hcju/isabelleps huggingface dataset:
barendregt: fixes y :: "'a"   and z :: "'a"   and L :: "'a trm"   and M :: "'a trm"   and N :: "'a trm" assumes "y \<noteq> z"   and "y \<notin> fvs L" shows "M[y ::= N][z ::= L] = M[z ::= L][y ::= N[z ::= L]]"
Equiv2WellDefined: fixes Ax :: "('a, 'b) Axioms"   and \<phi> :: "('a, 'b) Language" assumes "Axioms Ax"   and "\<phi> \<in> Axioms.Theory Ax" shows "Sig aS\<^bsub>Ax\<^esub> \<triangleright> \<phi>"
continuous_map_product_then_coordinatewise_1: fixes T1 :: "'d topology"   and T :: "'e \<Rightarrow> 'f topology"   and I :: "'e set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and i :: "'e" assumes "continuous_map T1 (product_topology T I) f"   and "i \<in> I" shows "continuous_map T1 (T i) (\<lambda>x. f x i)"
sum_relE: fixes x :: "'c + 'd"   and x' :: "'e + 'f"   and Rl :: "('c \<times> 'e) set"   and Rr :: "('d \<times> 'f) set" assumes "(x, x') \<in> \<langle>Rl, Rr\<rangle>sum_rel" obtains l :: "'c" and l' :: "'e" where "x = Inl l" and "x' = Inl l'" and "(l, l') \<in> Rl"   | r :: "'d" and r' :: "'f" where "x = Inr r" and "x' = Inr r'" and "(r, r') \<in> Rr"
sp_generalize_instr: fixes F :: "'n \<Rightarrow> 'o fundef option"   and x :: "('c, 'p, 'n, 'e, 'f, 'g, 'h, 'i) instr"   and \<Sigma> :: "Unboxed.type option list"   and \<Sigma>' :: "Unboxed.type option list" assumes "sp_instr F x \<Sigma> = Ok \<Sigma>'" shows "sp_instr F (generalize_instr x) (map Map.empty \<Sigma>) = Ok (map Map.empty \<Sigma>')"
pr_in_hom_2: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<guillemotleft>\<pp>\<^sub>1[a, b] : a \<otimes> b \<rightarrow> a\<guillemotright>"
times_pratt2_def: fixes x :: "pratt2"   and y :: "pratt2" shows "x \<cdot> y \<equiv> x +\<^sub>2 y"
insert_ops_appendD: fixes xs :: "('c \<times> 'c option) list"   and ys :: "('c \<times> 'c option) list" assumes "insert_ops (xs @ ys)" shows "insert_ops xs"
qpm_inf_term_axioms: shows "qpm_inf_term pair_of_term term_of_pair (\<preceq>) (\<prec>) (\<preceq>\<^sub>t) (\<prec>\<^sub>t)"
normal: fixes g :: "'a"   and k :: "'a" assumes "g \<in> G"   and "k \<in> L" shows "inverse g \<cdot> k \<cdot> g \<in> L"
functor_axioms: shows "functor (\<cdot>\<^sub>J) (\<cdot>) D"
Float_eq_zero_iff: fixes m :: "int"   and e :: "int" shows "(Float m e = 0) = (m = 0)"
not_eq_extremum: fixes a :: "'b list" shows "(a \<noteq> \<epsilon>) = (\<epsilon> <s a)"
observable_transitionI: fixes P :: "'state"   and Q :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and Q' :: "'state"   and P' :: "'state" assumes "P \<Rightarrow>\<^sub>S Q"   and "Q \<rightarrow>\<^sub>S \<langle>\<alpha>,Q'\<rangle>"   and "Q' \<Rightarrow>\<^sub>S P'" shows "P \<Rightarrow>{\<alpha>}\<^sub>S P'"
add_strict_increasing: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" 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)"
cnv_def: fixes x :: "'b"   and y :: "'b" shows "cnv x y \<equiv> x = y"
simps_1: fixes w\<^sub>b :: "nat"   and b :: "'a"   and a :: "'a" shows "sibling (Leaf w\<^sub>b b) a = a"
intro: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> \<not> less x x" shows "irreflexive A less"
region_unique: fixes v :: "'c \<Rightarrow> real"   and R :: "('c \<Rightarrow> real) set" assumes "v \<in> R"   and "R \<in> \<R>" shows "[v]\<^sub>\<R> = R"
DERIV_tan: fixes x :: "'b" assumes "cos x \<noteq> (0::'b)" shows "(tan has_field_derivative inverse ((cos x)\<^sup>2)) (at x)"
