hcju/isabelleps huggingface dataset:
Repr_SelfSubst_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c" shows "Repr_SelfSubst var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc S \<equiv> Encode var trm fmla Var FvarsT substT Fvars subst num enc \<and> Deduct2 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv \<and> Repr_SelfSubst_axioms var fmla Var FvarsT Fvars subst eql cnj imp all bprv enc S"
sp_one: fixes v1 :: "'a \<times> nat"   and e :: "('a \<times> nat) edge"   and v2 :: "'a \<times> nat" shows "SubRel.subpath g v1 [e] v2 subs = (sub_rel_of g subs \<and> (v1 = src e \<or> (v1, src e) \<in> subs) \<and> e \<in> edges g \<and> (tgt e = v2 \<or> (tgt e, v2) \<in> subs))"
domains_simp: fixes f :: "'s" assumes "arr f" shows "domains f = {local.dom f}"
in_borel_measurable_borel: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>S\<in>sets borel. f -` S \<inter> space M \<in> sets M)"
ssufD: fixes u :: "'b list"   and v :: "'b list" assumes "u <s v" shows "u \<le>s v \<and> u \<noteq> v"
assms_2: assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "T"
True: shows "a \<le> b \<and> poly p a = 0 \<and> p \<noteq> 0 \<or> a = b \<and> p = 0"
SeqWR_def: fixes s :: "hf"   and k :: "hf"   and y :: "hf" shows "SeqWR s k y \<equiv> LstSeq s k y \<and> app s 0 = 0 \<and> (\<forall>l\<^bold>\<in>k. app s (succ l) = q_Eats (app s l) (app s l))"
lem2: fixes e :: "real" assumes "0 < e" shows "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
zero_less_mult_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less zero (times a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
admissible_le1I: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and y :: "'a" assumes "cont lub ord Sup (\<le>) f" shows "ccpo.admissible lub ord (\<lambda>x. f x \<le> y)"
While: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'f set"   and b :: "'f set"   and c :: "('f, 'e, 'g) com"   and A :: "'f set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> b) c P,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P While b c (P \<inter> - b),A"
inverse_arrowsE: fixes f :: "'b"   and g :: "'b" assumes "B.inverse_arrows f g" obtains "B.ide (B g f)" and "B.ide (B f g)"
isPrefixPrefixToLevelLowerLevel: fixes level :: "nat"   and level' :: "nat"   and M :: "('a \<times> bool) list" assumes "level < level'" shows "isPrefix (prefixToLevel level M) (prefixToLevel level' M)"
simps_2: fixes uv :: "real" shows "(PInfty \<le> real_min_top.R uv) = False"
list_all2_Cons1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and ys :: "'b list" shows "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
simps_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and n :: "nat" assumes "class.wellorder less_eq less" shows "wellorder.enumerate less_eq S (Suc n) = wellorder.enumerate less_eq (S - {ord.Least less_eq (\<lambda>n. n \<in> S)}) n"
sum_eq_0_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<And>i. i \<in> A \<Longrightarrow> (0::'b) \<le> f i" shows "(sum f A = (0::'b)) = (\<forall>i\<in>A. f i = (0::'b))"
TER_plus_web_aux: fixes x :: "'v" assumes "x \<in> \<E> (TER f \<union> TER g)"   and "x \<in> SINK (g \<upharpoonleft> \<Gamma> / f)" shows "x \<in> TER (f \<frown> g)"
exhaust: fixes y :: "agent" obtains x :: "nat" where "y = Friend x"
count_const_lt_components_alt: fixes hs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> 'e) list" shows "count_const_lt_components hs = card (const_lt_component ` fst ` set hs - {None})"
bn_finite: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and bn :: "'b \<Rightarrow> atom set"   and \<alpha> :: "'b" assumes "class.bn permute bn" shows "finite (bn \<alpha>)"
fbox_simp: fixes x :: "'a"   and y :: "'a" shows " |x] y =  |x] d y"
lookupd_unicity: fixes y :: "name"   and a :: "coname"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and b :: "coname" assumes "lookupd y a \<theta>_c = Ax y b"   and "b \<sharp> \<theta>_c" shows "a = b"
