hcju/isabelleps huggingface dataset:
DERIV_chain: fixes f :: "'a \<Rightarrow> 'a"   and Da :: "'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and Db :: "'a"   and s :: "'a set" assumes "(f has_field_derivative Da) (at (g x))"   and "(g has_field_derivative Db) (at x within s)" shows "(f \<circ> g has_field_derivative Da * Db) (at x within s)"
simps_2: fixes delta :: "'f"   and inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs option" shows "qGood (qOp delta inp binp) = (liftAll qGood inp \<and> liftAll qGoodAbs binp \<and> |{i. inp i \<noteq> None}| <o |UNIV| \<and> |{i. binp i \<noteq> None}| <o |UNIV|)"
g_id: fixes f :: "int poly" assumes "f \<in># gs" shows "Mp f = f"
simps_1: fixes at :: "('a, 'b) atm" shows "varsL (Pos at) = varsA at"
r_inv: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.group G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
intro: 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" assumes "Contracts Xd Xh Pd Ch"   and "ContractsWithUnilateralSubstitutes_axioms Xd Ch" shows "ContractsWithUnilateralSubstitutes Xd Xh Pd Ch"
mismatchIdRight: fixes Rel :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name"   and P :: "pi" assumes "Id \<subseteq> Rel"   and "a \<noteq> b" shows "P \<leadsto>[Rel] [a\<noteq>b]P"
face_cycle_succ_no_arc: fixes G :: "('e, 'f) pre_digraph"   and M :: "'f pre_map"   and a :: "'f" assumes "digraph_map G M"   and "a \<notin> arcs G" shows "pre_digraph_map.face_cycle_succ M a = a"
prems: shows "ifex_sat_list (IF v t e) = Some u"
word_of_nat_eq_iff: fixes m :: "nat"   and n :: "nat" shows "(word_of_nat m = word_of_nat n) = (take_bit LENGTH('k) m = take_bit LENGTH('k) n)"
map_poly_pcompose: fixes f :: "real poly"   and g :: "real poly" shows "map_poly of_real (f \<circ>\<^sub>p g) = map_poly of_real f \<circ>\<^sub>p map_poly of_real g"
none1: fixes a :: "'a" assumes "x = None"   and "y = Some a" shows "thesis"
preserves_reflects_arr: fixes f :: "'a" shows "Span.arr (SPN f) = arr f"
mult_assoc: fixes A :: "rat mat"   and n :: "nat"   and B :: "rat mat"   and v :: "rat vec" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "dim_vec v = n" shows "A *\<^sub>v (B *\<^sub>v v) = A * B *\<^sub>v v"
rupture_surj_hom_1: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (A [X])"   and "p \<in> carrier (K [X]\<^bsub>A [X]\<^esub>)" shows "UP_A.rupture_surj K p \<in> ring_hom (K [X]\<^bsub>A [X]\<^esub>) (Rupt\<^bsub>A [X]\<^esub> K p)"
t_list_of_B_induct: fixes t :: "'b tree"   and n :: "nat" assumes "braun t"   and "height t \<le> n" shows "t_list_of_B t \<le> 3 * (n + 1) * size t"
splitAt_conv: fixes x :: "'c"   and xs :: "'c list" shows "splitAt x xs = (takeWhile (\<lambda>y. y \<noteq> x) xs, tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
in_br_conv: fixes c :: "'g"   and a :: "'h"   and \<alpha> :: "'g \<Rightarrow> 'h"   and I :: "'g \<Rightarrow> bool" shows "((c, a) \<in> br \<alpha> I) = (a = \<alpha> c \<and> I c)"
interaction_bounded_by_gpv_stop: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat" assumes "interaction_bounded_by consider gpv n" shows "interaction_bounded_by consider (gpv_stop gpv) n"
invertible_inverse_closed: fixes u :: "'a" assumes "additive.invertible u"   and "u \<in> R" shows "- u \<in> R"
spectral_radius_jnf_norm_bound_less_1: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "spectral_radius A < 1"   and "\<exists>n_as. jordan_nf A n_as" shows "\<exists>c. \<forall>k. norm_bound (A ^\<^sub>m k) c"
Rep_mod: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and x :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "Rep x mod int CARD('a) = Rep x"
empty_imp_not_in: fixes x :: "'a" shows "x \<notin> {}"
ideal_explicit: fixes S :: "'a set" shows "ideal_generated S = {y. \<exists>f U. finite U \<and> U \<subseteq> S \<and> (\<Sum>i\<in>U. f i * i) = y}"
