hcju/isabelleps huggingface dataset:
reds_cases: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr expr"   and es :: "'addr expr list"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and es' :: "'addr expr list"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" assumes "extTA,P,t \<turnstile> \<langle>e # es,s\<rangle> [-ta\<rightarrow>] \<langle>es',s'\<rangle>" obtains e' :: "'addr expr" where "es'a__ = e' # es" and "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>"   | es' :: "'addr expr list" and v :: "'addr val" where "e = Val v" and "es'a__ = Val v # es'" and "extTA,P,t \<turnstile> \<langle>es,s\<rangle> [-ta\<rightarrow>] \<langle>es',s'\<rangle>"
execute_parallel_operator_cons_equals_corollary: fixes s :: "'a \<Rightarrow> bool option"   and a :: "'a strips_operator"   and ops :: "'a strips_operator list" assumes "are_all_operators_applicable s (a # ops)" shows "execute_parallel_operator s (a # ops) = execute_parallel_operator (s \<then> a) ops"
word_and_le2: fixes a :: "'b word"   and y :: "'b word" shows "a AND y \<le> a"
strict_prefixE: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs <p ys" obtains "xs \<le>p ys" and "xs \<noteq> ys"
trans_trancl: fixes r :: "('a \<times> 'a) set" shows "trans (r\<^sup>+)"
choice: fixes Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x. \<exists>y. Q x y" shows "\<exists>f. \<forall>x. Q x (f x)"
UNlift_inj_on: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n) \<and> {} \<notin> F n ` P n"   and "\<And>n. n < length cl \<Longrightarrow> inj_on (F n) (P n)" shows "inj_on (UNlift cl dl P F) (UNpart cl P)"
first_baseE: fixes v :: "state \<Rightarrow> 'a"   and c :: "'a" assumes "basevars v" obtains x :: "nat \<Rightarrow> state" where "v (first x) = c"
one_is_same: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H\<rparr>)" shows "\<one>\<^bsub>G\<lparr>carrier := H, carrier := H\<rparr>\<^esub> = \<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>"
case_prodE2: fixes Q :: "'c \<Rightarrow> bool"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and z :: "'d \<times> 'e" assumes "Q (case z of (a, b) \<Rightarrow> P a b)" obtains x :: "'d" and y :: "'e" where "z = (x, y)" and "Q (P x y)"
finalGraph_face: fixes g :: "graph"   and f :: "face" assumes "final g"   and "f \<in> \<F> g" shows "final f"
step: fixes x :: "'a node \<times> 'a node set" assumes "((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)" shows "f x \<le> SPEC (expand_rslt_all \<xi> x)"
card_insert_le: fixes A :: "'c set"   and x :: "'c" assumes "finite A" shows "card A \<le> card (insert x A)"
finite_imp_bounded: fixes S :: "'b set" assumes "finite S" shows "bounded S"
vcgaIsMaximalAux1: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "\<exists>a. a \<in> maximalStrictAllocations N (set \<Omega>) b \<and> Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b))) = Outside' {seller} a \<and> a \<in> maximalStrictAllocations N (set \<Omega>) b"
induct: fixes x :: "color list"   and P :: "color list \<Rightarrow> bool" assumes "valid' x"   and "P []"   and "\<And>n. 3 \<le> n \<Longrightarrow> P (replicate n R)"   and "\<And>xs. \<lbrakk>valid' xs; P xs\<rbrakk> \<Longrightarrow> P (B # xs)"   and "\<And>xs n. \<lbrakk>valid' xs; P xs; 3 \<le> n\<rbrakk> \<Longrightarrow> P (replicate n R @ B # xs)" shows "P x"
