hcju/isabelleps huggingface dataset:
in_\<R>_dma': fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and \<R> :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and c\<^sub>A :: "'f"   and mds\<^sub>A :: "Mode \<Rightarrow> 'c set"   and mem\<^sub>A :: "'c \<Rightarrow> 'd"   and c\<^sub>C :: "'g"   and mds\<^sub>C :: "Mode \<Rightarrow> 'e set"   and mem\<^sub>C :: "'e \<Rightarrow> 'd"   and x\<^sub>A :: "'c" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of"   and "sifum_refinement.preserves_modes_mem var\<^sub>C_of \<R>"   and "(((c\<^sub>A, mds\<^sub>A), mem\<^sub>A), (c\<^sub>C, mds\<^sub>C), mem\<^sub>C) \<in> \<R>" shows "dma\<^sub>A mem\<^sub>A x\<^sub>A = dma\<^sub>C mem\<^sub>C (var\<^sub>C_of x\<^sub>A)"
simps_2: fixes x :: "'a" shows "extB.Bot + extB.Val x = extB.Val x"
lc_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "lc p = lookup p (lt p)"
list_dnf_finite: fixes \<phi> :: "'b ltln" shows "finite (list_dnf \<phi>)"
enum_alt_rel: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_alt :: "nat \<Rightarrow> 'b option" assumes "class.enumeration_both enum enum_all enum_ex enum_alt" shows "enum_alt = alt_from_ord enum"
ct_suffix_cons2: fixes xs :: "ct list"   and ys :: "ct list"   and zs :: "ct list" assumes "ct_suffix xs (ys @ zs)"   and "length xs = length zs" shows "ct_list_eq xs zs"
hyps: fixes xs :: "'a list" assumes "ram \<notin> set bs" shows "fst (splitAtRec ram xs bs) = xs @ fst (splitAtRec ram [] bs)"
spatial_weaken: fixes \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. \<phi> ts v \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> \<phi> ts v \<and> True)) \<and> True))"
assms_2: shows "rational_preference (lotteries_on outcomes) \<R>"
is_Zp_cauchy_imp_closed: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_cauchy s" shows "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
advantage3_def: fixes \<G> :: "'a cyclic_group"   and x :: "nat"   and \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" assumes "dis_log_alt \<G>" shows "Discrete_Log.dis_log_alt.advantage3 \<G> x \<A> \<equiv> spmf (Discrete_Log.dis_log_alt.dis_log3 \<G> x \<A>) True"
sets_into_space: fixes x :: "'a set" assumes "x \<in> {A \<in> sigma_sets \<Omega> G. P A}" shows "x \<subseteq> \<Omega>"
R_lin_independent_ConsD1: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b"   and ms :: "'b list" assumes "R_scalar_mult R"   and "R_scalar_mult.R_lin_independent R smult (m # ms)" shows "R_scalar_mult.R_lin_independent R smult ms"
is_pos_def: fixes literal :: "'a literal" shows "is_pos literal = (\<exists>x1. literal = Pos x1)"
simple_restrict_none: fixes i :: "nat"   and X :: "'b option list"   and A :: "nat set" assumes "i < length X"   and "i \<notin> A" shows "restrict A X ! i = None"
right_neutral: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and a :: "'a" assumes "Groups.monoid f z" shows "f a z = a"
component_of_def: fixes F :: "'b program"   and H :: "'b program" shows "F component_of H \<equiv> \<exists>G. F ok G \<and> F \<squnion> G = H"
induct: fixes P :: "'b list list \<Rightarrow> bool"   and a0 :: "'b list list" assumes "P []"   and "\<And>xs. P xs \<Longrightarrow> P ([] # xs)"   and "\<And>v va xs2. P xs2 \<Longrightarrow> P ((v # va) # xs2)" shows "P a0"
star_outer_increasing: fixes x :: "'a" shows "x \<le> top \<sqinter> x \<sqinter> top"
iMin_less_iff: fixes I :: "'a set"   and n :: "'a" assumes "I \<noteq> {}" shows "(iMin I < n) = (\<exists>a\<in>I. a < n)"
