hcju/isabelleps huggingface dataset:
simps_3: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc (Suc k) 0 = LocalLexing.\<pi> \<RR> (Suc k) {} (LocalLexing.\<I> \<TT> \<RR> \<SS> Lex Sel Doc k)"
elims: fixes x :: "dir list"   and xa :: "dir"   and xb :: "('c, 'd) tm"   and y :: "('c, 'd) tm" assumes "emb_step_at x xa xb = y" obtains t :: "('c, 'd) tm" and s :: "('c, 'd) tm" where "x = []" and "xa = dir.Left" and "xb = App t__ sa__" and "y = t__"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" where "x = []" and "xa = dir.Right" and "xb = App ta__ sb__" and "y = sb__"   | p :: "dir list" and dir :: "dir" and t :: "('c, 'd) tm" and s :: "('c, 'd) tm" where "x = dir.Left # p__" and "xa = dir__" and "xb = App tb__ sc__" and "y = App (emb_step_at p__ dir__ tb__) sc__"   | p :: "dir list" and dir :: "dir" and t :: "('c, 'd) tm" and s :: "('c, 'd) tm" where "x = dir.Right # p" and "xa = dir" and "xb = App t s" and "y = App t (emb_step_at p dir s)"   | uu_ :: "dir list" and uv_ :: "dir" and h :: "('c, 'd) hd" where "x = uu_" and "xa = uv_" and "xb = Hd h" and "y = Hd h"
continuous_map_subset_aux2: fixes X :: "'c topology"   and f :: "'c \<Rightarrow> 'd"   and Y :: "'d topology" assumes "\<forall>S\<subseteq>topspace X. f ` (X closure_of S) \<subseteq> Y closure_of f ` S" shows "continuous_map X Y f"
multi_member_this: fixes x :: "'d"   and XS :: "'d multiset" shows "x \<in># {#x#} + XS"
cterms_not_call: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "p \<in> cterms \<Gamma>" shows "not_call p"
prems_2: fixes i :: "'a"   and r :: "'a" assumes "(i, Some r) \<in> set (xs @ [x])" shows "r \<noteq> oid"
N_4: shows "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r"
non_adj_def: fixes v :: "'v"   and v' :: "'v" shows "non_adj v v' \<equiv> v \<in> V \<and> v' \<in> V \<and> v \<noteq> v' \<and> \<not> adjacent v v'"
slice_Exists: fixes S :: "'c list set"   and \<phi> :: "'c MFOTL.formula"   and \<sigma> :: "(char list \<times> 'c list) trace" shows "map_\<Gamma> (\<lambda>D. D \<inter> relevant_events (formula.Exists \<phi>) S) \<sigma> = map_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> (set_Cons UNIV S)) \<sigma>"
simps_16: fixes x2 :: "'a" shows "set_literal (Neg x2) = {x2}"
ele_n_prod: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set"   and a :: "'a" assumes "\<forall>k\<le>n. ideal R (J k)"   and "a \<in> i\<Pi>\<^bsub>R,n\<^esub> J" shows "\<forall>k\<le>n. a \<in> J k"
assms: shows "\<forall>i. p = Nom i \<or> p = (\<^bold>\<diamond> Nom i) \<longrightarrow> i \<in> A"
Monotone_def: fixes \<phi> :: "((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "Monotone \<phi> = (\<forall>\<Phi> \<Psi>. (\<forall>s t. \<Phi> (s, t) \<longrightarrow> \<Psi> (s, t)) \<longrightarrow> (\<forall>s t. \<phi> \<Phi> (s, t) \<longrightarrow> \<phi> \<Psi> (s, t)))"
redT_updTs_Some: fixes ts :: "'f \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and t :: "'f"   and xw :: "'d \<times> 'e \<Rightarrow>f nat"   and tas :: "('f, 'd, 'g) new_thread_action list" assumes "ts t = \<lfloor>xw\<rfloor>"   and "thread_oks ts tas" shows "redT_updTs ts tas t = \<lfloor>xw\<rfloor>"
hyps: fixes gc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and gc' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and bs :: "nat list" assumes "gc \<rightarrow>\<^bsub>as\<^esub> gc'"   and "gc' \<rightarrow>\<^bsub>bs\<^esub> gc''" shows "gc \<rightarrow>\<^bsub>as @ bs\<^esub> gc''"
