hcju/isabelleps huggingface dataset:
reachable_def: fixes xs :: "('a \<times> 'b \<times> 'a) set" shows "reachable xs \<equiv> {(x, y). y \<in> set (case x of (uu_, uua_, q) \<Rightarrow> map (\<lambda>\<nu>. (q, \<nu>, \<delta> q \<nu>)) \<Sigma>)}\<^sup>* `` xs"
desc_encode: fixes \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>"   and G :: "\<Pi>\<^sub>1" shows "[\<lbrace>\<^bold>\<iota>x. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F),G\<rbrace> \<^bold>\<equiv> \<phi> G in dw]"
recurrence_time_flip_sign_1: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and T :: "real"   and x :: "'b" assumes "auto_ll_on_open f X"   and "T \<in> auto_ll_on_open.existence_ivl0 f X x"   and "auto_ll_on_open.flow0 f X x T = x" shows "- T \<in> auto_ll_on_open.existence_ivl0 f X x"
l_def: shows "l = Lcm (Polynomial.lead_coeff ` P)"
degree_le: fixes n :: "nat"   and p :: "'a poly" assumes "\<forall>i>n. poly.coeff p i = (0::'a)" shows "Polynomial.degree p \<le> n"
app_execs: fixes cs :: "com list"   and s :: "state"   and cs' :: "com list"   and s' :: "state"   and cs2 :: "com list" assumes "(cs, s) \<rightarrow>\<^sup>* (cs', s')" shows "(cs @ cs2, s) \<rightarrow>\<^sup>* (cs' @ cs2, s')"
DBM_reset_int_preservation': fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and k :: "nat"   and d :: "'c"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "dbm_int M n"   and "DBM_reset M n k d M'"   and "d \<in> \<int>"   and "k \<le> n" shows "dbm_int M' n"
components_are_iso: fixes a :: "'a arrow_of_spans_data" assumes "ide a" shows "local.iso (trg a)"
disjunctiveD: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> disjunctive.disjunctive (\<squnion>) (\<squnion>) (*)" shows "x * (y \<squnion> z) = x * y \<squnion> x * z"
heapify_snd: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "snd (heapify n xs) = drop n xs"
merges_property1: fixes G :: "nat graph_rec"   and qs :: "(nat \<times> nat list) list"   and ss :: "nat list list" assumes "dom_sl G"   and "\<forall>(q, \<tau>)\<in>set qs. q < length ss \<and> \<tau> \<in> (rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G))"   and "set ss \<subseteq> (rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G))"   and "length ss = length (cfg_doms.nodes G)"   and "distinct (map fst qs)" shows "\<forall>(q, \<tau>)\<in>set qs. merges nodes_sup qs ss ! q = \<tau> \<squnion>\<^bsub>nodes_sup\<^esub> ss ! q"
simps_2: fixes a :: "('b, 'c) Combinators" shows "member a DenyAll = (a = DenyAll)"
obj_arr_len: fixes r :: "Value" assumes "isArrV r" shows "ref (arr_len r) = r"
clml: fixes l :: "'a" assumes "l \<in> collect_locks las" shows "may_lock (ls $ l) t"
convex_rel_interior_if2: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "z \<in> rel_interior S" shows "\<forall>x\<in>affine hull S. \<exists>e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
snd_Gauss_Jordan_in_ij_PA_eq: fixes P :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b" shows "snd (Gauss_Jordan_in_ij_PA (P, A) i j) = Gauss_Jordan_in_ij A i j"
top_def: fixes L :: "('c, 'd) gorder_scheme" shows "\<top>\<^bsub>L\<^esub> = (SOME x. greatest L x (carrier L))"
FunctorCompDef: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and A :: "('a, 'c, 'e) Category_scheme"   and B :: "('b, 'd, 'f) Category_scheme"   and f :: "'c"   and g :: "'c" assumes "Ftor F : A \<longrightarrow> B"   and "f \<approx>>\<^bsub>A\<^esub> g" shows "F ## f \<approx>>\<^bsub>B\<^esub> F ## g"
