hcju/isabelleps huggingface dataset:
region_set'_id: fixes R :: "('d \<Rightarrow> real) set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and v :: "'d \<Rightarrow> real"   and c :: "nat"   and r :: "'d list" assumes "R \<in> {region X I r |I r. valid_region X k I r}"   and "v \<in> R"   and "finite X"   and "0 \<le> c"   and "\<forall>x\<in>set r. c \<le> k x"   and "set r \<subseteq> X" shows "[[r\<rightarrow>real c]v]\<^sub>{region X I r |I r. valid_region X k I r} = region_set' R r (real c) \<and> [[r\<rightarrow>real c]v]\<^sub>{region X I r |I r. valid_region X k I r} \<in> {region X I r |I r. valid_region X k I r} \<and> [r\<rightarrow>real c]v \<in> [[r\<rightarrow>real c]v]\<^sub>{region X I r |I r. valid_region X k I r}"
assms: fixes x :: "nat" assumes "x \<in> set xs - set ys" shows "\<exists>y. y \<in> set ys \<and> x \<le> y"
openE: fixes m :: "'a llist set" assumes "m open" obtains "m \<in> T"
prob_space: fixes M :: "'a pmf" shows "measure_pmf.prob M (space (measure_pmf M)) = 1"
indep_card_eq_dim_span: fixes B :: "('a, 'b) vec set" assumes "vec.independent B" shows "finite B \<and> card B = vec.dim (vec.span B)"
distinct_disc_10: fixes event :: "('d, 'e, 'f) event" assumes "isRecv event" shows "\<not> isSend event"
client0_def: shows "client0 \<equiv> ''x'' ::= N 0;; WHILE Less (N 0) (V ''n'') DO (''x'' ::= Plus (V ''x'') (V ''n'');; ''n'' ::= V ''nondet'');; IF BEq (V ''x'') (N 2000000) THEN ''low'' ::= V ''high'' ELSE SKIP"
ex_loc_ass_def: shows "ex_loc_ass \<equiv> \<lambda>x. if x = home then {''Patient''} else if x = hospital then {''Doctor'', ''Eve''} else {}"
loB\<^sub>2: shows "falsifies\<^sub>l (B @ [d]) lo"
in_box_complex_iff: fixes x :: "complex"   and a :: "complex"   and b :: "complex" shows "(x \<in> box a b) = (Re x \<in> {Re a<..<Re b} \<and> Im x \<in> {Im a<..<Im b})"
intros_1: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option" shows "executionally_equivalent e1 s1 r1 e2 s2 r2 []"
hyps: assumes "0 < M" shows "(\<Sum>n = 1..2 ^ M. 1 / real n) = 1 + (\<Sum>m = 1..M. \<Sum>n = 2 ^ (m - 1) + 1..2 ^ m. 1 / real n)"
cd_trans0: fixes j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"   and "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j" shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"
two_chain_horizontal_boundary_is_boundary_chain: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set" shows "boundary_chain (two_chain_horizontal_boundary twoChain)"
execute: fixes p :: "'state"   and a :: "'action" assumes "p \<in> nodes"   and "en a p" shows "ex a p \<in> nodes"
winding_number_pos_meets: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and w :: "complex" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "1 \<le> Re (winding_number \<gamma> z)"   and "w \<noteq> z" shows "\<exists>a>0. z + complex_of_real a * (w - z) \<in> path_image \<gamma>"
scale_left_commute: fixes a :: "'c"   and b :: "'c"   and x :: "'c" shows "a * (b * x) = b * (a * x)"
set_intros_4: fixes yc :: "('c, 'd) Term.term"   and x22 :: "('c, 'd) Term.term list"   and ye :: "'d"   and x21 :: "'c" assumes "yc \<in> set x22"   and "ye \<in> fv yc" shows "ye \<in> fv (Fun x21 x22)"
assms_3: shows "\<FF>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
nth_map: fixes n :: "nat"   and xs :: "'g list"   and f :: "'g \<Rightarrow> 'h" assumes "n < length xs" shows "map f xs ! n = f (xs ! n)"
