hcju/isabelleps huggingface dataset:
hcomp_simps_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (H \<nu> \<mu>)" shows "trg (H \<nu> \<mu>) = trg \<nu>"
usubstappt_const: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and f :: "char"   and r :: "trm"   and U :: "variable set" assumes "SConst \<sigma> f = Aterm r"   and "FVT r \<inter> U = {}" shows "usubstappt \<sigma> U (Const f) = Aterm r"
T_bal: fixes n :: "nat"   and xs :: "'b list" shows "T_bal n xs = 2 * n + 1"
terminatess_impl_not_inf: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>cs,css \<Down> s" shows "\<not> AlternativeSmallStep.inf \<Gamma> cs css s"
of_nat_card_eq_0: shows "of_nat CARD('b) = 0"
of_int_dvd_of_int_gauss_int_iff: fixes m :: "int"   and n :: "int" shows "(of_int m dvd of_int n) = (m dvd n)"
esing_simps_1: fixes x :: "'d"   and n :: "'c" shows "(esing x\<cdot>n) x = n"
vec_mat_Tensor_elements: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and v :: "'b list"   and M :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "\<forall>i j. (i < length v * mult.row_length M \<and> j < length M) \<and> mat (mult.row_length M) (length M) M \<longrightarrow> mult.vec_mat_Tensor f v M ! j ! i = f (v ! (i div mult.row_length M)) (M ! j ! (i mod mult.row_length M))"
compl_sup: fixes x :: "'a"   and y :: "'a" shows "- (x + y) = - x \<cdot> - y"
spr_simVal_def: shows "spr_simVal \<equiv> envVal \<circ> sprLst"
matchs_reorder: fixes fps :: "(char list \<times> pat) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and \<Delta> :: "binding list"   and fs :: "(char list \<times> trm) list"   and ts :: "trm list" assumes "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>"   and "\<forall>(l, U)\<in>set fTs. \<exists>u. fs\<langle>l\<rangle>\<^sub>? = \<lfloor>u\<rfloor>"   and "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts" shows "\<turnstile> fps [\<rhd>] map (\<lambda>(l, T). (l, the (fs\<langle>l\<rangle>\<^sub>?))) fTs \<Rightarrow> ts"
aext_var: fixes x :: "'a \<Longrightarrow> 'c"   and a :: "'c \<Longrightarrow> 'b" shows "x \<oplus>\<^sub>p a = x ;\<^sub>L a"
shiftl_0: fixes n :: "nat" shows "0 << n = 0"
asSort_inj: fixes xs :: "'varSort"   and ys :: "'varSort" shows "(asSort xs = asSort ys) = (xs = ys)"
simplex_retraction2: fixes x :: "'a set" assumes "x \<in> f \<turnstile> X" shows "f ` x = x"
complex_Cauchy_convergent: fixes X :: "nat \<Rightarrow> complex" assumes "Cauchy X" shows "convergent X"
elims_3: fixes x :: "'a ifex" assumes "\<not> ifex_minimal x" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e" and "\<not> (t \<noteq> e \<and> ifex_minimal t \<and> ifex_minimal e)"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a XOR b) XOR c = a XOR b XOR c"
globs_def: shows "globs = case_st (\<lambda>g l. g)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) L local.map \<ll>"
all_possible_assignments_for_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem" shows "all_possible_assignments_for \<Psi> \<equiv> concat (map (possible_assignments_for \<Psi>) (\<Psi>\<^sub>\<V>\<^sub>+))"
g: fixes x :: "'a" assumes "x \<in> A'" shows "g x \<in> B x"
PX4d_linear: fixes d :: "D2" shows "Linear_order (PX4d d)"
non_zero: shows "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
left_neutral: fixes a :: "'c" shows "\<bottom> \<squnion> a = a"
body': shows "P,E(this \<mapsto> Class (last Ds), pns [\<mapsto>] Ts) \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2(this \<mapsto> Ref (a, Ds), pns [\<mapsto>] vs'))\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2 ++ l\<^sub>3)\<rangle>"
