UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

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lemma subterm_inj_on_subset: assumes "subterm_inj_on f A" and "B \<subseteq> A" shows "subterm_inj_on f B"
lemma gcollapse_groot_None [simp]: "groot_sym t = None \<Longrightarrow> gcollapse t = None" "fst (groot t) = None \<Longrightarrow> gcollapse t = None"
lemma single_Trg_last_in_targets: assumes "Arr T" shows "[\<Lambda>.Trg (last T)] \<in> targets T"
lemma permutes_bij_inv_into: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close> fixes A :: "'a set" and B :: "'b set" assumes "p permutes A" and "bij_betw f A B" shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"
lemma fromArr_toArr [simp]: assumes "arr f" and "\<not>ide f" shows "fromArr (toArr f) = f"
lemma Con_char: shows "\<T> \<lbrace>\<frown>\<rbrace> \<U> \<longleftrightarrow> N.is_Cong_class \<T> \<and> N.is_Cong_class \<U> \<and> (\<exists>t u. t \<in> \<T> \<and> u \<in> \<U> \<and> t \<frown> u)"
theorem main\<^sub>K: \<open>G \<TTurnstile>\<^sub>K p \<longleftrightarrow> G \<turnstile>\<^sub>K p\<close>
lemma farkas_minkowsky_weyl_theorem: "(\<exists> X. X \<subseteq> carrier_vec n \<and> finite X \<and> P = cone X) \<longleftrightarrow> (\<exists> A nr. A \<in> carrier_mat nr n \<and> P = polyhedral_cone A)"
lemma OclOr7[simp]: "(null or null) = null"
lemma subst_lconsts_empty_subst[simp]: "subst_lconsts empty_subst = {}"
lemma (in Ring) exact3_comp_0:"\<lbrakk>R module L; R module M; R module N; f \<in> mHom R L M; g \<in> mHom R M N; exact3 R L f M g N\<rbrakk> \<Longrightarrow> compos L g f = mzeromap L N"
lemma comp_ipresIGFreshAbs: assumes "ipresIGWlsAbs hA MOD MOD'" and "ipresIGFreshAbs hA MOD MOD'" and "ipresIGFreshAbs hA' MOD' MOD''" shows "ipresIGFreshAbs (hA' o hA) MOD MOD''"
lemma atom_fresh_perm: "\<lbrakk>x \<in> Vs; y \<in> Vs\<rbrakk> \<Longrightarrow> atom x \<sharp> p \<bullet> y"
lemma ccompare_set_code [code]: "CCOMPARE('a :: ccompare set) = (case ID CCOMPARE('a) of None \<Rightarrow> None \| Some _ \<Rightarrow> Some (comp_of_ords cless_eq_set cless_set))"
theorem th_3: "c_unfold (length ls) (c_fold ls) = ls"
lemma prim_glue: assumes last_neq: "us = \<epsilon> \<or> last us \<noteq> w" and unique_blocks: "\<And>bs\<^sub>1 bs\<^sub>2. glue_block w us bs\<^sub>1 \<Longrightarrow> glue_block w us bs\<^sub>2 \<Longrightarrow> concat bs\<^sub>1 = concat bs\<^sub>2 \<Longrightarrow> bs\<^sub>1 = bs\<^sub>2" shows "primitive us \<Longrightarrow> primitive (glue w us)"
lemma expands_to_divide: "trimmed G \<Longrightarrow> basis_wf basis \<Longrightarrow> (f expands_to F) basis \<Longrightarrow> (g expands_to G) basis \<Longrightarrow> ((\<lambda>x. f x / g x) expands_to F / G) basis"
lemma label_incr_0_rev [dest]: "\<lbrakk>n \<oplus> i = Label 0; i > 0\<rbrakk> \<Longrightarrow> False"
lemma (in dist_execution) state_is_associated_string: assumes "is_valid_state_id i" shows "is_certified_associated_string (set (received_messages i)) (state i)"
lemma DropToShift: fixes l i list assumes "l + i < length list" shows "(drop l list) ! i = list ! (l + i)"
lemma bconf_Try[iff]: "P,sh \<turnstile>\<^sub>b (try e\<^sub>1 catch(C V) e\<^sub>2,b) \<surd> \<longleftrightarrow> P,sh \<turnstile>\<^sub>b (e\<^sub>1,b) \<surd>"
lemma exec_Cons_1 [intro]: "P \<turnstile> (0,s,stk) \<rightarrow>* (j,t,stk') \<Longrightarrow> instr#P \<turnstile> (1,s,stk) \<rightarrow>* (1+j,t,stk')"
lemma outgoing_transitions_deterministic2: "(\<And>s a b ba aa bb bc. ((a, b), ba) \|\<in>\| outgoing_transitions e s \<Longrightarrow> ((aa, bb), bc) \|\<in>\| (outgoing_transitions e s) - {\|((a, b), ba)\|} \<Longrightarrow> b \<noteq> bb \<or> ba \<noteq> bc \<Longrightarrow> \<not>choice ba bc) \<Longrightarrow> deterministic e"