sum_not_less_zero: fixes s :: "'lbl" assumes "s < (0::'lbl)" shows "False"
hyps: shows "foldl (+) (0::'a) (map f (filter P xs)) + foldl (+) (0::'a) (map f (filter (\<lambda>s. \<not> P s) xs)) = foldl (+) (0::'a) (map f xs)"
to_stream_def: fixes X :: "nat \<Rightarrow> 'b" shows "to_stream X = smap X nats"
square_subset: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "square R S T U"   and "T \<le> T'" shows "square R S T' U"
asn_simps_1: fixes p :: "assertion" shows "asn (Ribbon p) = p"
const_imply4: fixes P :: "'a \<Rightarrow> 'b"   and Q :: "'a \<Rightarrow> 'c"   and R :: "'a \<Rightarrow> 'd"   and S :: "'a \<Rightarrow> 'e" assumes "\<And>\<tau> \<tau>'. \<lbrakk>P \<tau> = P \<tau>'; Q \<tau> = Q \<tau>'; R \<tau> = R \<tau>'\<rbrakk> \<Longrightarrow> S \<tau> = S \<tau>'"   and "const P"   and "const Q"   and "const R" shows "const S"
strict_sorted_appendE1: fixes xs :: "'a list"   and ys :: "'a list" assumes "Lists_Ex.strict_sorted (xs @ ys)" shows "Lists_Ex.strict_sorted xs \<and> Lists_Ex.strict_sorted ys"
is_borel_def: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "Borel_Space.is_borel f M = (f \<in> borel_measurable M)"
lfilled_collapse1: fixes n :: "nat"   and lholed :: "Lholed"   and vs :: "e list"   and es :: "e list"   and LI :: "e list"   and l :: "nat" assumes "Lfilled n lholed (vs @ es) LI"   and "const_list vs"   and "l \<le> length vs" shows "\<exists>lholed'. Lfilled n lholed' (drop (length vs - l) vs @ es) LI"
fold_insert_idem2: fixes A :: "'a set"   and z :: "'a option"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold MaxR_f z (insert x A) = Finite_Set.fold MaxR_f (MaxR_f x z) A"
set_like_def: fixes R :: "(ZF \<times> ZF) set" shows "set_like R \<equiv> \<forall>y. Ext R y \<in> range explode"
allB_atLeastLessThan_lower: fixes i :: "nat"   and j :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "i \<le> j"   and "\<forall>x\<in>{i..<n}. P x" shows "\<forall>x\<in>{j..<n}. P x"
n_inter_num_choose_design_inter: fixes b1 :: "'b set"   and b2 :: "'b set"   and n :: "nat" assumes "b1 \<in># \<B>'"   and "b2 \<in># \<B>'" shows "(b1 |\<inter>|\<^sub>n b2) = int (nat (b1 |\<inter>| b2) choose n)"
less_le_trans: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x <lex y"   and "y \<le>lex z" shows "x <lex z"
\<gamma>\<^sub>\<tau>: shows "\<guillemotleft>\<gamma>\<^sub>\<tau> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma>\<^sub>\<tau> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>\<^sub>\<tau>)"
simps: fixes v :: "'a"   and n :: "nat" shows "Time_Monad.val (??.Time_Monad.tm.TM v n) = v"
pred_eq_const1: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and c :: "'d" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "c \<in> space N \<Longrightarrow> {c} \<in> sets N" shows "pred M (\<lambda>x. f x = c)"
length_msort: fixes xs :: "'c list" shows "length (msort xs) = length xs"
ht_hash_replicate: fixes n :: "nat" shows "ht_hash (replicate n [])"
assms: shows "(x, y) \<in> (succsr succs)\<^sup>*"
add_pow_int_ge: fixes k :: "int"   and R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "0 \<le> k" shows "[k] \<cdot>\<^bsub>R\<^esub> a = [nat k] \<cdot>\<^bsub>R\<^esub> a"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c set \<Rightarrow> 'b"   and nres :: "'c nres" shows "P (case nres of FAILi \<Rightarrow> f1 | RES x \<Rightarrow> f2 x) = ((nres = FAILi \<longrightarrow> P f1) \<and> (\<forall>x2. nres = RES x2 \<longrightarrow> P (f2 x2)))"
shunt_mapping: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "id z \<sqinter> z \<le> top \<and> z \<sqinter> top = top" shows "(x \<le> z \<sqinter> y) = (id z \<sqinter> x \<le> y)"