vecsumlist': fixes xss :: "'b list list"   and i :: "nat" assumes "\<And>xs. xs \<in> set xss \<Longrightarrow> i \<le> length xs" shows "vecsumlist' i xss = map (\<lambda>i. \<Sum>xs\<leftarrow>xss. xs ! i) [0..<i]"
adj_cancel_eq1: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "f \<circ> g \<circ> f = f"
block_img: shows "image_mset ((`) \<pi>) \<B> = \<B>'"
fold_superset: fixes acc :: "(bool \<times> 'e \<times> 'f) list"   and c :: "'g \<Rightarrow> (bool \<times> 'e \<times> 'f) list \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ps :: "(bool \<times> 'g \<times> 'e \<times> 'f) list" shows "set acc \<subseteq> set (fold (\<lambda>(ic, q_in_bs, p, q) ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps acc)"
trapped_sol: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped x K" shows "rev.existence_ivl0 x = UNIV"
assms: shows "R[x\<turnstile>n>y] = ImpL <c>.M w.N u"
ex_in_conv: fixes A :: "'a set" shows "(\<exists>x. x \<in> A) = (A \<noteq> {})"
zcf_add: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "zcf (p \<oplus>\<^bsub>P\<^esub> q) = zcf p \<oplus> zcf q"
compE_def: fixes \<rho> :: "'a \<Rightarrow> 'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and x :: "'a" shows "compE \<rho> \<xi> x \<equiv> int \<xi> (\<rho> x)"
indep_vars_iff_distr_eq_PiM: fixes I :: "'d set"   and X :: "'d \<Rightarrow> ennreal \<Rightarrow> 'e"   and i :: "'b"   and M' :: "'d \<Rightarrow> 'e measure" assumes "I \<noteq> {}"   and "\<And>i. Y.random_variable i (M' i) (X i)" shows "Y.indep_vars i M' X I = (distr (distr M borel (Y i)) (Pi\<^sub>M I M') (\<lambda>x. \<lambda>i\<in>I. X i x) = Pi\<^sub>M I (\<lambda>i. distr (distr M borel (Y i)) (M' i) (X i)))"
synthetic_div_correct: fixes p :: "'a poly"   and c :: "'a" shows "p + smult c (synthetic_div p c) = pCons (poly p c) (synthetic_div p c)"
SNF_1x2_works: shows "\<forall>A\<in>carrier_mat 1 2. is_SNF A (1\<^sub>m 1, Smith_1x2 A)"
frechet_derivative_times_fun: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "f differentiable at a"   and "g differentiable at a" shows "frechet_derivative (f * g) (at a) = (\<lambda>x. f a * frechet_derivative g (at a) x + frechet_derivative f (at a) x * g a)"
id_converter_def: shows "id_converter \<equiv> corec_converter (\<lambda>x a. map_gpv (map_prod id (\<lambda>uu. Inr ())) id (Generative_Probabilistic_Value.Pause a (\<lambda>b. Generative_Probabilistic_Value.Done (b, ())))) ()"
ide_compE: fixes g :: "'d"   and f :: "'d" assumes "D.ide (g \<cdot>\<^sub>D f)" obtains "D.seq g f" and "D.seq f g" and "g \<cdot>\<^sub>D f = D.dom f" and "g \<cdot>\<^sub>D f = D.cod g"
Bpow_def: fixes r :: "('b \<times> 'b) set"   and A :: "'a set" shows "Bpow r A \<equiv> {X. X \<subseteq> A \<and> |X| \<le>o r}"
birkhoff_sum_1_1: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "A.birkhoff_sum f 0 x = (0::'b)"
funpow_neq_less_funpow_dist: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and m :: "nat"   and n :: "nat" assumes "y \<in> orbit f x"   and "m \<le> funpow_dist f x y"   and "n \<le> funpow_dist f x y"   and "m \<noteq> n" shows "(f ^^ m) x \<noteq> (f ^^ n) x"
SubsetGuards_sound: fixes c :: "('d, 'e, 'f) com"   and c' :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and P :: "'d set"   and Q :: "'d set"   and A :: "'d set"   and n :: "nat" assumes "c \<subseteq>\<^sub>g c'"   and "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P c' Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P c Q,A"