matrix_to_iarray_matrix_matrix_mult: fixes A :: "(('c, 'f) vec, 'e) vec"   and B :: "(('c, 'd) vec, 'f) vec" shows "matrix_to_iarray (A ** B) = matrix_to_iarray A **i matrix_to_iarray B"
square_free_m_prod_imp_coprime_m: fixes A :: "int poly"   and B :: "int poly" assumes "square_free_m (A * B)" shows "coprime_m A B"
rep_eq: fixes x :: "('d, 'e) fmap"   and xa :: "('d, 'e) fmap" shows "fmlookup (x ++\<^sub>f xa) = Map.map_add (fmlookup x) (fmlookup xa)"
M_def: shows "M = map_minsky (\<lambda>s. if s = 1 then 2 else if s = 2 then 1 else s) id N"
ffUnion_least: fixes A :: "'c fset fset"   and C :: "'c fset" assumes "fBall A (\<lambda>X. X |\<subseteq>| C)" shows "ffUnion A |\<subseteq>| C"
map_the_restrict: fixes i :: "nat"   and A :: "nat set"   and v :: "'a option list" assumes "i \<in> A" shows "map the (restrict A v) ! i = map the v ! i"
elims: fixes x :: "nat list"   and xa :: "nat"   and xb :: "nat list"   and y :: "nat list" assumes "inc x xa xb = y" obtains y :: "nat list" and i :: "nat" and u :: "nat list" where "x = y" and "xa = i" and "xb = u" and "ya__ = (if i < length y then if u ! i < y ! i then u[i := u ! i + 1] else inc y (Suc i) u else u)"
norm_imp_pos_and_ge: fixes x :: "'b"   and n :: "real" assumes "\<parallel>x\<parallel> \<equiv> n" shows "0 \<le> \<parallel>x\<parallel> \<and> \<parallel>x\<parallel> \<le> n"
has_integral_empty_eq: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b" shows "(f has_integral i) {} = (i = (0::'b))"
reduced_GB_unique_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and G :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "dickson_grading d"   and "finite (local.punit.component_of_term ` Keys F)"   and "F \<subseteq> punit.dgrad_p_set d m"   and "punit.is_reduced_GB G"   and "local.punit.pmdl G = local.punit.pmdl F" shows "punit.reduced_GB F = G"
H: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list" assumes "fst (iter M T) \<Longrightarrow> P M (snd (iter M T))" shows "P M T"
tr: fixes a :: "'a"   and b :: "'b"   and c :: "'c" assumes "P1 a b"   and "P2 b c" shows "P3 a c"
weave_complementary_nthss: fixes A :: "nat set"   and xs :: "'m list" shows "weave A (nths xs A) (nths xs (- A)) = xs"
projection_subset_eq_from_superset_eq: fixes xs :: "'b list"   and X :: "'b set"   and Y :: "'b set"   and ys :: "'b list" assumes "xs \<upharpoonleft> (X \<union> Y) = ys \<upharpoonleft> (X \<union> Y)" shows "xs \<upharpoonleft> X = ys \<upharpoonleft> X"
ide_def: fixes a :: "'c" shows "C.ide a \<equiv> C a a \<noteq> C.null \<and> (\<forall>f. (C f a \<noteq> C.null \<longrightarrow> C f a = f) \<and> (C a f \<noteq> C.null \<longrightarrow> C a f = f))"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Gt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f3"
tyenv_sec_mode_update: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and P'' :: "'BExp set"   and \<Gamma>'' :: "'Var \<Rightarrow> 'BExp set option"   and mem :: "'Var \<Rightarrow> 'Val"   and \<S> :: "'Var set \<times> 'Var set"   and mds :: "Mode \<Rightarrow> 'Var set"   and P :: "'BExp set"   and upd :: "'Var ModeUpd"   and \<S>'' :: "'Var set \<times> 'Var set" assumes "\<forall>x. to_total \<Gamma> x \<le>:\<^sub>P'' to_total \<Gamma>'' x"   and "pred P'' mem"   and "\<S> = (mds AsmNoWrite, mds AsmNoReadOrWrite)"   and "anno_type_sec \<Gamma> \<S> P upd"   and "\<S>'' = add_anno_stable \<S> upd"   and "type_stable \<S> P"   and "P'' = P |` {v. stable \<S>'' v}"   and "\<Gamma>'' = \<Gamma> \<oplus>\<^sub>\<S> upd"   and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem \<le> dma mem x" shows "\<forall>x\<in>dom \<Gamma>''. x \<notin> update_modes upd mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma>'' x)) mem \<le> dma mem x"
