (A : Type) : list A -> nat :=
fix length l :=
match l with
| nil => O
| _ :: l' => S (length l')
end.
Definition app (A : Type) : list A -> list A -> list A :=
fix app l m :=
match l with
| nil => m
| a :: l1 => a :: app l1 m
end.
Infix "++" := app (right associativity, at level 60) : list_scope.
Notation "[m: ]" := nil (format "[m: ]") : list_scope.
Notation "[m: x ]" := (cons x nil) : list_scope.
Import Strings.String.
Import NArith.BinNat.
Unset Implicit Arguments.
Inductive Exception : Type := exception : Exception.
Definition NotUnifiable {A} (x y : A) : Exception.
admit.
Defined.
Definition Failure (s : string) : Exception.
admit.
Defined.
Definition NotAGoal : Exception.
admit.
Defined.
Definition DoesNotMatch : Exception.
admit.
Defined.
Definition NoPatternMatches : Exception.
admit.
Defined.
Definition EmptyList : Exception.
admit.
Defined.
Definition NotCumul {A B} (x: A) (y: B) : Exception.
admit.
Defined.
Polymorphic Record dyn := Dyn { type : Type; elem :> type }.
Inductive redlist A := rlnil | rlcons : A -> redlist A -> redlist A.
Arguments rlnil {_}.
Arguments rlcons {_} _ _.
Notation "[rl: x ; .. ; y ]" := (rlcons x (.. (rlcons y rlnil) ..)).
Inductive RedFlags :=
| RedBeta | RedDelta | RedMatch | RedFix | RedZeta
| RedDeltaC | RedDeltaX
| RedDeltaOnly : redlist dyn -> RedFlags
| RedDeltaBut : redlist dyn -> RedFlags.
Inductive Reduction :=
| RedNone
| RedSimpl
| RedOneStep
| RedWhd : redlist RedFlags -> Reduction
| RedStrong : redlist RedFlags -> Reduction
| RedVmCompute.
Inductive Unification : Type :=
| UniCoq : Unification
| UniMatch : Unification
| UniMatchNoRed : Unification
| UniEvarconv : Unification.
Inductive Hyp : Type :=
| ahyp : forall {A}, A -> option A -> Hyp.
Record Case :=
mkCase {
case_ind : Type;
case_val : case_ind;
case_return : dyn;
case_branches : list dyn
}.
Definition reduce (r : Reduction) {A} (x : A) := x.
Notation rone_step := (reduce RedOneStep).
Inductive goal :=
| Goal : forall {A}, A -> goal
| AHyp : forall {A}, option A -> (A -> goal) -> goal
| HypRem : forall {A}, A -> goal -> goal.
Inductive pattern (M : Type -> Type) (A : Type) (B : A -> Type) (y : A) : Prop :=
| pbase : forall x : A, (y = x -> M (B x)) -> Unification -> pattern M A B y
| ptele : forall {C}, (forall x : C, pattern M A B y) -> pattern M A B y.
Arguments pbase {M A B y} _ _ _.
Arguments ptele {M A B y C} _.
Declare Scope pattern_scope.
Notation "[? x .. y ] ps" := (ptele (fun x => .. (ptele (fun y => ps)).. ))
(at level 202, x binder, y binder, ps at next level) : pattern_scope.
Notation "p => [ H ] b" := (pbase p%core (fun H => b%core) UniMatch)
(no associativity, at level 201, H at next level) : pattern_scope.
Notation "'_' => b " := (ptele (fun x=> pbase x (fun _ => b%core) UniMatch))
(at level 201, b at next level) : pattern_scope.
Notation "p '=n>' b" := (pbase p%core (fun _ => b%core) UniMatchNoRed)
(no associativity, at level 201) : pattern_scope.
Notation "p '=n>' [ H ] b" := (pbase p%core (fun H => b%core) UniMatchNoRed)
(no associativity, at level 201, H at next level) : pattern_scope.
Notation "p '=u>' b" := (pbase p%core (fun _ => b%core) UniCoq)
(no associativity, at level 201) : pattern_scope.
Notation "p '=u>' [ H ] b" := (pbase p%core (fun H => b%core) UniCoq)
(no associativity, at level 201, H at next level) : pattern_scope.
Delimit Scope pattern_scope with pattern.
Declare Scope with_pattern_scope.
Notation "'with' | p1 | .. | pn 'end'" :=
((@cons (pattern _ _ _ _) p1%pattern (.. (@cons (pattern _ _ _ _) pn%pattern nil) ..)))
(at level 91, p1 at level 210, pn at level 210) : with_pattern_scope.
Notation "'with' p1 | .. | pn 'end'" :=
((@cons (pattern _ _ _ _) p1%pattern (.. (@cons (pattern _ _ _ _) pn%pattern nil) ..)))
(at level 91, p1 at level 210, pn at level 210) : with_pattern_scope.
Delimit Scope with_pattern_scope with with_pattern.
Inductive t@{U1 U2 E1 L1 H1 O1} : Type@{U1} -> Prop :=
| ret : forall {A : Type@{U1}}, A -> t A
| bind : forall {A : Type@{U1}} {B : Type@{U1}},
t A -> (A -> t B) -> t B
| mtry' : forall {A : Type@{U1}}, t A -> (Exception@{E1} -> t A) -> t A
| raise : forall {A : Type@{U1}}, Exception@{E1} -> t A
| nu : forall {A : Type@{U1}} {B : Type@{U1}}, string -> option A -> (A -> t B) -> t B
| gen_evar : forall (A : Type@{U1}), option (list@{L1} Hyp@{H1}) -> t A
| unify {A : Type@{U2}} (x y : A) : Unification -> t (option@{O1} (x = y))
| unify_univ (A B : Type@{U1}) : Unification -> t (option (A -> B))
.
(* Inductive t : Type -> Prop := *)
(* | ret : forall {A : Type}, A -> t A *)
(* | bind : forall {A : Type} {B : Type}, *)
(* t A -> (A -> t B) -> t B *)
(* | mtry' : forall {A : Type}, t A -> (Exception -> t A) -> t A *)
(* | raise : forall {A : Type}, Exception -> t A *)
(* | nu : forall {A : Type} {B : Type}, string -> option A -> (A -> t B) -> t B *)
(* | gen_evar : forall (A : Type), option (list Hyp) -> t A *)
(* | unify {A : Type} (x y : A) : Unification -> t (option (x = y)) *)
(* | unify_univ (A B : Type) : Unification -> t (option (A -> B)) *)
(* . *)
Definition evar (A : Type) : t A := gen_evar A None.
Definition failwith {A} (s : string) : t A := raise (Failure s).
(* Definition print_term {A} (x : A) : t unit := *)
(* bind (pretty_print x) (fun s=> print s). *)
Module Export monad_notations.
Declare Scope M_scope.
Delimit Scope M_scope with MC.
Open Scope M_scope.
Notation "r '<-' t1 ';' t2" := (@bind _ _ t1 (fun r=> t2%MC))
(at level 81, right associativity, format "'[' r '<-' '[' t1 ; ']' ']' '/' t2 ") : M_scope.
Notation "t1 ';;' t2" := (bind t1 (fun _ => t2%MC))
(at level 81, right associativity, format "'[' '[' t1 ;; ']' ']' '/' t2 ") : M_scope.
Notation "t >>= f" := (bind t f) (at level 70) : M_scope.
Notation "'mif' b 'then' t 'else' u" :=
(cond <- b; if cond then t else u) (at level 200) : M_scope.
End monad_notations.
Fixpoint open_pattern {A P y} (p : pattern t A P y) : t (P y) :=
match p with
| pbase x f u =>
oeq <- unify x y u;
match oeq return t (P y) with
| Some eq =>
let h := reduce (RedStrong [rl:RedBeta;RedDelta;RedMatch]) (eq_sym eq) in
let 'eq_refl := eq in
let b := reduce (RedStrong [rl:RedBeta]) (f h) in b
| None => raise DoesNotMatch
end
| @ptele _ _ _ _ C f => e <- evar C; open_pattern (f e)
end.
Fixpoint mmatch' {A P} (y : A) (ps : list (pattern t A P y)) : t (P y) :=
match ps with
| [m:] => raise NoPatternMatches
| p :: ps' =>
mtry' (open_pattern p) (fun e =>
mif unify e DoesNotMatch UniMatchNoRed then mmatch' y ps' else raise e)
end.
Module Export notations.
Export monad_notations.
(* Notation "'mfix1' f ( x : A ) : 'M' T := b" := *)
(* (fix1 (fun x=>T%type) (fun f (x : A)=>b%MC)) *)
(* (at level 85, f at level 0, x at next level, format *)
(* "'[v ' 'mfix1' f '(' x ':' A ')' ':' 'M' T ':=' '/ ' b ']'") : M_scope. *)
Notation "'mmatch' x ls" :=
(@mmatch' _ (fun _ => _) x ls%with_pattern)
(at level 90, ls at level 91) : M_scope.
End notations.
Definition unify_cumul {A B} (x: A) (y: B) (u : Unification) : t bool :=
of <- unify_univ A B u;
match of with
| Some f =>
let fx := reduce RedOneStep (f x) in
oeq <- unify fx y u;
match oeq with Some _ => ret true | None => ret false end
| None => ret false
end.
