hcju/isabelleps huggingface dataset:
Reg_in_outPortsRule: fixes c :: "'form"   and r :: "'rule" shows "(Reg c |\<in>| outPortsRule r) = (c |\<in>| f_consequent r)"
totalValuationForClauseDefinesItsValue: fixes valuation :: "Literal list"   and clause :: "Literal list" assumes "CNF.total valuation (vars clause)" shows "clauseTrue clause valuation \<or> clauseFalse clause valuation"
verit_eq_simplify_6: fixes x :: "'h"   and y :: "'h" assumes "x = y" shows "x \<le> y"
Mp_in: shows "q.Mp (Polynomial.smult ilc f) =m Polynomial.smult ilc f"
IndP_graph_aux_def: shows "IndP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>IndP_aux m x. x1 = x \<and> x2 = SyntaxN.Ex m (OrdP (Var m) AND x EQ Q_Ind (Var m)) \<and> atom m \<sharp> x)"
intro: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and utp_unit :: "'a hrel" assumes "utp_theory_unital hcond utp_unit"   and "utp_theory_mono hcond" shows "utp_theory_mono_unital hcond utp_unit"
ofilter_underS_Field: fixes A :: "'b set" shows "s.ofilter A = ((\<exists>a\<in>Field s. A = s.underS a) \<or> A = Field s)"
Rep_quot_inject: fixes x :: "'d quot"   and y :: "'d quot" shows "(Rep_quot x = Rep_quot y) = (x = y)"
lprefix_ltake_same: fixes n :: "enat"   and xs :: "'c llist"   and m :: "enat" shows "ltake n xs \<sqsubseteq> ltake m xs = (n \<le> m \<or> llength xs \<le> m)"
ennreal_of_enat_def: fixes n :: "enat" shows "ennreal_of_enat n = (case n of enat n \<Rightarrow> of_nat n | \<infinity> \<Rightarrow> top)"
IH: fixes i :: "'sz" assumes "\<And>i f'. SFunctions \<sigma> i = Some f' \<Longrightarrow> dfree f'" shows "sterm_sem I (Tsubst (args i) \<sigma>) (fst \<nu>) = sterm_sem (local.adjoint I \<sigma> \<nu>) (args i) (fst \<nu>)"
countable_set_pmf: fixes p :: "'a pmf" shows "countable (set_pmf p)"
maxsimp_map_into_image: fixes x :: "'a set" assumes "folding_g.chamber x" shows "SimplicialComplex.maxsimp (g \<turnstile> X) (g ` x)"
crdt_ops_independent_suf: fixes xs :: "('c \<times> 'd) list"   and oid :: "'c"   and oper :: "'d"   and deps :: "'d \<Rightarrow> 'c set"   and ys :: "('c \<times> 'd) list"   and zs :: "('c \<times> 'd) list"   and op2 :: "'d"   and r :: "'c" assumes "spec_ops (xs @ [(oid, oper)]) deps"   and "crdt_ops (ys @ [(oid, oper)] @ zs) deps"   and "set (xs @ [(oid, oper)]) = set (ys @ [(oid, oper)] @ zs)"   and "op2 \<in> snd ` set zs"   and "r \<in> deps op2" shows "r \<noteq> oid"
order_OrdSum_def: fixes a :: "('a, 'b) OrdSum"   and b :: "('a, 'b) OrdSum" shows "a \<le> b \<equiv> a l\<rightarrow> b = 1"
lq_ne: fixes p :: "'b list"   and u :: "'b list" assumes "p \<le>p u \<cdot> p"   and "u \<noteq> \<epsilon>" shows "p\<inverse>\<^sup>>(u \<cdot> p) \<noteq> \<epsilon>"
dependent_choice: fixes r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool"   and x0 :: "'b" assumes "trans r"   and "P x0"   and "\<And>x. P x \<Longrightarrow> \<exists>y. P y \<and> (x, y) \<in> r" obtains f :: "nat \<Rightarrow> 'b" where "\<And>n. P (f n)" and "\<And>n m. n < m \<Longrightarrow> (f n, f m) \<in> r"
simps_12: fixes f :: "'d \<Rightarrow> 'c" shows "lmap f LNil = LNil"
antisym_conv2: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le>s y" shows "(\<not> x <s y) = (x = y)"
trivial_limit_eq: fixes net :: "'a filter" shows "(net = bot) = (\<forall>P. eventually P net)"
every_clause_is_a_resolvent: fixes S :: "'at Literal set set"   and A :: "'at"   and R :: "'at Literal set set"   and P1 :: "'at Literal set" assumes "all_fulfill (in_all_resolvents_upon S A) R"   and "all_fulfill (\<lambda>x. \<not> tautology x) S"   and "P1 \<in> S \<union> R" shows "in_all_resolvents_upon S A P1"
