hcju/isabelleps huggingface dataset:
to_nat_eq: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "(mod_type.to_nat Rep x = mod_type.to_nat Rep y) = (x = y)"
cut_bound_': fixes n :: "nat"   and m :: "nat"   and F :: "'b formula" shows "cut_bound n m F \<le> cut_bound' (max n m) F"
env_len_def: fixes env :: "'e list set"   and l :: "nat" shows "env_len env l = (\<forall>xs\<in>env. length xs = l)"
strict_implies_order: fixes b :: "'a"   and a :: "'a" assumes "b \<prec> a" shows "b \<preceq> a"
uminus_p_integer_def: fixes p :: "integer"   and x :: "integer" shows "uminus_p_integer p x = (if x = 0 then 0 else p - x)"
unit_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (G.unit a) = G.map\<^sub>0 a"
raw_filtermap_apply: fixes f :: "'f \<Rightarrow> 'c"   and F :: "('f \<Rightarrow> 'd) \<Rightarrow> 'e" shows "raw_filtermap f F = (\<lambda>P. F (\<lambda>x. P (f x)))"
subcls_is_class1: fixes P :: "'b prog"   and C :: "String.literal"   and D :: "String.literal" assumes "P \<turnstile> C \<preceq>\<^sup>* D"   and "is_class P D" shows "is_class P C"
divideC_field_splits_simps_1_2: fixes c :: "complex"   and b :: "'d"   and a :: "'d" shows "(b /\<^sub>C c = a) = (if c = 0 then a = (0::'d) else b = c *\<^sub>C a)"
squarefree_factorial_semiring: fixes n :: "'a" assumes "n \<noteq> (0::'a)" shows "squarefree n = (\<forall>p. prime p \<longrightarrow> \<not> p\<^sup>2 dvd n)"
finite_monom_mult_lookup_not_zero: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and f :: "'b \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "finite {t. MPoly_Type_Class.punit.monom_mult (lookup p t) 0 (subst_pp f t) \<noteq> 0}"
sum_is_submodule: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and N1 :: "'i set"   and N2 :: "'i set" assumes "module R M"   and "LinearCombinations.submodule R N1 M"   and "LinearCombinations.submodule R N2 M" shows "LinearCombinations.submodule R (LinearCombinations.module.submodule_sum M N1 N2) M"
drinks2_vend_insufficient: fixes r :: "nat \<Rightarrow>f value option" shows "possible_steps drinks2 1 r STR ''vend'' [] = {|(1, vend_nothing)|}"
disc_3: fixes x11 :: "'a"   and x12 :: "'a process" shows "\<not> isChoice (Action x11 x12)"
c: shows "0 < 1 + x + x\<^sup>2"
g: shows "emb ps' s = (emb ps s)(x := aval a (emb ps s))"
step_simulation: shows "\<turnstile> psi \<longrightarrow> init \<and> \<box>[m1 \<or> m2]_(x, y)"
order2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and p :: "'b list"   and a :: "'b" assumes "class.idom_char_0 one times plus zero minus uminus"   and "semiring_0.poly plus zero times p \<noteq> semiring_0.poly plus zero times []" shows "semiring_0.divides plus zero times (semiring_1.pexp one times plus zero [uminus a, one] (ring_1.order one times plus zero uminus a p)) p \<and> \<not> semiring_0.divides plus zero times (semiring_1.pexp one times plus zero [uminus a, one] (Suc (ring_1.order one times plus zero uminus a p))) p"
assms: shows "\<forall>A. A \<in> carrier_mat 1 2 \<longrightarrow> admits_triangular_reduction A"
not_eq_order_implies_strict: fixes a :: "'a list"   and b :: "'a list" assumes "\<not> a = b"   and "prefix a b" shows "strict_prefix a b"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (div_write_new_val i result temp_V s1)) \<and> t2 = snd (fst (div_write_new_val i result temp_V s2))"
