hcju/isabelleps huggingface dataset:
distributed_emeasure: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> ennreal"   and A :: "'d set" assumes "distributed M N X f"   and "A \<in> sets N" shows "emeasure M (X -` A \<inter> space M) = set_nn_integral N A f"
bin_last_conv_AND: fixes i :: "int" shows "odd i = (i AND 1 \<noteq> 0)"
simps_1: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c" shows "bts_dfs f [] = []"
zariski_open_is_subset: fixes U :: "'a set set" assumes "is_zariski_open U" shows "U \<subseteq> Spec"
cases: 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 C :: "char list"   and a :: "char list list \<times> (char list \<Rightarrow> val option)" assumes "init_obj P C a" obtains Cs :: "char list list" where "a = (Cs, init_class_fieldmap P (last Cs))" and "Subobjs P C Cs"
no_id: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b" assumes "valid_unMultigraph G" shows "(v, w, v) \<notin> edges G"
assms: shows "invalidate rt dests dip = Some (dsn, dsk, flag, hops, nhip, pre)"
inf_llist_def: fixes xs :: "'a llist"   and ys :: "'a llist" shows "xs \<sqinter> ys = unfold_llist (\<lambda>(xs, ys). xs \<noteq> LNil \<longrightarrow> ys \<noteq> LNil \<longrightarrow> lhd xs \<noteq> lhd ys) (lhd \<circ> snd) (map_prod ltl ltl) (xs, ys)"
sound_FAll: fixes u :: "nat"   and f :: "form"   and s :: "form list" assumes "u \<notin> set (sfv (FAll f # s))"   and "Svalid (s @ [finst f u])" shows "Svalid (FAll f # s)"
map_add_right_dom_eq: fixes m\<^sub>0 :: "'c \<Rightarrow> 'd option"   and m\<^sub>1 :: "'c \<Rightarrow> 'd option"   and m\<^sub>0' :: "'c \<Rightarrow> 'd option"   and m\<^sub>1' :: "'c \<Rightarrow> 'd option" assumes "m\<^sub>0 ++ m\<^sub>1 = m\<^sub>0' ++ m\<^sub>1'"   and "dom m\<^sub>1 = dom m\<^sub>1'" shows "m\<^sub>1 = m\<^sub>1'"
LPD_fv_regex_le: fixes ms :: "mregex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list" assumes "ms \<in> LPD mr" shows "fv_regex (from_mregex ms \<phi>s) \<subseteq> fv_regex (from_mregex mr \<phi>s)"
jm: shows "jm \<in> DeclConcepts.imethds G J sig"
measurable_left: fixes M :: "'d measure" shows "left \<in> tree_sigma M \<rightarrow>\<^sub>M tree_sigma M"
norm_cross_dot: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "(norm (x \<times> y))\<^sup>2 + (x \<bullet> y)\<^sup>2 = (norm x * norm y)\<^sup>2"
fun_upd_same: fixes f :: "'h \<Rightarrow> 'g"   and x :: "'h"   and y :: "'g" shows "(f(x := y)) x = y"
unadjoin_unique: fixes y :: "'a"   and h :: "nat" assumes "y \<in> H"   and "h < subgroup_indicator G H a" shows "unadjoin (y \<otimes> a [^] h) = (y, h)"
Un_assoc: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
x: shows "x \<in> {(hd vs, ram1), (ram1, ram2), (ram2, last vs)} \<union> Edges (rev vs)"
\<nu>\<upsilon>_\<omega>\<nu>_is_\<omega>\<upsilon>: fixes x :: "\<omega>" shows "\<nu>\<upsilon> (\<omega>\<nu> x) = \<omega>\<upsilon> x"
Ball_set_list_all: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
absorb_iff2: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c" shows "less_eq_prod b a = (inf_prod a b = b)"
FinalAllows_subseteq_in_doubt_allow: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs" shows "_Collect p (\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow) \<subseteq> _Collect p ((\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow)"