le2: shows "adjoint M1 * WS (W P k) * M1 \<le>\<^sub>L adjoint M1 * WS (W Q k) * M1"
augmentingFlow_def: fixes p :: "(nat \<times> nat) list" shows "augmentingFlow p \<equiv> \<lambda>(u, v). if (u, v) \<in> set p then resCap p else (0::'capacity)"
extending_maps_Union: fixes \<F> :: "'a set set"   and T :: "'b set"   and K :: "'a set"   and h :: "'a \<Rightarrow> 'b" assumes "finite \<F>"   and "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>g. continuous_on S g \<and> g ` S \<subseteq> T \<and> (\<forall>x\<in>S \<inter> K. g x = h x)"   and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"   and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>; \<not> X \<subseteq> Y; \<not> Y \<subseteq> X\<rbrakk> \<Longrightarrow> X \<inter> Y \<subseteq> K" shows "\<exists>g. continuous_on (\<Union> \<F>) g \<and> g ` \<Union> \<F> \<subseteq> T \<and> (\<forall>x\<in>\<Union> \<F> \<inter> K. g x = h x)"
has_white_path_to_def: fixes x :: "'d"   and y :: "'d" shows "x has_white_path_to y = (\<lambda>s. (\<lambda>x y. (x points_to y) s \<and> white y s)\<^sup>*\<^sup>* x y)"
k_frontier_def: fixes a :: "'a" shows "Clock.k_frontier a \<equiv> map (local.clock_simInit a \<circ> envObs a) envInit"
monoid_comm_monoidI: assumes "\<And>x y. \<lbrakk>x \<in> carrier (homology_group p X); y \<in> carrier (homology_group p X)\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>homology_group p X\<^esub> y = y \<otimes>\<^bsub>homology_group p X\<^esub> x" shows "Group.comm_monoid (homology_group p X)"
map_upds_fold_map_upd: fixes m :: "'a \<Rightarrow> 'b option"   and ks :: "'a list"   and vs :: "'b list" shows "m(ks [\<mapsto>] vs) = foldl (\<lambda>m (k, v). m(k \<mapsto> v)) m (zip ks vs)"
dim_row_of_dagger: fixes M :: "complex Matrix.mat" shows "dim_row M\<^sup>\<dagger> = dim_col M"
ALG_P'_rest: fixes n :: "nat"   and qs :: "'c list"   and Strat :: "(nat \<times> nat list) list"   and init :: "'c list"   and x :: "'c"   and y :: "'c" assumes "n < length qs"   and "n < length Strat" shows "ALG_P' Strat (take n qs @ [qs ! n]) init n x y = ALG_P' (take n Strat @ [Strat ! n]) (take n qs @ [qs ! n]) init n x y"
bind_assoc: fixes f :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'c \<Rightarrow> 'd measure"   and R :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "g \<in> N \<rightarrow>\<^sub>M subprob_algebra R" shows "M \<bind> f \<bind> g = M \<bind> (\<lambda>x. f x \<bind> g)"
closure_bounded_linear_image_subset: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "bounded_linear f" shows "f ` closure S \<subseteq> closure (f ` S)"
assms: fixes w :: "(real, 'n) vec" assumes "w \<in> S" shows "orthogonal v w"
CConcStart2: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "p \<noteq> []"   and "x \<notin> dom (Cp a)" shows "Cp (list2FWpolicy (p @ [a])) x = Cp (list2FWpolicy p) x"
Sup_apply: fixes A :: "('d \<Rightarrow> 'c) set"   and x :: "'d" shows "(\<Squnion> A) x = (\<Squnion>f\<in>A. f x)"
collinear_3_eq_affine_dependent: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "collinear {a, b, c} = (a = b \<or> a = c \<or> b = c \<or> affine_dependent {a, b, c})"
\<alpha>_simp: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.arr f"   and "C.arr g"   and "C.arr h" shows "\<a>\<^sub>C[f, g, h] = (f \<otimes>\<^sub>C g \<otimes>\<^sub>C h) \<cdot>\<^sub>C \<a>\<^sub>C[C.dom f, C.dom g, C.dom h]"