dyn_equiv_stat: shows "dynamically_complete_calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> local.Red_F_\<G> = statically_complete_calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> local.Red_F_\<G>"
possible_blues_seen_2: fixes n :: "nat"   and p :: "'person"   and w :: "'person \<Rightarrow> 'color"   and w' :: "'person \<Rightarrow> 'color"   and p' :: "'person" assumes "local.possible n p w w'"   and "w p' = blue"   and "p \<noteq> p'"   and "w' p \<noteq> blue" shows "card (local.blues_seen w p) = Suc (card (local.blues_seen w' p'))"
sp_imp_ex_se_star_succ: fixes rv1 :: "'a \<times> nat"   and res :: "('a \<times> nat) edge list"   and rv2 :: "'a \<times> nat" assumes "RedBlack prb"   and "SubRel.subpath (red prb) rv1 res rv2 (subs prb)" shows "\<exists>c. se_star (confs prb rv1) (trace (ui_es res) (labelling (black prb))) c"
le_cases3: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.linorder less_eq less" obtains "less_eq x y" and "less_eq y z"   | "less_eq y x" and "less_eq x z"   | "less_eq x z" and "less_eq z y"   | "less_eq z y" and "less_eq y x"   | "less_eq y z" and "less_eq z x"   | "less_eq z x" and "less_eq x y"
Lambert_W_le_minus_one: fixes x :: "real" assumes "x \<le> - exp (- 1)" shows "Lambert_W x = - 1"
member_rec_2: fixes y :: "'d" shows "List.member [] y = False"
bres_galois_imp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<le> z" shows "y \<le> x \<rightarrow> z"
Rep_perm_inverse: fixes x :: "perm" shows "Abs_perm (Rep_perm x) = x"
trans_def: fixes r :: "('a \<times> 'a) set" shows "trans r = (\<forall>x y z. x \<succeq>[r] y \<longrightarrow> y \<succeq>[r] z \<longrightarrow> x \<succeq>[r] z)"
simps_2: fixes a :: "'a"   and as :: "'a list"   and b :: "'a"   and x :: "'a" shows "nextElem (a # as) b x = (if x = a then case as of [] \<Rightarrow> b | a' # as' \<Rightarrow> a' else nextElem as b x)"
delta': fixes S :: "'b set"   and a :: "'b"   and b :: "'b \<Rightarrow> 'c" assumes "finite S" shows "(\<Prod>k\<in>S. if a = k then b k else (1::'c)) = (if a \<in> S then b a else (1::'c))"
lits_subseteq_imp_atms_subseteq: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "set_mset C \<subseteq> set_mset D" shows "atms_of C \<subseteq> atms_of D"
swap_commute: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "Fun.swap a b f = Fun.swap b a f"
insert_dens_def: fixes V :: "nat set"   and V' :: "nat set"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" shows "insert_dens V V' f \<delta> \<equiv> \<lambda>\<sigma>. \<delta> (remove_var \<sigma>) * f (remove_var \<sigma>) (\<sigma> 0)"
LEAST_FP_def: fixes L :: "('c, 'd) gorder_scheme"   and f :: "'c \<Rightarrow> 'c" shows "LFP\<^bsub>L\<^esub> f = \<Sqinter>\<^bsub>L\<^esub>{u \<in> carrier L. f u \<sqsubseteq>\<^bsub>L\<^esub> u}"
f_nconst: shows "\<exists>\<^sub>F w in at z. f w \<noteq> 0"
carrier_polynomial: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (poly_ring R)"   and "polynomial\<^bsub>poly_ring R\<^esub> K p" shows "polynomial\<^bsub>poly_ring R\<^esub> (carrier (poly_ring R)) p"
all_closedin_homeomorphic_image: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and P :: "'d set \<Rightarrow> bool" assumes "homeomorphic_map X Y f" shows "(\<forall>V. closedin Y V \<longrightarrow> P V) = (\<forall>U. closedin X U \<longrightarrow> P (f ` U))"
cardSuc_finite: fixes r :: "('a \<times> 'a) set" assumes "Card_order r" shows "finite (Field (cardSuc r)) = finite (Field r)"
AL_delete_idempotent: fixes M :: "('a \<times> 'b) list"   and x :: "'a" shows "delete M x = delete (delete M x) x"
