hcju/isabelleps huggingface dataset:
assms_1: shows "\<NN> : \<HH> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>' : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
pdevs_val_const_pdevs_of_list: fixes c :: "real"   and xs :: "'a list" shows "pdevs_val (\<lambda>_. c) (pdevs_of_list xs) = c *\<^sub>R sum_list xs"
topological_space_ow_axioms: shows "topological_space_ow U\<^sub>1 \<tau>\<^sub>1"
finite_occurrences_val_expr_4: fixes e :: "('r, 'l, 'v) expr" shows "finite (LID\<^sub>E e)"
mon_upclose_def: fixes u :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and m :: "nat" shows "mu u m = muh m u m"
iTILL_cut_greater: fixes n :: "nat"   and t :: "nat" shows "[\<dots>n] \<down>> t = (if n \<le> t then {} else [Suc t\<dots>,n - Suc t])"
locally_irrefl_def: fixes R :: "('a \<times> 'a) set"   and A :: "'a multiset" shows "locally_irrefl R A \<equiv> \<forall>t. t \<in># A \<longrightarrow> (t, t) \<notin> R"
cltn2_rep_abs: fixes A :: "((real, 3) vec, 3) vec" assumes "invertible A" shows "(A, cltn2_rep (cltn2_abs A)) \<in> invertible_proportionality"
finite_imp_closedin: fixes S :: "'b set"   and T :: "'b set" assumes "finite S"   and "S \<subseteq> T" shows "closedin (top_of_set T) S"
has_right_adjoint_functor: shows "right_adjoint_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
li_trans_r: fixes R :: "('a \<times> 'a) set"   and A :: "'a multiset"   and B :: "'a multiset" assumes "locally_irrefl R (A + B)" shows "locally_irrefl R B"
supp_fun_app: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "supp (f x) \<subseteq> supp f \<union> supp x"
fresh_star_supp_conv: fixes x :: "'f"   and y :: "'e" assumes "supp x \<sharp>* y" shows "supp y \<sharp>* x"
errMOD_igFreshIGAbs2STR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igAbsIPresIGWls MOD"   and "igFreshIGAbs2 MOD" shows "igFreshIGAbs2STR (errMOD MOD)"
lexord_lex: fixes x :: "'g list"   and y :: "'g list"   and r :: "('g \<times> 'g) set" shows "((x, y) \<in> lex r) = ((x, y) \<in> lexord r \<and> length x = length y)"
state_to_strips_state_range_is: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and v :: "'a"   and a :: "'b"   and s :: "'a \<Rightarrow> 'b option" assumes "is_valid_problem_sas_plus \<Psi>"   and "(v, a) \<in> dom (\<phi>\<^sub>S \<Psi> s)" shows "(\<phi>\<^sub>S \<Psi> s) (v, a) = Some (the (s v) = a)"
simps_2: fixes op :: "'b strips_operator"   and ops :: "'b strips_operator list" shows "are_all_operators_non_interfering (op # ops) = (list_all (\<lambda>op'. \<not> are_operators_interfering op op') ops \<and> are_all_operators_non_interfering ops)"
f: shows "\<forall>t. t \<in> set ts \<longrightarrow> vars\<^sub>t t \<subseteq> vars\<^sub>t\<^sub>s ts"
simps_75: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "'a"   and f6 :: "'a"   and f7 :: "'a"   and f8 :: "'b \<Rightarrow> 'a"   and f9 :: "('b, String.literal) phantom \<Rightarrow> 'a" shows "(case OclAny of OclAny \<Rightarrow> f1 | OclVoid \<Rightarrow> f2 | Boolean \<Rightarrow> f3 | Real \<Rightarrow> f4 | Integer \<Rightarrow> f5 | UnlimitedNatural \<Rightarrow> f6 | String \<Rightarrow> f7 | \<langle>x\<rangle>\<^sub>\<T> \<Rightarrow> f8 x | Enum x \<Rightarrow> f9 x) = f1"
