hcju/isabelleps huggingface dataset:
IH: shows "trunc_ell2 F x \<in> cspan (range ket)"
cases: fixes a1 :: "'a"   and a2 :: "'a" assumes "(a1, a2) \<in> hull_rel" obtains p :: "perm" and x :: "'a" where "a1 = p \<bullet> x" and "a2 = x"
CFGExit_wf_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b" shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit \<equiv> CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val \<and> CFGExit sourcenode targetnode kind valid_edge Entry Exit \<and> CFGExit_wf_axioms Def Use Exit"
invariant_def: fixes A :: "('c, 'd) ioa"   and P :: "'c \<Rightarrow> bool" shows "invariant A P \<equiv> \<forall>s. reachable A s \<longrightarrow> P s"
Zmax_geTr: fixes f :: "nat \<Rightarrow> int"   and n :: "nat" shows "f \<in> {j. j \<le> n} \<rightarrow> UNIV \<longrightarrow> (\<forall>j\<in>{j. j \<le> n}. f j \<le> Zmax n f)"
G'_fields_2: shows "g_V0 G' = f ` V0"
absorb_iff1: fixes b :: "'t"   and a :: "'t" shows "(b \<preceq>\<^sub>t a) = (ord_term_lin.max a b = a)"
keys_uminus: fixes p :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys (- p) = keys p"
sqrt_int_pos: fixes x :: "int"   and s :: "int"   and ms :: "int list" assumes "sqrt_int x = s # ms" shows "0 \<le> s"
simps_5: fixes F :: "'b formula" shows "nnf (\<^bold>\<not> (\<^bold>\<not> F)) = nnf F"
kinSuci: shows "z \<in> set (take (Suc i) cs)"
NextMem_def: fixes SR :: "(('c \<times> ('a \<Rightarrow> 'b)) \<times> 'c list \<times> 'c option \<times> ('a \<Rightarrow> 'b)) set"   and c :: "'c"   and m :: "'a \<Rightarrow> 'b" shows "WHATWHERE.NextMem SR c m \<equiv> THE m'. \<exists>p \<alpha>. ((c, m), \<alpha>, p, m') \<in> SR"
ta_bisim_init_fin_bisim_ThreadFinishAction: fixes final1 :: "'b \<Rightarrow> bool"   and final2 :: "'d \<Rightarrow> bool"   and bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool" shows "ta_bisim (FWbisimulation_base.init_fin_bisim final1 final2 bisim) \<lbrace>ThreadFinishAction\<rbrace> \<lbrace>ThreadFinishAction\<rbrace>"
ldropn_def: fixes n :: "nat"   and xs :: "'a llist" shows "ldropn n xs = (ltl ^^ n) xs"
bound_on_all_plans_bounds_problem_plan_bound_thesis: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and k :: "nat" assumes "finite PROB"   and "\<forall>as s. s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' \<le> k)" shows "problem_plan_bound PROB \<le> k"
comm_monoid_add_ow_axioms: shows "comm_monoid_add_ow U (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w"
index_one_mat_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "1\<^sub>m n $$ (i, j) = (if i = j then 1::'c else (0::'c))"
cyclic_moduli_def: shows "cyclic_moduli = {1, 2, 4} \<union> {p ^ k |p k. prime p \<and> odd p \<and> 0 < k} \<union> {2 * p ^ k |p k. prime p \<and> odd p \<and> 0 < k}"
list_id_1: shows "[0..<Suc (i - 1)] = [0..<i - 1] @ [i - 1]"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "at_begin_norm (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 rn. lm = lm1 @ lm2 \<and> length lm1 = s \<and> (if lm1 = [] then l = Bk # Bk # ires else l = Bk # <rev lm1> @ Bk # Bk # ires) \<and> r = <lm2> @ Bk \<up> rn)"
eq3: shows "a ^ n * \<integral> \<chi> {w. a \<le> f w} \<partial>M = \<integral> (\<lambda>t. a ^ n * \<chi> {w. a \<le> f w} t) \<partial>M"
assms: shows "s \<in> carrier R\<^bsup>\<omega>\<^esup>"
divide_geq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "rat"   and a :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less_eq b (scaleRat c a)"   and "0 < c" shows "less_eq (scaleRat (1 / c) b) a"