weakly_Pareto_optimal_for_men_def: fixes \<mu> :: "('m \<times> 'w) set" shows "weakly_Pareto_optimal_for_men \<mu> = (simple \<mu> \<and> (\<nexists>\<mu>'. simple \<mu>' \<and> weakly_preferred_by_men \<mu> \<mu>' \<and> strictly_preferred_by_a_man \<mu> \<mu>'))"
while_SE_def: fixes c :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> (unit \<times> 'a) option" shows "_while_SE c B \<equiv> Rel2Mon (lfp (\<Gamma> c (Mon2Rel B)))"
degen_run_find_change: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat"   and i :: "nat"   and j :: "nat"   and q :: "'Q"   and n :: "nat"   and q' :: "'Q"   and n' :: "nat" assumes "num_acc \<noteq> 0"   and "graph_defs.is_run (degeneralize_ext m) r"   and "i \<le> j"   and "r i = (q, n)"   and "r j = (q', n')"   and "n \<noteq> n'" obtains k :: "nat" and qk :: "'Q" where "i \<le> k" and "k < j" and "r k = (qk, n)" and "n \<in> acc qk"
negD: fixes x :: "int" assumes "x < 0" shows "\<exists>n. x = - int (Suc n)"
exL_exMap: fixes f :: "Label \<Rightarrow> sterm option"   and lz :: "Label \<Rightarrow> sterm option"   and f' :: "Label \<Rightarrow> sterm option"   and L1 :: "char list set"   and L2 :: "char list set" assumes "dom f = dom lz"   and "dom f' = dom f"   and "finite L1"   and "\<forall>l\<in>dom f. \<forall>s p. s \<notin> L1 \<and> p \<notin> L1 \<and> s \<noteq> p \<longrightarrow> (\<exists>t. (the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> (\<forall>z. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => z \<longrightarrow> (\<exists>u. t => u \<and> z => u))) \<and> the (f' l) = \<sigma>[s,p] t)"   and "finite L2"   and "\<forall>l\<in>dom lz. \<forall>s p. s \<notin> L2 \<and> p \<notin> L2 \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (lz l) = \<sigma>[s,p] t)" shows "\<exists>L'. finite L' \<and> (\<exists>lu. dom lu = dom f \<and> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L' \<and> p \<notin> L' \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (f' l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (lu l) = \<sigma>[s,p] t)) \<and> (\<forall>l\<in>dom f. body (the (f' l))) \<and> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L' \<and> p \<notin> L' \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (lz l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (lu l) = \<sigma>[s,p] t)) \<and> (\<forall>l\<in>dom f. body (the (lz l))))"
lcos_module_rev: fixes x :: "'a"   and x' :: "'a" assumes "Group.group G"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "inv x \<otimes> x' \<in> H" shows "x' \<in> x <# H"
mult_cancel: fixes s :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Z :: "'a multiset"   and Y :: "'a multiset" assumes "trans s"   and "irrefl s" shows "((X + Z, Y + Z) \<in> mult s) = ((X, Y) \<in> mult s)"
Hrew: shows "(take n xs @ drop (Suc n) xs) ! p = drop (Suc n) xs ! (p - n)"
residualInject'_6: fixes Q :: "('d, 'e, 'f) psi"   and \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" shows "(RTau Q = \<alpha> \<prec> P) = (\<alpha> = \<tau> \<and> P = Q)"
analyzed_def: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set" assumes "intruder_model arity Ana" shows "intruder_model.analyzed arity public Ana M \<equiv> \<forall>t. intruder_model.intruder_deduct arity public Ana M t = intruder_model.intruder_synth arity public M t"
isSome_simps_2: fixes x :: "'a option" shows "(\<not> isSome x) = (x = None)"
size_regex_estimation': fixes x :: "'a"   and r :: "'a Regex.regex"   and y :: "nat"   and f :: "'a \<Rightarrow> nat" assumes "x \<in> regex.atms r"   and "y \<le> f x" shows "y \<le> regex.size_regex f r"