combine_common_factor: fixes a :: "'a"   and e :: "'a"   and b :: "'a"   and c :: "'a" shows "a ; e + (b ; e + c) = (a + b) ; e + c"
retractionE: fixes g :: "'a \<times> 'a" assumes "VoV.retraction g" obtains f :: "'a \<times> 'a" where "VoV.ide (VoV.comp g f)"
mfinal1_inv_simulation: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "FWdelay_bisimulation_final_base final1 r1 final2 r2 bisim \<tau>move1 \<tau>move2"   and "FWbisimulation_base.mbisim bisim bisim_wait s1 s2" shows "\<exists>s2'. (\<tau>trsys.silent_move (multithreaded_base.redT final2 r2 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move2))\<^sup>*\<^sup>* s2 s2' \<and> FWbisimulation_base.mbisim bisim bisim_wait s1 s2' \<and> final_thread.final_threads final1 s1 \<subseteq> final_thread.final_threads final2 s2' \<and> shr s2' = shr s2"
norm_infsetsum_bound: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "norm (infsetsum f A) \<le> (\<Sum>\<^sub>ax\<in>A. norm (f x))"
simps_2: fixes x :: "'a set" shows "min_maxsimpchain [x] = maxsimp x"
length_Cons: fixes x :: "'a"   and xs :: "'a list" shows "|x # xs| = Suc |xs|"
card_of_Fpow_mono: fixes A :: "'c set"   and B :: "'d set" assumes "|A| \<le>o |B|" shows "|Fpow A| \<le>o |Fpow B|"
assoc_simps_5: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.cod \<a>\<^sub>D[f, g, h] = D.cod f \<star>\<^sub>D D.cod g \<star>\<^sub>D D.cod h"
pmdl_gb_red_aux: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "set (gb_red_aux bs ps) \<subseteq> pmdl (args_to_set ([], bs, ps))"
\<phi>: shows "local.iso \<phi> \<and> \<guillemotleft>\<phi> : a \<star> a \<Rightarrow> a\<guillemotright>"
intro: fixes p :: "int" assumes "prime p" shows "padic_integers p"
dividesI: fixes p2 :: "'a list"   and p1 :: "'a list"   and q :: "'a list" assumes "poly p2 = poly (p1 *** q)" shows "p1 divides p2"
simps_1: shows "ExcessNotAtRec [] = (\<lambda>g. 0)"
ppp: shows "p = int_of_uint64 (uint64_of_int p)"
many_strongly_connected_iff_12: fixes x :: "'a" assumes "is_p_fun x" shows "many_strongly_connected x = (x\<^sup>\<star> ; x ; x\<^sup>T \<le> x\<^sup>+)"
step_invariantD: fixes A :: "('e, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<times> 'f \<times> 'e \<Rightarrow> bool"   and s :: "'e"   and a :: "'f"   and s' :: "'e" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "P (s, a, s')"
meet_distr: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ddlat le" shows "dlat.meet le x (dlat.join le y z) = dlat.join le (dlat.meet le x y) (dlat.meet le x z)"
cod_inv: fixes f :: "'b" assumes "B.iso f" shows "B.cod (B.inv f) = B.dom f"
proj_lnil: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b" shows "dynamic_component.proj tCMP active c []\<^sub>l = []\<^sub>l"
log_neg_const: fixes x :: "real" assumes "x \<le> 0" shows "log b x = log b 0"
set3_FGl_def: fixes x :: "(('b, 'c, 'd, 'h, 'i, 'j, 'k) G, 'b, 'a, 'd, 'l, 'm, 'i, 'n, 'o, 'p) F" shows "set3_FGl x = set3_F x"
