Statement:
stringlengths 7
24.3k
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lemma while_left_plus_below:
"x * (x \<star> y) \<le> x \<star> y"
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lemma iam_iterateoi_impl:
"poly_map_iterateoi iam_\<alpha> iam_invar iam_iterateoi"
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lemma game_equiv_subst_eq: "game_equiv \<alpha> \<beta> \<Longrightarrow> P (game_sem I \<alpha> X) == P (game_sem I \<beta> X)"
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lemma diagonal_to_Smith_row_i_preserves_previous_diagonal:
fixes A::"'a:: {bezout_ring}^'b::mod_type^'c::mod_type"
assumes ib: "is_bezout_ext bezout"
and i_min: "i < min (nrows A) (ncols A)"
and a_notin: "to_nat a \<notin> set [i + 1..<min (nrows A) (ncols A)]"
and ab: "to_nat a = to_nat b"
and ai: "to_nat a \<noteq> i"
shows "Diagonal_to_Smith_row_i A i bezout $ a $ b = A $ a $ b"
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lemma "((sep_empty \<longrightarrow>* (not ((not A) ** sep_empty))) imp A)
(h::'a::heap_sep_algebra)"
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lemma restrict_on_source:
assumes "map f S T"
shows "restrict f S = f"
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lemma set_child_nodes_get_shadow_root: "\<forall>w \<in> set_child_nodes_locs ptr. (h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow>
(\<forall>r \<in> get_shadow_root_locs ptr'. r h h'))"
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lemma int_of_uint32_minus: "int_of_uint32 (x - y) = (int_of_uint32 x - int_of_uint32 y) mod 4294967296"
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lemma Reals_divide [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a / b \<in> \<real>"
for a b :: "'a::{real_field,field}"
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lemma scaleR_cont: "f \<in> bfun \<Longrightarrow> (\<lambda>x. a *\<^sub>R f x) \<in> bfun" for f :: "'a \<Rightarrow> 'b"
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lemma MutuallyDistinct_EmptySet [simp]:
"MutuallyDistinct {}"
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lemma weaken_rely:
assumes i_refsto_j: "i \<sqsubseteq> j"
shows "(c // j) \<sqsubseteq> (c // i)"
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lemma weaken_post:
"\<lbrakk> \<turnstile>\<^sub>1 {P} c {e \<Down> Q}; \<forall>l s. Q l s \<longrightarrow> Q' l s \<rbrakk> \<Longrightarrow> \<turnstile>\<^sub>1 {P} c {e \<Down> Q'}"
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lemma map_App_map_un_App1:
shows "\<lbrakk>Arr U; set U \<subseteq> Collect \<Lambda>.is_App; \<Lambda>.Ide b; \<Lambda>.un_App2 ` set U \<subseteq> {b}\<rbrakk> \<Longrightarrow>
map (\<lambda>t. \<Lambda>.App t b) (map \<Lambda>.un_App1 U) = U"
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lemma mcont2mcont_map_option[THEN option.mcont2mcont, simp, cont_intro]:
shows mcont_map_option: "mcont (flat_lub None) option_ord (flat_lub None) option_ord (map_option f)"
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lemma hom_decomp_exact:
assumes "valid_decomp X qs" and "standard_decomp k qs" and "hom_decomp qs"
shows "hom_decomp (exact k qs)"
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lemma meet_L_apx_isotone:
"x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> L \<sqsubseteq> y \<sqinter> L"
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lemma consec_un_defined:
"consec i j \<longrightarrow> (Rep_nat_int (i \<squnion> j) \<in> {S . (\<exists> (m::nat) n . {m..n }=S) })"
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lemma cc_n_minus_1: "cc (n - 1) = 2 * d"
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theorem dnf_eval :
"(\<exists>(al,fl)\<in>set (dnf \<phi>).
