hcju/isabelleps huggingface dataset:
\<pi>\<pi>'_1: shows "wfp UNIV \<pi>"
moebius_circline_cmat_cmat_def: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "moebius_circline_cmat_cmat M H = congruence (mat_inv M) H"
bind_eq_Some_conv: fixes f :: "'d option"   and g :: "'d \<Rightarrow> 'c option"   and x :: "'c" shows "(f \<bind> g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
l_inv: fixes x :: "int" assumes "x \<in> carrier \<Z>" shows "\<ominus>\<^bsub>\<Z>\<^esub> x \<oplus>\<^bsub>\<Z>\<^esub> x = \<zero>\<^bsub>\<Z>\<^esub>"
awalk_verts_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "digraph_isomorphism hom"   and "awalk u p v" shows "pre_digraph.awalk_verts (app_iso hom G) (iso_verts hom u) (map (iso_arcs hom) p) = map (iso_verts hom) (awalk_verts u p)"
eventually_rev_mp: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and Q :: "'a \<Rightarrow> bool" assumes "eventually P F"   and "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x" shows "eventually Q F"
pf_def: shows "pf = p div gcd p (pderiv p)"
f: shows "f = (\<lambda>x. if x \<in> B - B' then (THE X. x \<in> X \<and> X \<in> P') \<inter> C else undefined)"
prems: shows "x \<turnstile> xa \<rightarrow> xb"
existing_mutex_not_true: fixes G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" assumes "\<exists>x\<in>set G. \<exists>y\<in>set G. mutex x y" shows "\<not> apply_guards G s"
boolean_axiomsp_boolean_axioms_eq: shows "boolean_axiomsp = (\<lambda>x. x \<in> boolean_axioms)"
assms_3: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "f (p ^ k) = g (p ^ k)"
triangulize_preserves_dimensions_2: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "triangulize sel_fun mf A = (r', A')" shows "dim_col A' = dim_col A"
propa_property2: fixes qs :: "(nat \<times> nat list) list"   and ss :: "nat list list"   and w :: "nat list"   and q :: "nat" assumes "\<forall>(q, \<tau>)\<in>set qs. q < length ss \<and> \<tau> \<in> A"   and "Sorted_Less.sorted w"   and "set ss \<subseteq> A"   and "length ss = n"   and "distinct (map fst qs)"   and "q \<notin> set (map fst qs)"   and "q < length ss" shows "fst (propa f qs ss w) ! q = ss ! q"
complete_iff_height: fixes t :: "'b tree" shows "Tree.complete t = (min_height t = height t)"
list_nil_or_snoc: fixes xs :: "'c list" shows "xs = [] \<or> (\<exists>y ys. xs = ys @ [y])"
fold_product: fixes x :: "'b fset" shows "Finite_Set.fold (\<lambda>x. (\<otimes>) {{|x|}}) {{||}} (fset x) = {x}"
ext_inject: fixes om :: "'b \<Rightarrow> 'd"   and am :: "'c \<Rightarrow> 'e"   and more :: "'f"   and om' :: "'b \<Rightarrow> 'd"   and am' :: "'c \<Rightarrow> 'e"   and more' :: "'f" shows "(\<lparr>om = om, am = am, \<dots> = more\<rparr> = \<lparr>om = om', am = am', \<dots> = more'\<rparr>) = (om = om' \<and> am = am' \<and> more = more')"
update_convs_4: fixes mem' :: "mem list \<Rightarrow> mem list"   and inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.mem_update mem' \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem' mem, globs = globs, \<dots> = more\<rparr>"
inf0: fixes a :: "'d" shows "a \<equiv> a \<sqinter> \<top>"
pfa_append_def: shows "pfa_append \<equiv> \<lambda>(a, n) x. Array.upd n x a \<bind> (\<lambda>_. return (a, n + 1))"
minimal_inD1: fixes P :: "'a set"   and x :: "'a" assumes "minimal_in P x" shows "x \<in> P"
dom_runit: fixes a :: "'a" assumes "ide a" shows "local.dom \<r>[a] = a \<otimes> \<I>"
loop_no_start: fixes p :: "'a" assumes "is_point p" shows "start_points (p ; p\<^sup>T) = 0"
has_disc_negated_disj_split: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and neg :: "bool"   and m :: "'b match_expr" shows "has_disc_negated (\<lambda>a. P a \<or> Q a) neg m = (has_disc_negated P neg m \<or> has_disc_negated Q neg m)"
irsSp_Cons_sum: fixes n'' :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node"   and p :: "'pname" assumes "n'' irs-ns\<rightarrow>\<^sub>d* n'"   and "n s-p\<rightarrow>\<^bsub>sum\<^esub> n''" shows "n irs-n # ns\<rightarrow>\<^sub>d* n'"
