hcju/isabelleps huggingface dataset:
max_aform_err_def: fixes p :: "nat"   and a1 :: "(real \<times> real pdevs) \<times> real"   and a2 :: "(real \<times> real pdevs) \<times> real" shows "max_aform_err p a1 a2 = (let ivl1 = ivl_of_aform_err p a1; ivl2 = ivl_of_aform_err p a2 in if upper ivl1 < lower ivl2 then a2 else if upper ivl2 < lower ivl1 then a1 else ivl_err (real_interval (max_interval ivl1 ivl2)))"
eq_prop: fixes P :: "bool"   and Q :: "bool" assumes "P"   and "P = Q" shows "Q"
hfthm_sound: fixes H :: "fm set"   and A :: "fm"   and e :: "(name, hf) finfun" assumes "H \<turnstile> A"   and "\<forall>B\<in>H. eval_fm e B" shows "eval_fm e A"
s_1: fixes i :: "nat" shows "e \<le> infdist (flow0 x (s i)) (\<omega>_limit_set x)"
HNextPart_allInput: fixes s :: "state"   and s' :: "state" assumes "HNextPart s s'"   and "Inv2c s" shows "\<forall>p. inpt s' p \<in> allInput s'"
eval_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" obtains c :: "('Var, 'AExp, 'BExp) Stmt" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "ca__ = cxt_to_stmt E__ cb__" and "c'a__ = cxt_to_stmt E__ c'b__" and "mds' = mds" and "(cb__, mem) \<leadsto>\<^sub>s (c'b__, mem')"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" where "ca__ = c\<^sub>1 ;; c\<^sub>2" and "c'a__ = c\<^sub>1' ;; c\<^sub>2" and "\<langle>c\<^sub>1, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c\<^sub>1', mds', mem'\<rangle>\<^sub>w"   | c :: "('Var, 'AExp, 'BExp) Stmt" and mu :: "'Var ModeUpd" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "ca__ = cxt_to_stmt E (c@[mu])" and "c'a__ = cxt_to_stmt E c'" and "\<langle>c, update_modes mu mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w"
sep_conj_assoc: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" shows "((P \<and>* Q) \<and>* R) = (P \<and>* Q \<and>* R)"
continuous_map_pairwise: fixes Z :: "'a topology"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'a \<Rightarrow> 'b \<times> 'c" shows "continuous_map Z (prod_topology X Y) f = (continuous_map Z X (fst \<circ> f) \<and> continuous_map Z Y (snd \<circ> f))"
assms_2: shows "\<not> (\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0)"
sum_eq_def: fixes eql :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and eqr :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and s1 :: "'e + 'g"   and s2 :: "'f + 'h" shows "sum_eq eql eqr s1 s2 \<equiv> case (s1, s2) of (Inl x1, Inl x2) \<Rightarrow> eql x1 x2 | (Inl x1, Inr ba) \<Rightarrow> False | (Inr x1, Inl a) \<Rightarrow> False | (Inr x1, Inr x2) \<Rightarrow> eqr x1 x2"
ih_11: shows "InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state')"
msetext_dersh_trans: fixes zs :: "'b list"   and A :: "'b set"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "zs \<in> lists A"   and "ys \<in> lists A"   and "xs \<in> lists A"   and "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "msetext_dersh gt zs ys"   and "msetext_dersh gt ys xs" shows "msetext_dersh gt zs xs"
in_disconnected_nodes_no_parent: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr"   and owner_document :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "h \<turnstile> Shadow_DOM.get_parent node_ptr \<rightarrow>\<^sub>r None"   and "h \<turnstile> Shadow_DOM.get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r owner_document"   and "h \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r disc_nodes"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "node_ptr \<in> set disc_nodes"
\<mu>_sq_non_zero: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "prob_space_with_indicators M I A" shows "0 < (prob_space_with_indicators.\<mu> M I A)\<^sup>2"