PLemmas_8: fixes x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and y :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" shows "Cp (x \<oplus> y) = Cp y \<Oplus> Cp x"
sum_ideals_commute: fixes A :: "'a set"   and B :: "'a set" assumes "ideal R A"   and "ideal R B" shows "A \<minusplus> B = B \<minusplus> A"
fold_map: fixes g :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and f :: "'i \<Rightarrow> 'h"   and xs :: "'i list" shows "fold g (map f xs) = fold (g \<circ> f) xs"
VPow_le_VPow_iff: fixes a :: "V"   and b :: "V" shows "(VPow a \<subseteq>\<^sub>\<circ> VPow b) = (a \<subseteq>\<^sub>\<circ> b)"
rel_simps_92: shows "(0::'b) \<noteq> (1::'b)"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> set_mr"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> set_mr; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> set_mr; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 M k NM1 i. \<lbrakk>evs1 \<in> set_mr; P evs1; M = Merchant k; Nonce NM1 \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says M (CA i) \<lbrace>Agent M, Nonce NM1\<rbrace> # evs1)"   and "\<And>evs2 NCA i M NM1. \<lbrakk>evs2 \<in> set_mr; P evs2; Nonce NCA \<notin> used evs2; Gets (CA i) \<lbrace>Agent M, Nonce NM1\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says (CA i) M \<lbrace>sign (priSK (CA i)) \<lbrace>Agent M, Nonce NM1, Nonce NCA\<rbrace>, cert (CA i) (pubEK (CA i)) onlyEnc (priSK RCA), cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\<rbrace> # evs2)"   and "\<And>evs3 M k NM2 KM1 SKi X NM1 NCA i EKi. \<lbrakk>evs3 \<in> set_mr; P evs3; M = Merchant k; Nonce NM2 \<notin> used evs3; Key KM1 \<notin> used evs3; KM1 \<in> symKeys; Gets M \<lbrace>sign (invKey SKi) \<lbrace>Agent X, Nonce NM1, Nonce NCA\<rbrace>, cert (CA i) EKi onlyEnc (priSK RCA), cert (CA i) SKi onlySig (priSK RCA)\<rbrace> \<in> set evs3; Says M (CA i) \<lbrace>Agent M, Nonce NM1\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says M (CA i) \<lbrace>Crypt KM1 (sign (priSK M) \<lbrace>Agent M, Nonce NM2, Key (pubSK M), Key (pubEK M)\<rbrace>), Crypt EKi (Key KM1)\<rbrace> # Notes M \<lbrace>Key KM1, Agent (CA i)\<rbrace> # evs3)"   and "\<And>evs4 M k merSK merEK i KM1 NM2. \<lbrakk>evs4 \<in> set_mr; P evs4; M = Merchant k; merSK \<notin> symKeys; merEK \<notin> symKeys; Notes (CA i) (Key merSK) \<notin> set evs4; Notes (CA i) (Key merEK) \<notin> set evs4; Gets (CA i) \<lbrace>Crypt KM1 (sign (invKey merSK) \<lbrace>Agent M, Nonce NM2, Key merSK, Key merEK\<rbrace>), Crypt (pubEK (CA i)) (Key KM1)\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says (CA i) M \<lbrace>sign (priSK (CA i)) \<lbrace>Agent M, Nonce NM2, Agent (CA i)\<rbrace>, cert M merSK onlySig (priSK (CA i)), cert M merEK onlyEnc (priSK (CA i)), cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\<rbrace> # Notes (CA i) (Key merSK) # Notes (CA i) (Key merEK) # evs4)" shows "P x"
is_sorted_list_of_set_alt_def: fixes A :: "'b set"   and xs :: "'b list" shows "is_sorted_list_of_set A xs = (sorted xs \<and> distinct xs \<and> ran\<^sub>l xs = A)"
cong_diff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a"   and e :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo d e a" shows "unique_euclidean_semiring.cong modulo (minus b d) (minus c e) a"