Pruned_Port_Graph_axioms: shows "Pruned_Port_Graph nodes inPorts outPorts vertices nodeOf edges"
embranch_all0: fixes rcs :: "recf list"   and xs :: "nat list"   and rgs :: "recf list" assumes "\<forall>j<length rcs. rec_exec (rcs ! j) xs = 0"   and "length rgs = length rcs"   and "rcs \<noteq> []"   and "list_all (\<lambda>rf. primerec rf (length xs)) (rgs @ rcs)" shows "rec_exec (rec_embranch (zip rgs rcs)) xs = 0"
extended_ord_strict_def: fixes s :: "'a option \<Rightarrow>\<^sub>0 nat"   and t :: "'a option \<Rightarrow>\<^sub>0 nat" shows "extended_ord_strict s t = (restrict_indets_pp s \<prec> restrict_indets_pp t \<or> restrict_indets_pp s = restrict_indets_pp t \<and> lookup s None < lookup t None)"
split_inv: fixes t :: "('a \<times> 'b) tree"   and x :: "'a"   and l :: "('a \<times> 'b) tree"   and b :: "bool"   and r :: "('a \<times> 'b) tree" assumes "split t x = (l, b, r)"   and "inv t" shows "inv l \<and> inv r"
ys'_def: shows "ys' = take j ysa @ y # drop j ysa"
uint_of_bl_is_bl_to_bin_drop: fixes l :: "bool list" assumes "length (dropWhile Not l) \<le> LENGTH('b)" shows "uint (of_bl l) = bl_to_bin l"
eq: fixes m :: "'c multiset" shows "(indexed_eval_aux Ps i \<Otimes> i) m \<oplus> P m = (indexed_eval_aux Qs' i \<Otimes> i) m \<oplus> Q' m"
sep_conj_empty': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj (sep_algebra.sep_empty zero) P = P"
Nmlize_Nmlize: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
two_component_support_set: fixes a :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and b :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'d"   and y :: "'d" assumes "\<exists>n w. a n w \<noteq> 0"   and "\<exists>n w. b n w \<noteq> 0"   and "x \<noteq> y" shows "support_set ((\<lambda>x n w. 0)(x := a, y := b)) = {x, y}"
Eps_unique_transfer_lemma: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> bool"   and g :: "'e \<Rightarrow> bool"   and f' :: "'d \<Rightarrow> 'f"   and g' :: "'e \<Rightarrow> 'f" assumes "right_total A"   and "rel_fun A (=) f g"   and "rel_fun A (=) f' g'"   and "\<exists>x. Domainp A x \<and> f x"   and "\<And>x y. \<lbrakk>g x; g y\<rbrakk> \<Longrightarrow> g' x = g' y" shows "f' (SOME x. Domainp A x \<and> f x) = g' (Eps g)"
inference_saturated_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and S :: "'b eclause set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.inference_saturated trm_ord sel pos_ord filter_trms S = (\<forall>C P \<sigma> C' D \<theta> \<eta>. basic_superposition.derivable trm_ord sel pos_ord filter_trms C P S \<sigma> Ground C' \<longrightarrow> ground_clause (cl_ecl C) \<longrightarrow> grounding_set P \<sigma> \<longrightarrow> basic_superposition.derivable trm_ord sel pos_ord filter_trms D P S \<theta> FirstOrder C' \<longrightarrow> trms_subsumes D C \<eta> \<longrightarrow> \<sigma> \<doteq> \<theta> \<lozenge> \<eta> \<longrightarrow> basic_superposition.redundant_inference trm_ord (subst_ecl D \<eta>) S P \<sigma>)"
div_mult_cancel: fixes a :: "'b"   and b :: "'b" shows "a div b * b = a - a mod b"
finite_index_eq_nextl: shows "finite (UNIV // eq_nextl)"