notMTnMT: fixes a :: "'a"   and p :: "'a list"   and C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option" assumes "a \<in> set p"   and "none_MT_rules C p" shows "dom (C a) \<noteq> {}"
assign_eval\<^sub>w': fixes mds :: "Mode \<Rightarrow> 'Var set"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and mem :: "'Var \<Rightarrow> 'Val"   and x :: "'Var"   and e :: "'AExp" assumes "mds = mds'"   and "mem' = mem(x := ev\<^sub>A mem e)" shows "\<langle>x \<leftarrow> e, mds, mem\<rangle> \<leadsto> \<langle>Stop, mds', mem'\<rangle>"
assms_1: shows "\<forall>(q, \<tau>)\<in>set qs. q < length ss \<and> \<tau> \<in> A"
sup_aci_3: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) (sup (y::'b) (z::'b)) = sup y (sup x z)"
ASSERT_refine: fixes \<Phi>' :: "bool"   and \<Phi> :: "bool" assumes "\<Phi>' \<Longrightarrow> \<Phi>" shows "ASSERT \<Phi> \<le> \<Down> unit_rel (ASSERT \<Phi>')"
bij_betw_linorders_on: fixes A :: "'b set" assumes "finite A" shows "bij_betw linorder_of_list (permutations_of_set A) (linorders_on A)"
ge_p: shows "\<forall>m. m \<^bold>\<in> lan vn \<longrightarrow> p < m"
arg_min_nat_le: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and m :: "'b \<Rightarrow> nat" assumes "P x" shows "order.greater_eq (m x) (m (arg_min m P))"
l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool" shows "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_disconnected_nodes get_shadow_root get_tag_name heap_is_wellformed \<equiv> heap_is_wellformed = l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_heap_is_wellformed get_child_nodes get_disconnected_nodes get_shadow_root get_tag_name"
eq_fold: fixes A :: "'a set" shows "F A = (if finite A then Finite_Set.fold (\<^bold>*) \<^bold>\<top> A else \<^bold>\<bottom>)"
Ams: fixes n :: "nat" shows "A n \<in> measurable_sets M"
that_2: fixes Pa :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and Ea :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and es :: "char list exp list"   and T :: "ty"   and Ts :: "ty list" assumes "x = Pa"   and "xa = Ea"   and "xb = e # es"   and "xc = T # Ts"   and "Pa,Ea \<turnstile> e :: T"   and "Pa,Ea \<turnstile> es [::] Ts" shows "thesis"
set_encode_inverse: fixes A :: "nat set" assumes "finite A" shows "set_decode (set_encode A) = A"
path_component_of_canon_iff: fixes T :: "'a set"   and a :: "'a"   and b :: "'a" shows "path_component_of (top_of_set T) a b = path_component T a b"
contour_integral_reverse_linepath: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "continuous_on (closed_segment a b) f" shows "contour_integral (linepath a b) f = - contour_integral (linepath b a) f"
yun_factorization_def: fixes p :: "'a poly" shows "local.yun_factorization p = (if p = 0 then (0::'a, []) else let c = lead_coeff p; q = smult (inverse c) p in (c, yun_monic_factorization q))"
linorder_injI: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. x < y \<Longrightarrow> f x \<noteq> f y" shows "inj f"
asymptotics1: fixes x :: "real"   and i :: "nat" assumes "x\<^sub>0 \<le> x"   and "i < k" shows "1 + ln x powr (- e / 2) \<le> (1 - hb * inverse (bs ! i) * ln x powr - (1 + e)) powr p * (1 + ln (bs ! i * x + hb * x / ln x powr (1 + e)) powr (- e / 2))"