(* UNCOMMENT THE NEXT TWO COMMENTS TO MAKE IT COMPILE *)
(* Unset Universe Polymorphism. *)
Definition cumul_or_fail
{A B} (x: A) (y: B) : t unit :=
b <- unify_cumul x y UniCoq;
if b then ret tt else raise (NotCumul x y).
(* Set Universe Polymorphism. *)
Notation M := t.
Import notations.
Definition NotAProduct : Exception.
admit.
Defined.
Definition gtactic (A : Type) := goal -> M (list (A * goal)).
Notation tactic := (gtactic unit).
Definition exact {A} (x:A) : tactic := fun g =>
match g with
| Goal g => cumul_or_fail x g;; ret [m:]
| _ => raise NotAGoal
end.
Fail Definition intro_base {A B} (var : string) (t : A -> gtactic B) : gtactic B := fun g =>
mmatch g with
| [? P e] @Goal (forall x:A, P x) e =u>
@nu nat _ var None (fun x=>
exact nat g;;
raise exception)
| _ => raise NotAProduct
end.
|
example : (p ∨ p) → p := fun h => match
|
State Before: F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : CommApplicative F
α' β' : Type u
f : α' → F β'
α β : Type u_1
h : α → β
⊢ Functor.map h ∘ Coe.coe = Coe.coe ∘ Functor.map h State After: case h
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : CommApplicative F
α' β' : Type u
f : α' → F β'
α β : Type u_1
h : α → β
x✝ : List α
⊢ (Functor.map h ∘ Coe.coe) x✝ = (Coe.coe ∘ Functor.map h) x✝ Tactic: funext State Before: case h
F : Type u → Type u
inst✝¹ : Applicative F
inst✝ : CommApplicative F
α' β' : Type u
f : α' → F β'
α β : Type u_1
h : α → β
x✝ : List α
⊢ (Functor.map h ∘ Coe.coe) x✝ = (Coe.coe ∘ Functor.map h) x✝ State After: no goals Tactic: simp only [Function.comp_apply, Coe.coe, fmap_def, coe_map, List.map_eq_map] |
lemma prime_int_numeral_eq [simp]: "prime (numeral m :: int) \<longleftrightarrow> prime (numeral m :: nat)" |
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
! This file was ported from Lean 3 source module init.meta.environment
! leanprover-community/mathlib commit 1340477dccb7fbe0cf2146aa1f1995022c13cd30
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
prelude
import Leanbin.Init.Meta.Declaration
import Leanbin.Init.Meta.Exceptional
import Leanbin.Init.Data.Option.Basic
import Leanbin.Init.Meta.RbMap
/--
An __environment__ contains all of the declarations and notation that have been defined so far. -/
unsafe axiom environment : Type
#align environment environment
namespace Environment
/--
Consider a type `ψ` which is an inductive datatype using a single constructor `mk (a : α) (b : β) : ψ`.
Lean will automatically make two projection functions `a : ψ → α`, `b : ψ → β`.
Lean tags these declarations as __projections__.
This helps the simplifier / rewriter not have to expand projectors.
Eg `a (mk x y)` will automatically reduce to `x`.
If you `extend` a structure, all of the projections on the parent will also be created for the child.
Projections are also treated differently in the VM for efficiency.
Note that projections have nothing to do with the dot `mylist.map` syntax.
You can find out if a declaration is a projection using `environment.is_projection` which returns `projection_info`.
Data for a projection declaration:
- `cname` is the name of the constructor associated with the projection.
- `nparams` is the number of constructor parameters. Eg `and.intro` has two type parameters.
- `idx` is the parameter being projected by this projection.
- `is_class` is tt iff this is a typeclass projection.
### Examples:
- `and.right` is a projection with ``{cname := `and.intro, nparams := 2, idx := 1, is_class := ff}``
- `ordered_ring.neg` is a projection with ``{cname := `ordered_ring.mk, nparams := 1, idx := 5, is_class := tt}``.
-/
structure ProjectionInfo where
cname : Name
nparams : Nat
idx : Nat
isClass : Bool
#align environment.projection_info Environment.ProjectionInfo
/-- A marking on the binders of structures and inductives indicating
how this constructor should mark its parameters.
inductive foo
| one {} : foo -> foo -- relaxed_implicit
| two ( ) : foo -> foo -- explicit
| two [] : foo -> foo -- implicit
| three : foo -> foo -- relaxed implicit (default)
-/
inductive ImplicitInferKind
| implicit
| relaxed_implicit
| none
#align environment.implicit_infer_kind Environment.ImplicitInferKind
instance ImplicitInferKind.inhabited : Inhabited ImplicitInferKind :=
⟨ImplicitInferKind.implicit⟩
#align environment.implicit_infer_kind.inhabited Environment.ImplicitInferKind.inhabited
/-- One introduction rule in an inductive declaration -/
unsafe structure intro_rule where
constr : Name
type : expr
infer : ImplicitInferKind := ImplicitInferKind.implicit
#align environment.intro_rule environment.intro_rule
/-- Create a standard environment using the given trust level -/
unsafe axiom mk_std : Nat → environment
#align environment.mk_std environment.mk_std
/-- Return the trust level of the given environment -/
unsafe axiom trust_lvl : environment → Nat
#align environment.trust_lvl environment.trust_lvl
/-- Add a new declaration to the environment -/
unsafe axiom add : environment → declaration → exceptional environment
#align environment.add environment.add
/-- make declaration `n` protected -/
unsafe axiom mk_protected : environment → Name → environment
#align environment.mk_protected environment.mk_protected
/-- add declaration `d` and make it protected -/
unsafe def add_protected (env : environment) (d : declaration) : exceptional environment := do
let env ← env.add d
pure <| env d
#align environment.add_protected environment.add_protected
/-- check if `n` is the name of a protected declaration -/
unsafe axiom is_protected : environment → Name → Bool
#align environment.is_protected environment.is_protected
/-- Retrieve a declaration from the environment -/
unsafe axiom get : environment → Name → exceptional declaration
#align environment.get environment.get
unsafe def contains (env : environment) (d : Name) : Bool :=
match env.get d with
| exceptional.success _ => true
| exceptional.exception _ => false
#align environment.contains environment.contains
unsafe axiom add_defn_eqns (env : environment) (opt : options) (lp_params : List Name)
(params : List expr) (sig : expr) (eqns : List (List (expr false) × expr)) (is_meta : Bool) :
exceptional environment
#align environment.add_defn_eqns environment.add_defn_eqns
/-- Register the given name as a namespace, making it available to the `open` command -/
unsafe axiom add_namespace : environment → Name → environment
#align environment.add_namespace environment.add_namespace
/-- Mark a namespace as open -/
unsafe axiom mark_namespace_as_open : environment → Name → environment
#align environment.mark_namespace_as_open environment.mark_namespace_as_open
/-- Modify the environment as if `open %%name` had been parsed -/
unsafe axiom execute_open : environment → Name → environment
#align environment.execute_open environment.execute_open
/-- Retrieve all registered namespaces -/
unsafe axiom get_namespaces : environment → List Name
#align environment.get_namespaces environment.get_namespaces
/-- Return tt iff the given name is a namespace -/
unsafe axiom is_namespace : environment → Name → Bool
#align environment.is_namespace environment.is_namespace
/-- Add a new inductive datatype to the environment
name, universe parameters, number of parameters, type, constructors (name and type), is_meta -/
unsafe axiom add_inductive (env : environment) (n : Name) (levels : List Name) (num_params : Nat)
(type : expr) (intros : List (Name × expr)) (is_meta : Bool) : exceptional environment
#align environment.add_inductive environment.add_inductive
/-- Add a new general inductive declaration to the environment.
This has the same effect as a `inductive` in the file, including generating
all the auxiliary definitions, as well as triggering mutual/nested inductive
compilation, by contrast to `environment.add_inductive` which only adds the
core axioms supported by the kernel.
The `inds` argument should be a list of inductives in the mutual family.
The first argument is a pair of the name of the type being constructed
and the type of this inductive family (not including the params).
The second argument is a list of intro rules, specified by a name, an
`implicit_infer_kind` giving the implicitness of the params for this constructor,
and an expression with the type of the constructor (not including the params).
-/
unsafe axiom add_ginductive (env : environment) (opt : options) (levels : List Name)
(params : List expr) (inds : List ((Name × expr) × List intro_rule)) (is_meta : Bool) :
exceptional environment
#align environment.add_ginductive environment.add_ginductive
/-- Return tt iff the given name is an inductive datatype -/
unsafe axiom is_inductive : environment → Name → Bool
#align environment.is_inductive environment.is_inductive
/-- Return tt iff the given name is a constructor -/
unsafe axiom is_constructor : environment → Name → Bool
#align environment.is_constructor environment.is_constructor
/-- Return tt iff the given name is a recursor -/
unsafe axiom is_recursor : environment → Name → Bool
#align environment.is_recursor environment.is_recursor
/-- Return tt iff the given name is a recursive inductive datatype -/
unsafe axiom is_recursive : environment → Name → Bool
#align environment.is_recursive environment.is_recursive
/-- Return the name of the inductive datatype of the given constructor. -/
unsafe axiom inductive_type_of : environment → Name → Option Name
#align environment.inductive_type_of environment.inductive_type_of
/-- Return the constructors of the inductive datatype with the given name -/
unsafe axiom constructors_of : environment → Name → List Name
#align environment.constructors_of environment.constructors_of
/-- Return the recursor of the given inductive datatype -/
unsafe axiom recursor_of : environment → Name → Option Name
#align environment.recursor_of environment.recursor_of
/-- Return the number of parameters of the inductive datatype -/
unsafe axiom inductive_num_params : environment → Name → Nat
#align environment.inductive_num_params environment.inductive_num_params
/-- Return the number of indices of the inductive datatype -/
unsafe axiom inductive_num_indices : environment → Name → Nat
#align environment.inductive_num_indices environment.inductive_num_indices
/-- Return tt iff the inductive datatype recursor supports dependent elimination -/
unsafe axiom inductive_dep_elim : environment → Name → Bool
#align environment.inductive_dep_elim environment.inductive_dep_elim
/-- Functionally equivalent to `is_inductive`.
