hcju/isabelleps huggingface dataset:
Sup: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq M"   and "\<forall>x\<in>M. x \<in> ccpo.iterates Sup less_eq f" shows "Sup M \<in> ccpo.iterates Sup less_eq f"
OPT2_A: fixes x :: "'a"   and y :: "'a"   and qs :: "'a list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])" shows "T\<^sub>p [x, y] qs (OPT2 qs [x, y]) = 1"
sort_key_less_def: fixes x :: "'b sort_key"   and y :: "'b sort_key" shows "(x < y) = (sort_key_embedding x < sort_key_embedding y)"
collinear_affine_hull_collinear: fixes s :: "'a set" shows "collinear (affine hull s) = collinear s"
simps_5: fixes r :: "(bool \<times> 'b) rexp"   and s :: "(bool \<times> 'b) rexp" shows "final (Times r s) = (final s \<or> nullable s \<and> final r)"
bisim1CAS2: fixes P :: "'a prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and e2' :: "'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 e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e3 :: "'addr expr1"   and v :: "'addr val" assumes "P,e2,h \<turnstile> (e2', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3),h \<turnstile> (Val v\<bullet>compareAndSwap(D\<bullet>F, e2', e3), xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 e1) + pc, xcp)"
zero_less_measure_iff: fixes M :: "'c measure"   and A :: "'c set" shows "(0 < Sigma_Algebra.measure M A) = (Sigma_Algebra.measure M A \<noteq> 0)"
assms_1: shows "(p \<^bold>\<or> q) at a in (ps, a) # branch"
rank_link: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and t' :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.rank t = BinomialHeapStruc.rank t'" shows "BinomialHeapStruc.rank (BinomialHeapStruc.link t t') = BinomialHeapStruc.rank t + 1"
case_rews_1: fixes f1 :: "'r"   and f2 :: "'s \<rightarrow> 's tree \<rightarrow> 's tree \<rightarrow> 'r" shows "tree_case\<cdot>f1\<cdot>f2\<cdot>\<bottom> = \<bottom>"
sees_method_mono: fixes C' :: "char list"   and C :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> subtype P T' T"
and_mask: fixes w :: "'b word"   and n :: "nat" shows "w AND mask n = w << size w - n >> size w - n"
shrK_not_AKcryptSK: fixes evs :: "event list"   and K :: "nat"   and A :: "agent" assumes "evs \<in> kerbIV_gets" shows "\<not> AKcryptSK K (shrK A) evs"
conga3_suma__suma: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p"   and G' :: "'p"   and H' :: "'p"   and I' :: "'p" assumes "A B C D E F SumA G H I"   and "A B C CongA A' B' C'"   and "D E F CongA D' E' F'"   and "G H I CongA G' H' I'" shows "A' B' C' D' E' F' SumA G' H' I'"
add_mset_add_single: fixes a :: "'a"   and A :: "'a multiset" shows "a, A = A + (a, {#})"
u_simps_2: shows "hseq (tab\<^sub>0 g) f\<^sub>0g\<^sub>1.p\<^sub>0"
loops: shows "homotopic_loops S (p +++ reversepath q) (linepath a a)"
l_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier H'"   and "x \<in> carrier H'" shows "x <#\<^bsub>H'\<^esub> H \<subseteq> carrier H'"
matrix_right_invertible: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec" assumes "linear f" shows "(\<exists>B. matrix f ** B = mat 1) = (\<exists>g. linear g \<and> f \<circ> g = id)"
rel_fundefs_Some1: fixes f :: "'a \<Rightarrow> ('dyn, 'b, 'c, 'd, 'opinl) Inca.instr fundef option"   and g :: "'a \<Rightarrow> ('dyn, 'b, 'c, 'd, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef option"   and x :: "'a"   and y :: "('dyn, 'b, 'c, 'd, 'opinl) Inca.instr fundef" assumes "rel_fundefs f g"   and "f x = Some y" shows "\<exists>z. g x = Some z \<and> rel_fundef (\<lambda>x y. x = norm_instr y) y z"