crel_vs_return: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "Rel R x y" shows "Rel (crel_vs R) (Wrap x) \<langle>y\<rangle>"
dgrad_p_set_closed_except: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and S :: "'t set" assumes "p \<in> dgrad_p_set d m" shows "except p S \<in> dgrad_p_set d m"
rep_eq: fixes x :: "'b poly"   and xa :: "'b poly" shows "coeff (x + xa) = (\<lambda>n. coeff x n + coeff xa n)"
hd_lex: fixes ms :: "nat list"   and ns :: "nat list" assumes "hd ms < hd ns"   and "length ms = length ns"   and "ns \<noteq> []" shows "(ms, ns) \<in> lex less_than"
intro: fixes B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<exists>a b c. \<not> B a b c \<and> \<not> B b c a \<and> \<not> B c a b"   and "\<forall>p q a b c. p \<noteq> q \<and> C a p a q \<and> C b p b q \<and> C c p c q \<longrightarrow> B a b c \<or> B b c a \<or> B c a b" shows "tarski_absolute_axioms C B"
monotone_LCons: fixes x :: "'c" shows "monotone (\<sqsubseteq>) (\<sqsubseteq>) (LCons x)"
converse_nreachable_cases: fixes u :: "'a"   and n :: "nat"   and v :: "'a" assumes "u \<rightarrow>\<^bsup>n\<^esup> v" obtains "u = v" and "n = 0" and "u \<in> verts G"   | w :: "'a" and m :: "nat" where "u \<rightarrow> w" and "n = Suc m" and "w \<rightarrow>\<^bsup>m\<^esup> v"
mult_scalar_monomial_monomial: fixes t :: "'a"   and c :: "'c"   and v :: "'t"   and d :: "'c" shows "monomial c t \<odot> monomial d v = monomial (c * d) (t \<oplus> v)"
nts_snoc: shows "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [progress_regex \<sigma> P r j..<Suc j] (nts @ [\<tau> \<sigma> j])"
sylvester_mat_def: fixes p :: "'b poly"   and q :: "'b poly" shows "sylvester_mat p q \<equiv> sylvester_mat_sub (degree p) (degree q) p q"
div_ordinal_def: fixes x :: "ordinal"   and y :: "ordinal" shows "x div y = (if 0 < y then oInv ((*) y) x else 0)"
intros_5: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F, G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<and> G, \<Gamma> \<Rightarrow> \<Delta>"
assms_5: fixes x :: "'a" assumes "x \<in> S" shows "q x = x"
prime_elem_not_unit: fixes p :: "'a" assumes "prime_elem p" shows "\<not> p dvd (1::'a)"
fls_mult_nonzero_base_subdegree_eq: fixes f :: "'a fls"   and g :: "'a fls" assumes "fls_nth f (fls_subdegree f) * fls_nth g (fls_subdegree g) \<noteq> (0::'a)" shows "fls_subdegree (f * g) = fls_subdegree f + fls_subdegree g"
normalize_src_ips_def: shows "normalize_src_ips = normalize_primitive_extract (is_Src, src_sel) Src ipt_iprange_compress"
is_fl_dghm_def: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB> \<equiv> \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB> \<and> is_fl_dghm_axioms \<AA> \<BB> \<FF>"
dijkstra_def: shows "dijkstra \<equiv> snd dijkstra_loop"
lookup_plus_fun: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "lookup (s + t) = lookup s + lookup t"
root_divisor: fixes w :: "'b list"   and k :: "nat" assumes "periodN w k"   and "k dvd \<^bold>|w\<^bold>|" shows "w \<in> take k w*"
in_implied_frontier_alt_in_implication_frontier: fixes c :: "('loc, 't) configuration"   and b :: "'t"   and loc2 :: "'loc" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "worklists_vacant_to c b"   and "b \<in>\<^sub>A implied_frontier_alt c loc2" shows "b \<in>\<^sub>A frontier (c_imp c loc2)"