is_UNIV_def: fixes A :: "'b set" shows "is_UNIV A = (A = UNIV)"
monad_simps_1: fixes f :: "'b \<Rightarrow> 'c option" shows "gets_the f \<equiv> gets f >>= assert_opt"
ring_distribs_4: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * (b - c) = a * b - a * c"
rel_transfer: fixes S1a :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and S1c :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and S2a :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and S2c :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_fun (rel_fun S1a (rel_fun S1c (=))) (rel_fun (rel_fun S2a (rel_fun S2c (=))) (rel_fun (tllist_all2 S1a S2a) (rel_fun (tllist_all2 S1c S2c) (=)))) tllist_all2 tllist_all2"
vseq_implies_hpar_1: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.seq \<nu> \<mu>" shows "src\<^sub>D \<nu> = src\<^sub>D \<mu>"
circ_decompose_9: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<circ> * y\<^sup>\<circ>)\<^sup>\<circ> = x\<^sup>\<circ> * y\<^sup>\<circ> * (x\<^sup>\<circ> * y\<^sup>\<circ>)\<^sup>\<circ>"
prefix_length_prefix: fixes ps :: "'e list"   and xs :: "'e list"   and qs :: "'e list" assumes "prefix ps xs"   and "prefix qs xs"   and "length ps \<le> length qs" shows "prefix ps qs"
bij_is_inj: fixes f :: "'a \<Rightarrow> 'b" assumes "bij f" shows "inj f"
countable_basis: fixes p :: "'b set \<Rightarrow> bool"   and B :: "'b set set" assumes "countable_basis p B" shows "countable B"
F': shows "F' = derivation_shift (take (n - m) F) 0 (Suc j)"
h3_2: shows "zs = sort_key p (remove1 z xs)"
tagged_division: fixes d :: "(real \<times> real set) set"   and a :: "real"   and b :: "real" assumes "d tagged_division_of {a..b}" shows "(\<Sum>(uu_, l)\<in>d. if l = {} then 0::'a else f (interval_upperbound l) - f (interval_lowerbound l)) = (if {a..b} = {} then 0::'a else f (interval_upperbound {a..b}) - f (interval_lowerbound {a..b}))"
inj_eq: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "inj f" shows "(f x = f y) = (x = y)"
bot_ennreal_def: shows "\<bottom> \<equiv> e2ennreal 0"
scc_of_eq: fixes u :: "'a"   and v :: "'a" assumes "u \<in> S.scc_of v" shows "S.scc_of u = S.scc_of v"
A: fixes x :: "'a" obtains "x = (0::'a)"   | "x \<noteq> (0::'a)"
simps_1: fixes p :: "perm"   and tset :: "('b, 'c, 'd) Tree set['b]" shows "p \<bullet> tConj tset = tConj (map_bset (permute p) tset)"
less_hmset_zhmsetE: fixes M :: "zhmultiset"   and N :: "zhmultiset" assumes "M < N" obtains A :: "hmultiset" and B :: "hmultiset" and C :: "zhmultiset" where "M = zhmset_of A + C" and "N = zhmset_of B + C" and "A < B"
dropWhile_append3: fixes P :: "'b \<Rightarrow> bool"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "\<not> P y" shows "dropWhile P (xs \<cdot> y # ys) = dropWhile P xs \<cdot> y # ys"
Pd_singletons_for_ds_linear: fixes X :: "'a set"   and ds :: "'b set"   and d :: "'b" assumes "inj_on Xd X" shows "Linear_order (TruePref.Pd_singletons_for_ds X ds d)"
union_fold_mset_add_mset: fixes A :: "'b multiset"   and B :: "'b multiset" shows "A + B = fold_mset add_mset A B"
that: fixes x :: "'a" assumes "x \<in> set xs" shows "(0::'a) \<le> x"
aa: shows "snd (TSdet init h ((as @ x # bs) @ [x]) (Suc (length as + length bs))) = rev (take (Suc (length as + length bs)) ((as @ x # bs) @ [x])) @ h"
surj_on_image_vimage_eq: fixes s :: "'b set"   and f :: "'c \<Rightarrow> 'b" assumes "surj_on s f" shows "f ` f -` s = s"
