hcju/isabelleps huggingface dataset:
summable_imp_abs_convergent_prod: fixes f :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>i. norm (f i - (1::'a)))" shows "abs_convergent_prod f"
evnodd_def: fixes A :: "(nat \<times> nat) set"   and b :: "nat" shows "evnodd A b = A \<inter> {(i, j). (i + j) mod 2 = b}"
red_dfs_def: fixes E :: "('a \<times> 'a) set"   and onstack :: "'a set"   and V :: "'a set"   and u :: "'a" shows "red_dfs E onstack V u \<equiv> REC\<^sub>T (\<lambda>D (V, u). let V = insert u V in NDFS_SI_Statistics.vis_red_nres \<bind> (\<lambda>_. FOREACH\<^sub>C (E `` {u}) (\<lambda>brk. brk = None) (\<lambda>t _. if t \<in> onstack then RETURN (red_init_witness u t) else RETURN None) None \<bind> (\<lambda>brk. case brk of None \<Rightarrow> FOREACH\<^sub>C (E `` {u}) (\<lambda>(V, brk). brk = None) (\<lambda>t (V, uu_). if t \<notin> V then D (V, t) \<bind> (\<lambda>(V, brk). RETURN (V, prep_wit_red u brk)) else RETURN (V, None)) (V, None) | Some a \<Rightarrow> RETURN (V, brk)))) (V, u)"
advantage3_def: fixes \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" shows "local.dis_log_alt.advantage3 \<A> \<equiv> spmf (local.dis_log_alt.dis_log3 \<A>) True"
unilateral_substitutes_on_bilateral_substitutes_on: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.unilateral_substitutes_on Xd A f" shows "Contracts.bilateral_substitutes_on Xd A f"
normalized_vec_norm: fixes v :: "complex vec"   and n :: "nat" assumes "v \<in> carrier_vec n"   and "v \<noteq> 0\<^sub>v n" shows "inner_prod (vec_normalize v) (vec_normalize v) = 1"
res_lim_pushforward: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_continuous f"   and "is_Zp_cauchy s"   and "t = alt_seq s" shows "res_lim (f \<circ> t) = f (res_lim t)"
type_conf_Void: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> Void) = (P,E,h \<turnstile> e : Void)"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "f x = f' (p x)"
assms_3: shows "g ` {a..b} \<subseteq> {c..d}"
InvariantVarsQAfterAssertLiteral: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "InvariantConsistent (getM state @ [(literal, decision)])"   and "InvariantUniq (getM state @ [(literal, decision)])"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantVarsQ (getQ state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl" shows "let state' = assertLiteral literal decision state in InvariantVarsQ (getQ state') F0 Vbl"
kruskal_algo_def: fixes L :: "(nat \<times> int \<times> nat) list" shows "kruskal_algo L = kruskal (return L) (\<lambda>(u, w, v). return (u, v)) ()"
lens_indep_get: fixes x :: "'e \<Longrightarrow> 'f"   and y :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'g" assumes "x \<bowtie> y" shows "get\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) = get\<^bsub>x\<^esub> \<sigma>"
lin_indpt_list_fs': shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs')"
tree_imp_connected: fixes G :: "('c, 'd) pre_digraph" assumes "tree G" shows "Digraph_Component.connected G"
subst_to_var_is_var: fixes t :: "('a, 'c) Term.term"   and s :: "'c \<Rightarrow> ('a, 'b) Term.term"   and v :: "'b" assumes "t \<cdot> s = Var v" shows "\<exists>w. t = Var w"
set_generate'': fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "set (generate' A B a b) = {(x, y). (x, y) \<noteq> (zeroes (length a), zeroes (length b)) \<and> x \<le>\<^sub>v replicate (length a) A \<and> y \<le>\<^sub>v replicate (length b) B}"
card_image_le: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite A" shows "card (f ` A) \<le> card A"
necessity_1: assumes "\<forall>A. admits_diagonal_reduction A" shows "\<forall>A. admits_triangular_reduction A"
summable_scaleR_right: fixes X :: "nat \<Rightarrow> 'b"   and r :: "real" assumes "summable X" shows "summable (\<lambda>n. r *\<^sub>R X n)"
A_2: shows "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter> B \<noteq> {}"
