hcju/isabelleps huggingface dataset:
MGT_lemma: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and Z :: "'e"   and c :: "('e, 'd, 'f) com" assumes "\<forall>p\<in>dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>Call p \<down> Normal s} Call p {t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Abrupt t}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>c \<down> Normal s} c {t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
assms_1: fixes a :: "'a" shows "0 \<le> X a"
count_add_mset: fixes b :: "'a"   and A :: "'a multiset"   and a :: "'a" shows "count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"
assms: shows "((f, l), a, f', l') \<in> pr_algo_lts"
less_Suc0: fixes n :: "nat" shows "(n < Suc 0) = (n = 0)"
env_delete_env_restr_swap: fixes x :: "'a"   and S :: "'a set"   and e :: "'a \<Rightarrow> 'b" shows "env_delete x (e f|` S) = env_delete x e f|` S"
equiv_fun_def: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" shows "(f \<cong> g) = (\<exists>c\<^sub>1>0. \<exists>c\<^sub>2>0. \<exists>n. \<forall>m>n. f m \<le> c\<^sub>1 * g m \<and> g m \<le> c\<^sub>2 * f m)"
A: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> Cop.\<i> Cop.trg Cop.src f g \<eta> \<epsilon>"
FAccRedsThrow: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs},s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
set_3: fixes x1 :: "'c" shows "vars_term (Var x1) = {x1}"
islist_def: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref" shows "islist p h = (\<exists>ps. List p h ps)"
numeral_power_le_nat_cancel_iff: fixes x :: "num"   and n :: "nat"   and a :: "int" shows "(numeral x ^ n \<le> nat a) = (numeral x ^ n \<le> a)"
gen_lossless_gpv_lossless_spmfD: fixes co :: "bool"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "gen_lossless_gpv co \<I> gpv" shows "lossless_spmf (the_gpv gpv)"
arev_eq: fixes a :: "'b" assumes "a \<notin> arcs H" shows "rev_H' a = a"
rel_funD: fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and B :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'j"   and g :: "'i \<Rightarrow> 'k"   and x :: "'h"   and y :: "'i" assumes "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"
normalize_def'_2: fixes p :: "'a list" shows "local.normalize p = drop (length p - length (local.normalize p)) p"
simps_1: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a" shows "history_consistent \<theta> p [] = True"
Nsub_def: shows "(-\<^sub>N) = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
remdups_map_component_of_term_punit: fixes bs :: "(('a \<Rightarrow>\<^sub>0 'd) \<times> 'e) list" shows "remdups (map (\<lambda>_. ()) (punit.Keys_to_list (map fst bs))) = (if \<forall>b\<in>set bs. fst b = 0 then [] else [()])"
local_time_birkhoff: fixes A :: "'a set"   and n :: "nat"   and x :: "'a" shows "local_time A n x = birkhoff_sum (indicator A) n x"
return_pure: fixes x :: "'c"   and h :: "'a" shows "pure (return x) h"
lcm_div_unit1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "lcm (b div a) c = lcm b c"
initial_arrow_to_functor_axioms: shows "initial_arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y (F y) (\<eta> y)"
sets_PiM: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
char_cases: fixes k :: "nat" shows "\<chi> k \<in> {- 1, 0, 1}"
