hcju/isabelleps huggingface dataset:
diagonal_step_PQ_JNF_dim: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat m n"   and "diagonal_step_PQ_JNF A i j bezout = (P, Q)" shows "P \<in> carrier_mat m m \<and> Q \<in> carrier_mat n n"
presSwapAll_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresSwapAll h hA (fromMOD MOD)" shows "presSwapAll (fromIMor h) (fromIMorAbs hA) MOD"
eFreshBinp_simp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'b option"   and ys :: "'varSort"   and y :: "'var" assumes "igWlsBinp MOD delta binp" shows "eFreshBinp MOD ys y (OKI binp) = igFreshBinp MOD ys y binp"
conf: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) u = ([x, y], rev u @ h)"
assms_2: shows "AE x in M. incseq (\<lambda>n. f n x)"
nonempty: shows "space (dens_ctxt_measure \<Y> \<rho>) \<noteq> {}"
ereal_le_real: fixes x :: "ereal"   and y :: "ereal" assumes "\<And>z. x \<le> ereal z \<Longrightarrow> y \<le> ereal z" shows "y \<le> x"
\<eta>_3: shows "f' \<eta> = (f y - f z) / (y - z)"
subst_res_def: fixes \<sigma> :: "'d \<Rightarrow> 'd"   and x :: "'c \<Longrightarrow> 'd" shows "\<sigma> \<restriction>\<^sub>s x = (\<lambda>s. get\<^bsub>x\<^esub> (\<sigma> (create\<^bsub>x\<^esub> s)))"
assms: shows "f (lead_coeff p) \<noteq> (0::'a)"
inf_sup_ord_3: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.lattice (inf::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) ((sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) x (y::'b))"
reachable_rename_ioa: fixes C :: "('g set \<times> 'g set \<times> 'g set) \<times> 'f set \<times> ('f \<times> 'g \<times> 'f) set"   and g :: "'e \<Rightarrow> 'g option"   and s :: "'f" assumes "reachable (rename C g) s" shows "reachable C s"
dvd_smult: fixes p :: "'a poly"   and q :: "'a poly"   and a :: "'a" assumes "p dvd q" shows "p dvd smult a q"
compose_eq': fixes A :: "'a set"   and G :: "'d \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'd" shows "compose A G F = restrict (G \<circ> F) A"
PPs_INT: fixes X :: "'b set set" shows ".[\<Inter> X] = \<Inter> (PPs ` X)"
gcd_impl_start_def: shows "gcd_impl_start = subresultant_prs_impl fst"
M': shows "m = dfa_steps (binop_dfa f A B) 0 bs"
Max_insert: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq (insert x A) = ord.max less_eq x (linorder.Max less_eq A)"
rel_fun_conj: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> bool"   and f' :: "'b \<Rightarrow> bool"   and g :: "'a \<Rightarrow> bool"   and g' :: "'b \<Rightarrow> bool" assumes "rel_fun R (=) f f'"   and "rel_fun R (=) g g'" shows "rel_fun R (=) (\<lambda>x. f x \<and> g x) (\<lambda>y. f' y \<and> g' y)"
z_def: shows "z \<equiv> Variable (Suc (Suc 0))"
assoc_ide_simp: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "D'.src f = D'.trg g"   and "D'.src g = D'.trg h" shows "D'.\<a> f g h = \<a>\<^sub>D\<^sup>-\<^sup>1[h, g, f]"
unitary_zero: fixes A :: "'c mat" assumes "A \<in> carrier_mat 0 0" shows "unitary A"
inorder_2: shows "inorder t2 = [k + 1..j]"
test2: fixes g :: "nat list list" assumes "g \<in> set arch" shows "\<exists>h\<in>set_tries gs. iso_test g h"
mult_stalk_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and I :: "'d set set"   and x :: "'d" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "stalk.mult_stalk \<FF> \<rho> mult_str I \<equiv> direct_lim.mult_rel \<FF> \<rho> mult_str I"
mod_diff_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 uminus :: "'k \<Rightarrow> 'k"   and a :: "'k"   and c :: "'k"   and a' :: "'k"   and b :: "'k"   and b' :: "'k" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (minus a b) c = modulo (minus a' b') c"
