hcju/isabelleps huggingface dataset:
signed_less_numeral_iff: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and w :: "'b Word.word"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.signed one plus zero minus w) (numeral.numeral one plus n) = (sint w < numeral n)"
counit_in_vhom: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" assumes "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "partial_magma.in_hom V \<epsilon> (H f g) (src g)"
simps_4: fixes x :: "((int \<times> int) set set, int) Combinators"   and y :: "((int \<times> int) set set, int) Combinators" shows "CRotate (x \<oplus> y) = CRotate x \<Oplus> CRotate y"
wellformed_token_def: 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 t :: "symbol \<times> character list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.wellformed_token \<TT> t = CFG.is_terminal \<TT> (terminal_of_token t)"
analz_Un_HashSet: fixes H :: "msg set"   and G :: "msg set" assumes "Enc_keys_clean H"   and "G \<subseteq> - payload" shows "analz (HashSet G \<union> H) \<subseteq> HashSet G \<union> analz H"
list_of_dnelist_cases: fixes y :: "'a list" assumes "y \<in> {xs. distinct xs \<and> xs \<noteq> []}" obtains x :: "'a dnelist" where "y = list_of_dnelist x"
bij_betwI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> A"   and "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x"   and "\<And>y. y \<in> B \<Longrightarrow> f (g y) = y" shows "bij_betw f A B"
F1'_2: shows "set (map of_int_hom.vec_hom fs') \<subseteq> Rn"
gcd_impl_rec_def: shows "gcd_impl_rec = subresultant_prs_main_impl fst"
eq_vecI: fixes w :: "'b Matrix.vec"   and v :: "'b Matrix.vec" assumes "\<And>i. i < dim_vec w \<Longrightarrow> v $v i = w $v i"   and "dim_vec v = dim_vec w" shows "v = w"
d_ipurge_unwinding: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd" assumes "deterministic P" shows "secure P I D = d_weakly_future_consistent P I D (rel_ipurge P I D)"
ch2ch_snd: fixes Y :: "nat \<Rightarrow> 'c \<times> 'd" assumes "chain Y" shows "chain (\<lambda>i. snd (Y i))"
next_tame0_def: fixes p :: "nat"   and g :: "graph" shows "next_tame0\<^bsub>p\<^esub> g \<equiv> let fs = nonFinals g in if fs = [] then [] else let f = minimalFace fs; v = minimalVertex g f in concat (map (\<lambda>i. generatePolygonTame i v f g) (polysizes p g))"
subset_eq_imp_le_zmset: fixes M :: "'b zmultiset"   and N :: "'b zmultiset" assumes "M \<subseteq>#\<^sub>z N" shows "M \<le> N"
poly_subst: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and \<sigma> :: "'e \<Rightarrow> ('d monom \<times> 'c) list"   and p :: "('e monom \<times> 'c) list" shows "eval_poly \<alpha> (poly_subst \<sigma> p) = eval_poly (\<lambda>v. eval_poly \<alpha> (\<sigma> v)) p"
AI_def: fixes c :: "com" shows "AI c = pfp (step' \<top>) (bot c)"
assms_3: shows "ht_hash (l[bounded_hashcode_nat (length l) k := fst (ls_delete k (l ! bounded_hashcode_nat (length l) k))])"
af_append_congruent: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool"   and \<phi> :: "'a ltln"   and w :: "'a set list"   and \<psi> :: "'a ltln"   and w' :: "'a set list" assumes "af_congruent eq"   and "eq (af \<phi> w) (af \<psi> w)" shows "eq (af \<phi> (w @ w')) (af \<psi> (w @ w'))"