bicategory_def: 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" shows "bicategory V H \<a> \<i> src trg \<equiv> (horizontal_composition V H src trg \<and> natural_isomorphism (subcategory.comp (product_category.comp V (product_category.comp V V)) (\<lambda>\<tau>\<mu>\<nu>. partial_magma.arr V (fst \<tau>\<mu>\<nu>) \<and> partial_magma.arr (subcategory.comp (product_category.comp V V) (\<lambda>\<mu>\<nu>. partial_magma.arr V (fst \<mu>\<nu>) \<and> partial_magma.arr V (snd \<mu>\<nu>) \<and> src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>))) (snd \<tau>\<mu>\<nu>) \<and> src (fst \<tau>\<mu>\<nu>) = trg (fst (snd \<tau>\<mu>\<nu>)))) V (horizontal_composition.HoHV V H src trg) (horizontal_composition.HoVH V H src trg) (\<lambda>\<mu>\<nu>\<tau>. \<a> (fst \<mu>\<nu>\<tau>) (fst (snd \<mu>\<nu>\<tau>)) (snd (snd \<mu>\<nu>\<tau>)))) \<and> fully_faithful_functor V V (\<lambda>\<mu>. if partial_magma.arr V \<mu> then H (trg \<mu>) \<mu> else partial_magma.null V) \<and> fully_faithful_functor V V (\<lambda>\<mu>. if partial_magma.arr V \<mu> then H \<mu> (src \<mu>) else partial_magma.null V) \<and> bicategory_axioms V H \<a> \<i> src trg"
m_2: shows "m \<cdot> \<delta> = Var y"
Sup_distr: fixes X :: "'a set"   and y :: "'a" shows "\<Squnion>X \<cdot> y = \<Squnion>{x \<cdot> y |x. x \<in> X}"
adds_minus: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and s :: "'b"   and t :: "'b" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus s t" shows "plus (minus t s) s = t"
Phase1or2ReadThen_HInv3_pq: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Phase1or2ReadThen s s' p d q"   and "HInv3_L s' p q d"   and "p \<noteq> q"   and "Inv2b s" shows "HInv3_R s' p q d"
distinct_1: fixes x1 :: "'b"   and x2 :: "'c error_result" shows "Rval x1 \<noteq> Rerr x2"
continuous_compose_quotient_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "quotient_map X X' f"   and "continuous_map X X'' (g \<circ> f)" shows "continuous_map X' X'' g"
the_inv_f_f: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "inj f" shows "the_inv f (f x) = x"
spmf_rel_map2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c spmf"   and g :: "'e \<Rightarrow> 'd"   and y :: "'e spmf" shows "rel_spmf R x (map_spmf g y) = rel_spmf (\<lambda>x y. R x (g y)) x y"
one_dvd: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times one a"
cr_spmf_prob_optionTD: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "('c, 'c spmf) optionT"   and q :: "'d spmf" assumes "cr_spmf_prob_optionT A p q" shows "rel_spmf A (run_option p) q"
1: fixes a :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle 0 X {} a" shows "homologous_rel_set 0 X {} a \<in> range (\<lambda>n. homologous_rel_set 0 X {} (frag_cmul n (frag_of f)))"
R_G_div_cfg_def: fixes cfg :: "('a \<times> ('c \<Rightarrow> 'b) set) cfg" shows "R_G_div_cfg cfg \<equiv> AE \<omega> in MDP.MC.T cfg. \<R>_div (smap (snd \<circ> state) \<omega>)"
sound_ground_sound_\<Gamma>: shows "sound_inference_system ground_sound_\<Gamma>"
coprime_right_2_iff_odd: 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 a :: "'c" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "algebraic_semidom.coprime times one a (numeral.numeral one plus (num.Bit0 num.One)) = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
xs_def: shows "xs \<equiv> snd (normalise_decomp x y n)"
to_int_def: fixes arith_ops_record :: "'b arith_ops_record" shows "arith_ops_record.to_int arith_ops_record = (case arith_ops_record of Arith_Ops_Record x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 \<Rightarrow> x13)"