ik\<^sub>s\<^sub>t_is_rcv_set: fixes A :: "('d, 'e) strand_step list" shows "ik\<^sub>s\<^sub>t A = {t. receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set A}"
simps_7: fixes a :: "MIR.num" shows "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if v \<le> 0 then T else F | _ \<Rightarrow> Le (reducecoeff a'))"
IH1_3: fixes i :: "nat" assumes "i < lo" shows "xs'' ! i = xs' ! i"
Awhile_5: shows "lesvars upds \<inter> varacom ({A} WHILE b DO C) = {}"
rep_eq: shows "lookup 1 = (\<lambda>k. (1::'c) when k = (0::'b))"
simps_1: fixes t\<^sub>1 :: "pterm"   and t\<^sub>2 :: "pterm" shows "pre_strong_term_class.wellformed (t\<^sub>1 $\<^sub>p t\<^sub>2) = (pre_strong_term_class.wellformed t\<^sub>1 \<and> pre_strong_term_class.wellformed t\<^sub>2)"
i: shows "\<forall>(k, \<gamma>)\<in>subdiv. line_integral_exists F basis \<gamma>"
fcomp_id: fixes f :: "'d \<Rightarrow> 'e" shows "fcomp f id = f"
ord_modulus_mult_coprime: fixes m :: "nat"   and n :: "nat"   and x :: "nat" assumes "coprime m n" shows "ord (m * n) x = lcm (ord m x) (ord n x)"
simps_6: fixes f1 :: "'g"   and f2 :: "nat \<Rightarrow> 'g \<Rightarrow> 'g" shows "rec_nat f1 f2 0 = f1"
Collect_subset: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "{x::'b \<in> A::'b set. (P::'b \<Rightarrow> bool) x} \<subseteq> A"
find_lasso_tr_correct: fixes G_impl :: "('c \<Rightarrow> bool, 'c \<Rightarrow> 'c list, 'c list, 'c \<Rightarrow> integer, 'd) gen_igbg_impl_scheme"   and G :: "('e, 'f) igb_graph_rec_scheme"   and Re :: "('d \<times> 'f) set"   and node_rel :: "('c \<times> 'e) set"   and node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat"   and node_def_hash_size :: "nat" assumes "(G_impl, G) \<in> igbg_impl_rel_ext Re node_rel"   and "(node_eq_impl, (=)) \<in> node_rel \<rightarrow> node_rel \<rightarrow> bool_rel"   and "is_bounded_hashcode node_rel node_eq_impl node_hash_impl"   and "is_valid_def_hm_size TYPE('c) node_def_hash_size"   and "igb_fr_graph G" shows "RETURN (find_lasso_tr node_eq_impl node_hash_impl node_def_hash_size G_impl) \<le> \<Down> (fl_rel node_rel) (igb_graph.find_lasso_spec G)"
to_fun_monom: fixes c :: "'a"   and x :: "'a"   and n :: "nat" assumes "c \<in> carrier R"   and "x \<in> carrier R" shows "to_fun (monom P c n) x = c \<otimes> x [^] n"
ur1: shows "\<exists>!x. root_cond (p1, l1, r1) x"
changes_append_replicate_0: fixes xs :: "'b list"   and n :: "nat" shows "changes (xs @ replicate n (0::'b)) = changes xs"
C_wf: shows "list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) C"
bisim1_xcp_pcD: 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 a :: "'addr" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "pc < length (compE2 e)"
equiv_utrans_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "equiv_utrans t u = (le_utrans t u \<and> le_utrans u t)"
analz_valid: fixes H :: "msg set"   and x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> asym.valid \<union> range LtK \<union> Tags"   and "implem_asym (Chan x A B M) \<in> analz H" shows "implem_asym (Chan x A B M) \<in> H"
seqI': fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and c :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.in_hom f a b"   and "S.VV.in_hom g b c" shows "S.VV.seq g f"
R_consts_2: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "\<not> w \<Turnstile>\<^sub>n \<phi> R\<^sub>n false\<^sub>n"
unit_factor_lcm: fixes a :: "'a"   and b :: "'a" shows "unit_factor (lcm a b) = (if a = (0::'a) \<or> b = (0::'a) then 0::'a else (1::'a))"