simps_1: fixes s :: "'d stream" shows "[] @- s = s"
simps_2: fixes i :: "nat"   and x :: "'v"   and y :: "'v" shows "h_plus (Suc i) (x, y) = (if enum_v (Suc i) = x \<and> d_OUT (h_plus i) x < d_IN (h_plus i) x then let total = d_IN (h_plus i) x - d_OUT (h_plus i) x; share = g (x, y) - h_plus i (x, y); shares = d_OUT g x - d_OUT (h_plus i) x in h_plus i (x, y) + share * total / shares else h_plus i (x, y))"
m_def: shows "m \<equiv> t.max_fun_diff (local.rev_curr f) (local.rev_curr g)"
zero_less_iff_1_le_hmset: fixes n :: "hmultiset" shows "(0 < n) = (1 \<le> n)"
next_plane0_finals_subset: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)" shows "set (finals g) \<subseteq> set (finals g')"
nat_of_ord_ord_of: fixes n :: "nat" shows "nat_of_ord (ord_of n) = n"
check_poly_weak_mono_all_def: fixes p :: "('c monom \<times> 'd) list" shows "check_poly_weak_mono_all p \<equiv> list_all (\<lambda>(m, c). (0::'d) \<le> c) p"
sigma_algebra_eq_Int_stable: shows "sigma_algebra \<Omega> (Dynkin \<Omega> M) = Int_stable (Dynkin \<Omega> M)"
intv'_of_def: fixes l :: "int"   and u :: "int"   and v :: "real" shows "intv'_of l u v \<equiv> if real_of_int u < v then Greater' u else if v < real_of_int l then Smaller' l else if \<exists>x. real_of_int x = v then Const' \<lfloor>v\<rfloor> else Intv' \<lfloor>v\<rfloor>"
psubst_all: fixes \<phi> :: "'fmla"   and y :: "'var"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "y \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "y \<notin> snd ` set txs"   and "y \<notin> \<Union> (FvarsT ` fst ` set txs)"   and "distinct (map snd txs)" shows "psubst (all y \<phi>) txs = all y (psubst \<phi> txs)"
cs_split: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and xs :: "'n list"   and x :: "'n"   and ys :: "'n list" assumes "cs\<^bsup>\<pi>\<^esup> i = xs @ [x] @ ys @ [\<pi> i]" shows "\<exists>m. cs\<^bsup>\<pi>\<^esup> m = xs @ [x] \<and> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
update_preserves_le: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and mds\<^sub>1 :: "Mode \<Rightarrow> 'b set"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'b set"   and annos :: "'b ModeUpd list" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "mds\<^sub>1 \<sqsubseteq> mds\<^sub>2" shows "sifum_modes.update_annos mds\<^sub>1 annos \<sqsubseteq> sifum_modes.update_annos mds\<^sub>2 annos"
mat_of_cols_def: fixes n :: "nat"   and cs :: "'b vec list" shows "mat_of_cols n cs = mat n (length cs) (\<lambda>(i, j). cs ! j $ i)"
simplex_map_def: fixes p :: "nat"   and g :: "'d \<Rightarrow> 'c"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'd" shows "simplex_map p g c \<equiv> restrict (g \<circ> c) (standard_simplex p)"
CongruenceI: fixes a :: "'a"   and b :: "'a"   and k :: "'a" assumes "a = b \<cdot> k"   and "a \<in> G"   and "b \<in> G"   and "k \<in> K" shows "(a, b) \<in> Congruence"
leg1_commutes: shows "\<nu>.cod.leg1 \<cdot> \<nu>.chine = \<nu>.dom.leg1"
qPsubst_preserves_alpha1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and Y :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "qGoodEnv rho"   and "qGood X \<or> qGood Y"   and "X #= Y" shows "X #[[rho]] #= Y #[[rho]]"
