hcju/isabelleps huggingface dataset:
add_im_sheaf_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and S' :: "'f set"   and is_open' :: "'f set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f" assumes "im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "im_sheaf.add_im_sheaf S add_str f \<equiv> \<lambda>V. add_str (f \<^sup>\<inverse> S V)"
irrefl: fixes a :: "'d multiset" shows "\<not> a \<subset># a"
inverse_formula_renaming: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a set"   and L :: "'a Literal set set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.rename_formula A (propositional_atoms.rename_formula A L) = L"
ideD_3: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.cod a = a"
"2_2": fixes t :: "'t" assumes "0 < zcount M t" shows "\<exists>s\<le>t. s \<in>\<^sub>A frontier (c_imp c loc2)"
lex_two_compat: fixes ns1 :: "('b \<times> 'b) set"   and s1 :: "('b \<times> 'b) set"   and ns2 :: "('c \<times> 'c) set"   and s2 :: "('c \<times> 'c) set"   and ab1 :: "'b \<times> 'c"   and ab2 :: "'b \<times> 'c"   and ab3 :: "'b \<times> 'c" assumes "ns1 O s1 \<subseteq> s1"   and "s1 O ns1 \<subseteq> s1"   and "s1 O s1 \<subseteq> s1"   and "ns1 O ns1 \<subseteq> ns1"   and "ns2 O s2 \<subseteq> s2"   and "(ab1, ab2) \<in> lex_two s1 ns1 ns2"   and "(ab2, ab3) \<in> lex_two s1 ns1 s2" shows "(ab1, ab3) \<in> lex_two s1 ns1 s2"
simps_1: fixes c :: "rat"   and irr :: "rat poly list" shows "factorize_rat_poly_main c irr [] = (c, irr)"
in_gets: fixes r :: "'b"   and s' :: "'c"   and f :: "'c \<Rightarrow> 'b"   and s :: "'c" shows "((r, s') = fst (gets f s)) = (r = f s \<and> s' = s)"
matches_and_comm: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and m' :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (MatchAnd m m') a p = matches \<gamma> (MatchAnd m' m) a p"
1: fixes g :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G" shows "\<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> Dube (Suc (card X)) (maxdeg F) \<and> (f \<notin> F' \<longrightarrow> q f = 0))"
sgn_sgn: fixes a :: "'a" shows "sgn (sgn a) = sgn a"
shortest_augmenting_path: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity" assumes "NFlow c s t f"   and "\<exists>p. NPreflow.isAugmentingPath c s t f p" shows "Graph.isShortestPath (cf_of f) s (shortest_augmenting_path f) t"
hyps_3: assumes "S \<noteq> {}" shows "\<exists>f d. (\<Sum>a\<in>S. f a * a) = d \<and> (\<forall>a\<in>S. d dvd a) \<and> (\<forall>d'. (\<forall>a\<in>S. d' dvd a) \<longrightarrow> d' dvd d)"
cpt_def: fixes P :: "'c set"   and env :: "'c \<Rightarrow> nat"   and \<pi>l :: "'d list" shows "cpt P env \<pi>l \<equiv> \<forall>p\<in>P. env p < length \<pi>l"
check_types_def: fixes P :: "'b prog"   and mxs :: "nat"   and mxl :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option err list" shows "check_types P mxs mxl \<tau>s \<equiv> set \<tau>s \<subseteq> states P mxs mxl"
UNF_defs: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "UNF_on r A = (\<forall>a\<in>A. \<forall>b c. (a, b) \<in> r\<^sup>! \<and> (a, c) \<in> r\<^sup>! \<longrightarrow> b = c)"
fst_def: fixes prod :: "'e \<times> 'f" shows "fst prod = (case prod of (x1, x2) \<Rightarrow> x1)"
select_def: shows "Metis.select = (\<lambda>x. x)"
ah_il_def: fixes h1 :: "'b \<Rightarrow> 'b set"   and h2 :: "'b \<Rightarrow> 'b set" shows "h1 [*] h2 \<equiv> \<nexists>m1 m2. m1 \<in> h2 m2 \<and> m2 \<in> h1 m1"
refine_basic: fixes Tc :: "('b, 'c) TS_scheme"   and R :: "('d \<times> 'b) set"   and Ta :: "('d, 'e) TS_scheme" assumes "init Tc \<subseteq> R `` init Ta"   and "{R} TS.trans Ta, TS.trans Tc {> R}" shows "PO_refines R Ta Tc"
glb_def: fixes S :: "'a set" shows "\<Sqinter>S = (THE x. S >>| x)"
sets_def: fixes M :: "'b measure" shows "sets M = fst (snd (Rep_measure M))"
fix_transform: fixes g :: "'h \<rightarrow> 'g"   and f :: "'g \<rightarrow> 'h"   and F :: "'g \<rightarrow> 'g" assumes "\<And>x. g\<cdot>(f\<cdot>x) = x" shows "fix\<cdot>F = g\<cdot>(fix\<cdot>(f oo F oo g))"