fin_U_imp_wf: fixes U :: "'a set" assumes "finite U" shows "wf (gen_dfs_var U)"
leg1_simps_2: shows "arr fg\<^sub>0h\<^sub>1.p\<^sub>1"
le_funD: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f \<le> g" shows "f x \<le> g x"
rp: shows "(\<xi>, p) \<in> reachable A I'"
assms_2: shows "asEnv qrho = asEnv qrho'"
sp_Cons: fixes v1 :: "'a \<times> nat"   and e :: "('a \<times> nat) edge"   and es :: "('a \<times> nat) edge list"   and v2 :: "'a \<times> nat" shows "SubRel.subpath g v1 (e # es) v2 subs = (sub_rel_of g subs \<and> (v1 = src e \<or> (v1, src e) \<in> subs) \<and> e \<in> edges g \<and> SubRel.subpath g (tgt e) es v2 subs)"
glb_maximal: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" assumes "class.po below"   and "po.is_lb below S x"   and "x \<in> S" shows "po.glb below S = x"
rotate_topneg_kauff_mat: shows "kauff_mat (basic [vert, brick.under] \<circ> basic [cap, vert]) = kauff_mat (basic [over, vert] \<circ> basic [vert, cap])"
homotopic_with_equal: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and X :: "'b topology"   and Y :: "'c topology" assumes "P f"   and "P g"   and "continuous_map X Y f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = g x" shows "homotopic_with P X Y f g"
Sup_distr: fixes X :: "'a set"   and y :: "'a" shows "Sup X \<cdot> y = Sup {z. \<exists>x\<in>X. z = x \<cdot> y}"
litsim_def: fixes subsumeseq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a"   and t :: "'a" assumes "subsumable subsumeseq" shows "subsumable.litsim subsumeseq s t = (subsumeseq s t \<and> subsumeseq t s)"
distinct_filter: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "distinct xs" shows "distinct (filter P xs)"
complete_iff_size1: fixes t :: "'b tree" shows "Tree.complete t = (size1 t = 2 ^ height t)"
is_empty_def: fixes A :: "'g set" shows "Set.is_empty A = (A = {})"
uvV: shows "Vagree \<nu> \<omega> (- fst (usubstappp \<sigma> U \<alpha>))"
witness_l0': fixes Y :: "nat"   and M :: "nat" shows "witness' Y [M] = Pool (Conv (id_matrix Y M) (Input M)) (Conv (id_matrix Y M) (Input M))"
obj_simps_1: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "arr a"
init_fin_silent_move_RunningI: fixes r :: "('e, 'f, 'c, 'd, 'g, 'h) semantics"   and t :: "'f"   and \<tau>move :: "('e, 'f, 'c, 'd, 'g, 'h) \<tau>moves"   and x :: "'c"   and m :: "'d"   and x' :: "'c"   and m' :: "'d"   and final :: "'c \<Rightarrow> bool" assumes "\<tau>trsys.silent_move (r t) \<tau>move (x, m) (x', m')" shows "\<tau>trsys.silent_move (multithreaded_base.init_fin final r t) (\<tau>multithreaded.init_fin_\<tau>move \<tau>move) ((Running, x), m) ((Running, x'), m')"
t_def: shows "t \<equiv> THE x. abss x = s' \<and> x \<in> set_pmf (action cfg')"
ws: shows "weight s \<le> sum_list (map weight (scf_list (scf (f, length (bef @ s # aft))) (bef @ s # aft)))"
assms_1: shows "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v',s\<^sub>1,b\<^sub>1\<rangle>"
ZFfunApp: fixes x :: "ZF"   and A :: "ZF"   and B :: "ZF"   and f :: "ZF \<Rightarrow> ZF" assumes "x |\<in>| A" shows "ZFfun A B f |@| x = f x"
inside_complement_unbounded_connected_empty: fixes S :: "'a set" assumes "connected (- S)"   and "\<not> bounded (- S)" shows "inside S = {}"
lfinite_ltake: fixes n :: "enat"   and xs :: "'c llist" shows "lfinite (ltake n xs) = (lfinite xs \<or> n < \<infinity>)"
nonpos_Reals_inverse_iff: fixes a :: "'a" shows "(inverse a \<in> \<real>\<^sub>\<le>\<^sub>0) = (a \<in> \<real>\<^sub>\<le>\<^sub>0)"
idom_def: fixes g :: "'g"   and n :: "'node" shows "idom g n \<equiv> THE m. isIdom g n m"
