hcju/isabelleps huggingface dataset:
simps: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter"   and z :: "'c"   and w :: "'d Filters.filter" assumes "triple phi" shows "triple.pairs_inf (x, y) (z, w) = (x \<sqinter> z, y \<squnion> w)"
iTILL_iff: fixes x :: "nat"   and n :: "nat" shows "(x \<in> [\<dots>n]) = (x \<le> n)"
eq_closure_of_def: fixes S :: "('c, 'd) eq_object_scheme"   and A :: "'c set" shows "closure_of\<^bsub>S\<^esub> A = {y \<in> carrier S. y .\<in>\<^bsub>S\<^esub> A}"
nat_pow_0: fixes x :: "nat \<Rightarrow> 'a" shows "x [^]\<^bsub>P\<^esub> 0 = \<one>\<^bsub>P\<^esub>"
read_tmps_append: fixes xs :: "'b memref list"   and ys :: "'b memref list" shows "read_tmps (xs @ ys) = read_tmps xs \<union> read_tmps ys"
induct: fixes P :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c) \<Rightarrow> ('c \<Rightarrow> bool) \<Rightarrow> 'b match_expr \<Rightarrow> bool"   and a0 :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and a1 :: "'c \<Rightarrow> bool"   and a2 :: "'b match_expr" assumes "\<And>uu_ uv_. P uu_ uv_ MatchAny"   and "\<And>disc sel n P. P (disc, sel) n (Match P)"   and "\<And>disc sel n P. P (disc, sel) n (MatchNot (Match P))"   and "\<And>disc sel n m1 m2. \<lbrakk>P (disc, sel) n m1; P (disc, sel) n m2\<rbrakk> \<Longrightarrow> P (disc, sel) n (MatchAnd m1 m2)"   and "\<And>uw_ ux_ uy_ uz_. P uw_ ux_ (MatchNot (MatchAnd uy_ uz_))"   and "\<And>va_ vb_ vc_. P va_ vb_ (MatchNot (MatchNot vc_))"   and "\<And>vd_ ve_. P vd_ ve_ (MatchNot MatchAny)" shows "P a0 a1 a2"
part_eq: fixes x :: "i"   and y :: "i" shows "P x y = (PP x y \<or> x = y)"
shows_prec_arctic_delta_def: fixes d :: "nat"   and ari :: "'a arctic_delta" shows "shows_prec d ari = local.shows_arctic_delta ari"
simps_1: shows "add_match_MatchAnd_foldr [] = id"
elims: fixes x :: "trm option"   and xa :: "trm option"   and y :: "trm option" assumes "Timeso x xa = y" obtains \<theta> :: "trm" and \<eta> :: "trm" where "x = Aterm \<theta>" and "xa = Aterm \<eta>__" and "y = Aterm (Times \<theta> \<eta>__)"   | \<eta> :: "trm option" where "x = undeft" and "xa = \<eta>" and "y = undeft"   | v :: "trm" where "x = Aterm v" and "xa = undeft" and "y = undeft"
times_assoc_eqpoll: fixes A :: "'a set"   and B :: "'b set"   and C :: "'c set" shows "(A \<times> B) \<times> C \<approx> A \<times> B \<times> C"
strong_low_bisim_neval: fixes c\<^sub>1 :: "'Com\<^sub>A"   and mds\<^sub>1 :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem\<^sub>1 :: "'Var\<^sub>A \<Rightarrow> 'Val"   and n :: "nat"   and c\<^sub>n :: "'Com\<^sub>A"   and mds\<^sub>n :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem\<^sub>n :: "'Var\<^sub>A \<Rightarrow> 'Val"   and c\<^sub>1' :: "'Com\<^sub>A"   and mem\<^sub>1' :: "'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 \<langle>c\<^sub>1, mds\<^sub>1, mem\<^sub>1\<rangle>\<^sub>A n \<langle>c\<^sub>n, mds\<^sub>n, mem\<^sub>n\<rangle>\<^sub>A"   and "(\<langle>c\<^sub>1, mds\<^sub>1, mem\<^sub>1\<rangle>\<^sub>A, \<langle>c\<^sub>1', mds\<^sub>1, mem\<^sub>1'\<rangle>\<^sub>A) \<in> \<R>\<^sub>A"   and "abs.strong_low_bisim_mm \<R>\<^sub>A" shows "\<exists>c\<^sub>n' mem\<^sub>n'. abs.neval \<langle>c\<^sub>1', mds\<^sub>1, mem\<^sub>1'\<rangle>\<^sub>A n \<langle>c\<^sub>n', mds\<^sub>n, mem\<^sub>n'\<rangle>\<^sub>A \<and> (\<langle>c\<^sub>n, mds\<^sub>n, mem\<^sub>n\<rangle>\<^sub>A, \<langle>c\<^sub>n', mds\<^sub>n, mem\<^sub>n'\<rangle>\<^sub>A) \<in> \<R>\<^sub>A"