eval\<^sub>wp_eval\<^sub>w_eq: shows "eval\<^sub>wp = (\<leadsto>)"
simps_1: fixes uu :: "(name list \<times> 'a list) list" shows "caseListSeqSubst [] uu = []"
pcr_cr_eq: shows "list_R1.pcr_list (=) = list_R1.cr_list"
simps: fixes a1 :: "pi"   and a2 :: "pi set" shows "((a1, a2) \<in> sumComposeSet) = (a1 = \<zero> \<and> a2 = {} \<or> (\<exists>Q S P. a1 = P \<oplus> Q \<and> a2 = S \<and> Q \<in> S \<and> (P, S - {Q}) \<in> sumComposeSet))"
submonoid_invertible: fixes u :: "'a \<Rightarrow> 'a" assumes "transformation.sub.invertible u"   and "u \<in> translation ` G" shows "transformation.invertible u"
inf1: fixes A :: "'a"   and k :: "'a"   and a :: "'a"   and b :: "'a" assumes "A \<equiv> inf k a" shows "inf A b \<equiv> inf k (inf a b)"
assms_1: shows "M 0 (v c) \<le> \<one>"
elem_neq_classes: shows "P - {Y} - {Y - {elem}} = P - {Y}"
third_alt: fixes a :: "alt"   and b :: "alt" assumes "a \<noteq> b" shows "\<exists>c. distinct [a, b, c]"
Says_ticket_parts: fixes S :: "agent"   and A :: "agent"   and K :: "nat"   and SesKey :: "msg"   and B :: "msg"   and TimeStamp :: "msg"   and Ticket :: "msg"   and evs :: "event list" assumes "Says S A \<lbrace>Crypt K \<lbrace>SesKey, B, TimeStamp\<rbrace>, Ticket\<rbrace> \<in> set evs" shows "Ticket \<in> parts (knows Spy evs)"
intro: fixes G :: "('a, 'b) Graph_scheme"   and root :: "'a" assumes "Tree G"   and "RootedTree_axioms G root" shows "RootedTree G root"
alg_int: fixes x :: "complex" assumes "x \<in> Roots'" shows "algebraic_int (of_int l * x)"
cThin: fixes A' :: "triple set"   and C :: "triple set"   and A :: "triple set" assumes "A' |\<turnstile> C"   and "A' \<subseteq> A" shows "A |\<turnstile> C"
space_restr_to_subalg: fixes M :: "'c measure"   and F :: "'c measure" shows "space (restr_to_subalg M F) = space M"
finite_nat_set_bounded2: fixes N :: "nat set" assumes "finite N" shows "\<exists>n. \<forall>i\<in>N. i \<le> n"
0: shows "map2 ntsem (arOf f) (map (Ik.int \<xi>) Tl) = map (TE.int (eenv \<xi>) \<circ> tNN) Tl"
all_union_ts_iff: fixes c :: "'a"   and A :: "'b \<Rightarrow> 'a set"   and B :: "'a set"   and k :: "'b" shows "(c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k) = (c \<in> A k \<or> c \<in> B)"
nonneg_vel_ego: shows "0 \<le> v\<^sub>e"
complete_refl_on: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "complete A r"   and "refl_on A r"   and "x \<in> A"   and "y \<in> A" shows "x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x"
index_minus_vec_2: fixes v\<^sub>1 :: "'b vec"   and v\<^sub>2 :: "'b vec" shows "dim_vec (v\<^sub>1 - v\<^sub>2) = dim_vec v\<^sub>2"
hc: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) c = Cv"
foldSetDp_foldSetD_eq: fixes D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f" shows "foldSetDp (\<lambda>x. x \<in> D) f e = (\<lambda>x xa. (x, xa) \<in> foldSetD D f e)"
fold_combine_code: fixes g :: "'d \<Rightarrow> ('c, real) mapping"   and xs :: "'d list" shows "fold_combine_plus.combine.F g (set xs) = foldr (\<lambda>x. Mapping.combine (+) (g x)) (remdups xs) Mapping.empty"
cond_def: shows "cond \<equiv> \<Lambda> i t e. case i of ValF\<cdot>f \<Rightarrow> \<bottom> | ValTT \<Rightarrow> t | ValFF \<Rightarrow> e | ValN\<cdot>n \<Rightarrow> \<bottom>"