(\<forall>a\<in>set al. aEval a xs)
\<and> (\<forall>f\<in>set fl. eval f xs))
= eval \<phi> xs"
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lemma execn_Abrupt_end: assumes exec: "\<Gamma>\<turnstile>\<langle>c,s\<rangle> =n\<Rightarrow> t" and s: "s=Abrupt s'"
shows "t=Abrupt s'"
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lemma satA_o_subst:
"satA \<xi> o substA \<rho> = satA (int \<xi> o \<rho>)"
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lemma drop_bit_or [simp]:
\<open>drop_bit n (a OR b) = drop_bit n a OR drop_bit n b\<close>
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lemma DirProds_one_cong_sym: "(f G) \<cong> (DirProds f {G})"
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lemma prod_exp:
fixes x::real
shows "4*(x*y) = (x+y)^2 - (x-y)^2"
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lemma eval_fds_zeroD_aux:
fixes h :: "'a fds"
assumes conv: "fds_abs_converges h (s0 :: 'a)"
assumes freq: "frequently (\<lambda>s. eval_fds h s = 0) ((\<lambda>s. s \<bullet> 1) going_to at_top)"
shows "h = 0"
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lemma meta_spec_protect:
assumes g: "\<And>x. PROP P x"
shows "PROP Pure.prop (PROP P x)"
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lemma cnj_in_Ints_iff [simp]: "cnj x \<in> \<int> \<longleftrightarrow> x \<in> \<int>"
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lemma AA9: "\<turnstile> \<box>[P]_v \<and> \<diamond>\<langle>A\<rangle>_v \<longrightarrow> \<diamond>\<langle>[P]_v \<and> A\<rangle>_v"
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lemma separating_tensor:
fixes A :: \<open>'a::domain update set\<close> and B :: \<open>'b::domain update set\<close>
assumes [simp]: \<open>separating TYPE('c::domain) A\<close>
assumes [simp]: \<open>separating TYPE('c) B\<close>
shows \<open>separating TYPE('c) {a \<otimes>\<^sub>u b | a b. a\<in>A \<and> b\<in>B}\<close>
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lemma [sepref_fr_rules]:
"(\<lambda>cfi. return cfi, PR_CONST compute_rflow) \<in> (asmtx_assn N id_assn)\<^sup>d \<rightarrow>\<^sub>a is_rflow N"
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lemma sconjunctive_assert [simp]: "sconjunctive {.p.}"
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lemma match_fresh_mono:
assumes "\<turnstile> p \<rhd> t \<Rightarrow> \<theta>" and "(x::name) \<sharp> t"
shows "\<forall>(y, t)\<in>set \<theta>. x \<sharp> t"
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lemma keys_pm_of_idx_pm_subset: "keys (pm_of_idx_pm xs f) \<subseteq> set xs"
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lemma test_subid: "test p \<Longrightarrow> p \<le> 1"
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lemma hd_in_set_conv: "hd l \<in> set l \<longleftrightarrow> l\<noteq>[]"
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lemma low_mds_eq_from_conc_to_abs:
"conc.low_mds_eq mds mem mem' \<Longrightarrow> abs.low_mds_eq (mds\<^sub>A_of mds) (mem\<^sub>A_of mem) (mem\<^sub>A_of mem')"
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lemma well_com_mat_O:
"well_com (Utrans_P vars1 mat_O)"
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lemma prove_HRP: fixes A::fm shows "{} \<turnstile> HRP \<guillemotleft>A\<guillemotright> \<guillemotleft>\<guillemotleft>A\<guillemotright>\<guillemotright>"
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lemma bind_gpv_unfold [code]:
"r \<bind> f = GPV (
do {
generat \<leftarrow> the_gpv r;
case generat of Pure x \<Rightarrow> the_gpv (f x)
| IO out c \<Rightarrow> return_spmf (IO out (\<lambda>input. c input \<bind> f))
})"
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lemma sums_unique2: "f sums a \<Longrightarrow> f sums b \<Longrightarrow> a = b"