poly_card_roots_less_leq: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "card {x. a < x \<and> x \<le> b \<and> poly p x = 0} = count_roots_between p a b"
F: shows "range F \<subseteq> {A \<times> B |A B. A \<in> sets M1 \<and> B \<in> sets M2} \<and> incseq F \<and> \<Union> (range F) = space M1 \<times> space M2 \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
order_iff_strict: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (a < b \<or> a = b)"
avl_mkt_bal_l: fixes l :: "'b tree"   and r :: "'b tree"   and n :: "'b" assumes "avl l"   and "avl r"   and "height l = height r \<or> height l = height r + 1 \<or> height r = height l + 1 \<or> height l = height r + 2" shows "avl (mkt_bal_l n l r)"
lepoll_imp_Card_le: fixes a :: "V"   and b :: "V" assumes "a \<lesssim>\<^sub>\<circ> b" shows "vcard a \<subseteq>\<^sub>\<circ> vcard b"
norm_cauchy_schwarz_abs_eq: fixes x :: "'b"   and y :: "'b" shows "(\<bar>x \<bullet> y\<bar> = norm x * norm y) = (norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x)"
CapSS: fixes \<psi> :: "'fmla"   and f :: "'trm \<Rightarrow> 'trm" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {xx}"   and "f \<in> F" shows "ssap \<psi> (tenc (encF f)) = \<langle>inst \<psi> (encF f (tenc (encF f)))\<rangle>"
hcomp_simps\<^sub>W\<^sub>C_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "hseq \<nu> \<mu>"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_diff a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "head t >\<^sub>h\<^sub>d head s" and "is_Sym (head s)" and "chkchop (>\<^sub>t) t s"
openE: fixes m :: "'a set" assumes "m open\<^bsub>T\<^esub>" obtains "m \<in> T"
verit_comp_simplify1_3: fixes b' :: "'d"   and a' :: "'d" shows "(\<not> b' \<le> a') = (a' < b')"
mkarr_Assoc: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and u :: "'b local.term"   and v :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t"   and "monoidal_language.Arr C u"   and "monoidal_language.Arr C v" shows "free_monoidal_category.mkarr C \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = elementary_monoidal_category.\<alpha> (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.assoc\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.mkarr C t, free_monoidal_category.mkarr C u, free_monoidal_category.mkarr C v)"
simps_3: fixes l :: "dag"   and v\<^sub>1 :: "ref"   and r :: "dag"   and v\<^sub>2 :: "ref" shows "isLeaf (Node (Node l v\<^sub>1 r) v\<^sub>2 Tip) = False"
fconverseI: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "[b, a]\<^sub>\<circ> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet>"
Vset_universal: fixes x :: "hf" shows "\<exists>n. Ord n \<and> x \<^bold>\<in> Vset n"
path_append: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and u :: "'a"   and ls1 :: "enat list"   and ls2 :: "enat list"   and v :: "'a" shows "WGraph.path w u (ls1 @ ls2) v = (\<exists>w. WGraph.path w u ls1 w \<and> WGraph.path w w ls2 v)"
arcE: fixes e :: "'b"   and u :: "'a"   and v :: "'a" assumes "arc e (u, v)" obtains "e \<in> arcs G" and "tail G e = u" and "head G e = v"