T_def: shows "T \<equiv> \<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)"
upd_cs_Append: fixes cs :: "'edge list"   and as :: "'edge list"   and cs' :: "'edge list"   and as' :: "'edge list"   and cs'' :: "'edge list" assumes "upd_cs cs as = cs'"   and "upd_cs cs' as' = cs''" shows "upd_cs cs (as @ as') = cs''"
take_append: fixes n :: "nat"   and xs :: "'a list"   and ys :: "'a list" shows "take n (xs @ ys) = take n xs @ take (n - length xs) ys"
lnull_ltakeWhile: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "lnull (ltakeWhile P xs) = (\<not> lnull xs \<longrightarrow> \<not> P (lhd xs))"
N: shows "\<forall>x. N x \<in> null_sets M \<and> null_part M (S x) \<subseteq> N x"
order_refl: fixes less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "'x" assumes "class.preorder less_eq less" shows "less_eq x x"
distinct_1: fixes name :: "name"   and lam1' :: "lam"   and lam2' :: "lam" shows "Var name \<noteq> App lam1' lam2'"
Bfun_metric_def: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "Bfun f F = (\<exists>y K. 0 < K \<and> (\<forall>\<^sub>F x in F. dist (f x) y \<le> K))"
div_half_nat: fixes y :: "nat"   and x :: "nat" assumes "y \<noteq> 0" shows "(x div y, x mod y) = (let q = 2 * (x div 2 div y); r = x - q * y in if y \<le> r then (q + 1, r - y) else (q, r))"
compat2: fixes y :: "nat"   and x :: "nat"   and z :: "nat" assumes "y < x"   and "z \<le> y" shows "z < x"
res: shows "dmu_array_row_main fs (fs !! i) i (iarray_append dmus (IArray [fs !! i \<bullet> fs !! 0])) 0 = res"
invar_merge: fixes merge :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> 'b tree"   and q1 :: "'b tree"   and q2 :: "'b tree" assumes "Heap_Merge merge"   and "heap q1"   and "heap q2" shows "heap (merge q1 q2)"
union: shows "{x \<in> T. 0 < norm (\<mu> x)} = \<Union> (range S)"
Collect_mono: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
conj_xor_distrib: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "conj x (boolean_algebra.xor conj disj compl y z) = boolean_algebra.xor conj disj compl (conj x y) (conj x z)"
AE_I_eq_1: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "prob_space M"   and "emeasure M {x \<in> space M. P x} = 1"   and "{x \<in> space M. P x} \<in> sets M" shows "almost_everywhere M P"
index_sum: fixes s :: "'c list"   and q :: "'c" assumes "distinct s"   and "q \<in> set s" shows "index s q = (\<Sum>e\<in>set s. if e < q in s then 1 else 0)"
stc_approx_self: fixes x :: "complex star" assumes "x \<in> HFinite" shows "stc x \<approx> x"
project_axioms: shows "project (set \<circ> \<sigma> \<Sigma>) \<pi>"
l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes to_tree_order_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog" shows "l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order_si get_child_nodes get_shadow_root \<equiv> to_tree_order_si = l_to_tree_order_si\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_to_tree_order_si get_child_nodes get_shadow_root"
mult_delta_left: fixes b :: "bool"   and x :: "'a"   and y :: "'a" shows "(if b then x else (0::'a)) * y = (if b then x * y else (0::'a))"
trivial_group_subset: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "trivial_group G = (carrier G \<subseteq> {\<one>\<^bsub>G\<^esub>})"