hazard: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and t :: "Term.term"   and Q :: "'b assn" shows "G,A\<turnstile>{P \<and>. Not \<circ> type_ok G t} t\<succ> {Q}"
elims_2: fixes x :: "'a local.term" assumes "Diag x" obtains "x = \<^bold>\<I>"   | f :: "'a" where "x = \<^bold>\<langle>f__\<^bold>\<rangle>" and "C.arr f__"   | f :: "'a" and u :: "'a local.term" where "x = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> u" and "C.arr f \<and> Diag u \<and> u \<noteq> \<^bold>\<I>"
finite_rderivs: fixes r :: "'b rexp" shows "finite {\<guillemotleft>rderivs xs r\<guillemotright> |xs. True}"
pd_refl: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob" shows "IFC_def.is_pd prob x x"
alloc_def: fixes x :: "'b"   and h :: "heap" shows "Ref.alloc x h = (let l = lim h; r = Ref l in (r, Ref.set r x (h\<lparr>lim := l + 1\<rparr>)))"
f_def: shows "f = (\<lambda>x. x powr - s)"
update_bounds_id_1: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x :: "nat"   and c :: "'a" assumes "Update update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<B>\<^sub>i (update x c s) = \<B>\<^sub>i s"
evals_final: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "finals es'"
CT_holds_def: fixes g :: "'sf"   and \<theta> :: "('sf, 'sz) trm"   and \<theta>' :: "('sf, 'sz) trm" shows "CT_holds g \<theta> \<theta>' \<equiv> valid (Equals \<theta> \<theta>') \<longrightarrow> valid (Equals ($f g (singleton \<theta>)) ($f g (singleton \<theta>')))"
add_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E" shows "x + y \<in> E"
idem_overrider_def: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "idem_overrider F \<equiv> overrider F \<and> idem_overrider_axioms F"
zero_stalk_def: fixes V :: "'a set" shows "stx.zero_stalk V \<equiv> stx.class_of V \<zero>\<^bsub>V\<^esub>"
simps: fixes a1 :: "'a literal set set"   and a2 :: "'a literal set" shows "(a1 \<turnstile> a2) = ((\<exists>C S. a1 = S \<and> a2 = C \<and> C \<in> S) \<or> (\<exists>S C D k. a1 = S \<and> a2 = C - {k\<^sup>+} \<union> (D - {k\<inverse>}) \<and> (S \<turnstile> C) \<and> (S \<turnstile> D) \<and> k\<^sup>+ \<in> C \<and> k\<inverse> \<in> D))"
cnt_mono: fixes a :: "'b"   and b :: "'b"   and xs :: "'b list"   and c :: "'b" shows "cnt a (b # xs) \<le> cnt a (b # c # xs)"
combine_matrix_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "combine_matrix f \<equiv> \<lambda>A B. Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))"
lem: fixes a :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and b :: "'a" assumes "\<forall>y. (a, y) \<in> r\<^sup>+ \<longrightarrow> P a \<longrightarrow> P y" shows "((b, a) \<in> {(y, x). P x \<and> (x, y) \<in> r}\<^sup>+) = ((b, a) \<in> {(y, x). P x \<and> (x, y) \<in> r\<^sup>+})"
reds: shows "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e'',s''\<rangle>"
bi_uniqueDr: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g"   and y :: "'h"   and z :: "'h" assumes "bi_unique A"   and "A x y"   and "A x z" shows "y = z"
SUP_sup_distrib: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "sup (Sup (f ` A)) (Sup (g ` A)) = Sup ((\<lambda>a. sup (f a) (g a)) ` A)"
isTrivial_the_trivial: fixes g :: "'g"   and v :: "'val"   and vs :: "'val list"   and v' :: "'val" assumes "phi g v = Some vs"   and "isTrivialPhi g v v'" shows "the_trivial v vs = Some v'"