circle_circline: fixes cl :: "complex set"   and \<mu> :: "complex"   and r :: "real" assumes "cl = circle \<mu> r"   and "0 \<le> r" shows "cl = circline 1 (- \<mu>) ((cmod \<mu>)\<^sup>2 - r\<^sup>2)"
elims_1: fixes x :: "'d"   and xa :: "'e"   and xb :: "'d set list"   and xc :: "('d, 'e) ta_rule"   and y :: "bool" assumes "r_matchc x xa xb xc = y" obtains q :: "'d" and l :: "'e" and Qs :: "'d set list" and qr :: "'d" and lr :: "'e" and qsr :: "'d list" where "x = q" and "xa = l" and "xb = Qs" and "xc = qr \<rightarrow> lr qsr" and "y = (q = qr \<and> l = lr \<and> r_match Qs qsr)"
lunit_simps_5: fixes f :: "'a" assumes "ide f" shows "cod \<l>[f] = f"
hom_power: fixes x :: "'a poly"   and n :: "nat"   and a :: "'a" shows "poly (x ^ n) a = poly x a ^ n"
quotientI: fixes x :: "'b"   and A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "x \<in> A" shows "r `` {x} \<in> A // r"
minus_fold_remove: fixes A :: "'g set"   and B :: "'g set" assumes "finite A" shows "B - A = Finite_Set.fold Set.remove B A"
reset: fixes l :: "'c list"   and u' :: "'c \<Rightarrow> 'a"   and u :: "'c \<Rightarrow> 'a" assumes "set l \<subseteq> \<X>"   and "u' = [l\<rightarrow>(0::'a)]u" shows "local.trans u u'"
fixpt_dist_nodes: fixes M :: "nat bdd list \<times> bool list"   and v :: "nat"   and p :: "nat"   and q :: "nat" assumes "wf_dfa M v"   and "dfa_is_node M p"   and "dfa_is_node M q" shows "tr_lookup (fixpt M (init_tr M)) p q = (\<exists>n. dist_nodes M n v p q)"
trisuma_perm_231: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TriSumA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.TriSumA Bet Cong B C A D E F"
diff_le_0_iff_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (minus a b) zero = less_eq a b"
filterlim_filtercomap: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'c filter" shows "filterlim f F (filtercomap f F)"
factors_unique: fixes fs :: "'a list"   and a :: "'a"   and fs' :: "'a list" assumes "factors G fs a"   and "factors G fs' a"   and "a \<in> carrier G"   and "a \<notin> Units G"   and "set fs \<subseteq> carrier G"   and "set fs' \<subseteq> carrier G" shows "essentially_equal G fs fs'"
rtranclp_subset: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "R \<le> S"   and "S \<le> R\<^sup>*\<^sup>*" shows "S\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
begin_partial_correctness: fixes n :: "nat"   and stp :: "nat" assumes "is_final (steps0 (1, [], Oc \<up> n) tcopy_begin stp)"   and "0 < n" shows "{inv_begin1 n} tcopy_begin {inv_begin0 n}"
cobounded1: fixes a :: "'c"   and b :: "'c" shows "a \<le> a + b"
call_of_return_nodes_det: fixes n :: "'node"   and n' :: "'node"   and n'' :: "'node" assumes "call_of_return_node n n'"   and "call_of_return_node n n''" shows "n' = n''"
unitary_inv: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary M" shows "unitary (mat_inv M)"
polychain_of_add: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "polychain_of (x + y) xs = map ((+) (y, y)) (polychain_of x xs)"
simps_2: fixes rr :: "('c, 'd) routing_rule_scheme"   and rtbl :: "('c, 'd) routing_rule_scheme list" shows "unambiguous_routing_code (rr # rtbl) = (list_all (\<lambda>ra. routing_match rr \<noteq> routing_match ra \<or> routing_rule_sort_key rr \<noteq> routing_rule_sort_key ra) rtbl \<and> unambiguous_routing_code rtbl)"
homeomorphism_cont1: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "homeomorphism S T f g" shows "continuous_on S f"
berlekamp_hensel_def: fixes p :: "int"   and n :: "nat"   and f :: "int poly" shows "berlekamp_hensel p n f = (case finite_field_factorization_int p f of (x, fs) \<Rightarrow> hensel_lifting p n f fs)"
WTrtWhile: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and c :: "char list exp"   and T :: "ty" assumes "P,E,h \<turnstile> e : Boolean"   and "P,E,h \<turnstile> c : T" shows "P,E,h \<turnstile> while (e) c : Void"