comp_char: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" shows "g \<cdot> f = (if C1.seq (fst g) (fst f) \<and> C2.seq (snd g) (snd f) then (fst g \<cdot>\<^sub>1 fst f, snd g \<cdot>\<^sub>2 snd f) else (C1.null, C2.null))"
select_convs_2: fixes V :: "'b set"   and N :: "'b set"   and C :: "'b set"   and more :: "'c" shows "N \<lparr>V = V, N = N, C = C, \<dots> = more\<rparr> = N"
\<delta>_def: shows "\<delta> = Inf (r ` S0)"
I1: shows "cc (s i) \<le> k \<and> (\<forall>t ka. t < N \<and> has_ticket (tts (s i) t) ka \<longrightarrow> cc (s i) \<le> ka \<and> ka < k \<and> (\<forall>t' k'. t' < N \<and> has_ticket (tts (s i) t') k' \<and> t \<noteq> t' \<longrightarrow> ka \<noteq> k') \<and> (\<forall>k. k \<noteq> cc (s i) \<longrightarrow> \<not> locks_ticket (tts (s i) t) k))"
inf_left_commute: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
numeral_nat_1: shows "Numeral1 = Suc 0"
n_dist_var1': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x + !y) \<cdot> (t x + !z) = t x \<cdot> !y + !x \<cdot> !z"
exhaust_sel: fixes list :: "'f list" obtains "list = []"   | "list = hd list # tl list"
eU: fixes x :: "nat \<Rightarrow> real" assumes "x \<in> standard_simplex p" shows "0 < e x \<and> U x \<in> \<C> \<and> x \<in> U x"
borel_measurable_nn_integral: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> ennreal"   and N :: "'c measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. integral\<^sup>N M (f x)) \<in> borel_measurable N"
left_nesting_simps_1: fixes name :: "name" shows "left_nesting (const name) = 0"
stab_def: fixes s :: "'b"   and a :: "'a" shows "(s \<in> Stab a) = (s \<circ> a = a)"
n: shows "\<forall>x. eval r_phi [n, x] = eval r_phi [the (eval k [n]), x]"
lesss_Cons: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and ys :: "'b list" shows "lesss R x (y # ys) = (if (y, x) \<in> R \<and> (x, y) \<notin> R then y # lesss R x ys else lesss R x ys)"
derived_set_of_empty: fixes X :: "'b topology" shows "X derived_set_of {} = {}"
minus_minus: fixes x :: "'a" assumes "x \<in> V" shows "- (- x) = x"
correctCompositionLoc_def: fixes x :: "specID" shows "correctCompositionLoc x \<equiv> loc x = \<Union> (ins ` subcomponents x) \<inter> \<Union> (out ` subcomponents x)"
2: shows "(Ze2, Re2) = poincare_circle_euclidean (of_complex z2) r2"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "seq (\<I> \<otimes> g) (\<I> \<otimes> f)"
s2_def: shows "s2 = nat \<bar>s1\<bar>"
ipp_cond1_minus: fixes a :: "'e"   and i :: "'e \<Rightarrow> 'e" assumes "ipp_cond1 {a} i" shows "ipp_cond1 {\<^sub>i- a} i"
lfp_trans_lowerbound: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "le_trans (T t) t"   and "\<And>P. sound P \<Longrightarrow> sound (t P)" shows "le_trans (lfp_trans T) t"
assms: fixes \<sigma> :: "'a"   and s :: "'b"   and a :: "'c"   and \<sigma>' :: "'a"   and s' :: "'b" assumes "(\<sigma>, s) \<in> oreachable A S U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "P ((\<sigma>, s), a, \<sigma>', s')"
fresh_1: fixes a :: "name"   and x2 :: "name"   and x1 :: "name" shows "a \<sharp> x2[x1] = (a \<sharp> x2 \<and> a \<sharp> x1)"