simps_2: fixes a :: "int"   and n :: "nat" shows "gen_lucas_lehmer_sequence a (Suc n) = (gen_lucas_lehmer_sequence a n)\<^sup>2 - 2"
eval_hom: fixes K :: "'a set"   and a :: "'a"   and p :: "'a list" assumes "subring K R"   and "a \<in> carrier R"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)" shows "h (Hom.R.eval p a) = Hom.S.eval (map h p) (h a)"
sim_A_J: shows "similar_mat (of_real_hom.mat_hom A) J"
split_block4_decreases_dim_row: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and D :: "'a mat"   and E :: "'a mat" assumes "(A, B, C, D) = split_block E 1 1"   and "1 < dim_row E"   and "1 < dim_col E" shows "dim_row D < dim_row E"
IH: fixes e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
word_less_no: fixes a :: "num"   and b :: "num" shows "(numeral a < numeral b) = (uint (numeral a) < uint (numeral b))"
assms_14: shows "\<forall>t\<in>R_I. \<forall>y. (y \<in> genericJoin J (newQuery J Q_J_pos (I, t)) (newQuery J Q_J_neg (I, t))) = (wf_tuple n J y \<and> (\<forall>(A, X)\<in>newQuery J Q_J_pos (I, t). restrict A y \<in> X) \<and> (\<forall>(A, X)\<in>newQuery J Q_J_neg (I, t). restrict A y \<notin> X))"
prod_list_map_filter: fixes f :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "prod_list (map g (filter f xs)) * prod_list (map g (filter (\<lambda>x. \<not> f x) xs)) = prod_list (map g xs)"
BD: shows "subset_dfs A (nfa_startnode A) = (bd, ls)"
iterSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "iterSTR MOD \<equiv> Equiv_Relation2.univ (qInit MOD)"
B: shows "independent B \<and> B \<subseteq> S \<and> S \<subseteq> span B \<and> card B = dim S"
no_abs_cases: fixes t :: "'a" assumes "no_abs t" obtains name :: "name" where "t = free name__"   | name :: "name" where "t = const name"   | t\<^sub>1 :: "'a" and t\<^sub>2 :: "'a" where "t = app t\<^sub>1 t\<^sub>2" and "no_abs t\<^sub>1" and "no_abs t\<^sub>2"
fin_succ_hd_b: shows "finite (succs (hd b))"
prems_1: shows "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1} \<Gamma>',\<S>',P'"
u_max_def: shows "u_max = u (ego2.t_stop + \<delta>)"
phull_row_echelon: fixes ps :: "('c \<Rightarrow>\<^sub>0 'd) list"   and ts :: "'c list" assumes "Keys (set ps) \<subseteq> set ts"   and "distinct ts" shows "phull (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = phull (set ps)"
k4If: fixes upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set) list"   and LVQ :: "char list set"   and C1 :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and C2 :: "acom"   and b :: "bexp" assumes "K4 upds LVQ C1 Q"   and "K4 upds LVQ C2 Q" shows "K4 upds LVQ (IF b THEN C1 ELSE C2) Q"
Sup1_E: fixes A :: "('b \<Rightarrow> bool) set"   and a :: "'b" assumes "Sup A a" obtains P :: "'b \<Rightarrow> bool" where "P \<in> A" and "P a"
it_simr: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> x \<cdot> y" shows "y \<cdot> it x \<le> it x \<cdot> y"
fmpt_null_space: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "emeasure M (space M) = 0"   and "T \<in> M \<rightarrow>\<^sub>M M" shows "fmpt M T"
rep_eq: fixes x :: "rat" shows "fpxs_nth (fpxs_X_power x) = (\<lambda>n. if n = x then 1::'b else (0::'b))"
foldli_snoc: fixes xs :: "'e list"   and x :: "'e"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldli (xs @ [x]) c f \<sigma> = (if c (foldli xs c f \<sigma>) then f x (foldli xs c f \<sigma>) else foldli xs c f \<sigma>)"
per_rootI: fixes u :: "'b list"   and r :: "'b list" assumes "u \<le>p r \<cdot> u"   and "r \<noteq> \<epsilon>" shows "u \<le>p r\<^sup>\<omega>"
Z: fixes b :: "bool"   and w :: "int" shows "bin_sc 0 b w = of_bool b + 2 * (w div 2)"