rep_eq: fixes x :: "Linear_Aggregation_Algebras.unit"   and xa :: "Linear_Aggregation_Algebras.unit" shows "(x \<le> xa) = (Linear_Aggregation_Algebras.unit.Rep_unit x \<le> Linear_Aggregation_Algebras.unit.Rep_unit xa)"
accepts_conc: fixes L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "NAe.accepts (RegExp2NAe.conc L R) w = (\<exists>u v. w = u @ v \<and> NAe.accepts L u \<and> NAe.accepts R v)"
single_multiset_eq_non_empty: fixes A :: "'a"   and \<Delta> :: "'a multiset"   and \<Delta>' :: "'a multiset" assumes "\<LM> A  \<RM> = \<Delta> + \<Delta>'"   and "\<Delta> \<noteq> \<Empt>" shows "\<Delta>' = \<Empt> \<and> \<Delta> = \<LM> A  \<RM>"
lsorted_lprefixD: fixes xs :: "'c llist"   and ys :: "'c llist"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "xs \<sqsubseteq> ys"   and "ord.lsorted less_eq ys" shows "ord.lsorted less_eq xs"
ms_of_greek_elem: fixes x :: "accent \<times> 'a"   and xs :: "(accent \<times> 'a) list"   and ys :: "(accent \<times> 'a) list" assumes "(x, xs) \<in># ms_of_greek ys" shows "x \<in> set ys"
sim_config_emptyE: fixes ts\<^sub>s\<^sub>b :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "\<forall>i p is xs sb \<D> \<O> \<R>. i < length ts\<^sub>s\<^sub>b \<longrightarrow> ts\<^sub>s\<^sub>b ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> sb = []"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)" shows "\<S> = \<S>\<^sub>s\<^sub>b \<and> m = m\<^sub>s\<^sub>b \<and> length ts = length ts\<^sub>s\<^sub>b \<and> (\<forall>i<length ts\<^sub>s\<^sub>b. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts\<^sub>s\<^sub>b ! i in ts ! i = (p, is, \<theta>, (), \<D>, \<O>, \<R>))"
nonneg_cexpr_subset: fixes V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and V' :: "nat set" assumes "nonneg_cexpr V \<Gamma> e"   and "V \<subseteq> V'"   and "free_vars e \<subseteq> V" shows "nonneg_cexpr V' \<Gamma> e"
Pi'_cong: fixes I :: "'d set"   and f :: "'d \<Rightarrow> 'e set"   and g :: "'d \<Rightarrow> 'e set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i = g i" shows "Pi' I f = Pi' I g"
iterm_to_fterm_cancel: fixes t :: "fterm" shows "iterm_to_fterm (fterm_to_iterm t) = t"
hyps: fixes c :: "('p, 's, 'm) configuration" assumes "trace c (t @ [ev]) c'" shows "\<exists>c''. trace c t c'' \<and> c'' \<turnstile> ev \<mapsto> c'"
FieldI2: fixes i :: "'a"   and j :: "'a"   and R :: "('a \<times> 'a) set" assumes "(i, j) \<in> R" shows "j \<in> Field R"
dg_def: shows "\<AA> = [\<AA>\<lparr>Obj\<rparr>, \<AA>\<lparr>Arr\<rparr>, \<AA>\<lparr>Dom\<rparr>, \<AA>\<lparr>Cod\<rparr>]\<^sub>\<circ>"
cases: fixes a1 :: "'a OCL_Types.type"   and a2 :: "'a OCL_Types.type" assumes "a1 \<sqsubset> a2" obtains \<tau> :: "'a basic_type" and \<sigma> :: "'a basic_type" where "a1 = \<tau>__[1]" and "a2 = \<sigma>__[1]" and "\<tau>__ \<sqsubset>\<^sub>B \<sigma>__"   | \<tau> :: "'a basic_type" and \<sigma> :: "'a basic_type" where "a1 = \<tau>'__[?]" and "a2 = \<sigma>'__[?]" and "\<tau>'__ \<sqsubset>\<^sub>B \<sigma>'__"   | \<tau> :: "'a basic_type" where "a1 = \<tau>''__[1]" and "a2 = \<tau>''__[?]"   | "a1 = OclAny[?]" and "a2 = OclSuper"   | \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "a1 = Collection \<tau>'''__" and "a2 = Collection \<sigma>''__" and "\<tau>'''__ \<sqsubset> \<sigma>''__"   | \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "a1 = Set \<tau>''''__" and "a2 = Set \<sigma>'''__" and "\<tau>''''__ \<sqsubset> \<sigma>'''__"   | \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "a1 = OrderedSet \<tau>'''''__" and "a2 = OrderedSet \<sigma>''''__" and "\<tau>'''''__ \<sqsubset> \<sigma>''''__"   | \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "a1 = Bag \<tau>''''''__" and "a2 = Bag \<sigma>'''''__" and "\<tau>''''''__ \<sqsubset> \<sigma>'''''__"   | \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "a1 = Sequence \<tau>'''''''__" and "a2 = Sequence \<sigma>" and "\<tau>'''''''__ \<sqsubset> \<sigma>"   | \<tau> :: "'a OCL_Types.type" where "a1 = Set \<tau>''''''''__" and "a2 = Collection \<tau>''''''''__"   | \<tau> :: "'a OCL_Types.type" where "a1 = OrderedSet \<tau>'''''''''__" and "a2 = Collection \<tau>'''''''''__"   | \<tau> :: "'a OCL_Types.type" where "a1 = Bag \<tau>''''''''''__" and "a2 = Collection \<tau>''''''''''__"   | \<tau> :: "'a OCL_Types.type" where "a1 = Sequence \<tau>" and "a2 = Collection \<tau>"   | "a1 = Collection OclSuper" and "a2 = OclSuper"   | \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" and \<xi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" where "a1 = Tuple \<pi>__" and "a2 = Tuple \<xi>" and "strict_subtuple (\<lambda>\<tau> \<sigma>. \<tau> \<sqsubset> \<sigma> \<or> \<tau> = \<sigma>) \<pi>__ \<xi>"   | \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" where "a1 = Tuple \<pi>" and "a2 = OclSuper"
nz: fixes z :: "complex" assumes "z \<in> eball 0 r" shows "eval_fps g z \<noteq> 0"
intro: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and z :: "'c" assumes "\<And>m. n < m \<Longrightarrow> f m = z" shows "bound z n f"
integral_map_pmf: fixes g :: "'b \<Rightarrow> real"   and M :: "'b pmf"   and f :: "real \<Rightarrow> real" shows "measure_pmf.expectation (map_pmf g M) f = measure_pmf.expectation M (\<lambda>x. f (g x))"
fundchamber_vertex_stabilizer1: fixes t :: "'a permutation"   and w :: "'a permutation" assumes "t \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"   and "w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"   and "w \<rightarrow> fundantivertex t = fundantivertex t" shows "w \<in> \<langle>(\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) - {t}\<rangle>"
set_intros_2: fixes y :: "'b"   and x22 :: "'b list"   and x21 :: "'b" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
LeftDerivation_append_suffix: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and c :: "symbol list" assumes "LeftDerivation a D b"   and "is_sentence c" shows "LeftDerivation (a @ c) D (b @ c)"
cones_functor_def: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and S :: "'ai \<Rightarrow> 'ai \<Rightarrow> 'ai"   and \<iota> :: "('ag \<Rightarrow> 'ah) \<Rightarrow> 'ai" shows "cones_functor J C D S \<iota> \<equiv> (category C \<and> dual_category C) \<and> category J \<and> diagram J C D \<and> replete_concrete_set_category S UNIV \<iota>"
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) ===> rel_set B\<^sub>2 ===> (=)"
one_hom_axioms: shows "one_hom (inv (inv (inv hom)))"
itrm_hom: fixes f :: "dB"   and x :: "dB" shows "Pure f \<diamondop> Pure x \<simeq> Pure (f \<degree> x)"