Technically, this works by checking if the name is in the ginductive environment
extension which is outside the kernel, whereas `is_inductive` works by looking at the kernel extension.
But there are no `is_inductive`s which are not `is_ginductive`.
-/
unsafe axiom is_ginductive : environment → Name → Bool
#align environment.is_ginductive environment.is_ginductive
/-- See the docstring for `projection_info`. -/
unsafe axiom is_projection : environment → Name → Option ProjectionInfo
#align environment.is_projection environment.is_projection
/-- Fold over declarations in the environment. -/
unsafe axiom fold {α : Type} : environment → α → (declaration → α → α) → α
#align environment.fold environment.fold
/-- `relation_info env n` returns some value if n is marked as a relation in the given environment.
the tuple contains: total number of arguments of the relation, lhs position and rhs position. -/
unsafe axiom relation_info : environment → Name → Option (Nat × Nat × Nat)
#align environment.relation_info environment.relation_info
/-- `refl_for env R` returns the name of the reflexivity theorem for the relation R -/
unsafe axiom refl_for : environment → Name → Option Name
#align environment.refl_for environment.refl_for
/-- `symm_for env R` returns the name of the symmetry theorem for the relation R -/
unsafe axiom symm_for : environment → Name → Option Name
#align environment.symm_for environment.symm_for
/-- `trans_for env R` returns the name of the transitivity theorem for the relation R -/
unsafe axiom trans_for : environment → Name → Option Name
#align environment.trans_for environment.trans_for
/-- `decl_olean env d` returns the name of the .olean file where d was defined.
The result is none if d was not defined in an imported file. -/
unsafe axiom decl_olean : environment → Name → Option String
#align environment.decl_olean environment.decl_olean
/-- `decl_pos env d` returns the source location of d if available. -/
unsafe axiom decl_pos : environment → Name → Option Pos
#align environment.decl_pos environment.decl_pos
/-- `decl_pos env d` returns the name of a declaration that d inherits
noncomputability from, or `none` if it is computable.
Note that this also returns `none` on `axiom`s and `constant`s. These can be detected by using
`environment.get_decl` and `declaration.is_axiom` and `declaration.is_constant`. -/
unsafe axiom decl_noncomputable_reason : environment → Name → Option Name
#align environment.decl_noncomputable_reason environment.decl_noncomputable_reason
/-- Return the fields of the structure with the given name, or `none` if it is not a structure -/
unsafe axiom structure_fields : environment → Name → Option (List Name)
#align environment.structure_fields environment.structure_fields
/-- `get_class_attribute_symbols env attr_name` return symbols
occurring in instances of type classes tagged with the attribute `attr_name`.
Example: [algebra] -/
unsafe axiom get_class_attribute_symbols : environment → Name → name_set
#align environment.get_class_attribute_symbols environment.get_class_attribute_symbols
/--
The fingerprint of the environment is a hash formed from all of the declarations in the environment. -/
unsafe axiom fingerprint : environment → Nat
#align environment.fingerprint environment.fingerprint
/-- Gets the equation lemmas for the declaration `n`. -/
unsafe axiom get_eqn_lemmas_for (env : environment) (n : Name) : List Name
#align environment.get_eqn_lemmas_for environment.get_eqn_lemmas_for
/-- Gets the equation lemmas for the declaration `n`, including lemmas for match statements, etc. -/
unsafe axiom get_ext_eqn_lemmas_for (env : environment) (n : Name) : List Name
#align environment.get_ext_eqn_lemmas_for environment.get_ext_eqn_lemmas_for
/-- Adds the equation lemma `n`.
It is added for the declaration `t.pi_codomain.get_app_fn.const_name` where `t` is the type of the equation lemma.
-/
unsafe axiom add_eqn_lemma (env : environment) (n : Name) : environment
#align environment.add_eqn_lemma environment.add_eqn_lemma
open Expr
unsafe axiom unfold_untrusted_macros : environment → expr → expr
#align environment.unfold_untrusted_macros environment.unfold_untrusted_macros
unsafe axiom unfold_all_macros : environment → expr → expr
#align environment.unfold_all_macros environment.unfold_all_macros
unsafe def is_constructor_app (env : environment) (e : expr) : Bool :=
is_constant (get_app_fn e) && is_constructor env (const_name (get_app_fn e))
#align environment.is_constructor_app environment.is_constructor_app
unsafe def is_refl_app (env : environment) (e : expr) : Option (Name × expr × expr) :=
match refl_for env (const_name (get_app_fn e)) with
| some n => if get_app_num_args e ≥ 2 then some (n, app_arg (app_fn e), app_arg e) else none
| none => none
#align environment.is_refl_app environment.is_refl_app
/-- Return true if 'n' has been declared in the current file -/
unsafe def in_current_file (env : environment) (n : Name) : Bool :=
(env.decl_olean n).isNone && env.contains n &&
n ∉ [`` Quot, `` Quot.mk, `` Quot.lift, `` Quot.ind]
#align environment.in_current_file environment.in_current_file
unsafe def is_definition (env : environment) (n : Name) : Bool :=
match env.get n with
| exceptional.success (declaration.defn _ _ _ _ _ _) => true
| _ => false
#align environment.is_definition environment.is_definition
end Environment
unsafe instance : Repr environment :=
⟨fun e => "[environment]"⟩
unsafe instance : Inhabited environment :=
⟨environment.mk_std 0⟩
|
/- ANCHOR: doc -/
open Lean in
macro "begin " ts:tactic,*,? "end"%i : term => do
-- preserve position of the last token, which is used
-- as the error position in case of an unfinished proof
`(by { $[$ts:tactic]* }%$i)
theorem ex1 (x : Nat) : x + 0 = 0 + x :=
begin
rw [Nat.zero_add],
rw [Nat.add_zero],
end
/- ANCHOR_END: doc -/
theorem ex2 (x : Nat) : x + 0 = 0 + x :=
begin
rw [Nat.zero_add]
end -- error should be shown here
|
theory DemoITP
imports "$HIPSTER_HOME/IsaHipster"
begin
(* Normal list datatype *)
datatype 'a Lst =
Nil
| Cons "'a" "'a Lst" (infix ";" 65)
(* The append function. Syntactic sugar: we use +++ instead of Haskell's ++ *)
fun app :: "'a Lst \<Rightarrow> 'a Lst \<Rightarrow> 'a Lst" (infix "+++" 60)
where
"Nil +++ xs = xs"
| "(x;xs) +++ ys = x;(xs +++ ys)"
(* The reverse function *)
fun rev :: "'a Lst \<Rightarrow> 'a Lst"
where
"rev Nil = Nil"
| "rev (x;xs) = (rev xs) +++ (x;Nil)"
(* Datatype for binary trees from your exercises *)
datatype 'a Tree =
Empty
| Node "'a" "'a Tree" "'a Tree"
(* The swap function: swaps the left and right subtree. *)
fun swap :: "'a Tree => 'a Tree"
where
"swap Empty = Empty"
| "swap (Node data l r) = Node data (swap r) (swap l)"
(* The flatten function: turn a tree into a list *)
fun flatten :: "'a Tree \<Rightarrow> 'a Lst"
where
"flatten Empty = Nil"
| "flatten (Node data l r) = ((flatten l) +++ (data;Nil)) +++ (flatten r)"
(* Last week's exercise 10 *)
theorem exercise10: "flatten (swap p) = rev (flatten p)"
apply hipster_induct
(* A tail recursive reverse function *)
fun qrev :: "'a Lst \<Rightarrow> 'a Lst \<Rightarrow> 'a Lst"
where
"qrev Nil acc = acc"
| "qrev (x;xs) acc = qrev xs (x;acc)"
(* Hard exercise (optional) *)
theorem hardExercise: "rev xs = qrev xs Nil"
(* The spine function: turns a list into a tree. *)
(* fun spine :: "'a Lst \<Rightarrow> 'a Tree"
where
"spine Nil = Empty"
| "spine (x;xs) = Node x Empty (spine xs)"
hipster spine flatten
lemma lemma_af [thy_expl]: "flatten (spine y) = y"
apply (induct y)
by simp_all
*)
end |
/-
Copyright (c) 2021 OpenAI. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f
-/
import mathzoo.imports.miniF2F
open_locale nat rat real big_operators topological_space
theorem mathd_algebra_346
(f g : ℝ → ℝ)
(h₀ : ∀ x, f x = 2 * x - 3)
(h₁ : ∀ x, g x = x + 1) :
g (f 5 - 1) = 7 :=
begin
rw [h₀, h₁],
norm_num,
end |
axiom f {α : Type} : List α → List α
theorem t (a : α) (as : List α) : f (a :: as) = as :=
sorry
theorem tt {a : α} {as : List α} : f (a :: as) = as :=
by simp [t _ as]
|
(*
Copyright 2016 University of Luxembourg
This file is part of our formalization of Platzer's
"A Complete Uniform Substitution Calculus for Differential Dynamic Logic"
available here: http://arxiv.org/pdf/1601.06183.pdf.