distinct_203: fixes x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term"   and x12 :: "Syntax.term" shows "Syntax.Let x81 x82 x83 \<noteq> Prj2 x12"
le_neq_trans: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subseteq># b"   and "a \<noteq> b" shows "a \<subset># b"
add_rel_carrier: fixes X :: "('a set \<times> 'b) set"   and Y :: "('a set \<times> 'b) set" assumes "X \<in> stk1.carrier_direct_lim"   and "Y \<in> stk1.carrier_direct_lim" shows "stk1.add_rel X Y \<in> stk1.carrier_direct_lim"
part_to_full': fixes C :: "com"   and ps :: "char list \<Rightarrow> int option"   and m :: "nat"   and ps' :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int" assumes "(C, ps) \<Rightarrow>\<^sub>A m \<Down> ps'" shows "(C, emb ps s) \<Rightarrow> m \<Down> emb ps' s"
set_of_ivl_def: fixes x :: "'e \<times> 'e" shows "set_of_ivl x = {fst x..snd x}"
create_factor_tree: fixes xs :: "int poly list" assumes "xs \<noteq> []" shows "factors_of_factor_tree (create_factor_tree xs) = mset xs"
Contingent_def: fixes x :: "e" shows "Contingent x \<equiv> \<^bold>\<diamond>E! x \<^bold>\<and> \<^bold>\<not>Necessary x"
is_interval_oc: fixes r :: "real"   and s :: "real" shows "is_interval {r<..s}"
map_comp: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and v :: "'f multiset" shows "image_mset g (image_mset f v) = image_mset (g \<circ> f) v"
divmod_int_def: fixes n :: "int"   and m :: "int" shows "divmod_int n m = (n div m, n mod m)"
type_def: shows "\<exists>Rep Abs. type_definition Rep Abs {0..<m}"
Insert_equal: fixes e1 :: "'a \<times> 'b \<times> bool"   and e2 :: "'a \<times> 'b \<times> bool"   and i1 :: "'a"   and n1 :: "'a option"   and i :: "nat"   and i2 :: "'a"   and n2 :: "'a option"   and j :: "nat" assumes "fst e1 = fst e2"   and "Broadcast (i1, Insert e1 n1) \<in> set (history i)"   and "Broadcast (i2, Insert e2 n2) \<in> set (history j)" shows "Insert e1 n1 = Insert e2 n2"
\<tau>move2_BinOp1D: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e1 :: "'addr expr1"   and bop :: "bop"   and e2 :: "'addr expr1"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "\<tau>move2 P h stk (e1 \<guillemotleft>bop\<guillemotright> e2) pc' xcp'"   and "pc' < length (compE2 e1)" shows "\<tau>move2 P h stk e1 pc' xcp'"
notin_remove_ident_fset: fixes S :: "'b fset"   and x :: "'b" assumes "x |\<notin>| S" shows "remove_fset x S = S"
signed_arith_sint_1: fixes a :: "'a Word.word"   and b :: "'a Word.word" assumes "- (2 ^ (size a - 1)) \<le> sint a + sint b \<and> sint a + sint b \<le> 2 ^ (size a - 1) - 1" shows "sint (a + b) = sint a + sint b"
index_eq: shows "cindex_pathE (part_circlepath z r st tt) z0 = cindexE st tt h"
not_Some_eq: fixes x :: "'h option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
plus_left_distributivity: fixes a :: "int poly fract"   and b :: "int poly fract"   and c :: "int poly fract" shows "rat_poly_times a (rat_poly_plus b c) = rat_poly_plus (rat_poly_times a b) (rat_poly_times a c)"
powser_sums_zero: fixes a :: "nat \<Rightarrow> 'a" shows "(\<lambda>n. a n * (0::'a) ^ n) sums a 0"
intuitionisticD: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a"   and h' :: "'a" assumes "intuitionistic P"   and "P h"   and "h \<preceq> h'" shows "P h'"
Interleaves_equal_nil_all: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list"   and zs :: "'a list" assumes "xs \<cong> {[], zs, P}" shows "xs = zs"
sums_split_initial_segment: fixes f :: "nat \<Rightarrow> 'b"   and s :: "'b"   and n :: "nat" assumes "f sums s" shows "(\<lambda>i. f (i + n)) sums (s - sum f {..<n})"
even_two_times_div_two: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (numeral.numeral one plus (num.Bit0 num.One))) = a"
vimage_ident: fixes Y :: "'c set" shows "(\<lambda>x. x) -` Y = Y"
union_list'_language: fixes AA :: "('c, 'd) dca list" assumes "\<Inter> (dca.alphabet ` set AA) = \<Union> (dca.alphabet ` set AA)" shows "DGCA.language (union_list' AA) = \<Union> (DCA.language ` set AA)"