theorem zeta_neg_of_nat: "zeta (-of_nat n) = -of_real (bernoulli' (Suc n)) / of_nat (Suc n)"
lemma ferrack: assumes qf: "qfree p" shows "qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)" (is "_ \<and> ?rhs = ?lhs")
lemma disj_gtt_states_disj_snd_ta_states: assumes dist_st: "gtt_states \<G>\<^sub>1 \|\<inter>\| gtt_states \<G>\<^sub>2 = {\|\|}" shows "\<Q> (snd \<G>\<^sub>1) \|\<inter>\| \<Q> (snd \<G>\<^sub>2) = {\|\|}"
lemma "syzygy_basis DRLEX [Vec\<^sub>0 0 (X * Y - Z), Vec\<^sub>0 0 (X * Z - Y), Vec\<^sub>0 0 (Y * Z - X)] = [ Vec\<^sub>0 0 (- X * Z + Y) + Vec\<^sub>0 1 (X * Y - Z), Vec\<^sub>0 0 (- Y * Z + X) + Vec\<^sub>0 2 (X * Y - Z), Vec\<^sub>0 1 (- Y * Z + X) + Vec\<^sub>0 2 (X * Z - Y), Vec\<^sub>0 0 (Y - Y * Z ^ 2) + Vec\<^sub>0 1 (Y ^ 2 * Z - Z) + Vec\<^sub>0 2 (Y ^ 2 - Z ^ 2) ]"
lemma (in vmc_path) strategy_attracts_lset: assumes "strategy_attracts p \<sigma> A W" "v0 \<in> A" shows "lset P \<inter> W \<noteq> {}"
lemma degree_le: assumes d: "\<forall>j \<ge> d. pdevs_apply x j = 0" shows "degree x \<le> d"
lemma ternary_lift5: "eval_ternary_Not tv = TernaryUnknown \<longleftrightarrow> tv = TernaryUnknown"
lemma word_plus_mcs_4: "\<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> (w::'a::len word)"
lemma table_of_mapf_SomeD [dest!]: "table_of (map (\<lambda>(k,x). (k, f x)) t) k = Some z \<Longrightarrow> (\<exists>y\<in>table_of t k: z=f y)"
lemma srs: "r2s \<circ> s2r = id"
lemma bigo_add_commute_imp: "f \<in> g +o O(h) \<Longrightarrow> g \<in> f +o O(h)"
lemma word_upto_set_eq: "a \<le> b \<Longrightarrow> x \<in> set (word_upto a b) \<longleftrightarrow> a \<le> x \<and> x \<le> b"
lemma smap_union_empty2[simp]: "m \<union>. {}. = m"
lemma openin_contains_ball: "openin (top_of_set T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x \<in> S. \<exists>e. 0 < e \<and> ball x e \<inter> T \<subseteq> S)" (is "?lhs = ?rhs")
lemma "dom(map_of system_UploadDroid) \<subseteq> set (nodesL policy)"
lemma swap_cs_Trans_Snapshot: assumes "c \<turnstile> ev \<mapsto> d" and "d \<turnstile> ev' \<mapsto> e" and "isTrans ev" and "isSnapshot ev'" and "c \<turnstile> ev' \<mapsto> d'" and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
lemma wtA_gA[simp]: "Ik.wtA at \<Longrightarrow> GE.wtA (gA at)"
lemma rk_ABPRa : assumes "rk {A, B, P} = 3" and "rk {A, B, C, A', B', C', P} = 3" and "rk {P, Q, R} = 2" and "rk {P, R} = 2" and "rk {A', B', P, Q} = 4" shows "rk {A, B, P, R, a} \<ge> 4"
lemma ptrm_alpha_equiv_fvs: assumes "X \<approx> Y" shows "ptrm_fvs X = ptrm_fvs Y"
lemma keys_lower: "keys (lower p v) = {u\<in>keys p. u \<prec>\<^sub>t v}"
lemma satisfies_tableau_Cons: "v \<Turnstile>\<^sub>t t \<Longrightarrow> v \<Turnstile>\<^sub>e e \<Longrightarrow> v \<Turnstile>\<^sub>t (e # t)"
lemma permutations_of_set_singleton [simp]: "permutations_of_set {x} = {[x]}"
lemma fsm_transition_output[intro]: "\<And> t . t \<in> (transitions M) \<Longrightarrow> t_output t \<in> outputs M"
lemma merkle_interface_unit: "merkle_interface hash_unit blinding_of_unit merge_unit"
lemma tr_sem_equiv': assumes "\<forall>(t,t') \<in> set D. (fv t \<union> fv t') \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}" and "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}" and "ground M" and \<I>: "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>" shows "\<lbrakk>M; set D \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>; A\<rbrakk>\<^sub>s \<I> \<longleftrightarrow> (\<exists>A' \<in> set (tr A D). \<lbrakk>M; A'\<rbrakk>\<^sub>d \<I>)" (is "?P \<longleftrightarrow> ?Q")
lemma PO_l2_inv4 [iff]: "reach l2 \<subseteq> l2_inv4"
lemma oprod_monoR: assumes "ozero <o r" "s <o t" shows "r o s <o r o t" (is "?L <o ?R")
lemma sup_right_dist_test_set: "test_set A \<Longrightarrow> test_set { x \<squnion> -p \| x . x \<in> A }"