spmf_bind: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf"   and y :: "'b" shows "spmf (p \<bind> f) y = LINT x|measure_spmf p. spmf (f x) y"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "partial_magma.ide C (Chn \<mu>)" shows "identity_arrow_of_spans_axioms C \<mu>"
Range_iff: fixes a :: "'c"   and r :: "('d \<times> 'c) set" shows "(a \<in> Range r) = (\<exists>y. (y, a) \<in> r)"
1: shows "\<Up>{y. \<exists>z\<in>F. x \<sqinter> z = y} \<noteq> {}"
to_tree_order_node_ptrs: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "ptr' \<noteq> ptr"   and "ptr' \<in> set nodes" shows "is_node_ptr_kind ptr'"
L_init: fixes r :: "'a rexp" shows "L (init r) = lang r"
includes_nhip: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). \<forall>dip\<in>kD (rt \<xi>). the (nhop (rt \<xi>) dip) \<in> kD (rt \<xi>))"
snd_abs: fixes x :: "'b \<times> 'a" shows "snd \<bar>x\<bar> = \<bar>snd x\<bar>"
simps_9: fixes opt :: "atom fm \<Rightarrow> atom fm"   and step :: "nat \<Rightarrow> nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and \<phi> :: "atom fm" shows "QE_dnf opt step (ExN 0 \<phi>) = QE_dnf opt step \<phi>"
k_def: fixes i :: "nat" assumes "i < 2" shows "suffix k w \<Turnstile>\<^sub>n af (if i = 0 then \<phi> else \<psi>) (prefix k w)[X]\<^sub>\<nu>"
f_def: shows "\<forall>u\<in>L. u \<in> f u \<and> f u \<in> P"
sublens_trans: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and Z :: "'g \<Longrightarrow> 'e" assumes "X \<subseteq>\<^sub>L Y"   and "Y \<subseteq>\<^sub>L Z" shows "X \<subseteq>\<^sub>L Z"
lucas_lehmer_mult_commute: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "lucas_lehmer_mult m x y = lucas_lehmer_mult m y x"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(Ref x2 = Ref y2) = (x2 = y2)"
affine_dependent_imp_dependent2: fixes S :: "'a set" assumes "affine_dependent (insert (0::'a) S)" shows "dependent S"
rec_1: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "rec_latin_type f1 f2 f3 Row = f1"
subchain: fixes n :: "nat"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "\<forall>i>n. \<exists>j>i. P (f i) (f j)" shows "\<exists>\<phi>. (\<forall>i j. i < j \<longrightarrow> \<phi> i < \<phi> j) \<and> (\<forall>i. P (f (\<phi> i)) (f (\<phi> (Suc i))))"
simps_2: fixes l :: "'b tree23"   and a :: "'b"   and r :: "'b tree23" shows "treeI (OF l a r) = \<langle>l, a, r\<rangle>"
elims_3: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> 'i" assumes "\<not> P1 x" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "'i" where "x = (s1, s2)" and "\<not> (\<forall>p. status s1 p \<in> {Pending, Aborted} \<longrightarrow> fst (pending s1 p) = p)"
some_eq_imp: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "Eps P = a"   and "P b" shows "P a"
kk_def: shows "kk = {t \<in> {0..1}. ball t (ee t / 3) \<in> C'}"
transitive: fixes P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "P \<sim> Q"   and "Q \<sim> R" shows "P \<sim> R"
\<ll>_ide_simp: 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 f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "transformation_by_components.map V V (\<lambda>\<mu>. if partial_magma.arr V \<mu> then H (trg \<mu>) \<mu> else partial_magma.null V) (bicategory.lunit V H \<a> \<i> src trg) f = bicategory.lunit V H \<a> \<i> src trg f"
rank_of_Ord: fixes i :: "V" assumes "Ord i" shows "rank i = i"
lens_source_def: fixes X :: "'c \<Longrightarrow> 'b" shows "\<S>\<^bsub>X\<^esub> = {s. \<exists>v s'. s = put\<^bsub>X\<^esub> s' v}"