exE_realizer': fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<times> 'b" assumes "P (snd p) (fst p)" obtains x :: "'c" and y :: "'b" where "P y x"
conjoin_non_aborting: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "conj_distrib (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)"   and "(chaos::'b::refinement_lattice) \<sqsubseteq> (c::'b::refinement_lattice)" shows "(d::'b::refinement_lattice) \<sqsubseteq> (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) d (c::'b::refinement_lattice)"
in_MPLS_leq_2_pow_n: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and x :: "nat" assumes "finite PROB"   and "x \<in> MPLS PROB" shows "x \<le> 2 ^ card (prob_dom PROB) - 1"
Suc_1: fixes A :: "(('a, 'n) vec, 'n) vec"   and P :: "(('a, 'n) vec, 'n) vec" assumes "n < nrows A" shows "det (fst (row_add_iterate_PA (P, A) n i j)) = det P"
min_bot2: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.order_bot bot less_eq less" shows "ord.min less_eq x bot = bot"
A: shows "b \<in> arcs (pre_digraph.del_arc (del_arc (arev a)) a)"
simps_4: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and b :: "bexp"   and C :: "acom" shows "varacom ({(I, S, E)} WHILE b DO C) = support I \<union> varacom C"
realpow_pos_nth: fixes n :: "nat"   and a :: "real" assumes "0 < n"   and "0 < a" shows "\<exists>r>0. r ^ n = a"
validInput: fixes a :: "name"   and x :: "name"   and P :: "pi" shows "valid (a<x>.P) = valid P"
i_def: shows "i \<equiv> \<lambda>x. ((1 - x) ^ (n - 1) / fact (n - 1)) *\<^sub>R nth_derivative n f (X + x *\<^sub>R H) H"
list: shows "root \<Midarrow>(x # xs)\<Rightarrow> root''"
choose_square_sum: fixes n :: "nat" shows "(\<Sum>k\<le>n. (n choose k)\<^sup>2) = 2 * n choose n"
secret_object4_def: shows "\<delta> = \<beta> \<squnion> (- (\<alpha> \<squnion> - \<beta>) \<squnion> - (\<alpha> \<squnion> - \<beta>))"
plus_comm: fixes a :: "int poly fract"   and b :: "int poly fract" shows "rat_poly_plus a b = rat_poly_plus b a"
vars_Var: fixes x :: "nat" shows "vars (Abstract_Linear_Poly.Var x) = {x}"
simps_5: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b"   and e :: "'b" shows "median_le_5 [a, b, c, d, e] = median_5 a b c d e"
sets_kernel: fixes K :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and N :: "'b measure"   and a :: "'a" assumes "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "a \<in> space M" shows "sets (K a) = sets N"
big_small_step: fixes \<rho> :: "(nat \<times> bval) list"   and e :: "exp"   and w :: "bval"   and \<rho>' :: "(nat \<times> exp) list" assumes "\<rho> \<turnstile> e \<Down> w"   and "bs_env \<rho> \<rho>'"   and "FV e \<subseteq> set (map fst \<rho>)" shows "\<exists>v. psubst \<rho>' e \<longrightarrow>* v \<and> is_val v \<and> bs_val w v"
dbm_entry_dbm_min3: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "dbm_entry_val u (Some c) None (min a b)" shows "dbm_entry_val u (Some c) None b"
replicate_mset_msubseteq_iff: fixes m :: "nat"   and a :: "'k"   and n :: "nat"   and b :: "'k" shows "(replicate_mset m a \<subseteq># replicate_mset n b) = (m = 0 \<or> a = b \<and> m \<le> n)"
rev_OrderingSetMap: shows "OrderingSetMap (\<^bold>\<le>*) (\<^bold><*) (\<^bold>\<le>) (\<^bold><) (f ` P) (the_inv_into P f)"
wf_imethds_hiding_objmethdsD: fixes G :: "prog"   and sig :: "sig"   and old :: "qtname \<times> methd"   and I :: "qtname"   and new :: "qtname \<times> mhead" assumes "methd G Object sig = Some old"   and "is_iface G I"   and "wf_prog G"   and "accmodi old \<noteq> Private"   and "new \<in> DeclConcepts.imethds G I sig" shows "G\<turnstile>resTy new\<preceq>resTy old \<and> is_static new = is_static old"