fst_def: fixes prod :: "'a \<times> 'b" shows "fst prod = (case prod of (x1, x2) \<Rightarrow> x1)"
disc_eq_case_1: fixes "typ" :: "typ" shows "is_Ty typ = (case typ of Ty uu_ uua_ \<Rightarrow> True | Tv uu_ uua_ \<Rightarrow> False)"
assoc_coherence: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "F \<a>\<^sub>C[f, g, h] \<cdot>\<^sub>D \<Phi> (f \<star>\<^sub>C g, h) \<cdot>\<^sub>D (\<Phi> (f, g) \<star>\<^sub>D F h) = \<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h]"
mat_of_cols_mult_mat_vec_permute_list: fixes f :: "nat \<Rightarrow> nat"   and ss :: "'b vec list"   and v :: "'b list"   and nr :: "nat" assumes "f permutes {..<length ss}"   and "length ss = length v" shows "mat_of_cols nr (permute_list f ss) *\<^sub>v vec_of_list (permute_list f v) = mat_of_cols nr ss *\<^sub>v vec_of_list v"
dim_mono: fixes V :: "('b, 'c) vec set"   and W :: "('b, 'c) vec set" assumes "V \<subseteq> vec.span W" shows "vec.dim V \<le> vec.dim W"
sum0: shows "(\<Sum>k\<in>{0..CARD('a)} - {0} - {CARD('a)}. of_nat (CARD('a) choose k) * x ^ k * y ^ (CARD('a) - k)) = 0"
one_cinner_one: shows "\<langle>1::'b, 1::'b\<rangle> = 1"
has_start_end_points_iff: fixes x :: "'a" shows "has_start_end_points x = (has_start_points x \<and> has_end_points x)"
Rep_Abs_mod: fixes m :: "int" shows "Rep (Abs (m mod int CARD('a))) = m mod int CARD('a)"
jumpF_polyR_def: fixes q :: "real poly"   and p :: "real poly"   and a :: "real" shows "jumpF_polyR q p a = jumpF (\<lambda>x. poly q x / poly p x) (at_right a)"
simps_2: fixes i :: "String.literal"   and h :: "tm"   and t :: "tm list" shows "occurs_in_list i (h # t) = (occurs_in i h \<or> occurs_in_list i t)"
eval_cases_14: 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 e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   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 "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" where "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1__\<rangle>" and "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>1__\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   | s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" where "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>" and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   | e'a :: "expr" where "e' = throw e'a" and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw e'a,s'\<rangle>"
generate_is_subgroup: fixes H :: "'c set" assumes "H \<subseteq> carrier H" shows "subgroup (generate H H) H"
pair_digraph_def: fixes G :: "'b pair_pre_digraph" shows "pair_digraph G \<equiv> pair_fin_digraph G \<and> pair_loopfree_digraph G"
Carmichael_dvd_mono_coprime: fixes m :: "nat"   and n :: "nat" assumes "coprime m n"   and "1 < m"   and "1 < n" shows "Carmichael m dvd Carmichael (m * n)"
gen_mono_on_def: fixes f :: "'c \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'c set" shows "gen_mono_on f R S = (\<forall>i\<in>S. \<forall>j\<in>S. i \<le> j \<longrightarrow> R (f i) (f j))"
StrongEq_L_subst3: fixes P :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "cp P"   and "\<tau> \<Turnstile> x \<triangleq> y" shows "(\<tau> \<Turnstile> P x) = (\<tau> \<Turnstile> P y)"
vproduct_vpair: fixes a :: "V"   and b :: "V"   and f :: "V \<Rightarrow> V" shows "([a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> vproduct (2\<^sub>\<nat>) f) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> f (0\<^sub>\<nat>) \<times>\<^sub>\<circ> f (1\<^sub>\<nat>))"
intros_4: fixes a1 :: "aexp"   and s :: "char list \<Rightarrow> val"   and i1 :: "int"   and a2 :: "aexp"   and i2 :: "int" assumes "taval a1 s (Iv i1)"   and "taval a2 s (Iv i2)" shows "taval (Plus a1 a2) s (Iv (i1 + i2))"
max2_underS: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and z :: "'a Bonk_Schramm_extension_unfolded"   and y :: "'a Bonk_Schramm_extension_unfolded" assumes "x \<in> wo.underS z"   and "y \<in> wo.underS z" shows "wo.max2 x y \<in> wo.underS z"
