hcju/isabelleps huggingface dataset:
induct: fixes P :: "('c, 'd) FingerTreeStruc \<Rightarrow> bool"   and a0 :: "('c, 'd) FingerTreeStruc" assumes "P Empty"   and "\<And>a. P (Single a)"   and "\<And>uu_ pr m a b. P (Deep uu_ pr m (Two a b))"   and "\<And>uv_ pr m a b c. P (Deep uv_ pr m (Three a b c))"   and "\<And>uw_ pr m a b c d. P (Deep uw_ pr m (Four a b c d))"   and "\<And>ux_ pr m a. P m \<Longrightarrow> P (Deep ux_ pr m (One a))" shows "P a0"
lem_e1: fixes x :: "real" assumes "x \<in> {x. poly p x = 0}" shows "card {s \<in> consistent_sign_vec_copr qs ` {x. poly p x = 0} \<inter> {s. z I s = 1}. consistent_sign_vec_copr qs x = s} = (if 0 < poly (prod_list (retrieve_polys qs I)) x then 1 else 0)"
intro: fixes get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "get_host = l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_host get_shadow_root"   and "get_host_locs = l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_host_locs get_shadow_root_locs" shows "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_shadow_root get_shadow_root_locs get_host get_host_locs"
has_integral_sum: fixes T :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and i :: "'b \<Rightarrow> 'd"   and S :: "'c set" assumes "finite T"   and "\<And>a. a \<in> T \<Longrightarrow> (f a has_integral i a) S" shows "((\<lambda>x. \<Sum>a\<in>T. f a x) has_integral sum i T) S"
mod_2_eq_odd: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "modulo a (numeral.numeral one plus (num.Bit0 num.One)) = zero_neq_one.of_bool one zero (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
UINF_as_Sup: fixes \<P> :: "('a, 'b) uexpr set" shows "(\<Sqinter> P \<in> \<P> \<bullet> P) = \<Sqinter>\<P>"
soundness: fixes r :: "nat rexp"   and s :: "nat rexp" assumes "Equivalence_Checking.check_eqv r s" shows "lang r = lang s"
lzip_simps_1: fixes ys :: "'d llist" shows "lzip LNil ys = LNil"
set_mode_ok: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root_mode :: "shadow_root_mode" assumes "ShadowRootClass.type_wf h"   and "shadow_root_ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok set_mode shadow_root_ptr shadow_root_mode"
x2: shows "x2 \<in> poincare_circle (of_complex z1) r1 \<inter> poincare_circle (of_complex z2) r2"
chain_rel_prodD1: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Y :: "('c \<times> 'd) set" assumes "Complete_Partial_Order.chain (rel_prod orda ordb) Y" shows "Complete_Partial_Order.chain orda (fst ` Y)"
collapse: fixes rbt :: "('a, 'b) rbt" assumes "rbt \<noteq> Leaf" shows "Node (lsub rbt) (cl rbt) (key rbt) (val rbt) (rsub rbt) = rbt"
IH: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length (p\<^sub>2 # p\<^sub>3 # ps)"   and "(c\<^sub>0, c\<^sub>1) = closest_pair_bf (p\<^sub>2 # p\<^sub>3 # ps)" shows "sparse (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (set (p\<^sub>2 # p\<^sub>3 # ps))"
assms_1: fixes \<eta> :: "nat" shows "\<I>_common' \<eta>, \<I>_common \<eta> \<turnstile>\<^sub>C conv \<eta> \<surd>"
list_update_le_listI: fixes xs :: "'a list"   and ys :: "'a list"   and p :: "nat"   and x :: "'a" assumes "set xs \<subseteq> A"   and "set ys \<subseteq> A"   and "xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys"   and "p < length xs"   and "x \<sqsubseteq>\<^bsub>r\<^esub> ys ! p"   and "x \<in> A" shows "xs[p := x \<squnion>\<^bsub>f\<^esub> xs ! p] [\<sqsubseteq>\<^bsub>r\<^esub>] ys"
RTranCl_induct: fixes h :: "'a"   and h' :: "'a"   and succs :: "'a \<Rightarrow> 'a list"   and P :: "'a \<Rightarrow> bool" assumes "h [succs]\<rightarrow>* h'"   and "P h"   and "\<And>g g'. \<lbrakk>g' \<in> set (succs g); P g\<rbrakk> \<Longrightarrow> P g'" shows "P h'"
