hcju/isabelleps huggingface dataset:
rComm1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P (M\<lparr>N\<rparr>) P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q (K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>) Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "distinct xvec"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
inducts: fixes k :: "nat"   and x :: "nat set"   and P :: "nat set \<Rightarrow> bool" assumes "i_set_multp k x"   and "\<And>n. P [n\<dots>]"   and "\<And>n. P [\<dots>n]"   and "\<And>n d. P [n\<dots>,d]"   and "\<And>r m. P [ r, mod m * k ]"   and "\<And>r m c. P [ r, mod m * k, c ]" shows "P x"
st_projs_compat: shows "\<infinity>-smooth_compat st_proj1_chart st_proj2_chart"
index_transpose_mat_3: fixes A :: "'a mat" shows "dim_col A\<^sup>T = dim_row A"
preserves_hhom: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "\<guillemotleft>F \<mu> : F.map\<^sub>0 (src\<^sub>D \<mu>) \<rightarrow>\<^sub>C F.map\<^sub>0 (trg\<^sub>D \<mu>)\<guillemotright>"
ipresWls_wlsInp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'sort, 'h, 'i, 'd, 'j, 'k) model_scheme" assumes "wlsInp delta inp"   and "ipresWls h MOD" shows "igWlsInp MOD delta (lift h inp)"
unwind_cont_def: fixes \<Delta> :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and \<Delta>s :: "('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) set" shows "unwind_cont \<Delta> \<Delta>s \<equiv> \<forall>s vl s1 vl1. reachNT s \<and> reach s1 \<and> \<Delta> s vl s1 vl1 \<longrightarrow> iaction (disjAll \<Delta>s) s vl s1 vl1 \<or> (vl \<noteq> [] \<or> vl1 = []) \<and> reaction (disjAll \<Delta>s) s vl s1 vl1"
simps: fixes callee :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and c :: "('a, 'c, 'd) gpv"   and s :: "'b" shows "exec_gpv callee c s = bind_spmf (the_gpv c) (case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. bind_spmf (callee s out) (\<lambda>(x, y). exec_gpv callee (c x) y)))"
sorted_rev_subset_len_lt: fixes a :: "'b list"   and b :: "'b list" assumes "Sorted_Less.sorted (rev a)"   and "Sorted_Less.sorted (rev b)"   and "set a \<subset> set b" shows "length a < length b"
preserves_hom: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "\<guillemotleft>\<chi>' f : \<chi>'.A.map a \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] D b\<guillemotright>"
simps_1: fixes r :: "'rule"   and h :: "'form"   and a :: "('form, 'var) antecedent" shows "hyps (Rule r) (Hyp h a) = (if a |\<in>| f_antecedent r \<and> h |\<in>| a_hyps a then Some a else None)"
elims_2: fixes x :: "cmd" assumes "final x" obtains "x = Skip"
hyps_1: fixes x :: "nat"   and y :: "nat" assumes "c = card (older_seniors x n)"   and "y \<in> older_seniors x n"   and "older_seniors x n = older_seniors x (Suc n)" shows "older_seniors y n = older_seniors y (Suc n)"
abs_Tree\<^sub>p_def: shows "abs_Tree\<^sub>p \<equiv> quot_type.abs hull_relp Abs_Tree\<^sub>p"
assms_3: fixes a :: "'c"   and b :: "'f" assumes "(S1 OO S2) a b" shows "S' a b"
p_pow_rep: fixes n :: "nat"   and k :: "nat" shows "([p] \<cdot> \<one> [^] n) k = p ^ n mod p ^ k"
asym: fixes b :: "'a"   and a :: "'a" assumes "b \<sqsubset> a"   and "a \<sqsubset> b" shows "False"
image_empty: fixes f :: "'c \<Rightarrow> 'a" shows "f ` {} = {}"