set_pps_to_list: fixes S :: "'a set" assumes "finite S" shows "set (punit.pps_to_list S) = S"
wf_graph_def: fixes G :: "'b graph" shows "wf_graph G \<equiv> (fst ` edges G \<subseteq> nodes G \<and> snd ` edges G \<subseteq> nodes G) \<and> finite (edges G) \<and> finite (nodes G)"
UTM_uhalt_lemma_pre: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and args :: "nat list" assumes "tm_wf0 tp"   and "\<forall>stp. \<not> TSTD (steps0 (Suc 0, Bk \<up> l, <args>) tp stp)"   and "args \<noteq> []" shows "\<forall>stp. \<not> is_final (steps0 (Suc 0, [], <code tp # args>) UTM_pre stp)"
gt_tpoly_irrefl: fixes p :: "('v pvar, hmultiset) tpoly" shows "\<not> p >\<^sub>p p"
gfp_F_stable_pair_on: fixes ds :: "'b set" shows "stable_pair_on ds (gfp_F ds)"
interchange: fixes h :: "'c local.term set"   and g :: "'c local.term set"   and h' :: "'c local.term set"   and g' :: "'c local.term set" assumes "\<F>C.seq h g"   and "\<F>C.seq h' g'" shows "\<F>C.tensor h h' \<cdot> \<F>C.tensor g g' = \<F>C.tensor (h \<cdot> g) (h' \<cdot> g')"
assms_1: shows "q \<in> R\<^sup>* `` Q0"
Iff_refl: fixes H :: "fm set"   and A :: "fm" shows "H \<turnstile> A IFF A"
intros: fixes p :: "'c \<times> 'd" shows "fst p \<in> Basic_BNFs.fsts p"
successively_map: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "successively P (map f xs) = successively (\<lambda>x y. P (f x) (f y)) xs"
c2_3: shows "C2 \<in> carrier_mat n2 nc1"
derivation_def: fixes X :: "'a set" shows "derivation X \<equiv> X \<subseteq> A \<and> well_ordered_set X (\<sqsubseteq>) \<and> (\<forall>x\<in>X. let Y = {y \<in> X. y \<sqsubset> x} in (\<exists>y. extreme Y (\<sqsubseteq>) y \<and> x = f y) \<or> f ` Y \<subseteq> Y \<and> extreme_bound A (\<sqsubseteq>) Y x)"
top_semantics: fixes A :: "'b \<Rightarrow> bool" shows "A \<Turnstile> \<top>"
mix_pmf_preferred_independence: fixes r :: "'a pmf"   and \<alpha> :: "real"   and p :: "'a pmf"   and q :: "'a pmf" assumes "r \<in> local.\<P>"   and "\<alpha> \<in> {0..1}"   and "p \<succeq>[\<R>] q" shows "mix_pmf \<alpha> p r \<succeq>[\<R>] mix_pmf \<alpha> q r"
Number_Enc: fixes X :: "nat"   and X' :: "msg"   and Y' :: "msg" shows "Number X \<noteq> Enc X' Y'"
eq: shows "ell_field_class.add a p\<^sub>1 p\<^sub>2 = ell_field_class.add a p\<^sub>1 p\<^sub>3"
path_change: fixes u :: "'a"   and S :: "'a set"   and v :: "'a"   and g :: "'a ugraph"   and p :: "('a \<times> 'a) list" assumes "u \<in> S"   and "v \<notin> S"   and "path g u p v"   and "simple p" obtains x :: "'a" and y :: "'a" and p1 :: "('a \<times> 'a) list" and p2 :: "('a \<times> 'a) list" where "(x, y) \<in> set p" and "x \<in> S" and "y \<notin> S" and "path (restrict_edges g (- {(x, y), (y, x)})) u p1 x" and "path (restrict_edges g (- {(x, y), (y, x)})) y p2 v"
zero_not_succ: fixes n :: "V" shows "0 \<noteq> ZFC_in_HOL.succ n"
minus_bit0_def: fixes x :: "'b bit0"   and y :: "'b bit0" shows "x - y = Abs_bit0' (Rep_bit0 x - Rep_bit0 y)"
atomicsR_Throw: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and a :: "('b, 'a, 'c) ann"   and r :: "'b set \<times> ('b, 'a, 'c) com" shows "atomicsR \<Gamma> \<Theta> a Throw r = False"
summable: shows "summable (\<lambda>n. \<chi> n * of_nat n powr - s)"
to_fun_res: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a k = b k" shows "(f \<bullet> a) k = (f \<bullet> b) k"
interval_integral_Icc: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b" assumes "a \<le> b" shows "interval_lebesgue_integral lborel (ereal a) (ereal b) f = set_lebesgue_integral lborel {a..b} f"