map_inv2_1: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and t :: "('a, 'c) rbt" shows "inv2 (RBT_Impl.map f t) = inv2 t"
inducts: fixes x :: "'a set \<Rightarrow> 'a set"   and P :: "('a set \<Rightarrow> 'a set) \<Rightarrow> bool" assumes "CK_nf_pos x"   and "P id"   and "P I"   and "P K"   and "P (I \<circ> K)"   and "P (K \<circ> I)"   and "P (I \<circ> K \<circ> I)"   and "P (K \<circ> I \<circ> K)" shows "P x"
a: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>s t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
revis_superexpansion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A" shows "K \<^bold>* (\<phi> .\<and>. \<psi>) \<subseteq> (K \<^bold>* \<phi>) \<oplus> \<psi>"
add_mset_remove_trivial: fixes x :: "'d"   and M :: "'d multiset" shows "add_mset x M - {#x#} = M"
set_eq_pairI: fixes x :: "'a"   and x' :: "'a"   and y :: "'a" assumes "x .= x'"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "y \<in> carrier L" shows "{x, y} {.=} {x', y}"
subst_compose_upd: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f \<Rightarrow> 'f"   and \<rho> :: "'g \<Rightarrow> 'f"   and v :: "('e, 'g) uexpr" assumes "x \<sharp> \<sigma>" shows "\<sigma> \<circ> \<rho>(x \<mapsto>\<^sub>s v) = (\<sigma> \<circ> \<rho>)(x \<mapsto>\<^sub>s v)"
RPDs_Test_3: fixes \<phi> :: "nat" shows "RPDs (MTestNeg \<phi>) \<subseteq> {MTestNeg \<phi>}"
abs_eq: shows "o_x_axis = abs_ocircline x_axis_clmat"
prob_space_uniform_measure: fixes M :: "'a measure"   and A :: "'a set" assumes "emeasure M A \<noteq> 0"   and "emeasure M A \<noteq> \<infinity>" shows "prob_space (uniform_measure M A)"
sinvar_all_edges_normal_form_not_refl_SR_def: fixes P :: "'a \<Rightarrow> 'v \<Rightarrow> 'a \<Rightarrow> 'v \<Rightarrow> bool" shows "sinvar_all_edges_normal_form_not_refl_SR P \<equiv> \<forall>G nP. sinvar G nP = (\<forall>(s, r)\<in>edges G. s \<noteq> r \<longrightarrow> P (nP s) s (nP r) r)"
mset_del_min: fixes q :: "'a tree" assumes "heap q"   and "q \<noteq> \<langle>\<rangle>" shows "mset_tree (del_min q) = mset_tree q - {#value q#}"
monoid_axioms: shows "monoid G' (\<cdot>') \<one>'"
prod_prime_factors: fixes n :: "nat" assumes "0 < n" shows "g' n = (\<Prod>p\<in>prime_factors n. g' (p ^ multiplicity p n))"
cauchy_tendsto_in_subseq: fixes u :: "nat \<Rightarrow> 'c"   and N :: "'c quasinorm"   and r :: "nat \<Rightarrow> nat"   and x :: "'c" assumes "\<And>n. u n \<in> space\<^sub>N N"   and "cauchy_in\<^sub>N N u"   and "strict_mono r"   and "tendsto_in\<^sub>N N (u \<circ> r) x" shows "tendsto_in\<^sub>N N u x"
norm_lt_square: fixes x :: "'c"   and a :: "real" shows "(norm x < a) = (0 < a \<and> inner x x < a\<^sup>2)"
bind_eq_Some_conv: fixes f :: "'g option"   and g :: "'g \<Rightarrow> 'f option"   and x :: "'f" shows "(f \<bind> g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
h_def: fixes a :: "nat"   and b :: "nat" shows "h a b = a div b"
invariant_2_realI: fixes plr :: "int poly \<times> rat \<times> rat"   and x :: "real" assumes "root_cond plr x"   and "sgn (rai_lb plr) = sgn (rai_ub plr)"   and "\<exists>!x. root_cond plr x"   and "poly_cond (poly_real_alg_1 plr)" shows "invariant_2 (real_alg_2 plr) \<and> real_of_2 (real_alg_2 plr) = x"
absorb_iff2: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" shows "(b \<preceq> a) = (linordered_monoid.min a b = b)"
topo_open: fixes t :: "'a set" shows "(t open) = (t \<in> topo B)"
smap_cong: fixes xs :: "[:'a:]"   and xs' :: "[:'a:]"   and f :: "'a \<rightarrow> 'b"   and f' :: "'a \<rightarrow> 'b" assumes "xs = xs'"   and "\<And>x. x \<in> sset xs \<Longrightarrow> f\<cdot>x = f'\<cdot>x" shows "smap\<cdot>f\<cdot>xs = smap\<cdot>f'\<cdot>xs'"