axioms_2: fixes step1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and step2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and final1 :: "'d \<Rightarrow> bool"   and final2 :: "'e \<Rightarrow> bool"   and load1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and load2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and order :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and match :: "'h \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and compile :: "'f \<Rightarrow> 'g option" assumes "compiler step1 step2 final1 final2 load1 load2 order match compile" shows "language step2 final2"
sel_3: fixes x21 :: "(Term_Class.term \<times> sterm) list"   and x22 :: "(name, value) fmap" shows "un_Vabs1 (Vabs x21 x22) = x21"
effect_apply_eqvt_aux: fixes p :: "perm" shows "p \<bullet> effect_apply = effect_apply"
simps_5: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi>\<^sub>1 :: "'a formula"   and \<phi>\<^sub>2 :: "'a formula" shows "((w, I) \<Turnstile> FOr \<phi>\<^sub>1 \<phi>\<^sub>2) = ((w, I) \<Turnstile> \<phi>\<^sub>1 \<or> (w, I) \<Turnstile> \<phi>\<^sub>2)"
holomorphic_on_extend_bounded: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex" assumes "f holomorphic_on S - {\<xi>}"   and "\<xi> \<in> interior S" shows "(\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S - {\<xi>}. g z = f z)) = (\<exists>B. \<forall>\<^sub>F z in at \<xi>. cmod (f z) \<le> B)"
tick_def: fixes v :: "'a" shows "tick v = TM v 1"
fdia_def: fixes x :: "'a"   and y :: "'a" shows " |x\<rangle> y = ad (ad (x \<cdot> y))"
in_Inf_from_imp_to_F_in_Inf_from: fixes \<iota> :: "('f \<times> 'l) inference"   and N :: "('f \<times> 'l) set" assumes "\<iota> \<in> Inf_from N" shows "to_F \<iota> \<in> no_labels.Inf_from (fst ` N)"
rep_eq: fixes x :: "'d set" shows "space_as_set (ccspan x) = closure (cspan x)"
lesso_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f <o g = (\<lambda>x. max (f x - g x) (0::'b))"
HStartBallot_HInv4c_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "HInv4c s q"   and "p \<noteq> q" shows "HInv4c s' q"
span_superset: fixes S :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "S \<subseteq> pmdl S"
that: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "x \<noteq> y" shows "thesis"
map_poly_pCons_hom: fixes a :: "'c"   and p :: "'c poly" shows "map_poly (\<lambda>a. [:a:]) (pCons a p) = pCons [:a:] (map_poly (\<lambda>a. [:a:]) p)"
iSince_imp: fixes I :: "nat set"   and P1 :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool"   and P2 :: "nat \<Rightarrow> bool" assumes "P1 t1. t1 \<S> t2 I. Q t2"   and "\<box> t I. P1 t \<longrightarrow> P2 t" shows "P2 t1. t1 \<S> t2 I. Q t2"
map\<^sub>0_in_hom_2: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<rho>\<^sub>0 a : \<rho>\<^sub>0 a \<Rightarrow>\<^sub>D \<rho>\<^sub>0 a\<guillemotright>"
hcomp_simps_4: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.hseq \<nu> \<mu>" shows "D.cod (\<nu> \<star>\<^sub>D \<mu>) = D.cod \<nu> \<star>\<^sub>D D.cod \<mu>"
null_sets_outer_le: fixes S :: "'a set"   and M :: "'a measure" shows "(S \<in> null_sets (completion M)) = (\<forall>e>0. \<exists>T\<in>fmeasurable (completion M). S \<subseteq> T \<and> Sigma_Algebra.measure (completion M) T \<le> e)"
keys_minus: fixes a :: "'b \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.keys (- a) = Poly_Mapping.keys a"