simps_2: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and uu :: "color"   and l :: "('b, 'a) rbt"   and x :: "'b"   and y :: "'a"   and r :: "('b, 'a) rbt"   and k :: "'b" shows "ord.rbt_lookup less (Branch uu l x y r) k = (if less k x then ord.rbt_lookup less l k else if less x k then ord.rbt_lookup less r k else Some y)"
indaut_order2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g \<circ> OpposedThinChamberComplexFoldings.induced_automorphism X f g = id"
id2: shows "as \<inter> UNIV \<times> {0..<Suc n} = as \<inter> UNIV \<times> {n} \<union> as \<inter> UNIV \<times> {0..<n}"
residualAlpha: fixes p :: "(name \<times> name) list"   and \<alpha> :: "'a action"   and P :: "('a, 'b, 'c) psi" assumes "bn (p \<bullet> \<alpha>) \<sharp>* object \<alpha>"   and "bn (p \<bullet> \<alpha>) \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn (p \<bullet> \<alpha>) \<sharp>* subject \<alpha>"   and "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))" shows "\<alpha> \<prec> P = p \<bullet> \<alpha> \<prec> p \<bullet> P"
share_unchanged: fixes a :: "nat"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "a \<notin> all_shared sb \<union> all_unshared sb \<union> all_acquired sb" shows "share sb \<S> a = \<S> a"
to_list_def: fixes it :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<Rightarrow> 'g list \<Rightarrow> 'g list) \<Rightarrow> 'h list \<Rightarrow> 'd"   and s :: "'e" shows "to_list it s \<equiv> it s top (#) []"
axiom4_F: fixes f1 :: "'o \<Rightarrow> 'q"   and f2 :: "'p \<Rightarrow> 'r" shows "F'set3 \<circ> Fmap f1 f2 = (`) f1 \<circ> F'set3"
maddux_3_3: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<squnion> - (x \<squnion> - y) = - x"
tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "dom \<theta> \<union> load_tmps is = {i. i < t}"
limiting_parallels: fixes a :: "complex_homo"   and x1 :: "complex_homo"   and x2 :: "complex_homo" assumes "a \<in> unit_disc"   and "x1 \<in> unit_disc"   and "x2 \<in> unit_disc"   and "\<not> poincare_on_line a x1 x2" shows "\<exists>a1\<in>unit_disc. \<exists>a2\<in>unit_disc. \<not> poincare_on_line a a1 a2 \<and> \<not> poincare_ray_meets_line a a1 x1 x2 \<and> \<not> poincare_ray_meets_line a a2 x1 x2 \<and> (\<forall>a'\<in>unit_disc. poincare_in_angle a' a1 a a2 \<longrightarrow> poincare_ray_meets_line a a' x1 x2)"
rreq_dip_in_vD_dip_eq_ip: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:18..PRreq-:21} \<longrightarrow> dip \<xi> \<in> vD (rt \<xi>)) \<and> (l \<in> {PRreq-:6, PRreq-:7} \<longrightarrow> dip \<xi> = ip \<xi>) \<and> (l \<in> {PRreq-:17..PRreq-:21} \<longrightarrow> dip \<xi> \<noteq> ip \<xi>))"
sel: fixes x :: "'d" shows "run_writer (WriterT x) = x"
sup_state_opt_refl: fixes P :: "'b prog"   and s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> s \<le>' s"
retraction_refl: fixes S :: "'a set" shows "retraction S S (\<lambda>x. x)"
1: shows "\<forall>x. x \<in> set A \<longrightarrow> length x = nr"
power_0_left: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero n = (if n = 0 then one else zero)"
Edges_append: fixes xs :: "nat list"   and ys :: "nat list" shows "Edges (xs @ ys) = (if xs = [] then Edges ys else if ys = [] then Edges xs else Edges xs \<union> Edges ys \<union> {(last xs, hd ys)})"
returns_heap_eq: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and h' :: "'a"   and h'' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h \<turnstile> f \<rightarrow>\<^sub>h h''" shows "h' = h''"