We refer to this formalization as DdlCoq here.
DdlCoq is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
DdlCoq is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with DdlCoq. If not, see <http://www.gnu.org/licenses/>.
authors:
Vincent Rahli
Marcus Völp
Ivana Vukotic
*)
Require Export deriv_util.
Require Export multiplication.
(**
This file introduces lemma which checks if nth derivative of division of two functions exists.
*)
(* Regarding division:
https://www.physicsforums.com/threads/quotient-rule-for-higher-order-derivatives.289320/
*)
(** does derivative of two functions exists *)
Lemma ex_derive_n_div :
forall (n : nat) (f g : R -> R) (pt : R),
(forall pt, g pt <> 0)
-> (forall pt k, (k <= n)%nat -> ex_derive_n f k pt)
-> (forall pt k, (k <= n)%nat -> ex_derive_n g k pt)
-> ex_derive_n (fun x => (f x) / (g x)) n pt.
Proof.
induction n as [n IHn] using comp_ind; introv d loc1 loc2; destruct n;[simpl;auto|].
simpl.
apply ex_derive_n_S_if; auto.
{
apply ex_derive_div; auto.
{ pose proof (loc1 pt 1%nat) as q1; simpl in q1; apply q1; try omega. }
{ pose proof (loc2 pt 1%nat) as q2; simpl in q2; apply q2; try omega. }
}
{
apply (@ex_derive_n_ext (fun x => (Derive f x * g x - f x * Derive g x) / g x ^ 2) _).
{ introv.
rewrite (Derive_div f g); auto; simpl.
{ pose proof (loc1 t 1%nat) as q1; simpl in q1; apply q1; try omega. }
{ pose proof (loc2 t 1%nat) as q2; simpl in q2; apply q2; try omega. }
}
{
pose proof (IHn n) as h; clear IHn.
repeat (autodimp h hyp).
pose proof (h (fun x => (Derive f x * g x - f x * Derive g x))
(fun x => g x ^ 2)
pt) as q; clear h.
repeat (autodimp q hyp).
{
introv.
apply pow_nonzero; auto.
}
{
introv w.
apply ex_derive_n_minus.
{
exists R1pos; introv w1 w2.
apply ex_derive_n_mult_gen; introv w3.
{
pose proof (loc1 y0 (S k1)) as q.
autodimp q hyp; try omega.
apply ex_derive_n_S_implies; auto.
pose proof (loc1 y0 1%nat) as z; apply z; try omega.
}
{
pose proof (loc2 y0 k1) as q.
autodimp q hyp; try omega.
}
}
{
exists R1pos; introv w1 w2.
apply ex_derive_n_mult_gen; introv w3.
{
pose proof (loc1 y0 k1) as q.
autodimp q hyp; try omega.
}
{
pose proof (loc2 y0 (S k1)) as q.
autodimp q hyp; try omega; auto.
apply ex_derive_n_S_implies; auto.
pose proof (loc2 y0 1%nat) as z; apply z; try omega.
}
}
}
{
introv w.
apply ex_derive_n_pow.
introv w1.
apply loc2; try omega.
}
}
}
Qed.
|
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ sinh (dist z w / 2) = dist ↑z ↑w / (2 * sqrt (im z * im w))
[PROOFSTEP]
rw [dist_eq, mul_div_cancel_left (arsinh _) two_ne_zero, sinh_arsinh]
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ cosh (dist z w / 2) = dist (↑z) (↑(starRingEnd ℂ) ↑w) / (2 * sqrt (im z * im w))
[PROOFSTEP]
have H₁ : (2 ^ 2 : ℝ) = 4 := by norm_num1
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ 2 ^ 2 = 4
[PROOFSTEP]
norm_num1
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H₁ : 2 ^ 2 = 4
⊢ cosh (dist z w / 2) = dist (↑z) (↑(starRingEnd ℂ) ↑w) / (2 * sqrt (im z * im w))
[PROOFSTEP]
have H₂ : 0 < z.im * w.im := mul_pos z.im_pos w.im_pos
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H₁ : 2 ^ 2 = 4
H₂ : 0 < im z * im w
⊢ cosh (dist z w / 2) = dist (↑z) (↑(starRingEnd ℂ) ↑w) / (2 * sqrt (im z * im w))
[PROOFSTEP]
have H₃ : 0 < 2 * sqrt (z.im * w.im) := mul_pos two_pos (sqrt_pos.2 H₂)
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H₁ : 2 ^ 2 = 4
H₂ : 0 < im z * im w
H₃ : 0 < 2 * sqrt (im z * im w)
⊢ cosh (dist z w / 2) = dist (↑z) (↑(starRingEnd ℂ) ↑w) / (2 * sqrt (im z * im w))
[PROOFSTEP]
rw [← sq_eq_sq (cosh_pos _).le (div_nonneg dist_nonneg H₃.le), cosh_sq', sinh_half_dist, div_pow, div_pow,
one_add_div (pow_ne_zero 2 H₃.ne'), mul_pow, sq_sqrt H₂.le, H₁]
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H₁ : 2 ^ 2 = 4
H₂ : 0 < im z * im w
H₃ : 0 < 2 * sqrt (im z * im w)
⊢ (4 * (im z * im w) + dist ↑z ↑w ^ 2) / (4 * (im z * im w)) = dist (↑z) (↑(starRingEnd ℂ) ↑w) ^ 2 / (4 * (im z * im w))
[PROOFSTEP]
congr 1
[GOAL]
case e_a
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H₁ : 2 ^ 2 = 4
H₂ : 0 < im z * im w
H₃ : 0 < 2 * sqrt (im z * im w)
⊢ 4 * (im z * im w) + dist ↑z ↑w ^ 2 = dist (↑z) (↑(starRingEnd ℂ) ↑w) ^ 2
[PROOFSTEP]
simp only [Complex.dist_eq, Complex.sq_abs, Complex.normSq_sub, Complex.normSq_conj, Complex.conj_conj, Complex.mul_re,
Complex.conj_re, Complex.conj_im, coe_im]
[GOAL]
case e_a
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H₁ : 2 ^ 2 = 4
H₂ : 0 < im z * im w
H₃ : 0 < 2 * sqrt (im z * im w)
⊢ 4 * (im z * im w) + (↑Complex.normSq ↑z + ↑Complex.normSq ↑w - 2 * ((↑z).re * (↑w).re - im z * -im w)) =
↑Complex.normSq ↑z + ↑Complex.normSq ↑w - 2 * ((↑z).re * (↑w).re - im z * im w)
[PROOFSTEP]
ring
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ tanh (dist z w / 2) = dist ↑z ↑w / dist (↑z) (↑(starRingEnd ℂ) ↑w)
[PROOFSTEP]
rw [tanh_eq_sinh_div_cosh, sinh_half_dist, cosh_half_dist, div_div_div_comm, div_self, div_one]
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ 2 * sqrt (im z * im w) ≠ 0
[PROOFSTEP]
exact (mul_pos (zero_lt_two' ℝ) (sqrt_pos.2 <| mul_pos z.im_pos w.im_pos)).ne'
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ exp (dist z w / 2) = (dist ↑z ↑w + dist (↑z) (↑(starRingEnd ℂ) ↑w)) / (2 * sqrt (im z * im w))
[PROOFSTEP]
rw [← sinh_add_cosh, sinh_half_dist, cosh_half_dist, add_div]
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ cosh (dist z w) = 1 + dist ↑z ↑w ^ 2 / (2 * im z * im w)
[PROOFSTEP]
rw [dist_eq, cosh_two_mul, cosh_sq', add_assoc, ← two_mul, sinh_arsinh, div_pow, mul_pow,
sq_sqrt (mul_pos z.im_pos w.im_pos).le, sq (2 : ℝ), mul_assoc, ← mul_div_assoc, mul_assoc,
mul_div_mul_left _ _ (two_ne_zero' ℝ)]
[GOAL]
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ sinh ((dist a b + dist b c) / 2) =
(dist ↑a ↑b * dist (↑c) (↑(starRingEnd ℂ) ↑b) + dist ↑b ↑c * dist (↑a) (↑(starRingEnd ℂ) ↑b)) /
(2 * sqrt (im a * im c) * dist (↑b) (↑(starRingEnd ℂ) ↑b))
[PROOFSTEP]
simp only [add_div _ _ (2 : ℝ), sinh_add, sinh_half_dist, cosh_half_dist, div_mul_div_comm]
[GOAL]
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ dist ↑a ↑b * dist (↑b) (↑(starRingEnd ℂ) ↑c) / (2 * sqrt (im a * im b) * (2 * sqrt (im b * im c))) +
dist (↑a) (↑(starRingEnd ℂ) ↑b) * dist ↑b ↑c / (2 * sqrt (im a * im b) * (2 * sqrt (im b * im c))) =
(dist ↑a ↑b * dist (↑c) (↑(starRingEnd ℂ) ↑b) + dist ↑b ↑c * dist (↑a) (↑(starRingEnd ℂ) ↑b)) /
(2 * sqrt (im a * im c) * dist (↑b) (↑(starRingEnd ℂ) ↑b))
[PROOFSTEP]
rw [← add_div, Complex.dist_self_conj, coe_im, abs_of_pos b.im_pos, mul_comm (dist (b : ℂ) _), dist_comm (b : ℂ),
Complex.dist_conj_comm, mul_mul_mul_comm, mul_mul_mul_comm _ _ _ b.im]
[GOAL]