bind_put: fixes s :: "'b"   and m :: "'m"   and f :: "'s \<Rightarrow> 'm" shows "bind (put s m) f = put s (bind m f)"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a' \<rightarrow>\<^sub>A a\<guillemotright>"   and "B.iso (F f)" shows "A.iso f"
simps_1: shows "eval_ternary_Imp TernaryTrue TernaryTrue = TernaryTrue"
submodule_sc_closed: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and a :: "'f"   and h :: "'e" assumes "R module M"   and "submodule R M H"   and "a \<in> carrier R"   and "h \<in> H" shows "a \<cdot>\<^sub>s\<^bsub>M\<^esub> h \<in> H"
brnL_mono2: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n1 :: "nat"   and n2 :: "nat" assumes "properL cl"   and "n1 < n2"   and "n2 \<le> length cl" shows "brnL cl n1 < brnL cl n2"
space_uniform_count_measure: fixes A :: "'b set" shows "space (uniform_count_measure A) = A"
CCexp_pre_simps_1: fixes x :: "Vars.var"   and n :: "Arity" shows "ccExp_syn n (Var x) = \<bottom>"
mem_fgs_l_comb: fixes K :: "'a set"   and x :: "'a" assumes "K \<noteq> {}"   and "K \<subseteq> carrier M"   and "x \<in> fgs R M K" shows "\<exists>n. \<exists>f\<in>{j. j \<le> n} \<rightarrow> K. \<exists>s\<in>{j. j \<le> n} \<rightarrow> carrier R. x = l_comb R M n s f"
return_app_return: fixes f :: "'f \<Rightarrow> ['d| 'e]"   and x :: "'f" shows "\<langle>f\<rangle> . \<langle>x\<rangle> = f x"
leI: fixes x :: "'c"   and y :: "'c" assumes "\<not> x < y" shows "y \<le> x"
rel_fset_alt: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d fset"   and b :: "'e fset" shows "rel_fset R a b = ((\<forall>t\<in>fset a. \<exists>u\<in>fset b. R t u) \<and> (\<forall>t\<in>fset b. \<exists>u\<in>fset a. R u t))"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> b \<le> a"
univalent_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "univalent x"   and "univalent y" shows "univalent (x * y)"
terminal_char1: fixes t :: "'s" shows "S.terminal t = (S.ide t \<and> (\<exists>!x. x \<in> S.set t))"
set_mult_ker_hom_1: fixes I :: "'a set" assumes "I \<subseteq> carrier G" shows "h ` (I <#> kernel G H h) = h ` I"
farkas_coefficients_atoms_tableau_unsat: fixes as :: "'a atom set"   and t :: "(nat \<times> linear_poly) list"   and C :: "(rat \<times> 'a atom) list" assumes "farkas_coefficients_atoms_tableau as t C" shows "\<nexists>v. v \<Turnstile>\<^sub>t t \<and> v \<Turnstile>\<^sub>a\<^sub>s as"
components_are_iso: fixes a :: "'b \<times> 'b" assumes "B.VV.ide a" shows "C.iso (\<Phi>\<^sub>F a)"
ConjI: fixes A0 :: "formula"   and Gamma :: "formula list"   and A :: "(formula list \<times> formula list set) set"   and A1 :: "formula"   and Delta :: "formula list" assumes "A0 # Gamma \<in> deductions A"   and "A1 # Delta \<in> deductions A"   and "Conjs \<subseteq> A" shows "FConj Pos A0 A1 # Gamma @ Delta \<in> deductions A"
n_ok: shows "n = 1 + length (rev (tl (minus_poly_rev_list (map ((*) lc) (rev r)) (map ((*) (hd (rev r))) (rev d))))) - length d"
assms_2: shows "P \<turnstile> T' \<le> T"