sim_rfunc_adv: fixes p :: "'s_rest spmf"   and q :: "'s_rest' spmf"   and ia :: "'iadv_rest"   and s1 :: "'s_rest"   and s2 :: "'s_rest'"   and oa :: "'oadv_rest \<times> 'event list"   and s1' :: "'s_rest"   and s2' :: "'s_rest'" assumes "S p q"   and "ia \<in> IA"   and "s1 \<in> set_spmf p"   and "s2 \<in> set_spmf q"   and "(oa, s1') \<in> set_spmf (rfunc_adv rest1 s1 ia)"   and "(oa, s2') \<in> set_spmf (rfunc_adv rest2 s2 ia)" shows "S (cond_spmf_fst (p \<bind> (\<lambda>s1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. rfunc_adv rest2 s2 ia)) oa)"
simps: fixes s :: "name"   and v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and k :: "name" assumes "atom s \<sharp> (v, i, x, x', k)"   and "atom k \<sharp> (v, i, x, x')" shows "SubstFormP v i x x' = VarP v AND TermP i AND SyntaxN.Ex s (SyntaxN.Ex k (SeqSubstFormP v i x x' (Var s) (Var k)))"
S\<^sub>1_def: shows "S\<^sub>1 = \<lbrakk>\<lbrakk>tab\<^sub>1 s\<rbrakk>\<rbrakk>"
M5exist_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x \<parallel> y"   and "y \<parallel> z"   and "z \<parallel> w" shows "\<exists>t. x \<parallel> t \<and> t \<parallel> w"
map_getGMUserCom_surj: fixes s :: "('St, 'U, 'C) state"   and ucl :: "('U \<times> 'C) list" assumes "isState s" shows "\<exists>sl. wffp sl \<and> list_all isState sl \<and> hd sl = s \<and> map getGMUserCom (tl sl) = ucl"
upper_inv_eq: fixes y :: "'b" assumes "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y) = y"
monadic_WHILEIT_arities: fixes I :: "'a \<Rightarrow> bool" shows "PR_CONST (monadic_WHILEIT I) \<equiv> \<lambda>x. (#\<lambda>xa. (#\<lambda>xb. (#SP (PR_CONST (monadic_WHILEIT I)) $ (\<lambda>xa. (#x $ xa#)) $ (\<lambda>x. (#xa $ x#)) $ xb#)#)#)"
fmrestrict_fset_null: fixes m :: "('a, 'b) fmap" shows "fmrestrict_fset {||} m = fmempty"
ps': shows "ps' = filter (\<lambda>p. real_of_int l - dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < real_of_int (fst p) \<and> real_of_int (fst p) < real_of_int l + dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps"
range_eq: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a \<le> b" shows "({a..b} = {c..d}) = (a = c \<and> b = d)"
mkArr_eqI': fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's"   and F' :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "mkArr A B F = mkArr A B F'"
run_grun: fixes A :: "('c, 'd) nba"   and k :: "nat"   and w :: "'c stream"   and r :: "'d stream"   and p :: "'d" assumes "run A (sdrop k w ||| r) p" shows "gurun A w (fromN (Suc k) ||| r) (k, p)"
borel_measurable_nn_integral: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> ennreal"   and N :: "'c measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M N)" shows "(\<lambda>x. integral\<^sup>N N (f x)) \<in> borel_measurable N"
derivation_well_ordered: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set" assumes "fixed_point_proof A f"   and "fixed_point_proof.derivation A less_eq f P" shows "well_ordered_set P less_eq"
append_eq_append_conv2: fixes xs :: "'g list"   and ys :: "'g list"   and zs :: "'g list"   and ts :: "'g list" shows "(xs @ ys = zs @ ts) = (\<exists>us. xs = zs @ us \<and> us @ ys = ts \<or> xs @ us = zs \<and> ys = us @ ts)"
assms_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and c :: "'a"   and exp :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* c"   and "\<not> nonrec \<Gamma>"   and "\<And>c x. x \<in> domA \<Gamma> \<Longrightarrow> P c (the (map_of \<Gamma> x))"   and "\<And>c. P c exp" shows "P c (Let \<Gamma> exp)"