aLeq2: shows "- 9223372039002259455 \<le> - (2 ^ 31)"
cnj_minus: fixes x :: "quat" shows "Quaternions.cnj (- x) = - Quaternions.cnj x"
eq_transfer: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "bi_unique A" shows "(A ===> A ===> (=)) (=) (=)"
hyperprime_factor_exists: fixes n :: "nat star" assumes "1 < n" shows "\<exists>k\<in>starprime. k dvd n"
progress: fixes s :: "nat \<Rightarrow> nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and t :: "nat"   and i :: "nat" assumes "A.is_fair_run s"   and "t < N"   and "is_WAIT (tts (s i) t)" shows "\<exists>j>i. is_HOLD (tts (s j) t)"
domains_def: fixes f :: "'a2" shows "C2.domains f \<equiv> {a. C2.ide a \<and> f \<cdot>\<^sub>2 a \<noteq> C2.null}"
lossless_resource_mono: fixes \<I>' :: "('c, 'd) \<I>"   and res :: "('c, 'd) resource"   and \<I> :: "('c, 'd) \<I>" assumes "lossless_resource \<I>' res"   and "outs_\<I> \<I> \<subseteq> outs_\<I> \<I>'" shows "lossless_resource \<I> res"
distinct_10: fixes x4 :: "Vars.var"   and x2 :: "Vars.var" shows "Dummy x4 \<noteq> Arg x2"
weaken_proves: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and A :: "Core.term" assumes "\<Theta>,\<Gamma> \<turnstile> B"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "finite \<Gamma>" shows "\<Theta>,insert A \<Gamma> \<turnstile> B"
multi_member_last: fixes x :: "'b" shows "x \<in># x, {#}"
not_empty_eq_Iic_eq_empty: fixes b :: "'b"   and h :: "'b" shows "{} \<noteq> flat_ord.atMost b h"
not_None_eq: fixes x :: "'l option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
pivot_tableau_valuated: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "Pivot eq_idx_for_lvar pivot"   and "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)"   and "\<nabla> s" shows "\<nabla> (pivot x\<^sub>i x\<^sub>j s)"
eval_tm_inject: fixes e :: "(name, hf) finfun"   and t :: "tm"   and u :: "tm" shows "(\<lbrakk>\<guillemotleft>t\<guillemotright>\<rbrakk>e = \<lbrakk>\<guillemotleft>u\<guillemotright>\<rbrakk>e) = (t = u)"
word_unat_eq_iff: fixes v :: "'d word"   and w :: "'d word" shows "(v = w) = (unat v = unat w)"
ab: shows "normalise_decomp (x div y) y n = (a, b)"
rPar1: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'" shows "Prop \<Psi> (P \<parallel> !P) M N (P' \<parallel> !P)"
that_2: fixes r :: "'a" assumes "x = dRETURN r" shows "thesis"
in_hhomE: fixes V :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and src :: "'r \<Rightarrow> 'r"   and trg :: "'r \<Rightarrow> 'r"   and \<mu> :: "'r"   and a :: "'r"   and b :: "'r" assumes "horizontal_homs V src trg"   and "horizontal_homs.in_hhom V src trg \<mu> a b" obtains "partial_magma.arr V \<mu>" and "horizontal_homs.obj V src a" and "horizontal_homs.obj V src b" and "src \<mu> = a" and "trg \<mu> = b"
ncoplanar_perm_7: 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 C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet B A D C"
le_less_trans: fixes x :: "'b Word.word"   and y :: "'b Word.word"   and z :: "'b Word.word" assumes "x \<le>s y"   and "y <s z" shows "x <s z"
covector_conv_vector: fixes v :: "'a" shows "covector v = vector (v\<^sup>T)"
x': shows "x = \<lbrakk>quot_dbfm B'\<rbrakk>e"
elims: fixes x :: "'a extB"   and y :: "'a extB" assumes "x\<^sup>T = y" obtains x :: "'a extB" where "xa__ = x" and "y = x"
bicategory_axioms: shows "bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
z_def: shows "z = Octo 0 1 0 1 1 1 0 0"
preserves_cod: fixes f :: "'a" assumes "B.arr f" shows "B.cod (B.\<rr> f) = B.map (B.cod f)"