rel_refl_strong: fixes x :: "('a, 'b, 'c) generat"   and R1a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R3a :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> generat_pures x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> generat_outs x \<Longrightarrow> R2a z2 z2"   and "\<And>z3. z3 \<in> generat_conts x \<Longrightarrow> R3a z3 z3" shows "rel_generat R1a R2a R3a x x"
preciseD: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> assn"   and a :: "'b"   and p :: "'c"   and F :: "assn"   and a' :: "'b"   and F' :: "assn"   and h :: "heap \<times> nat set" assumes "precise R"   and "h \<Turnstile> R a p * F \<and>\<^sub>A R a' p * F'" shows "a = a'"
acc_le_optI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "acc r" shows "acc (Opt.le r)"
induct: fixes x :: "'val"   and g :: "'g"   and \<phi> :: "'val"   and P :: "'val \<Rightarrow> bool" assumes "x \<in> reachable g \<phi>"   and "unnecessaryPhi g \<phi> \<Longrightarrow> P \<phi>"   and "\<And>\<phi>' \<phi>'''. \<lbrakk>\<phi>' \<in> reachable g \<phi>; P \<phi>'; phiArg g \<phi>' \<phi>'''; unnecessaryPhi g \<phi>'''\<rbrakk> \<Longrightarrow> P \<phi>'''" shows "P x"
convergent_limsup_cl: fixes X :: "nat \<Rightarrow> 'b" assumes "convergent X" shows "limsup X = lim X"
mat_of_rows_index: fixes i :: "nat"   and rs :: "'a vec list"   and j :: "nat"   and n :: "nat" assumes "i < length rs"   and "j < n" shows "mat_of_rows n rs $$ (i, j) = rs ! i $ j"
Fun_comp: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and g :: "'b"   and f :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (S g f)" shows "set_category_given_img.Fun S img (S g f) = restrict (set_category_given_img.Fun S img g \<circ> set_category_given_img.Fun S img f) (set_category_data.set S img (partial_magma.dom S f))"
g_bounded: shows "\<forall>\<^sub>F a in at_top. \<forall>b>a. \<exists>c. \<forall>x\<in>{a..b}. g' x \<le> c"
additive_wp_Bind: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. Transformers.additive (wp (a (f x)))" shows "Transformers.additive (wp (Bind f a))"
composite_simps_4: shows "trg ((g \<star> f) \<star> g) = trg g"
not_square_less_zero: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "\<not> less (times a a) zero"
uminus_l_inv_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "- A + A = 0\<^sub>m nr nc"
simps_2: fixes v :: "char"   and va :: "char list" shows "(Iface (v # va) \<le> Iface []) = False"
minim_isMinim: fixes r :: "('d \<times> 'd) set"   and B :: "'d set" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "B \<noteq> {}" shows "isMinim r B (minim r B)"
cov: fixes i :: "'a" assumes "i \<in> K" shows "\<exists>j. j \<in> C \<and> \<not> disjnt (cball (a i) (r i)) (cball (a j) (r j)) \<and> cball (a i) (r i) \<subseteq> ball (a j) (5 * r j)"
preserves_dom: fixes f :: "'c" assumes "C.arr f" shows "D.dom (G\<epsilon>'_\<eta>G.map f) = G (C.dom f)"
unconst_const: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unconst (const name) = Some name"
sign_changes_inf_def: fixes ps :: "'a poly list" shows "sign_changes_inf ps = length (remdups_adj (list_neq (map poly_inf ps) (0::'a))) - 1"
comp_galcon_right_unit: fixes F :: "('d, 'e, unit, unit) galcon" shows "F \<circ>\<^sub>g I\<^sub>g \<X>\<^bsub>F\<^esub> = F"
binop_dfa_accepts: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and B :: "nat bdd list \<times> bool list"   and bss :: "bool list list"   and f :: "bool \<Rightarrow> bool \<Rightarrow> bool" assumes "wf_dfa A n"   and "wf_dfa B n"   and "list_all (is_alph n) bss" shows "dfa_accepts (binop_dfa f A B) bss = f (dfa_accepts A bss) (dfa_accepts B bss)"