Cons: fixes x :: "'m"   and xs :: "'m list" shows "prod_list (x # xs) = x * prod_list xs"
prems_1: shows "tp = concat (take (Suc as) (tms_of ap))"
set_diff_eqvt: fixes pi :: "(name \<times> name) list"   and X :: "'a set"   and Y :: "'a set" shows "pi \<bullet> (X - Y) = pi \<bullet> X - pi \<bullet> Y"
is_weakly_scdom_component_safe_def: shows "Shadow_DOM_SC_DOM_Components.is_weakly_scdom_component_safe \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_scdom_component_safe Shadow_DOM.get_owner_document get_disconnected_nodes Shadow_DOM.to_tree_order"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (F f) = A.arr f"
\<phi>_nonneg: fixes p :: "nat \<times> int"   and x :: "real" shows "0 \<le> \<phi> p x"
bounded_iff: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "order.greater_eq a (sup b c) = (order.greater_eq a b \<and> order.greater_eq a c)"
card_of_Times_cong1: fixes A :: "'d set"   and B :: "'e set"   and C :: "'f set" assumes "|A| =o |B|" shows "|A \<times> C| =o |B \<times> C|"
fps_lr_inverse_power_2: fixes x :: "'a"   and a :: "'a fps"   and n :: "nat" assumes "x * fps_nth a 0 = (1::'a)"   and "fps_nth a 0 * x = (1::'a)" shows "fps_right_inverse (a ^ n) (x ^ n) = fps_right_inverse a x ^ n"
create_gba_from_nodes_precond_def: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set" shows "create_gba_from_nodes_precond \<phi> qs \<equiv> inres (create_graph \<phi>) qs"
continuous_openin_preimage_gen: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "open T" shows "openin (top_of_set S) (S \<inter> f -` T)"
outstanding_volatile_writes_unowned_by_others: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb\<^sub>i = {}"
ground\<^sub>l\<^sub>s_vars\<^sub>l\<^sub>s: fixes L :: "fterm literal set" assumes "ground\<^sub>l\<^sub>s L" shows "vars\<^sub>l\<^sub>s L = {}"
ran_lm_\<alpha>_Assoc_List_set: fixes v :: "('c, 'b) assoc_list" shows "ran (lm.\<alpha> v) = snd ` Assoc_List.set v"
le_zero_eq: fixes n :: "'a multiset" shows "(n \<subseteq># {#}) = (n = {#})"
iso_cancel_left: fixes f :: "'a"   and g :: "'a"   and g' :: "'a" assumes "B.iso f"   and "f \<cdot>\<^sub>B g = f \<cdot>\<^sub>B g'"   and "B.seq f g" shows "g = g'"
sum_abs_ge_zero: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(0::'a) \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
DiaN: fixes a :: "'d"   and ps :: "('c, 'd) fm list"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and p :: "('c, 'd) fm"   and i :: "'d"   and A :: "'d set"   and n :: "nat" assumes "(\<^bold>\<not> (\<^bold>\<diamond> p)) at a in (ps, a) # branch"   and "(\<^bold>\<diamond> Nom i) at a in (ps, a) # branch"   and "new (\<^bold>\<not> (\<^bold>@ i p)) a ((ps, a) # branch)"   and "A, Suc n \<turnstile> ((\<^bold>\<not> (\<^bold>@ i p)) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
simps_7: fixes c :: "int"   and e :: "Cooper.num" shows "\<beta> (Gt (CN 0 c e)) = [Neg e]"
eventually_Abs_filter: fixes F :: "('c \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
andfold_MatchExp_not_discI: fixes ms :: "'b match_expr list"   and disc :: "'b \<Rightarrow> bool" assumes "\<forall>m\<in>set ms. \<not> has_disc disc m" shows "\<not> has_disc disc (andfold_MatchExp ms)"