mk_minsky_add2: fixes u :: "nat"   and v :: "nat"   and w :: "nat" assumes "u \<noteq> v"   and "u \<noteq> w"   and "v \<noteq> w" shows "mk_minsky (\<lambda>vs vs'. vs' = (\<lambda>x. if x = u then 0 else if x = v then vs u + vs v else if x = w then vs u + vs w else vs x))"
reflexive_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "reflexive B (\<sqsubseteq>)"
assms_3: shows "\<NN>\<lparr>NTMap\<rparr> = \<NN>'\<lparr>NTMap\<rparr>"
f_accessible_alt: fixes \<delta> :: "('a, 'b) ta_rule set"   and Q0 :: "'a set" shows "f_accessible \<delta> Q0 = f_accessible_alt \<delta> Q0"
mono: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "f x \<le> f y"
lattice_pair_ow_def: fixes U\<^sub>1 :: "'a set"   and inf\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and U\<^sub>2 :: "'b set"   and inf\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "lattice_pair_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2 \<equiv> lattice_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 \<and> lattice_ow U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2"
lists_UNION_nlists: fixes A :: "'a set" shows "lists A = \<Union> (range (nlists A))"
simps_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" shows "efficient_power f y x (Suc 0) = f x y"
simps_8: fixes a :: "'a"   and x :: "'a"   and b :: "'a"   and B :: "'a tree"   and A :: "'a tree"   and CD :: "'a tree" assumes "a < x"   and "x < b"   and "B \<noteq> \<langle>\<rangle>" shows "splay x \<langle>\<langle>A, a, B\<rangle>, b, CD\<rangle> = (case splay x B of \<langle>B1, b', B2\<rangle> \<Rightarrow> \<langle>\<langle>A, a, B1\<rangle>, b', \<langle>B2, b, CD\<rangle>\<rangle>)"
relpow_Suc_I2: fixes x :: "'c"   and y :: "'c"   and R :: "('c \<times> 'c) set"   and z :: "'c"   and n :: "nat" assumes "(x, y) \<in> R"   and "(y, z) \<in> R ^^ n" shows "(x, z) \<in> R ^^ Suc n"
assms_1: shows "(\<^bold>\<not> (p \<^bold>\<or> q)) at a in (ps, a) # branch"
nextnat_empty_neq: fixes s :: "nat \<Rightarrow> 'b" assumes "notemptyseq s" shows "s (nextnat s) \<noteq> s 0"
map_filter_M_pure_E: fixes h :: "'a"   and f :: "'d \<Rightarrow> ('a, 'b, 'c option) prog"   and xs :: "'d list"   and ys :: "'c list"   and y :: "'c" assumes "h \<turnstile> map_filter_M f xs \<rightarrow>\<^sub>r ys"   and "y \<in> set ys"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (f x) h" obtains x :: "'d" where "h \<turnstile> f x \<rightarrow>\<^sub>r Some y" and "x \<in> set xs"
invariant_factor_decomposition_dirprod: obtains gs :: "'a list" where "set gs \<subseteq> carrier G" and "cycle gs" and "DirProds (\<lambda>g. G\<lparr>carrier := generate G {g}\<rparr>) (set gs) \<cong> G" and "successively (dvd) (map ord gs)" and "card (set gs) \<le> card gen" and "compl_fam (\<lambda>g. generate G {g}) (set gs)" and "\<one> \<notin> set gs"
PPs_mono: fixes X :: "'b set"   and Y :: "'b set" assumes "X \<subseteq> Y" shows ".[X] \<subseteq> .[Y]"
filterlim_at: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and b :: "'b"   and s :: "'b set"   and F :: "'c filter" assumes "class.topological_space open" shows "filterlim f (topological_space.at_within open b s) F = ((\<forall>\<^sub>F x in F. f x \<in> s \<and> f x \<noteq> b) \<and> filterlim f (topological_space.nhds open b) F)"