R_plus: fixes x :: "'a"   and y :: "'a" assumes "x \<in> R"   and "y \<in> R" shows "x + y \<in> R"
hseq_leg\<^sub>0: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and w :: "'b" assumes "arrow_of_spans_of_maps V H \<a> \<i> src trg r\<^sub>0 r\<^sub>1 s\<^sub>0 s\<^sub>1 w" shows "partial_magma.arr V (H s\<^sub>0 w)"
order_linear: fixes a :: "'c" shows "order a [:- a, 1::'c:] = Suc 0"
IH: shows "\<forall>M. (\<forall>N. (N, M) \<in> sub_nmset \<longrightarrow> P N) \<longrightarrow> P M"
comp: fixes f :: "'f \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'g"   and h :: "'h \<Rightarrow> 'd"   and i :: "'e \<Rightarrow> 'i" shows "map_fun f g \<circ> map_fun h i = map_fun (h \<circ> f) (g \<circ> i)"
mult_base: fixes m :: "hmultiset" assumes "m \<in># hmsetmset M" shows "base ^ local.decode 0 m div base ^ e = base * (base ^ local.decode 0 m div (base * base ^ e))"
degree_pdev_upd_degree: fixes b :: "'a pdevs"   and n :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" assumes "degree b \<le> Suc n"   and "degree b \<le> Suc (degree_aform_err X)"   and "degree_aform_err X \<le> n" shows "degree (pdev_upd b (degree_aform_err X) (0::'a)) \<le> n"
filter_append: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" shows "filter P (xs @ ys) = filter P xs @ filter P ys"
map_cong: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "list.map f xs = list.map g ys"
product_concat_map: fixes xs :: "'g list"   and ys :: "'h list" shows "List.product xs ys = concat (map (\<lambda>x. map (Pair x) ys) xs)"
prj_simps_3: shows "C.arr trg_trg_f.Prj\<^sub>0"
emeasure_PiM: fixes I :: "nat set"   and A :: "nat \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M.events" shows "emeasure (Pi\<^sub>M I (\<lambda>i. M)) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure M (A i))"
rel_with_source_impl_SRel_preserves_barbs: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and SRel :: "('procS \<times> 'procS) set" assumes "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a (STCalWB SWB TWB))"   and "\<forall>S1 S2. (S1, S2) \<in> SRel \<longrightarrow> (SourceTerm S1, SourceTerm S2) \<in> Rel" shows "rel_preserves_binary_pred SRel (\<lambda>a. hasBarb a SWB)"
mult_cong_r: fixes G :: "('c, 'd) monoid_scheme"   and b :: "'c"   and b' :: "'c"   and a :: "'c" assumes "Group.monoid G"   and "b \<sim>\<^bsub>G\<^esub> b'"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b' \<in> carrier G" shows "a \<otimes>\<^bsub>G\<^esub> b \<sim>\<^bsub>G\<^esub> a \<otimes>\<^bsub>G\<^esub> b'"
assms_2: fixes c :: "real"   and x :: "'a" shows "f (c *\<^sub>R x) = c *\<^sub>R f x"
UN_in_lattice: fixes L :: "'b set set"   and I :: "'c set"   and r :: "'c \<Rightarrow> 'b set" assumes "lattice L"   and "\<And>i. i \<in> I \<Longrightarrow> r i \<in> L" shows "\<Union> (r ` I) \<in> L"
AddSA_Chi: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "Chi (A [++] (S, SA)) S = States SA \<union> Chi A S"
jump_cong: fixes x :: "real"   and y :: "real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "x = y"   and "\<forall>\<^sub>F x in at x. f x = g x" shows "jump f x = jump g y"
UP\<^sub>0_def: fixes a :: "'a" shows "UP\<^sub>0 a \<equiv> if B.obj a then MkIde \<^bold>\<langle>a\<^bold>\<rangle>\<^sub>0 else null"
Max_eq_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and m :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(linorder.Max less_eq A = m) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq a m))"