subterm_reduction_aux_def: fixes S :: "'a eclause set"   and t :: "'a trm" shows "subterm_reduction_aux S t = (if (trm_rep (lhs t) S \<cdot> trm_rep (rhs t) S, t) \<in> trm_ord then trm_rep (trm_rep (lhs t) S \<cdot> trm_rep (rhs t) S) S else t)"
permutation_self_in_orbit: fixes f :: "'b \<Rightarrow> 'b"   and s :: "'b" assumes "permutation f" shows "s \<in> orbit f s"
sumE: fixes u :: "hf"   and A :: "hf"   and B :: "hf" assumes "u \<^bold>\<in> A + B" obtains x :: "hf" where "x \<^bold>\<in> A" and "u = HF.Inl x"   | y :: "hf" where "y \<^bold>\<in> B" and "u = HF.Inr y"
hyps: shows "eval_poly f p = eval_poly g (poly_subst q p)"
\<beta>\<^sub>h_def: shows "\<beta>\<^sub>h = \<a>[tab\<^sub>1 h, g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (g\<^sub>0h\<^sub>1.\<phi> \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1, f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine] \<cdot> (tab\<^sub>0 g \<star> \<gamma>\<^sub>g) \<cdot> \<a>[tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> local.inv fg\<^sub>0h\<^sub>1.\<phi>"
minus_p_def: fixes p :: "int"   and x :: "int"   and y :: "int" shows "minus_p p x y \<equiv> if y \<le> x then x - y else x + p - y"
ltl_prop_unfold_equiv_gt_and_lt_1: shows "(\<sim>\<^sub>C) \<le> (\<sim>\<^sub>Q)"
foo_cases2: fixes x1a :: "'b list"   and x2a :: "'c list" shows "foo x1a x2a = (case x1a of [] \<Rightarrow> case x2a of [] \<Rightarrow> 3::'a | x2fa # x1fa \<Rightarrow> 1::'a | x2ea # x1ea \<Rightarrow> case x2a of [] \<Rightarrow> 0::'a | x4a # x3a \<Rightarrow> foo [] [])"
normal_moment_even: fixes \<sigma> :: "real"   and \<mu> :: "real"   and k :: "nat" assumes "0 < \<sigma>" shows "has_bochner_integral lborel (\<lambda>x. normal_density \<mu> \<sigma> x * (x - \<mu>) ^ (2 * k)) (fact (2 * k) / ((2 / \<sigma>\<^sup>2) ^ k * fact k))"
Y1: fixes y :: "'a vec" assumes "y \<in> Y1" shows "y $ n = (1::'a)"
exhaust: fixes y :: "'a atom" obtains x11 :: "nat" and x12 :: "'a" where "y = Leq x11 x12"   | x21 :: "nat" and x22 :: "'a" where "y = Geq x21 x22"
scopeExtPar: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<parallel> Q) \<sim>\<^sup>s P \<parallel> <\<nu>x>Q"
select_convs_3: fixes mstate_i :: "nat"   and mstate_m :: "('a, 'b) mformula"   and mstate_n :: "nat"   and more :: "'c" shows "mstate_n \<lparr>mstate_i = mstate_i, mstate_m = mstate_m, mstate_n = mstate_n, \<dots> = more\<rparr> = mstate_n"
esssup_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" shows "esssup M f = (if f \<in> borel_measurable M then Limsup (ae_filter M) f else top)"
p: shows "processes prog !! sidx = (sidx', start, argDecls, decls)"
order_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "order r = ((\<forall>x. x \<sqsubseteq>\<^bsub>r\<^esub> x) \<and> (\<forall>x y. x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> x \<longrightarrow> x = y) \<and> (\<forall>x y z. x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> z))"
ex_in_conv: fixes A :: "'d set" shows "(\<exists>x. x \<in> A) = (A \<noteq> {})"
steps: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>+ (c', s')"
comp_arr_inv: fixes f :: "'a"   and g :: "'a" assumes "inverse_arrows f g" shows "f \<cdot> g = local.dom g"
naturality_singular_subdivision: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and g :: "'c \<Rightarrow> 'd" assumes "singular_chain p X c" shows "singular_subdivision p (chain_map p g c) = chain_map p g (singular_subdivision p c)"