AE_measure_singleton: fixes M :: "'b measure"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "Sigma_Algebra.measure M {x} \<noteq> 0"   and "almost_everywhere M P" shows "P x"
refine: fixes mrel :: "('c \<times> 'd) set"   and node_rel :: "('e \<times> 'f) set"   and node_eq_impl :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'e \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('e \<Rightarrow> bool, 'e \<Rightarrow> 'e list, 'e list, 'e \<Rightarrow> integer, 'c) gen_igbg_impl_scheme"   and G :: "('f, 'd) igb_graph_rec_scheme" assumes "impl_lasso_loc mrel node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G" shows "(find_ce_code node_eq_impl node_hash_impl node_def_hash_size G_impl, igb_fr_graph.find_ce_impl G) \<in> \<langle>ce_rel node_rel\<rangle>nres_rel"
assms_1: assumes "X = {}" shows "d \<le> SPEC P"
le_cases3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" obtains "x \<sqsubseteq> y" and "y \<sqsubseteq> z"   | "y \<sqsubseteq> x" and "x \<sqsubseteq> z"   | "x \<sqsubseteq> z" and "z \<sqsubseteq> y"   | "z \<sqsubseteq> y" and "y \<sqsubseteq> x"   | "y \<sqsubseteq> z" and "z \<sqsubseteq> x"   | "z \<sqsubseteq> x" and "x \<sqsubseteq> y"
exp_first_term: fixes x :: "'a" shows "exp x = (1::'a) + (\<Sum>n. x ^ Suc n /\<^sub>R fact (Suc n))"
add_vert_simps_1: fixes u :: "'a" shows "verts (add_vert u) = insert u (verts G)"
th: shows "\<forall>x. Ifm vs (x # bs) (minusinf p)"
labeled_tiebreaker_lifting: fixes q :: "'q" assumes "q \<in> Q" shows "tiebreaker_lifting Bot_FL Inf_FL Bot_G (entails_q q) (Inf_G_q q) (Red_I_q q) (Red_F_q q) (\<G>_F_L_q q) (\<G>_I_L_q q) (\<lambda>g. (\<sqsubset>))"
sbintrunc_numeral: fixes k :: "num"   and x :: "int" shows "signed_take_bit (numeral k) x = of_bool (odd x) + 2 * signed_take_bit (pred_numeral k) (x div 2)"
hpair_def: fixes a :: "hf"   and b :: "hf" shows "\<langle>a, b\<rangle> = \<lbrace>\<lbrace>a\<rbrace>, \<lbrace>a, b\<rbrace>\<rbrace>"
substitutes_on_def2: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "substitutes_on A f = (\<forall>B\<subseteq>A. \<forall>a\<in>A. \<forall>b\<in>A. b \<notin> f (insert b B) \<longrightarrow> b \<notin> f (insert a (insert b B)))"
dvd_div_eq_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semidom_divide divide plus minus zero times one"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "(divide a c = divide b c) = (a = b)"
arccos_arctan: fixes x :: "real" assumes "- 1 < x"   and "x < 1" shows "arccos x = pi / 2 - arctan (x / sqrt (1 - x\<^sup>2))"
count_image_mset_ge_count: fixes A :: "'b multiset"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "count A b \<le> count (image_mset f A) (f b)"
prems_2: shows "lin_ts_mmuaux aux = (ts # tss) @ tss'"
suc_greater: fixes B :: "'a set"   and b :: "'a" assumes "B \<subseteq> Field r"   and "local.AboveS B \<noteq> {}"   and "b \<in> B" shows "suc B \<noteq> b \<and> (b, suc B) \<in> r"
isVal_eta_expand: fixes e :: "exp"   and n :: "nat" assumes "isVal e" shows "isVal (eta_expand n e)"