lemma Cons_shuffles_subset1: "(#) x ` shuffles xs ys \<subseteq> shuffles (x # xs) ys"
lemma (in fin_digraph) pair_fin_digraph_mk_symmetric[intro]: "pair_fin_digraph (mk_symmetric G)"
lemma get_root_node_si_is_component_unsafe: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "h \<turnstile> get_root_node_si ptr' \<rightarrow>\<^sub>r root" shows "set \|h \<turnstile> get_scdom_component ptr'\|\<^sub>r = set \|h \<turnstile> get_scdom_component root\|\<^sub>r \<or> set \|h \<turnstile> get_scdom_component ptr'\|\<^sub>r \<inter> set \|h \<turnstile> get_scdom_component root\|\<^sub>r = {}"
lemma usc_imp_limsup: fixes f :: "'a::metric_space \<Rightarrow> ereal" assumes "usc_at x0 f" assumes "x \<longlonglongrightarrow> x0" shows "f x0 \<ge> limsup (f \<circ> x)"
lemma sorted_simps: "sorted [] = True" "sorted (x # ys) = ((\<forall>y \<in> set ys. x\<le>y) \<and> sorted ys)"
lemma Oops_range_spies1: "\<lbrakk> Says Kas A (Crypt KeyA \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace>) \<in> set evs ; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> authK \<notin> range shrK \<and> authK \<in> symKeys"
lemma card_union_disjoint_fset: shows "xs \|\<inter>\| ys = {\|\|} \<Longrightarrow> card_fset (xs \|\<union>\| ys) = card_fset xs + card_fset ys"
lemma arg_uminus: assumes "z \<noteq> 0" shows "Arg (-z) = \<downharpoonright>Arg z + pi\<downharpoonleft>"
lemma seq_hoare_inv_r_3 [hoare]: "\<lbrakk> \<lbrace>p\<rbrace>Q\<^sub>1\<lbrace>p\<rbrace>\<^sub>u ; \<lbrace>p\<rbrace>Q\<^sub>2\<lbrace>q\<rbrace>\<^sub>u \<rbrakk> \<Longrightarrow> \<lbrace>p\<rbrace>Q\<^sub>1 ;; Q\<^sub>2\<lbrace>q\<rbrace>\<^sub>u"
lemma ovalidNF_wp_comb1 [wp_comb]: "\<lbrakk> ovalidNF P' f Q; ovalidNF P f Q'; \<And>s. P s \<Longrightarrow> P' s \<rbrakk> \<Longrightarrow> ovalidNF P f (\<lambda>r s. Q r s \<and> Q' r s)"
lemma idiv_ile_mono2: "\<lbrakk> 0 < m; m \<le> (n::enat) \<rbrakk> \<Longrightarrow> k div n \<le> k div m"
theorem freshEnv_idEnv: "freshEnv xs x idEnv"
lemma sublist_CONS1_E: fixes l1 l2 assumes "subseq (h # l1) l2" shows "subseq l1 l2"
lemma CARD_prod [card_simps]: "CARD('a * 'b) = card_prod CARD('a) CARD('b)"
lemma seqE_cons: assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r#rs, s\<rangle> \<Rightarrow> t" obtains ti where "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r],s\<rangle> \<Rightarrow> ti" "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs,ti\<rangle> \<Rightarrow> t"
lemma unitary11_unitary11_gen [simp]: assumes "unitary11 M" shows "unitary11_gen M"
lemma "maybe\<cdot>x\<cdot>ID = fromMaybe\<cdot>x"
lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
lemma Farkas_Lemma: fixes A :: "'a mat" and b :: "'a vec" assumes A: "A \<in> carrier_mat n nr" and b: "b \<in> carrier_vec n" shows "(\<exists> x. x \<ge> 0\<^sub>v nr \<and> A \<^sub>v x = b) \<longleftrightarrow> (\<forall> y. y \<in> carrier_vec n \<longrightarrow> A\<^sup>T \<^sub>v y \<ge> 0\<^sub>v nr \<longrightarrow> y \<bullet> b \<ge> 0)"
lemma bij_finite_const_subst_exists': assumes "finite (S::'v set)" "finite (T::('f,'v) terms)" "infinite (U::'f set)" shows "\<exists>\<sigma>::('f,'v) subst. subst_domain \<sigma> = S \<and> bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>) \<and> subst_range \<sigma> \<subseteq> ((\<lambda>c. Fun c []) ` U) - T"
lemma k_steps_to_rtrancl: assumes steps: "\<forall>i<k. \<Gamma>\<turnstile> p i \<rightarrow> p (Suc i)" shows "\<Gamma>\<turnstile>p 0\<rightarrow>\<^sup>* p k"
lemma iT_Plus_neg_Plus_ge_cut_eq: " b \<le> a \<Longrightarrow> (I \<oplus>- a) \<oplus> b = (I \<down>\<ge> a) \<oplus>- (a - b)"
lemma analz_insert_freshCryptK: "\<lbrakk>evs \<in> orb; Key K \<notin> analz (knows Spy evs); Seskey \<notin> range shrK\<rbrakk> \<Longrightarrow> (Crypt K X \<in> analz (insert (Key Seskey) (knows Spy evs))) = (Crypt K X \<in> analz (knows Spy evs))"