tendsto_ine_in: fixes u :: "nat \<Rightarrow> 'a"   and N :: "'a quasinorm"   and x :: "'a" assumes "\<And>n. u n \<in> space\<^sub>N N"   and "x \<in> space\<^sub>N N" shows "tendsto_ine\<^sub>N N u x = tendsto_in\<^sub>N N u x"
S_def: fixes base :: "'b"   and elements :: "'a list"   and n :: "nat" shows "S base elements n \<equiv> {foldr f ys base |ys. subseq_of_length n elements ys}"
assms_1: fixes i :: "nat" assumes "i < n" shows "\<exists>j. suffix (i + j) w \<Turnstile>\<^sub>n af \<phi> (w [i \<rightarrow> j + i])[X]\<^sub>\<nu>"
S_subset: shows "\<forall>i\<in>{1..m}. S i \<subseteq> U"
circ_sup_mult_zero: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<circ> * y = (x \<squnion> y * bot)\<^sup>\<circ> * y"
positive_antisym: fixes A :: "complex mat" assumes "positive A"   and "positive (- A)" shows "A = 0\<^sub>m (dim_col A) (dim_col A)"
Sup_closed_set_def: fixes X :: "'b set" shows "Sup_closed_set X = (\<forall>Y\<subseteq>X. \<Squnion> Y \<in> X)"
inj_IMG: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "inj_on IMG (hom unity a)"
inputAlpha: fixes P :: "pi"   and a :: "name"   and u :: "name"   and P' :: "pi"   and r :: "name" assumes "P \<longmapsto> a<u> \<prec> P'"   and "u \<sharp> P"   and "r \<sharp> P'" shows "P \<longmapsto> a<r> \<prec> [(u, r)] \<bullet> P'"
comp_ide_self: fixes a :: "'a" assumes "S.ide a" shows "S.comp a a = a"
closure_invariant_complete: fixes r :: "'b rexp"   and s :: "'b rexp" assumes "lang n r = lang n s"   and "local.wf n r"   and "local.wf n s" shows "\<exists>ps N. closure_invariant ([([], init r, init s)], [], {(post (init r), post (init s))}) = Some ([], ps, N)"
supp_alpha_supp_rel: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" shows "supp t\<^sub>\<alpha> = supp_rel (=\<^sub>\<alpha>) (rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
intro: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr" assumes "\<And>\<alpha> \<beta> e x e'. \<R>\<^sub>E \<alpha> \<beta> (subst e x e') = subst (\<R>\<^sub>E \<alpha> \<beta> e) x (\<R>\<^sub>E \<alpha> \<beta> e')"   and "\<And>e x e'. RID\<^sub>E (subst e x e') \<subseteq> RID\<^sub>E e \<union> RID\<^sub>E e'"   and "\<And>e x e'. LID\<^sub>E (subst e x e') \<subseteq> LID\<^sub>E e \<union> LID\<^sub>E e'" shows "substitution subst"
assms: fixes x :: "'b" assumes "x \<in> A" shows "fds_abs_converges (f x) s"
atLeastAtMost_nat_numeral: fixes m :: "nat"   and k :: "num" shows "{m..numeral k} = (if m \<le> numeral k then insert (numeral k) {m..pred_numeral k} else {})"
uint_word_of_int_eq: fixes k :: "int" shows "uint (word_of_int k) = take_bit LENGTH('b) k"
case_1: fixes f1 :: "nat \<Rightarrow> 'c"   and f2 :: "'c"   and nat :: "nat" shows "(case enat nat of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f1 nat"
c: shows "(c, s) \<Rightarrow> s\<^sub>1'"
DF_subset: fixes A :: "'b set"   and B :: "'b set" assumes "A \<noteq> {}"   and "A \<subseteq> B" shows "DF B \<sqsubseteq>\<^sub>F\<^sub>D DF A"
assms_4: shows "circline_type (of_ocircline H2) < 0"
update_by_fun_pair_eq_update_by_pair: fixes xs :: "('a \<times> 'c) list"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv_raw xs" shows "update_by_fun_pair k f xs = update_by_pair (k, f (lookup_pair xs k)) xs"
map_pred_def: fixes f :: "'f \<Rightarrow> 'e" shows "map_pred f = f ---> id"