minimal_good_strategy_def: fixes v :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "minimal_good_strategy v \<sigma> \<equiv> \<sigma> \<in> good v \<and> (\<forall>\<sigma>'. (\<sigma>', \<sigma>) \<in> r - Id \<longrightarrow> \<sigma>' \<notin> good v)"
dWHILET_def: shows "dWHILET \<equiv> iWHILET (\<bind>) dRETURN"
llist_le_anti_sym: fixes s :: "'b llist"   and t :: "'b llist" assumes "s \<le> t"   and "t \<le> s" shows "s = t"
\<tau>inf1': shows "s1 -\<tau>1-sstls1\<rightarrow>*t \<infinity>"
quot_to_fract_def: shows "quot_to_fract = (\<lambda>(a, b). Fraction_Field.Fract a b)"
alpha_abs_res_stronger1: fixes as :: "atom set"   and x :: "'a"   and p' :: "perm"   and as' :: "atom set"   and x' :: "'a" assumes "(as, x) \<approx>res (=) supp p' (as', x')" shows "\<exists>p. (as, x) \<approx>res (=) supp p (as', x') \<and> supp p \<subseteq> as \<union> as'"
simps_6: fixes v :: "'b poly"   and va :: "'b poly" shows "isnpolyh (Mul v va) = (\<lambda>k. False)"
power_inject_exp: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a" shows "(power.power one times a m = power.power one times a n) = (m = n)"
Swap: fixes B :: "('a, 'b) form"   and A :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "\<stileturn> B # A # G" shows "\<stileturn> A # B # G"
maxsimpchainD_maxsimp: fixes xs :: "'a set list"   and x :: "'a set" assumes "maxsimpchain xs"   and "x \<in> set xs" shows "maxsimp x"
injfD: fixes x :: "'b"   and y :: "'b" assumes "f x = f y"   and "x \<in> Field s"   and "y \<in> Field s" shows "x = y"
associatedE2: fixes a :: "'a"   and b :: "'a"   and thesis :: "bool" assumes "a \<sim> b"   and "\<And>u. \<lbrakk>a = b \<otimes> u; u \<in> Units G\<rbrakk> \<Longrightarrow> thesis"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "thesis"
vreal_inverse_transfer: shows "rel_fun cr_vreal cr_vreal vreal_inverse_app inverse"
def_lfp_unfold: fixes h :: "'d"   and f :: "'d \<Rightarrow> 'd" assumes "h \<equiv> \<nu> f"   and "mono f" shows "h = f h"
polytope_imp_convex: fixes S :: "'b set" assumes "polytope S" shows "convex S"
assms_6: shows "\<forall>i. xs !! i \<in> V"
vdomainE: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> vdomain A" obtains r :: "V" where "x = \<langle>r, \<D>\<^sub>\<circ> r\<rangle>" and "r \<in>\<^sub>\<circ> A"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
manipulable_profile_def: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" shows "manipulable_profile R i Ri' = sds R \<prec>[SD (R i)] sds (R(i := Ri'))"
atLeast0_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum g {0..Suc n} = g 0 + sum (g \<circ> Suc) {0..n}"
wf_world_model_has_atoms: fixes f :: "object atom formula"   and M :: "object atom formula set" assumes "f \<in> M"   and "wf_world_model M" shows "is_predAtom f"
1: shows "A \<subseteq> Univ \<and> |A| <o \<AA> \<and> B \<subseteq> Univ \<and> |B| <o \<AA> \<and> F \<in> A \<rightarrow> B"
False: shows "\<not> k \<le> mod_type_class.to_nat i"
nonneg: shows "nonneg_cexpr V (\<Gamma>(x := t1)) e"
enumerator_def: fixes inner :: "nat"   and outer :: "nat" shows "enumerator inner outer \<equiv> let nmax = outer - 2; k = inner - 3 in map (\<lambda>is. [0] @ is @ [outer - 1]) ((enumAppend nmax ^^ k) (enumBase nmax))"
util_imp_total: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and u :: "'a \<Rightarrow> real" assumes "ordinal_utility carrier relation u" shows "total_on carrier relation"