ext_inject: fixes c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e"   and c_records' :: "'d zmultiset"   and c_temp' :: "'c \<Rightarrow> 'd zmultiset"   and c_msg' :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob' :: "'c \<Rightarrow> 'd zmultiset"   and more' :: "'e" shows "(\<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = \<lparr>c_records = c_records', c_temp = c_temp', c_msg = c_msg', c_glob = c_glob', \<dots> = more'\<rparr>) = (c_records = c_records' \<and> c_temp = c_temp' \<and> c_msg = c_msg' \<and> c_glob = c_glob' \<and> more = more')"
WT_elim_cases_9: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and C :: "char list"   and M :: "char list"   and ps :: "expr list"   and T :: "ty" assumes "P,E \<turnstile> Call e \<lfloor>C\<rfloor> M ps :: T" obtains C' :: "char list" and Ts :: "ty list" and m :: "char list list \<times> expr" and Cs :: "char list list" and Ts' :: "ty list" where "P,E \<turnstile> e :: Class C'" and "P \<turnstile> Path C' to C unique" and "P \<turnstile> C has least M = (Ts, T, m) via Cs" and "P,E \<turnstile> ps [::] Ts'" and "P \<turnstile> Ts' [\<le>] Ts"
cod_char: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "dual_category C" shows "partial_magma.cod (dual_category.comp C) f = partial_magma.dom C f"
Pi_split_domain: fixes x :: "'g \<Rightarrow> 'h"   and I :: "'g set"   and J :: "'g set"   and X :: "'g \<Rightarrow> 'h set" shows "(x \<in> Pi (I \<union> J) X) = (x \<in> Pi I X \<and> x \<in> Pi J X)"
l11: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - y = - (- - x \<sqinter> - - y)"
filters_filtered: fixes X :: "'a set" assumes "filterp X" shows "filtered X"
simps: fixes z :: "name"   and t :: "tm"   and u :: "tm" assumes "atom z \<sharp> (t, u)" shows "t SUBS u = All2 z t (Var z IN u)"
can_Ide_self: 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 t :: "'b bicategorical_language.term" assumes "bicategory V H \<a> \<i> src trg"   and "bicategorical_language.Ide V src trg t" shows "bicategory.can V H \<a> \<i> src trg t t = Coherence.evaluation_map.eval V H \<a> \<i> src trg (\<lambda>\<mu>. if partial_magma.arr V \<mu> then \<mu> else partial_magma.null V) t"
emeasure_spmf_single: fixes p :: "'a spmf"   and x :: "'a" shows "emeasure (measure_spmf p) {x} = ennreal (spmf p x)"
updEnv_commute: fixes xs :: "'varSort"   and ys :: "'varSort"   and x :: "'var"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "xs \<noteq> ys \<or> x \<noteq> y" shows "rho [x \<leftarrow> X]_xs [y \<leftarrow> Y]_ys = rho [y \<leftarrow> Y]_ys [x \<leftarrow> X]_xs"
q_has_vector_derivative: fixes t :: "real"   and u :: "real set" shows "(ego.q has_vector_derivative ego2.q' t) (at t within u)"
simps_18: fixes x41 :: "nat"   and x42 :: "recf"   and x43 :: "recf list" shows "Cn x41 x42 x43 \<noteq> S"
tnil: fixes b :: "'a" shows "monotone orda ordb (\<lambda>f. tnil f b)"
lspasl_e_der: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a" assumes "h1,h2\<triangleright>h0" shows "(h1,h2\<triangleright>h0) \<and> (h2,h1\<triangleright>h0)"
terminates_zip_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and h :: "'g \<Rightarrow> ('h, 'g) step" assumes "terminates g"   and "terminates h" shows "terminates (zip_raw g h)"
map_semantics_ltlc_aux: fixes f :: "'b \<Rightarrow> 'c"   and APs :: "'b set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc" assumes "inj_on f APs"   and "\<Union> (range w) \<subseteq> APs"   and "atoms_ltlc \<phi> \<subseteq> APs" shows "w \<Turnstile>\<^sub>c \<phi> = (\<lambda>i. f ` w i) \<Turnstile>\<^sub>c map_ltlc f \<phi>"
of_int_1: shows "of_int 1 = (1::'b)"
simple_path_sym: fixes p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "pathfinish p = pathstart q"   and "pathfinish q = pathstart p" shows "simple_path (p +++ q) = simple_path (q +++ p)"
run_fail_option: fixes return :: "'b option \<Rightarrow> 'a" shows "run_option (fail_option return) = return None"
aforms_err_Nil: fixes e :: "nat \<Rightarrow> real" shows "aforms_err e [] = {[]}"
thmD12: fixes k :: "nat"   and u :: "nat" assumes "items_le k (\<J> k u) = Gen (paths_le k (\<P> k u))"   and "\<T> k u = \<Z> k u" shows "Gen (paths_le k (\<P> k (Suc u))) \<subseteq> items_le k (\<J> k (Suc u))"