reduced_row_echelon_form_upt_k_intro: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'b) vec" assumes "\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)"   and "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'c)) = (1::'c)"   and "\<forall>i. i < i + (1::'b) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'b)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'c)) < (LEAST n. A $ (i + (1::'b)) $ n \<noteq> (0::'c))"   and "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'c)) = (0::'c))" shows "reduced_row_echelon_form_upt_k A k"
hd_add: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Add $$ e \<Rightarrow> (\<lambda>\<sigma> z. \<integral>\<^sup>+ x. f \<sigma> <|x, op_sem Add <|z, op_sem Minus x|>|> \<partial>stock_measure (val_type z))"
assms_2: shows "vwalk (p \<oplus> q) G"
Inf_pres_radj_aux: fixes g :: "'b \<Rightarrow> 'c" assumes "Inf_pres g" shows "ladj g \<stileturn> g"
bisim_final: fixes s1 :: "'s1"   and s2 :: "'s2" assumes "s1 \<approx> s2" shows "final1 s1 = final2 s2"
lta__lea: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F"
inv_ide: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "VV.inv a = a"
bl_or_aux_bin: fixes n :: "nat"   and v :: "int"   and bs :: "bool list"   and w :: "int"   and cs :: "bool list" shows "map2 (\<or>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = bin_to_bl_aux n (v OR w) (map2 (\<or>) bs cs)"
hrestrict_def: fixes r :: "hf"   and A :: "hf" shows "hrestrict r A = \<lbrace>z \<^bold>\<in> r. \<exists>x\<^bold>\<in>A. \<exists>y. z = \<langle>x, y\<rangle>\<rbrace>"
representation_basis: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set"   and b :: "'d" assumes "module scale"   and "\<not> module.dependent scale basis"   and "b \<in> basis" shows "module.representation scale basis b = (\<lambda>v. if v = b then 1::'c else (0::'c))"
simps: fixes v :: "'d option"   and m :: "('c \<times> ('c, 'd) Trie.trie) list" shows "is_empty_trie (Trie.trie.Trie v m) = (v = None \<and> m = [])"
integrable_restrict_UNIV: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. if x \<in> s then f x else (0::'b)) integrable_on UNIV) = (f integrable_on s)"
diagonalize_2x2_works: fixes P :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and D :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and Q :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec"   and A :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" assumes "(P, D, Q) = diagonalize_2x2 A" shows "D = P ** A ** Q \<and> invertible P \<and> invertible Q \<and> isDiagonal D"
not_le_imp_less: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c" assumes "class.linorder less_eq less"   and "\<not> less_eq y x" shows "less x y"
assign_doesnt_read: fixes y :: "'Var"   and e :: "'AExp"   and x :: "'Var"   and annos :: "'Var ModeUpd list" assumes "y \<notin> aexp_vars e" shows "doesnt_read ((x \<leftarrow> e) \<otimes> annos) y"
Rep_bit1_def: fixes x :: "'a bit1" shows "Rep x = Rep_bit1 x"
e: shows "map_of (effect_to_assignments op) = map_of as\<^sub>2 ++ map_of as\<^sub>1"
set3_F_map: fixes l1 :: "'n \<Rightarrow> 'y"   and l2 :: "'o \<Rightarrow> 'z"   and l3 :: "'p \<Rightarrow> 'x" shows "set3_F \<circ> mapl_F l1 l2 l3 = (`) l3 \<circ> set3_F"
\<Sigma>_protocols_base_axioms: shows "\<Sigma>_protocols_base Rel valid_pub"
induct: fixes P :: "('a list \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a list \<Rightarrow> bool"   and a1 :: "nat" assumes "\<And>Pa n. (\<And>x. n \<noteq> 0 \<Longrightarrow> P (\<lambda>xs. Pa (x # xs)) (n - 1)) \<Longrightarrow> P Pa n" shows "P a0 a1"