super: assumes "OFCLASS('a, Gcd_class)" shows "OFCLASS('a, gcd_class)"
match_cong_strong: fixes i :: "nat"   and j :: "nat"   and r :: "'a Regex.regex"   and t :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and t' :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>k z. \<lbrakk>k \<in> {i..<j + 1}; z \<in> regex.atms r\<rbrakk> \<Longrightarrow> t k z = t' k z" shows "Regex.match t r i j = Regex.match t' r i j"
run_state_set: fixes s' :: "'a"   and s :: "'a" shows "run_state (State_Monad.set s') s = ((), s')"
sup_compl_top_left2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup x (sup (uminus x) y) = top"
expectation_gpv_cong_fail: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and fail :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and fail' :: "ennreal" assumes "colossless_gpv \<I> gpv" shows "expectation_gpv fail \<I> f gpv = expectation_gpv fail' \<I> f gpv"
mset_le_mono_add_single: fixes a :: "'a"   and ys :: "'a multiset"   and b :: "'a"   and ws :: "'a multiset" assumes "a \<in># ys"   and "b \<in># ws" shows "{#a, b#} \<subseteq># ys + ws"
no_pending_imp_succ_discovered: fixes u :: "'v"   and v :: "'v" assumes "u \<in> dom (discovered s)"   and "pending s `` {u} = {}"   and "v \<in> E `` {u}" shows "v \<in> dom (discovered s)"
pick_def: fixes A :: "'b quot" shows "pick A = (SOME a. A = \<lfloor>a\<rfloor>)"
delblock: fixes x :: "'a list" shows "delblock x = delblockhelp x 512"
word_add_not: fixes x :: "'b word" shows "x + NOT x = - 1"
R'_S: fixes y :: "'a" assumes "y \<in> S" shows "(\<mu> \<rho>'. (\<rho> ++\<^bsub>S\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>S\<^esub> e1\<cdot>\<rho>''))(x := e2\<cdot>(\<mu> \<rho>''. \<rho>' ++\<^bsub>S\<^esub> e1\<cdot>\<rho>''))) y = (e1\<cdot>(\<mu> \<rho>''. (\<mu> \<rho>'. (\<rho> ++\<^bsub>S\<^esub> (\<mu> \<rho>''. \<rho>' ++\<^bsub>S\<^esub> e1\<cdot>\<rho>''))(x := e2\<cdot>(\<mu> \<rho>''. \<rho>' ++\<^bsub>S\<^esub> e1\<cdot>\<rho>''))) ++\<^bsub>S\<^esub> e1\<cdot>\<rho>'')) y"
simps_5: fixes bs :: "real list"   and a :: "Ferrack.num" shows "Ifm bs (Le a) = (Inum bs a \<le> 0)"
simps_2: fixes Q :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool"   and cf :: "('c \<Rightarrow> 'd option) \<times> 'e"   and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" shows "local.CFG.transfer (Q)\<^sub>\<surd> (cf # cfs) = cf # cfs"
powr_realpow: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "x powr real n = x ^ n"
zero_notin_hpair: fixes x :: "hf"   and y :: "hf" shows "0 \<^bold>\<notin> \<langle>x, y\<rangle>"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_loop_start m lm (l, r) = (wprepare_loop_start_on_rightmost m lm (l, r) \<or> wprepare_loop_start_in_middle m lm (l, r))"
in_image_mset: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" shows "(y \<in># image_mset f M) = (y \<in> f ` set_mset M)"
dnso6: shows "ans_d (zero_class.zero::'a) = (zero_class.zero::'a)"
card_ge_0_finite: fixes A :: "'b set" assumes "order.greater (card A) 0" shows "finite A"
alpha_Tree_eqvt': fixes t1 :: "('a, 'b, 'c) Formula.Tree"   and t2 :: "('a, 'b, 'c) Formula.Tree"   and p :: "perm" shows "(t1 =\<^sub>\<alpha> t2) = (p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2)"
apath_ends: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "S.apath u p v"   and "S.apath u' p v'" shows "p \<noteq> [] \<and> u \<noteq> v \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
split_sum: shows "(\<Sum>j<n. d j * base ^ j) = (\<Sum>j<i. d j * base ^ j) + base ^ i * (\<Sum>j = i..<n. d j * base ^ (j - i))"