cong: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "comm_monoid_add.Sum_any plus zero \<equiv> comm_monoid_add.Sum_any plus zero"
env_restr_below_self: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "f f|` S \<sqsubseteq> f"
lookup_combine_with_key: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m1 :: "('c, 'b) mapping"   and m2 :: "('c, 'b) mapping"   and x :: "'c" shows "lookup (combine_with_key f m1 m2) x = combine_options (f x) (lookup m1 x) (lookup m2 x)"
AddSA_CompFun_ran3_IFF: fixes SA1 :: "('a, 'b, 'c) seqauto"   and A :: "('a, 'b, 'c) hierauto"   and SA2 :: "('a, 'b, 'c) seqauto"   and SA3 :: "('a, 'b, 'c) seqauto"   and S :: "'a"   and T :: "'a"   and X :: "('a, 'b, 'c) seqauto set set" assumes "States SA1 \<inter> HAStates A = {}"   and "States SA2 \<inter> (HAStates A \<union> States SA1) = {}"   and "States SA3 \<inter> (HAStates A \<union> States SA1 \<union> States SA2) = {}"   and "S \<in> HAStates A"   and "T \<in> States SA1"   and "insert {} (insert {SA3, SA2} (ran (CompFun (A [++] (S, SA1))))) = X" shows "ran (CompFun (A [++] (S, SA1) [++] (T, SA2) [++] (T, SA3))) = X"
le_supI2: fixes b :: "'a"   and x :: "'a"   and a :: "'a" assumes "b \<le> x" shows "a \<sqinter> b \<le> x"
mk_total_program_def: fixes args :: "'c set \<times> ('c \<times> 'c) set set \<times> ('c \<times> 'c) set set" shows "mk_total_program args \<equiv> totalize (mk_program args)"
partial_magma_axioms: shows "partial_magma (\<cdot>\<^sub>C)"
GodIsEssential: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<forall>xa. \<P> xa w \<longrightarrow> xa x) \<longrightarrow> (\<forall>xa. \<P> xa w \<longrightarrow> xa x) \<and> (\<forall>xa. xa x \<longrightarrow> (\<^bold>\<box>(\<lambda>v. \<forall>x. existsAt x v \<longrightarrow> (\<forall>xa. \<P> xa w \<longrightarrow> xa x) \<longrightarrow> xa x)) w))\<rfloor>"
strong_supplementation: fixes x :: "i"   and y :: "i" assumes "\<not> P x y" shows "\<exists>z. P z x \<and> \<not> O z y"
functions_of: fixes A :: "'c set"   and B :: "'d set"   and P :: "'c set set" assumes "finite A"   and "finite B"   and "partition_on A P"   and "card P \<le> card B" shows "Twelvefold_Way_Entry7.functions_of P A B \<in> (A \<rightarrow>\<^sub>E B) // range_permutation A B"
compat_tr_compat: fixes NS :: "('a \<times> 'a) set"   and S :: "('a \<times> 'b) set" assumes "NS O S \<subseteq> S" shows "NS\<^sup>* O S \<subseteq> S"
Suc_reindex_ivl: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "prod g {m..n} * g (Suc n) = g m * (\<Prod>i = m..n. g (Suc i))"
preserves_dom: fixes f :: "'a" assumes "S.arr f" shows "S.dom (S.trg f) = S.trg (S.dom f)"
degree_of_int: fixes k :: "int" shows "degree (of_int k) = 0"
hom_F: fixes f' :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "f' \<in> F" shows "homogeneous f'"
N_prime: fixes p :: "int" assumes "prime p" shows "N p = N 1 - 1 / q ^ nat p"
ins_inv1_inv2_1: fixes t :: "('b, 'c) RBT_Impl.rbt"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'b"   and x :: "'c" assumes "inv1 t"   and "inv2 t" shows "inv2 (ord.rbt_ins less f k x t)"
Lcm_fin_least: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "finite A"   and "\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times b a" shows "dvd.dvd times (semiring_gcd.Lcm_fin one zero lcm A) a"