distinct_1: fixes vrs :: "vrs"   and ty :: "ty"   and tyvrs' :: "tyvrs"   and ty' :: "ty" shows "VarB vrs ty \<noteq> TVarB tyvrs' ty'"
Aexp_correct: fixes e :: "exp"   and a :: "Arity"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" assumes "eq\<rho> (\<A>\<^bsub>a\<^esub> e) \<rho>1 \<rho>2" shows "eq a (\<lbrakk> e \<rbrakk>\<^bsub>\<rho>1\<^esub>) (\<lbrakk> e \<rbrakk>\<^bsub>\<rho>2\<^esub>)"
deduct_if_synth: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "intruder_model.intruder_synth arity public M t" shows "intruder_model.intruder_deduct arity public Ana M t"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>\<ll>'.map f : local.map a \<Rightarrow> L b\<guillemotright>"
mset_set_subseteq_mset_set: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "A \<subseteq> B" shows "mset_set A \<subseteq># mset_set B"
simps_3: fixes r1 :: "'b rexp"   and r2 :: "'b rexp"   and c :: "'b"   and v2 :: "'b val" shows "injval (Plus r1 r2) c (val.Right v2) = val.Right (injval r2 c v2)"
0: shows "f (nat (padic_val p f) + 1) \<noteq> \<zero>\<^bsub>residue_ring (p ^ (nat (padic_val p f) + 1))\<^esub>"
eq_inf_word: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream"   and p :: "'state" assumes "w\<^sub>1 =\<^sub>I w\<^sub>2"   and "run w\<^sub>1 p" shows "run w\<^sub>2 p"
m: shows "maintainedA {reflexivity_rule I, transitive_rule I, symmetry_rule I} G"
assms_1: shows "(interval.upper \<circ> X) \<longlonglongrightarrow> x"
par_comp_hom_deduct_2: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and M :: "'lbl \<Rightarrow> ('fun, 'var) Term.term set"   and Discl :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "par_comp \<A> Sec"   and "\<forall>l. \<forall>m\<in>M l. homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec"   and "\<forall>l. M l \<subseteq> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>))"   and "\<forall>l. Discl \<subseteq> M l"   and "Discl \<subseteq> Sec \<union> {m. {} \<turnstile>\<^sub>c m}"   and "\<forall>l. \<forall>s\<in>Sec - Discl. \<not> \<langle>M l;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m s"   and "\<langle>\<Union> (range M);\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t" shows "\<forall>l. t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) \<longrightarrow> \<langle>M l;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t"
disc2_noC: shows "\<forall>a. \<not> disc2 (C a)"
a_2: shows "Pi' (domain x) a \<subseteq> O'"
lift_lower: fixes p :: "'a pfm" assumes "static p" shows "lift (lower p) = p"
bspec: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>x\<in>A. P x"   and "x \<in> A" shows "P x"
continuous_def: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'e" shows "continuous F f = (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
IH2: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "s \<in> reachable (pnet np p2) TT"   and "(s, i:deliver(d), s') \<in> automaton.trans (pnet np p2)" shows "i \<in> net_ips s"
seq_doesnt_read: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var"   and c' :: "('Var, 'AExp, 'BExp) Stmt" assumes "doesnt_read_or_modify c x" shows "doesnt_read_or_modify (c ;; c') x"
shift1_nz: fixes p :: "poly" shows "shift1 p \<noteq> 0\<^sub>p"
lucas_lehmer_hom_cong': fixes a :: "int"   and b :: "int"   and m :: "nat"   and c :: "int"   and d :: "int" assumes "[a = b] (mod int m)"   and "[c = d] (mod int m)" shows "lucas_lehmer_hom m (a, c) = lucas_lehmer_hom m (b, d)"