lebesgue_integral_const: fixes M :: "'d measure"   and a :: "'c" shows "LINT x|M. a = Sigma_Algebra.measure M (space M) *\<^sub>R a"
kat_2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<cdot> ad z \<le> ad x \<cdot> y" shows "d x \<cdot> y \<cdot> ad z = (zero_class.zero::'a)"
Domainp_fun: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "left_unique A" shows "Domainp (A ===> B) = (\<lambda>f. f ` Collect (Domainp A) \<subseteq> Collect (Domainp B))"
A1: shows "\<forall>a b. a b \<equiv> b a"
simps_2: fixes uu :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and uv :: "'c \<times> 'd"   and c :: "tcolor"   and uw :: "'c \<times> 'd"   and ux :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "color \<langle>uu, (uv, c, uw), ux\<rangle> = c"
sep_disj_commuteI: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "sep_disj y x"
flush_all_until_volatile_write_Ghost\<^sub>s\<^sub>b_commute: fixes i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "'c"   and \<O>' :: "'d"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, Ghost\<^sub>s\<^sub>b A L R W # sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p', is', \<theta>', sb, \<D>', \<O>', \<R>')]) m = flush_all_until_volatile_write ts m"
assms: shows "(u, v) \<in> parcs (list_digraph G_list)"
beta_eta_norm_preserves_proves: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "\<Theta>,\<Gamma> \<turnstile> t"   and "beta_eta_norm t = Some u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> u"
is_parser_return: fixes x :: "'a" shows "is_parser (Parser_Monad.return x)"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "b \<preceq> a" shows "b \<prec> a"
comp_simp: fixes g :: "'d"   and f :: "'d" assumes "D'.seq g f" shows "D'.comp g f = g \<cdot>\<^sub>D f"
connected_space_iff_components_eq: fixes X :: "'b topology" shows "connected_space X = (\<forall>C\<in>connected_components_of X. \<forall>C'\<in>connected_components_of X. C = C')"
nth_zip: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'b list" assumes "i \<sqsubset> length xs"   and "i \<sqsubset> length ys" shows "zip xs ys ! i = (xs ! i, ys ! i)"
seq_extract_Nil: fixes A :: "nat set" shows "A \<upharpoonleft>\<^sub>l [] = []"
that_1: shows "num_params (tm_poly t1) \<le> X"
result: shows "finite_field_factorization_int p f = (c, fs)"
mono_neutral_cong_right: fixes T :: "'d set"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = \<^bold>1"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "F g T = F h S"
hyps_1: assumes "{$$} \<turnstile> e, eP, eV : \<tau>" shows "\<Gamma> \<turnstile> e, eP, eV : \<tau>"
simps_1: fixes s :: "'state_t option"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "run 0 s execs = s"
bisimilarity_implies_equivalence: fixes P :: "'state"   and Q :: "'state" assumes "P \<sim>\<cdot>\<^sub>S Q" shows "S_transform.logically_equivalent P Q"
eball_conv_UNION_balls: fixes z :: "'a"   and r :: "ereal" shows "eball z r = \<Union> (ball z ` {r'. ereal r' < r})"
length_drop: fixes n :: "nat"   and xs :: "'a list" shows "|drop n xs| = |xs| - n"
cnt_distinct_intro: fixes xs :: "'b list" assumes "\<forall>x\<in>set xs. cnt x xs \<le> 1" shows "distinct xs"
create_reservation_def: fixes ts :: "traffic"   and c :: "cars"   and ts' :: "traffic" shows "ts \<^bold>\<midarrow>r( c ) \<^bold>\<rightarrow> ts' \<equiv> pos ts' = pos ts \<and> res ts' = (res ts)(c := res ts c \<squnion> clm ts c) \<and> dyn ts' = dyn ts \<and> clm ts' = (clm ts)(c := bot) \<and> physical_size ts' = physical_size ts \<and> braking_distance ts' = braking_distance ts"
continuous_on_power: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. f x ^ n)"
unitary_def: fixes P :: "'c \<Rightarrow> real" shows "unitary P = (sound P \<and> bounded_by 1 P)"