emb_prj_emb: fixes x :: "'a" assumes "DEFL('a) \<sqsubseteq> DEFL('b)" shows "EMB('b)\<cdot>(PRJ('b)\<cdot>(EMB('a)\<cdot>x)) = EMB('a)\<cdot>x"
rank_meldUniq: fixes bq1 :: "('c, 'd) SkewBinomialTree list"   and bq2 :: "('c, 'd) SkewBinomialTree list" assumes "rank_invar bq1"   and "rank_invar bq2" shows "rank_invar (meldUniq bq1 bq2)"
acc_lengths_append: fixes acc :: "nat"   and xs :: "'a list list"   and ys :: "'a list list" shows "acc_lengths acc (xs @ ys) = acc_lengths acc xs @ acc_lengths (acc + sum_list (map length xs)) ys"
finite_Collect_subsets: fixes A :: "'a set" assumes "finite A" shows "finite {B. B \<subseteq> A}"
submap_imp_state_succ_submap_b: fixes s1 :: "('a, 'b) fmap"   and s2 :: "('a, 'b) fmap"   and s0 :: "('a, 'b) fmap" assumes "s1 \<subseteq>\<^sub>f s2" shows "s0 ++ s1 \<subseteq>\<^sub>f s0 ++ s2"
ring_distribs_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> (b + c) = a \<cdot> b + a \<cdot> c"
bot_stone_phi_pair_def: shows "bot \<equiv> Abs_stone_phi_pair stone_phi.pairs_bot"
condition1_part3: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and a :: "'f"   and b :: "'f"   and m :: "'f" assumes "echelon_form_upt_k A k"   and "is_bezout_ext bezout"   and "is_zero_row_upt_k a (Suc k) (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k))"   and "a < b"   and "\<forall>m>mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1). A $ m $ mod_type_class.from_nat k = (0::'d)"   and "(if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) \<noteq> nrows A"   and "mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) \<le> m"   and "A $ m $ mod_type_class.from_nat k \<noteq> (0::'d)" shows "is_zero_row_upt_k b (Suc k) A"
test_bit_cong: fixes xa :: "'c"   and y :: "'c"   and x :: "nat" assumes "xa = y" shows "xa !! x = y !! x"
nn_integral_pmf_of_set: fixes f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf local.pmf_of_set) f = sum f S / of_nat (card S)"
closure_Un: fixes S :: "'a set"   and T :: "'a set" shows "closure (S \<union> T) = closure S \<union> closure T"
evaluate_net_Conv_id: fixes m :: "real mat convnet"   and input :: "real vec list"   and j :: "nat"   and nr :: "nat" assumes "valid_net' m"   and "input_sizes m = map dim_vec input"   and "j < nr" shows "evaluate_net (Conv (id_matrix nr (output_size' m)) m) input $ j = (if j < output_size' m then evaluate_net m input $ j else 0)"
ring_hom_memE_2: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
assms_1: shows "on_triple R `` E \<subseteq> G"
assms_2: shows "diff_invariant (\<lambda>s. \<mu> s < \<nu> s) f U S t\<^sub>0 G"
merge_eq_val_merge_tm: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list" shows "Time_Monad.val (merge_tm f xs ys) = Common.merge f xs ys"
comp_Rel_components_1: fixes S :: "V"   and T :: "V" shows "(S \<circ>\<^sub>R\<^sub>e\<^sub>l T)\<lparr>ArrVal\<rparr> = S\<lparr>ArrVal\<rparr> \<circ>\<^sub>\<circ> T\<lparr>ArrVal\<rparr>"
irreflPE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "irreflP r" obtains "\<forall>a. \<not> r a a"
delete_point_proper: fixes p :: "'a" assumes "\<And>bl. \<lbrakk>bl \<in># \<B>; p \<in> bl\<rbrakk> \<Longrightarrow> 2 \<le> card bl" shows "proper_design (del_point p) (del_point_blocks p)"
nAct_less: fixes k :: "nat"   and t :: "cnf llist"   and n :: "enat"   and n' :: "enat"   and c :: "'id" assumes "enat k < llength t"   and "n \<le> enat k"   and "enat k < n'"   and "\<parallel>c\<parallel>\<^bsub>lnth t k\<^esub>" shows "\<langle>c #\<^bsub>n\<^esub>t\<rangle> < \<langle>c #\<^bsub>n'\<^esub>t\<rangle>"
iwlsFSbSwTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "iwlsFSbSwTR MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWlsSTR MOD \<and> igSubstAllIPresIGWlsAllSTR MOD \<and> igFreshClsSTR MOD \<and> igSubstClsSTR MOD \<and> igAbsRenSTR MOD"
ennreal_mult_divide_eq: fixes b :: "ennreal"   and a :: "ennreal" assumes "b \<noteq> 0"   and "b \<noteq> \<top>" shows "a * b / b = a"
PSQ2: shows "(Mod_Type_Connect.to_hma\<^sub>m P, Mod_Type_Connect.to_hma\<^sub>m S, Mod_Type_Connect.to_hma\<^sub>m Q) = diagonal_to_Smith_PQ (Mod_Type_Connect.to_hma\<^sub>m A) bezout"
preserves_src: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "src\<^sub>C (F \<mu>) = F.map\<^sub>0 (src\<^sub>D \<mu>)"
eq: shows "null_sets N \<subseteq> null_sets (distr M N f)"
rtranclE: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> r\<^sup>*" obtains "a = b"   | y :: "'a" where "(a, y) \<in> r\<^sup>*" and "(y, b) \<in> r"
dim_subset_UNIV: fixes S :: "('b, 'c) vec set" shows "vec.dim S \<le> finite_dimensional_vector_space.dimension cart_basis"
hom_one: shows "h \<one> = \<one>\<^bsub>H\<lparr>carrier := h ` carrier G\<rparr>\<^esub>"
distinct_57: fixes x5 :: "'a OCL_Types.type"   and x8 :: "'a OCL_Types.type" shows "Set x5 \<noteq> Sequence x8"
fresh_subst_fm_if: fixes j :: "atom"   and A :: "fm"   and i :: "name"   and x :: "tm" shows "j \<sharp> A(i::=x) = (atom i \<sharp> A \<and> j \<sharp> A \<or> j \<sharp> x \<and> (j \<sharp> A \<or> j = atom i))"
R_map: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and r :: "'a"   and m :: "'b" assumes "RModuleHom R smult M smult' T"   and "r \<in> R"   and "m \<in> M" shows "T (smult r m) = smult' r (T m)"
surj_fun_eq: fixes f :: "'r \<Rightarrow> 'q"   and X :: "'r set"   and g1 :: "'q \<Rightarrow> 's"   and g2 :: "'q \<Rightarrow> 's" assumes "f ` X = UNIV"   and "\<forall>x\<in>X. (g1 \<circ> f) x = (g2 \<circ> f) x" shows "g1 = g2"
cr_omega_def: fixes a :: "V"   and b :: "nat" shows "cr_omega a b = (a = b\<^sub>\<nat>)"
is_inf_greatest: fixes x :: "'b"   and y :: "'b"   and inf :: "'b"   and z :: "'b" assumes "is_inf x y inf"   and "z \<sqsubseteq> x"   and "z \<sqsubseteq> y" shows "z \<sqsubseteq> inf"
Trans_fst_sndI: fixes s :: "'state"   and a :: "'act" shows "valid [Trans s a (fst (step s a)) (snd (step s a))]"
filter_inf_right_dist_sup: fixes F :: "'a set"   and G :: "'a set"   and H :: "'a set" assumes "ord_class.filter F"   and "ord_class.filter G"   and "ord_class.filter H" shows "filter_sup F G \<inter> H = filter_sup (F \<inter> H) (G \<inter> H)"
hyps_1: shows "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> M\<lparr>N\<rparr> \<prec> C"
abs_eq_iff: fixes x :: "'f llist \<times> 'g"   and y :: "'f llist \<times> 'g" shows "((case x of (xs, a) \<Rightarrow> tllist_of_llist a xs) = (case y of (xs, a) \<Rightarrow> tllist_of_llist a xs)) = (case x of (xs, a) \<Rightarrow> \<lambda>(ys, b). xs = ys \<and> (lfinite ys \<longrightarrow> a = b)) y"
A3: shows "\<bar>spmf (R1_14_interm2 M (c0, c1) \<bind> D) True - spmf (S1_14 M out \<bind> D) True\<bar> = \<bar>spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). R1_OT12 (m0, m1) c1 \<bind> (\<lambda>view. A3 view (m0, m1)))) True - spmf (pair_spmf coin_spmf coin_spmf \<bind> (\<lambda>(m0, m1). S1_OT12 (m0, m1) () \<bind> (\<lambda>view. A3 view (m0, m1)))) True\<bar>"