typ_of_eq_ax_5: shows "typ_of eq_elim_ax = Some propT"
\<R>\<^sub>S_id: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option" shows "\<R>\<^sub>S id id \<sigma> = \<sigma>"
i_not_k: shows "i \<noteq> mod_type_class.from_nat (Suc k)"
H\<^sub>R_def: fixes f :: "'a" shows "H\<^sub>R f \<equiv> \<lambda>\<mu>. \<mu> \<star> f"
call_return_edges: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and a' :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "a' \<in> get_return_edges a" shows "\<exists>Q' f'. kind a' = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"
assms: shows "of_int x / of_int y \<in> \<int>"
interior_inside_frontier: fixes S :: "'a set" assumes "bounded S" shows "interior S \<subseteq> inside (frontier S)"
inline_parametric: fixes S :: "'a \<Rightarrow> 'f \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'g \<Rightarrow> bool"   and C' :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and A :: "'i \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun C (rel_gpv (rel_prod (=) S) C'))) (rel_fun (rel_gpv A C) (rel_fun S (rel_gpv (rel_prod A S) C'))) inline inline"
rar_ar: fixes x :: "'a" shows "r (ar x) = ar x"
one_integer_mod_group: fixes n :: "nat" shows "\<one>\<^bsub>Z n\<^esub> = 0"
weakCongSimulation_def: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q \<equiv> \<forall>Q'. \<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q' \<longrightarrow> (\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel)"
Subject_reductions: 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 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "P,E,s \<turnstile> e : T \<surd>" shows "P,E,hp s' \<turnstile> e' :\<^bsub>NT\<^esub> T"
divides_mult: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a dvd c"   and "b dvd c"   and "coprime a b" shows "a * b dvd c"
eval_wp_Embed: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "wp (Embed t) = t"
erH_mult: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and h :: "'a \<Rightarrow> 'e"   and p :: "'a"   and q :: "'a" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B"   and "p \<in> carrier R"   and "q \<in> carrier R" shows "erH R S X A B Y h (p \<cdot>\<^sub>r q) = erH R S X A B Y h p \<cdot>\<^sub>r\<^bsub>A\<^esub> erH R S X A B Y h q"
THE_DRES_def: fixes xi :: "'a option" shows "THE_DRES xi \<equiv> case xi of None \<Rightarrow> dSUCCEED | Some x \<Rightarrow> dRETURN x"
assms_4: shows "\<forall>z\<in>S. dist a x \<le> dist a z"
cdiv_in_iff': fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "((\<lambda>x. f x / c) \<in> O[F](g)) = (f \<in> O[F](g))"
sum_is_submodule: fixes N1 :: "'c set"   and N2 :: "'c set" assumes "LinearCombinations.submodule R N1 M"   and "LinearCombinations.submodule R N2 M" shows "LinearCombinations.submodule R (submodule_sum N1 N2) M"
orientable_symmetric: fixes x :: "'a" assumes "bounded_distrib_allegory.orientable (\<sqinter>) (\<squnion>) bot id x" shows "local.comp_inf.symmetric x"
All_rev_conv: fixes P :: "'b list \<Rightarrow> bool" shows "(\<forall>x. P (rev x)) = (\<forall>y. P y)"
callee: fixes x :: "'b"   and s :: "'a" assumes "x \<in> outs_\<I> \<I>" shows "lossless_spmf (callee s x)"
rel_gpv'_def: fixes A :: "'e \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_gpv' A = rel_gpv A (=)"
