hcju/isabelleps huggingface dataset:
Ne: fixes e :: "real" assumes "0 < e"   and "e \<le> 1" shows "ennreal (1 - e) < emeasure M {x \<in> space M. \<forall>B\<ge>Ne e. real (card ({n. \<exists>l\<in>{Ne e..n}. u n x - u (n - l) x \<le> - e * real l} \<inter> {..<B})) < e * real B}"
cgenideal_def: fixes R :: "('c, 'd) monoid_scheme"   and a :: "'c" shows "PIdl\<^bsub>R\<^esub> a = {x \<otimes>\<^bsub>R\<^esub> a |x. x \<in> carrier R}"
div_mult_mod_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (times (divide a b) b) (modulo a b) = a"
atLeast_iff: fixes i :: "'b \<times> 'c"   and k :: "'b \<times> 'c" shows "(i \<in> prod.atLeast k) = less_eq_prod k i"
is_continuation_base: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> \<P> k u"   and "charslength p = k" shows "is_continuation k u p []"
npD': fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>np v" shows "u \<noteq> \<epsilon>"
replay'_code: fixes thy :: "exetheory"   and vs :: "(variable \<times> typ) list"   and ns :: "variable set"   and Hs :: "Core.term list"   and P :: "proofterm" assumes "exe_wf_theory' thy" shows "replay' (translate_theory thy) vs ns Hs P = exereplay' thy vs ns Hs P"
reads_writes_preserved: fixes SR :: "('a \<Rightarrow> 'a \<Rightarrow> bool) set"   and getter :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a"   and SW :: "('a, 'd, 'e) prog set"   and setter :: "('a, 'd, 'f) prog"   and x :: "'c" assumes "reads SR getter h h'"   and "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h'. \<forall>w\<in>SW. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>SR. r h h')" shows "h \<turnstile> getter \<rightarrow>\<^sub>r x = h' \<turnstile> getter \<rightarrow>\<^sub>r x"
intro: fixes \<phi> :: "'a MFOTL.formula" assumes "MFOTL.future_reach \<phi> \<noteq> \<infinity>" shows "future_bounded_mfotl \<phi>"
insert_not_empty: fixes a :: "'e"   and A :: "'e set" shows "insert a A \<noteq> {}"
split_beta': fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
AE_space: fixes M :: "'d measure" shows "AE x in M. x \<in> space M"
mod_by_0: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a zero = a"
not_eq_order_implies_strict: fixes a :: "'b list"   and b :: "'b list" assumes "a \<noteq> b"   and "a \<le>p b" shows "a <p b"
minus_hf_def: fixes A :: "hf"   and B :: "hf" shows "A - B = \<lbrace>x \<^bold>\<in> A. x \<^bold>\<notin> B\<rbrace>"
rev_take: fixes i :: "nat"   and xs :: "'g list" shows "rev (take i xs) = drop (length xs - i) (rev xs)"
simps: fixes a1 :: "'a"   and a2 :: "[:'a:]" shows "slistmem a1 a2 = ((\<exists>x xs. a1 = x \<and> a2 = x :# xs \<and> x \<noteq> \<bottom> \<and> xs \<noteq> \<bottom>) \<or> (\<exists>x xs y. a1 = x \<and> a2 = y :# xs \<and> slistmem x xs \<and> y \<noteq> \<bottom>))"
assms_2: shows "bld \<in># \<B>\<^sup>D"
set_cases_2: fixes e :: "'a"   and a :: "('b, 'a) Term.term" assumes "e \<in> vars_term a" obtains "a = Var e"   | z1 :: "'b" and z2 :: "('b, 'a) Term.term list" and x :: "('b, 'a) Term.term" where "a = Fun z1 z2" and "x \<in> set z2" and "e \<in> vars_term x"
finite_incidence_system_axioms: shows "finite_incidence_system \<V> \<B>\<^sup>C"
hr_form_lemma: fixes evs :: "event list"   and hr :: "msg"   and R :: "agent"   and S2TTP :: "msg"   and pwd :: "msg" assumes "evs \<in> certified_mail"   and "hr \<notin> synth (analz (knows Spy evs))"   and "Notes TTP \<lbrace>Agent R, Agent TTP, S2TTP, pwd, hr\<rbrace> \<in> set evs" shows "\<exists>clt q S em. hr = Hash \<lbrace>Number clt, Nonce q, response S R q, em\<rbrace>"