intros: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t1 :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m"   and t2 :: "('c \<times> ('c, 'd) rose_tree\<^sub>m list, 'd \<times> 'd rose_tree\<^sub>h list) blindable\<^sub>m" assumes "blinding_of_blindable (map_prod h (map (hash_tree h))) (rel_prod bo (list_all2 (blinding_of_tree h bo))) t1 t2" shows "blinding_of_tree h bo (Tree\<^sub>m t1) (Tree\<^sub>m t2)"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p. (\<And>x. P p) \<Longrightarrow> P (E p)"   and "\<And>p. (\<And>x. P (NOT p)) \<Longrightarrow> P (A p)"   and "\<And>p. (\<And>x. P p) \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "P T"   and "P F"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Le v)"   and "\<And>v. P (Gt v)"   and "\<And>v. P (Ge v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)" shows "P a0"
Inf_flat_complete_lattice_def: fixes A :: "'a flat_complete_lattice set" shows "Inf A = (if A = {} \<or> A = {flat_complete_lattice.Top} then flat_complete_lattice.Top else if \<exists>v. A - {flat_complete_lattice.Top} = {flat_complete_lattice.Value v} then flat_complete_lattice.Value (THE v. A - {flat_complete_lattice.Top} = {flat_complete_lattice.Value v}) else flat_complete_lattice.Bot)"
valofd_eq: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "valofd x = valofd y" shows "fraction x = fraction y"
intros: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "\<And>x. \<lbrakk>x \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub>; x \<notin> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> KIR f x" shows "web_flow \<Gamma> f"
map_snd_zip_take: fixes xs :: "'d list"   and ys :: "'c list" shows "map snd (zip xs ys) = take (min (length xs) (length ys)) ys"
exhaust: fixes y :: "'a array" obtains x :: "nat" where "y = Array x"
rank_Gauss_Jordan_eq: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A = (let A' = Gauss_Jordan A in card {row i A' |i. row i A' \<noteq> 0})"
iter: fixes P :: "'b \<Rightarrow> bool"   and s :: "'a" assumes "Measurable.pred N P" shows "emeasure (M s) {x \<in> space N. F P x} = f (\<lambda>s. emeasure (M s) {x \<in> space N. P x}) s"
convex_hull_subset: fixes s :: "'a set"   and t :: "'a set" assumes "s \<subseteq> convex hull t" shows "convex hull s \<subseteq> convex hull t"
interpret_floatarith_plus: fixes f :: "floatarith"   and g :: "floatarith"   and xs :: "real list" shows "interpret_floatarith (f + g) xs = interpret_floatarith f xs + interpret_floatarith g xs"
eval_monom1: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and s :: "'e" assumes "UP_pre_univ_prop R S h"   and "s \<in> carrier S" shows "eval R S h s (monom (UP R) \<one>\<^bsub>R\<^esub> 1) = s"
index_swap_if_distinct: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat"   and x :: "'b" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "index (xs[i := xs ! j, j := xs ! i]) x = (if x = xs ! i then j else if x = xs ! j then i else index xs x)"
prv_imp_sdsj: fixes \<chi> :: "'fmla"   and F :: "'fmla set"   and \<phi> :: "'fmla" assumes "\<chi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> F"   and "prv (imp \<chi> \<phi>)" shows "prv (imp \<chi> (sdsj F))"
that: fixes m :: "nat" assumes "f ^ m \<in> F" shows "thesis"
fls_of_fpxs_const: fixes c :: "'b" shows "fls_of_fpxs (fpxs_const c) = fls_const c"
arcsin_sin: fixes x :: "real" assumes "- (pi / 2) \<le> x"   and "x \<le> pi / 2" shows "arcsin (sin x) = x"
hyps_4: fixes ba :: "lam"   and bb :: "lam"   and b :: "name" assumes " ba \<longrightarrow>\<^sub>1 bb" shows " P[b::=ba] \<longrightarrow>\<^sub>1 P[b::=bb]"
open_Collect_less: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "open {x. f x < g x}"