less_top: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and top :: "'d"   and a :: "'d" assumes "class.order_top less_eq less top" shows "(a \<noteq> top) = less a top"
lex_comp_aux_def: shows "lex_comp_aux = (\<lambda>x y. case comp_of_ord lex_pp (fst (rep_nat_term x)) (fst (rep_nat_term y)) of Eq \<Rightarrow> comparator_of (snd (rep_nat_term x)) (snd (rep_nat_term y)) | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
factors_def: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list"   and a :: "'c" shows "factors G fs a = ((\<forall>x\<in>set fs. irreducible G x) \<and> foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> = a)"
ideal_prod_def: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and J :: "'c set" shows "I \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> J \<equiv> \<Inter> {L. ideal R L \<and> {x. \<exists>i\<in>I. \<exists>j\<in>J. x = i \<cdot>\<^sub>r\<^bsub>R\<^esub> j} \<subseteq> L}"
min_ennreal: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "min (ennreal x) (ennreal y) = ennreal (min x y)"
par_col2_par: fixes E :: "'p"   and F :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "E \<noteq> F"   and "A B Par C D"   and "Col C D E"   and "Col C D F" shows "A B Par E F"
assms: shows "\<forall>s\<in>set ss. \<exists>v. s = flat v \<and> \<turnstile> v : r"
HomFtorContraMor_1: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "HomC\<^bsub>C\<^esub>[f,X] \<in> mor\<^bsub>SET\<^esub>"
ext: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "CCC.comp g f \<noteq> CCC.null" shows "CCC.seq g f"
meet_ac_2: fixes La :: "('c, 'd) gorder_scheme"   and L :: "('e, 'f) gorder_scheme"   and x :: "'e"   and y :: "'e" assumes "bounded_lattice La" shows "x \<sqinter>\<^bsub>L\<^esub> y = y \<sqinter>\<^bsub>L\<^esub> x"
comp_id: fixes f :: "'b \<Rightarrow> 'c" shows "f \<circ> id = f"
bij_imp_ex1_iff: fixes P :: "'b \<Rightarrow> bool" shows "(\<exists>!x. P (f x)) = (\<exists>!y. P y)"
assms_15: shows "winding_number (c1 +++ reversepath c) z = d"
radicalI: fixes f :: "'c"   and m :: "nat"   and F :: "'c set" assumes "f ^ m \<in> F" shows "f \<in> \<surd>F"
divergences_def: fixes P :: "'b process" shows "divergences P \<equiv> snd (Rep_process P)"
cancel_nonneg_pos_add_multI: fixes c :: "real"   and x :: "real" assumes "0 \<le> c"   and "0 \<le> 1 + x" shows "0 \<le> c + c * x"
simps_45: fixes x1 :: "'b" shows "vars (Var x1) = {x1}"
Assign_Locals: fixes \<pi> :: "char list \<Rightarrow> com option"   and l :: "char list \<Rightarrow> int \<Rightarrow> int"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (Assign_Locals l, s) \<Rightarrow> <l|s>"
const_finfun: fixes a :: "'b" shows "(\<lambda>x. a) \<in> finfun"
add_nth_pref: fixes u :: "'b list"   and w :: "'b list" assumes "u <p w" shows "u \<cdot> [w ! \<^bold>|u\<^bold>|] \<le>p w"
listrel_app_same_size: fixes xs1 :: "'b list"   and ys1 :: "'c list"   and xs2 :: "'b list"   and ys2 :: "'c list"   and r :: "('b \<times> 'c) set" assumes "length xs1 = length ys1"   and "length xs2 = length ys2" shows "((xs1 @ xs2, ys1 @ ys2) \<in> listrel r) = ((xs1, ys1) \<in> listrel r \<and> (xs2, ys2) \<in> listrel r)"
simplices_map: shows "order.greater_eq X (f \<turnstile> X)"
space_restrict_space: fixes M :: "'d measure"   and \<Omega> :: "'d set" shows "space (restrict_space M \<Omega>) = \<Omega> \<inter> space M"