ParTL: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>t s'" shows "(c1 | c2, s) \<rightarrow>c (c2, s')"
E1_def: shows "E1 = (\<Sum>i<N. \<bar>u 1 ((T ^^ i) x)\<bar>)"
region_cover_spec: fixes u :: "'c \<Rightarrow> real" assumes "\<forall>x\<in>X. 0 \<le> u x" shows "\<exists>R. R \<in> \<R> \<and> u \<in> R"
rbt_bulkload_is_rbt: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "('b \<times> 'c) list" assumes "class.linorder less_eq less" shows "ord.is_rbt less (ord.rbt_bulkload less xs)"
ALength1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a\<bullet>length,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
axioms_2: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "reducedly_statically_complete_calculus Bot Inf entails Red_I Red_F" shows "reducedly_statically_complete_calculus_axioms Bot Inf entails Red_I Red_F"
stable_on_allocation: fixes ds :: "'b set"   and X :: "'a set" assumes "Singleton_for_d.stable_on ds X" shows "inj_on Xd X"
swap_simp: fixes x :: "'d"   and y :: "'c" shows "prod.swap (x, y) = (y, x)"
andfold_MatchExp_not_disc_negated_mapMatch: fixes disc :: "'c \<Rightarrow> bool"   and C :: "'d \<Rightarrow> 'c"   and ls :: "'d list" shows "\<not> has_disc_negated disc False (andfold_MatchExp (map (Match \<circ> C) ls))"
IH: shows "Abs_fps (eval_polyexp xs) = polyexp_fps xs"
v11_vimage_vpsubset_neq: fixes r :: "V"   and A :: "V"   and B :: "V" assumes "v11 r"   and "A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "B \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "A \<noteq> B" shows "r `\<^sub>\<circ> A \<noteq> r `\<^sub>\<circ> B"
gba_impl_rel_eext_internal_def: fixes Rm :: "('c \<times> 'f) set"   and Rv :: "('a \<times> 'd) set"   and Rl :: "('b \<times> 'e) set" shows "gba_impl_rel_eext Rm Rv Rl \<equiv> \<langle>Rm, Rv \<rightarrow> Rl \<rightarrow> bool_rel\<rangle>gen_gba_impl_rel_eext"
set_mult_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and K :: "'e set" shows "H <#>\<^bsub>G\<^esub> K = (\<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes>\<^bsub>G\<^esub> k})"
index_of_r_to_l_lm: fixes x :: "nat"   and y :: "nat"   and z :: "nat" shows "nat_to_pr index_of_r_to_l (c_pair x (c_pair y z)) = c_pair (c_pair x y) z"
vp_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant" shows "vp K v = {x \<in> carrier (Vr K v). 0 < v x}"
le_ccInf_iff: fixes A :: "'a set"   and b :: "'a" assumes "countable A" shows "(b \<le> Inf A) = (\<forall>a\<in>A. b \<le> a)"
list_all2_refl': fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b list" assumes "equivp R" shows "list_all2 R r r"
nat_mod_cong: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and n :: "nat" assumes "a = b + c"   and "a mod n = b mod n" shows "c mod n = 0"
exp_approx': fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 \<le> x"   and "x \<le> 1" shows "\<bar>exp x - (\<Sum>k\<le>n. x ^ k / fact k)\<bar> \<le> x ^ n / fact n"
snoc_1: fixes t3s :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> xs : t2s _> t3s" shows "\<S>\<bullet>\<C> \<turnstile> es @ xs : t1s _> t3s"
simps_2: fixes j :: "nat" shows "sig_of_pair (Inr j) = term_of_pair (0::'a, j)"
msetext_dersh_compat_list: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and xs :: "'b list"   and xs' :: "'b list" assumes "gt y x" shows "msetext_dersh gt (xs @ y # xs') (xs @ x # xs')"