sin_converges: fixes x :: "'a" shows "(\<lambda>n. sin_coeff n *\<^sub>R x ^ n) sums sin x"
is_volatile_Read\<^sub>s\<^sub>b_def: fixes r :: "'b memref" shows "is_volatile_Read\<^sub>s\<^sub>b r = (case r of Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> volatile | _ \<Rightarrow> False)"
assms: shows "\<forall>i. \<exists>N. \<forall>n\<ge>N. P i n"
cmod_1_plus_mult_le: fixes z :: "complex"   and w :: "complex" shows "cmod (1 + z * w) \<le> sqrt ((1 + (cmod z)\<^sup>2) * (1 + (cmod w)\<^sup>2))"
sup_bot_right: fixes sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and bot :: "'g"   and x :: "'g" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
integrity_is_local: fixes \<sigma> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and \<tau> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and dec :: "'g \<Rightarrow> 'h option"   and p :: "'f"   and v :: "'h"   and vals :: "'i \<Rightarrow> 'h"   and n :: "nat" assumes "locally_similar \<sigma> \<tau>"   and "\<And>n. dec (\<sigma> n p) = Some v \<Longrightarrow> v \<in> range vals"   and "dec (\<tau> n p) = Some v" shows "v \<in> range vals"
simps_1: fixes \<mu> :: "'a" shows "Obj \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = obj \<mu>"
enum_all_latin_type_def: fixes P :: "latin_type \<Rightarrow> bool" shows "enum_class.enum_all P = (P Row \<and> P Col \<and> P Num)"
range_ordering_lemma: fixes x :: "ordinal" shows "\<forall>y. y \<in> A \<longrightarrow> y < OrdinalVeblen.ordering A x \<longrightarrow> y \<in> range (OrdinalVeblen.ordering A)"
finite_Psegments_congE: fixes Q :: "real \<Rightarrow> bool"   and lb :: "real"   and ub :: "real"   and P :: "real \<Rightarrow> bool" assumes "finite_Psegments Q lb ub"   and "\<And>t. \<lbrakk>lb < t; t < ub\<rbrakk> \<Longrightarrow> Q t = P t" shows "finite_Psegments P lb ub"
cases: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'b"   and a3 :: "'a" assumes "converse3p r a1 a2 a3" obtains a :: "'a" and b :: "'b" and c :: "'c" where "a1 = c" and "a2 = b" and "a3 = a" and "r a b c"
nat_into_Card: fixes \<alpha> :: "V" assumes "\<alpha> \<in>\<^sub>\<circ> \<omega>" shows "Card \<alpha>"
P_Ana: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list"   and k :: "('fun, 'var) Term.term" assumes "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec \<and> t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"   and "Ana t = (K, T)"   and "k \<in> set K" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t k \<A> Sec \<and> k \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
eucl_of_list_map_plus: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "length xs = DIM('b)" shows "eucl_of_list (map (\<lambda>x. f x + g x) xs) = eucl_of_list (map f xs) + eucl_of_list (map g xs)"
sym_inverse: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "inverse_arrows \
[a, b] \
[b, a]"
GreenThm_typeII_divisible: assumes "gen_division s (cubeImage ` two_chain_typeII)" shows "integral s (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) = one_chain_line_integral F {j} (two_chain_boundary two_chain_typeII)"
emeasure_PiM_emb: fixes J :: "'i set"   and X :: "'i \<Rightarrow> 'a set" assumes "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> events i" shows "emeasure (Pi\<^sub>M I M) (emb I J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. emeasure (M i) (X i))"