cs_split': fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and xs :: "'n list"   and x :: "'n"   and x' :: "'n"   and ys :: "'n list" assumes "cs\<^bsup>\<pi>\<^esup> i = xs @ [x, x'] @ ys" shows "\<exists>m. cs\<^bsup>\<pi>\<^esup> m = xs @ [x] \<and> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
lincomb_list_def: fixes c :: "nat \<Rightarrow> 'a"   and vs :: "'a vec list" shows "lincomb_list c vs = M.sumlist (map (\<lambda>i. c i \<cdot>\<^sub>v vs ! i) [0..<length vs])"
euclid_ext2_def: fixes a :: "'b"   and b :: "'b" shows "euclid_ext2 a b = (fst (bezout_coefficients a b), snd (bezout_coefficients a b), - b div gcd a b, a div gcd a b, gcd a b)"
iso_presheaves_of_rings_axioms_def: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" shows "iso_presheaves_of_rings_axioms X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> \<exists>\<psi>. morphism_presheaves_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<FF> \<rho> b add_str mult_str zero_str one_str \<psi> \<and> (\<forall>U. is_open U \<longrightarrow> (\<forall>x\<in>\<FF>' U. (fam_morphisms U \<circ> \<psi> U) x = x) \<and> (\<forall>x\<in>\<FF> U. (\<psi> U \<circ> fam_morphisms U) x = x))"
sums_sum: fixes x :: "'i \<Rightarrow> 'a" assumes "\<And>i. i \<in> I \<Longrightarrow> f i sums x i" shows "(\<lambda>n. \<Sum>i\<in>I. f i n) sums sum x I"
label_incr_Exit: fixes n :: "label"   and i :: "nat" assumes "n \<oplus> i = Exit" shows "n = Exit"
that: fixes n :: "nat" assumes "0 < n"   and "\<bar>x - ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi\<bar> < 1 / 2" shows "thesis"
c2c: shows "path_image c2 \<inter> path_image c = {a, b}"
toArr_fromArr: fixes x :: "'arr" assumes "x \<in> Arr" shows "toArr (fromArr x) = x"
good_item_simps_12: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and xsa :: "'varSort"   and xa :: "'var"   and y :: "'var" assumes "goodAbs A" shows "goodAbs (A $[xa \<and> y]_xsa)"
div_add1_eq: 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"   and b :: "'c"   and c :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (plus a b) c = plus (plus (divide a c) (divide b c)) (divide (plus (modulo a c) (modulo b c)) c)"
hyps_2: shows "treach \<tau> s@@k (ltype l)"
simps: fixes a :: "char list \<Rightarrow> 'b option"   and b :: "char list \<Rightarrow> 'b option" shows "Env a+Env b = (if ok (Env a) \<and> ok (Env b) \<and> env_dom (Env b) \<inter> env_dom (Env a) = {} then Env (a ++ b) else Malformed)"
rel_cases: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d Monomorphic_Monad.id"   and b :: "'e Monomorphic_Monad.id" assumes "rel_id R a b" obtains xa :: "'d" and ya :: "'e" where "a = return_id xa" and "b = return_id ya" and "R xa ya"
ListRed1: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e # es,s\<rangle> [\<rightarrow>] \<langle>e' # es,s'\<rangle>"
lowner_le_inner_prod_le: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and v :: "complex vec" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "v \<in> carrier_vec n"   and "A \<le>\<^sub>L B" shows "inner_prod v (A *\<^sub>v v) \<le> inner_prod v (B *\<^sub>v v)"
prv_scnj_mono: fixes F1 :: "'fmla set"   and F2 :: "'fmla set" assumes "F1 \<subseteq> fmla"   and "F2 \<subseteq> F1"   and "finite F1" shows "prv (imp (scnj F1) (scnj F2))"