poly_diff: fixes p :: "'b poly"   and q :: "'b poly"   and x :: "'b" shows "poly (p - q) x = poly p x - poly q x"
hfd: fixes x :: "complex"   and n :: "nat" assumes "x \<in> S" shows "f n field_differentiable at x"
insert_sub: fixes A :: "'h set"   and B :: "'h set"   and a :: "'h" assumes "A \<subseteq> B"   and "a \<in> B" shows "insert a A \<subseteq> B"
not_choice_simps_2: fixes l :: "'d"   and fa :: "'a \<Rightarrow> 'a"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_choice ({l}\<lbrakk>fa\<rbrakk> p)"
abs_int_poly_0: shows "abs_int_poly 0 = 0"
valid_def: shows "(\<Turnstile>) \<equiv> (id ---> Rep_formula ---> id) valid_Tree\<^sub>\<alpha>"
div0_uint_def: fixes x :: "uint" shows "div0_uint x = undefined (div) x 0"
padic_mult_def: fixes p :: "int"   and f :: "nat \<Rightarrow> int"   and g :: "nat \<Rightarrow> int" shows "padic_mult p f g \<equiv> \<lambda>n. f n \<otimes>\<^bsub>residue_ring (p ^ n)\<^esub> g n"
distinct_5: fixes x1 :: "'b" shows "Match x1 \<noteq> MatchAny"
abs_eq: fixes x :: "'a \<Rightarrow> 'b option" shows "pfun_app (Abs_pfun x) = (\<lambda>xa. if xa \<in> dom x then the (x xa) else undefined)"
scc_ofI_reachable': fixes v :: "'a"   and u :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> u" shows "u \<in> G.scc_of v"
ctr_1: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and g22 :: "'c \<Rightarrow> 'c" assumes "p a" shows "unfold_llist p g21 g22 a = LNil"
LLL_invD_4: fixes upw :: "bool"   and ia :: "nat"   and fs :: "int vec list"   and i :: "nat" assumes "LLL_invariant upw ia fs"   and "i < m" shows "fs ! i \<in> carrier_vec n"
r_result1_prim: shows "prim_recfn 3 r_result1"
evalHeap_subst_exp: fixes eval :: "'e \<Rightarrow> 'd"   and e :: "'e"   and e' :: "'e"   and x :: "'f"   and \<Gamma> :: "('f \<times> 'e) list" assumes "eval e = eval e'" shows "evalHeap ((x, e) # \<Gamma>) eval = evalHeap ((x, e') # \<Gamma>) eval"
intros: fixes pq :: "('b \<times> 'c) pmf"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'c pmf" assumes "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"   and "map_pmf fst pq = p"   and "map_pmf snd pq = q" shows "rel_pmf R p q"
least_consequence_graph_def: fixes t :: "'e itself"   and Rs :: "(('f, 'g) labeled_graph \<times> ('f, 'g) labeled_graph) set"   and S :: "('f, 'h) labeled_graph"   and G :: "('f, 'h) labeled_graph" shows "least_consequence_graph t Rs S G \<equiv> consequence_graph Rs G \<and> least t Rs S G"
simps_1: fixes ys :: "'a list" shows "shuffles [] ys = {ys}"
bvars\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map (set \<circ> bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p) S))"
len_lexext_rev_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "len_lexext_rev gt (xs @ [x]) xs"
tc_extract_function: fixes p :: "'a set"   and c :: "'a com"   and q :: "'a set" assumes "[p]   c   [q]" shows "\<exists>f. \<forall>s. s \<in> p \<longrightarrow> f s \<in> q"
intro: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and backward_slice :: "'b set \<Rightarrow> 'b set"   and sem :: "'f \<Rightarrow> 'c \<Rightarrow> 'f \<Rightarrow> 'c \<Rightarrow> bool"   and identifies :: "'b \<Rightarrow> 'f \<Rightarrow> bool" assumes "BackwardSlice sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice"   and "CFG_semantics_wf sourcenode targetnode kind valid_edge Entry sem identifies" shows "BackwardSlice_wf sourcenode targetnode kind valid_edge Entry Def Use state_val backward_slice sem identifies"
inorder_ins: fixes t :: "'b tree23"   and x :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (treeI (ins x t)) = ins_list x (inorder t)"
af_homomorphism: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "pure f \<diamondop> pure x = pure (f x)"