infinite_hyp_changes_not_Lim: fixes f :: "nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f \<in> U"   and "\<forall>n. \<exists>m\<^sub>1>n. \<exists>m\<^sub>2>n. s (f \<triangleright> m\<^sub>1) \<noteq> s (f \<triangleright> m\<^sub>2)" shows "\<not> learn_lim \<psi> U s"
not_zero_g: shows "(GREATEST m. \<not> is_zero_row_upt_k m k A) + (1::'m) \<noteq> (0::'m)"
col_inter: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and E :: "Points"   and F :: "Points" assumes "distinct6 A B C D E F"   and "col A B C"   and "col D E F" shows "Pascal_Property.inter (line B C) (line E F) = Pascal_Property.inter (line A B) (line D E)"
map_of_concat: fixes xss :: "('a \<times> 'b) list list" shows "map_of (concat xss) = foldr (\<lambda>xs f. f ++ map_of xs) xss Map.empty"
M: shows "field (L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>)"
veqpoll_pair: fixes a :: "V"   and b :: "V"   and c :: "V"   and d :: "V" assumes "a \<noteq> b"   and "c \<noteq> d" shows "set {a, b} \<approx>\<^sub>\<circ> set {c, d}"
Abs_freelist_rev_append_alternating_list_in_Q: fixes s :: "'w"   and t :: "'w"   and n :: "nat" assumes "s \<in> S"   and "t \<in> S" shows "Abs_freelist (rev (alternating_list n s t) @ alternating_list n s t) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
order_impl_root_order: fixes k :: "nat"   and t :: "'b btree" assumes "0 < k"   and "order k t" shows "root_order k t"
path_connectedin_discrete_topology: fixes U :: "'a set"   and S :: "'a set" shows "path_connectedin (discrete_topology U) S = (S \<subseteq> U \<and> (\<exists>a. S \<subseteq> {a}))"
hom_0: fixes x :: "'a" assumes "hom x = (0::'b)" shows "x = (0::'a)"
stutter_freeI: fixes \<sigma> :: "nat \<Rightarrow> 'b" assumes "\<And>k n. \<lbrakk>\<sigma> (Suc k) = \<sigma> k; k < n\<rbrakk> \<Longrightarrow> \<sigma> n = \<sigma> k" shows "stutter_free \<sigma>"
yun_relD_7: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly" assumes "yun_rel F c f" shows "(f = 1) = (F = 1)"
pdevs_val_of_list_of_pdevs2: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and X :: "'a pdevs" assumes "e \<in> UNIV \<rightarrow> I" obtains e' :: "nat \<Rightarrow> real" where "pdevs_val e X = pdevs_val e' (pdevs_of_list (map snd (list_of_pdevs X)))" and "e' \<in> UNIV \<rightarrow> I"
decisive_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "decisive_on A f = (\<forall>B\<subseteq>A. B \<noteq> {} \<longrightarrow> f B \<noteq> {})"
prod: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> real" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> R" shows "(\<lambda>x. \<Prod>i\<in>I. f i x) \<in> R"
asym: fixes a :: "'a"   and b :: "'a" assumes "lt a b"   and "lt b a" shows "False"
assms_2: shows "\<not> ps (S t i) p \<noteq> None"
intros_45: fixes E :: "char list \<Rightarrow> ty 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' :: "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 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 es :: "expr list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e # es,s\<rangle> [\<rightarrow>] \<langle>e' # es,s'\<rangle>"
in_upd_image: fixes A :: "nat set"   and i :: "nat" assumes "A \<subseteq> {..<n}"   and "i < n" shows "(upd i \<in> upd ` A) = (i \<in> A)"
f_Exec_Nil: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d" shows "f_Exec_Comp trans_fun [] c = c"