strong_low_bisim_neval': fixes c\<^sub>1 :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and n :: "nat"   and c\<^sub>n :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and c\<^sub>1' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set" assumes "abs.neval c\<^sub>1 n c\<^sub>n"   and "(c\<^sub>1, c\<^sub>1') \<in> \<R>\<^sub>A"   and "snd (fst c\<^sub>1) = snd (fst c\<^sub>1')"   and "abs.strong_low_bisim_mm \<R>\<^sub>A" shows "\<exists>c\<^sub>n''. abs.neval c\<^sub>1' n c\<^sub>n'' \<and> (c\<^sub>n, c\<^sub>n'') \<in> \<R>\<^sub>A \<and> snd (fst c\<^sub>n) = snd (fst c\<^sub>n'')"
4: shows "\<guillemotleft>local.inv \<xi> \<star> f : trg f \<star> f \<Rightarrow> f \<star> g \<star> f\<guillemotright>"
select_convs_3: fixes I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b" shows "N \<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = N"
distinct_5: fixes x2 :: "'c" shows "Lt x2 \<noteq> \<infinity>"
finite_subset_chain: fixes Y :: "nat \<Rightarrow> 'b set"   and S :: "'b set" assumes "chain Y"   and "S \<subseteq> \<Union> (range Y)"   and "finite S" shows "\<exists>i. S \<subseteq> Y i"
c: fixes v :: "'variable" assumes "v \<in> set (precondition_of op')" shows "s v = Some True"
left_commute: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and b :: "'k"   and a :: "'k"   and c :: "'k" assumes "abel_semigroup f" shows "f b (f a c) = f a (f b c)"
SMP_wf_trm: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set" assumes "typed_model arity public Ana \<Gamma>"   and "t \<in> typed_model.SMP arity Ana \<Gamma> M"   and "\<forall>t\<in>M. intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t" shows "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t"
rel_eq: shows "rel = foldr (\<union>) (concat (map (\<lambda>(t, rel). if left I \<le> \<tau> \<sigma> ne - t then [rel] else []) aux')) {}"
weakTransitionStatEq: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "\<Psi> \<simeq> \<Psi>'" shows "\<Psi>' : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"
fmfilter_false: fixes m :: "('b, 'c) fmap"   and P :: "'b \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> \<not> P x" shows "fmfilter P m = fmempty"
cart_basis_Rm: shows "cart_basis_Rm \<equiv> cart_basis'"
pdevs_apply_pdev_upd: fixes X :: "'b pdevs"   and n :: "nat"   and x :: "'b" shows "pdevs_apply (pdev_upd X n x) = (pdevs_apply X)(n := x)"
simps_1: fixes ys :: "'k list" shows "subset_eq_mset_impl [] ys = Some (ys \<noteq> [])"
assms_2: shows "p_iiface p \<in> set ifs"
less: shows "\<bar>real_of_int t - real u * x\<bar> < 1 / real s"
better_LAxR_intro: fixes M :: "trm"   and a :: "coname"   and x :: "name"   and b :: "coname" assumes "fic M a" shows "Cut <a>.M x.Ax x b \<longrightarrow>\<^sub>l M[a\<turnstile>c>b]"
right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier \<G>"   and "y \<in> carrier \<G>"   and "z \<in> carrier \<G>" shows "(y \<otimes> x = z \<otimes> x) = (y = z)"
comp_null_1: fixes f :: "'a" shows "null \<cdot> f = null"
consistent_extend: fixes S :: "('a, 'b) fm set"   and f :: "nat \<Rightarrow> ('a, 'b) fm"   and n :: "nat" assumes "consistent S"   and "finite (params S)"   and "infinite UNIV" shows "consistent (extend S f n)"
delete_invalid_block_eq: fixes b :: "'a set" assumes "b \<notin># \<B>" shows "del_block b = \<B>"