l_cancellation: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice" assumes "dist_over_inf (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "(x::'b::refinement_lattice) \<sqsubseteq> ((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) \<circ> F\<^sup>\<flat>) x"
Image_iff: fixes b :: "'b"   and r :: "('c \<times> 'b) set"   and A :: "'c set" shows "(b \<in> r `` A) = (\<exists>x\<in>A. (x, b) \<in> r)"
to_mu_repr: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "state = (f, mu, ds)" shows "mu_repr mu fs"
f: fixes x :: "real" assumes "a < ereal x"   and "ereal x < b" shows "isCont f x"
case_cong_weak: fixes generat :: "('d, 'e, 'f) generat"   and generat' :: "('d, 'e, 'f) generat"   and f1 :: "'d \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "generat = generat'" shows "(case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (case generat' of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa)"
wf_upb_step: fixes p1 :: "nat \<times> nat" assumes "\<And>p2. (p2, p1) \<in> lex_p \<Longrightarrow> c_assoc_have_key (pr_gr (loc_upb (fst p2) (snd p2))) (c_pair (fst p2) (snd p2)) = 0" shows "c_assoc_have_key (pr_gr (loc_upb (fst p1) (snd p1))) (c_pair (fst p1) (snd p1)) = 0"
A_in: fixes x :: "'v"   and y :: "'v" assumes "x \<in> A \<Gamma>" shows "\<not> edge \<Gamma> y x"
numeral_le_floor: fixes v :: "num"   and x :: "'a" shows "(numeral v \<le> \<lfloor>x\<rfloor>) = (numeral v \<le> x)"
intros_1: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream" assumes "\<phi> xs" shows "ev \<phi> xs"
match_Field_1: fixes \<mu> :: "('m \<times> 'w) set"   and m :: "'m"   and w :: "'w" assumes "match \<mu>"   and "(m, w) \<in> \<mu>" shows "w \<in> Field (Pm m)"
assms_2: shows "\<forall>x\<in>s. \<forall>y. dist x y < d \<longrightarrow> f y = g y"
map_comp0: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "map_coF (g \<circ> f) = map_coF g \<circ> map_coF f"
SUP_eq_const: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and 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 bot :: "'b"   and top :: "'b"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> f i = x" shows "Sup (f ` I) = x"
prems: shows "invar \<langle>l, (a, lv), r\<rangle>"
map_rec: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "map f xs = rec_list [] (\<lambda>x _. (#) (f x)) xs"
execute_def: fixes h :: "'c"   and p :: "('c, 'a, 'b) prog" shows "h \<turnstile> p = the_prog p h"
inv_comp_right_2: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.iso (VV.comp g f)"   and "VV.iso f" shows "VV.iso g"
intros_5: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm" assumes "A >> A'"   and "B >> B'" shows "SimplyTyped.Pair A B >> SimplyTyped.Pair A' B'"
comp_res_aux: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>T ; - (x ; y) \<cdot> y = 0"
Lambert_W_times_exp_self: fixes x :: "real" assumes "- exp (- 1) \<le> x" shows "Lambert_W x * exp (Lambert_W x) = x"
bisim_Val_pc_not_Invoke: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and v :: "'addr val"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and M :: "String.literal"   and n' :: "nat" assumes "P,e,h \<turnstile> (Val v, xs) \<leftrightarrow> (stk, loc, pc, xcp)"   and "pc < length (compE2 e)" shows "compE2 e ! pc \<noteq> Invoke M n'"
TopoS_modelLibrary_def: fixes m :: "('a, 'b) TopoS_packed"   and sinvar_spec :: "'a graph \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" shows "TopoS_modelLibrary m sinvar_spec \<equiv> 0 < length (nm_name m) \<and> TopoS_List_Impl (nm_default m) sinvar_spec (nm_sinvar m) (nm_receiver_violation m) (nm_offending_flows m) (nm_node_props m) (nm_eval m)"
discrete: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and a :: "'c"   and b :: "'c" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "less a b = less_eq (plus a one) b"