lex_ext_stri_imp_nstri: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and m :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "fst (lex_ext f m xs ys)" shows "snd (lex_ext f m xs ys)"
exec_gpv_Fail: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and s :: "'e" shows "exec_gpv callee Fail s = return_pmf None"
get_odd_def: fixes n :: "nat" shows "get_odd n = (if odd n then n else Suc n)"
list_before_rel_alt: fixes l :: "'b list" shows "list_before_rel l = {(l ! i, l ! j) |i j. i < j \<and> j < length l}"
assms: shows "Arr \<^bold>\<langle>f\<^bold>\<rangle>"
intro: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" assumes "sat_preserving_inference_system \<Gamma>"   and "redundancy_criterion \<Gamma> Rf Ri" shows "sat_preserving_redundancy_criterion \<Gamma> Rf Ri"
monofun_cfun_fun: fixes f :: "'c \<rightarrow> 'd"   and g :: "'c \<rightarrow> 'd"   and x :: "'c" assumes "f \<sqsubseteq> g" shows "f\<cdot>x \<sqsubseteq> g\<cdot>x"
inj_on_map_eq_map: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and ys :: "'b list" assumes "inj_on f (ran\<^sub>l xs \<union> ran\<^sub>l ys)" shows "(map f xs = map f ys) = (xs = ys)"
a_consistentE: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, e, S)" obtains "edom ae \<subseteq> domA \<Gamma> \<union> upds S" and "Astack S \<sqsubseteq> a" and "(AEstack as S \<squnion> (\<A>\<^bsub>a\<^esub> e \<squnion> ABinds \<Gamma>\<cdot>ae)) f|` (domA \<Gamma> \<union> upds S) \<sqsubseteq> ae" and "stack_consistent as S"
Pd_linear: shows "\<forall>d. Linear_order (Pd d)"
consistent_list_set: fixes s :: "(nat \<times> nat) list" assumes "wf_partial_state s" shows "list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') s) s"
Un_iff: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
calculate_data_satisfies_properties: fixes p :: "real poly"   and qs :: "real poly list" shows "p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> satisfies_properties p qs (get_subsets (calculate_data p qs)) (get_signs (calculate_data p qs)) (get_matrix (calculate_data p qs))"
Sup_llist_LNil: shows "Sup_llist LNil = {}"
monom_mult_zero_right: fixes c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" shows "local.punit.monom_mult c t 0 = 0"
rel_gpv''D: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and gpv :: "('b, 'e, 'g) gpv"   and gpv' :: "('d, 'f, 'h) gpv" assumes "rel_gpv'' A C R gpv gpv'" shows "rel_spmf (rel_generat A C (rel_fun R (rel_gpv'' A C R))) (the_gpv gpv) (the_gpv gpv')"
real_sqrt_le_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "sqrt x \<le> sqrt y"
arrE: fixes f :: "'a" assumes "S.arr f" shows "C.arr f \<and> Ide (C.dom f) \<and> Ide (C.cod f)"
distinct_50: fixes x81 :: "nat"   and x82 :: "msg"   and x5 :: "nat" shows "Crypt x81 x82 \<noteq> Key x5"
bind_nondet_def: fixes bind :: "'e \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and merge :: "'f \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and m :: "('d, 'e) nondetT"   and f :: "'d \<Rightarrow> ('d, 'e) nondetT" shows "nondetM_base.bind_nondet bind merge m f = NondetT (bind (run_nondet m) (\<lambda>A. merge A (run_nondet \<circ> f)))"
cong_def: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c a = (modulo b a = modulo c a)"
PTrans: shows "P \<Longrightarrow>rev perm \<bullet> \<alpha> \<prec> P'"
is_normal_Float_normal: fixes x :: "('a, 'b) IEEE.float" assumes "is_normal x" shows "is_normal_Float TYPE(('a, 'b) IEEE.float) (Float (normal_mantissa x) (normal_exponent x))"
ntrp_mon_increasing: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list el_step list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> trcl (ntrp fg)" shows "mon_s fg s \<subseteq> mon_s fg s' \<and> mon_c fg c \<subseteq> mon_c fg c'"
N: fixes n :: "nat" assumes "N \<le> n" shows "C < ereal (u n)"
vector_smult_assoc: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) vec" shows "a *s (b *s x) = a * b *s x"