simps_2: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "max_stutter_sampler \<sigma> (Suc n) = (let prev = max_stutter_sampler \<sigma> n in if \<forall>k>prev. \<sigma> k = \<sigma> prev then Suc prev else LEAST k. prev < k \<and> \<sigma> k \<noteq> \<sigma> prev)"
div_by_0: 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 a :: "'b" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a zero = zero"
tranclp_into_rtranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "r\<^sup>+\<^sup>+ a b" shows "r\<^sup>*\<^sup>* a b"
select_convs_3: fixes precondition_of :: "'a list"   and add_effects_of :: "'a list"   and delete_effects_of :: "'a list"   and more :: "'b" shows "delete_effects_of \<lparr>precondition_of = precondition_of, add_effects_of = add_effects_of, delete_effects_of = delete_effects_of, \<dots> = more\<rparr> = delete_effects_of"
substituitionslemma: fixes E :: "nat rexp"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" shows "L (substL (lang E) \<sigma>) = lang (subst E \<sigma>)"
eval_cases_13: 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\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   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 "P,E \<turnstile> \<langle>e\<^sub>1;; e\<^sub>2,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains v :: "val" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" where "P,E \<turnstile> \<langle>e\<^sub>1,s\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>" and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   | e :: "expr" where "e' = throw e" and "P,E \<turnstile> \<langle>e\<^sub>1,s\<rangle> \<Rightarrow> \<langle>throw e,s'\<rangle>"
da_elim_cases_47: fixes Env :: "env"   and B :: "lname set"   and C :: "qtname"   and sig :: "sig"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Methd C sig\<rangle>\<guillemotright> A" obtains m :: "qtname \<times> methd" where "methd (prg Env) C sig = Some m" and "Env\<turnstile> B \<guillemotright>\<langle>Body (declclass m) (stmt (mbody (mthd m)))\<rangle>\<guillemotright> A"
step: fixes s :: "'s" assumes "guard s"   and "I s" shows "p \<le> spmf (map_spmf (\<lambda>s'. f s' < f s) (body s)) True"
intros_1: fixes x :: "char list" shows "Fvar x => Fvar x"
endpow0: shows "endpow 0 = id \<down> V"
poly_0_coeff_0: fixes p :: "'b poly" shows "poly p (0::'b) = constant_term p"
star_of_simps_9: shows "star_of (0::'a) = 0"
base: fixes stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "\<tau>exec_moves ci P t es h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "Q stk' loc' pc' xcp'"
xs: shows "rev (sort_key snd (map (\<lambda>f. (f, degree f)) xs)) \<noteq> []"
fps_mult_fps_X_nonzero_2: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "f * fps_X \<noteq> 0"
prems_3: shows "pairwise (\<lambda>x y. coprime (card (S x)) (card (S y))) (insert x I)"
pos_invariant_iff_compl_neg_invariant: fixes M :: "'a set" shows "positively_invariant M = negatively_invariant (X - M)"
filter_def: fixes F :: "'b set" shows "ord_class.filter F \<equiv> F \<noteq> {} \<and> (\<forall>x\<in>F. \<forall>y\<in>F. \<exists>z\<in>F. z \<le> x \<and> z \<le> y) \<and> is_up_set F"
A_sub_aug_pm_set: fixes A :: "'e set"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e" shows "A \<subseteq> aug_pm_set z i A"
S_o_alpha_type_scheme: fixes S :: "nat \<Rightarrow> typ"   and alpha :: "nat \<Rightarrow> nat"   and sch :: "type_scheme" shows "$ (S \<circ> alpha) sch = $ S ($ (\<lambda>x. TVar (alpha x)) sch)"
int_o_subst: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm" shows "int \<xi> \<circ> subst \<rho> = int (int \<xi> \<circ> \<rho>)"
xmin_def: shows "xmin = (LEAST x. 0 < x \<and> \<not> (s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x)))"
reachable_target_elim: fixes q :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and p :: "'d" assumes "q \<in> reachable A p" obtains r :: "(('e \<times> 'f) \<times> 'd) list" where "path A r p" and "q = target r p"