poincare_circle_in_disc_3: fixes r :: "real"   and z :: "complex_homo"   and ze :: "complex"   and re :: "real" assumes "0 < r"   and "z \<in> unit_disc"   and "(ze, re) = poincare_circle_euclidean z r" shows "\<forall>x\<in>circle ze re. cmod x < 1"
extremum_uniqueI: fixes a :: "'d" assumes "a \<le> \<bottom>" shows "a = \<bottom>"
trms_of_deg_leq_cfs: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "f \<in> carrier P" shows "trms_of_deg_leq n f k = (if k \<le> n then f k else \<zero>)"
assms_2: shows "pred_stream (\<lambda>s. s \<in> S) xs"
dom_restrict_idem: fixes p :: "'a \<Rightarrow> 'b decision option" shows "dom p \<triangleleft> p = p"
prime_power_iff: fixes p :: "'a"   and n :: "nat" shows "prime (p ^ n) = (prime p \<and> n = 1)"
Sigma_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'c set"   and D :: "'b \<Rightarrow> 'c set" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sigma A C = Sigma B D"
coeff_def: fixes p :: "'a mpoly" shows "MPoly_Type.coeff p = lookup (mapping_of p)"
divisor_count_altdef: fixes n :: "nat" shows "divisor_count n = divisor_sigma 0 n"
apartment_standard_uniqueness_isomorphs: fixes X :: "'a set set"   and W :: "'b set set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and C :: "'b set" assumes "X \<in> \<A>"   and "ChamberComplexIsomorphism W X f"   and "ChamberComplexIsomorphism W X g"   and "SimplicialComplex.maxsimp W C"   and "fun_eq_on f g C" shows "fun_eq_on f g (\<Union> W)"
denote_while_split: fixes M :: "nat \<Rightarrow> complex mat"   and S :: "com"   and \<rho> :: "complex mat" assumes "well_com (While M S)"   and "\<rho> \<in> density_states" shows "denote (While M S) \<rho> = M 0 * \<rho> * adjoint (M 0) + denote (While M S) (denote S (M 1 * \<rho> * adjoint (M 1)))"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map Adj.GF.map Adj.\<eta>"
length_Lev2': fixes kl :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and a :: "'e" assumes "kl \<in> Lev2 s1 s2 n a" shows "kl \<in> Lev2 s1 s2 (length kl) a"
mult_inverse: fixes a :: "'a"   and v :: "'a Matrix.vec"   and w :: "'a Matrix.vec" assumes "a \<in> UNIV"   and "v \<in> carrier_vec n"   and "a \<cdot>\<^sub>v v = w"   and "a \<noteq> (0::'a)" shows "v = (if a = (0::'a) then div0 else inverse a) \<cdot>\<^sub>v w"
thetaSeqWT_WbisT: shows "thetaSeqWT \<subseteq> WbisT"
mkId4: fixes l :: "Location"   and h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "l \<in> Dom h" shows "(l, l) \<in> mkId h"
hpair_neq_Ord: fixes k :: "hf"   and x :: "hf"   and y :: "hf" assumes "Ord k" shows "\<langle>x, y\<rangle> \<noteq> k"
arev_neq: fixes a :: "'b" assumes "a \<in> arcs G" shows "edge_rev M a \<noteq> a"
and_propag_on_create_graph: fixes \<mu> :: "'a ltlr"   and \<eta> :: "'a ltlr" shows "create_graph \<phi> \<le> SPEC (\<lambda>nds. \<forall>nd\<in>nds. \<mu> and\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd)"
and_eq_not_not_or: fixes a :: "'a"   and b :: "'a" shows "a AND b = NOT (NOT a OR NOT b)"
is_type_simps: fixes P :: "'a prog"   and C :: "char list" shows "is_type P Void \<and> is_type P Boolean \<and> is_type P Integer \<and> is_type P NT \<and> is_type P (Class C) = is_class P C"
Reals_cnj_iff: fixes z :: "complex" shows "(z \<in> \<real>) = (cnj z = z)"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "subtract (x # xs) ys = subtract xs (removeAll x ys)"
assms_1: shows "rga_ops (pre @ suf @ [(oid, ref)])"
seq_char: fixes f :: "(bool, unit) arr"   and g :: "(bool, unit) arr" shows "seq f g = (arr f \<and> f = g)"
poly_smult_zero_iff: fixes a :: "'b"   and p :: "'b poly"   and x :: "'b" shows "(poly (smult a p) x = (0::'b)) = (a = (0::'b) \<or> poly p x = (0::'b))"