lemma while_denest_6: "(w * (x \<star> y)) \<star> z = z \<squnion> w * ((x \<squnion> y * w) \<star> (y * z))"
lemma evalcomb_PR_CONST[sepref_monadify_comb]: "EVAL$(PR_CONST x) \<equiv> SP (RETURN$(PR_CONST x))"
lemma mirror_elem_Min: " \<lbrakk> finite I; I \<noteq> {} \<rbrakk> \<Longrightarrow> mirror_elem (iMin I) I = Max I"
lemma fls_fresh [simp]: "a \<sharp> fls"
theorem invertible_commutes_with_inverse: "\<lbrakk> invertible a; a \<in> M \<rbrakk> \<Longrightarrow> \<eta> (inverse a) = inverse' (\<eta> a)"
lemma Par2F: fixes Q :: pi and \<alpha> :: freeRes and Q' :: pi assumes QTrans: "Q \<Longrightarrow>\<^sub>l\<alpha> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<alpha> \<prec> (P \<parallel> Q')"
lemma abc_Hoare_plus_unhalt1: "\<lbrace>P\<rbrace> (A::abc_prog) \<up> \<Longrightarrow> \<lbrace>P\<rbrace> (A [+] B) \<up>"
lemma set_ord_eq_linorder[intro?]: "eq_linorder cmp \<Longrightarrow> eq_linorder (cmp_set cmp)"
lemma fronts_cons: "fronts (x#xs) = ((#) x) ` fronts xs \<union> {[]}" (is "?l = ?r")
lemma differentiable_on_openD: "f differentiable at x" if "f differentiable_on X" "open X" "x \<in> X"
lemma lunionM_lUnionM1: "lunionM (lUnionM A) x = foldr lunionM A x" for A x
lemma bi_unique_poly_rel[transfer_rule]: "bi_unique (mat_rel R)"
lemma splitset_finite [simp]: "finite (splitset xs)"
lemma list_augment_twice: "list_augment (list_augment xs i u) j v = (list_pad_out xs (max i j))[i:=u, j:=v]"
lemma get_attribute_is_l_get_attribute [instances]: "l_get_attribute type_wf get_attribute get_attribute_locs"
lemma sliden_surj:"i < j \<Longrightarrow> surj_to (sliden i) (nset i j) {k. k \<le> (j - i)}"
lemma IVT_two_functions: fixes f :: "('a::{linear_continuum_topology, real_vector}) \<Rightarrow> ('b::{linorder_topology,real_normed_vector,ordered_ab_group_add})" assumes conts: "continuous_on {a..b} f" "continuous_on {a..b} g" and ahyp: "f a < g a" and bhyp: "g b < f b " and "a \<le> b" shows "\<exists>x\<in>{a..b}. f x = g x"
lemma (in prob_space) sum_indep_random_variable_lborel: assumes ind: "indep_var borel X borel Y" assumes [simp, measurable]: "random_variable lborel X" assumes [simp, measurable]:"random_variable lborel Y" shows "distr M lborel (\<lambda>x. X x + Y x) = convolution (distr M lborel X) (distr M lborel Y)"
lemma lspasl_a_inv: "Gamma \<and> (h1,h2\<triangleright>h0) \<and> (h3,h4\<triangleright>h1) \<longrightarrow> Delta \<Longrightarrow> (\<exists>h5. Gamma \<and> (h3,h5\<triangleright>h0) \<and> (h2,h4\<triangleright>h5) \<and> (h1,h2\<triangleright>h0) \<and> (h3,h4\<triangleright>h1)) \<longrightarrow> Delta"
lemma "(a \<and> b) = (\<not> (\<not> a \<or> \<not> b))"
lemma lcm_unique: "a dvd d \<and> b dvd d \<and> normalize d = d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"
lemma cartesian_product: fixes f::"('c \<Rightarrow> real)" fixes g::"('d \<Rightarrow> real)" assumes "finite A" assumes "finite B" shows "(\<Sum>b\<in>B. g b) * (\<Sum>a\<in>A. f a) = (\<Sum>ab\<in>A\<times>B. f (fst ab) * g (snd ab))"
lemma Status_EmptySet: "(Abs_hierauto ((@ x . True), {Abs_seqauto ({ @ x . True}, (@ x . True), {}, {})}, {}, Map.empty(@ x . True \<mapsto> {})), {@x. True},{}, @x. True) \<in> {(HA,C,E,D) \| HA C E D. Status HA C E D}"
lemma upd_bruijn_pres_distinct: assumes "sorted xs" "distinct xs" shows "distinct (upd_bruijn xs)"
lemma pass_io_set_from_pass_separator : assumes "is_separator M q1 q2 A t1 t2" and "pass_separator_ATC S A s1 t2" and "observable M" and "observable S" and "q1 \<in> states M" and "s1 \<in> states S" and "(inputs S) = (inputs M)" shows "pass_io_set (from_FSM S s1) (atc_to_io_set (from_FSM M q1) A)"
lemma union_grounding_of_cls_ground: "is_ground_clss (\<Union> (grounding_of_cls ` N))"