arr_def: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "A_B.arr f \<equiv> A_B.domains f \<noteq> {} \<or> A_B.codomains f \<noteq> {}"
simps_2: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x1 :: "'d"   and x2 :: "'e" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
Mp_coeff: fixes m :: "int"   and f :: "int poly"   and i :: "nat" shows "poly.coeff (poly_mod.Mp m f) i = poly_mod.M m (poly.coeff f i)"
zero_neq_neg_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.ring_char_0 one times plus zero minus uminus" shows "zero \<noteq> uminus one"
VPow_vinsert: fixes a :: "V"   and A :: "V" shows "VPow (vinsert a A) = VPow A \<union>\<^sub>\<circ> VLambda (VPow A) (vinsert a) `\<^sub>\<circ> VPow A"
pdevs_apply_zero_pdevs: fixes i :: "nat" shows "pdevs_apply zero_pdevs i = (0::'d)"
component_JN: fixes i :: "'c"   and I :: "'c set"   and F :: "'c \<Rightarrow> 'd program" assumes "i \<in> I" shows "F i \<le> JOIN I F"
subset_UnE: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "C \<subseteq> A \<union> B" obtains A' :: "'b set" and B' :: "'b set" where "A' \<subseteq> A" and "B' \<subseteq> B" and "C = A' \<union> B'"
that: fixes f :: "'a \<Rightarrow> 'b" assumes "continuous_on U f"   and "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b"   and "\<And>x. x \<in> U \<Longrightarrow> (f x = a) = (x \<in> S)"   and "\<And>x. x \<in> U \<Longrightarrow> (f x = b) = (x \<in> T)" shows "thesis"
axioms_3: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "vsv (\<CC>\<lparr>Dom\<rparr>)"
dom: shows "dom (f'(l \<mapsto> t')) = dom (f(l \<mapsto> t))"
inf_left_unit: fixes x :: "'a" shows "bot \<sqinter> - x = bot"
alw_ev_conjI: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream"   and Q :: "'b \<Rightarrow> bool" assumes "alw (holds P) xs"   and "alw (ev (holds Q)) xs" shows "alw (ev (holds (\<lambda>x. P x \<and> Q x))) xs"
dec_seqn: fixes a :: "int"   and A :: "int set"   and n :: "nat" assumes "a \<in> A"   and "A \<subseteq> Zset"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)" shows "dec_seq A a (Suc n) < dec_seq A a n"
ipurge_tr_aux_nil: fixes u :: "'c"   and U :: "'c set"   and xs :: "'d list"   and D :: "'d \<Rightarrow> 'c"   and I :: "('c \<times> 'c) set" assumes "u \<in> U"   and "\<And>x. x \<in> set xs \<Longrightarrow> (u, D x) \<in> I" shows "ipurge_tr_aux I D U xs = []"
partial_function_definitions_axioms: shows "partial_function_definitions sum_bot_ord (flat_lub Bottom)"
simps_2: fixes cs :: "'edge list"   and a :: "'edge"   and as :: "'edge list" shows "same_level_path_aux cs (a # as) = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> same_level_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> False | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> same_level_path_aux cs' as | _ \<Rightarrow> same_level_path_aux cs as)"
Pxy: shows "finite_entropy (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
arrI: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "arr f"
normal_subgroup_intersect: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and N :: "'c set" assumes "Group.group G"   and "M \<lhd> G"   and "N \<lhd> G" shows "M \<inter> N \<lhd> G"
b_nz: shows "\<forall>\<^sub>F x in at_top. 0 < b x"