tendsto_zero_mult_left_iff: fixes c :: "'a"   and a :: "nat \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "(\<lambda>n. c * a n) \<longlonglongrightarrow> (0::'a) = a \<longlonglongrightarrow> (0::'a)"
function_of_sequence_of: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "function_of A enum (sequence_of A enum f) = f"
of_complex_minus: fixes x :: "complex" shows "of_complex (- x) = - of_complex x"
at_most_one_mset_mset_diff: fixes a :: "'c"   and M :: "'c multiset" assumes "a \<notin># M - {#a#}" shows "set_mset (M - {#a#}) = set_mset M - {a}"
Units_def: fixes G :: "('d, 'e) monoid_scheme" shows "Units G = {y \<in> carrier G. \<exists>x\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>}"
RestrictedP_Zero: fixes H :: "fm set"   and k :: "tm" shows "H \<turnstile> RestrictedP Zero k Zero"
pid_system_automaton'_simps_4: shows "sa_L pid_system_automaton' = pid_label"
lim_inverse_n: shows "(\<lambda>n. inverse (of_nat n)) \<longlonglongrightarrow> (0::'e)"
sprefix_shift: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c stream" assumes "length xs \<le> length ys" shows "sprefix xs (ys @- zs) = prefix xs ys"
order_iff: fixes a :: "'a::semilattice_inf"   and b :: "'a::semilattice_inf" shows "((a::'a::semilattice_inf) \<le> (b::'a::semilattice_inf)) = (a = a \<sqinter> b)"
openin_prod_topology_alt: fixes X :: "'c topology"   and Y :: "'d topology"   and S :: "('c \<times> 'd) set" shows "openin (prod_topology X Y) S = (\<forall>x y. (x, y) \<in> S \<longrightarrow> (\<exists>U V. openin X U \<and> openin Y V \<and> x \<in> U \<and> y \<in> V \<and> U \<times> V \<subseteq> S))"
truncate_down_float: fixes p :: "nat"   and x :: "real" shows "truncate_down p x \<in> float"
of_int_0: shows "\<guillemotleft>0\<guillemotright> = \<zero>"
right_inverse: fixes a :: "'c" shows "a + - a = (0::'c)"
valid_path_polynomial_function: fixes p :: "real \<Rightarrow> 'a" assumes "polynomial_function p" shows "valid_path p"
fps_inverse_compose: fixes b :: "'a fps"   and a :: "'a fps" assumes "fps_nth b 0 = (0::'a)"   and "fps_nth a 0 \<noteq> (0::'a)" shows "inverse a oo b = inverse (a oo b)"
olv_round_refines: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and D :: "process \<Rightarrow> val option"   and Ob :: "process set" shows "{olv_ref_rel \<inter> (OV_inv2 \<inter> OV_inv3 \<inter> OV_inv4) \<times> UNIV} obsv_round r S v D Ob, olv_round r S v D Ob {> olv_ref_rel}"
terminates_on_conv_dom_terminates_within: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" shows "terminates_on g = dom (terminates_within g)"
inv_ipp_minus: fixes a :: "'e"   and i :: "'e \<Rightarrow> 'e"   and z :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "ipp_cond1 {a} i"   and "inv_ipp z i f {a}" shows "inv_ipp z i f {\<^sub>i- a}"
1: shows "A' \<subset> Field r \<and> |A'| <o r"
Prim_Impl_axioms_def: fixes G_invar :: "'a \<Rightarrow> bool"   and g :: "'a" shows "Prim_Impl_axioms G_invar g \<equiv> G_invar g"
val_fun_le_refl: fixes n :: "nat" shows "\<forall>v t. n = val_size v + fun_size t \<longrightarrow> v \<sqsubseteq> v \<and> t \<lesssim> t"
order_def: fixes a :: "'a G"   and b :: "'a G" shows "a \<le> b \<equiv> a l\<rightarrow> b = 1"
nasc_seq_mem: fixes a :: "nat"   and A :: "nat set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)" shows "nasc_seq A a n \<in> A"