eval_finalId: fixes e :: "expr"   and 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 s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "final e" shows "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e,s\<rangle>"
simps_2: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and l :: "'c btree"   and a :: "'c"   and r :: "'c btree" assumes "split split" shows "height_up\<^sub>i (Up\<^sub>i l a r) = max (height l) (height r)"
eval\<^sub>w_def: shows "eval\<^sub>w \<equiv> {(((xa, x), xb), (xd, xc), xe). eval\<^sub>wp xa x xb xd xc xe}"
ActualDisjI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<psi> in v]" shows "[\<^bold>\<A>(\<phi> \<^bold>\<or> \<psi>) in v]"
components_are_iso: fixes a :: "'a \<times> 'a \<times> 'a" assumes "C.CCC.ide a" shows "C.iso (C.\<alpha>' a)"
iv1: shows "height_tree (Node e1 a1 r ts1) = r"
better_worse_good_mix_preferred: fixes b :: "'a pmf"   and w :: "'a pmf"   and \<alpha> :: "real"   and \<beta> :: "real" assumes "b \<succeq>[\<R>] w"   and "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "\<beta> \<le> \<alpha>" shows "mix_pmf \<alpha> b w \<succeq>[\<R>] mix_pmf \<beta> b w"
ps2_P_d2: shows "ps2_P.d2 = K"
that: fixes h :: "nat \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "\<And>p. h p 0 = 0"   and "\<And>p c1 c2. h p (c1 - c2) = h p c1 - h p c2"   and "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (h p c)"   and "\<And>p X c. singular_chain p X c \<Longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c) = singular_subdivision p c - c" shows "thesis"
kuratowski_def: fixes X :: "'a set" shows "kuratowski X = (\<not> extremally_disconnected X \<and> \<not> open_unresolvable X)"
inverse_nonzero_iff_nonzero: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(inverse a = zero) = (a = zero)"
distinct_2: fixes name1 :: "name"   and name2 :: "name" shows "\<tau> \<noteq> name1[name2]"
two_chain_valid_valid_cubes: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeI_chain i j F two_chain s" shows "\<forall>two_cube\<in>two_chain. valid_two_cube two_cube"
is_invI_2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "a \<otimes> b = \<one>" shows "inv a = b"
vector_add_component: fixes x :: "('d, 'e) vec"   and y :: "('d, 'e) vec"   and i :: "'e" shows "(x + y) $ i = x $ i + y $ i"
OrdP_linear_lemma: fixes j :: "name"   and i :: "name" assumes "atom j \<sharp> i" shows "{OrdP (Var i)} \<turnstile> SyntaxN.All j (OrdP (Var j) IMP Var i IN Var j OR Var i EQ Var j OR Var j IN Var i)"
segment_a_a'_ne': shows "segment G.face_cycle_succ a a' \<noteq> {}"
ldistinct_lfilterE: fixes P :: "'b \<Rightarrow> bool"   and lst :: "'b llist"   and e :: "'b"   and i :: "nat"   and j :: "nat" assumes "ldistinct (lfilter P lst)"   and "e = lst $ i"   and "e = lst $ j"   and "enat i < llength lst"   and "enat j < llength lst"   and "P e" shows "i = j"
all_ideals: fixes R :: "('e, 'f) ring_scheme" assumes "field R" shows "{I. ideal I R} = {{\<zero>\<^bsub>R\<^esub>}, carrier R}"
ramanujan_sum_def: fixes k :: "nat"   and n :: "nat" shows "c k n = (\<Sum>m | m \<in> {1..k} \<and> coprime m k. unity_root k (int (m * n)))"
independent_and_consistent_imp_card_1: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "vec.dim (solution_set A 0) = 0"   and "consistent A b" shows "card (solution_set A b) = 1"