one_fls_def: shows "1 \<equiv> Abs_fls (\<lambda>k. if k = 0 then 1::'b else (0::'b))"
less_le_not_le: fixes x :: "'a"   and y :: "'a" shows "(x \<prec> y) = (x \<preceq> y \<and> \<not> y \<preceq> x)"
if_dens_def: fixes \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and b :: "bool" shows "if_dens \<delta> f b \<equiv> \<lambda>\<sigma>. \<delta> \<sigma> * f \<sigma> (BoolVal b)"
Constrains_def: fixes A :: "'c set"   and B :: "'c set" shows "A Co B \<equiv> {F. F \<in> reachable F \<inter> A co B}"
cqt_1: fixes \<phi> :: "'a \<Rightarrow> \<o>"   and \<alpha> :: "'a" shows "[[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> \<phi> \<alpha>]]"
A: shows "Pri_AgrK x \<in> analz (A \<union> spies evsFR5) \<longrightarrow> Pri_AgrK x \<in> A"
unit_simps_3: shows "cod \<eta>' = g' \<star> f"
simp_num_pair_def: shows "simp_num_pair = (\<lambda>(t, n). if n = 0 then (C 0, 0) else let t' = simpnum t; g = numgcd t' in if 1 < g then let g' = gcd n g in if g' = 1 then (t', n) else (reducecoeffh t' g', n div g') else (t', n))"
plossless_inline: fixes gpv :: "('g, 'a, 'b) gpv"   and \<I>' :: "('h, 'i) \<I>"   and s :: "'s" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. \<lbrakk>I s; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> plossless_gpv \<I>' (lift_spmf (callee s x))"   and "I s" shows "plossless_gpv \<I>' (inline (\<lambda>s x. lift_spmf (callee s x)) gpv s)"
left_adjoint_is_ide: fixes f :: "'a" assumes "is_left_adjoint f" shows "ide f"
maxR_rangeD_2: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "MaxR r"   and "MaxR.maxR r x y \<noteq> y" shows "MaxR.maxR r x y = x"
meet_def: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y = (if \<exists>z. {x, y} >>| z then \<Sqinter>{x, y} else x)"
Diff_insert2: fixes A :: "'b set"   and a :: "'b"   and B :: "'b set" shows "(A::'b set) - ((a::'b) \<triangleright> (B::'b set)) = A - {a} - B"
LeftDerives1_append_replace_in_left: fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and \<beta> :: "symbol list" assumes "LeftDerives1 (\<alpha> @ \<delta>) i r \<beta>"   and "i < length \<alpha>" shows "\<exists>\<alpha>'. \<beta> = \<alpha>' @ \<delta> \<and> LeftDerives1 \<alpha> i r \<alpha>' \<and> i + length (snd r) \<le> length \<alpha>'"
pfa_extend_def: shows "pfa_extend \<equiv> \<lambda>(a, n) (b, m). blit b 0 a n m \<bind> (\<lambda>_. return (a, n + m))"
param_id: fixes R :: "('a \<times> 'b) set" shows "(id, id) \<in> R \<rightarrow> R"
\<Phi>_simp: fixes yx :: "'d \<times> 'c" assumes "DopxC.ide yx" shows "\<phi>\<psi>.\<Phi> yx = set_category_given_img.mkArr replete_setcat.comp (set_category.some_img replete_setcat.comp (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set (\<cdot>\<^sub>C) \<phi>\<psi>.\<phi>C (F (fst yx), snd yx)) (hom_functor.set (\<cdot>\<^sub>D) \<phi>\<psi>.\<phi>D (fst yx, G (snd yx))) (\<phi>\<psi>.inD \<circ> \<phi> (fst yx) \<circ> hom_functor.\<psi> (\<cdot>\<^sub>C) \<phi>\<psi>.\<phi>C (F (fst yx), snd yx))"
hom: shows "graph_homomorphism (snd R) (graph_of (X (Suc x'))) (extend (fst (X x')) R f)"
intros_12: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F \<^bold>\<rightarrow> G"   and "\<Gamma> \<turnstile> F" shows "\<Gamma> \<turnstile> G"
Iod_def: fixes D :: "('h, 'i) Order_scheme"   and T :: "'h set" shows "Iod D T = D\<lparr>carrier := T, rel := {(a, b). (a, b) \<in> rel D \<and> a \<in> T \<and> b \<in> T}\<rparr>"