fconj1: fixes n :: "nat"   and m :: "nat"   and f :: "form"   and g :: "form"   and xs :: "(nat \<times> form) list"   and nfs :: "(nat \<times> form) list" assumes "(n, (m, FConj f g) # xs) \<in> deriv nfs"   and "\<not> is_axiom (s_of_ns ((m, FConj f g) # xs))" shows "(Suc n, xs @ [(0, f)]) \<in> deriv nfs"
fresh_star_empty: fixes bs :: "'a" shows "{} \<sharp>* bs"
stasbs_2: shows "ts = as @ pair_relator_halflist t s @ bs"
squarefree_part_dvd: fixes n :: "nat" shows "\<Prod>(squarefree_part n) dvd n"
tendsto_mult_real_ereal: fixes u :: "'b \<Rightarrow> ereal"   and l :: "real"   and F :: "'b filter"   and v :: "'b \<Rightarrow> ereal"   and m :: "real" assumes "(u \<longlongrightarrow> ereal l) F"   and "(v \<longlongrightarrow> ereal m) F" shows "((\<lambda>n. u n * v n) \<longlongrightarrow> ereal l * ereal m) F"
master_theorem_restricted: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> = (\<exists>X\<subseteq>subformulas\<^sub>\<mu> \<phi> \<inter> restricted_subformulas \<phi>. \<exists>Y\<subseteq>subformulas\<^sub>\<nu> \<phi> \<inter> restricted_subformulas \<phi>. \<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[X]\<^sub>\<nu> \<and> (\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>[Y]\<^sub>\<mu>)) \<and> (\<forall>\<psi>\<in>Y. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[X]\<^sub>\<nu>)))"
simps: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'b" shows "r\<inverse>\<inverse> a1 a2 = (\<exists>a b. a1 = b \<and> a2 = a \<and> r a b)"
assms_2: assumes "vwalk p G"   and "distinct p" shows "P"
VLambda_eq_D2: fixes A :: "V"   and f :: "V \<Rightarrow> V"   and g :: "V \<Rightarrow> V"   and x :: "V" assumes "VLambda A f = VLambda A g"   and "x \<in>\<^sub>\<circ> A" shows "f x = g x"
nat_of_hatom_hatom_of_nat: fixes n :: "nat" shows "nat_of_hatom (hatom_of_nat n) = n"
eulerian_poly'': fixes k :: "nat" assumes "0 < k" shows "Abs_fps (\<lambda>n. of_nat n ^ k) = fps_of_poly (pCons (0::'b) (eulerian_poly k)) / (1 - fps_X) ^ (k + 1)"
12393: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> - y) = - x \<sqinter> - y"
that: shows "\<forall>q\<in>outs_\<I> \<I>. x \<in> responses_\<I> \<I> q"
pcompose_0': fixes p :: "'b poly" shows "p \<circ>\<^sub>p 0 = [:coeff p 0:]"
fst_splitAt_rev: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs"   and "x \<in> set xs" shows "fst (splitAt x (rev xs)) = rev (snd (splitAt x xs))"
poly_one: fixes \<alpha> :: "'d \<Rightarrow> 'c" shows "eval_poly \<alpha> one_poly = (1::'c)"
starfun_exp_minus: fixes x :: "'a star" shows "(*f* exp) (- x) = inverse ((*f* exp) x)"
SUP_mono': fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` A))"
prems_2: shows "a # tuv setinterleaves ((tu, v), A)"
LBset_def: fixes z :: "ant" shows "LBset z = {x. z \<le> x}"
the1I2: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<exists>!x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
encode_problem_with_operator_interference_exclusion_valid: fixes \<Pi> :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and sv :: "sat_plan_variable"   and h :: "nat" assumes "sat_solve_sasp \<Pi>"   and "sv \<in> atoms (\<Phi>\<^sub>\<forall> \<phi> prob_with_noop (ast_problem.abs_prob \<Pi>)  h)" shows "cnf_to_dimacs.valid_state_var (Suc h) (Suc (length (ast_problem.ast\<delta> \<Pi>))) sv"
sinh_def: fixes x :: "'b" shows "sinh x = (exp x - exp (- x)) /\<^sub>R 2"