TokensAt_elem_in_\<W>: fixes P :: "(symbol \<times> character list) list set"   and u :: "symbol \<times> character list"   and k :: "nat" assumes "P \<subseteq> \<PP>"   and "u \<in> TokensAt k (Gen P)" shows "u \<in> \<W> P k"
basic_trans_rules_32: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "a = b"   and "b \<in> A" shows "a \<in> A"
nhip_is_sip: fixes \<xi> :: "state"   and \<xi>' :: "state"   and ip :: "nat"   and sn :: "nat"   and hops :: "nat"   and sip :: "nat" assumes "\<forall>dip\<in>kD (rt \<xi>). the (nhop (rt \<xi>) dip) \<in> kD (rt \<xi>)"   and "\<xi>' = \<xi>\<lparr>rt := update (rt \<xi>) ip (sn, kno, val, Suc hops, sip)\<rparr>"   and "sip \<in> kD (rt \<xi>)" shows "(the (nhop (update (rt \<xi>) ip (sn, kno, val, Suc hops, sip)) ip) = ip \<or> the (nhop (update (rt \<xi>) ip (sn, kno, val, Suc hops, sip)) ip) \<in> kD (rt \<xi>)) \<and> (\<forall>dip\<in>kD (rt \<xi>). the (nhop (update (rt \<xi>) ip (sn, kno, val, Suc hops, sip)) dip) = ip \<or> the (nhop (update (rt \<xi>) ip (sn, kno, val, Suc hops, sip)) dip) \<in> kD (rt \<xi>))"
swap_lassocr_def: shows "swap_lassocr \<equiv> rassocl\<^sub>C \<odot> (swap\<^sub>C |\<^sub>= 1\<^sub>C) \<odot> lassocr\<^sub>C"
semiring_hom_axioms: fixes x :: "'c"   and y :: "'c" shows "semiring_hom (\<lambda>p. poly2 p x y)"
simps_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Le x1 = Le y1) = (x1 = y1)"
H0: shows "H = LCoset_rel H `` {0::'g}"
fmap_neq: fixes s :: "('a, bool) fmap"   and s' :: "('a, bool) fmap" assumes "fmdom' s = fmdom' s'" shows "(s \<noteq> s') = (\<exists>v\<in>fmdom' s. fmlookup s v \<noteq> fmlookup s' v)"
le_less_trans: 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.preorder less_eq less"   and "less_eq x y"   and "less y z" shows "less x z"
index_eqI: fixes i :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "i \<le> length xs"   and "\<forall>j<i. xs ! j \<noteq> x"   and "i < length xs \<Longrightarrow> xs ! i = x" shows "index xs x = i"
fresh_3: fixes a :: "W.var"   and x1 :: "W.tvar"   and x2 :: "tyS" shows "a \<sharp> \<forall>[x1].x2 = a \<sharp> [x1].x2"
b: shows "\<forall>x\<in>E. \<bar>g x\<bar> \<le> p x"
unit_prod: 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"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one"   and "dvd.dvd times b one" shows "dvd.dvd times (times a b) one"
less_dres_def: fixes a :: "'b dres"   and b :: "'b dres" shows "(a < b) = (a \<le> b \<and> \<not> b \<le> a)"
IH: fixes a :: "real"   and b :: "real" assumes "a \<le> b"   and "\<And>x. x \<in> {a<..<b} - S \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "continuous_on {a..b} f" shows "(f' has_integral f b - f a) {a..b}"
all_zero: shows "\<forall>m. is_zero_row_upt_k m k A"
resZeroTrans: fixes x :: "name"   and Rs :: "residual" assumes "<\<nu>x>\<zero> \<longmapsto> Rs" shows "False"
sys_block_sizes_def: shows "target.sys_block_sizes \<equiv> {int (card bl) |bl. bl \<in># \<B>'}"
simps: fixes a1 :: "nat"   and a2 :: "nat" shows "residue_primroot a1 a2 = (\<exists>n g. a1 = n \<and> a2 = g \<and> 0 < n \<and> coprime n g \<and> ord n g = totient n)"
SUP_const: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and A :: "'f set"   and f :: "'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Sup ((\<lambda>i. f) ` A) = f"
rep_eq: fixes x :: "('c, 'b) sq_matrix" shows "to_fun (Square_Matrix.transpose x) = (\<lambda>i j. to_fun x j i)"
clinear_compose: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f" assumes "clinear f"   and "clinear g" shows "clinear (g \<circ> f)"
SN_on_Image_rtrancl_iff: fixes R :: "('a \<times> 'a) set"   and X :: "'a set" shows "SN_on R (R\<^sup>* `` X) = SN_on R X"
simps_5: fixes x2 :: "'a"   and x1 :: "'a" shows "Abrupt x2 \<noteq> Normal x1"
assms_5: shows "\<forall>z\<in>S. dist a x \<le> dist a z"
residue_of_diff': fixes b :: "nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and k :: "nat" assumes "b \<in> carrier Zp" shows "(a \<ominus> b) k = (a k - b k) mod p ^ k"