iT_Div_cut_ge: fixes k :: "nat"   and I :: "nat set"   and t :: "nat" assumes "0 < k" shows "(I \<oslash> k) \<down>\<ge> t = I \<down>\<ge> (t * k) \<oslash> k"
Gromov_completion_rel_def: fixes u :: "nat \<Rightarrow> 'b"   and v :: "nat \<Rightarrow> 'b" shows "Gromov_completion_rel u v = (Gromov_converging_at_boundary u \<and> Gromov_converging_at_boundary v \<and> (\<forall>a. (\<lambda>x. ereal (Gromov_product_at a (u x) (v x))) \<longlonglongrightarrow> \<infinity>) \<or> (\<forall>n m. u n = v m \<and> u n = u m \<and> v n = v m))"
surj_card_le: fixes A :: "'e set"   and B :: "'f set"   and f :: "'e \<Rightarrow> 'f" assumes "finite A"   and "B \<subseteq> f ` A" shows "card B \<le> card A"
set_empty_eq: fixes xo :: "'a option" shows "(set_option xo = {}) = (xo = None)"
whileLoop_results_simps_valid: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool" shows "((Some x, Some y) \<in> whileLoop_results C B) = ((\<exists>r s. x = (r, s) \<and> y = (r, s) \<and> \<not> C r s) \<or> (\<exists>r s. x = (r, s) \<and> C r s \<and> (\<exists>r' s'. (r', s') \<in> fst (B r s) \<and> (Some (r', s'), Some y) \<in> whileLoop_results C B)))"
oghoare_DynCom: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and c' :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (DynCom c') Q, A" shows "\<exists>r ad. P = AnnRec r ad \<and> r \<subseteq> pre ad \<and> (\<forall>s\<in>r. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> ad (c' s) Q, A)"
Tgs_never_says_newer_timestamp: fixes evs :: "event list"   and T :: "nat"   and X :: "msg" assumes "CT evs \<le> T"   and "Number T \<in> parts {X}"   and "evs \<in> kerbV" shows "\<forall>A. Says Tgs A X \<notin> set evs"
that: fixes f :: "V" assumes "v11 f"   and "\<D>\<^sub>\<circ> f = A"   and "\<R>\<^sub>\<circ> f = B" shows "thesis"
Rep_inverse: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and x :: "'d" assumes "type_definition Rep Abs A" shows "Abs (Rep x) = x"
vnm_utility_def: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b \<Rightarrow> real" shows "vnm_utility carrier le u \<equiv> finite_total_preorder_on carrier le \<and> vnm_utility_axioms carrier le u"
p_antitone_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<le> - y) = (y \<le> - x)"
elims_1: fixes G :: "('a, 'b) cfg"   and x :: "('c + 'b) list"   and y :: "bool" assumes "\<oo>\<^sub>P G x = y" obtains "x = []" and "y"   | uu_ :: "'c" and uv_ :: "('c + 'b) list" where "x = Inl uu_ # uv_" and "\<not> y"   | N :: "'b" and \<alpha> :: "('c + 'b) list" where "x = Inr N # \<alpha>" and "y = ([] |\<in>| cfg.prod G N \<and> \<oo>\<^sub>P G \<alpha>)"
wf_chain_add_match: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and m :: "'c match_expr" assumes "wf_chain \<Gamma> rs" shows "wf_chain \<Gamma> (add_match m rs)"
inducts: fixes P :: "nat \<Rightarrow> bool"   and nat :: "nat" assumes "P 0"   and "\<And>nat. P nat \<Longrightarrow> P (Suc nat)" shows "P nat"
surjec_def: fixes F :: "('d, 'e) aGroup_scheme"   and G :: "('f, 'g) aGroup_scheme"   and f :: "'d \<Rightarrow> 'f" shows "surjec\<^bsub>F,G\<^esub> f = (f \<in> aHom F G \<and> surj_to f (carrier F) (carrier G))"
neg_le_0_iff_le: fixes a :: "'a" shows "(- a \<le> (0::'a)) = ((0::'a) \<le> a)"
intros_2: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and H :: "nat set"   and K :: "nat set"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and t :: "'d net_state"   and R :: "nat set"   and t' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), H\<not>K:arrive(m), \<sigma>', s') \<in> S"   and "((\<sigma>, t), R:*cast(m), \<sigma>', t') \<in> T"   and "H \<subseteq> R"   and "K \<inter> R = {}" shows "((\<sigma>, SubnetS s t), R:*cast(m), \<sigma>', SubnetS s' t') \<in> opnet_sos S T"