RA2_parts_spies: fixes A :: "agent"   and B :: "agent"   and c :: "msg"   and evs :: "event list" assumes "Says A B c \<in> set evs" shows "c \<in> parts (knows Spy evs)"
add_rng_of_frac_fundamental_lemma: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel" shows "(r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (s' \<otimes> r \<oplus> s \<otimes> r' |\<^bsub>rel\<^esub> s \<otimes> s')"
mult_succ: fixes x :: "V"   and y :: "V" shows "x * ZFC_in_HOL.succ y = x * y + x"
assms: shows "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"
fps_nonzero_nth_minimal: fixes f :: "'b fps" shows "(f \<noteq> 0) = (\<exists>n. f $ n \<noteq> (0::'b) \<and> (\<forall>m<n. f $ m = (0::'b)))"
is_bezout_ext_def: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "class.bezout_ring times one plus zero minus uminus" shows "bezout_ring.is_bezout_ext times plus uminus bezout = (\<forall>a b. let (p, q, u, v, gcd_a_b) = bezout a b in plus (times p a) (times q b) = gcd_a_b \<and> dvd.dvd times gcd_a_b a \<and> dvd.dvd times gcd_a_b b \<and> (\<forall>d'. dvd.dvd times d' a \<and> dvd.dvd times d' b \<longrightarrow> dvd.dvd times d' gcd_a_b) \<and> times gcd_a_b u = uminus b \<and> times gcd_a_b v = a)"
vdomain_vdoubleton: fixes a :: "V"   and b :: "V"   and c :: "V"   and d :: "V" shows "\<D>\<^sub>\<circ> (set {\<langle>a, b\<rangle>, \<langle>c, d\<rangle>}) = set {a, c}"
closed_segment_subset_interval: fixes T :: "'c set"   and a :: "'c"   and b :: "'c" assumes "is_interval T"   and "a \<in> T"   and "b \<in> T" shows "{a--b} \<subseteq> T"
finprod_Un_disjoint: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier G"   and "g \<in> B \<rightarrow> carrier G" shows "finprod G g (A \<union> B) = finprod G g A \<otimes> finprod G g B"
l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs DocumentClass.type_wf create_character_data DocumentClass.known_ptr"
abs_eq: fixes x :: "'b \<times> 'b" assumes "eq_onp (\<lambda>x. x \<in> {(a, b). a \<le> b}) x x" shows "upper (Interval x) = snd x"
weak_distributivity1: fixes A1 :: "int poly fract list list"   and A2 :: "int poly fract list list"   and B1 :: "int poly fract list list" assumes "weak_matrix_match A1 A2 B1" shows "rat_poly.matrix_mult A1 A2 \<otimes> B1 = rat_poly.matrix_mult (A1 \<otimes> B1) A2"
prems_1: shows "\<rho>\<^sub>1 \<in> carrier_mat d d"
counit_simps_4: shows "S.src \<xi>' = S.trg (S.UP f)"
simps: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and F :: "atom fm" shows "linear_substitution_fm var b c F = linear_substitution_fm_helper var b c F 0"
Hermite_Lindemann: fixes I :: "'a set"   and \<alpha> :: "'a \<Rightarrow> complex"   and \<beta> :: "'a \<Rightarrow> complex" assumes "finite I"   and "\<And>x. x \<in> I \<Longrightarrow> algebraic (\<alpha> x)"   and "\<And>x. x \<in> I \<Longrightarrow> algebraic (\<beta> x)"   and "inj_on \<alpha> I"   and "(\<Sum>x\<in>I. \<beta> x * exp (\<alpha> x)) = 0" shows "\<forall>x\<in>I. \<beta> x = 0"
fsubset_funion_eq: fixes A :: "'a fset"   and B :: "'a fset" shows "(A |\<subseteq>| B) = (A |\<union>| B = B)"
invimage_vrange: fixes r :: "V" shows "r -`\<^sub>\<circ> \<R>\<^sub>\<circ> r = \<D>\<^sub>\<circ> r"
Var\<^sub>0_power: fixes v :: "'b"   and n :: "nat" shows "Var\<^sub>0 v ^ n = monomial (1::'c) (monomial n v)"
succ_neq_hpair: fixes k :: "hf"   and x :: "hf"   and y :: "hf" shows "succ k \<noteq> \<langle>x, y\<rangle>"