rank_nullity: assumes "V.fin_dim" shows "im.dim + Ker.dim = V.dim"
src_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src\<^sub>B (src\<^sub>B \<mu>) = src\<^sub>B \<mu>"
backflows_filternew_disjunct_flows_fix: fixes \<T> :: "'a stateful_policy" shows "\<forall>b\<in>backflows (filternew_flows_state \<T>). b \<notin> flows_fix \<T>"
prime_elem_dvd_mult_iff: fixes p :: "'a"   and a :: "'a"   and b :: "'a" assumes "prime_elem p" shows "(p dvd a * b) = (p dvd a \<or> p dvd b)"
fix_eq4: fixes f :: "'d"   and F :: "'d \<rightarrow> 'd" assumes "f = fix\<cdot>F" shows "f = F\<cdot>f"
orbits_list_conv_impl: fixes f :: "'b \<Rightarrow> 'b"   and xs :: "'b list" assumes "permutation f" shows "orbits_list f xs = orbits_list_impl f xs"
domain_eq: shows "Domainp (pcr_poly_mapping (=) (=)) = (\<lambda>f. finite {x. f x \<noteq> (0::'b)})"
half_segments_of_aform_strict: fixes e :: "nat \<Rightarrow> real"   and seg :: "(real \<times> real) \<times> real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1<..<1}"   and "seg \<in> set (half_segments_of_aform X)"   and "length (half_segments_of_aform X) \<noteq> 1" shows "ccw' (fst seg) (snd seg) (aform_val e X)"
finite_imageI: fixes F :: "'h set"   and h :: "'h \<Rightarrow> 'i" assumes "finite F" shows "finite (h ` F)"
null_sets_completion_iff: fixes N :: "'b set"   and M :: "'b measure" assumes "N \<in> sets M" shows "(N \<in> null_sets (completion M)) = (N \<in> null_sets M)"
abs_fun_supp_ineq: fixes a :: "'c"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "cp TYPE('a) TYPE('b) TYPE('c)"   and "disjoint TYPE('c) TYPE('b)" shows "supp ([a].x) = supp x"
"case": fixes f :: "char list option \<Rightarrow> char list option \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'c"   and x1 :: "char list option"   and x2 :: "char list option"   and x3 :: "'d"   and x4 :: "'e" shows "(case prod.Prod x1 x2 x3 x4 of prod.Prod x xa xb xc \<Rightarrow> f x xa xb xc) = f x1 x2 x3 x4"
ccbrt_def: fixes z :: "complex" shows "ccbrt z = (SOME w. w ^ 3 = z)"
finite_RPDs: fixes r :: "mregex" shows "finite (RPDs r)"
ats_val_process_adapted: fixes p :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "trading_strategy p"   and "support_adapt Mkt p" shows "borel_adapt_stoch_proc F (val_process Mkt p)"
inverse_functors_axioms: shows "inverse_functors (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) local.inv G"
trans: fixes subsumeseq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "subsumable subsumeseq"   and "subsumeseq x y"   and "subsumeseq y z" shows "subsumeseq x z"
suminf_geometric: fixes c :: "'c" assumes "norm c < 1" shows "(\<Sum>n. c ^ n) = (1::'c) / ((1::'c) - c)"
diagonal_to_Smith_aux_dvd: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and a :: "'b"   and b :: "'c"   and c :: "nat"   and k :: "nat"   and A :: "(('a, 'c) vec, 'b) vec" assumes "is_bezout_ext bezout"   and "to_nat a = to_nat b"   and "c < k"   and "c \<le> to_nat a"   and "k < min (nrows A) (ncols A)" shows "diagonal_to_Smith_aux A [0..<k] bezout $ from_nat c $ from_nat c dvd diagonal_to_Smith_aux A [0..<k] bezout $ a $ b"
os_is_empty_impl: shows "imp_list_is_empty os_list os_is_empty"
lasttoshow: fixes d :: "nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> int"   and s :: "nat"   and q :: "int"   and A :: "'a"   and b :: "nat \<Rightarrow> int" assumes "\<forall>n. 1 < d n"   and "\<forall>n. 0 < a n"   and "0 < s"   and "0 < q"   and "(1::'a) < A"   and "\<forall>n. 0 < b n"   and "\<forall>m\<ge>s. \<exists>n\<ge>m. d (n + 1) * (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j))) \<le> real_of_int (a (n + 1)) powr (1 / real_of_int (2 ^ (n + 1)))"   and "LIM n sequentially. d n ^ 2 ^ n / real_of_int (b n) :> at_top"   and "\<forall>\<^sub>F n in sequentially. (\<Sum>j. real_of_int (b (n + j)) / real_of_int (a (n + j))) \<le> real_of_int (2 * b n) / real_of_int (a n)" shows "\<exists>n. real_of_int q * (\<Prod>j = 1..n. real_of_int (a j)) * (\<Sum>j. real_of_int (b (j + n + 1)) / real_of_int (a (j + n + 1))) < 1"