const_is_polynomial: fixes a :: "'c"   and K :: "'c set" assumes "a \<in> K - {\<zero>\<^bsub>A\<^esub>}" shows "polynomial\<^bsub>A\<^esub> K [a]"
strategy_proof_def: fixes ds :: "'d set"   and \<phi> :: "('d \<Rightarrow> ('x \<times> 'x) set) \<Rightarrow> ('h \<Rightarrow> 'x set \<Rightarrow> 'x set) \<Rightarrow> 'd set \<Rightarrow> 'x set" shows "strategy_proof ds \<phi> = (\<forall>Pd Ch. ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch \<longrightarrow> \<not> (\<exists>d\<in>ds. \<exists>Pd'. ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (Pd(d := Pd')) Ch \<and> (\<exists>y\<in>\<phi> (Pd(d := Pd')) Ch ds. y \<in> AboveS (Pd d) (dX (\<phi> Pd Ch ds) d))))"
assms_11: shows "InvariantNoDecisionsWhenConflict (butlast (getF state)) (getM state) (currentLevel (getM state))"
rec_next: fixes ls :: "nat"   and key :: "nat" shows "g_rec ls key = ls \<or> c_tl (g_rec ls key) = ls"
insort_remove1_same_eq: fixes cmp :: "'b comparator"   and xs :: "'b list"   and x :: "'b" assumes "Sorting_Algorithms.sorted cmp xs"   and "x \<in> set xs"   and "hd (filter (\<lambda>y. compare cmp x y = Equiv) xs) = x" shows "Sorting_Algorithms.insort cmp x (remove1 x xs) = xs"
list_int_nat: fixes k :: "nat"   and x :: "'b"   and xs :: "'b list" shows "0 < k \<longrightarrow> (x # xs) ! k = xs ! nat (int k + - 1)"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "\<^bold>\<lfloor>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<^bold>\<rfloor> = (\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>v\<^bold>\<rfloor>"
reversepath_joinpaths: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "pathfinish g1 = pathstart g2" shows "reversepath (g1 +++ g2) = reversepath g2 +++ reversepath g1"
ListUpdate_E: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "('a \<times> 'b \<times> 'c) list" shows "ListUpdateE f xs = ListUpdate f xs (\<lambda>e x. e)"
assms: assumes "nofail a" shows "hn_refine \<Gamma> c \<Gamma>' R a"
twinAmicPair_def: fixes n :: "nat"   and m :: "nat" shows "twinAmicPair n m = (n Amic m \<and> (\<nexists>k l. Min {n, m} < k \<and> k < Max {n, m} \<and> k Amic l))"
subtype_acyclic: shows "acyclicP (\<sqsubset>)"
simps: fixes lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "ccpo.compact lub ord x = (ccpo.admissible lub ord (\<lambda>y. \<not> ord x y) \<and> ccpo.admissible lub ord ((\<noteq>) x))"
Gr_range: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "Range (BNF_Def.Gr A f) = f ` A"
Step_CE: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and \<psi> :: "('sf, 'sc, 'sz) formula"   and \<sigma> :: "('sf, 'sc, 'sz) subst" shows "step_result (SG, C) (i, CE \<phi> \<psi> \<sigma>) = (closeI SG i, C)"
integ1: shows "integrable M1 (\<lambda>x. LINT y|M2. norm (f (x, y)))"
"2_1": fixes x :: "'a"   and gs :: "'a vec list"   and mus :: "'a list"   and g' :: "'a vec" assumes "x = f \<bullet> g / n_g"   and "adjuster_wit (x # mus_acc) f gs = (mus, g')"   and "n_gs = map (\<lambda>x. (x, \<parallel>x\<parallel>\<^sup>2)) gs"   and "f \<in> carrier_vec n"   and "- x \<cdot>\<^sub>v g + acc \<in> carrier_vec n"   and "set gs \<subseteq> carrier_vec n" shows "mus_adjuster f n_gs (x # mus_acc) (- x \<cdot>\<^sub>v g + acc) = (mus, - x \<cdot>\<^sub>v g + acc + g')"
completeness_game_def: fixes h :: "'g \<times> 'a"   and w :: "'h \<times> 'b"   and e :: "'e" shows "\<Sigma>_AND.completeness_game h w e = init_AND h w \<bind> (\<lambda>(r, a). response_AND r w e \<bind> (\<lambda>z. return_spmf (check_AND h a e z)))"