elims_3: fixes x :: "nat"   and xa :: "'person"   and xb :: "'person \<Rightarrow> 'color" assumes "\<not> local.leaves x xa xb" obtains n :: "nat" and p :: "'person" and w :: "'person \<Rightarrow> 'color" where "x = n" and "xa = p" and "xb = w" and "\<not> (\<forall>w'. local.possible n p w w' \<longrightarrow> w' p = w p)"
distinctFrame: fixes A\<^sub>F :: "name list"   and C :: "'d"   and \<Psi>\<^sub>F :: "'e" assumes "A\<^sub>F \<sharp>* C" obtains A\<^sub>F' :: "name list" where "\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle> = \<langle>A\<^sub>F', \<Psi>\<^sub>F\<rangle>" and "distinct A\<^sub>F'" and "A\<^sub>F' \<sharp>* C"
isSepaux: fixes p :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "p \<noteq> []"   and "noDenyAll (a # p)"   and "separated (a # p)"   and "x \<in> dom (C (DenyAllFromTo (first_srcNet a) (first_destNet a) \<oplus> DenyAllFromTo (first_destNet a) (first_srcNet a) \<oplus> a))" shows "x \<notin> dom (C (list2FWpolicy p))"
bz: shows "(p, q, u, v, d) = bezout a b"
direct_decomp_unique: fixes A :: "'a set"   and ss :: "'a set list"   and qs :: "'a list"   and qs' :: "'a list" assumes "direct_decomp A ss"   and "qs \<in> listset ss"   and "qs' \<in> listset ss"   and "sum_list qs = sum_list qs'" shows "qs = qs'"
word_log_esimps_3: fixes x :: "'b word" shows "x || 0 = x"
separation_normal: fixes S :: "'b set"   and T :: "'b set" assumes "closed S"   and "closed T"   and "S \<inter> T = {}" obtains U :: "'b set" and V :: "'b set" where "open U" and "open V" and "S \<subseteq> U" and "T \<subseteq> V" and "U \<inter> V = {}"
simps_1: fixes C :: "acom"   and l :: "'a \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "preList [] C l s = True"
vfinite_vimage: fixes A :: "V"   and Aa :: "V"   and f :: "V \<Rightarrow> V" assumes "vfinite A" shows "vfinite (VLambda Aa f `\<^sub>\<circ> A)"
simps_17: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x1 :: "('d, 'e) Term.term" shows "(case send\<langle>x1\<rangle>\<^sub>s\<^sub>t of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = f1 x1"
ide_char: fixes f :: "'s" assumes "S.arr f" shows "S.ide f = (S.Dom f = S.Cod f \<and> S.Fun f = (\<lambda>x\<in>S.Dom f. x))"
suma_distincts: 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"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I" shows "A \<noteq> B \<and> B \<noteq> C \<and> D \<noteq> E \<and> E \<noteq> F \<and> G \<noteq> H \<and> H \<noteq> I"
af_congruent_axioms: shows "af_congruent (\<sim>\<^sub>C)"
ts_process_next_events: fixes T :: "'c list set"   and x :: "'c"   and xs :: "'c list" assumes "trace_set T" shows "(x \<in> next_events (ts_process T) xs) = (xs @ [x] \<in> T)"
PP_def: shows "PP \<equiv> (B\<^sub>1 ===> B\<^sub>1 ===> B\<^sub>1) ===> B\<^sub>1 ===> (B\<^sub>1 ===> B\<^sub>1 ===> B\<^sub>1) ===> (B\<^sub>1 ===> B\<^sub>1) ===> ((=) ===> B\<^sub>1 ===> B\<^sub>1) ===> rel_set B\<^sub>1 ===> (B\<^sub>2 ===> B\<^sub>2 ===> B\<^sub>2) ===> B\<^sub>2 ===> (B\<^sub>2 ===> B\<^sub>2 ===> B\<^sub>2) ===> (B\<^sub>2 ===> B\<^sub>2) ===> ((=) ===> B\<^sub>2 ===> B\<^sub>2) ===> (=)"
symmetric_def: fixes G :: "('b, 'c) pre_digraph" shows "symmetric G \<equiv> sym (arcs_ends G)"
lconjby_uminus: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus (plus x (uminus y)) x = uminus (minus (plus x y) x)"
xt1_7: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<in> U"   and "x \<in> U"   and "z \<in> U"   and "y <\<^sub>o\<^sub>w x"   and "z \<le>\<^sub>o\<^sub>w y" shows "z <\<^sub>o\<^sub>w x"