lemma subterm_inj_on_subset: assumes "subterm_inj_on f A" and "B \<subseteq> A" shows "subterm_inj_on f B"

lemma gcollapse_groot_None [simp]: "groot_sym t = None \<Longrightarrow> gcollapse t = None" "fst (groot t) = None \<Longrightarrow> gcollapse t = None"

lemma single_Trg_last_in_targets: assumes "Arr T" shows "[\<Lambda>.Trg (last T)] \<in> targets T"

lemma permutes_bij_inv_into: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close> fixes A :: "'a set" and B :: "'b set" assumes "p permutes A" and "bij_betw f A B" shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"

lemma fromArr_toArr [simp]: assumes "arr f" and "\<not>ide f" shows "fromArr (toArr f) = f"

lemma Con_char: shows "\<T> \<lbrace>\<frown>\<rbrace> \ \<longleftrightarrow> N.is_Cong_class \<T> \<and> N.is_Cong_class \ \<and> (\<exists>t u. t \<in> \<T> \<and> u \<in> \ \<and> t \<frown> u)"

theorem main\<^sub>K: \<open>G \<TTurnstile>\<^sub>K p \<longleftrightarrow> G \<turnstile>\<^sub>K p\<close>

lemma farkas_minkowsky_weyl_theorem: "(\<exists> X. X \<subseteq> carrier_vec n \<and> finite X \<and> P = cone X) \<longleftrightarrow> (\<exists> A nr. A \<in> carrier_mat nr n \<and> P = polyhedral_cone A)"

lemma OclOr7[simp]: "(null or null) = null"

lemma subst_lconsts_empty_subst[simp]: "subst_lconsts empty_subst = {}"

lemma (in Ring) exact3_comp_0:"\<lbrakk>R module L; R module M; R module N; f \<in> mHom R L M; g \<in> mHom R M N; exact3 R L f M g N\<rbrakk> \<Longrightarrow> compos L g f = mzeromap L N"

lemma comp_ipresIGFreshAbs: assumes "ipresIGWlsAbs hA MOD MOD'" and "ipresIGFreshAbs hA MOD MOD'" and "ipresIGFreshAbs hA' MOD' MOD''" shows "ipresIGFreshAbs (hA' o hA) MOD MOD''"

lemma atom_fresh_perm: "\<lbrakk>x \<in> Vs; y \<in> Vs\<rbrakk> \<Longrightarrow> atom x \<sharp> p \<bullet> y"

lemma ccompare_set_code [code]: "CCOMPARE('a :: ccompare set) = (case ID CCOMPARE('a) of None \<Rightarrow> None | Some _ \<Rightarrow> Some (comp_of_ords cless_eq_set cless_set))"

theorem th_3: "c_unfold (length ls) (c_fold ls) = ls"

lemma prim_glue: assumes last_neq: "us = \<epsilon> \<or> last us \<noteq> w" and unique_blocks: "\<And>bs\<^sub>1 bs\<^sub>2. glue_block w us bs\<^sub>1 \<Longrightarrow> glue_block w us bs\<^sub>2 \<Longrightarrow> concat bs\<^sub>1 = concat bs\<^sub>2 \<Longrightarrow> bs\<^sub>1 = bs\<^sub>2" shows "primitive us \<Longrightarrow> primitive (glue w us)"

lemma expands_to_divide: "trimmed G \<Longrightarrow> basis_wf basis \<Longrightarrow> (f expands_to F) basis \<Longrightarrow> (g expands_to G) basis \<Longrightarrow> ((\<lambda>x. f x / g x) expands_to F / G) basis"

lemma label_incr_0_rev [dest]: "\<lbrakk>n \<oplus> i = Label 0; i > 0\<rbrakk> \<Longrightarrow> False"

lemma (in dist_execution) state_is_associated_string: assumes "is_valid_state_id i" shows "is_certified_associated_string (set (received_messages i)) (state i)"

lemma DropToShift: fixes l i list assumes "l + i < length list" shows "(drop l list) ! i = list ! (l + i)"

lemma bconf_Try[iff]: "P,sh \<turnstile>\<^sub>b (try e\<^sub>1 catch(C V) e\<^sub>2,b) \<surd> \<longleftrightarrow> P,sh \<turnstile>\<^sub>b (e\<^sub>1,b) \<surd>"

lemma exec_Cons_1 [intro]: "P \<turnstile> (0,s,stk) \<rightarrow>* (j,t,stk') \<Longrightarrow> instr#P \<turnstile> (1,s,stk) \<rightarrow>* (1+j,t,stk')"

lemma outgoing_transitions_deterministic2: "(\<And>s a b ba aa bb bc. ((a, b), ba) |\<in>| outgoing_transitions e s \<Longrightarrow> ((aa, bb), bc) |\<in>| (outgoing_transitions e s) - {|((a, b), ba)|} \<Longrightarrow> b \<noteq> bb \<or> ba \<noteq> bc \<Longrightarrow> \<not>choice ba bc) \<Longrightarrow> deterministic e"

theorem zeta_neg_of_nat: "zeta (-of_nat n) = -of_real (bernoulli' (Suc n)) / of_nat (Suc n)"

lemma ferrack: assumes qf: "qfree p" shows "qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)" (is "_ \<and> ?rhs = ?lhs")