num_syms_subst: fixes t :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term" shows "num_syms t \<le> num_syms (t \<cdot> \<sigma>)"
prj_in_hom_2: shows "\<guillemotleft>\<nu>\<pi>\<rho>.Prj\<^sub>0\<^sub>1 : HH\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<pi>.apex\<guillemotright>"
FinGroupRepresentation_axioms_def: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" shows "FinGroupRepresentation_axioms G smult V \<equiv> of_nat (card G) \<noteq> (0::'b) \<and> (\<exists>ms. set ms \<subseteq> V \<and> R_scalar_mult.RSpan UNIV (aezfun_scalar_mult.fsmult smult) ms = V)"
enables_def: fixes I :: "infrastructure"   and l :: "location"   and a :: "actor"   and a' :: "action" shows "enables I l a a' \<equiv> \<exists>(p, e)\<in>delta I (graphI I) l. a' \<in> e \<and> p a"
domains_def: fixes f :: "'a" shows "C.domains f \<equiv> {a. C.ide a \<and> f \<cdot>\<^sub>C a \<noteq> C.null}"
fgmodule_carr: fixes R :: "('f, 'g) Ring_scheme"   and A :: "'e set"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'f \<Rightarrow> 'e \<Rightarrow> 'e" shows "carrier (fgmodule R A z i f s) = addition_set f (aug_pm_set z i (s_set R s A))"
dk: fixes k :: "nat" assumes "k < n" shows "adjoint (M k) * P k * M k \<in> carrier_mat d d"
IC_rate1: fixes i :: "nat" shows "rho_bound1 (\<lambda>p. IC p i)"
tensor\<^sub>F\<^sub>M\<^sub>C_def: fixes f :: "'c local.term set"   and g :: "'c local.term set" shows "f \<otimes> g \<equiv> if \<F>C.arr f \<and> \<F>C.arr g then \<F>C.mkarr (\<F>C.rep f \<^bold>\<otimes> \<F>C.rep g) else \<F>C.null"
degree_zero_imp_splitted: fixes p :: "'a list list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "degree p = 0" shows "UP.splitted p"
elims_2: fixes x :: "nat \<times> nat list"   and xa :: "nat \<times> cell list \<times> cell list"   and xb :: "cell list" assumes "inv_on_left_moving x xa xb" obtains as :: "nat" and lm :: "nat list" and s :: "nat" and l :: "cell list" and r :: "cell list" and ires :: "cell list" where "x = (as, lm)" and "xa = (s, l, r)" and "xb = ires" and "inv_on_left_moving_norm (as, lm) (s, l, r) ires \<or> inv_on_left_moving_in_middle_B (as, lm) (s, l, r) ires"
arev_arev: fixes a :: "'b" shows "rev_H (rev_H a) = a"
simps_2: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and u :: "'a"   and l :: "enat"   and ls :: "enat list"   and v :: "'a" shows "WGraph.path w u (l # ls) v = (\<exists>uh. l = w (u, uh) \<and> WGraph.path w uh ls v)"
not_le: fixes x :: "'t"   and y :: "'t" shows "(\<not> x \<preceq>\<^sub>t y) = (y \<prec>\<^sub>t x)"
map2_nth: fixes n :: "nat"   and xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "n < length xs"   and "n < length ys" shows "map2 f xs ys ! n = f (xs ! n) (ys ! n)"
promela_props_def: fixes g :: "gState" shows "promela_props g = {e. exprArith g emptyProc e \<noteq> 0}"
infinite_var_regular_INNER: shows "infinite UNIV \<and> regular |UNIV|"
assms_12: shows "InvariantNoDecisionsWhenUnit [getC state] (getM state) (getBackjumpLevel state)"
collapse_1: fixes step :: "('c, 'd) step" assumes "is_Skip step" shows "Skip (un_Skip step) = step"
standard_simplex_0: shows "standard_simplex 0 = {\<lambda>j. if j = 0 then 1 else 0}"
partial_order_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
only_one_SOME_dist_edge: fixes a :: "'edge"   and n' :: "'node"   and x :: "nat" assumes "valid_edge a"   and "intra_kind (kind a)"   and "distance (targetnode a) n' x" shows "\<exists>!a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') n' x \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = (SOME nx. \<exists>a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') n' x \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = nx)"
is_functor: shows "functor (\<cdot>\<^sub>A\<^sup>o\<^sup>p) (\<cdot>\<^sub>A\<^sup>o\<^sup>p) Aop.map"