chine_simps_3: shows "C.cod trg_\<mu>.chine = trg_\<mu>.cod.apex"
simps_6: fixes v :: "'d"   and va :: "('c, 'd) Digit"   and vb :: "('c, 'd) FingerTreeStruc"   and vc :: "('c, 'd) Digit"   and xs :: "('c, 'd) Node list"   and x :: "('c, 'd) Node" shows "app3 (Deep v va vb vc) xs (Single x) = nrcons (rconsNlist (Deep v va vb vc) xs) x"
plus: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ring_ops ops R" shows "(R ===> R ===> R) (arith_ops_record.plus ops) (+)"
globs_upd_gobj_new: fixes s :: "st"   and r :: "oref"   and n :: "vn"   and v :: "val" assumes "globs s r = None" shows "globs (upd_gobj r n v s) = globs s"
stmt_step_load_tmps_range': fixes \<theta> :: "nat \<Rightarrow> nat option"   and s :: "stmt"   and t :: "nat"   and s' :: "stmt"   and t' :: "nat"   and "is" :: "instr list" assumes "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is)" shows "load_tmps is = {i. t \<le> i \<and> i < t'}"
elims: fixes x :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and y :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "Subx.generalize_fundef x = y" obtains xs :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list" and ar :: "nat" where "x = Fundef xs ar" and "y = Fundef (map Subx.generalize_instr xs) ar"
star_1l: fixes x :: "'a" shows "x \<cdot> x\<^sup>\<dagger> \<le> x\<^sup>\<dagger>"
assoc'_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "\<a>\<^sup>-\<^sup>1[a, b, c] \<equiv> \<langle>\<langle>\<pp>\<^sub>1[a, b \<otimes> c], \<pp>\<^sub>1[b, c] \<cdot> \<pp>\<^sub>0[a, b \<otimes> c]\<rangle>, \<pp>\<^sub>0[b, c] \<cdot> \<pp>\<^sub>0[a, b \<otimes> c]\<rangle>"
ltlc_next_free_iff: fixes \<phi> :: "'b ltlc" shows "next_free (ltlc_to_pltl \<phi>) = ltlc_next_free \<phi>"
is_red_implies_0_red_finite: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "finite B"   and "pmdl B \<subseteq> pmdl A"   and "\<And>q. \<lbrakk>q \<in> pmdl A; q \<noteq> 0\<rbrakk> \<Longrightarrow> is_red B q"   and "p \<in> pmdl A" shows "(red B)\<^sup>*\<^sup>* p 0"
e: shows "e \<in> UNIV \<rightarrow> {- 1..1}"
divide_le_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide b c) a = (if less zero c then less_eq b (times a c) else if less c zero then less_eq (times a c) b else less_eq zero a)"
intro: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure" assumes "Filtration.filtration M F" shows "filtrated_prob_space_axioms M F"
2: shows "t \<le> v * v\<^sup>T \<sqinter> - - g"
bounded_pos_less: fixes S :: "'a set" shows "bounded S = (\<exists>b>0. \<forall>x\<in>S. norm x < b)"
prefix_finI_item: fixes a :: "'a"   and b :: "'a"   and u :: "'a list"   and v :: "'a list" assumes "a = b"   and "u \<le> v" shows "a # u \<le> b # v"
blaschke_preserve_distance_formula: fixes k :: "complex"   and u :: "complex_homo"   and v :: "complex_homo" assumes "of_complex k \<in> unit_disc"   and "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance_formula (to_complex (moebius_pt (blaschke k) u)) (to_complex (moebius_pt (blaschke k) v)) = poincare_distance_formula (to_complex u) (to_complex v)"
simps_5: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_WbisT tval aval indis (c1 | c2) = (PL_Indis.SC_WbisT tval aval indis c1 \<and> PL_Indis.SC_WbisT tval aval indis c2 \<or> noWhile (c1 | c2) \<and> PL_Indis.SC_discr aval indis (c1 | c2) \<or> PL_Indis.SC_siso tval aval indis (c1 | c2))"
ind_space_is_top_space: shows "topological_space U it.ind_is_open"