for a b :: 'a
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lemma "p2 = bot"
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lemma homeomorphic_maps_imp_map:
"homeomorphic_maps X Y f g \<Longrightarrow> homeomorphic_map X Y f"
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lemma bl_bin_bk_oc[simp]:
"bl_bin (xs @ [Bk, Oc]) =
bl_bin xs + 2*2^(length xs)"
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lemma step_dom_mono_aux:
fixes \<tau> p \<tau>' a b
assumes sorted: "sorted (rev (transf p \<tau>)) "
and a_b_step: "(a, b) \<in> set (step p \<tau>) "
and "\<tau> \<in> A " and " p < n " and " \<tau> \<sqsubseteq>\<^bsub>r\<^esub> \<tau>'"
shows "\<exists>\<tau>''. (a, \<tau>'') \<in> set (step p \<tau>') \<and> b \<sqsubseteq>\<^bsub>r\<^esub> \<tau>''"
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lemma before:
assumes a:"\<I> i"
shows "BEGIN i \<lless> END i"
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lemma seq_cases:
assumes "seq t u"
shows "(is_Var t \<and> is_Var u) \<or>
(is_Lam t \<and> is_Lam u) \<or>
(is_App t \<and> is_App u) \<or>
(is_App t \<and> is_Beta u \<and> is_Lam (un_App1 t)) \<or>
(is_App t \<and> is_Beta u \<and> is_Beta (un_App1 t)) \<or>
is_Beta t"
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lemma finite_imp_pfinite_gpv:
assumes "finite_gpv \<I> gpv" "\<I> \<turnstile>g gpv \<surd>"
shows "pfinite_gpv \<I> gpv"
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lemma if_not_P': "\<lbrakk> \<not> P; y = z \<rbrakk> \<Longrightarrow> (if P then x else y) = z"
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lemma ifex_sat_list_SomeD: "ifex_no_twice i \<Longrightarrow> ifex_sat_list i = Some u \<Longrightarrow> ass = update_assignment u ass' \<Longrightarrow> val_ifex i ass = True"
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lemma cblinfun_right_adj_apply[simp]: \<open>cblinfun_right* *\<^sub>V \<psi> = snd \<psi>\<close>
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lemma (in \<Z>) smc_Set_obj_initial: "obj_initial (smc_Set \<alpha>) A \<longleftrightarrow> A = 0"
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lemma reach_PublicV_imples_FriendV[simp]:
assumes "reach s"
and "vis s pid \<noteq> PublicV"
shows "vis s pid = FriendV"
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lemma monom_mult_sum_right: "monom_mult c t (sum f P) = (\<Sum>p\<in>P. monom_mult c t (f p))"
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lemma Abs_finfun_inject_finite_class:
fixes x y :: "('a :: finite) \<Rightarrow> 'b"
shows "Abs_finfun x = Abs_finfun y \<longleftrightarrow> x = y"
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lemma fps_shift_nth [simp]: "fps_shift n f $ i = f $ (i + n)"
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lemma foldr_prs_aux:
assumes a: "Quotient3 R1 abs1 rep1"
and b: "Quotient3 R2 abs2 rep2"
shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"
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lemma path_append_conv[simp]: "epath E u (p@q) v \<longleftrightarrow> (\<exists>w. epath E u p w \<and> epath E w q v)"
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lemma Abs_res_fcb2:
fixes as bs :: "atom set"
and x y :: "'b :: fs"
and c::"'c::fs"
assumes eq: "[as]res. x = [bs]res. y"
and fin: "finite as" "finite bs"
and fcb1: "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c"
and fresh1: "as \<sharp>* c"
and fresh2: "bs \<sharp>* c"
and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f (as \<inter> supp x) x c) = f (p \<bullet> (as \<inter> supp x)) (p \<bullet> x) c"
and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f (bs \<inter> supp y) y c) = f (p \<bullet> (bs \<inter> supp y)) (p \<bullet> y) c"