corthogonalD: fixes vs :: "'b vec list"   and i :: "nat"   and j :: "nat" assumes "corthogonal vs"   and "i < length vs"   and "j < length vs" shows "(vs ! i \<bullet>c vs ! j = (0::'b)) = (i \<noteq> j)"
mult_dvd_mono: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b"   and "dvd.dvd times c d" shows "dvd.dvd times (times a c) (times b d)"
simps: fixes xs :: "'c list" shows "C_msort xs = (let n = length xs; ys = take (n div 2) xs; zs = drop (n div 2) xs in if n \<le> 1 then 0 else C_msort ys + C_msort zs + C_merge (msort ys) (msort zs))"
fresh_2: fixes a :: "name"   and x2 :: "name"   and x1 :: "name" shows "a \<sharp> x2[x1] = (a \<sharp> x2 \<and> a \<sharp> x1)"
simps_2: fixes v :: "'b" shows "is_disj (Atom v) = is_lit_plus (Atom v)"
xt: fixes x :: "'a" assumes "x \<in> dom" shows "t \<in> existence_ivl0 x"
sgn_div_norm: fixes x :: "'a" shows "sgn x = x /\<^sub>R norm x"
len_lexext_rev_iff_len_lexext: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" shows "len_lexext_rev gt ys xs = len_lexext gt (rev ys) (rev xs)"
NotL_in_SNa: fixes M :: "trm"   and a :: "coname"   and x :: "name" assumes "SNa M" shows "SNa (NotL <a>.M x)"
subgame_def: fixes V' :: "'a set" shows "G'.subgame V' \<equiv> subgame V'\<lparr>verts := V\<^bsub>subgame V'\<^esub> \<inter> V', arcs := Restr E\<^bsub>subgame V'\<^esub> V', player0 := V0\<^bsub>subgame V'\<^esub> \<inter> V'\<rparr>"
intros_3: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and c :: "('a, 'c, 'd) Stmt"   and mu :: "'a ModeUpd"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and E :: "('a, 'c, 'd) Stmt list" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((c, sifum_lang_no_dma.update_modes mu mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" shows "(((sifum_lang_no_dma.cxt_to_stmt E (c@[mu]), mds), mem), (sifum_lang_no_dma.cxt_to_stmt E c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
msgsCJ'_def: fixes m :: "('p, 'v) message" shows "msgsCJ' m = (if m = <pJ, inM False> \<or> m = <pJ, inM True> then 0 else msgs cJ m)"
simps_2: fixes i :: "'b memref"   and "is" :: "'b memref list" shows "instrs (i # is) = (case i of Write\<^sub>s\<^sub>b volatile a sop v A L R W \<Rightarrow> Write volatile a sop A L R W # instrs is | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> Read volatile a t # instrs is | Prog\<^sub>s\<^sub>b p1 p2 list \<Rightarrow> instrs is | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> Ghost A L R W # instrs is)"
p_gt_1: fixes p :: "nat" assumes "euler_witness_context p" shows "1 < p"
integrable_subinterval: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and c :: "real"   and d :: "real" assumes "integrable (lebesgue_on {a..b}) f"   and "{c..d} \<subseteq> {a..b}" shows "integrable (lebesgue_on {c..d}) f"
prj_simps_6: shows "C.dom \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0 = HH\<mu>H\<nu>\<pi>\<rho>.chine"
char_poly_def: fixes A :: "'b mat" shows "char_poly A = Determinant.det (char_poly_matrix A)"
stake_invert_Nil: fixes n :: "nat"   and s :: "'a stream" shows "(stake n s = []) = (n = 0)"
distinct_1: fixes x11 :: "nat"   and x12 :: "nat"   and x21 :: "nat"   and x22 :: "nat" shows "Pkt x11 x12 \<noteq> Newpkt x21 x22"
E: shows "xs @ y # ipurge_tr I D (D y) (zs @ [z]) \<in> traces P"
trans: fixes x :: "('a, 'b) pp \<times> nat"   and y :: "('a, 'b) pp \<times> nat"   and z :: "('a, 'b) pp \<times> nat" assumes "nat_term_compare cmp x y = Lt"   and "nat_term_compare cmp y z = Lt" shows "nat_term_compare cmp x z = Lt"
simps: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list" shows "Rep_Pre M2 M1 vs xs = (\<exists>xs1 xs2. prefix xs1 xs2 \<and> prefix xs2 xs \<and> xs1 \<noteq> xs2 \<and> (\<exists>s2. io_targets M2 (initial M2) (vs @ xs1) = {s2} \<and> io_targets M2 (initial M2) (vs @ xs2) = {s2}) \<and> (\<exists>s1. io_targets M1 (initial M1) (vs @ xs1) = {s1} \<and> io_targets M1 (initial M1) (vs @ xs2) = {s1}))"