mapi_block_add_oob: fixes ps :: "('b, 'c) fm list"   and v' :: "nat"   and v :: "nat"   and xs :: "(nat \<times> nat) set"   and f :: "('b, 'c) fm \<Rightarrow> ('b, 'c) fm"   and i :: "'c" assumes "length ps \<le> v'" shows "mapi_block (mapper f ({(v, v')} \<union> xs) v) (ps, i) = mapi_block (mapper f xs v) (ps, i)"
natural_isomorphism_axioms: shows "natural_isomorphism B.VVV.comp (\<cdot>\<^sub>B) B.HoHV B.HoVH (\<lambda>\<mu>\<nu>\<tau>. \<a>\<^sub>B[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)])"
jlength2: fixes f :: "nat \<Rightarrow> real" shows "j < dim_vec (evaluate_net (insert_weights s m2 (if s then \<lambda>i. f (i + a) else (\<lambda>i. f (i + count_weights s m1 + a)))) (drop (length (input_sizes (insert_weights s m1 (\<lambda>i. f (i + a))))) inputs))"
R3: fixes x :: "'b"   and ts :: "'b bintree list"   and X :: "'b t_type" assumes "\<lparr>t_type.folding = False, item = x, subtrees = Leaf # ts\<rparr> \<in> t_ins_set X" shows "\<lparr>t_type.folding = True, item = x, subtrees = Branch x Leaf Leaf # ts\<rparr> \<in> t_ins_set X"
return_time_pos: fixes f :: "'c \<Rightarrow> 'c"   and X :: "'c set"   and P :: "'c set"   and x :: "'c" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x"   and "closed P" shows "0 < auto_ll_on_open.return_time f X P x"
Catch: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c\<^sub>1' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and c\<^sub>2 :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')" shows "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, s) \<rightarrow> (Catch c\<^sub>1' c\<^sub>2, s')"
i: shows "(LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) = i"
ll_on_open_it_axioms: shows "ll_on_open_it T f X"
elims_3: fixes x :: "nat \<Rightarrow> 'a"   and xa :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)" assumes "\<not> x \<Turnstile>\<^sub>b xa" obtains v :: "nat \<Rightarrow> 'a" and b :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)" where "x = v" and "xa = b" and "\<not> (\<forall>x. in_bounds x v b)"
lists_sing_map_concat_ident: fixes xs :: "'b list list"   and B :: "'b set" assumes "xs \<in> lists (sings B)" shows "xs = map (\<lambda>x. [x]) (concat xs)"
None: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b option" shows "ord_option ord None x"
star_decompose_3: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.left_kleene_algebra star one times sup less_eq less bot" shows "star (times (star x) (star y)) = times (star x) (star (times y (star x)))"
assoc_in_homs_1: shows "\<guillemotleft>\<mu> \<star> \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr> : \<mu> \<star> (\<nu> \<star> \<pi>) \<star> \<rho> \<Rightarrow> \<mu> \<star> \<nu> \<star> \<pi> \<star> \<rho>\<guillemotright>"
Atom_def: shows "Atom \<equiv> \<lambda>x. {Abs_Node (\<lambda>k. Inr 0, x)}"
mod_mult_cong: fixes divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and euclidean_size :: "'k \<Rightarrow> nat"   and a :: "'k"   and c :: "'k"   and a' :: "'k"   and b :: "'k"   and b' :: "'k" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (times a b) c = modulo (times a' b') c"
1: fixes y :: "(char list multiset \<times> int) list"   and s :: "(char list list \<times> int) list" assumes "(p, y) \<in> \<langle>unsorted_term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel"   and "p' = mset y"   and "map (\<lambda>(a, y). (mset a, y)) (rev p) = map (\<lambda>(a, y). (mset a, y)) s"   and "\<forall>x\<in>set s. sorted_wrt var_order (fst x)" shows "(s, map (\<lambda>(a, y). (mset a, y)) s) \<in> \<langle>term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel"