lcs_plus_left: fixes u :: "'a"   and s :: "'a"   and t :: "'a" shows "lcs (u + s) (u + t) = u + lcs s t"
same_upto_trans: fixes n :: "nat"   and j :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and C :: "'b mat" assumes "same_upto n j A B"   and "same_upto n j B C" shows "same_upto n j A C"
rel_Grp: fixes A1 :: "'f set"   and f1 :: "'f \<Rightarrow> 'i"   and A2 :: "'g set"   and f2 :: "'g \<Rightarrow> 'j"   and A3 :: "'h set"   and f3 :: "'h \<Rightarrow> 'k" shows "F1rel (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) (BNF_Def.Grp A3 f3) = BNF_Def.Grp (F1in A1 A2 A3) (F1map f1 f2 f3)"
simps_5: fixes x :: "octo"   and y :: "octo" shows "Im4 (x + y) = Im4 x + Im4 y"
covering_space_lift: fixes C :: "'a set"   and p :: "'a \<Rightarrow> 'b"   and S :: "'b set"   and U :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "covering_space C p S"   and "simply_connected U"   and "locally path_connected U"   and "continuous_on U f"   and "f ` U \<subseteq> S" obtains g :: "'c \<Rightarrow> 'a" where "continuous_on U g" and "g ` U \<subseteq> C" and "\<And>y. y \<in> U \<Longrightarrow> p (g y) = f y"
simps: fixes bnd :: "'a"   and todo :: "(int poly \<times> root_info) list"   and old :: "(int poly \<times> root_info) list"   and l :: "rat"   and r :: "rat"   and n :: "nat" shows "local.select_correct_factor_main bnd todo old l r n = (case todo of [] \<Rightarrow> if n = 1 then (hd old, l, r) else let bnd' = bnd_update bnd in case bnd_get bnd' of (l, r) \<Rightarrow> local.select_correct_factor_main bnd' old [] l r 0 | (p, ri) # todo \<Rightarrow> let m = root_info.l_r ri l r in if m = 0 then local.select_correct_factor_main bnd todo old l r n else local.select_correct_factor_main bnd todo ((p, ri) # old) l r (n + m))"
elims: fixes x :: "'a factor_tree"   and y :: "'a factor_tree set" assumes "sub_trees x = y" obtains i :: "'a" and xa :: "int poly" where "x = Factor_Leaf i__ xa" and "y = {Factor_Leaf i__ xa}"   | i :: "'a" and l :: "'a factor_tree" and r :: "'a factor_tree" where "x = Factor_Node i l r" and "y = insert (Factor_Node i l r) (sub_trees l \<union> sub_trees r)"
circ_back_loop_fixpoint: fixes z :: "'a"   and y :: "'a" shows "z * y\<^sup>\<star> * y \<squnion> z = z * y\<^sup>\<star>"
A: shows "\<Psi>.FUN a' (\<iota> \<chi>') \<in> Hom.set (a', a)"
def_homo: fixes x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>" shows "\<delta> f x y = (\<upsilon> x and \<upsilon> y)"
CallRedsFinal: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and es :: "char list exp list"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and l\<^sub>2' :: "sfields"   and ef :: "char list exp"   and h\<^sub>3 :: "heap"   and l\<^sub>3 :: "sfields"   and sh\<^sub>3 :: "sheap"   and b\<^sub>3 :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1,b\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = (pns, body) in D"   and "length vs = length pns"   and "l\<^sub>2' = [this \<mapsto> Addr a, pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2),b\<^sub>2\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3, sh\<^sub>3),b\<^sub>3\<rangle>"   and "final ef" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2, sh\<^sub>3),b\<^sub>3\<rangle>"