subgroup_axioms: shows "subgroup (\<eta> ` R) R' (+') \<zero>'"
fst_eqvt: fixes p :: "perm"   and x :: "'a \<times> 'b" shows "p \<bullet> fst x = fst (p \<bullet> x)"
assms_2: shows "(\<box>(\<lambda>s. Q s \<longrightarrow> (\<diamond>R) s)) \<sigma>"
VUnionI: fixes x :: "V"   and A :: "V"   and a :: "V" assumes "x \<in>\<^sub>\<circ> A"   and "a \<in>\<^sub>\<circ> x" shows "a \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> A"
eq_span_insert_eq: fixes x :: "'a"   and y :: "'a"   and S :: "'a set" assumes "x - y \<in> ideal S" shows "ideal (insert x S) = ideal (insert y S)"
le_add_same_cancel1: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subseteq># a + b) = ({#} \<subseteq># b)"
assms_4: shows "vfinite (\<D>\<^sub>\<circ> r)"
Sup_mult_distr: fixes X :: "'a set"   and y :: "'a" shows "Sup X \<cdot> y = Sup {x \<cdot> y |x. x \<in> X}"
term_variants_pred_inv_Var_1: fixes P :: "'a \<Rightarrow> 'a list"   and x :: "'b"   and t :: "('a, 'b) Term.term" shows "term_variants_pred P (Var x) t = (t = Var x)"
of_nat_times_\<omega>_exp: fixes n :: "nat"   and m :: "hmultiset" shows "of_nat n * \<omega>^ m = HMSet (replicate_mset n m)"
derivation_infl: shows "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"
WTrtFAcc: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P,E,h,sh \<turnstile> e : Class C"   and "P \<turnstile> C has F,NonStatic:T in D" shows "P,E,h,sh \<turnstile> e\<bullet>F{D} : T"
GF_advice_a2: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "X \<subseteq> \<G>\<F> \<phi> w"   and "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>" shows "w \<Turnstile>\<^sub>n \<phi>"
Sbis_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> Sretr (theta \<union> Sbis)" shows "theta \<subseteq> Sbis"
psubstt_closed_2: fixes i :: "nat"   and f :: "'b \<Rightarrow> 'a"   and ts :: "'b FOL_Fitting.term list" shows "closedts i (psubstts f ts) = closedts i ts"
atms_empty_progress: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat" assumes "safe_regex m g r"   and "atms r = {}" shows "progress_regex \<sigma> P r j = j"
LT18: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and B :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and C :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and D :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (A \<leadsto> B \<or> C) \<longrightarrow> (B \<leadsto> D) \<longrightarrow> (C \<leadsto> D) \<longrightarrow> A \<leadsto> D"
MN_imp_dfa: fixes L :: "'b list set"   and R :: "('b list \<times> 'b list) set" assumes "MyhillNerode L R" obtains M :: "'b dfa" where "dfa M" and "dfa.language M = L" and "card (dfa.states M) = card (UNIV // R)"
valid: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> cf.E" shows "l u \<le> l v + 1"
coprime_normalize_right_iff: fixes a :: "'a"   and b :: "'a" shows "coprime a (normalize b) = coprime a b"
r_inv: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<oplus> \<ominus> x = \<zero>"
fetch_action_eq: fixes b :: "cell"   and r :: "nat"   and tp :: "(action \<times> nat) list"   and st :: "nat"   and nact :: "action"   and ns :: "nat" assumes "block_map b = scan r"   and "fetch tp st b = (nact, ns)"   and "st \<le> length tp div 2" shows "actn (code tp) st r = action_map nact"