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ (dist ↑a ↑b * dist (↑c) (↑(starRingEnd ℂ) ↑b) + dist (↑a) (↑(starRingEnd ℂ) ↑b) * dist ↑b ↑c) /
(2 * 2 * (sqrt (im a * im b) * sqrt (im b * im c))) =
(dist ↑a ↑b * dist (↑c) (↑(starRingEnd ℂ) ↑b) + dist (↑a) (↑(starRingEnd ℂ) ↑b) * dist ↑b ↑c) /
(2 * 2 * (sqrt (im a * im c) * im b))
[PROOFSTEP]
congr 2
[GOAL]
case e_a.e_a
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ sqrt (im a * im b) * sqrt (im b * im c) = sqrt (im a * im c) * im b
[PROOFSTEP]
rw [sqrt_mul, sqrt_mul, sqrt_mul, mul_comm (sqrt a.im), mul_mul_mul_comm, mul_self_sqrt, mul_comm]
[GOAL]
case e_a.e_a
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 ≤ im b
[PROOFSTEP]
exact (im_pos _).le
[GOAL]
case e_a.e_a.hx
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 ≤ im a
[PROOFSTEP]
exact (im_pos _).le
[GOAL]
case e_a.e_a.hx
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 ≤ im b
[PROOFSTEP]
exact (im_pos _).le
[GOAL]
case e_a.e_a.hx
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 ≤ im a
[PROOFSTEP]
exact (im_pos _).le
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ dist z w = dist w z
[PROOFSTEP]
simp only [dist_eq, dist_comm (z : ℂ), mul_comm]
[GOAL]
z w : ℍ
r R : ℝ
⊢ dist z w ≤ r ↔ dist ↑z ↑w / (2 * sqrt (im z * im w)) ≤ sinh (r / 2)
[PROOFSTEP]
rw [← div_le_div_right (zero_lt_two' ℝ), ← sinh_le_sinh, sinh_half_dist]
[GOAL]
z w : ℍ
r R : ℝ
⊢ dist z w = r ↔ dist ↑z ↑w / (2 * sqrt (im z * im w)) = sinh (r / 2)
[PROOFSTEP]
rw [← div_left_inj' (two_ne_zero' ℝ), ← sinh_inj, sinh_half_dist]
[GOAL]
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ dist z w = r ↔ dist ↑z ↑w ^ 2 / (4 * im z * im w) = sinh (r / 2) ^ 2
[PROOFSTEP]
rw [dist_eq_iff_eq_sinh, ← sq_eq_sq, div_pow, mul_pow, sq_sqrt, mul_assoc]
[GOAL]
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ dist ↑z ↑w ^ 2 / (2 ^ 2 * (im z * im w)) = sinh (r / 2) ^ 2 ↔ dist ↑z ↑w ^ 2 / (4 * (im z * im w)) = sinh (r / 2) ^ 2
[PROOFSTEP]
norm_num
[GOAL]
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ 0 ≤ im z * im w
[PROOFSTEP]
exact (mul_pos z.im_pos w.im_pos).le
[GOAL]
case ha
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ 0 ≤ dist ↑z ↑w / (2 * sqrt (im z * im w))
[PROOFSTEP]
exact div_nonneg dist_nonneg (mul_nonneg zero_le_two <| sqrt_nonneg _)
[GOAL]
case hb
z w : ℍ
r R : ℝ
hr : 0 ≤ r
⊢ 0 ≤ sinh (r / 2)
[PROOFSTEP]
exact sinh_nonneg_iff.2 (div_nonneg hr zero_le_two)
[GOAL]
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ dist a c ≤ dist a b + dist b c
[PROOFSTEP]
rw [dist_le_iff_le_sinh, sinh_half_dist_add_dist, div_mul_eq_div_div _ _ (dist _ _), le_div_iff, div_mul_eq_mul_div]
[GOAL]
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ dist ↑a ↑c * dist (↑b) (↑(starRingEnd ℂ) ↑b) / (2 * sqrt (im a * im c)) ≤
(dist ↑a ↑b * dist (↑c) (↑(starRingEnd ℂ) ↑b) + dist ↑b ↑c * dist (↑a) (↑(starRingEnd ℂ) ↑b)) /
(2 * sqrt (im a * im c))
[PROOFSTEP]
exact
div_le_div_of_le (mul_nonneg zero_le_two (sqrt_nonneg _))
(EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist (a : ℂ) b c (conj (b : ℂ)))
[GOAL]
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ 0 < dist (↑b) (↑(starRingEnd ℂ) ↑b)
[PROOFSTEP]
rw [dist_comm, dist_pos, Ne.def, Complex.conj_eq_iff_im]
[GOAL]
z w : ℍ
r R : ℝ
a b c : ℍ
⊢ ¬(↑b).im = 0
[PROOFSTEP]
exact b.im_ne_zero
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ dist z w ≤ dist ↑z ↑w / sqrt (im z * im w)
[PROOFSTEP]
rw [dist_le_iff_le_sinh, ← div_mul_eq_div_div_swap, self_le_sinh_iff]
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ 0 ≤ dist ↑z ↑w / (2 * sqrt (im z * im w))
[PROOFSTEP]
exact div_nonneg dist_nonneg (mul_nonneg zero_le_two (sqrt_nonneg _))
[GOAL]
z✝ w : ℍ
r R : ℝ
z : ℍ
⊢ dist z z = 0
[PROOFSTEP]
rw [dist_eq, dist_self, zero_div, arsinh_zero, mul_zero]
[GOAL]
z w : ℍ
r R : ℝ
x✝¹ x✝ : ℍ
⊢ (fun x y =>
↑{
val :=
2 *
arsinh
(↑(let src := Complex.abs;
{
toAddGroupSeminorm :=
{ toFun := src.toFun, map_zero' := Complex.instNormedAddCommGroupComplex.proof_1,
add_le' :=
(_ :
∀ (x y : ℂ),
MulHom.toFun src.toMulHom (x + y) ≤
MulHom.toFun src.toMulHom x + MulHom.toFun src.toMulHom y),
neg' := Complex.instNormedAddCommGroupComplex.proof_2 },
eq_zero_of_map_eq_zero' :=
Complex.instNormedAddCommGroupComplex.proof_3 }).toAddGroupSeminorm
(↑x - ↑y) /
(2 * sqrt (im x * im y))),
property :=
(_ :
0 ≤
2 *
arsinh
(↑(let src := Complex.abs;
{
toAddGroupSeminorm :=
{ toFun := src.toFun, map_zero' := Complex.instNormedAddCommGroupComplex.proof_1,
add_le' :=
(_ :
∀ (x y : ℂ),
MulHom.toFun src.toMulHom (x + y) ≤
MulHom.toFun src.toMulHom x + MulHom.toFun src.toMulHom y),
neg' := Complex.instNormedAddCommGroupComplex.proof_2 },
eq_zero_of_map_eq_zero' :=
Complex.instNormedAddCommGroupComplex.proof_3 }).toAddGroupSeminorm
(↑x - ↑y) /
(2 * sqrt (im x * im y)))) })
x✝¹ x✝ =
ENNReal.ofReal (dist x✝¹ x✝)
[PROOFSTEP]
exact ENNReal.coe_nnreal_eq _
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
h : dist z w = 0
⊢ z = w
[PROOFSTEP]
simpa [dist_eq, Real.sqrt_eq_zero', (mul_pos z.im_pos w.im_pos).not_le, ext_iff] using h
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ Real.cosh (dist z w) = ((re z - re w) ^ 2 + im z ^ 2 + im w ^ 2) / (2 * im z * im w)
[PROOFSTEP]
have H : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H : 0 < 2 * im z * im w
⊢ Real.cosh (dist z w) = ((re z - re w) ^ 2 + im z ^ 2 + im w ^ 2) / (2 * im z * im w)
[PROOFSTEP]
field_simp [cosh_dist, Complex.dist_eq, Complex.sq_abs, normSq_apply, H, H.ne']
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
H : 0 < 2 * im z * im w
⊢ 2 * im z * im w + ((re z - re w) * (re z - re w) + (im z - im w) * (im z - im w)) =
(re z - re w) ^ 2 + im z ^ 2 + im w ^ 2
[PROOFSTEP]
ring
[GOAL]
z✝ w : ℍ
r R : ℝ
z : ℍ
⊢ im (center z 0) = im z
[PROOFSTEP]
rw [center_im, Real.cosh_zero, mul_one]
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
⊢ dist ↑z ↑(center w r) ^ 2 = 2 * im z * im w * (Real.cosh (dist z w) - Real.cosh r) + (im w * Real.sinh r) ^ 2
[PROOFSTEP]
have H : 2 * z.im * w.im ≠ 0 := by apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero]
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
⊢ 2 * im z * im w ≠ 0
[PROOFSTEP]
apply_rules [mul_ne_zero, two_ne_zero, im_ne_zero]
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
H : 2 * im z * im w ≠ 0
⊢ dist ↑z ↑(center w r) ^ 2 = 2 * im z * im w * (Real.cosh (dist z w) - Real.cosh r) + (im w * Real.sinh r) ^ 2
[PROOFSTEP]
simp only [Complex.dist_eq, Complex.sq_abs, normSq_apply, coe_re, coe_im, center_re, center_im, cosh_dist',
mul_div_cancel' _ H, sub_sq z.im, mul_pow, Real.cosh_sq, sub_re, sub_im, mul_sub, ← sq]
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
H : 2 * im z * im w ≠ 0
⊢ (re z - re w) ^ 2 + (im z ^ 2 - 2 * im z * (im w * Real.cosh r) + im w ^ 2 * (Real.sinh r ^ 2 + 1)) =