add_opp_double_opp: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "local.add a p\<^sub>1 p\<^sub>2 = opp p\<^sub>1" shows "p\<^sub>2 = local.add a (opp p\<^sub>1) (opp p\<^sub>1)"
same_root_imp_not_coprime: fixes p :: "'a poly"   and x :: "'a"   and q :: "'a poly" assumes "poly p x = (0::'a)"   and "poly q x = (0::'a)" shows "\<not> coprime p q"
intro: fixes U :: "'a set"   and one :: "'a"   and zero :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "zero_neq_one_ow U one zero"   and "semiring_0_ow U plus zero times"   and "monoid_mult_ow U one times" shows "semiring_1_ow U one times plus zero"
domID: 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" assumes "p \<noteq> [] \<and> x \<in> dom (Cp (list2FWpolicy p))" shows "x \<in> dom (Cp (list2FWpolicy (insertDenies p)))"
bin_cat_in_uintsI: fixes a :: "int"   and l :: "nat"   and n :: "nat"   and m :: "nat"   and b :: "int" assumes "a \<in> uints l"   and "l + n \<le> m" shows "concat_bit n b a \<in> uints m"
forget: fixes x :: "name"   and e :: "trm"   and e' :: "trm" assumes "x \<sharp> e" shows "e[x::=e'] = e"
0: shows "rawpsubstT (Var y) (zip (map Var us) (map snd txs)) = Var (us ! i)"
bounded_blinfun_apply_image: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'c"   and s :: "'d set"   and g :: "'d \<Rightarrow> 'b" assumes "bounded (f ` s)"   and "bounded (g ` s)" shows "bounded ((\<lambda>x. blinfun_apply (f x) (g x)) ` s)"
inf_concat_simple_add: fixes f :: "nat \<Rightarrow> nat"   and k :: "nat"   and i :: "nat"   and j :: "nat"   and l :: "nat" assumes "inf_concat_simple f k = (i, j)"   and "j + l < f i" shows "inf_concat_simple f (k + l) = (i, j + l)"
poly_eq_poly_eval': fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and x :: "'a"   and a :: "'b" assumes "p \<in> P[{x}]" shows "poly (poly_of_pm x p) a = poly_eval (\<lambda>_. a) p"
inject_4: fixes x4 :: "Vars.var"   and y4 :: "Vars.var" shows "(Dummy x4 = Dummy y4) = (x4 = y4)"
long_divisionI: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list"   and b :: "'a list"   and r :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])"   and "q \<noteq> []"   and "long_divides p q (b, r)" shows "(b, r) = (p pdiv q, p pmod q)"
multTensor2_bis: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat" assumes "dim_row A = 2"   and "dim_col A = 1"   and "dim_row B = 2"   and "dim_col B = 1" shows "mult.Tensor (*) (mat_to_cols_list A) (mat_to_cols_list B) = [[A $$ (0, 0) * B $$ (0, 0), A $$ (0, 0) * B $$ (1, 0), A $$ (1, 0) * B $$ (0, 0), A $$ (1, 0) * B $$ (1, 0)]]"
of_nat_times_monomial: fixes p :: "nat"   and i :: "'a"   and c :: "nat" shows "of_nat p * monomial c i = monomial (p * c) i"
lcms_def: fixes xs :: "'a list" shows "lcms xs = lcm_list xs"
envDetJKBP': fixes t :: "('a, 'es, 'as) BEState Trace"   and n :: "nat"   and act :: "'aAct"   and a :: "'a" assumes "t \<in> SPR.jkbpCn n"   and "act \<in> set (jAction (SPR.MCn n) t a)" shows "jAction (SPR.MCn n) t a = [act]"
Prefix_Suffix: fixes L :: "'c list set" shows "Prefix L = rev ` Suffix (rev ` L)"
lift_IntVal_def: fixes f :: "int \<Rightarrow> int" shows "lift_IntVal f \<equiv> \<lambda>x. case x of IntVal v \<Rightarrow> IntVal (f v) | _ \<Rightarrow> IntVal (f 0)"