a_coset_join1: fixes H :: "'a set"   and x :: "'a" assumes "H +> x = H"   and "x \<in> carrier R"   and "subgroup H (add_monoid R)" shows "x \<in> H"
wf_rexp_of_alt': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "local.wf n (rexp_of_alt' n \<phi>)"
empty_neutral_1: fixes x :: "'c multiset" shows "{#} + x = x"
constr_rep_1: fixes x3 :: "name"   and x1 :: "name"   and x2 :: "pi" shows "Rep_residual (x3<\<nu>x1> \<prec> x2) = BoundOutputR_Rep x3 ([x1].x2)"
indep_in_sub_cong: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and \<E>' :: "'b set"   and X :: "'b set" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier"   and "\<E>' \<subseteq> \<E>" shows "indep_system.indep_in (indep_system.indep_in indep \<E>) \<E>' X = indep_system.indep_in indep \<E>' X"
S'_def: shows "S' \<equiv> {x \<in> S. det (matrix (g' x)) * f (g x) \<noteq> 0}"
big_step_induct: fixes x1a :: "com"   and x1b :: "(char list \<Rightarrow> int option) option"   and x2a :: "(char list \<Rightarrow> int option) option"   and P :: "com \<Rightarrow> (char list \<Rightarrow> int option) option \<Rightarrow> (char list \<Rightarrow> int option) option \<Rightarrow> bool" assumes "(x1a, x1b) \<Rightarrow> x2a"   and "\<And>c. P c None None"   and "\<And>s. P SKIP s s"   and "\<And>a s x. Def_Init_Exp.aval a s = None \<Longrightarrow> P (x ::= a) (Some s) None"   and "\<And>a s i x. Def_Init_Exp.aval a s = Some i \<Longrightarrow> P (x ::= a) (Some s) (Some (s(x \<mapsto> i)))"   and "\<And>c\<^sub>1 s\<^sub>1 s\<^sub>2 c\<^sub>2 s\<^sub>3. \<lbrakk>(c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2; P c\<^sub>1 s\<^sub>1 s\<^sub>2; (c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3; P c\<^sub>2 s\<^sub>2 s\<^sub>3\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) s\<^sub>1 s\<^sub>3"   and "\<And>b s c\<^sub>1 c\<^sub>2. Def_Init_Exp.bval b s = None \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (Some s) None"   and "\<And>b s c\<^sub>1 s' c\<^sub>2. \<lbrakk>Def_Init_Exp.bval b s = Some True; (c\<^sub>1, Some s) \<Rightarrow> s'; P c\<^sub>1 (Some s) s'\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (Some s) s'"   and "\<And>b s c\<^sub>2 s' c\<^sub>1. \<lbrakk>Def_Init_Exp.bval b s = Some False; (c\<^sub>2, Some s) \<Rightarrow> s'; P c\<^sub>2 (Some s) s'\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (Some s) s'"   and "\<And>b s c. Def_Init_Exp.bval b s = None \<Longrightarrow> P (WHILE b DO c) (Some s) None"   and "\<And>b s c. Def_Init_Exp.bval b s = Some False \<Longrightarrow> P (WHILE b DO c) (Some s) (Some s)"   and "\<And>b s c s' s''. \<lbrakk>Def_Init_Exp.bval b s = Some True; (c, Some s) \<Rightarrow> s'; P c (Some s) s'; (WHILE b DO c, s') \<Rightarrow> s''; P (WHILE b DO c) s' s''\<rbrakk> \<Longrightarrow> P (WHILE b DO c) (Some s) s''" shows "P x1a x1b x2a"
Zero_not_eq_One: shows "J.Zero \<noteq> J.One"
sup_id_star2: fixes x :: "'a" assumes "(1::'a) \<le> x" shows "x\<^sup>\<star> \<cdot> x = x\<^sup>\<star>"
i_cut_set_restriction_disj: fixes cut_op :: "'a set \<Rightarrow> 'a \<Rightarrow> 'a set"   and f :: "'a set \<Rightarrow> 'a set"   and t :: "'a" assumes "cut_op = (\<down><) \<or> cut_op = (\<down>\<le>) \<or> cut_op = (\<down>>) \<or> cut_op = (\<down>\<ge>)"   and "f = (\<lambda>I. cut_op I t)" shows "set_restriction f"
pure_assn_def: fixes b :: "bool" shows "\<up> b = Abs_assn (Assn (\<lambda>h. addrOf h = {} \<and> b))"
R\<^sub>0_in_hom: shows "Maps.in_hom rs.R\<^sub>0 (Maps.MkIde (src (tab\<^sub>0 r))) (Maps.MkIde (src r))"