(re z - re w) ^ 2 + im z ^ 2 + im w ^ 2 - 2 * im z * im w * Real.cosh r + im w ^ 2 * Real.sinh r ^ 2
[PROOFSTEP]
ring
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
⊢ dist ↑z ↑(center w r) = sqrt (2 * im z * im w * (Real.cosh (dist z w) - Real.cosh r) + (im w * Real.sinh r) ^ 2)
[PROOFSTEP]
rw [← sqrt_sq dist_nonneg, dist_coe_center_sq]
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)
[PROOFSTEP]
letI := metricSpaceAux
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)
[PROOFSTEP]
cases' lt_or_le r 0 with hr₀ hr₀
[GOAL]
case inl
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : r < 0
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)
[PROOFSTEP]
trans Ordering.gt
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : r < 0
⊢ cmp (dist z w) r = Ordering.gt
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : r < 0
⊢ Ordering.gt = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)
[PROOFSTEP]
exacts [(hr₀.trans_le dist_nonneg).cmp_eq_gt,
((mul_neg_of_pos_of_neg w.im_pos (sinh_neg_iff.2 hr₀)).trans_le dist_nonneg).cmp_eq_gt.symm]
[GOAL]
case inr
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : 0 ≤ r
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)
[PROOFSTEP]
have hr₀' : 0 ≤ w.im * Real.sinh r := mul_nonneg w.im_pos.le (sinh_nonneg_iff.2 hr₀)
[GOAL]
case inr
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : 0 ≤ r
hr₀' : 0 ≤ im w * Real.sinh r
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)
[PROOFSTEP]
have hzw₀ : 0 < 2 * z.im * w.im := mul_pos (mul_pos two_pos z.im_pos) w.im_pos
[GOAL]
case inr
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : 0 ≤ r
hr₀' : 0 ≤ im w * Real.sinh r
hzw₀ : 0 < 2 * im z * im w
⊢ cmp (dist z w) r = cmp (dist ↑z ↑(center w r)) (im w * Real.sinh r)
[PROOFSTEP]
simp only [← cosh_strictMonoOn.cmp_map_eq dist_nonneg hr₀, ←
(@strictMonoOn_pow ℝ _ _ two_pos).cmp_map_eq dist_nonneg hr₀', dist_coe_center_sq]
[GOAL]
case inr
z✝ w✝ : ℍ
r✝ R : ℝ
z w : ℍ
r : ℝ
this : MetricSpace ℍ := metricSpaceAux
hr₀ : 0 ≤ r
hr₀' : 0 ≤ im w * Real.sinh r
hzw₀ : 0 < 2 * im z * im w
⊢ cmp (Real.cosh (dist z w)) (Real.cosh r) =
cmp (2 * im z * im w * (Real.cosh (dist z w) - Real.cosh r) + (im w * Real.sinh r) ^ 2) ((im w * Real.sinh r) ^ 2)
[PROOFSTEP]
rw [← cmp_mul_pos_left hzw₀, ← cmp_sub_zero, ← mul_sub, ← cmp_add_right, zero_add]
[GOAL]
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ dist ↑z ↑(center z r) = im z * (Real.cosh r - 1)
[PROOFSTEP]
rw [dist_of_re_eq (z.center_re r).symm, dist_comm, Real.dist_eq, mul_sub, mul_one]
[GOAL]
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ |(↑(center z r)).im - (↑z).im| = im z * Real.cosh r - im z
[PROOFSTEP]
exact abs_of_nonneg (sub_nonneg.2 <| le_mul_of_one_le_right z.im_pos.le (one_le_cosh _))
[GOAL]
z w : ℍ
r R : ℝ
h : re z = re w
⊢ dist z w = dist (log (im z)) (log (im w))
[PROOFSTEP]
have h₀ : 0 < z.im / w.im := div_pos z.im_pos w.im_pos
[GOAL]
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist z w = dist (log (im z)) (log (im w))
[PROOFSTEP]
rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero, sinh_log h₀,
dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div] <;>
[skip; exact h]
[GOAL]
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist z w = dist (log (im z)) (log (im w))
[PROOFSTEP]
rw [dist_eq_iff_dist_coe_center_eq, Real.dist_eq, ← abs_sinh, ← log_div z.im_ne_zero w.im_ne_zero, sinh_log h₀,
dist_of_re_eq, coe_im, coe_im, center_im, cosh_abs, cosh_log h₀, inv_div]
[GOAL]
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist (im z) (im w * ((im z / im w + im w / im z) / 2)) = im w * |(im z / im w - im w / im z) / 2|
[PROOFSTEP]
skip
[GOAL]
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ (↑z).re = (↑(center w |log (im z / im w)|)).re
[PROOFSTEP]
exact h
[GOAL]
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist (im z) (im w * ((im z / im w + im w / im z) / 2)) = im w * |(im z / im w - im w / im z) / 2|
[PROOFSTEP]
nth_rw 4 [← abs_of_pos w.im_pos]
[GOAL]
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ dist (im z) (im w * ((im z / im w + im w / im z) / 2)) = |im w| * |(im z / im w - im w / im z) / 2|
[PROOFSTEP]
simp only [← _root_.abs_mul, coe_im, Real.dist_eq]
[GOAL]
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ |im z - im w * ((im z / im w + im w / im z) / 2)| = |im w * ((im z / im w - im w / im z) / 2)|
[PROOFSTEP]
congr 1
[GOAL]
case e_a
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ im z - im w * ((im z / im w + im w / im z) / 2) = im w * ((im z / im w - im w / im z) / 2)
[PROOFSTEP]
field_simp [z.im_pos, w.im_pos, z.im_ne_zero, w.im_ne_zero]
[GOAL]
case e_a
z w : ℍ
r R : ℝ
h : re z = re w
h₀ : 0 < im z / im w
⊢ im z * (im w * im z * 2) - im w * (im z * im z + im w * im w) = im w * (im z * im z - im w * im w)
[PROOFSTEP]
ring
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ dist ↑(mk { re := 0, im := im z } (_ : 0 < im z)) ↑(mk { re := 0, im := im w } (_ : 0 < im w)) ≤ dist ↑z ↑w
[PROOFSTEP]
simpa [sqrt_sq_eq_abs] using Complex.abs_im_le_abs (z - w)
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ im z ≤ im w * Real.exp (dist z w)
[PROOFSTEP]
rw [← div_le_iff' w.im_pos, ← exp_log z.im_pos, ← exp_log w.im_pos, ← Real.exp_sub, exp_le_exp]
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ log (im z) - log (im w) ≤ dist z w
[PROOFSTEP]
exact (le_abs_self _).trans (dist_log_im_le z w)
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ dist ↑z ↑(center w (dist z w)) + dist ↑w ↑(center w (dist z w)) = im w * (Real.exp (dist z w) - 1)
[PROOFSTEP]
rw [dist_center_dist, dist_self_center, ← mul_add, ← add_sub_assoc, Real.sinh_add_cosh]
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ im w * (1 - Real.exp (-dist z w)) = dist ↑z ↑(center w (dist z w)) - dist ↑w ↑(center w (dist z w))
[PROOFSTEP]
rw [dist_center_dist, dist_self_center, ← Real.cosh_sub_sinh]
[GOAL]
z✝ w✝ : ℍ
r R : ℝ
z w : ℍ
⊢ im w * (1 - (Real.cosh (dist z w) - Real.sinh (dist z w))) =
im w * Real.sinh (dist z w) - im w * (Real.cosh (dist z w) - 1)
[PROOFSTEP]
ring
[GOAL]
z w : ℍ
r R : ℝ
⊢ instTopologicalSpaceUpperHalfPlane = UniformSpace.toTopologicalSpace
[PROOFSTEP]
refine' le_antisymm (continuous_id_iff_le.1 _) _
[GOAL]
case refine'_1
z w : ℍ
r R : ℝ
⊢ Continuous id
[PROOFSTEP]
refine' (@continuous_iff_continuous_dist ℍ ℍ metricSpaceAux.toPseudoMetricSpace _ _).2 _
[GOAL]
case refine'_1
z w : ℍ
r R : ℝ
⊢ Continuous fun x => dist (id x.fst) (id x.snd)
[PROOFSTEP]
have : ∀ x : ℍ × ℍ, 2 * Real.sqrt (x.1.im * x.2.im) ≠ 0 := fun x =>
mul_ne_zero two_ne_zero (Real.sqrt_pos.2 <| mul_pos x.1.im_pos x.2.im_pos).ne'
[GOAL]
case refine'_1
z w : ℍ
r R : ℝ
this : ∀ (x : ℍ × ℍ), 2 * sqrt (im x.fst * im x.snd) ≠ 0