lemma disj_gtt_states_disj_snd_ta_states: assumes dist_st: "gtt_states \<G>\<^sub>1 |\<inter>| gtt_states \<G>\<^sub>2 = {||}" shows "\<Q> (snd \<G>\<^sub>1) |\<inter>| \<Q> (snd \<G>\<^sub>2) = {||}"

lemma "syzygy_basis DRLEX [Vec\<^sub>0 0 (X * Y - Z), Vec\<^sub>0 0 (X * Z - Y), Vec\<^sub>0 0 (Y * Z - X)] = [ Vec\<^sub>0 0 (- X * Z + Y) + Vec\<^sub>0 1 (X * Y - Z), Vec\<^sub>0 0 (- Y * Z + X) + Vec\<^sub>0 2 (X * Y - Z), Vec\<^sub>0 1 (- Y * Z + X) + Vec\<^sub>0 2 (X * Z - Y), Vec\<^sub>0 0 (Y - Y * Z ^ 2) + Vec\<^sub>0 1 (Y ^ 2 * Z - Z) + Vec\<^sub>0 2 (Y ^ 2 - Z ^ 2) ]"

lemma (in vmc_path) strategy_attracts_lset: assumes "strategy_attracts p \<sigma> A W" "v0 \<in> A" shows "lset P \<inter> W \<noteq> {}"

lemma degree_le: assumes d: "\<forall>j \<ge> d. pdevs_apply x j = 0" shows "degree x \<le> d"

lemma ternary_lift5: "eval_ternary_Not tv = TernaryUnknown \<longleftrightarrow> tv = TernaryUnknown"

lemma word_plus_mcs_4: "\<lbrakk>v + x \<le> w + x; x \<le> v + x\<rbrakk> \<Longrightarrow> v \<le> (w::'a::len word)"

lemma table_of_mapf_SomeD [dest!]: "table_of (map (\<lambda>(k,x). (k, f x)) t) k = Some z \<Longrightarrow> (\<exists>y\<in>table_of t k: z=f y)"

lemma srs: "r2s \<circ> s2r = id"

lemma bigo_add_commute_imp: "f \<in> g +o O(h) \<Longrightarrow> g \<in> f +o O(h)"

lemma word_upto_set_eq: "a \<le> b \<Longrightarrow> x \<in> set (word_upto a b) \<longleftrightarrow> a \<le> x \<and> x \<le> b"

lemma smap_union_empty2[simp]: "m \<union>. {}. = m"

lemma openin_contains_ball: "openin (top_of_set T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x \<in> S. \<exists>e. 0 < e \<and> ball x e \<inter> T \<subseteq> S)" (is "?lhs = ?rhs")

lemma "dom(map_of system_UploadDroid) \<subseteq> set (nodesL policy)"

lemma swap_cs_Trans_Snapshot: assumes "c \<turnstile> ev \<mapsto> d" and "d \<turnstile> ev' \<mapsto> e" and "isTrans ev" and "isSnapshot ev'" and "c \<turnstile> ev' \<mapsto> d'" and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"

lemma wtA_gA[simp]: "Ik.wtA at \<Longrightarrow> GE.wtA (gA at)"

lemma rk_ABPRa : assumes "rk {A, B, P} = 3" and "rk {A, B, C, A', B', C', P} = 3" and "rk {P, Q, R} = 2" and "rk {P, R} = 2" and "rk {A', B', P, Q} = 4" shows "rk {A, B, P, R, a} \<ge> 4"

lemma ptrm_alpha_equiv_fvs: assumes "X \<approx> Y" shows "ptrm_fvs X = ptrm_fvs Y"

lemma keys_lower: "keys (lower p v) = {u\<in>keys p. u \<prec>\<^sub>t v}"

lemma satisfies_tableau_Cons: "v \<Turnstile>\<^sub>t t \<Longrightarrow> v \<Turnstile>\<^sub>e e \<Longrightarrow> v \<Turnstile>\<^sub>t (e # t)"

lemma permutations_of_set_singleton [simp]: "permutations_of_set {x} = {[x]}"

lemma fsm_transition_output[intro]: "\<And> t . t \<in> (transitions M) \<Longrightarrow> t_output t \<in> outputs M"

lemma merkle_interface_unit: "merkle_interface hash_unit blinding_of_unit merge_unit"

lemma tr_sem_equiv': assumes "\<forall>(t,t') \<in> set D. (fv t \<union> fv t') \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}" and "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}" and "ground M" and \: "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \" shows "\<lbrakk>M; set D \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \; A\<rbrakk>\<^sub>s \ \<longleftrightarrow> (\<exists>A' \<in> set (tr A D). \<lbrakk>M; A'\<rbrakk>\<^sub>d \)" (is "?P \<longleftrightarrow> ?Q")

lemma PO_l2_inv4 [iff]: "reach l2 \<subseteq> l2_inv4"

lemma oprod_monoR: assumes "ozero <o r" "s <o t" shows "r *o s <o r *o t" (is "?L <o ?R")

lemma sup_right_dist_test_set: "test_set A \<Longrightarrow> test_set { x \<squnion> -p | x . x \<in> A }"

lemma Cons_shuffles_subset1: "(#) x ` shuffles xs ys \<subseteq> shuffles (x # xs) ys"

lemma (in fin_digraph) pair_fin_digraph_mk_symmetric[intro]: "pair_fin_digraph (mk_symmetric G)"