P2SQ2: shows "(P2, S', Q2) = diagonal_to_Smith_PQ D euclid_ext2"
sub_7: shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>"
simps_6: fixes i :: "'a"   and p :: "'a pfm" shows "static (K\<^sub>! i p) = static p"
simps_3: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and uu :: "'c list" shows "zip_with f uu [] = []"
nullhomotopic_into_contractible_space: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_map X Y f"   and "contractible_space Y" obtains c :: "'c" where "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
assms: fixes \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n f \<phi> = w \<Turnstile>\<^sub>n \<phi>"
induct: fixes x1 :: "'b formula multiset"   and x2 :: "'b formula multiset"   and P :: "'b formula multiset \<Rightarrow> 'b formula multiset \<Rightarrow> bool" assumes "x1 \<Rightarrow> x2"   and "\<And>\<Gamma> \<Delta>. \<bottom> \<in># \<Gamma> \<Longrightarrow> P \<Gamma> \<Delta>"   and "\<And>k \<Gamma> \<Delta>. \<lbrakk>Atom k \<in># \<Gamma>; Atom k \<in># \<Delta>\<rbrakk> \<Longrightarrow> P \<Gamma> \<Delta>"   and "\<And>\<Gamma> F \<Delta>. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta>; P \<Gamma> (F, \<Delta>)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> F, \<Gamma>) \<Delta>"   and "\<And>F \<Gamma> \<Delta>. \<lbrakk>F, \<Gamma> \<Rightarrow> \<Delta>; P (F, \<Gamma>) \<Delta>\<rbrakk> \<Longrightarrow> P \<Gamma> (\<^bold>\<not> F, \<Delta>)"   and "\<And>F G \<Gamma> \<Delta>. \<lbrakk>F, G, \<Gamma> \<Rightarrow> \<Delta>; P (F, G, \<Gamma>) \<Delta>\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G, \<Gamma>) \<Delta>"   and "\<And>\<Gamma> F \<Delta> G. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta>; P \<Gamma> (F, \<Delta>); \<Gamma> \<Rightarrow> G, \<Delta>; P \<Gamma> (G, \<Delta>)\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<and> G, \<Delta>)"   and "\<And>F \<Gamma> \<Delta> G. \<lbrakk>F, \<Gamma> \<Rightarrow> \<Delta>; P (F, \<Gamma>) \<Delta>; G, \<Gamma> \<Rightarrow> \<Delta>; P (G, \<Gamma>) \<Delta>\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G, \<Gamma>) \<Delta>"   and "\<And>\<Gamma> F G \<Delta>. \<lbrakk>\<Gamma> \<Rightarrow> F, G, \<Delta>; P \<Gamma> (F, G, \<Delta>)\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<or> G, \<Delta>)"   and "\<And>\<Gamma> F \<Delta> G. \<lbrakk>\<Gamma> \<Rightarrow> F, \<Delta>; P \<Gamma> (F, \<Delta>); G, \<Gamma> \<Rightarrow> \<Delta>; P (G, \<Gamma>) \<Delta>\<rbrakk> \<Longrightarrow> P (F \<^bold>\<rightarrow> G, \<Gamma>) \<Delta>"   and "\<And>F \<Gamma> G \<Delta>. \<lbrakk>F, \<Gamma> \<Rightarrow> G, \<Delta>; P (F, \<Gamma>) (G, \<Delta>)\<rbrakk> \<Longrightarrow> P \<Gamma> (F \<^bold>\<rightarrow> G, \<Delta>)" shows "P x1 x2"
V_simps_2: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" assumes "\<not> \<phi> trn" shows "V (trn # tr) = V tr"
empty_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.valid_node sourcenode targetnode valid_edge n" shows "CFG.path sourcenode targetnode valid_edge n [] n"
retract_axioms_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "retract_axioms G \<equiv> \<forall>x. x \<in> carrier \<X>\<^bsub>G\<^esub> \<longrightarrow> \<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> x) \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> x"
consistency_on_Aizerman_on: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" assumes "consistency_on A f" shows "Aizerman_on A f"