inverse_arrows_sym: fixes f :: "'a"   and g :: "'a" shows "inverse_arrows f g = inverse_arrows g f"
bf_ifex_rel_consts_1: shows "(bf_True, Trueif) \<in> bf_ifex_rel"
is_image_rev: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and P' :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Reflect Bet Cong P P' A B" shows "Tarski_neutral_dimensionless.Reflect Bet Cong P P' B A"
equivalence_in_bicategory_axioms: shows "internal_equivalence (f' \<star> f) (g \<star> g') \<eta> \<epsilon>"
reduction_words_fin: fixes q :: "'state"   and w :: "'action list" assumes "q \<in> nodes"   and "R.path w q" shows "path w q"
cond_conj_not: fixes b :: "'a upred"   and P :: "'a upred"   and Q :: "'a upred" shows "(P \<triangleleft> b \<triangleright> Q \<and> \<not> b) = (Q \<and> \<not> b)"
closedin_derived_set: fixes X :: "'c topology"   and T :: "'c set"   and S :: "'c set" shows "closedin (subtopology X T) S = (S \<subseteq> topspace X \<and> S \<subseteq> T \<and> (\<forall>x. x \<in> X derived_set_of S \<and> x \<in> T \<longrightarrow> x \<in> S))"
thread_ok_ts_change: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ts' :: "'f \<Rightarrow> ('i \<times> 'j \<Rightarrow>f nat) option"   and ta :: "('f, 'k, 'l) new_thread_action" assumes "\<And>t. (ts t = None) = (ts' t = None)" shows "thread_ok ts ta = thread_ok ts' ta"
remdups_adj_Cons_alt: fixes x :: "'a"   and xs :: "'a list" shows "x # tl (remdups_adj (x # xs)) = remdups_adj (x # xs)"
try_spmf_lossless: fixes p :: "'d spmf"   and q :: "'d spmf" assumes "lossless_spmf p" shows "TRY p ELSE q = p"
Skip_is_action: shows "R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> \<not> wait A' \<and> alpha_rp.more A = alpha_rp.more A') \<in> {p. is_CSP_process p}"
NFlowCut_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and k :: "nat set" shows "NFlowCut c s t f k \<equiv> NFlow c s t f \<and> NCut c s t k"
f'_def: shows "f' \<equiv> \<lambda>x. \<integral>\<^sup>+ y. f <|x, y|> \<partial>stock_measure t2"
Sup_nat_less_sets_singleton: fixes T :: "nat set"   and n :: "nat" assumes "Sup T < n"   and "finite T" shows "T \<lless> {n}"
min_dep_imp_supset_circuit: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set"   and C :: "'b set"   and x :: "'b" assumes "matroid carrier indep"   and "indep X"   and "indep_system.circuit carrier indep C"   and "C \<subseteq> insert x X" shows "x \<in> C"
\<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_simps: shows "\<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G () = {()-:0}receive(\<lambda>msg msgs. msgs @ [msg]) . call(()) \<oplus> {()-:0}\<langle>\<lambda>msgs. if msgs \<noteq> [] then {msgs} else {}\<rangle> ({()-:1}send(hd) . ({()-:2}\<lbrakk>tl\<rbrakk> call(()) \<oplus> {()-:2}receive(\<lambda>msg msgs. tl msgs @ [msg]) . call(())) \<oplus> {()-:1}receive(\<lambda>msg msgs. msgs @ [msg]) . call(()))"
congs_refl: fixes xs :: "'a list" shows "xs \<cong> xs"
Hermite_of_upt_row_preserves_zero_rows_ge: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and k :: "nat"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "is_zero_row i A"   and "k \<le> nrows A"   and "k \<le> mod_type_class.to_nat i" shows "is_zero_row i (Hermite_of_upt_row_i A k ass res)"
times_hom_axioms: shows "times_hom (map_poly to_fract)"
AtSeqExpr2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a2 :: "('e, 'd, 'f) ann"   and c2 :: "('e, 'd, 'f) com"   and a :: "'e set \<times> ('e, 'd, 'f) com"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com" assumes "atomicsR \<Gamma> \<Theta> a2 c2 a" shows "atomicsR \<Gamma> \<Theta> (AnnComp a1 a2) (Seq c1 c2) a"