is_Done_cases: fixes r :: "('f, 'g, 'h) Resumption.resumption" assumes "Resumption.resumption.is_Done r" obtains x :: "'f" where "r = DONE x"   | "r = ABORT"
simps: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "wpo s t = (case s of Var x \<Rightarrow> (False, case t of Var xa \<Rightarrow> x = xa | Fun g ts \<Rightarrow> (s, t) \<in> UNIV \<and> \<sigma> (g, length ts) = [] \<and> prl (g, length ts)) | Fun f ss \<Rightarrow> if (s, t) \<in> {} then (True, True) else if (s, t) \<in> UNIV then if \<exists>i\<in>set (\<sigma> (f, length ss)). wpo_ns (ss ! i) t then (True, True) else case t of Var x \<Rightarrow> (False, False \<and> False) | Fun g ts \<Rightarrow> case prc (f, length ss) (g, length ts) of (prs, prns) \<Rightarrow> if prns \<and> (\<forall>j\<in>set (\<sigma> (g, length ts)). wpo_s s (ts ! j)) then if prs then (True, True) else let ss' = map ((!) ss) (\<sigma> (f, length ss)); ts' = map ((!) ts) (\<sigma> (g, length ts)); cf = Lex; cg = Lex in if cf = Lex \<and> cg = Lex then lex_ext wpo n ss' ts' else if cf = Mul \<and> cg = Mul then mul_ext wpo ss' ts' else (length ss' \<noteq> 0 \<and> length ts' = 0, length ts' = 0) else (False, False) else (False, False))"
max_coeff_bound: fixes a :: "nat list"   and b :: "nat list"   and xs :: "nat list"   and ys :: "nat list" assumes "hlde a b"   and "(xs, ys) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b" shows "(\<forall>x\<in>set xs. x \<le> maxne0 ys b) \<and> (\<forall>y\<in>set ys. y \<le> maxne0 xs a)"
segment_a'_a_ne': shows "segment G.face_cycle_succ a' a \<noteq> {}"
real_floor_code: fixes x :: "rat" shows "\<lfloor>Ratreal x\<rfloor> = \<lfloor>x\<rfloor>"
asym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "ordering less_eq less"   and "less a b"   and "less b a" shows "False"
P_mod_mult_monomial: fixes I :: "'a set"   and p :: "'a"   and m :: "nat" assumes "ideal S I"   and "p \<in> carrier R" shows "P_mod R S X I p = P_mod R S X I (p \<cdot>\<^sub>r X^\<^bsup>R m\<^esup>)"
between_inner_transitivity: 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 D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B D"   and "Bet B C D" shows "Bet A B C"
wls_fresh_swap_exchange1: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "wls s X" shows "fresh xs x2 (X #[x1 \<and> x2]_xs) = fresh xs x1 X"
prems_2: fixes S' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> U <: S"   and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'"   and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'" shows "P"
eq_conga_out: fixes A :: "'p"   and B :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B A CongA D E F" shows "E Out D F"
prob1_deutsch_algo_balanced: assumes "is_balanced" shows "prob1 2 deutsch_algo 0 = 1"
split_conv: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and a :: "'c"   and b :: "'d" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
of_nat_catalan_closed_form: fixes n :: "nat" shows "of_nat (catalan n) = of_nat (2 * n choose n) / of_nat (Suc n)"
simps_5: fixes C :: "ty"   and D :: "ty"   and X :: "ctrm set" shows "NEGn (C IMP D) X = AXIOMSn (C IMP D) \<union> BINDINGn (C IMP D) X \<union> IMPLEFT (C IMP D) (\<parallel><C>\<parallel>) (\<parallel>(D)\<parallel>)"
intros: fixes f :: "'fun"   and T :: "('fun, 'var) Term.term list" assumes "public f"   and "arity f = length T"   and "\<And>t. t \<in> set T \<Longrightarrow> public_ground_wf_term t" shows "public_ground_wf_term (Fun f T)"
at_least_as_good_def: fixes x :: "'b"   and B :: "'b set"   and P :: "('b \<times> 'b) set" shows "at_least_as_good x B P = {y \<in> B. y \<succeq>[P] x}"
vwb_impl_idem_scene: fixes X :: "'a \<Longrightarrow> 'b" assumes "vwb_lens X" shows "idem_scene \<lbrakk>X\<rbrakk>\<^sub>\<sim>"
A_dim_row_not1: shows "dim_row A \<noteq> 1"
v0: shows "v \<noteq> 0\<^sub>v (m + n)"
max_ipv6_addr_def: shows "max_ipv6_addr \<equiv> ipv6addr_of_nat (2 ^ 128 - 1)"
semiconfluentp_imp_confluentp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "semiconfluentp r" shows "confluentp r"