in_funswapid_image_iff: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and S :: "'a set" shows "(x \<in> (a \<rightleftharpoons>\<^sub>F b) ` S) = ((a \<rightleftharpoons>\<^sub>F b) x \<in> S)"
dvdm_factor: fixes f :: "int poly"   and g :: "int poly"   and h :: "int poly" assumes "f dvdm g" shows "f dvdm g * h"
div_divide_to_fract: fixes x :: "'b fract"   and y :: "'b fract"   and z :: "'b fract"   and x' :: "'b"   and y' :: "'b"   and z' :: "'b" assumes "x \<in> range to_fract"   and "x = y / z"   and "x' = y' div z'"   and "y = to_fract y'"   and "z = to_fract z'" shows "x = to_fract x'"
simps_10: fixes v :: "'c"   and va :: "'d"   and vb :: "('b, 'c, 'd) Tree" shows "(tPred v =\<^sub>\<alpha> tAct va vb) = False"
fmap_cfcomp: fixes f :: "'d \<rightarrow> 'c"   and g :: "'a \<rightarrow> 'd" shows "fmap\<cdot>(f oo g) = fmap\<cdot>f oo fmap\<cdot>g"
altc_env_def: fixes altc :: "'f cset \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and C :: "'f cset"   and f :: "'f \<Rightarrow> ('d, 'e) envT" shows "altc_env altc C f = EnvT (\<lambda>r. altc C (\<lambda>c. run_env (f c) r))"
FunctorMFunctor: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme" assumes "Functor F" shows "FunctorM F"
atLeast0_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {0..Suc n} = g 0 + sum (g \<circ> Suc) {0..n}"
that: fixes T :: "'a set" assumes "open T"   and "S = T \<inter> closure S" shows "thesis"
strict_trans2: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "strict_prefix b a"   and "prefix c b" shows "strict_prefix c a"
term_ok_imp_typ_ok_pre: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and Ts :: "typ list"   and ty :: "typ" assumes "is_std_sig \<Sigma>"   and "wf_term \<Sigma> t"   and "list_all (typ_ok_sig \<Sigma>) Ts"   and "typ_of1 Ts t = Some ty" shows "typ_ok_sig \<Sigma> ty"
sum_ideals_la2: fixes R :: "('c, 'd) Ring_scheme"   and I1 :: "'c set"   and I2 :: "'c set" assumes "Ring R"   and "ideal R I1"   and "ideal R I2" shows "I2 \<subseteq> I1 \<minusplus>\<^bsub>R\<^esub> I2"
Collect_disj_eq: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "_Collect x (P x \<or> Q x) = _Collect x (P x) \<union> _Collect x (Q x)"
word_unat_less_le: fixes a :: "'a word"   and b :: "nat" assumes "a \<le> word_of_nat b" shows "unat a \<le> b"
Rep_fs_set_cases: fixes y :: "'b set" assumes "y \<in> {x. finite (supp x)}" obtains x :: "'b fs_set" where "y = Rep_fs_set x"
NF': shows "nofail (Sup {filter_nb cond m |m. \<exists>n. m = bounded_WHILE n cond step m0})"
in_dom_in_ran_m: fixes i :: "'c"   and N :: "('c, 'd) fmap" assumes "i \<in># dom_m N" shows "the (fmlookup N i) \<in># ran_m N"
simps_2: fixes cs :: "(Term_Class.term \<times> sterm) list" shows "pre_strong_term_class.wellformed (Sabs cs) = wellformed_clauses cs"
abs_triangle_ineq2_sym: fixes abs :: "'b \<Rightarrow> '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 a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (minus (abs a) (abs b)) (abs (minus b a))"
init_fin_FWdelay_bisimulation_lift_aux: shows "FWdelay_bisimulation_lift_aux r1.init_fin_final r1.init_fin r2.init_fin_final r2.init_fin r1.init_fin_\<tau>move r2.init_fin_\<tau>move"
llist_all2_LNil_LNil: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "llist_all2 R LNil LNil"
arcsin_eq_Re_Arcsin: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "arcsin x = Re (Arcsin (complex_of_real x))"
set_shadow_root_get_tag_name_is_l_set_shadow_root_get_tag_name: shows "l_set_shadow_root_get_tag_name set_shadow_root_locs get_tag_name_locs"
distr_cong: fixes M :: "'a measure"   and K :: "'a measure"   and N :: "'b measure"   and L :: "'b measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = K"   and "sets N = sets L"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = g x" shows "distr M N f = distr K L g"
correctCompositionDiffLevelsA4: shows "correctCompositionDiffLevels sA4"