obtain_qFresh: fixes V :: "'var set"   and XS :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort" assumes "finite V \<or> |V| <o |UNIV|"   and "finite XS \<or> |XS| <o |UNIV|"   and "Ball XS qGood" shows "\<exists>x. x \<notin> V \<and> (\<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
tuple_decode_encode: fixes xs :: "nat list" assumes "0 < length xs" shows "tuple_decode (length xs - 1) (tuple_encode (length xs - 1) xs) = xs"
that: assumes "\<Z>_sgrp_basis \<alpha> \<SS>"   and "\<And>a b c. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c)" shows "thesis"
sgn_cblinfun_def: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "sgn x = complex_of_real (inverse (norm x)) *\<^sub>C x"
assms_2: shows "u \<in> iso_verts hom ` verts G"
homotopic_imp_homologous_rel_chain_maps: fixes T :: "'b set"   and V :: "'c set"   and S :: "'b topology"   and U :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "homotopic_with (\<lambda>h. h ` T \<subseteq> V) S U f g"   and "singular_relcycle p S T c" shows "homologous_rel p U V (chain_map p f c) (chain_map p g c)"
sg: shows "\<forall>i. simple_bochner_integrable M (sg i)"
dens_ctxt_measure_def: shows "dens_ctxt_measure \<equiv> \<lambda>(V, V', \<Gamma>, \<delta>) \<rho>. density (state_measure' V V' \<Gamma> \<rho>) \<delta>"
set_to_map_simp: fixes S :: "('d \<times> 'e) set"   and k :: "'d"   and v :: "'e" assumes "inj_on fst S" shows "(set_to_map S k = Some v) = ((k, v) \<in> S)"
64241: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> - - x \<sqinter> - y) = - x"
inv_mult_group: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> (x \<oplus>\<^bsub>K [X]\<^esub> y) = \<ominus>\<^bsub>K [X]\<^esub> y \<oplus>\<^bsub>K [X]\<^esub> \<ominus>\<^bsub>K [X]\<^esub> x"
linear_order_cases: fixes x :: "'c"   and y :: "'c" obtains "x \<sqsubseteq> y"   | "y \<sqsubseteq> x"
subcls1_induct: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and Q :: "char list \<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"
insert_union: fixes x :: "'e"   and X :: "'e set" shows "insert x X = X \<union> {x}"
Suc_k_less_ncols': shows "Suc k < ncols (snd (snd f))"
strict_prefix_lexord_rel: fixes xs :: "'b list"   and ys :: "'b list"   and R :: "('b \<times> 'b) set" assumes "strict_prefix xs ys" shows "(xs, ys) \<in> lexord R"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'a"   and f3 :: "'a"   and action :: "'b action" shows "P (case action of NormalAction x \<Rightarrow> f1 x | InitialThreadAction \<Rightarrow> f2 | ThreadFinishAction \<Rightarrow> f3) = ((\<forall>x1. action = NormalAction x1 \<longrightarrow> P (f1 x1)) \<and> (action = InitialThreadAction \<longrightarrow> P f2) \<and> (action = ThreadFinishAction \<longrightarrow> P f3))"
e: shows "1 < e \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
measure_subadditive_finite: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "finite I"   and "A ` I \<subseteq> sets M"   and "\<And>i. i \<in> I \<Longrightarrow> emeasure M (A i) \<noteq> \<infinity>" shows "Sigma_Algebra.measure M (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. Sigma_Algebra.measure M (A i))"
set_annul_privilege: fixes s' :: "'b sparc_state"   and b :: "bool"   and s :: "'b sparc_state" assumes "s' = snd (fst (set_annul b s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
gcd_diff_r: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "Arith2.gcd m n = Arith2.gcd m (n - m)"
MtransC_induct: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and phi :: "('test, 'atom) com \<times> 'state \<Rightarrow> ('test, 'atom) com \<times> 'state \<Rightarrow> bool" assumes "cf \<rightarrow>*c cf'"   and "\<And>cf. phi cf cf"   and "\<And>cf cf' cf''. \<lbrakk>cf \<rightarrow>*c cf'; phi cf cf'; cf' \<rightarrow>c cf''\<rbrakk> \<Longrightarrow> phi cf cf''" shows "phi cf cf'"