Dom_norm: fixes t :: "'c local.term" assumes "Arr t" shows "Dom \<^bold>\<parallel>t\<^bold>\<parallel> = Dom t"
p'_stop_zero: fixes a :: "real"   and v :: "real" shows "movement.p' a v (movement.t_stop a v) = (if a = 0 then v else 0)"
tiny_tdghm_in_Vset: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<NN> \<in>\<^sub>\<circ> Vset \<alpha>"
drop_all: fixes xs :: "'g list"   and n :: "nat" assumes "length xs \<le> n" shows "drop n xs = []"
IH: fixes z :: "'a" assumes "z + (1::'a) \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "z gchoose n = inverse ((z + (1::'a)) * Beta (z - of_nat n + (1::'a)) (of_nat n + (1::'a)))"
assms: shows "\<forall>x\<in>vars p. (0 < coeff p x \<longrightarrow> (0::'a) \<le> v x) \<and> (coeff p x < 0 \<longrightarrow> v x \<le> (0::'a))"
diag_mult: fixes x :: "'c"   and A :: "('c, 'd) sq_matrix" shows "diag x * A = x *\<^sub>S A"
Domainp_mod_ring_rel_integer: shows "Domainp mod_ring_rel_integer = (\<lambda>x. 0 \<le> x \<and> x < pp)"
exhaust: fixes y :: "'d\<cdot>'e writer" obtains "y = \<bottom>"   | w :: "'e" and a :: "'d" where "y = Writer\<cdot>w\<cdot>a"
sym_simps_3: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "cod (local.sym a b) = local.prod b a"
mod_minus_minus: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (uminus a) (uminus b) = uminus (modulo a b)"
comp_arr_dom: fixes f :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr" assumes "S.arr f"   and "S.dom f = a" shows "f \<cdot>\<^sub>S a = f"
neq_l: fixes a :: "MIR.num" shows "isrlfm (rsplit neq a)"
rel_set_refl_strong: fixes S :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x. x \<in> S \<Longrightarrow> P x x" shows "rel_set P S S"
approx_add_left_iff: fixes a :: "'a star"   and b :: "'a star"   and c :: "'a star" shows "(a + b \<approx> a + c) = (b \<approx> c)"
seqI: fixes f :: "'a"   and g :: "'a" assumes "Left.arr f"   and "Left.arr g"   and "Left.dom g = Left.cod f" shows "Left.seq g f"
Y: shows "Y = snd \<circ> (\<lambda>x. (X x, Y x))"
nullhomotopic_from_contractible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "contractible S" obtains c :: "'b" where "homotopic_with_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
tendsto_finmap: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow>\<^sub>F 'c"   and g :: "'b \<Rightarrow>\<^sub>F 'c" assumes "\<And>n. Fin_Map.domain (f n) = Fin_Map.domain g"   and "\<And>i. i \<in> Fin_Map.domain g \<Longrightarrow> (\<lambda>n. (f n)\<^sub>F i) \<longlonglongrightarrow> (g)\<^sub>F i" shows "f \<longlonglongrightarrow> g"
INIT2: shows "m < n \<and> 0 \<le> t \<and> t \<le> h"
solvable_def: fixes G :: "('e, 'f) monoid_scheme" shows "solvable G = solvable_seq G (carrier G)"
update_convs_9: fixes more' :: "'d \<Rightarrow> 'd"   and counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and more :: "'d" shows "state.more_update more' \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more' more\<rparr>"
A0: shows "OpposedThinChamberComplexFoldings X f0 g0 A0"
arr_iff_in_hom: fixes f :: "'a" shows "C.arr f = \<guillemotleft>f : C.dom f \<rightarrow> C.cod f\<guillemotright>"
inter_neg: fixes X :: "'b set"   and L :: "'b set" shows "X \<inter> - L = X - L"