sccs_verts_impl_def: fixes G :: "'b list \<times> ('b \<times> 'b) list" shows "sccs_verts_impl G \<equiv> set ` (\<lambda>x. RTLI.rtrancl_i (snd G) [x]) ` set (fst G)"
coprime_rat_poly_iff_coprimereal_poly: fixes p :: "rat poly"   and q :: "rat poly" shows "comm_monoid_mult_class.coprime p q = comm_monoid_mult_class.coprime (real_of_rat_poly p) (real_of_rat_poly q)"
Abs_binding_cases: fixes x :: "binding" obtains y :: "binding_Rep" where "x = Abs_binding y" and "y \<in> Collect binding_Rep_set"
dvd_mod_imp_dvd: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c dvd a mod b"   and "c dvd b" shows "c dvd a"
inject_10: fixes x111 :: "char list \<Rightarrow> com option"   and x112 :: "com"   and y111 :: "char list \<Rightarrow> com option"   and y112 :: "com" shows "(PScope x111 x112 = PScope y111 y112) = (x111 = y111 \<and> x112 = y112)"
finite_UN_I: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (\<Union> (B ` A))"
Live_def: shows "Live \<equiv> WF(m1)_(x, y) \<and> WF(m2)_(x, y)"
set_vwalk_arcs_cons: fixes p :: "'a list"   and u :: "'a" shows "set (vwalk_arcs p) \<subseteq> set (vwalk_arcs (u # p))"
simps_6: fixes a :: "Ferrack.num" shows "bound0 (Ge a) = numbound0 a"
assms_2: shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
stalk_at_prime_is_iso_to_local_ring_at_prime_aux: fixes V :: "'a set set"   and \<phi> :: "('a set set \<times> ('a set \<Rightarrow> ('a \<times> 'a) set)) set \<Rightarrow> ('a \<times> 'a) set" assumes "is_zariski_open V"   and "\<pp> \<in> V"   and "ring_homomorphism \<phi> st.carrier_stalk st.add_stalk st.mult_stalk (st.zero_stalk V) (st.one_stalk V) pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at"   and "\<forall>U\<in>top.neighborhoods \<pp>. \<forall>s\<in>\<O> U. (\<phi> \<circ> st.canonical_fun U) s = local.key_map U s" shows "ring_isomorphism \<phi> st.carrier_stalk st.add_stalk st.mult_stalk (st.zero_stalk V) (st.one_stalk V) pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at"
ennreal_add_less_top: fixes a :: "ennreal"   and b :: "ennreal" shows "(a + b < \<top>) = (a < \<top> \<and> b < \<top>)"
headn_nz: fixes p :: "poly"   and n0 :: "nat"   and m :: "nat" assumes "isnpolyh p n0" shows "(headn p m = 0\<^sub>p) = (p = 0\<^sub>p)"
Card_order_infinite_not_under: fixes r :: "('c \<times> 'c) set" assumes "Card_order r"   and "infinite (Field r)" shows "\<nexists>a. Field r = under r a"
absolute_liveness_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "absolute_liveness P = ((\<exists>\<sigma>. P \<sigma>) \<and> (\<forall>\<sigma> \<alpha>. P \<sigma> \<longrightarrow> P (\<alpha> @- \<sigma>)))"
seq_char: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" shows "seq g f = (C1.seq (fst g) (fst f) \<and> C2.seq (snd g) (snd f))"
ipurge_tr_rev_t_trace: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and u :: "'d" assumes "secure P I D"   and "xs \<in> traces P" shows "ipurge_tr_rev_t I D u xs \<in> traces P"
80: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<squnion> x"
K5_contractedI: fixes G :: "'b pair_pre_digraph"   and H :: "'b pair_pre_digraph" assumes "subdivision_pair G H"   and "K\<^bsub>5\<^esub> (with_proj G)" shows "K\<^bsub>5\<^esub> (with_proj (gen_contr_graph (with_proj H) (verts3 (with_proj H))))"
case_prod_app: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'd"   and x :: "'e \<times> 'f"   and y :: "'g" shows "(case x of (x, xa) \<Rightarrow> f x xa) y = (case x of (l, r) \<Rightarrow> f l r y)"
deg_one_eval: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> int" assumes "g \<in> carrier (UP Zp)"   and "deg Zp g = 1"   and "t \<in> carrier Zp" shows "g \<bullet> t = g 0 \<oplus> g 1 \<otimes> t"