Rc_def: shows "Rc = {(V', V''). (V', V'') \<in> Rc' \<and> set (PPV V') \<subseteq> set (PPc c) \<and> set (PPV V'') \<subseteq> set (PPc c)}"
interfree_aux_right_def: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set" shows "interfree_aux_right \<Gamma> \<Theta> F \<equiv> \<lambda>(q, cmd, ann). \<forall>aa ac. atomicsR \<Gamma> \<Theta> ann cmd (aa, ac) \<longrightarrow> \<Gamma> \<Turnstile>\<^bsub>/F\<^esub> (q \<inter> aa) ac q, q"
simps: fixes n :: "nat"   and st :: "'b status" shows "steps n st = (exec ^^ n) st"
assms_5: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "I (\<Gamma>, App e x, S)" shows "P (\<Gamma>, App e x, S) (\<Gamma>, e, Arg x # S)"
cong: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and h :: "'d \<Rightarrow> 'e" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sum g A = sum h B"
distinct_5: fixes x2 :: "'g"   and x3 :: "'h" shows "sum_bot.Left x2 \<noteq> sum_bot.Right x3"
extend_preserves: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and G :: "'c program"   and v :: "'c \<Rightarrow> 'f" assumes "Extend h" shows "(extend h G \<in> preserves (v \<circ> (\<lambda>z. fst (inv h z)))) = (G \<in> preserves v)"
real_of_float: fixes x :: "float" shows "real_of_float x \<in> float"
rk_Aa: fixes A :: "Points"   and B :: "Points"   and P :: "Points"   and A' :: "Points"   and Q :: "Points"   and a :: "Points"   and C :: "Points"   and B' :: "Points"   and C' :: "Points" assumes "rk {A, B, P} = 3"   and "rk {A, A'} = 2"   and "rk {A, A', P} = 2"   and "rk {Q, A', a} = 2"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {A, B, C, A', B', C'} \<le> 3" shows "rk {A, a} = 2"
pseudonaturally_equivalent_def: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd" shows "pseudonaturally_equivalent V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<equiv> \<exists>\<tau>\<^sub>0 \<tau>\<^sub>1. pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
homologous_rel_singular_chain: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S a b" shows "singular_chain p X a = singular_chain p X b"
carr: shows "set (basis_completion v) \<subseteq> carrier_vec n"
finite_nat_seg: fixes n :: "nat" assumes "class.countable TYPE('a)" shows "finite {x. countable.elem_index x < n}"
mem_restr_def: fixes A :: "'a list set"   and x :: "'a option list" shows "mem_restr A x = (\<exists>y\<in>A. list_all2 (\<lambda>a b. a \<noteq> None \<longrightarrow> a = Some b) x y)"
absorb_def: fixes d :: "('k, 'l) distinguisher"   and conv :: "('k, 'l, 'i, 'j) converter" shows "absorb d conv = map_gpv fst id (inline run_converter d conv)"
yys_gt_xxs: shows "msetext_huet gt (y # ys) (x # xs)"
sublocale_sas_plus_finite_domain_representation_ii_2: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>op\<in>set (\<Psi>\<^sub>\<O>\<^sub>+). is_valid_operator_sas_plus \<Psi> op"
simps: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list" shows "scanl1 f xs = map (\<lambda>k. foldl1 f (take k xs)) [1..<length xs + 1]"
diff_delta_arctan_upper_33_def: shows "diff_delta_arctan_upper_33 \<equiv> \<lambda>x. 64 * x ^ 10 / ((15 * x ^ 4 + 70 * x\<^sup>2 + 63)\<^sup>2 * (1 + x\<^sup>2))"
mat_eq_iff: fixes x :: "'b mat"   and y :: "'b mat" shows "(x = y) = (dim_row x = dim_row y \<and> dim_col x = dim_col y \<and> (\<forall>i j. i < dim_row y \<longrightarrow> j < dim_col y \<longrightarrow> x $$ (i, j) = y $$ (i, j)))"
assms_4: shows "W \<subseteq> U \<inter> V"
pigeonhole_infinite: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "infinite A"   and "finite (f ` A)" shows "\<exists>a0\<in>A. infinite {a \<in> A. f a = f a0}"
reachable_nodes_step'_2: fixes u :: "'a"   and g :: "'a ugraph"   and r :: "'a"   and v :: "'a" assumes "u \<in> reachable_nodes g r"   and "(u, v) \<in> edges g" shows "(u, v) \<in> edges (component_of g r)"
drop_all: fixes xs :: "'b list"   and n :: "nat" assumes "\<^bold>|xs\<^bold>| \<le> n" shows "drop n xs = \<epsilon>"