C: shows "SesKey SK \<in> spied s \<longrightarrow> (\<exists>S A C. fst SK = Some S \<and> A \<in> fst (snd SK) \<and> C \<in> snd (snd SK) \<and> {PriKey S, PriKey A, PriKey C} \<subseteq> spied s)"
add_0_right: fixes m :: "nat" shows "m + zero_class.zero = m"
iadd_Suc: fixes m :: "eint"   and n :: "eint" shows "m + 1 + n = m + n + 1"
D: shows "P y \<longrightarrow> (\<exists>u v. (x, u) \<in> r\<^sup>* \<and> (u, v) \<in> r \<and> (v, y) \<in> r\<^sup>* \<and> \<not> P u \<and> P v)"
Constrains_weaken_L: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set" assumes "F \<in> A Co A'"   and "B \<subseteq> A" shows "F \<in> B Co A'"
dagger_denest: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<dagger> = (x\<^sup>\<dagger> \<cdot> y)\<^sup>\<dagger> \<cdot> x\<^sup>\<dagger>"
Edges_def: fixes vs :: "nat list" shows "Edges vs \<equiv> {(a, b). is_sublist [a, b] vs}"
run_ipath: fixes r :: "nat \<Rightarrow> 'a" assumes "a.is_run r" shows "ipath a.E r"
simps_1: fixes k :: "nat" shows "[] \<odot>\<^bsub>fSuc\<^esub> k = []"
not_take_bit_negative: fixes n :: "nat"   and k :: "int" shows "\<not> take_bit n k < 0"
Abs_rename_res: fixes p :: "perm"   and bs :: "atom set"   and x :: "'a" assumes "(p \<bullet> bs) \<sharp>* x" shows "\<exists>q. [bs]res. x = [p \<bullet> bs]res. q \<bullet> x \<and> q \<bullet> bs = p \<bullet> bs"
Rep_regular_inject: fixes x :: "'c regular"   and y :: "'c regular" shows "(Rep_regular x = Rep_regular y) = (x = y)"
resize_vec_carrier: fixes n :: "nat"   and v :: "'c vec" shows "resize_vec n v \<in> carrier_vec n"
assms: shows "\<forall>t. (M \<turnstile> t) = (M' \<turnstile> t)"
replicate_mset_Suc: fixes n :: "nat"   and x :: "'b" shows "replicate_mset (Suc n) x = add_mset x (replicate_mset n x)"
prems_2: assumes "vs0 (ad, al) = \<lfloor>(v, b)\<rfloor>" shows "\<exists>wa. wa \<in> set (obs @ [ob]) \<and> is_write_action wa \<and> (ad, al) \<in> action_loc_aux P wa \<and> (b \<longrightarrow> \<not> is_new_action wa)"
fmlookup_drop: fixes a :: "'d"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmdrop a m $$ x = (if x \<noteq> a then m $$ x else None)"
card_of_Card_order: fixes A :: "'d set" shows "Card_order (card_of A)"
w_reach_1: shows "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w"
cases: fixes x :: "'a Trace" obtains s :: "'a" where "x = tInit s__"   | t :: "'a Trace" and s :: "'a" where "x = t \<leadsto> s"
monom_independent: fixes p :: "nat" shows "poly_vs.independent {monom (1::'b) x |x. x \<le> p}"
mpoly_base_conv_3: fixes x :: "'a" shows "x = poly [:0::'a, 1::'a:] x"
accfield_fields: fixes G :: "prog"   and S :: "qtname"   and C :: "qtname"   and fn :: "vname"   and f :: "qtname \<times> Decl.field" assumes "accfield G S C fn = Some f" shows "table_of (DeclConcepts.fields G C) (fn, declclass f) = Some (fld f)"
szip_shift: fixes u :: "'b list"   and s :: "'c list"   and v :: "'b stream"   and t :: "'c stream" assumes "length u = length s" shows "u @- v ||| s @- t = (u || s) @- (v ||| t)"
abs_eq: fixes xb :: "'a \<Rightarrow> 'b option"   and xa :: "'a"   and x :: "'b" shows "Abs_pfun xb(xa \<mapsto> x)\<^sub>p = Abs_pfun (xb(xa \<mapsto> x))"