moves: shows "s1 -\<tau>1\<rightarrow>* s1'"
natural_isomorphism_axioms: shows "natural_isomorphism Right_a.comp Right_a.comp Right_a.R (H\<^sub>R a') (H\<^sub>R \<phi>)"
abs_nn: fixes x :: "'a" shows "(0::'a) \<le> \<bar>x\<bar>"
exec_stop_vs_control_independence: fixes upd :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c control_state_scheme \<Rightarrow> 'c control_state_scheme"   and f :: "'b \<Rightarrow> 'b"   and \<sigma> :: "'c control_state_scheme" assumes "\<sharp> upd" shows "exec_stop (upd f \<sigma>) = exec_stop \<sigma>"
ext_funcset_to_sing_iff: fixes A :: "'e set"   and a :: "'f" shows "A \<rightarrow>\<^sub>E {a} = {\<lambda>x\<in>A. a}"
fundamental_theorem_of_linear_inequalities_full_dim_2: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n"   and "finite A"   and "local.span A = carrier_vec n" shows "cone A = {b \<in> carrier_vec n. \<nexists>B c. B \<subseteq> A \<and> c \<in> {normal_vector B, - normal_vector B} \<and> Suc (card B) = n \<and> lin_indpt B \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)}"
runit_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and trm :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "elementary_cartesian_category C pr0 pr1 one trm"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (pr1 a one) (elementary_category_with_binary_products.prod C pr0 pr1 a one) a"
func_le_def: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "f \<preceq> g = (\<forall>x\<in>dom f. f x = g x)"
simps_2: fixes A :: "'a"   and m :: "'a convnet" shows "input_sizes (Conv A m) = input_sizes m"
invariance_of_dimension: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "open S"   and "inj_on f S"   and "S \<noteq> {}" shows "DIM('a) \<le> DIM('b)"
delay_bisimulation_final_base: fixes \<tau>move1 :: "'s1 \<Rightarrow> 'tl1 \<Rightarrow> 's1 \<Rightarrow> bool"   and \<tau>move2 :: "'s2 \<Rightarrow> 'tl2 \<Rightarrow> 's2 \<Rightarrow> bool" shows "delay_bisimulation_final_base trsys1 trsys2 bisim \<tau>move1 \<tau>move2 final1 final2"
simps_1: fixes c1 :: "'a acom"   and c2 :: "'a acom" shows "sub\<^sub>2 (c1;; c2) = c2"
Sup_subset_mono: fixes A :: "('a \<times> 'b) set"   and B :: "('a \<times> 'b) set" assumes "A \<subseteq> B" shows "less_eq_prod (Sup_prod A) (Sup_prod B)"
stake_0_2: fixes xs :: "[:'a:]" assumes "xs \<noteq> \<bottom>" shows "stake\<cdot>(MkI\<cdot>0)\<cdot>xs = [::]"
finite_Lim_measure_incseq: fixes A :: "nat \<Rightarrow> real set"   and n :: "nat" assumes "range A \<subseteq> Mn.events n"   and "incseq A" shows "(\<lambda>i. Mn.prob n (A i)) \<longlonglongrightarrow> Mn.prob n (\<Union> (range A))"
connected_component_eq_eq: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" shows "(connected_component_set S x = connected_component_set S y) = (x \<notin> S \<and> y \<notin> S \<or> x \<in> S \<and> y \<in> S \<and> connected_component S x y)"
\<beta>_dim_1: shows "\<beta> \<in> carrier_vec N"
assms: shows "g = (\<lambda>i. blinfun_apply (f i))"
card_B: shows "|{p \<bullet> x |p. supp P \<sharp>* p}| <o natLeq +c |UNIV|"
is_rendered_commutative_by_cone: fixes a :: "'c"   and \<chi> :: "J.Arr \<Rightarrow> 'c" assumes "local.cone a \<chi>" shows "is_rendered_commutative_by (\<chi> J.AA) (\<chi> J.BB)"
down_raw: fixes Y :: "V set"   and x :: "V" assumes "Y \<subseteq> elts x" shows "Y \<in> range elts"
OO_def: fixes R :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "R OO S = (\<lambda>x z. \<exists>y. R x y \<and> S y z)"
assms: shows "T \<in> M \<rightarrow>\<^sub>M M"