\<tau>_def: shows "\<tau> = gauss_sum 1 / complex_of_real (sqrt (real n))"
bounded_clinear_intro: fixes f :: "'b \<Rightarrow> 'c"   and K :: "real" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>r x. f (r *\<^sub>C x) = r *\<^sub>C f x"   and "\<And>x. norm (f x) \<le> norm x * K" shows "bounded_clinear f"
scaleR_poly_mapping_def: fixes r :: "real"   and x :: "'a \<Rightarrow>\<^sub>0 'b" shows "r *\<^sub>R x \<equiv> Abs_poly_mapping (\<lambda>i. r *\<^sub>R poly_mapping.lookup x i)"
consec_un_element2: fixes i :: "nat_int"   and j :: "nat_int"   and n :: "nat" shows "consec i j \<and> n \<^bold>\<in> j \<longrightarrow> n \<^bold>\<in> i \<squnion> j"
left_neutral: fixes a :: "'b" shows "\<bottom> \<squnion> a = a"
measure_times: fixes A :: "'d \<Rightarrow> 'e set" assumes "\<And>i. i \<in> Basis \<Longrightarrow> A i \<in> sets lborel" shows "emeasure (Pi\<^sub>M Basis (\<lambda>_. lborel)) (Pi\<^sub>E Basis A) = (\<Prod>i\<in>Basis. emeasure lborel (A i))"
UP_is_fully_faithful_functor: shows "fully_faithful_functor (\<cdot>) (\<cdot>\<^sub>S) S.UP"
tab_def: shows "tab = fst (prod_dfs A B (0, 0))"
inv_QP: shows "invertible (Q ** row_add (mat (1::'b)) (mod_type_class.from_nat n) i ((res (A $ i $ j) (A $ mod_type_class.from_nat n $ j) - A $ mod_type_class.from_nat n $ j) div A $ i $ j))"
extract_var_finite_set: fixes p :: "'a mpoly"   and S :: "(nat \<Rightarrow>\<^sub>0 nat) set"   and v :: "nat" assumes "{m'. coeff p m' \<noteq> (0::'a)} \<subseteq> S"   and "finite S" shows "extract_var p v = (\<Sum>m\<in>S. monom (remove_key v m) (monom (monomial (lookup m v) v) (coeff p m)))"
is_extensional: fixes f :: "'a" assumes "\<not> FG.A.arr f" shows "G f = FG.B.null"
disjoint_ballI: fixes r :: "real"   and s :: "real"   and x :: "'a"   and y :: "'a" assumes "r + s \<le> dist x y" shows "ball x r \<inter> ball y s = {}"
components_pmdl: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and B :: "('a \<Rightarrow>\<^sub>0 'd) set" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.component_of_term pair_of_term ` Keys (module.span (term_powerprod.mult_scalar pair_of_term term_of_pair) B) = term_powerprod.component_of_term pair_of_term ` Keys B"
coprime_prod_list: fixes ps :: "real poly list"   and q :: "real poly"   and p :: "real poly" assumes "\<And>p. p \<in> set ps \<Longrightarrow> p \<noteq> 0"   and "comm_monoid_mult_class.coprime (prod_list ps) q"   and "p \<in> set ps" shows "comm_monoid_mult_class.coprime p q"
max_tstp_idem': fixes x :: "nat + nat"   and y :: "nat + nat" shows "max_tstp x (max_tstp x y) = max_tstp x y"
range_inj_infinite: fixes f :: "nat \<Rightarrow> 'b" assumes "inj f" shows "infinite (range f)"
matrix_to_iarray_det_euclidean_ring: fixes A :: "(('b, 'c) vec, 'c) vec" shows "det A = det_iarrays_rings (matrix_to_iarray A)"
dist_triangle2: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.metric_space dist uniformity open" shows "dist x y \<le> dist x z + dist y z"
poly_to_mpoly_def: fixes v :: "nat"   and p :: "'b poly" shows "poly_to_mpoly v p = MPoly (Abs_poly_mapping (\<lambda>m. poly.coeff p (lookup m v) when keys m \<subseteq> {v}))"
simps: fixes a1 :: "'node"   and a2 :: "'var option"   and a3 :: "'edge list"   and a4 :: "'node" shows "DynPDG_edge a1 a2 a3 a4 = ((\<exists>n n' as. a1 = n \<and> a2 = None \<and> a3 = as \<and> a4 = n' \<and> n controls n' via as) \<or> (\<exists>n V n' as. a1 = n \<and> a2 = Some V \<and> a3 = as \<and> a4 = n' \<and> n influences V in n' via as))"