tab_in_hom: 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 r :: "'b"   and \<rho> :: "'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b" assumes "composite_tabulation_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1" shows "partial_magma.in_hom V (composite_tabulation_in_maps.tab V H \<a> \<i> src trg r \<rho> r\<^sub>0 s \<sigma> s\<^sub>0 s\<^sub>1) (H r\<^sub>1 (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg r\<^sub>0) s\<^sub>1))) (H (H r s) (H s\<^sub>0 (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (H (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg r\<^sub>0) s\<^sub>1))))"
IH: shows "degree (\<Prod>a\<leftarrow>as. [:f a, 1::'a:]) = length as"
is_root_merge: fixes h1 :: "'b heap"   and h2 :: "'b heap" assumes "is_root h1"   and "is_root h2" shows "is_root (merge h1 h2)"
strict_prefixE': fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_prefix xs ys" obtains z :: "'a" and zs :: "'a list" where "ys = xs @ z # zs"
v11_vrange_ex1_eq: fixes b :: "V" shows "(b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> f) = (\<exists>!a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f \<and> f\<lparr>a\<rparr> = b)"
mono_sup: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b"   and B :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "order.mono less_eq f" shows "sup (f A) (f B) \<le> f (sup A B)"
eq: shows "path_image ((linepath (a + complex_of_real kde) (a + complex_of_real e) +++ p +++ linepath (a - complex_of_real d) (a - complex_of_real kde)) +++ linepath (a - complex_of_real kde) (a + complex_of_real kde)) = path_image (p +++ linepath (a - complex_of_real d) (a + complex_of_real e))"
assms: shows "is_idirprod A (\<lambda>g. generate G {g}) gs"
equiv_up_toD1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int" assumes "P \<Turnstile> c \<sim> c'"   and "(c, s) \<Rightarrow> s'"   and "P s" shows "(c', s) \<Rightarrow> s'"
assms_3: shows "f \<star> g \<cong>\<^sub>B h"
assms: shows "\<guillemotleft>\<mu>' : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" assumes "l_get_attribute type_wf get_attribute get_attribute_locs"   and "l_set_attribute type_wf set_attribute set_attribute_locs"   and "l_set_attribute_get_attribute_axioms get_attribute set_attribute" shows "l_set_attribute_get_attribute type_wf get_attribute get_attribute_locs set_attribute set_attribute_locs"
map_a_and: fixes \<gamma> :: "'b \<Rightarrow> bool"   and a :: "'b negation_type"   and ds :: "'b negation_type list list" shows "dnf_to_bool \<gamma> (map ((#) a) ds) = (dnf_to_bool \<gamma> [[a]] \<and> dnf_to_bool \<gamma> ds)"
C1_def: shows "C1 \<equiv> sqrt (2 * exp 1)"
mkSPRSingleAuto_def: shows "mkSPRSingleAuto \<equiv> \<lambda>kbp envInit envAction envTrans envVal envObs. mkAlgAuto trie_odlist_MapOps trans_MapOps (SPRViewSingle.spr_simObs envObs) (SPRViewSingle.spr_simInit envInit envObs) (SPRViewSingle.spr_simTrans kbp envAction envTrans envVal envObs) (SPRViewSingle.spr_simAction kbp envVal) (\<lambda>a. map (SPRViewSingle.spr_simInit envInit envObs a \<circ> envObs a) envInit)"
homotopic_Borsuk_maps_in_bounded_component: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "compact S"   and "a \<notin> S"   and "b \<notin> S"   and "bounded (connected_component_set (- S) a)"   and "homotopic_with_canon (\<lambda>x. True) S (sphere (0::'a) 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) (\<lambda>x. (x - b) /\<^sub>R norm (x - b))" shows "connected_component (- S) a b"
valid_in_def: fixes U :: "nat set"   and p :: "fm" shows "valid_in U p \<equiv> \<forall>i. range i \<subseteq> domain U \<longrightarrow> eval i p = Det True"