assms: shows "x \<^bold>\<in> ide_to_hf a"
of_int_0: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 0 = zero"
2: shows "Number_Theory_Aux.inverse x q * int x + snd (bezw x q) * int q = 1"
Abs_uint32_inverse: fixes y :: "32 word" assumes "y \<in> UNIV" shows "Rep_uint32 (Abs_uint32 y) = y"
InvariantWatchListsCharacterizationNotifyWatchesLoop: fixes state :: "State"   and Wl :: "nat list"   and literal :: "Literal"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "\<forall>c l. l \<noteq> literal \<longrightarrow> (c \<in> set (getWatchList state l)) = (Some l = getWatch1 state c \<or> Some l = getWatch2 state c)"   and "\<forall>c. (c \<in> set newWl \<or> c \<in> set Wl) = (Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c)"   and "set Wl \<inter> set newWl = {}"   and "uniq Wl"   and "uniq newWl" shows "let state' = notifyWatches_loop literal Wl newWl state in InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchListsUniq (getWatchList state')"
SubobjsR_Base: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" assumes "is_class P C" shows "Subobjs\<^sub>R P C [C]"
convex_hull_indexed: fixes S :: "'c set" shows "convex hull S = {y. \<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<and> (\<Sum>i = 1..k. u i *\<^sub>R x i) = y}"
closure_of_restrict: fixes X :: "'b topology"   and S :: "'b set" shows "X closure_of S = X closure_of (topspace X \<inter> S)"
comp_null_2: fixes f :: "'a" shows "Left.comp f Left.null = Left.null"
simps_1: fixes nodes :: "'a set"   and edges :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
PiE_eq: fixes I :: "'d set"   and S :: "'d \<Rightarrow> 'e set"   and T :: "'d \<Rightarrow> 'e set" shows "(Pi\<^sub>E I S = Pi\<^sub>E I T) = (Pi\<^sub>E I S = {} \<and> Pi\<^sub>E I T = {} \<or> (\<forall>i\<in>I. S i = T i))"
vec_lambda_beta: fixes g :: "'e \<Rightarrow> 'd"   and i :: "'e" shows "vec_lambda g $ i = g i"
strong_induct: fixes P :: "'a \<Rightarrow> tm \<Rightarrow> bool"   and c :: "'a"   and tm :: "tm" assumes "\<And>c. P c Zero"   and "\<And>x c. P c (Var x)"   and "\<And>x1 x2a c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2a\<rbrakk> \<Longrightarrow> P c (Eats x1 x2a)" shows "P c tm"
Pow_insert: fixes a :: "'c"   and A :: "'c set" shows "Pow (insert a A) = Pow A \<union> insert a ` Pow A"
mod_ring_eq64: shows "(mod_ring_rel64 ===> mod_ring_rel64 ===> (=)) (=) (=)"
foldl_un_empty_eq: fixes i :: "'d set"   and ww :: "'d set list" shows "foldl (\<union>) i ww = i \<union> foldl (\<union>) {} ww"
diff_frac_eq: fixes y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "y \<noteq> (0::'a)"   and "z \<noteq> (0::'a)" shows "x / y - w / z = (x * z - w * y) / (y * z)"
Order_adjunct_ord: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D"   and "a \<notin> carrier D" shows "Order (adjunct_ord D a)"
prod_charts_def: shows "prod_charts = {prod_chart c1 c2 |c1 c2. c1 \<in> charts1 \<and> c2 \<in> charts2}"
lookup_monom_mult_plus: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (monom_mult c t p) (t \<oplus> v) = c * lookup p v"
f_nonpos: fixes x :: "real" assumes "x \<le> 0" shows "f x = 0"
take_all: fixes xs :: "'d list"   and n :: "nat" assumes "length xs \<le> n" shows "take n xs = xs"
finite_dom_subst_exists: fixes S :: "'a set" assumes "finite S" shows "\<exists>\<sigma>. subst_domain \<sigma> = S"