eval_prime_sum_upto: fixes f :: "nat \<Rightarrow> 'a"   and x :: "real" shows "prime_sum_upto f x = sum_list (map f (map fst (filter snd (List.enumerate 2 (sieve 1 (replicate (nat \<lfloor>x\<rfloor> - 1) True))))))"
fconst_onD: fixes a :: "V"   and c :: "V"   and A :: "V" assumes "[a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> fconst_on A c" shows "a \<in>\<^sub>\<circ> A"
c: shows "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
pairs_def: fixes r :: "V" shows "pairs r \<equiv> {(x, y). \<langle>x, y\<rangle> \<in> elts r}"
real_inverse_le_1_iff: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "(1 / x \<le> 1) = (x = 1 \<or> x = 0)"
cont_def: fixes f :: "'b \<Rightarrow> 'c" shows "cont f \<equiv> \<forall>C. C \<noteq> {} \<longrightarrow> f (Sup C) = Sup (f ` C)"
least_contains': fixes A :: "'b set" assumes "A \<noteq> {}" shows "least A \<in> A"
simps_2: fixes x :: "Vars.var"   and e :: "exp" shows "isLam (Lam [x]. e) = True"
hta_lookup_s_def: fixes q :: "'a"   and H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_lookup_s q H \<equiv> hll_idx.lookup q (the (hta_idx_s H))"
linear_diff: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and x :: "('b, 'c) vec"   and y :: "('b, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f (x - y) = f x - f y"
simps_2: fixes pat :: "Term_Class.term"   and rhs :: "sterm"   and cs :: "(Term_Class.term \<times> sterm) list"   and t :: "value" shows "vfind_match ((pat, rhs) # cs) t = (case vmatch (mk_pat pat) t of None \<Rightarrow> vfind_match cs t | Some env \<Rightarrow> Some (env, pat, rhs))"
extreme_points_of_convex_hull: fixes S :: "'b set" shows "{x. x extreme_point_of convex hull S} \<subseteq> S"
Sum1: fixes P :: "pi"   and V :: "Early_Semantics.residual"   and Q :: "pi" assumes "P \<longmapsto>\<^sub>e V" shows "P \<oplus> Q \<longmapsto>\<^sub>e V"
cdot_square_norm: fixes x :: "'a" shows "x \<bullet>\<^sub>C x = complex_of_real ((norm x)\<^sup>2)"
dom_comp: fixes g :: "'s"   and f :: "'s" assumes "seq g f" shows "local.dom (g \<cdot> f) = local.dom f"
not_mod_2_eq_0_eq_1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(modulo a (numeral.numeral one plus (num.Bit0 num.One)) \<noteq> zero) = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = one)"
GreenThm_typeI_divisible: assumes "gen_division s (cubeImage ` two_chain)" shows "integral s (\<lambda>x. - partial_vector_derivative (\<lambda>a. F a \<bullet> i) j x) = one_chain_line_integral F {i} (two_chain_boundary two_chain)"
SeqSubstFormP_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm"   and s :: "tm"   and s' :: "tm"   and k :: "tm"   and k' :: "tm"   and v :: "tm"   and i :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'"   and "H \<turnstile> s EQ s'"   and "H \<turnstile> k EQ k'" shows "H \<turnstile> SeqSubstFormP v i t u s k IFF SeqSubstFormP v i t' u' s' k'"
simps_2: fixes x21 :: "'a"   and x22 :: "'a convnet"   and y21 :: "'a"   and y22 :: "'a convnet" shows "(Conv x21 x22 = Conv y21 y22) = (x21 = y21 \<and> x22 = y22)"
mat_adj_mult_mm: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_adj (A *\<^sub>m\<^sub>m B) = mat_adj B *\<^sub>m\<^sub>m mat_adj A"
fas: shows "slp_of_fas' fas M1 slp1 = (M', slp')"
f1: shows "4 \<le> rk {A, B, C, R, a, b, c}"
transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_fun A (rel_fun A (pcr_uprod A)) Pair Upair"
dil_inverse_def: fixes f :: "nat \<Rightarrow> nat" shows "dil_inverse f \<equiv> \<lambda>n. Max {i. f i \<le> n}"
vdistinct_def: fixes xs :: "V" shows "vdistinct xs = distinct (vlist_of_vfsequence xs)"
ga_en_def: shows "ga_en \<equiv> stutter_extend_en ga_gen"
minus_sin_cos_eq: fixes x :: "'a" shows "- sin x = cos (x + of_real pi / (2::'a))"
lit_zero: shows "\<guillemotleft>0::'c\<guillemotright> = 0"
unit_mult_right_cancel: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "(times b a = times c a) = (b = c)"
unit_closed: shows "\<zero>\<^bsub>U\<^esub> \<in> \<FF> U"
assms_1: shows "(\<sigma>, p) \<in> reachable OA I"
r_inv_ex: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "\<exists>y\<in>carrier (relative_homology_group (int n) (nsphere n) (equator n)). x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> y = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub>"
simps_1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "option_assn P None None = emp"
nyinitcls_abrupt_cong: fixes G :: "prog"   and f :: "abopt \<Rightarrow> abopt"   and x :: "abopt"   and y :: "st" shows "nyinitcls G (f x, y) = nyinitcls G (x, y)"
simps_4: fixes v :: "nat"   and va :: "real mpoly"   and L :: "atom list" shows "find_lucky_eq v (Leq va # L) = find_lucky_eq v L"
f: shows "bij_betw f (set_spmf p) (set_spmf q)"
assms_2: shows "T \<in> M \<rightarrow>\<^sub>M M"
assms_2: shows "Range (R \<inter> I \<times> J) \<subseteq> K"
fun_pair_subst: fixes d :: "('fun, 'e) Term.term \<times> ('fun, 'e) Term.term"   and \<delta> :: "'e \<Rightarrow> ('fun, 'd) Term.term" shows "pair d \<cdot> \<delta> = pair (d \<cdot>\<^sub>p \<delta>)"
E_ss_cfinvE: shows "E \<subseteq> f.cf.E \<union> f.cf.E\<inverse>"
pm_of_idx_pm_of_pm: fixes f :: "'c \<Rightarrow>\<^sub>0 'd"   and xs :: "'c list" assumes "keys f \<subseteq> set xs" shows "pm_of_idx_pm xs (idx_pm_of_pm xs f) = f"
ssubst_insert2: fixes V :: "name set"   and i :: "name"   and t :: "tm" assumes "finite V"   and "i \<notin> V" shows "ssubst t (insert i V) F = ssubst (subst i (F i) t) V F"
cone_carrier: fixes A :: "'b vec set"   and n :: "nat" assumes "A \<subseteq> carrier_vec n" shows "gram_schmidt.cone n A \<subseteq> carrier_vec n"
has_codomain_char: fixes f :: "'c \<times> 'c" shows "(CopxC.codomains f \<noteq> {}) = (Cop.arr (fst f) \<and> C.arr (snd f))"
swap_cancel_2: fixes a :: "atom"   and b :: "atom" shows "(a \<rightleftharpoons> b) + (b \<rightleftharpoons> a) = 0"
t_lPar: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "(x, T) \<in> set \<Gamma>" shows " \<Gamma> \<turnstile> lPar x : T "
prems_2: fixes f :: "'a match_expr \<Rightarrow> 'a match_expr option"   and m :: "'a match_expr" assumes "f \<in> set (f__ # fs)"   and "normalized_nnf_match m"   and "f m = None" shows "\<not> matches \<gamma> m a p"
psiFreshVec_3: fixes xvec :: "name list"   and C :: "('d, 'e, 'f) psiCase" shows "xvec \<sharp>* Case C = xvec \<sharp>* C"
term_variants_pred_funs_term: fixes P :: "'a \<Rightarrow> 'a list"   and s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and f :: "'a" assumes "term_variants_pred P s t"   and "f \<in> funs_term t" shows "f \<in> funs_term s \<or> (\<exists>g\<in>funs_term s. f \<in> set (P g))"