w_cmpser_pre: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "w_cmpser G (Suc n) f" shows "w_cmpser G n f"
repr1: fixes A :: "'a" shows "(A, f A) \<in> F"
partially_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "partially_ordered_set A less_eq \<equiv> reflexive A less_eq \<and> antisymmetric A less_eq \<and> transitive A less_eq"
nat_pow_Suc2: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier (UP Zp)" shows "[Suc n] \<cdot>\<^bsub>UP Zp\<^esub> x = x \<oplus>\<^bsub>UP Zp\<^esub> [n] \<cdot>\<^bsub>UP Zp\<^esub> x"
subset_nodes_inpow: fixes G :: "nat graph_rec"   and xs :: "nat list" assumes "cfg_doms G"   and "Sorted_Less.sorted (rev xs)"   and "set xs \<subseteq> set (cfg_doms.nodes G)" shows "xs \<in> (rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G))"
Nil_is_map_conv: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "([] = map f xs) = (xs = [])"
Id_in_Hom: fixes A :: "'obj" assumes "A \<in> Obj" shows "A \<in> (if A = A then {A} else {})"
distinct_161: fixes x81 :: "expr"   and x82 :: "char list"   and x83 :: "char list list"   and x121 :: "expr"   and x122 :: "expr" shows "x81\<bullet>x82{x83} \<noteq> x121;; x122"
order_antisym: fixes x :: "'b"   and y :: "'b" assumes "order.greater_eq y x"   and "order.greater_eq x y" shows "x = y"
count_mset_set_le_one: fixes A :: "'a set"   and x :: "'a" shows "count (mset_set A) x \<le> 1"
v0_v1_notin_second_vertices_1: shows "v0 \<notin> second_vertices"
assms_2: shows "\<C> \<turnstile> es' : t2s _> t3s"
iso_is_retraction: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.retraction f"
infinite_coin_toss_space_def: fixes p :: "real"   and M :: "bool stream measure" shows "infinite_coin_toss_space p M \<equiv> 0 \<le> p \<and> p \<le> 1 \<and> M = bernoulli_stream p"
simps_3: fixes t :: "'a E.term"   and u :: "'a E.term" shows "E.Trg (t \<^bold>\<star> u) = E.Trg t"
l_null: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "semiring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
fconst_onE: fixes x :: "V"   and A :: "V"   and c :: "V" assumes "x \<in>\<^sub>\<circ> fconst_on A c" obtains a :: "V" where "a \<in>\<^sub>\<circ> A" and "x = [a, c]\<^sub>\<circ>"
prems_1: shows "v \<in> wfrestrictedvars\<^sub>s\<^sub>t (S @ [send\<langle>Var w\<rangle>\<^sub>s\<^sub>t])"
simps_1: fixes t :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>t\<^sub>p (send\<langle>t\<rangle>\<^sub>s\<^sub>t) = fv t"
frontier_subset_eq: fixes S :: "'b set" shows "(frontier S \<subseteq> S) = closed S"
simple_function_compose: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f" assumes "simple_function M f" shows "simple_function M (g \<circ> f)"
even_iff_mod_2_eq_zero: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = zero)"
size_fset_from_list: fixes l :: "'b list" assumes "distinct l" shows "size (fset_from_list l) = length l"
nonintSfmla_nonintSI: fixes st0 :: "'a"   and do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and out :: "'a \<Rightarrow> 'b \<Rightarrow> 'd"   and GH :: "'b set"   and GL :: "'b set" shows "GM_sec_model.nonintSfmla st0 do out GH GL [] = GM_sec_model.nonintSI st0 do out GH GL"
appl: shows "applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>"