ipresIGSwapAbs_def: fixes hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'ae, 'w, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ai, 'aj, 'ad, 'ak, 'x, 'al) model_scheme" shows "ipresIGSwapAbs hA MOD MOD' \<equiv> \<forall>zs z1 z2 us s A. igWlsAbs MOD (us, s) A \<longrightarrow> hA (igSwapAbs MOD zs z1 z2 A) = igSwapAbs MOD' zs z1 z2 (hA A)"
continuous_def: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "continuous F f = (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
vreal_of_real_transfer: fixes a :: "real" shows "cr_vreal (a\<^sub>\<real>) a"
canonical_empty_zone_spec: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" assumes "canonical M n" shows "([M]\<^bsub>v,n\<^esub> = {}) = (\<exists>i\<le>n. M i i < \<one>)"
discrete_topology_unique_derived_set: fixes U :: "'c set"   and X :: "'c topology" shows "(discrete_topology U = X) = (topspace X = U \<and> X derived_set_of U = {})"
continuous_on_chart_2: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and c :: "('c, 'b) chart" assumes "continuous_on s f"   and "f ` s \<subseteq> codomain c" shows "continuous_on s (\<lambda>x. inv_chart c (f x))"
toplevel_inters_lang_INT: fixes n :: "nat"   and s :: "'b rexp" shows "lang n s = \<Inter> (lang n ` toplevel_inters s)"
grid_Start: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set" assumes "p \<in> grid b ds"   and "level p = level b" shows "p = b"
simps_1: fixes q :: "'b list"   and r :: "'b list"   and d :: "'b list"   and n :: "nat" shows "divmod_poly_one_main_list q r d (Suc n) = (let a = hd r; qqq = cCons a q; rr = tl (if a = (0::'b) then r else minus_poly_rev_list r (map ((*) a) d)) in divmod_poly_one_main_list qqq rr d n)"
eq_refl: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>P \<phi>"
that: fixes T :: "real" assumes "0 < T"   and "\<And>t xa. flow0 x (t + T * real xa) = flow0 x t" shows "thesis"
same_upto_def: fixes j :: "nat"   and A :: "'a mat"   and B :: "'a mat" shows "local.same_upto j A B \<equiv> \<forall>i' j'. i' < n \<longrightarrow> j' < j \<longrightarrow> A $$ (i', j') = B $$ (i', j')"
frameChainEqSuppEmpty: fixes xvec :: "name list"   and \<Psi> :: "'d"   and yvec :: "name list"   and \<Psi>' :: "'d" assumes "\<langle>xvec, \<Psi>\<rangle> = \<langle>yvec, \<Psi>'\<rangle>"   and "supp \<Psi> = {}" shows "\<Psi> = \<Psi>'"
bind_sample1: fixes p :: "int pmf"   and f :: "int \<Rightarrow> 'm"   and g :: "int \<Rightarrow> 'm" shows "bind (sample p f) g = sample p (\<lambda>x. bind (f x) g)"
simps_1: fixes x21 :: "'f"   and x22 :: "'f list"   and y21 :: "'f"   and y22 :: "'f list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
sgivesu: fixes \<xi> :: "'a"   and \<xi>' :: "'a" assumes "S \<xi> \<xi>'" shows "U \<xi> \<xi>'"
invc2_baldR: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c \<times> 'd" assumes "invc l"   and "invc2 r" shows "invc2 (baldR l x r)"
general_product_character: fixes F :: "i \<Rightarrow> bool" assumes "GEM"   and "\<exists>z. \<forall>y. F y \<longrightarrow> P z y" shows "\<forall>y. O y (\<pi>x. F x) = (\<exists>z. (\<forall>y. F y \<longrightarrow> P z y) \<and> O y z)"
d: shows "Esigma (g * M) = Esigma g * Esigma M"
that: fixes e' :: "real" assumes "0 < e'"   and "\<And>y. y \<in> ball x e' \<Longrightarrow> returns_to {x \<in> S. s x = 0} y"   and "\<And>y. y \<in> ball x e' \<Longrightarrow> s (flow0 y (return_time {x \<in> S. s x = 0} y)) = 0"   and "continuous_on (ball x e') (return_time {x \<in> S. s x = 0})"   and "\<And>y. y \<in> ball x e' \<Longrightarrow> Ds (poincare_map {x \<in> S. s x = 0} y) o\<^sub>L flowderiv y (return_time {x \<in> S. s x = 0} y) o\<^sub>L embed2_blinfun \<in> invertibles_blinfun"   and "\<And>U v sa. \<lbrakk>\<And>sa. sa \<in> U \<Longrightarrow> s (flow0 sa (v sa)) = 0; return_time {x \<in> S. s x = 0} x = v x; continuous_on U v; sa \<in> U; x \<in> U; U \<subseteq> ball x e'; connected U; open U\<rbrakk> \<Longrightarrow> return_time {x \<in> S. s x = 0} sa = v sa"   and "(return_time {x \<in> S. s x = 0} has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)))) o\<^sub>L (Ds (poincare_map {x \<in> S. s x = 0} x) o\<^sub>L Dflow x (return_time {x \<in> S. s x = 0} x)))) (at x)" shows "thesis"