lemma get_root_node_si_is_component_unsafe: assumes "heap_is_wellformed h" and "type_wf h" and "known_ptrs h" assumes "h \<turnstile> get_root_node_si ptr' \<rightarrow>\<^sub>r root" shows "set |h \<turnstile> get_scdom_component ptr'|\<^sub>r = set |h \<turnstile> get_scdom_component root|\<^sub>r \<or> set |h \<turnstile> get_scdom_component ptr'|\<^sub>r \<inter> set |h \<turnstile> get_scdom_component root|\<^sub>r = {}"

lemma usc_imp_limsup: fixes f :: "'a::metric_space \<Rightarrow> ereal" assumes "usc_at x0 f" assumes "x \<longlonglongrightarrow> x0" shows "f x0 \<ge> limsup (f \<circ> x)"

lemma sorted_simps: "sorted [] = True" "sorted (x # ys) = ((\<forall>y \<in> set ys. x\<le>y) \<and> sorted ys)"

lemma Oops_range_spies1: "\<lbrakk> Says Kas A (Crypt KeyA \<lbrace>Key authK, Peer, Ta, authTicket\<rbrace>) \<in> set evs ; evs \<in> kerbIV \<rbrakk> \<Longrightarrow> authK \<notin> range shrK \<and> authK \<in> symKeys"

lemma card_union_disjoint_fset: shows "xs |\<inter>| ys = {||} \<Longrightarrow> card_fset (xs |\<union>| ys) = card_fset xs + card_fset ys"

lemma arg_uminus: assumes "z \<noteq> 0" shows "Arg (-z) = \<downharpoonright>Arg z + pi\<downharpoonleft>"

lemma seq_hoare_inv_r_3 [hoare]: "\<lbrakk> \<lbrace>p\<rbrace>Q\<^sub>1\<lbrace>p\<rbrace>\<^sub>u ; \<lbrace>p\<rbrace>Q\<^sub>2\<lbrace>q\<rbrace>\<^sub>u \<rbrakk> \<Longrightarrow> \<lbrace>p\<rbrace>Q\<^sub>1 ;; Q\<^sub>2\<lbrace>q\<rbrace>\<^sub>u"

lemma ovalidNF_wp_comb1 [wp_comb]: "\<lbrakk> ovalidNF P' f Q; ovalidNF P f Q'; \<And>s. P s \<Longrightarrow> P' s \<rbrakk> \<Longrightarrow> ovalidNF P f (\<lambda>r s. Q r s \<and> Q' r s)"

lemma idiv_ile_mono2: "\<lbrakk> 0 < m; m \<le> (n::enat) \<rbrakk> \<Longrightarrow> k div n \<le> k div m"

theorem freshEnv_idEnv: "freshEnv xs x idEnv"

lemma sublist_CONS1_E: fixes l1 l2 assumes "subseq (h # l1) l2" shows "subseq l1 l2"

lemma CARD_prod [card_simps]: "CARD('a * 'b) = card_prod CARD('a) CARD('b)"

lemma seqE_cons: assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r#rs, s\<rangle> \<Rightarrow> t" obtains ti where "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r],s\<rangle> \<Rightarrow> ti" "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs,ti\<rangle> \<Rightarrow> t"

lemma unitary11_unitary11_gen [simp]: assumes "unitary11 M" shows "unitary11_gen M"

lemma "maybe\<cdot>x\<cdot>ID = fromMaybe\<cdot>x"

lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"

lemma Farkas_Lemma: fixes A :: "'a mat" and b :: "'a vec" assumes A: "A \<in> carrier_mat n nr" and b: "b \<in> carrier_vec n" shows "(\<exists> x. x \<ge> 0\<^sub>v nr \<and> A *\<^sub>v x = b) \<longleftrightarrow> (\<forall> y. y \<in> carrier_vec n \<longrightarrow> A\<^sup>T *\<^sub>v y \<ge> 0\<^sub>v nr \<longrightarrow> y \<bullet> b \<ge> 0)"

lemma bij_finite_const_subst_exists': assumes "finite (S::'v set)" "finite (T::('f,'v) terms)" "infinite (U::'f set)" shows "\<exists>\<sigma>::('f,'v) subst. subst_domain \<sigma> = S \<and> bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>) \<and> subst_range \<sigma> \<subseteq> ((\<lambda>c. Fun c []) ` U) - T"

lemma k_steps_to_rtrancl: assumes steps: "\<forall>i<k. \<Gamma>\<turnstile> p i \<rightarrow> p (Suc i)" shows "\<Gamma>\<turnstile>p 0\<rightarrow>\<^sup>* p k"

lemma iT_Plus_neg_Plus_ge_cut_eq: " b \<le> a \<Longrightarrow> (I \<oplus>- a) \<oplus> b = (I \<down>\<ge> a) \<oplus>- (a - b)"

lemma analz_insert_freshCryptK: "\<lbrakk>evs \<in> orb; Key K \<notin> analz (knows Spy evs); Seskey \<notin> range shrK\<rbrakk> \<Longrightarrow> (Crypt K X \<in> analz (insert (Key Seskey) (knows Spy evs))) = (Crypt K X \<in> analz (knows Spy evs))"