bezout_coefficients_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b = fst (euclidean_ring_gcd.euclid_ext_aux divide minus zero times one modulo unit_factor normalize one zero zero one a b)"
one: fixes R :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "\<one>\<^bsub>R\<^esub> \<in> generate_ring R H"
flip_altsublist_adjacentI: fixes ss :: "'w list"   and as :: "'w list"   and s :: "'w"   and t :: "'w"   and bs :: "'w list"   and ts :: "'w list" assumes "ss = as @ pair_relator_halflist s t @ bs"   and "ts = as @ pair_relator_halflist t s @ bs" shows "flip_altsublist_adjacent ss ts"
simps_5: fixes b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "atom_com (Cond b c\<^sub>1 c\<^sub>2) = (atom_com c\<^sub>1 \<and> atom_com c\<^sub>2)"
FS_FMT_1: shows "eel = LOC (LCall p # w)"
op3_def: shows "op3 \<equiv> prod.fst (prod.snd (fix_syn (\<lambda>(root3, op3, rep3, grep3). (\<Lambda> pat. rep3\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root3\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op3\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. grep3\<cdot>pat\<cdot>(case cfst\<cdot>usvs of [::] \<Rightarrow> Null | u :# us \<Rightarrow> sfoldl\<cdot>(op3\<cdot>pat)\<cdot>(root3\<cdot>pat)\<cdot>us)\<cdot>usvs, \<Lambda> pat l usvs. Node\<cdot>usvs\<cdot>(next\<cdot>(csnd\<cdot>usvs)\<cdot>l)\<cdot>(case csnd\<cdot>usvs of [::] \<Rightarrow> Null | v :# vs \<Rightarrow> rep3\<cdot>pat\<cdot>(cfst\<cdot>usvs :@ [:v:], vs))))))"
bin_rsplit_all: fixes nw :: "nat"   and n :: "nat"   and w :: "int" assumes "0 < nw"   and "nw \<le> n" shows "bin_rsplit n (nw, w) = [take_bit n w]"
untame_negFin: fixes g :: "graph" assumes "Invariants.inv g"   and "final g"   and "tame g" shows "is_tame g"
nth_Cons_simp_2: fixes n :: "Integer"   and x :: "'b"   and xs :: "['b]" assumes "eq\<cdot>n\<cdot>0 = FF" shows "(x : xs) !! n = xs !! (n - 1)"
BS2: fixes f :: "'a"   and g :: "'a" assumes "C.is_left_adjoint f"   and "C.is_left_adjoint g"   and "src\<^sub>C f = src\<^sub>C g" shows "\<exists>r \<rho>. tabulation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C r \<rho> f g"
prod_eq_iff: fixes s :: "'c \<times> 'd"   and t :: "'c \<times> 'd" shows "(s = t) = (fst s = fst t \<and> snd s = snd t)"
syzygy_moduleD: fixes s :: "'b \<Rightarrow>\<^sub>0 'a"   and B :: "'b set" assumes "s \<in> syzygy_module B" shows "represents B s (0::'b)"
if_refine: fixes b :: "bool"   and b' :: "bool"   and S1 :: "'c nres"   and R :: "('c \<times> 'd) set"   and S1' :: "'d nres"   and S2 :: "'c nres"   and S2' :: "'d nres" assumes "b = b'"   and "\<lbrakk>b; b'\<rbrakk> \<Longrightarrow> S1 \<le> \<Down> R S1'"   and "\<lbrakk>\<not> b; \<not> b'\<rbrakk> \<Longrightarrow> S2 \<le> \<Down> R S2'" shows "(if b then S1 else S2) \<le> \<Down> R (if b' then S1' else S2')"
r_find_k_total: shows "Partial_Recursive.total r_find_k"
assoc_subst: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "a \<sim> b"   and "\<forall>a b. a \<in> carrier G \<and> b \<in> carrier G \<and> a \<sim> b \<longrightarrow> f a \<in> carrier G \<and> f b \<in> carrier G \<and> f a \<sim> f b"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "f a \<sim> f b"
ACI_norm_wf: fixes wf_atom :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat"   and r :: "'a rexp" shows "alphabet.wf wf_atom n \<guillemotleft>r\<guillemotright> = alphabet.wf wf_atom n r"