semiring_norm_14: fixes m :: "num"   and n :: "num" shows "num.Bit0 m * num.Bit1 n = num.Bit0 (m * num.Bit1 n)"
map_simp_1: fixes g :: "'c"   and a :: "'c" assumes "C.arr g"   and "C.ide a" shows "Hom.map (g, a) = S.mkArr (Hom.set (C.cod g, a)) (Hom.set (C.dom g, a)) (\<phi> (C.dom g, a) \<circ> (\<cdot>\<^sup>o\<^sup>p) g \<circ> \<psi> (C.cod g, a))"
assms_6: shows "poincare_distance x u = poincare_distance x v"
traceAB: shows "trace (P * B * Q) = trace (Q * P * B)"
normal_form_2: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "\<not> punit.is_red (punit.reduced_GB F) (normal_form F p)"
unitary_bound: fixes P :: "'a \<Rightarrow> real" assumes "unitary P" shows "bounded_by 1 P"
weight_spmf_eq_nn_integral_spmf: fixes p :: "'c spmf" shows "ennreal (weight_spmf p) = \<integral>\<^sup>+ x. ennreal (spmf p x) \<partial>count_space UNIV"
arrI: fixes f1 :: "'c"   and f2 :: "'c" assumes "Cop.arr f1"   and "C.arr f2" shows "CopxC.arr (f1, f2)"
nat_cut_less_le_conv: fixes t :: "nat"   and I :: "nat set" assumes "0 < t" shows "I \<down>< t = I \<down>\<le> (t - Suc 0)"
rqI: fixes z :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "z \<cdot> u = v" shows "v\<^sup><\<inverse>u  = z"
simps_4: fixes v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Plus v va) Zero = False"
homogeneous_def: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "homogeneous p = (\<forall>s\<in>keys p. \<forall>t\<in>keys p. deg_pm s = deg_pm t)"
clo0TB: shows "closedin (top_of_set ({0..1} \<times> T)) B"
de_morgan2: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "!p \<cdot> !q = !(p + q)"
that: fixes u :: "'a"   and e :: "'b" assumes "\<mu> c s v = \<mu> c s u + ereal (c e)"   and "arc e (u, v)" shows "thesis"
assms_1: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j"   and "j < n" shows "a i \<le> a j"
prems_2: shows "di + Suc len \<le> length ldst"
outside: fixes S :: "'b set" shows "outside S = {x. \<not> bounded (connected_component_set (- S) x)}"
inject_2: fixes x21 :: "'i"   and x22 :: "'j"   and y21 :: "'i"   and y22 :: "'j" shows "(IO x21 x22 = IO y21 y22) = (x21 = y21 \<and> x22 = y22)"
simps: fixes a :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "asymp a = (\<exists>R. a = R \<and> (\<forall>x xa. R x xa \<longrightarrow> \<not> R xa x))"
inverse_eq_divide_numeral: fixes w :: "num" shows "inverse (numeral w) = (1::'b) / numeral w"
HomContraMor: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e"   and X :: "'d" assumes "LSCategory C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "(Hom\<^bsub>C\<^esub>[\<midarrow>,X]) ## f = HomC\<^bsub>C\<^esub>[f,X]"
Static: assumes "dynmethd G statC dynC sig = Some statM" shows "thesis"
recursive_def: fixes h' :: "'b \<Rightarrow> 'b"   and b :: "'b" shows "recursive h' b \<equiv> h' b = b"
is_non_overlapping_def: fixes P :: "'b set set" shows "is_non_overlapping P = (\<forall>X\<in>P. \<forall>Y\<in>P. (X \<inter> Y \<noteq> {}) = (X = Y))"
changes_poly_at_pders_0: fixes p :: "real poly" shows "changes_poly_at (pders p) 0 = changes (coeffs p)"