i_shrink_i_take_mult: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd message_af" assumes "0 < k" shows "f \<Down> (n * k) \<div> k = f \<div> k \<Down> n"
conj_commute: fixes x :: "'a"   and y :: "'a" shows "x AND y = y AND x"
assms_2: shows "u \<le>p t \<^sup>@ k"
WT_elim_cases_8: 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 e :: "expr"   and M :: "char list"   and ps :: "expr list"   and T :: "ty" assumes "P,E \<turnstile> Call e None M ps :: T" obtains C :: "char list" and Ts :: "ty list" and m :: "char list list \<times> expr" and Cs :: "char list list" and Ts' :: "ty list" where "P,E \<turnstile> e :: Class C" and "P \<turnstile> C has least M = (Ts, T, m) via Cs" and "P,E \<turnstile> ps [::] Ts'" and "P \<turnstile> Ts' [\<le>] Ts"
m_o_widen: fixes X :: "char list set"   and S2 :: "ivl Abs_State.st option"   and S1 :: "ivl Abs_State.st option" assumes "finite X"   and "domo S2 \<subseteq> X"   and "\<not> S2 \<sqsubseteq> S1" shows "m_o (m_st m_ivl) (2 * card X) (S1 \<nabla> S2) < m_o (m_st m_ivl) (2 * card X) S1"
tl_assoc: fixes x :: "real"   and z :: "real"   and y :: "real" assumes "x \<in> {0..1}"   and "z \<in> {0..1}" shows "local.tl (local.tl x y) z = local.tl x (local.tl y z)"
poly_deg_poly_subst_eq_zeroI: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and f :: "'b \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>x. x \<in> indets p \<Longrightarrow> poly_deg (f x) = 0" shows "poly_deg (poly_subst f p) = 0"
power_mult_distrib: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.comm_monoid_mult times one" shows "power.power one times (times a b) n = times (power.power one times a n) (power.power one times b n)"
elims: fixes x :: "nat"   and y :: "nat" assumes "x! = y" obtains "x = 0" and "y = 1"   | xa :: "nat" where "x = Suc xa" and "y = Suc xa * xa!"
acts_MapOps_update_def: shows "acts_MapOps_update \<equiv> \<lambda>(cec, aec) pAct m. trie_update_with' (map fst (toList cec)) m empty_trie (\<lambda>m. trie_update_with' (map snd (toList cec)) m empty_trie (\<lambda>m. trie_update_with' (map fst (toList aec)) m empty_trie (trie_update (map snd (toList aec)) pAct)))"
in_sort_edge: fixes L :: "'a list"   and x :: "'a" assumes "sorted_wrt (\<lambda>e1 e2. weight e1 \<le> weight e2) L" shows "sorted_wrt (\<lambda>e1 e2. weight e1 \<le> weight e2) (in_sort_edge x L)"
cs: shows "closed_segment X (X + H) \<subseteq> G"
lcoeff_nonzero: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<noteq> \<zero>\<^bsub>P\<^esub>"   and "p \<in> carrier P" shows "lcoeff p \<noteq> \<zero>"
LCoset_rel_quotientE: fixes X :: "'g set"   and uu :: "'g set" assumes "X \<in> G // LCoset_rel uu" obtains x :: "'g" where "X = LCoset_rel uu `` {x}" and "x \<in> G"
name_swap: fixes a :: "name"   and b :: "name"   and x :: "'a" shows "[(a, b)] \<bullet> x = [(b, a)] \<bullet> x"
vsv_VLambda_cong: fixes f :: "V \<Rightarrow> V" assumes "\<And>a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>) \<Longrightarrow> \<BB>\<lparr>Cod\<rparr>\<lparr>a\<rparr> = f a" shows "VLambda (\<D>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>)) f = \<BB>\<lparr>Cod\<rparr>"
simps_5: fixes t :: "'a local.term" shows "Dom \<^bold>\<l>\<^bold>[t\<^bold>] = Trg t \<^bold>\<star> Dom t"