obj_at_fun_upd: fixes P :: "('a, 'b, 'c) object \<Rightarrow> bool"   and s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and fld :: "'d process_name"   and val :: "('a, 'd, 'b, 'c) Model.local_state"   and r :: "'c" assumes "map_option P (sys_heap (s(fld := val)) r) = map_option P (sys_heap s r)" shows "obj_at P r (s(fld := val)) = obj_at P r s"
mod_hom_axioms_def: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and N :: "('h, 'l, 'm) module_scheme"   and f :: "'j \<Rightarrow> 'l" shows "mod_hom_axioms R M N f \<equiv> f \<in> module_hom R M N"
bind_sample1: fixes p :: "'p pmf"   and f :: "'p \<Rightarrow> 'm"   and g :: "'a option \<Rightarrow> 'm" shows "bind (sample p f) g = sample p (\<lambda>x. bind (f x) g)"
valid_graph: shows "valid_graph (del_unEdge v e v' G)"
lang\<^sub>M\<^sub>2\<^sub>L_rexp_of'_rexp_of'': fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "lang n (rexp_of' n \<phi>) - {[]} = lang n (rexp_of'' n \<phi>) - {[]}"
ct_suffix_refl: fixes ts :: "ct list" shows "ct_suffix ts ts"
2: shows "set (map (\<lambda>i. SOME j. j < length ls \<and> ls ! j = i) ss) \<subseteq> {..<length ls}"
simps_2: fixes literal :: "Literal"   and clause :: "Literal list"   and valuation :: "Literal list" shows "clauseTrue (literal # clause) valuation = (literalTrue literal valuation \<or> clauseTrue clause valuation)"
assms_2: shows "theta\<inverse> \<subseteq> Wretr (theta\<inverse> \<union> Wbis)"
assms_8: shows "quicksort_post R h (Suc p) hi xs'' xs'''"
ideD_2: fixes a :: "(bool, unit) arr" assumes "ide a" shows "local.dom a = a"
Un_iff: fixes c :: "'ap"   and A :: "'ap set"   and B :: "'ap set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
std_lookup_action: fixes ast_op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "is_standard_operator' ast_op"   and "ast_op \<in> set ast\<delta>" shows "\<exists>ast_op'. lookup_action (abs_ast_operator ast_op) = Some ast_op'"
rank_skew_rank_invar: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" assumes "rank_skew_invar (t # bq)" shows "rank_invar bq"
nth_Cons_pos: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" assumes "0 < n" shows "(x # xs) ! n = xs ! (n - 1)"
ccpo: shows "class.ccpo spmf.lub_fun spmf.le_fun (mk_less spmf.le_fun)"
disj_of_clause_is_nnf: fixes F :: "'b literal list" shows "is_nnf (disj_of_clause F)"
contour_integral_local_primitive: fixes g :: "real \<Rightarrow> complex"   and S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g"   and "path_image g \<subseteq> S"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>d h. 0 < d \<and> (\<forall>y. cmod (y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within S))" shows "f contour_integrable_on g"
f: shows "eval f [i, e] \<down>= e_length e"
measure_stretch: fixes S :: "(real, 'a) vec set"   and m :: "'a \<Rightarrow> real" assumes "S \<in> lmeasurable" shows "Sigma_Algebra.measure lebesgue ((\<lambda>x. \<chi>k. m k * x $ k) ` S) = \<bar>prod m UNIV\<bar> * Sigma_Algebra.measure lebesgue S"
congruent_def: fixes r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c" shows "f respects r = (\<forall>(y, z)\<in>r. f y = f z)"
nand_sim: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> (F \<^bold>\<and> G)" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> F \<^bold>\<or> \<^bold>\<not> G"
Inf_Sup: 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 A :: "'b set set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (Sup ` A) = Sup (Inf ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})"