Union_upper: fixes B :: "'c set"   and A :: "'c set set" assumes "B \<in> A" shows "B \<subseteq> \<Union> A"
permutationInvariance: fixes l :: "'b list" shows "\<forall>n. set (perm2 l n) = set l"
IH: fixes c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and d :: "real"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and x :: "nat \<Rightarrow> real"   and y :: "nat \<Rightarrow> real" assumes "simplicial_chain p S c"   and "\<And>f x y. \<lbrakk>f \<in> Poly_Mapping.keys c; x \<in> standard_simplex p; y \<in> standard_simplex p\<rbrakk> \<Longrightarrow> \<bar>f x k - f y k\<bar> \<le> d"   and "f \<in> Poly_Mapping.keys (simplicial_subdivision p c)"   and "x \<in> standard_simplex p"   and "y \<in> standard_simplex p" shows "\<bar>f x k - f y k\<bar> \<le> real p / real (Suc p) * d"
phiDefs_def: fixes g :: "'g"   and n :: "'node" shows "braun_ssa.phiDefs g n \<equiv> {v. (n, v) \<in> dom (phis' g)}"
simps_2: fixes r :: "real"   and n :: "nat" shows "det_discount r (Suc n) = inverse (1 + r) * det_discount r n"
map_indets_sum: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and A :: "'e set" shows "map_indets f (sum g A) = (\<Sum>a\<in>A. map_indets f (g a))"
less_transition_ext_def: fixes t1 :: "'a transition_scheme"   and t2 :: "'a transition_scheme" shows "(t1 < t2) = ((Label t1, Arity t1, Guards t1, Outputs t1, Updates t1, more t1) < (Label t2, Arity t2, Guards t2, Outputs t2, Updates t2, more t2))"
A_2: fixes n :: "nat" shows "\<exists>N. A \<inter> {N..} \<subseteq> C n"
lookup_default_def: fixes d :: "'d"   and m :: "('e, 'd) mapping"   and k :: "'e" shows "Mapping.lookup_default d m k = (case Mapping.lookup m k of None \<Rightarrow> d | Some v \<Rightarrow> v)"
prime_elem_associated: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "comm_semiring_1.prime_elem times one zero q"   and "dvd.dvd times q p" shows "normalize q = normalize p"
pass12_merge_pairs: fixes hs :: "'b heap list" shows "pass\<^sub>2 (pass\<^sub>1 hs) = merge_pairs hs"
inv_solve_right: fixes G :: "('d, 'e) monoid_scheme"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(a = b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
linepath_cong: fixes a :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "a = a'"   and "b = b'" shows "linepath a b = linepath a' b'"
var_not_in_eval3: fixes var :: "nat"   and \<phi> :: "atom fm"   and xs' :: "real list"   and L :: "real list" assumes "freeIn var \<phi>"   and "length xs' = var" shows "(\<exists>x. eval \<phi> (xs' @ x # L)) = (\<forall>x. eval \<phi> (xs' @ x # L))"
bound_of_least: fixes b :: "real"   and P :: "'b \<Rightarrow> real" assumes "bounded_by b P" shows "bound_of P \<le> b"
simps_1: fixes a :: "int"   and b :: "int" shows "Pc a \<langle>+\<rangle> Pc b = Pc (a + b)"
smallo: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "g \<in> o(\<lambda>x. exp ((1 + \<epsilon>) * x))"
g''_def: shows "g'' = snd (snd (splitFace g' v a f' ws))"
relevant_vars_subt: fixes t :: "fterm"   and \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm" assumes "\<forall>x\<in>vars\<^sub>t t. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x" shows "t \<cdot>\<^sub>t \<sigma>\<^sub>1 = t \<cdot>\<^sub>t \<sigma>\<^sub>2"
D: shows "(xs @ ipurge_tr I D (D y) ys, {}) \<in> failures P"
dehomogenize_extend_indets: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "dehomogenize None (extend_indets p) = extend_indets p"
assms: fixes a :: "'a poly" assumes "a \<in># as" shows "monic a"
Sup_less_assert: fixes p :: "'b \<Rightarrow> 'a"   and w :: "'b" shows "Sup_less (\<lambda>w. {. p w .}) w = {. Sup_less p w .}"
contf': shows "continuous_on (S \<inter> f -` C) f"
Collect_empty_eq: fixes P :: "'b \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"