lemma while_denest_6: "(w * (x \<star> y)) \<star> z = z \<squnion> w * ((x \<squnion> y * w) \<star> (y * z))"

lemma evalcomb_PR_CONST[sepref_monadify_comb]: "EVAL$(PR_CONST x) \<equiv> SP (RETURN$(PR_CONST x))"

lemma mirror_elem_Min: " \<lbrakk> finite I; I \<noteq> {} \<rbrakk> \<Longrightarrow> mirror_elem (iMin I) I = Max I"

lemma fls_fresh [simp]: "a \<sharp> fls"

theorem invertible_commutes_with_inverse: "\<lbrakk> invertible a; a \<in> M \<rbrakk> \<Longrightarrow> \<eta> (inverse a) = inverse' (\<eta> a)"

lemma Par2F: fixes Q :: pi and \<alpha> :: freeRes and Q' :: pi assumes QTrans: "Q \<Longrightarrow>\<^sub>l\<alpha> \<prec> Q'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l\<alpha> \<prec> (P \<parallel> Q')"

lemma abc_Hoare_plus_unhalt1: "\<lbrace>P\<rbrace> (A::abc_prog) \<up> \<Longrightarrow> \<lbrace>P\<rbrace> (A [+] B) \<up>"

lemma set_ord_eq_linorder[intro?]: "eq_linorder cmp \<Longrightarrow> eq_linorder (cmp_set cmp)"

lemma fronts_cons: "fronts (x#xs) = ((#) x) ` fronts xs \<union> {[]}" (is "?l = ?r")

lemma differentiable_on_openD: "f differentiable at x" if "f differentiable_on X" "open X" "x \<in> X"

lemma lunionM_lUnionM1: "lunionM (lUnionM A) x = foldr lunionM A x" for A x

lemma bi_unique_poly_rel[transfer_rule]: "bi_unique (mat_rel R)"

lemma splitset_finite [simp]: "finite (splitset xs)"

lemma list_augment_twice: "list_augment (list_augment xs i u) j v = (list_pad_out xs (max i j))[i:=u, j:=v]"

lemma get_attribute_is_l_get_attribute [instances]: "l_get_attribute type_wf get_attribute get_attribute_locs"

lemma sliden_surj:"i < j \<Longrightarrow> surj_to (sliden i) (nset i j) {k. k \<le> (j - i)}"

lemma IVT_two_functions: fixes f :: "('a::{linear_continuum_topology, real_vector}) \<Rightarrow> ('b::{linorder_topology,real_normed_vector,ordered_ab_group_add})" assumes conts: "continuous_on {a..b} f" "continuous_on {a..b} g" and ahyp: "f a < g a" and bhyp: "g b < f b " and "a \<le> b" shows "\<exists>x\<in>{a..b}. f x = g x"

lemma (in prob_space) sum_indep_random_variable_lborel: assumes ind: "indep_var borel X borel Y" assumes [simp, measurable]: "random_variable lborel X" assumes [simp, measurable]:"random_variable lborel Y" shows "distr M lborel (\<lambda>x. X x + Y x) = convolution (distr M lborel X) (distr M lborel Y)"

lemma lspasl_a_inv: "Gamma \<and> (h1,h2\<triangleright>h0) \<and> (h3,h4\<triangleright>h1) \<longrightarrow> Delta \<Longrightarrow> (\<exists>h5. Gamma \<and> (h3,h5\<triangleright>h0) \<and> (h2,h4\<triangleright>h5) \<and> (h1,h2\<triangleright>h0) \<and> (h3,h4\<triangleright>h1)) \<longrightarrow> Delta"

lemma "(a \<and> b) = (\<not> (\<not> a \<or> \<not> b))"

lemma lcm_unique: "a dvd d \<and> b dvd d \<and> normalize d = d \<and> (\<forall>e. a dvd e \<and> b dvd e \<longrightarrow> d dvd e) \<longleftrightarrow> d = lcm a b"

lemma cartesian_product: fixes f::"('c \<Rightarrow> real)" fixes g::"('d \<Rightarrow> real)" assumes "finite A" assumes "finite B" shows "(\<Sum>b\<in>B. g b) * (\<Sum>a\<in>A. f a) = (\<Sum>ab\<in>A\<times>B. f (fst ab) * g (snd ab))"

lemma Status_EmptySet: "(Abs_hierauto ((@ x . True), {Abs_seqauto ({ @ x . True}, (@ x . True), {}, {})}, {}, Map.empty(@ x . True \<mapsto> {})), {@x. True},{}, @x. True) \<in> {(HA,C,E,D) | HA C E D. Status HA C E D}"

lemma upd_bruijn_pres_distinct: assumes "sorted xs" "distinct xs" shows "distinct (upd_bruijn xs)"

lemma pass_io_set_from_pass_separator : assumes "is_separator M q1 q2 A t1 t2" and "pass_separator_ATC S A s1 t2" and "observable M" and "observable S" and "q1 \<in> states M" and "s1 \<in> states S" and "(inputs S) = (inputs M)" shows "pass_io_set (from_FSM S s1) (atc_to_io_set (from_FSM M q1) A)"

lemma union_grounding_of_cls_ground: "is_ground_clss (\<Union> (grounding_of_cls ` N))"