eq_resource_on_trans: fixes A :: "'y set"   and res :: "('y, 'z) resource"   and res' :: "('y, 'z) resource"   and res'' :: "('y, 'z) resource" assumes "A \<turnstile>\<^sub>R res \<sim> res'"   and "A \<turnstile>\<^sub>R res' \<sim> res''" shows "A \<turnstile>\<^sub>R res \<sim> res''"
psp_stable: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto> A'"   and "F \<in> stable B" shows "F \<in> A \<inter> B \<longmapsto> A' \<inter> B"
hm_iteratei_impl: fixes m :: "('b, 'c) hashmap" shows "map_iterator (hm_iteratei m) (hm_\<alpha> m)"
assms_5: shows "(ir, Rem is e2) \<in> set (node_deliver_messages xs)"
comp_null_2: fixes f :: "'a arrow_of_spans_data" shows "f \<bullet> V.null = V.null"
strict_trans2: fixes b :: "'g"   and a :: "'g"   and c :: "'g" assumes "b < a"   and "c \<le> b" shows "c < a"
stat: shows "if stat then is_static f else \<not> is_static f"
distinct_ExcessTable: fixes vs :: "nat list"   and g :: "graph" assumes "distinct vs" shows "distinct (map fst (ExcessTable g vs))"
restrict_ext: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "restrict f A = restrict g A"
inv_closed: fixes x :: "nat" assumes "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<in> carrier G"
ord_iff_2: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (rep_nat x < rep_nat y)"
eq_inf_concat_end: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w :: "'a stream" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2" shows "traces.eq_inf ind (w\<^sub>1 @- w) (w\<^sub>2 @- w)"
simps_1: fixes cx :: "int"   and cy :: "int"   and black :: "bool"   and more :: "'b" shows "black \<lparr>cx = cx, cy = cy, black = black, \<dots> = more\<rparr> = black"
even_zero: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) zero"
map_def: fixes gf :: "'c \<times> 'c" shows "HomC.map gf = (if CopxC.arr gf then S.mkArr (HomC.set (CopxC.dom gf)) (HomC.set (CopxC.cod gf)) (\<phi>C (CopxC.cod gf) \<circ> (\<lambda>h. snd gf \<cdot>\<^sub>C h \<cdot>\<^sub>C fst gf) \<circ> \<psi>C (CopxC.dom gf)) else S.null)"
singleton_set_some: fixes X :: "'c set" shows "(\<exists>Y. Y \<in> singleton_set X) = (\<exists>x. x \<in> X)"
l3_inv1I: fixes x :: "l3_state" assumes "parts (ik x) \<inter> range LtK \<subseteq> ik x" shows "x \<in> l3_inv1"
skip_id: fixes I :: "interp"   and X :: "(variable \<Rightarrow> real) set" shows "game_sem I Skip X = X"
abs_eq: fixes x :: "complex" shows "blaschke x = abs_moebius (blaschke_mmat x)"
atm_iff_pos_or_neg_lit: fixes A :: "'a"   and L :: "'a literal multiset" shows "(A \<in> atms_of L) = (Pos A \<in># L \<or> Neg A \<in># L)"
JNF_5: shows "iP * P = 1\<^sub>m n"
deg_le_sect_pp_aux_2: fixes t :: "('x, nat) pp"   and n :: "nat"   and xs :: "'x list" assumes "deg_pp t \<le> n"   and "keys_pp t \<subseteq> set xs" shows "t \<in> keys (deg_le_sect_pp_aux xs n)"
less_suc_p: shows "\<forall>m. m \<^bold>\<in> lan vn \<longrightarrow> m < p + 1"
index_outer_prod: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec"   and m :: "nat"   and i :: "nat"   and j :: "nat" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec m"   and "i < n"   and "j < m" shows "outer_prod v w $$ (i, j) = v $ i * conjugate (w $ j)"
enum_all_prod_def: fixes P :: "'a \<times> 'b \<Rightarrow> bool" shows "enum_all P = enum_all (\<lambda>x. enum_all (\<lambda>y. P (x, y)))"
hom1: shows "(\<lambda>a. (a, [])) \<in> ring_hom R (RDirProd R nil_ring)"
wt_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" assumes "I.wtE \<xi>"   and "wt T" shows "intT (tpOf T) (I.int \<xi> T)"