objI_trg: fixes a :: "'c" assumes "C.arr a"   and "trg\<^sub>C a = a" shows "C.obj a"
execl_all_sub: fixes P :: "instr list"   and P' :: "'a \<Rightarrow> instr list"   and x :: "'a"   and P'' :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list"   and k :: "nat"   and Q :: "'a \<Rightarrow> int \<times> (char list \<Rightarrow> int) \<times> int list \<Rightarrow> int \<times> (char list \<Rightarrow> int) \<times> int list \<Rightarrow> bool" assumes "P @ P' x @ P'' \<Turnstile> cfs\<box>"   and "k < length cfs"   and "fst (cfs ! k) = size P"   and "\<And>cfs. P' x \<Turnstile> cfs\<box> \<Longrightarrow> Q x (cfs ! 0) (cfs ! (length cfs - 1))" shows "\<exists>k'<length cfs. Q x (off P (cfs ! k)) (off P (cfs ! k'))"
1: shows "to_conjugate_space ` closure X \<subseteq> closure (to_conjugate_space ` X)"
Blockchain_axioms: shows "Blockchain active cmp pin pout bc mining honest cb"
ex_minimum: fixes D :: "('c, 'd) Order_scheme" assumes "Worder D" shows "\<forall>X. X \<subseteq> carrier D \<and> X \<noteq> {} \<longrightarrow> (\<exists>x. minimum_elem D X x)"
InvariantConflictFlagCharacterizationAfterApplyUnitPropagate: fixes state :: "State" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "\<not> getConflictFlag state"   and "getQ state \<noteq> []" shows "let state' = applyUnitPropagate state in InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
rg_succ_ref: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'a"   and am :: "nat \<Rightarrow> nat list"   and u :: "nat" assumes "RGraph c s t cf"   and "Graph.is_adj_map c am"   and "u \<in> Graph.V c" shows "Network.rg_succ2 c am cf u \<le> (spec l. (l, Graph.E cf `` {u}) \<in> \<langle>nat_rel\<rangle>list_set_rel)"
eCons_simps_2: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and xs :: "'d"   and s :: "'e"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model"   and X :: "'g"   and x :: "'a" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.isInBar Delta (xs, s)"   and "igWls MOD s X" shows "igAbs (FixSyn.errMOD Delta MOD) xs x (OK X) = OK (igAbs MOD xs x X)"
Dflow_zero: fixes x :: "'a" assumes "x \<in> Y" shows "Y.Dflow x 0 = 1\<^sub>L"
confTs_hext: fixes P :: "'a prog"   and h :: "heap"   and loc :: "val list"   and LT :: "ty err list"   and h' :: "heap" assumes "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT"   and "h \<unlhd> h'" shows "P,h' \<turnstile> loc [:\<le>\<^sub>\<top>] LT"
ord_spmf_compp: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "ord_spmf (A OO B) = ord_spmf A OO ord_spmf B"
op'_strict_1: fixes pat :: "[:'r:]" shows "op'\<cdot>pat\<cdot>\<bottom> = \<bottom>"
prems: shows "partial_density_operator \<rho> \<and> \<rho> \<in> carrier_mat d d"
list_set_rel_range: shows "Range (\<langle>Rv\<rangle>list_set_rel) = {S. finite S \<and> S \<subseteq> Range Rv}"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>a p. (p \<noteq> 0 \<Longrightarrow> P p) \<Longrightarrow> P (pCons a p)" shows "P a0"
distinct_1: fixes x1 :: "'c"   and x2 :: "'c" shows "Broadcast x1 \<noteq> Deliver x2"
bex_simps_6: fixes Q :: "'d \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" shows "(\<exists>x\<in>Collect Q. P x) = (\<exists>x. Q x \<and> P x)"