red_mult_scalar_leading_monomial: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "(red {f})\<^sup>*\<^sup>* (p \<odot> monomial (lc f) (lt f)) (- p \<odot> tail f)"
assms: shows "h \<turnstile> get_mode shadow_root_ptr \<rightarrow>\<^sub>r children"
z2: shows "\<forall>x. x \<sqsubset> z2 \<longrightarrow> P2 x = P2'"
supI1: fixes A :: "'a pred"   and x :: "'a"   and B :: "'a pred" assumes "eval A x" shows "eval (A \<squnion> B) x"
ltakeWhile_LNil: fixes P :: "'c \<Rightarrow> bool" shows "ltakeWhile P LNil = LNil"
invariantI: fixes A :: "('c, 'd) automaton"   and P :: "'c \<Rightarrow> bool"   and I :: "'d \<Rightarrow> bool" assumes "\<And>s. s \<in> init A \<Longrightarrow> P s"   and "\<And>s a s'. \<lbrakk>s \<in> reachable A I; P s; (s, a, s') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P s'" shows "A \<TTurnstile> (I \<rightarrow>) P"
Fun_hfun_to_arr: fixes B :: "hf"   and C :: "hf"   and F :: "hf" assumes "hfun B C F" shows "Fun (hfun_to_arr B C F) = (\<lambda>x\<^bold>\<in>UP ` hfset B. UP (happ F (DOWN x)))"
d0_def: shows "st.d0 = prod_list st.dims0"
sound_modes_no_read: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and x :: "'Var"   and i :: "nat" assumes "sound_mode_use (cms, mem)"   and "x \<in> (map snd cms ! i) GuarNoReadOrWrite"   and "i < length cms" shows "doesnt_read_or_modify (fst (cms ! i)) x"
nfv_def: fixes \<phi> :: "'c MFOTL.formula" shows "MFOTL.nfv \<phi> = Max (insert 0 (Suc ` fv \<phi>))"
simps_1: fixes E_ES :: "'a set"   and I_ES :: "'a set"   and O_ES :: "'a set"   and Tr_ES :: "'a list set"   and more :: "'b" shows "E_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = E_ES"
state_io_relD: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('b, 'c) state"   and s :: "'b"   and a :: "'c"   and s' :: "'b" assumes "state_io_rel P m"   and "run_state m s = (a, s')" shows "P s s'"
C2_a: fixes x :: "'a" shows "(0::'a) l\<rightarrow> x = (1::'a)"
inres_simps_3: fixes x :: "'b" shows "inres (RETURN x) = (=) x"
new_actionsI: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and adal :: "'b \<times> addr_loc"   and P :: "'c prog" assumes "a \<in> actions E"   and "adal \<in> action_loc P E a"   and "is_new_action (action_obs E a)" shows "a \<in> new_actions_for P E adal"
map_def: fixes f :: "'j" shows "Dom_f.map f = (if J.arr f then C.dom f__ else C.null)"
CondT: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
prems_4: assumes "f \<noteq> 0"   and "\<not> lt f adds\<^sub>t u" shows "thesis"
null_def: fixes xs :: "'b list" shows "null xs = (xs = [])"
int_of_uint64_0_iff: fixes n :: "uint64" shows "(int_of_uint64 n = 0) = (n = 0)"
Det_is_action: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "R (\<not> action.relation_of P\<^sup>f\<^sub>f \<and> \<not> action.relation_of Q\<^sup>f\<^sub>f \<turnstile> action.relation_of P\<^sup>t\<^sub>f \<and> action.relation_of Q\<^sup>t\<^sub>f \<triangleleft> \<lambda>(A, A'). tr A = tr A' \<and> wait A' \<triangleright>  action.relation_of P\<^sup>t\<^sub>f \<or> action.relation_of Q\<^sup>t\<^sub>f) \<in> {p. is_CSP_process p}"
f_analytically_valid: shows "analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j"
3: fixes m :: "int" shows "poly_mod_prime_type TYPE('a) m = (m = int CARD('a))"
abs_eq: fixes x :: "complex_homo_coords" shows "to_complex (abs_complex_homo x) = to_complex_homo_coords x"