not_authKeys_not_AKcryptSK: fixes K :: "nat"   and evs :: "event list" assumes "K \<notin> authKeys evs"   and "K \<notin> range shrK"   and "evs \<in> kerbIV_gets" shows "\<forall>SK. \<not> AKcryptSK K SK evs"
I_int: fixes N :: "nat"   and K :: "real"   and e :: "real" shows "integrable M (I N K e)"
ABORT_bind: fixes f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "ABORT \<bind> f = ABORT"
assms: shows "is_exec_frag_of A (cons_exec e p)"
assms_1: shows "Ball (set ms\<^sub>1) valid_node"
abs_eq: fixes x :: "int" shows "num_ivl x = [Fin x, Fin x]"
check_eqv_def: shows "check_eqv \<equiv> WS1S_Alt.check_eqv"
poly_mapping_eqI: fixes f :: "'c \<Rightarrow>\<^sub>0 'd"   and g :: "'c \<Rightarrow>\<^sub>0 'd" assumes "\<And>k. lookup f k = lookup g k" shows "f = g"
bisim1Try: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and C :: "String.literal"   and V :: "nat"   and e2 :: "'addr expr1" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,try e catch(C V) e2,h \<turnstile> (try e' catch(C V) e2, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
seq_lident: fixes x :: "'a" shows "(1::'a) ; x = x"
Trans: fixes A :: "'c set"   and B :: "'c set"   and F :: "'c program"   and C :: "'c set" assumes "(A, B) \<in> leads F"   and "(B, C) \<in> leads F" shows "(A, C) \<in> leads F"
Fix_map_dual_var: fixes f :: "'a \<Rightarrow> 'a" shows "Fix (\<partial>\<^sub>F f) = \<partial> ` Fix f"
Repl_Rep_game_Abs_game: fixes z :: "ZF" assumes "\<forall>e. Elem e z \<longrightarrow> e \<in> games_lfp" shows "Repl z (Rep_game \<circ> Abs_game) = z"
filter_pdevs_raw_nonzeros: fixes s :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" shows "{i. filter_pdevs_raw s f i \<noteq> (0::'b)} = {i. f i \<noteq> (0::'b)} \<inter> {x. s x (f x)}"
dynmethd_member_in: fixes G :: "prog"   and statC :: "qtname"   and dynC :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "dynmethd G statC dynC sig = Some m"   and "is_class G statC"   and "wf_prog G" shows "G \<turnstile>Methd sig m member_in dynC"
sym: fixes x :: "'a"   and y :: "'a" assumes "x .=\<^bsub>L\<^esub> y"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "y .=\<^bsub>L\<^esub> x"
multiplicity_gt_zero_iff: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "(0 < factorial_semiring.multiplicity times one p x) = dvd.dvd times p x"
assms_3: shows "\<forall>P. local_peak ars P \<longrightarrow> (\<exists>\<sigma>' \<tau>'. DD ars r (fst P, snd P, \<sigma>', \<tau>'))"
ball_subset_ball_iff: fixes a :: "'a"   and r :: "real"   and a' :: "'a"   and r' :: "real" shows "(ball a r \<subseteq> ball a' r') = (dist a a' + r \<le> r' \<or> r \<le> 0)"
next_performop'_def: fixes c0 :: "('c, 'd) configuration"   and c1 :: "('c, 'd) configuration"   and p :: "'c"   and c :: "'d multiset"   and r :: "'d multiset" shows "next_performop' c0 c1 p c r = (let \<Delta> = zmset_of r - zmset_of c in (\<forall>t. int (count c t) \<le> zcount (c_records c0) t) \<and> upright \<Delta> \<and> c1 = c0\<lparr>c_records := c_records c0 + \<Delta>, c_temp := (c_temp c0)(p := c_temp c0 p + \<Delta>)\<rparr>)"
disc_1: fixes xs :: "('a, 'b) tllist" assumes "is_TNil xs" shows "lnull (llist_of_tllist xs)"
abs_eq: fixes xa :: "perm"   and x :: "('b, 'c, 'd, 'e) Tree" shows "xa \<bullet> abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> (xa \<bullet> x)"
mod_ring_minus32: shows "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (minus_p32 pp) (-)"
csn: shows "cs\<^bsup>\<pi> \<guillemotleft> k\<^esup> n = cs\<^bsup>\<pi>' \<guillemotleft> k'\<^esup> n'"