x_def: shows "x \<equiv> getFr [xx, yy, zz, z] [t1, t2] []"
snd_def: fixes p :: "('d, 'c) Tagged_Prod_Sum.prod" shows "Tagged_Prod_Sum.snd p = (case p of prod.Prod x xa xb b \<Rightarrow> b)"
I_def: fixes n :: "nat" shows "I n = {fst (ivl n)..snd (ivl n)}"
red_stat_is_stat_red: shows "reducedly_statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F = statically_complete_calculus Bot Inf (\<Turnstile>) reduc_calc.Red_Red_I Red_F"
UINF_false: fixes P :: "'e \<Rightarrow> 'd upred" shows "UINF P (\<lambda>i. false) = false"
complete_digraph_def: fixes n :: "nat"   and G :: "('c, 'd) pre_digraph" shows "K\<^bsub>n\<^esub> G \<equiv> graph G \<and> card (verts G) = n \<and> arcs_ends G = {(u, v). (u, v) \<in> verts G \<times> verts G \<and> u \<noteq> v}"
simps_3: fixes n :: "'a"   and v :: "'a"   and va :: "'a list" shows "tape_of_nat_list (n # v # va) = <n> @ Bk # tape_of_nat_list (v # va)"
insert_list_def: fixes x :: "'e"   and xs :: "'e list" shows "insert_list x xs = (if x \<in> set xs then xs else x # xs)"
mult_mat_vec_resize: fixes v :: "'c vec"   and A :: "'c mat" shows "v \<^sub>v* A = resize_vec (dim_row A) v \<^sub>v* A"
simps_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Rval x1 = Rval y1) = (x1 = y1)"
intros_22: shows "\<lparr>[$Unreachable]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
arr_dom: fixes f :: "'j" assumes "J.arr f" shows "J.arr (J.dom f)"
seq_comp_secure_aux_2_case_2: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and P :: "'d option process"   and Q :: "'d option process"   and xs :: "'d option list"   and zs :: "'d option list"   and X :: "'d option set"   and Y :: "'d option set"   and y :: "'d option" assumes "secure_termination I D"   and "sequential P"   and "secure P I D"   and "secure Q I D"   and "xs @ zs \<in> sentences P"   and "(xs @ zs, X) \<in> failures P"   and "([], Y) \<in> failures Q"   and "(xs @ [y], {}) \<in> seq_comp_failures P Q" shows "(xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs (insert None X \<inter> Y)) \<in> seq_comp_failures P Q"
completely_multiplicativeI: assumes "\<And>p k. \<lbrakk>prime p; 0 < k\<rbrakk> \<Longrightarrow> \<chi> (p ^ k) = \<chi> p ^ k" shows "completely_multiplicative_function \<chi>"
pi0: fixes b :: "'a set" assumes "b \<in># remove1_mset bl \<B>" shows "f b index {(x, ix), (y, iy)} = 0"
fdisjnt_contrI: fixes M :: "'b fset"   and N :: "'b fset" assumes "\<And>x. \<lbrakk>x |\<in>| M; x |\<in>| N\<rbrakk> \<Longrightarrow> False" shows "fdisjnt M N"
restrict_restrict: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" shows "restrict (restrict f A) B = restrict f (A \<inter> B)"
eq0: shows "punit.lc (rep_list m) = punit.lc (rep_list p)"
eq_iff: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" shows "(a = b) = (a \<preceq>\<^sub>p b \<and> b \<preceq>\<^sub>p a)"
finite_generat_outs: fixes generat :: "('h, 'g, 'i) generat" shows "finite (generat_outs generat)"
perfect_sec_P2_def: fixes m1 :: "'a \<times> 'a"   and m2 :: "bool" shows "sim_def.perfect_sec_P2 m1 m2 \<equiv> R2 m1 m2 = funct_OT_12 m1 m2 \<bind> (\<lambda>(s1, s2). S2 m2 s2)"