parts_mpair_1: fixes X :: "msg"   and Y :: "msg" shows "parts {\<lbrace>X, Y\<rbrace>} \<subseteq> insert \<lbrace>X, Y\<rbrace> (parts {X} \<union> parts {Y})"
P11: shows "B' Midpoint C' D'' \<and> Cong B C' B D''"
inf_absorb1: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "x \<subseteq># y" shows "x \<inter># y = x"
zero_less_divide_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide one a) = less zero a"
card_lists_length_eq: fixes B :: "'b set"   and n :: "nat" shows "card {xs. set xs \<subseteq> B \<and> length xs = n} = card B ^ n"
borel_measurable_iff_ge: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>a. {w \<in> space M. a \<le> f w} \<in> sets M)"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (x OR xa) = int_of_integer x OR int_of_integer xa"
conductor_leI: fixes d :: "nat" assumes "induced_modulus d" shows "conductor \<le> d"
node_order_is_total: fixes e1 :: "'evt"   and i :: "nat"   and e2 :: "'evt" assumes "e1 \<in> set (history i)"   and "e2 \<in> set (history i)"   and "e1 \<noteq> e2" shows "e1 \<sqsubset>\<^sup>i e2 \<or> e2 \<sqsubset>\<^sup>i e1"
refine: fixes mrel :: "('c \<times> 'd) set"   and node_rel :: "('e \<times> 'f) set"   and node_eq_impl :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'e \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('e \<Rightarrow> bool, 'e \<Rightarrow> 'e list, 'e list, 'e \<Rightarrow> integer, 'c) gen_igbg_impl_scheme"   and G :: "('f, 'd) igb_graph_rec_scheme" assumes "impl_lasso_loc mrel node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G" shows "(find_lasso_code node_eq_impl node_hash_impl node_def_hash_size G_impl, igb_fr_graph.find_lasso G) \<in> \<langle>fl_rel node_rel\<rangle>nres_rel"
elims_2: fixes x :: "label"   and xa :: "label" assumes "x \<sqsubset>l xa" obtains l :: "label" where "x = Old" and "xa = l__" and "l__ \<noteq> Old"   | l :: "label" where "x = Processed" and "xa = l" and "l = New"
starfun_pow: fixes r :: "real"   and N :: "nat star" shows "(*f* (^) r) N = hypreal_of_real r pow N"
axioms: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" assumes "ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
pairs_inf_closed: fixes x :: "'b regular"   and y :: "'b dense Filters.filter"   and z :: "'b regular"   and w :: "'b dense Filters.filter" assumes "(x, y) \<in> stone_phi.pairs"   and "(z, w) \<in> stone_phi.pairs" shows "stone_phi.pairs_inf (x, y) (z, w) \<in> stone_phi.pairs"
o_assoc: fixes f :: "'e \<Rightarrow> 'b"   and g :: "'f \<Rightarrow> 'e"   and h :: "'a \<Rightarrow> 'f" shows "f \<circ> (g \<circ> h) = f \<circ> g \<circ> h"
perron_frobenius_def: fixes A :: "((real, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "perron_frobenius A \<equiv> pf_nonneg_mat A \<and> perron_frobenius_axioms A"
consistent_def: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>s\<^sub>y\<^sub>m :: "'c \<times> nat \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" shows "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m s \<equiv> \<forall>v. \<sigma>\<^sub>s\<^sub>y\<^sub>m (symvar v s) = \<sigma> v"
power2_commute: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "power.power one times (minus x y) 2 = power.power one times (minus y x) 2"
prems_1: fixes i :: "nat" assumes "find_first x (y # ys) = Some i"   and "i < n" shows "thesis"
dist_commute: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open" shows "dist x y = dist y x"
poly_lift_0_iff: fixes p :: "'a poly" shows "(poly_lift p = 0) = (p = 0)"
H: shows "\<exists>hs. rev v @ x # h = [y, y] @ hs"
token_succeeds_def: fixes x :: "nat" shows "\<MM>.token_succeeds x = (\<exists>n. \<MM>.token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
pm_of_poly_pCons: fixes x :: "'c"   and c :: "'d"   and p :: "'d poly" shows "pm_of_poly x (pCons c p) = monomial c 0 + punit.monom_mult (1::'d) (monomial 1 x) (pm_of_poly x p)"