less_le_trans: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k"   and z :: "'k" assumes "class.preorder less_eq less"   and "less x y"   and "less_eq y z" shows "less x z"
linear_exists_left_inverse_on: fixes f :: "'b \<Rightarrow> 'c"   and V :: "'b set" assumes "linear (*a) (*b) f"   and "vs1.subspace V"   and "inj_on f V" shows "\<exists>g. range g \<subseteq> V \<and> linear (*b) (*a) g \<and> (\<forall>v\<in>V. g (f v) = v)"
simps_3: fixes x :: "nat"   and v :: "exp"   and y :: "nat"   and e :: "exp" shows "subst x v (ELam y e) = (if x = y then ELam y e else ELam y (subst x v e))"
induct: fixes P :: "('t \<Rightarrow>\<^sub>0 'c) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and a0 :: "('t \<Rightarrow>\<^sub>0 'c) list"   and a1 :: "'t \<Rightarrow>\<^sub>0 'c"   and a2 :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>fs p r. \<lbrakk>\<lbrakk>p \<noteq> 0; find_adds fs (lt p) = None\<rbrakk> \<Longrightarrow> P fs (tail p) (r + monomial (lc p) (lt p)); \<And>x2. \<lbrakk>p \<noteq> 0; find_adds fs (lt p) = Some x2\<rbrakk> \<Longrightarrow> P fs (tail p - monom_mult (lc p / lc x2) (lp p - lp x2) (tail x2)) r\<rbrakk> \<Longrightarrow> P fs p r" shows "P a0 a1 a2"
init_def: fixes y :: "'a"   and w :: "nat \<times> nat" shows "init y w = (let (x1, x2) = w in sample_uniform (order \<G>) \<bind> (\<lambda>r1. sample_uniform (order \<G>) \<bind> (\<lambda>r2. return_spmf ((r1, r2), \<^bold>g [^] r1 \<otimes> g' [^] r2))))"
assms_3: shows "\<forall>x. g x = 0 \<longrightarrow> f x = (0::'a)"
image_rev_lists: fixes S :: "'b set" shows "rev ` lists S = lists S"
supp_some: fixes p :: "perm" shows "supp (SOME x. weak_formula x \<and> supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) \<subseteq> supp (p \<bullet> P)"
v'_def: shows "interp.v' \<equiv> \<lambda>i. if i \<le> n then THE c. c \<in> clk_set A \<and> v c = i else x"
apsnd_compose: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and x :: "'c \<times> 'f" shows "apsnd f (apsnd g x) = apsnd (f \<circ> g) x"
paste_sub_Un: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" shows "P +* Q \<subseteq> P \<union> Q"
stable_translation_length_le_translation_length: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "real n * stable_translation_length f \<le> translation_length (f ^^ n)"
simps: fixes xs :: "'a list"   and x :: "'a" shows "iarray_append (IArray xs) x = IArray (xs @ [x])"
simps_2: fixes sys :: "'b \<Rightarrow> ('a, 'c) process_term"   and p :: "'a process" shows "PREF sys (PROC p) = prefOf p"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "C.ide (g \<cdot> f)" obtains "C.seq g f" and "C.seq f g" and "g \<cdot> f = C.dom f" and "g \<cdot> f = C.cod g"
upper_def: shows "upper \<equiv> cindex_pathE (part_circlepath z r 0 pi) z0"
rep_eq: fixes x :: "'d" shows "to_fun (diag x) = (\<lambda>i j. if i = j then x else (0::'d))"
RPD_ok: fixes m :: "nat"   and mr :: "mregex"   and ms :: "mregex" assumes "ok m mr"   and "ms \<in> RPD mr" shows "ok m ms"
updates_2: fixes k :: "nat" assumes "k < m" shows "\<parallel>gs2.gso k\<parallel>\<^sup>2 = (if k = i - 1 then \<parallel>gs1.gso i\<parallel>\<^sup>2 + gs1.\<mu> i (i - 1) * gs1.\<mu> i (i - 1) * \<parallel>gs1.gso (i - 1)\<parallel>\<^sup>2 else if k = i then \<parallel>gs1.gso i\<parallel>\<^sup>2 * \<parallel>gs1.gso (i - 1)\<parallel>\<^sup>2 / \<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 else \<parallel>gs1.gso k\<parallel>\<^sup>2)"