select_convs_1: fixes NTDom :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NTCod :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NatTransMap :: "'g \<Rightarrow> 'j"   and more :: "'m" shows "NTDom \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NTDom"
in_rel: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (\<exists>z. z \<in> UNIV \<and> id z = a \<and> id z = b)"
dom_eqI': fixes a :: "'s"   and f :: "'s" assumes "a \<in> domains f" shows "a = local.dom f"
Rep_phi_inverse: fixes x :: "('c, 'd) phi" shows "Abs_phi (Rep_phi x) = x"
ennreal_le_iff: fixes y :: "real"   and x :: "real" assumes "0 \<le> y" shows "(ennreal x \<le> ennreal y) = (x \<le> y)"
det_code_equation: fixes A :: "(('b, 'c) vec, 'c) vec" shows "det A = (let A' = Gauss_Jordan_det_P A in (\<Prod>i\<in>UNIV. snd A' $ i $ i) / fst A')"
last_tl: fixes xs :: "'b list" assumes "xs = [] \<or> tl xs \<noteq> []" shows "last (tl xs) = last xs"
simps_1: fixes name :: "name" shows "sterm_to_pterm (Sconst name) = Pconst name"
sigma_sets_single: fixes A :: "'c set" shows "sigma_sets A {A} = {{}, A}"
trace_parallel_plan_strips_operator_preconditions: fixes k :: "nat"   and I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>) - 1" shows "are_all_operators_applicable (trace_parallel_plan_strips I \<pi> ! k) (\<pi> ! k) \<and> are_all_operator_effects_consistent (\<pi> ! k)"
ipresIGSubstAbs_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('l, 'm, 'varSort, 'sort, 'n, 'o, 'h, 'j, 'p) model_scheme"   and MOD' :: "('q, 'r, 'varSort, 's, 't, 'o, 'i, 'k, 'u) model_scheme" shows "ipresIGSubstAbs h hA MOD MOD' \<equiv> \<forall>ys Y y us s A. igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A \<longrightarrow> hA (igSubstAbs MOD ys Y y A) = igSubstAbs MOD' ys (h Y) y (hA A)"
aEval_leq: fixes p :: "real mpoly"   and L :: "real list" shows "aEval (Leq p) L = (aEval (Less p) L \<or> aEval (Eq p) L)"
elims_3: fixes cenum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'a list \<Rightarrow> bool"   and xa :: "nat" assumes "\<not> ex_n_lists cenum_ex x xa" obtains P :: "'a list \<Rightarrow> bool" and n :: "nat" where "x = P" and "xa = n" and "\<not> (if n = 0 then P [] else cenum_ex (\<lambda>x. ex_n_lists cenum_ex (\<lambda>xs. P (x # xs)) (n - 1)))"
ideal_chain_is_trivial: fixes C :: "'a set set" assumes "C \<noteq> {}"   and "subset.chain {I. ideal I R} C" shows "\<Union> C \<in> C"
lrev_lrev_ident: fixes xs :: "'b llist" assumes "xs \<in> UNIV\<^sup>\<star>" shows "lrev (lrev xs) = xs"
exhaust_sel: fixes labeled_graph :: "('d, 'e) labeled_graph" obtains "labeled_graph = LG (edges labeled_graph) (vertices labeled_graph)"
that: fixes x :: "hf"   and y :: "hf" assumes "x \<^bold>\<in> A"   and "y \<^bold>\<in> B"   and "c = \<langle>x, y\<rangle>" shows "thesis"
finite_card_of_iff_card2: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "finite B" shows "(|A| \<le>o |B|) = (card A \<le> card B)"
coherent_Assoc_Ide: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src\<^sub>C :: "'a \<Rightarrow> 'a"   and trg\<^sub>C :: "'a \<Rightarrow> 'a"   and V :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i> :: "'d \<Rightarrow> 'd"   and src :: "'d \<Rightarrow> 'd"   and trg :: "'d \<Rightarrow> 'd"   and E :: "'a \<Rightarrow> 'd"   and a :: "'a local.term"   and b :: "'a local.term"   and c :: "'a local.term" assumes "Coherence.evaluation_map C src\<^sub>C trg\<^sub>C V H \<a> \<i> src trg E"   and "bicategorical_language.Ide C src\<^sub>C trg\<^sub>C