Parallel: fixes Ts :: "('a ann_com option \<times> 'a set) list" assumes "\<forall>i<length Ts. \<exists>c q. Ts ! i = (Some c, q) \<and> \<turnstile> c  q"   and "interfree Ts" shows "\<parallel>- (\<Inter>i\<in>{i. i < length Ts}. pre (the (OG_Hoare.com (Ts ! i)))) Parallel Ts (\<Inter>i\<in>{i. i < length Ts}. OG_Hoare.post (Ts ! i))"
intro: fixes G :: "('c, 'd) graph" assumes "\<forall>v\<in>nodes G. \<forall>v'\<in>nodes G. nodes_connected G v v'" shows "connected_graph_axioms G"
continuous_on_imp_continuous_within: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'd"   and t :: "'b set"   and x :: "'b" assumes "continuous_on s f"   and "t \<subseteq> s"   and "x \<in> s" shows "continuous (at x within t) f"
valid_W_inv_def: shows "valid_W_inv = (\<lambda>s. (\<forall>x xa. xa \<in> W (s x) \<or> sys_mem_lock s \<noteq> Some x \<and> xa \<in> ghost_honorary_grey (s x) \<longrightarrow> marked xa s) \<and> (\<forall>x xa. x \<noteq> xa \<longrightarrow> WL x s \<inter> WL xa s = {}) \<and> (\<forall>x xa xb. \<not> (xb \<in> ghost_honorary_grey (s x) \<and> xb \<in> W (s xa))) \<and> sys_ghost_honorary_grey s = {} \<and> (\<forall>x xa xb. mw_Mark xa xb \<in> set (sys_mem_store_buffers x s) \<longrightarrow> xb = sys_fM s \<and> xa \<in> ghost_honorary_grey (s x) \<and> sys_mem_lock s = Some x \<and> white xa s \<and> filter is_mw_Mark (sys_mem_store_buffers x s) = [mw_Mark xa xb]))"
ang_vec_plus_pi1: fixes z1 :: "complex"   and z2 :: "complex" assumes "0 < \<angle> z1 z2" shows "\<downharpoonright>\<angle> z1 z2 + pi\<downharpoonleft> = \<angle> z1 z2 - pi"
finite_set_in: fixes x :: "'m"   and FS :: "'m fset" shows "(x \<in> fset FS) = (x |\<in>| FS)"
negligible_def: fixes s :: "'a set" shows "negligible s = (\<forall>a b. (indicat_real s has_integral 0) (cbox a b))"
real_complex_eq_def: fixes r :: "real"   and c :: "complex" shows "real_complex_eq r c = (complex_of_real r = c)"
set_eq_fold: fixes x :: "'b"   and xs :: "'b list" shows "Min (set (x # xs)) = fold min xs x"
sgn_minus: fixes a :: "'a" shows "sgn (- a) = - sgn a"
eqvt: fixes x1 :: "lam"   and x2 :: "lam"   and pi :: "(name \<times> name) list" assumes "x1 \<rightarrow>\<^sub>s x2" shows "pi \<bullet> x1 \<rightarrow>\<^sub>s pi \<bullet> x2"
termMOD_igWlsAbsDisj: shows "igWlsAbsDisj termMOD"
mul\<^sub>O: fixes n :: "Ord"   and m :: "Ord" shows "\<langle>mulO n m\<rangle>\<^sub>O = (\<lambda>z s l. \<langle>m\<rangle>\<^sub>O z (\<lambda>m. \<langle>n\<rangle>\<^sub>O m s l) l)"
ltakeWhile_all_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "(ltakeWhile P xs = xs) = (lset xs \<subseteq> {x. P x})"
unitary11_def: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary11 M = (congruence M (1, 0, 0, - 1) = (1, 0, 0, - 1))"
nres_rel_def_internal: fixes R :: "('b \<times> 'c) set" shows "nres_rel R \<equiv> {(c, a). c \<le> \<Down> R a}"
asymp_equiv_at_top_transfer: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "f \<sim>[F] g"   and "filterlim f at_top F" shows "filterlim g at_top F"
v11E: fixes r :: "V" assumes "v11 r" obtains "vsv r" and "vsv (r\<inverse>\<^sub>\<circ>)"
hcomp_simps_3: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (H \<nu> \<mu>)" shows "partial_magma.dom V (H \<nu> \<mu>) = H (partial_magma.dom V \<nu>) (partial_magma.dom V \<mu>)"
chain: shows "Complete_Partial_Order.chain (rel_prod (\<sqsubseteq>) (\<sqsubseteq>)) Y"
b3: shows "(\<Sum>a\<in>t. r a *\<^sub>C a) = x"