shows "f (as \<inter> supp x) x c = f (bs \<inter> supp y) y c"
|
lemma replicate_append_same:
"replicate i x @ [x] = x # replicate i x"
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lemma bigo_pos_const:
"(\<exists>c::'a::linordered_idom. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>) \<longleftrightarrow>
(\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
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lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D"
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lemma five: "5 \<times> x = x \<squnion> x \<squnion> x \<squnion> x \<squnion> x"
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lemma vector_top_closed:
"vector top"
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lemma Pd_singletons_for_ds_range:
shows "Field (Pd_singletons_for_ds X ds d) \<subseteq> {x. Xd x = d}"
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lemma avl_ins_case: "avl t \<Longrightarrow> case ins x t of
Same t' \<Rightarrow> avl t' \<and> height t' = height t |
Diff t' \<Rightarrow> avl t' \<and> height t' = height t + 1 \<and>
(\<forall>l a r. t' = Node l (a,Bal) r \<longrightarrow> a = x \<and> l = Leaf \<and> r = Leaf)"
|
lemma comm_groupE:
fixes G (structure)
assumes "comm_group G"
shows "\<And>x y. \<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G"
and "\<one> \<in> carrier G"
and "\<And>x y z. \<lbrakk> x \<in> carrier G; y \<in> carrier G; z \<in> carrier G \<rbrakk> \<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
and "\<And>x y. \<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"
and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x"
and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
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lemma bw_diff_id [simp]: "bw_diff (\<lambda>x. x) = (\<lambda>_. 1)"
|
lemma Domain_set_zip[simp]:
assumes "length A = length B"
shows "Domain (set (zip A B)) = set A"
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lemma ct_eq_common_tsome: "ct_eq x y = (\<exists>t. ct_eq x (TSome t) \<and> ct_eq (TSome t) y)"
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lemma (in cring) trivialideals_fieldI:
assumes carrnzero: "carrier R \<noteq> {\<zero>}"
and haveideals: "{I. ideal I R} = {{\<zero>}, carrier R}"
shows "field R"
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lemma (in vfsequence) vfsequence_vinsert:
"vfsequence (vinsert \<langle>vcard xs, a\<rangle> xs)"
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lemma GreenThm_typeI_typeII_divisible_region':
assumes
only_vertical_division:
"only_vertical_division one_chain_typeI two_chain_typeI"
"boundary_chain one_chain_typeI" and
only_horizontal_division:
"only_horizontal_division one_chain_typeII two_chain_typeII"
"boundary_chain one_chain_typeII" and
typeI_and_typII_one_chains_have_gen_common_subdiv:
"common_sudiv_exists one_chain_typeI one_chain_typeII"
shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i x - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeI"
"integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. (F x) \<bullet> j) i x - partial_vector_derivative (\<lambda>x. (F x) \<bullet> i) j x) = one_chain_line_integral F {i, j} one_chain_typeII"
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lemma set_restriction_fun_is_set_restriction: "
set_restriction (set_restriction_fun P)"
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lemma ln_minus_ln_floor_bound:
assumes "x \<ge> 2"
shows "ln x - ln (floor x) \<in> {0..<1 / (x - 1)}"
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lemma dtree_from_list_eq_dverts: "dverts (dtree_from_list r xs) = insert r (fst ` set xs)"
|
lemma igSubstIGVar2_fromMOD[simp]:
"gSubstGVar2 MOD \<Longrightarrow> igSubstIGVar2 (fromMOD MOD)"
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theorem regular_star:
assumes S: "regular S" shows "regular (star S)"
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lemma (in domain) zero_is_irreducible_iff_field:
shows "irreducible R \<zero> \<longleftrightarrow> field R"
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lemma clop_idem_var [simp]: "cl_op (cl_op x) = cl_op x"
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lemma l9_30:
assumes "\<not> Coplanar A B C P" and
"\<not> Col D E F" and
"Coplanar D E F P" and
"Coplanar A B C X" and
"Coplanar A B C Y" and
"Coplanar A B C Z" and
"Coplanar D E F X" and
"Coplanar D E F Y" and
"Coplanar D E F Z"
shows "Col X Y Z"
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lemma step_1_correct:
"spec\<^sub>1 p \<Longrightarrow> spec\<^sub>0 p"
|
lemma wnf_lemma_1:
"(n(p * L) * n(q * L) \<squnion> an(p * L) * an(r * L)) * n(p * L) = n(p * L) * n(q * L)"
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lemma remove_child_pointers_preserved:
assumes "w \<in> remove_child_locs ptr owner_document"
assumes "h \<turnstile> w \<rightarrow>\<^sub>h h'"
shows "object_ptr_kinds h = object_ptr_kinds h'"
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lemma InvCapsNonneg_imp_InvRecordsNonneg: "InvCapsNonneg c \<Longrightarrow> InvRecordsNonneg c"
|
lemma integr_index: "integrable (measure_pmf (config'' (BIT_init, BIT_step) qs init n))
(\<lambda>(s, is). real (Suc (index s (qs ! n))))"
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lemma tt_zero [simp]: "tt 0 = min_term"
|
lemma acyclic_3a_3b:
"acyclic_3a x \<longleftrightarrow> acyclic_3b x"
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lemma lexordp_iff:
"lexordp xs ys \<longleftrightarrow> (\<exists>x vs. ys = xs @ x # vs) \<or> (\<exists>us a b vs ws. a < b \<and> xs = us @ a # vs \<and> ys = us @ b # ws)"
(is "?lhs = ?rhs")
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lemma c_k_with_K: "i < j \<Longrightarrow> c i j = c_k i j (K i j)"
|
lemma edges_ins_edge'[simp]:
"u\<noteq>v \<Longrightarrow> edges (ins_edge (u,v) g) = {(u,v),(v,u)} \<union> edges g"
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lemma P_lnull_ltl_deadend_v0: "lnull (ltl P) \<Longrightarrow> deadend v0"
|
lemma splitAt_take[simp]: "distinct ls \<Longrightarrow> i < length ls \<Longrightarrow> fst (splitAt (ls!i) ls) = take i ls"
|
theorem wt_prog_comp: "wf_java_prog G \<Longrightarrow> wt_jvm_prog (comp G) (compTp G)"
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lemma uminus_zero_vec[simp]: "- (0\<^sub>v n) = (0\<^sub>v n :: 'a :: group_add vec)"
|
lemma o_substA:
"substA \<pi>1 o substA \<pi>2 = substA (subst \<pi>1 o \<pi>2)"
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lemma sum_over_permutations_insert:
assumes fS: "finite S"
and aS: "a \<notin> S"
shows "sum f {p. p permutes (insert a S)} =
sum (\<lambda>b. sum (\<lambda>q. f (transpose a b \<circ> q)) {p. p permutes S}) (insert a S)"
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theorem (in semiring_of_sets) caratheodory:
assumes pos: "positive M \<mu>" and ca: "countably_additive M \<mu>"
shows "\<exists>\<mu>' :: 'a set \<Rightarrow> ennreal. (\<forall>s \<in> M. \<mu>' s = \<mu> s) \<and> measure_space \<Omega> (sigma_sets \<Omega> M) \<mu>'"
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lemma Gamma_series_Weierstrass_nonpos_Ints:
"eventually (\<lambda>k. Gamma_series_Weierstrass (- of_nat n) k = 0) sequentially"
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lemma (in pre_digraph) adj_mono:
assumes "u \<rightarrow>\<^bsub>H\<^esub> v" "subgraph H G"
shows "u \<rightarrow> v"
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