d: fixes u :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "u \<in> singular_relcycle_set p X S" shows "d (homologous_rel_set p X S u) = \<theta> u"
eq_refl: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" assumes "x = y" shows "rel_prod orda ordb x y"
span_legs_eq_3: shows "SPN_fgh.\<nu>.leg0 = Tfg.S\<^sub>0"
simps_16: fixes prec :: "nat"   and x :: "nat"   and a :: "floatarith"   and vs :: "float interval option list" shows "isDERIV_approx prec x (Floor a) vs = (isDERIV_approx prec x a vs \<and> (case approx prec a vs of None \<Rightarrow> False | Some ivl \<Rightarrow> real_of_int \<lfloor>real_of_float (upper ivl)\<rfloor> < real_of_float (lower ivl) \<and> real_of_float (upper ivl) < real_of_int \<lceil>real_of_float (lower ivl)\<rceil>))"
divide_ennreal_def: fixes x :: "ennreal"   and y :: "ennreal" shows "x / y = x * inverse y"
r_nf_t_0: fixes y :: "nat"   and i :: "nat"   and x :: "nat" assumes "eval r_result [pdec1 y, pdec12 y, pdec22 y] \<down>\<noteq> 0"   and "pdec2 y = prod_encode (i, x)" shows "eval r_nf_t [y, i, x] \<down>= 0"
boundOutputSupportDerivative: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> P" shows "supp P' - {x} \<subseteq> supp P"
abs_eq: shows "0 = abs_fract (0::'a, 1::'a)"
f3: shows "4 \<le> rk {A, B, P, a}"
Hermite_of_upt_row_i_iarrays_def: fixes A :: "'a iarray iarray"   and i :: "nat"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "Hermite_of_upt_row_i_iarrays A i ass res = foldl (Hermite_of_row_i_iarray ass res) A [0..<i]"
swap_edges: fixes G :: "('c, 'd) graph"   and a :: "'c"   and w :: "'d"   and b :: "'c"   and v :: "'c"   and v' :: "'c"   and w' :: "'d" assumes "valid_graph G"   and "nodes_connected (add_edge a w b G) v v'"   and "a \<in> nodes G"   and "b \<in> nodes G"   and "\<not> nodes_connected G v v'" shows "nodes_connected (add_edge v w' v' G) a b"
ct_suffix_shared: fixes cts :: "ct list"   and ts :: "t list"   and cts' :: "ct list" assumes "ct_suffix cts (to_ct_list ts)"   and "ct_suffix cts' (to_ct_list ts)" shows "ct_suffix cts cts' \<or> ct_suffix cts' cts"
ipset_from_cidr_alt2: fixes base :: "'a word"   and len :: "nat" shows "ipset_from_cidr base len = ipset_from_netmask base (~~ mask (LENGTH('a) - len))"
measurable_vimage_algebra2: fixes g :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and X :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and M :: "'e measure" assumes "g \<in> space N \<rightarrow> X"   and "(\<lambda>x. f (g x)) \<in> N \<rightarrow>\<^sub>M M" shows "g \<in> N \<rightarrow>\<^sub>M vimage_algebra X f M"
assms_2: shows "v' \<equiv> \<lambda>i. THE c. c \<in> X \<and> v c = i"
composite_simps_3: shows "trg (f \<star> g \<star> h) = trg f"
awalk_conv: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "awalk u p v = (set (awalk_verts u p) \<subseteq> verts G \<and> set p \<subseteq> arcs G \<and> awhd u p = u \<and> awlast u p = v \<and> cas u p v)"
poly_rat_1: fixes x :: "rat" shows "ipoly (poly_rat x) (of_rat x) = (0::'a)"
u_def: fixes n :: "nat" shows "u n = (SOME u. u\<up>f n \<and> (\<forall>m. \<exists>a. a \<in> sfis (u m) M))"
iMOD_inext_if: fixes t :: "nat"   and r :: "nat"   and m :: "nat" shows "inext t [ r, mod m ] = (if t \<in> [ r, mod m ] then t + m else t)"