regularCard_UNION: fixes r :: "('a \<times> 'a) set"   and As :: "'a \<Rightarrow> 'b set"   and B :: "'b set" assumes "Card_order r"   and "regularCard r"   and "relChain r As"   and "B \<subseteq> \<Union> (As ` Field r)"   and "|B| <o r" shows "\<exists>i\<in>Field r. B \<subseteq> As i"
PolysI_alt: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set" assumes "indets p \<subseteq> X" shows "p \<in> P[X]"
con_comp_ipurge_tr_filter: fixes I :: "('f \<times> 'f) set"   and D :: "'g \<Rightarrow> 'f"   and E :: "'h \<Rightarrow> 'f"   and p :: "'g \<Rightarrow> 'e"   and q :: "'h \<Rightarrow> 'e"   and u :: "'f option"   and xs :: "'e list" shows "ipurge_tr (con_comp_pol I) (con_comp_map D E p q) u (filter (\<lambda>x. x \<in> range p \<union> range q) xs) = ipurge_tr (con_comp_pol I) (con_comp_map D E p q) u xs"
causal_program_history_suffix: fixes is\<^sub>s\<^sub>b :: "instr list"   and sb :: "'b memref list"   and sb' :: "'b memref list" assumes "causal_program_history is\<^sub>s\<^sub>b (sb @ sb')" shows "causal_program_history is\<^sub>s\<^sub>b sb'"
WT_calleeI: fixes \<I> :: "('a, 'b) \<I>"   and callee :: "'a \<Rightarrow> ('b \<times> 'c) spmf" assumes "\<And>call ret s. \<lbrakk>call \<in> outs_\<I> \<I>; (ret, s) \<in> set_spmf (callee call)\<rbrakk> \<Longrightarrow> ret \<in> responses_\<I> \<I> call" shows "\<I> \<turnstile>c callee \<surd>"
pre_ranks_finite: fixes n :: "nat"   and \<nu> :: "'a"   and q :: "'b" shows "finite (pre_ranks (\<lambda>q. state_rank q n) \<nu> q)"
big_circle_def: shows "big_circle = part_circlepath 0 R 0 pi"
ws_update_foreach_def: fixes E :: "('c \<times> 'b) set"   and u :: "'c"   and p :: "'c list"   and V :: "'b set"   and ws :: "'b \<Rightarrow> 'c list option" shows "ws_update_foreach E u p V ws \<equiv> FOREACH (LIST_SET_REV_TAG (E `` {u})) (\<lambda>v (V, ws). if v \<in> V then RETURN (V, ws) else ASSERT (v \<notin> dom ws) \<bind> (\<lambda>_. RETURN (insert v V, ws(v \<mapsto> u # p)))) (V, ws)"
Fvars_subst: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "Fvars (subst \<phi> t x) = Fvars \<phi> - {x} \<union> (if x \<in> Fvars \<phi> then FvarsT t else {})"
eq: shows "set (fold (\<lambda>x. f (g x)) xs (f (g x) ys)) = (\<Union>x\<in>set xs. set (g x)) \<union> set (f (g x) ys)"
VLambdaI: fixes a :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> A" shows "\<langle>a, f a\<rangle> \<in>\<^sub>\<circ> VLambda A f"
cball_divide_subset_numeral: fixes x :: "'a"   and e :: "real"   and w :: "num" shows "cball x (e / numeral w) \<subseteq> cball x e"
assms: shows "\<Inter> (dca.alphabet ` set AA) = \<Union> (dca.alphabet ` set AA)"
merge_single2: fixes A :: "'a cset" shows "merge A (\<lambda>x. return (csingle x)) = return A"
CallParamsThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and es :: "char list exp list"   and vs :: "val list"   and ex :: "char list exp"   and es' :: "char list exp list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs @ throw ex # es',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw ex,s\<^sub>2\<rangle>"
a5: shows "\<T> = \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set (filter_compliant_stateful_ACS G M edgesList)\<rparr>"
H_I: fixes g :: "'a" assumes "g \<in> carrier G"   and "M1 #> g = M1" shows "g \<in> H"
poly_at_top_or_bot_at_bot: fixes p :: "real poly" assumes "1 \<le> degree p"   and "0 < lead_coeff p" shows "filterlim (poly p) (if even (degree p) then at_top else at_bot) at_bot"