⊢ Continuous fun x => dist (id x.fst) (id x.snd)
[PROOFSTEP]
apply_rules [Continuous.div, Continuous.mul, continuous_const, Continuous.arsinh, Continuous.dist, continuous_coe.comp,
continuous_fst, continuous_snd, Real.continuous_sqrt.comp, continuous_im.comp]
[GOAL]
case refine'_2
z w : ℍ
r R : ℝ
⊢ UniformSpace.toTopologicalSpace ≤ instTopologicalSpaceUpperHalfPlane
[PROOFSTEP]
letI : MetricSpace ℍ := metricSpaceAux
[GOAL]
case refine'_2
z w : ℍ
r R : ℝ
this : MetricSpace ℍ := metricSpaceAux
⊢ UniformSpace.toTopologicalSpace ≤ instTopologicalSpaceUpperHalfPlane
[PROOFSTEP]
refine' le_of_nhds_le_nhds fun z => _
[GOAL]
case refine'_2
z✝ w : ℍ
r R : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
⊢ 𝓝 z ≤ 𝓝 z
[PROOFSTEP]
rw [nhds_induced]
[GOAL]
case refine'_2
z✝ w : ℍ
r R : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
⊢ 𝓝 z ≤ comap Subtype.val (𝓝 ↑z)
[PROOFSTEP]
refine' (nhds_basis_ball.le_basis_iff (nhds_basis_ball.comap _)).2 fun R hR => _
[GOAL]
case refine'_2
z✝ w : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
⊢ ∃ i, 0 < i ∧ ball z i ⊆ Subtype.val ⁻¹' ball (↑z) R
[PROOFSTEP]
have h₁ : 1 < R / im z + 1 := lt_add_of_pos_left _ (div_pos hR z.im_pos)
[GOAL]
case refine'_2
z✝ w : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / im z + 1
⊢ ∃ i, 0 < i ∧ ball z i ⊆ Subtype.val ⁻¹' ball (↑z) R
[PROOFSTEP]
have h₀ : 0 < R / im z + 1 := one_pos.trans h₁
[GOAL]
case refine'_2
z✝ w : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / im z + 1
h₀ : 0 < R / im z + 1
⊢ ∃ i, 0 < i ∧ ball z i ⊆ Subtype.val ⁻¹' ball (↑z) R
[PROOFSTEP]
refine' ⟨log (R / im z + 1), Real.log_pos h₁, _⟩
[GOAL]
case refine'_2
z✝ w : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / im z + 1
h₀ : 0 < R / im z + 1
⊢ ball z (log (R / im z + 1)) ⊆ Subtype.val ⁻¹' ball (↑z) R
[PROOFSTEP]
refine' fun w hw => (dist_coe_le w z).trans_lt _
[GOAL]
case refine'_2
z✝ w✝ : ℍ
r R✝ : ℝ
this : MetricSpace ℍ := metricSpaceAux
z : ℍ
R : ℝ
hR : 0 < R
h₁ : 1 < R / im z + 1
h₀ : 0 < R / im z + 1
w : ℍ
hw : w ∈ ball z (log (R / im z + 1))
⊢ im z * (Real.exp (dist w z) - 1) < R
[PROOFSTEP]
rwa [← lt_div_iff' z.im_pos, sub_lt_iff_lt_add, ← Real.lt_log_iff_exp_lt h₀]
[GOAL]
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
h : dist w ↑(center z r) ≤ im z * Real.sinh r
⊢ dist (↑(center z r)) w ≤ im z * Real.sinh r
[PROOFSTEP]
rwa [dist_comm]
[GOAL]
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' closedBall z r = closedBall (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
ext w
[GOAL]
case h
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r ↔ w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
constructor
[GOAL]
case h.mp
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r → w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
rintro ⟨w, hw, rfl⟩
[GOAL]
case h.mp.intro.intro
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℍ
hw : w ∈ closedBall z r
⊢ ↑w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
exact dist_le_iff_dist_coe_center_le.1 hw
[GOAL]
case h.mpr
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ closedBall (↑(center z r)) (im z * Real.sinh r) → w ∈ UpperHalfPlane.coe '' closedBall z r
[PROOFSTEP]
intro hw
[GOAL]
case h.mpr
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
hw : w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' closedBall z r
[PROOFSTEP]
lift w to ℍ using im_pos_of_dist_center_le hw
[GOAL]
case h.mpr.intro
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℍ
hw : ↑w ∈ closedBall (↑(center z r)) (im z * Real.sinh r)
⊢ ↑w ∈ UpperHalfPlane.coe '' closedBall z r
[PROOFSTEP]
exact mem_image_of_mem _ (dist_le_iff_dist_coe_center_le.2 hw)
[GOAL]
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' ball z r = ball (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
ext w
[GOAL]
case h
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' ball z r ↔ w ∈ ball (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
constructor
[GOAL]
case h.mp
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' ball z r → w ∈ ball (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
rintro ⟨w, hw, rfl⟩
[GOAL]
case h.mp.intro.intro
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℍ
hw : w ∈ ball z r
⊢ ↑w ∈ ball (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
exact dist_lt_iff_dist_coe_center_lt.1 hw
[GOAL]
case h.mpr
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ ball (↑(center z r)) (im z * Real.sinh r) → w ∈ UpperHalfPlane.coe '' ball z r
[PROOFSTEP]
intro hw
[GOAL]
case h.mpr
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
hw : w ∈ ball (↑(center z r)) (im z * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' ball z r
[PROOFSTEP]
lift w to ℍ using im_pos_of_dist_center_le (ball_subset_closedBall hw)
[GOAL]
case h.mpr.intro
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℍ
hw : ↑w ∈ ball (↑(center z r)) (im z * Real.sinh r)
⊢ ↑w ∈ UpperHalfPlane.coe '' ball z r
[PROOFSTEP]
exact mem_image_of_mem _ (dist_lt_iff_dist_coe_center_lt.2 hw)
[GOAL]
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ UpperHalfPlane.coe '' sphere z r = sphere (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
ext w
[GOAL]
case h
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' sphere z r ↔ w ∈ sphere (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
constructor
[GOAL]
case h.mp
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ UpperHalfPlane.coe '' sphere z r → w ∈ sphere (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
rintro ⟨w, hw, rfl⟩
[GOAL]
case h.mp.intro.intro
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℍ
hw : w ∈ sphere z r
⊢ ↑w ∈ sphere (↑(center z r)) (im z * Real.sinh r)
[PROOFSTEP]
exact dist_eq_iff_dist_coe_center_eq.1 hw
[GOAL]
case h.mpr
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
⊢ w ∈ sphere (↑(center z r)) (im z * Real.sinh r) → w ∈ UpperHalfPlane.coe '' sphere z r
[PROOFSTEP]
intro hw
[GOAL]
case h.mpr
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℂ
hw : w ∈ sphere (↑(center z r)) (im z * Real.sinh r)
⊢ w ∈ UpperHalfPlane.coe '' sphere z r
[PROOFSTEP]
lift w to ℍ using im_pos_of_dist_center_le (sphere_subset_closedBall hw)
[GOAL]
case h.mpr.intro
z✝ w✝ : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
w : ℍ
hw : ↑w ∈ sphere (↑(center z r)) (im z * Real.sinh r)
⊢ ↑w ∈ UpperHalfPlane.coe '' sphere z r
[PROOFSTEP]
exact mem_image_of_mem _ (dist_eq_iff_dist_coe_center_eq.2 hw)
[GOAL]
z w : ℍ
r R : ℝ
⊢ ProperSpace ℍ
[PROOFSTEP]
refine' ⟨fun z r => _⟩
[GOAL]
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ IsCompact (closedBall z r)
[PROOFSTEP]
rw [← inducing_subtype_val.isCompact_iff (f := ((↑) : ℍ → ℂ)), image_coe_closedBall]