a"   and "bicategorical_language.Ide C src\<^sub>C trg\<^sub>C b"   and "bicategorical_language.Ide C src\<^sub>C trg\<^sub>C c"   and "bicategorical_language.Src src\<^sub>C a = bicategorical_language.Trg trg\<^sub>C b"   and "bicategorical_language.Src src\<^sub>C b = bicategorical_language.Trg trg\<^sub>C c" shows "V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Cod C src\<^sub>C trg\<^sub>C \<^bold>\<a>\<^bold>[a, b, c\<^bold>]))) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E \<^bold>\<a>\<^bold>[a, b, c\<^bold>]) = V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.Nmlize C \<^bold>\<a>\<^bold>[a, b, c\<^bold>])) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Dom C src\<^sub>C trg\<^sub>C \<^bold>\<a>\<^bold>[a, b, c\<^bold>])))"
vp_split_1: fixes n :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as @ a # as'\<rightarrow>\<^sub>\<surd>* n'" shows "n -as\<rightarrow>\<^sub>\<surd>* sourcenode a"
has_as_equalizer_def: fixes f0 :: "'s"   and f1 :: "'s"   and e :: "'s" shows "has_as_equalizer f0 f1 e \<equiv> par f0 f1 \<and> limit_cone (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) (\<cdot>) (parallel_pair_diagram.map (\<cdot>) f0 f1) (local.dom e) (parallel_pair_diagram.mkCone (\<cdot>) f0 e)"
elims: fixes x :: "prop_rule"   and y :: "(char list, prop_funs) pform list" assumes "consequent x = y" obtains "x = andI" and "y = [Fun and [X, Y]]"   | "x = andE" and "y = [X, Y]"   | "x = impI" and "y = [Fun imp [X, Y]]"   | "x = impE" and "y = [Y]"
hta_mem'_def: fixes t :: "'a tree"   and H :: "('b, 'a, 'c) hashedTa_scheme" shows "hta_mem' t H \<equiv> \<not> lh_set_xx.g_disjoint (faccs' H t) (hta_Qi H)"
example_nodes_2: shows "nodes M\<^sub>I = {0, 1}"
ddis: shows "disjnt K (frontier (cbox (- (b + d *\<^sub>R One)) (b + d *\<^sub>R One)))"
counit_simps_4: shows "src\<^sub>C \<epsilon> = trg\<^sub>C f"
Complete_set_residues_def: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a set" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.Complete_set_residues times plus minus g = (\<exists>f. ring_1.res_function times plus minus f \<and> (\<forall>c. pairwise (\<lambda>a b. \<not> ring_1.cong times minus a b c) (range (f c)) \<and> g c = range (f c)))"
trss_c_no_mon: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" shows "mon_c fg c' = mon_c fg c"
root_cond_cong: shows "root_cond (p, l, r) = root_cond (q, l, r)"
simps_10: fixes F :: "'f set" shows "strip_guards F Throw = Throw"
pterm_to_sterm_wellformed: fixes t :: "pterm" assumes "pre_strong_term_class.wellformed t" shows "pre_strong_term_class.wellformed (pterm_to_sterm t)"
subsequence_def: fixes w :: "nat \<Rightarrow> 'c"   and i :: "nat"   and j :: "nat" shows "w [i \<rightarrow> j] \<equiv> map w [i..<j]"
Exe2I: fixes F :: "\<Pi>\<^sub>2"   and x :: "\<kappa>"   and y :: "\<kappa>"   and v :: "i" assumes "\<exists>r o\<^sub>1 o\<^sub>2. Some r = d\<^sub>2 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> (o\<^sub>1, o\<^sub>2) \<in> ex2 r v" shows "[\<lparr>F,x,y\<rparr> in v]"
connected_component_sym: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "connected_component S x y" shows "connected_component S y x"
poly_decompose: fixes p :: "'a poly" assumes "\<not> constant (poly p)" shows "\<exists>k a q. a \<noteq> (0::'a) \<and> k \<noteq> 0 \<and> psize q + k + 1 = psize p \<and> (\<forall>z. poly p z = poly p (0::'a) + z ^ k * poly (pCons a q) z)"