rtrancl_mono_proof: fixes x :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "\<And>a b. x a b \<longrightarrow> y a b" shows "x\<^sup>*\<^sup>* a b \<longrightarrow> y\<^sup>*\<^sup>* a b"
lt_mono: fixes x :: "real"   and bs :: "real list" shows "\<forall>a n s. Inum (x # bs) a = Inum (x # bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x # bs) (lt n s) = Ifm (x # bs) (Lt a)"
degree_least_nonzero: fixes f :: "'a pdevs" assumes "degree f \<noteq> 0" shows "pdevs_apply f (degree f - 1) \<noteq> (0::'a)"
power_one_right: fixes a :: "'a \<times> 'b" shows "sd_power a 1 = a"
comparable: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> c \<or> c \<le> a"   and "b \<le> c \<or> c \<le> b" shows "a \<le> b \<or> b \<le> a"
assms_1: shows "A0 B0 C0 Cong3 A1 B1 C1"
rel_FGcontra_pos_distr_imp: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co4 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co4' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co5 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co5' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'y \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 'af \<Rightarrow> bool"   and Contra3 :: "'aa \<Rightarrow> 'ag \<Rightarrow> bool"   and Contra4 :: "'ab \<Rightarrow> 'ah \<Rightarrow> bool"   and Co2 :: "'ac \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra1' :: "'ae \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra2' :: "'af \<Rightarrow> 'ak \<Rightarrow> bool"   and Contra3' :: "'ag \<Rightarrow> 'al \<Rightarrow> bool"   and Contra4' :: "'ah \<Rightarrow> 'am \<Rightarrow> bool"   and Co2' :: "'ai \<Rightarrow> 'an \<Rightarrow> bool"   and Contra5 :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and Contra5' :: "'ap \<Rightarrow> 'aq \<Rightarrow> bool"   and tytok_F :: "('ar \<times> 'as \<times> 'at \<times> 'p \<times> 'q \<times> 'r \<times> 'au \<times> 'av \<times> 'aw \<times> 'ax) itself"   and tytok_G :: "('y \<times> 'ae \<times> 'aj \<times> 'z \<times> 'af \<times> 'ak \<times> 'ad) itself"   and Co3 :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and Co3' :: "'av \<Rightarrow> 'aw \<Rightarrow> bool"   and tytok_FGcontra :: "('ar \<times> 'as \<times> 'at \<times> 'ad \<times> 'ax) itself" assumes "rel_F_pos_distr_cond Co1 Co1' Co4 Co4' Co5 Co5' (rel_G Contra1 Contra2 Contra3 Contra4 Co1 Co2) (rel_G Contra1' Contra2' Contra3' Contra4' Co1' Co2') Contra1 Contra1' Contra5 Contra5' tytok_F"   and "rel_G_neg_distr_cond Contra3 Contra3' Contra4 Contra4' Co1 Co1' Co2 Co2' tytok_G" shows "rel_FGcontra_pos_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Co4 Co4' Co5 Co5' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' Contra4 Contra4' Contra5 Contra5' tytok_FGcontra"
inverse_inverse: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "group f z inverse" shows "inverse (inverse a) = a"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G' \<Phi>\<^sub>G' G \<Phi>\<^sub>G \<tau>'.map\<^sub>0 \<tau>'.map\<^sub>1"
mkPoint_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and t :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "t \<in> local.set a" shows "\<guillemotleft>mkPoint a t : unity \<rightarrow> a\<guillemotright>"
last_conv_nth: fixes xs :: "'f list" assumes "xs \<noteq> []" shows "last xs = xs ! (length xs - 1)"
Ke0_iff_inj: shows "inj_on T (carrier V) = (carrier (V.vs kerT) = {\<zero>\<^bsub>V\<^esub>})"
K_conjoin_imply: fixes G :: "'b fm list"   and p :: "'b fm" assumes "\<turnstile> (\<^bold>\<not> conjoin G (\<^bold>\<not> p))" shows "\<turnstile> imply G p"