simps_2: fixes a :: "'a"   and v :: "'a"   and va :: "'a list" shows "insertl 0 a (v # va) = a # v # va"
1: shows "(\<lambda>(is1, is2). is1 @ is2) ` ({is1. is1 \<lhd> ds1} \<times> {is2. is2 \<lhd> ds2}) = {is. is \<lhd> ds1 @ ds2}"
HF_axioms_hold: fixes A :: "fm"   and e :: "(name, hf) finfun" assumes "A \<in> HF_axioms" shows "eval_fm e A"
IH: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "hpsafe a"   and "(\<nu>, \<omega>) \<in> prog_sem I a" shows "Vagree \<nu> \<omega> (- BVP a)"
that_1: assumes "c"   and "effect t h h' r" shows "thesis"
c3: fixes x :: "'a set"   and y :: "'a set" assumes "closed_csubspace x"   and "closed_csubspace y"   and "x \<subseteq> y" shows "y \<subseteq> x +\<^sub>M orthogonal_complement x \<inter> y"
rep_eq: fixes x :: "\<o>"   and xa :: "\<o>" shows "eval\<o> (x \<^bold>\<rightarrow> xa) = (\<lambda>s w. s = dj \<and> (eval\<o> x dj w \<longrightarrow> eval\<o> xa dj w) \<or> s \<noteq> dj \<and> I_IMPL s (eval\<o> x s) (eval\<o> xa s) w)"
CHAIN': fixes i :: "nat" shows "(f' i, f' (Suc i)) \<in> S"
mono_mprefix_T_process: fixes P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process"   and A :: "'a set" assumes "\<forall>x. P x \<sqsubseteq>\<^sub>T Q x" shows "Mprefix A P \<sqsubseteq>\<^sub>T Mprefix A Q"
diff_def: fixes M :: "'a multiset"   and S :: "'a set" shows "M -s S = {#x \<in># M. x \<notin> S#}"
c: shows "carrier (V.vs kerT) = {\<zero>\<^bsub>V\<^esub>}"
subcls1_induct: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and Q :: "String.literal \<Rightarrow> bool" assumes "wf_prog wf_md P"   and "is_class P C"   and "Q Object"   and "\<And>C D. \<lbrakk>C \<noteq> Object; P \<turnstile> C \<prec>\<^sup>1 D; is_class P D; Q D\<rbrakk> \<Longrightarrow> Q C" shows "Q C"
assms_1: shows "B \<subseteq> U\<^sub>M\<^sub>_\<^sub>1"
assms_18: shows "InvariantClCurrentLevel (getCl state) (getM state)"
countable_image_inj_on: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "countable (f ` A)"   and "inj_on f A" shows "countable A"
domains_char: fixes f :: "'c" shows "Cop.domains f = C.codomains f"
TERMINAL_def: fixes a :: "'h" shows "TERMINAL a = In0 (Old_Datatype.Leaf (Inr (Inr a)))"
gcd_proj2_iff: fixes m :: "'b"   and n :: "'b" shows "(normalization_euclidean_semiring_class.gcd m n = normalize n) = (n dvd m)"
isDiagonal_diagonal_to_Smith_aux: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "is_bezout_ext bezout"   and "isDiagonal A"   and "k < min (nrows A) (ncols A)" shows "isDiagonal (diagonal_to_Smith_aux A [0..<k] bezout)"
rep_eq: fixes x :: "('a, 'b) mapping"   and xa :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and xb :: "'a set" shows "mapping.rep (upd_keys_on x xa xb) = (\<lambda>a. case Mapping.lookup x a of None \<Rightarrow> None | Some b \<Rightarrow> Some (if a \<in> xb then xa a b else b))"
divisor_count_upper_bound: fixes n :: "nat" shows "divisor_count n \<le> 2 * nat \<lfloor>sqrt (real n)\<rfloor>"
sup_admS_def: fixes f :: "'a admS"   and g :: "'a admS" shows "sup f g = mklr (unlr f \<union> unlr g)"
ge_gr: fixes A :: "'b set set"   and f :: "'b \<Rightarrow> real" assumes "sigma_algebra A"   and "\<forall>a. {w. a \<le> f w} \<in> A" shows "\<forall>a. {w. a < f w} \<in> A"
int_of_integer_numeral: fixes k :: "num" shows "int_of_integer (numeral k) = numeral k"