space_uniform_measure: fixes M :: "'a measure"   and A :: "'a set" shows "space (uniform_measure M A) = space M"
eq_or_less_helperD: fixes n :: "nat"   and m :: "nat" assumes "n = unat (2 ^ m - 1) \<or> n < unat (2 ^ m - 1)"   and "m < LENGTH('b)" shows "n < 2 ^ m"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (x AND xa) = int_of_integer x AND int_of_integer xa"
assms_6: shows "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
simps_4: fixes s :: "db"   and t :: "db"   and k :: "nat" shows "freedb (DBApp s t) k = (freedb s k \<or> freedb t k)"
ibind_strict_1: fixes f :: "unit \<Rightarrow> 'b nres" shows "SUCCEED \<bind> f = SUCCEED"
balance_inv2: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "inv2 l"   and "inv2 r"   and "bheight l = bheight r" shows "inv2 (balance l k v r)"
add_strict_increasing: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_cancel_comm_monoid_add plus minus zero less_eq less"   and "less zero a"   and "less_eq b c" shows "less b (plus a c)"
not_le: fixes x :: "'b word"   and y :: "'b word" shows "(\<not> x \<le>s y) = (y <s x)"
array_foldl_def: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a"   and b :: "'a"   and a :: "'b array" shows "array_foldl f b a = foldl (\<lambda>b (k, v). f k b v) b (assoc_list_of_array a)"
space_return: fixes M :: "'c measure"   and x :: "'c" shows "space (return M x) = space M"
homotopic_loops_sym: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_loops s p q" shows "homotopic_loops s q p"
aff_dim_affine_Int_hyperplane: fixes S :: "'c set"   and a :: "'c"   and b :: "real" assumes "affine S" shows "aff_dim (S \<inter> {x. a \<bullet> x = b}) = (if S \<inter> {v. a \<bullet> v = b} = {} then - 1 else if S \<subseteq> {v. a \<bullet> v = b} then aff_dim S else aff_dim S - 1)"
strict_border_def: fixes xs :: "'b list"   and ys :: "'b list" shows "strict_border xs ys = (border xs ys \<and> length xs < length ys)"
word_less_nat_alt: fixes a :: "'a word"   and b :: "'a word" shows "(a < b) = (unat a < unat b)"
assms_2: assumes "x \<in>\<^sub>\<circ> A"   and "x \<notin>\<^sub>\<circ> B" shows "P"
Rep_filter_inverse: fixes x :: "'c Filters.filter" shows "Filters.filter.Abs_filter (Filters.filter.Rep_filter x) = x"
Lt_def: 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" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Lt Bet Cong A B C D \<equiv> Tarski_neutral_dimensionless.Le Bet Cong A B C D \<and> \<not> Cong A B C D"
mem_Sigma_iff: fixes a :: "'d"   and b :: "'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" shows "((a, b) \<in> Sigma A B) = (a \<in> A \<and> b \<in> B a)"
timpls_transformable_to'_inv'_5: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to' TI (Fun f T) t"   and "\<not> is_Abs f" shows "f = the_Fun t"
ocompleteTEs_4: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and d :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" assumes "((\<sigma>, s), i:deliver(d), \<sigma>', s') \<in> ocnet_sos S" obtains "((\<sigma>, s), i:deliver(d), \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"
numeral_mult_ennreal: fixes x :: "real"   and b :: "num" assumes "0 \<le> x" shows "numeral b * ennreal x = ennreal (numeral b * x)"
Enc_keys_clean_payload: fixes NI :: "msg set" assumes "NI \<subseteq> payload" shows "Enc_keys_clean NI"
assms_2: shows "alw (\<lambda>xs. \<phi> xs \<longrightarrow> \<psi> xs) xs"
add_diff_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.group_add minus plus zero uminus" shows "plus a (minus b c) = minus (plus a b) c"
step: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k"   and ys :: "'k list"   and z :: "'k" assumes "r x y"   and "rtrancl_path r y ys z" shows "rtrancl_path r x (y # ys) z"
s_not_ex: fixes x :: "nat" assumes "x \<in> I"   and "\<nexists>n. I \<rightarrow> n = x" shows "x \<in> S"
vle_imp_odiff_eq: fixes x :: "V"   and y :: "V" assumes "x \<unlhd> y" shows "x + odiff y x = y"
T_def: fixes r :: "real"   and d :: "'a"   and x :: "'a" shows "T r d x \<equiv> r *\<^sub>R x + d"