[GOAL]
z✝ w : ℍ
r✝ R : ℝ
z : ℍ
r : ℝ
⊢ IsCompact (closedBall (↑(center z r)) (im z * Real.sinh r))
[PROOFSTEP]
apply isCompact_closedBall
[GOAL]
z w : ℍ
r R a : ℝ
⊢ Isometry fun y => mk { re := a, im := Real.exp y } (_ : 0 < Real.exp y)
[PROOFSTEP]
refine' Isometry.of_dist_eq fun y₁ y₂ => _
[GOAL]
z w : ℍ
r R a y₁ y₂ : ℝ
⊢ dist (mk { re := a, im := Real.exp y₁ } (_ : 0 < Real.exp y₁))
(mk { re := a, im := Real.exp y₂ } (_ : 0 < Real.exp y₂)) =
dist y₁ y₂
[PROOFSTEP]
rw [dist_of_re_eq]
[GOAL]
z w : ℍ
r R a y₁ y₂ : ℝ
⊢ dist (log (im (mk { re := a, im := Real.exp y₁ } (_ : 0 < Real.exp y₁))))
(log (im (mk { re := a, im := Real.exp y₂ } (_ : 0 < Real.exp y₂)))) =
dist y₁ y₂
z w : ℍ
r R a y₁ y₂ : ℝ
⊢ re (mk { re := a, im := Real.exp y₁ } (_ : 0 < Real.exp y₁)) =
re (mk { re := a, im := Real.exp y₂ } (_ : 0 < Real.exp y₂))
[PROOFSTEP]
exacts [congr_arg₂ _ (log_exp _) (log_exp _), rfl]
[GOAL]
z w : ℍ
r R a : ℝ
y₁ y₂ : ℍ
⊢ dist ((fun x x_1 => x +ᵥ x_1) a y₁) ((fun x x_1 => x +ᵥ x_1) a y₂) = dist y₁ y₂
[PROOFSTEP]
simp only [dist_eq, coe_vadd, vadd_im, dist_add_left]
[GOAL]
z w : ℍ
r R : ℝ
a : { x // 0 < x }
⊢ Isometry ((fun x x_1 => x • x_1) a)
[PROOFSTEP]
refine' Isometry.of_dist_eq fun y₁ y₂ => _
[GOAL]
z w : ℍ
r R : ℝ
a : { x // 0 < x }
y₁ y₂ : ℍ
⊢ dist ((fun x x_1 => x • x_1) a y₁) ((fun x x_1 => x • x_1) a y₂) = dist y₁ y₂
[PROOFSTEP]
simp only [dist_eq, coe_pos_real_smul, pos_real_im]
[GOAL]
z w : ℍ
r R : ℝ
a : { x // 0 < x }
y₁ y₂ : ℍ
⊢ 2 * arsinh (dist (↑a • ↑y₁) (↑a • ↑y₂) / (2 * sqrt (↑a * im y₁ * (↑a * im y₂)))) =
2 * arsinh (dist ↑y₁ ↑y₂ / (2 * sqrt (im y₁ * im y₂)))
[PROOFSTEP]
congr 2
[GOAL]
case e_a.e_x
z w : ℍ
r R : ℝ
a : { x // 0 < x }
y₁ y₂ : ℍ
⊢ dist (↑a • ↑y₁) (↑a • ↑y₂) / (2 * sqrt (↑a * im y₁ * (↑a * im y₂))) = dist ↑y₁ ↑y₂ / (2 * sqrt (im y₁ * im y₂))
[PROOFSTEP]
rw [dist_smul₀, mul_mul_mul_comm, Real.sqrt_mul (mul_self_nonneg _), Real.sqrt_mul_self_eq_abs, Real.norm_eq_abs,
mul_left_comm]
[GOAL]
case e_a.e_x
z w : ℍ
r R : ℝ
a : { x // 0 < x }
y₁ y₂ : ℍ
⊢ |↑a| * dist ↑y₁ ↑y₂ / (|↑a| * (2 * sqrt (im y₁ * im y₂))) = dist ↑y₁ ↑y₂ / (2 * sqrt (im y₁ * im y₂))
[PROOFSTEP]
exact mul_div_mul_left _ _ (mt _root_.abs_eq_zero.1 a.2.ne')
[GOAL]
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
⊢ Isometry fun x => g • x
[PROOFSTEP]
have h₀ : Isometry (fun z => ModularGroup.S • z : ℍ → ℍ) :=
Isometry.of_dist_eq fun y₁ y₂ =>
by
have h₁ : 0 ≤ im y₁ * im y₂ := mul_nonneg y₁.property.le y₂.property.le
have h₂ : Complex.abs (y₁ * y₂) ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
simp only [dist_eq, modular_S_smul, inv_neg, neg_div, div_mul_div_comm, coe_mk, mk_im, div_one, Complex.inv_im,
Complex.neg_im, coe_im, neg_neg, Complex.normSq_neg, mul_eq_mul_left_iff, Real.arsinh_inj, bit0_eq_zero,
one_ne_zero, or_false_iff, dist_neg_neg, mul_neg, neg_mul, dist_inv_inv₀ y₁.ne_zero y₂.ne_zero, ←
AbsoluteValue.map_mul, ← Complex.normSq_mul, Real.sqrt_div h₁, ← Complex.abs_apply, mul_div (2 : ℝ),
div_div_div_comm, div_self h₂, Complex.norm_eq_abs]
[GOAL]
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
⊢ dist (ModularGroup.S • y₁) (ModularGroup.S • y₂) = dist y₁ y₂
[PROOFSTEP]
have h₁ : 0 ≤ im y₁ * im y₂ := mul_nonneg y₁.property.le y₂.property.le
[GOAL]
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ im y₁ * im y₂
⊢ dist (ModularGroup.S • y₁) (ModularGroup.S • y₂) = dist y₁ y₂
[PROOFSTEP]
have h₂ : Complex.abs (y₁ * y₂) ≠ 0 := by simp [y₁.ne_zero, y₂.ne_zero]
[GOAL]
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ im y₁ * im y₂
⊢ ↑Complex.abs (↑y₁ * ↑y₂) ≠ 0
[PROOFSTEP]
simp [y₁.ne_zero, y₂.ne_zero]
[GOAL]
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
y₁ y₂ : ℍ
h₁ : 0 ≤ im y₁ * im y₂
h₂ : ↑Complex.abs (↑y₁ * ↑y₂) ≠ 0
⊢ dist (ModularGroup.S • y₁) (ModularGroup.S • y₂) = dist y₁ y₂
[PROOFSTEP]
simp only [dist_eq, modular_S_smul, inv_neg, neg_div, div_mul_div_comm, coe_mk, mk_im, div_one, Complex.inv_im,
Complex.neg_im, coe_im, neg_neg, Complex.normSq_neg, mul_eq_mul_left_iff, Real.arsinh_inj, bit0_eq_zero, one_ne_zero,
or_false_iff, dist_neg_neg, mul_neg, neg_mul, dist_inv_inv₀ y₁.ne_zero y₂.ne_zero, ← AbsoluteValue.map_mul, ←
Complex.normSq_mul, Real.sqrt_div h₁, ← Complex.abs_apply, mul_div (2 : ℝ), div_div_div_comm, div_self h₂,
Complex.norm_eq_abs]
[GOAL]
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
⊢ Isometry fun x => g • x
[PROOFSTEP]
by_cases hc : g 1 0 = 0
[GOAL]
case pos
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
hc : ↑g 1 0 = 0
⊢ Isometry fun x => g • x
[PROOFSTEP]
obtain ⟨u, v, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_eq_zero g hc
[GOAL]
case pos.intro.intro
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
hc : ↑g 1 0 = 0
u : { x // 0 < x }
v : ℝ
h : (fun x x_1 => x • x_1) g = (fun z => v +ᵥ z) ∘ fun z => u • z
⊢ Isometry fun x => g • x
[PROOFSTEP]
dsimp only at h
[GOAL]
case pos.intro.intro
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
hc : ↑g 1 0 = 0
u : { x // 0 < x }
v : ℝ
h : (fun x => g • x) = (fun z => v +ᵥ z) ∘ fun z => u • z
⊢ Isometry fun x => g • x
[PROOFSTEP]
rw [h]
[GOAL]
case pos.intro.intro
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
hc : ↑g 1 0 = 0
u : { x // 0 < x }
v : ℝ
h : (fun x => g • x) = (fun z => v +ᵥ z) ∘ fun z => u • z
⊢ Isometry ((fun z => v +ᵥ z) ∘ fun z => u • z)
[PROOFSTEP]
exact (isometry_real_vadd v).comp (isometry_pos_mul u)
[GOAL]
case neg
z w : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
hc : ¬↑g 1 0 = 0
⊢ Isometry fun x => g • x
[PROOFSTEP]
obtain ⟨u, v, w, h⟩ := exists_SL2_smul_eq_of_apply_zero_one_ne_zero g hc
[GOAL]
case neg.intro.intro.intro
z w✝ : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
hc : ¬↑g 1 0 = 0
u : { x // 0 < x }
v w : ℝ
h :
(fun x x_1 => x • x_1) g =
(fun x x_1 => x +ᵥ x_1) w ∘
(fun x x_1 => x • x_1) ModularGroup.S ∘ (fun x x_1 => x +ᵥ x_1) v ∘ (fun x x_1 => x • x_1) u
⊢ Isometry fun x => g • x
[PROOFSTEP]
dsimp only at h
[GOAL]
case neg.intro.intro.intro
z w✝ : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
hc : ¬↑g 1 0 = 0
u : { x // 0 < x }
v w : ℝ
h : (fun x => g • x) = (fun x => w +ᵥ x) ∘ (fun x => ModularGroup.S • x) ∘ (fun x => v +ᵥ x) ∘ fun x => u • x
⊢ Isometry fun x => g • x
[PROOFSTEP]
rw [h]
[GOAL]
case neg.intro.intro.intro
z w✝ : ℍ
r R : ℝ
g : SL(2, ℝ)
h₀ : Isometry fun z => ModularGroup.S • z
hc : ¬↑g 1 0 = 0
u : { x // 0 < x }
v w : ℝ
h : (fun x => g • x) = (fun x => w +ᵥ x) ∘ (fun x => ModularGroup.S • x) ∘ (fun x => v +ᵥ x) ∘ fun x => u • x
⊢ Isometry ((fun x => w +ᵥ x) ∘ (fun x => ModularGroup.S • x) ∘ (fun x => v +ᵥ x) ∘ fun x => u • x)
[PROOFSTEP]
exact (isometry_real_vadd w).comp (h₀.comp <| (isometry_real_vadd v).comp <| isometry_pos_mul u)
|
Subsets and Splits