imp_compat_eqv: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla"   and "prv (eqv \<phi>1 \<phi>2)"   and "prv (eqv \<chi>1 \<chi>2)" shows "prv (eqv (imp \<phi>1 \<chi>1) (imp \<phi>2 \<chi>2))"
const_of_f64: fixes cs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context" assumes "const_list cs"   and "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [T_f64]" shows "\<exists>c. cs = [$C ConstFloat64 c]"
diff_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
nonprincipal_mangoldt_bound: assumes "\<chi>\<^sub>0 \<noteq> \<chi>\<^sub>0" shows "sum_upto (\<lambda>k. \<chi>\<^sub>0 k * mangoldt k / of_nat k) \<in> O(\<lambda>_. 1)"
pwd_spy_enc_mult_def: fixes n :: "nat" shows "pwd_spy_enc_mult n \<equiv> let (s, S, A, B, C, D) = pwd_spy_enc_pubk n; SK = (Some S, {A, B}, {C, D}) in (insert (Spy, Crypt (SesK SK) (Auth_PriK n \<otimes> B)) s, S, A, B, C, D)"
CondReds2F: fixes e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>2 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>false,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2\<rangle>"
a_is_gen: shows "\<forall>x>0. x \<le> p - 2 \<longrightarrow> [a ^ x \<noteq> 1] (mod p)"
Range_insert: fixes a :: "'b"   and b :: "'a"   and r :: "('b \<times> 'a) set" shows "Range (insert (a, b) r) = insert b (Range r)"
sigma_sets_subseteq: fixes A :: "'b set set"   and B :: "'b set set"   and X :: "'b set" assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
A1: fixes x :: "i"   and y :: "i" assumes "AxiomsSet7" shows "(dom x \<cong> cod y \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E (x \<cdot> y) \<^bold>\<leftarrow> dom x \<cong> cod y)"
k: shows "godel_code' ps (Suc 0) = godel_code' (butlast ps) (Suc 0) * Pi (length ps) ^ last ps"
subst_del_upd_diff: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c"   and \<sigma> :: "'c \<Rightarrow> 'c"   and v :: "('d, 'c) uexpr" assumes "x \<bowtie> y" shows "\<sigma>(y \<mapsto>\<^sub>s v) -\<^sub>s x = (\<sigma> -\<^sub>s x)(y \<mapsto>\<^sub>s v)"
pure_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.pure P \<equiv> \<forall>h h'. P h = P h'"
noipdi: shows "\<forall>l\<in>{0..j}. \<pi> l \<noteq> ipd (\<pi> 0)"
SCF_R3: fixes xs :: "'a option list"   and P :: "'a option process"   and ys :: "'a option list"   and Y :: "'a option set"   and Q :: "'a option process" assumes "xs \<in> sentences P"   and "(ys, Y) \<in> failures Q"   and "ys \<noteq> []" shows "(xs @ ys, Y) \<in> seq_comp_failures P Q"
diff_minus_eq_add: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "minus a (uminus b) = plus a b"
feasible_fixes_bot: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "feasible t" shows "t (\<lambda>s. 0) = (\<lambda>s. 0)"
fcompose_def: fixes p :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" shows "fcompose p q r = fst (fold_coeffs (\<lambda>a (c, d). (d * [:a:] + q * c, r * d)) p (0, 1))"
aeT_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis" shows "PL_Indis.aeT aval tval cval indis c = (\<forall>s. AE bT in MC_syntax.T (PL_Indis.trans aval tval cval) (c, s). sfirst (holds (PL_Indis.discrCf aval tval cval indis)) ((c, s) ## bT) < \<infinity>)"
asumTr: fixes a :: "'a"   and n :: "nat"   and m :: "nat" assumes "a \<in> carrier A" shows "aSum A (n + m) a = aSum A n a \<plusminus> aSum A m a"