abs_eq: fixes xa :: "hmultiset zmultiset"   and x :: "hmultiset zmultiset" shows "(ZHMSet xa \<le> ZHMSet x) = (xa \<le> x)"
fsymsPB_def: fixes \<Phi> :: "('b, 'c) lit list set" shows "fsymsPB \<Phi> = \<Union> {fsymsC c |c. c \<in> \<Phi>}"
assms_1: shows "h \<turnstile> f \<rightarrow>\<^sub>h h'"
distinct_split2: fixes xs :: "'a list"   and y :: "'a list"   and r :: "'a"   and z :: "'a list" assumes "distinct xs"   and "xs = y @ [r] @ z" shows "r \<notin> set z"
atom_map_top_pres: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "bounded_lattice.atom_map less_eq less bot top = Collect (bounded_lattice.atom less bot)"
discrete_market_of_def: fixes S :: "'d set"   and A :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" shows "discrete_market_of S A = Abs_discrete_market (if stk_strict_subs S then S else {}, A)"
gamma_f_Top: shows "\<gamma>\<^sub>f \<top> = UNIV"
innermost_intvl: fixes \<phi> :: "atom fm"   and xs :: "'a list"   and l :: "'a"   and x :: "'a"   and y :: "'a" assumes "nqfree \<phi>"   and "nolb \<phi> xs l x"   and "l < x"   and "x \<notin> EQ \<phi> xs"   and "DLO.I \<phi> (x # xs)"   and "l < y"   and "y \<le> x" shows "DLO.I \<phi> (y # xs)"
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 H \<Phi>\<^sub>H \<rho>\<^sub>0 \<rho>\<^sub>1"
universe_absorbing: fixes x :: "i" assumes "CEMU" shows "x \<oplus> u = u"
MakeFaceFinal_faces_subset: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "faces_subset (makeFaceFinal f g)"
ideI: fixes a :: "'a" assumes "\<guillemotleft>a : trg e \<rightarrow> trg (trg f)\<guillemotright>"   and "ide a" shows "hom.ide a"
affine_unop: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and d :: "real"   and p :: "nat" assumes "x \<in> aform_err e X"   and "\<bar>f x - (a * x + b)\<bar> \<le> d"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "f x \<in> aform_err e (affine_unop p a b d X)"
ir_While_backwards_both: fixes P :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c :: "com"   and c' :: "com" assumes "\<And>n. ir_hoare (\<lambda>s s'. P n s s' \<and> bval b s \<and> bval b' s') c c' (P (Suc n))" shows "ir_hoare (P 0) (WHILE b DO c) (WHILE b' DO c') (\<lambda>s s'. \<exists>n. P n s s' \<and> \<not> bval b s \<and> \<not> bval b' s')"
iteratei_rule_P: assumes "set_iterator_genord it S0 (\<lambda>_ _. True)"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>True; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S - {x}. True; \<forall>y\<in>S0 - S. True\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> True; I S \<sigma>; \<forall>x\<in>S. \<forall>y\<in>S0 - S. True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
dgrad_sig_setD_lp: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and d :: "'a \<Rightarrow> nat" assumes "p \<in> dgrad_sig_set' j d" shows "d (lp p) \<le> dgrad_max d"
mgenerator0_1: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K (Suc n) P"   and "ideal (O\<^bsub>K P Suc n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P Suc n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P Suc n\<^esub>)"   and "j \<le> Suc n" shows "(\<nu>\<^bsub>K P j\<^esub>) (mprod_exp K (mL K P I) (Kb\<^bsub>K Suc n P\<^esub>) (Suc n)) = (\<nu>\<^bsub>K P j\<^esub>) (Zl_mI K P I j)"