intro: fixes B :: "'b set"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>x y z. \<lbrakk>x \<in> B; y \<in> B; z \<in> D\<rbrakk> \<Longrightarrow> f x (f y z) = f y (f x z)"   and "\<And>x y. \<lbrakk>x \<in> B; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D" shows "LCD B D f"
smult_r_null: fixes a :: "'a" assumes "a \<in> carrier R" shows "a \<odot>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
associated_polynomials_imp_same_is_root: fixes p :: "'a list list"   and q :: "'a list list"   and x :: "'a list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "q \<in> carrier (poly_ring (poly_ring R))"   and "p \<sim>\<^bsub>poly_ring (poly_ring R)\<^esub> q" shows "UP.is_root p x = UP.is_root q x"
quasinorm_on_def: fixes F :: "'b set"   and C :: "real"   and N :: "'b \<Rightarrow> ennreal" shows "quasinorm_on F C N = ((\<forall>x y. x \<in> F \<and> y \<in> F \<longrightarrow> x + y \<in> F \<and> N (x + y) \<le> ennreal C * N x + ennreal C * N y) \<and> (\<forall>c x. x \<in> F \<longrightarrow> c *\<^sub>R x \<in> F \<and> N (c *\<^sub>R x) = ennreal \<bar>c\<bar> * N x) \<and> 1 \<le> C \<and> (0::'b) \<in> F)"
pref_keeps_root: fixes u :: "'b list"   and r :: "'b list"   and v :: "'b list" assumes "u \<le>p r \<cdot> u"   and "v \<le>p u" shows "v \<le>p r \<cdot> v"
subdivision_pair_induct: fixes G :: "'b pair_pre_digraph"   and x :: "'b pair_pre_digraph"   and P :: "'b pair_pre_digraph \<Rightarrow> bool" assumes "subdivision_pair G x"   and "pair_bidirected_digraph G \<Longrightarrow> P G"   and "\<And>e w H. \<lbrakk>e \<in> parcs H; w \<notin> pverts H; subdivision_pair G H; P H\<rbrakk> \<Longrightarrow> P (subdivide H e w)" shows "P x"
prefix_never_empty: fixes d :: "'b prefix_match" shows "\<not> wordinterval_empty (prefix_to_wordinterval d)"
positive_add: fixes x :: "rat"   and y :: "rat" assumes "Rat.positive x"   and "Rat.positive y" shows "Rat.positive (x + y)"
fun_pair_eq_subst: fixes d :: "('fun, 'f) Term.term \<times> ('fun, 'f) Term.term"   and \<delta> :: "'f \<Rightarrow> ('fun, 'e) Term.term"   and d' :: "('fun, 'g) Term.term \<times> ('fun, 'g) Term.term"   and \<theta> :: "'g \<Rightarrow> ('fun, 'e) Term.term" shows "(pair d \<cdot> \<delta> = pair d' \<cdot> \<theta>) = (d \<cdot>\<^sub>p \<delta> = d' \<cdot>\<^sub>p \<theta>)"
finterD2: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" assumes "c |\<in>| A |\<inter>| B" shows "c |\<in>| B"
S\<^sub>e_nS2: shows "S\<^sub>e \<inter> S2 = {}"
freeword_inverse: fixes x :: "'b freeword" shows "Abs_freeword (freeword x) = x"
simps_2: fixes l :: "'a tree"   and x :: "'a"   and r :: "'a tree" shows "\<Phi> \<langle>l, x, r\<rangle> = 6 * e * real (imbal \<langle>l, x, r\<rangle>) + \<Phi> l + \<Phi> r"
out_arcs_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'a" assumes "digraph_isomorphism hom"   and "u \<in> verts G" shows "out_arcs (app_iso hom G) (iso_verts hom u) = iso_arcs hom ` out_arcs G u"
in_indetsE: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "x \<in> indets p" obtains t :: "'b \<Rightarrow>\<^sub>0 nat" where "t \<in> keys p" and "x \<in> keys t"
mod_eq_Suc_0_conv: fixes k :: "nat"   and x :: "nat" assumes "Suc 0 < k" shows "((x + k - Suc 0) mod k = 0) = (x mod k = Suc 0)"
Mu: shows "\<forall>t\<in>U. t \<sqsubseteq> linorder.Max (\<sqsubseteq>) U"