hcju/isabelleps huggingface dataset:
neighborhoods_eq: shows "top.neighborhoods \<pp> = {U. is_zariski_open U \<and> \<pp> \<in> U}"
preserves_ide: fixes a :: "'a1" assumes "A1.ide a" shows "A2_B.ide (G a)"
has_typ_def: fixes t :: "Core.term"   and T :: "typ" shows "\<turnstile>\<^sub>\<tau> t : T = [] \<turnstile>\<^sub>\<tau> t : T"
scp: shows "simplicial_chain p (standard_simplex p) (frag_of (restrict id (standard_simplex p)))"
LeftDerivationFixOrIntro: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and \<gamma> :: "symbol list"   and j :: "nat" assumes "LeftDerivation a D \<gamma>"   and "is_sentence \<gamma>"   and "j < length \<gamma>" shows "(\<exists>i. LeftDerivationFix a i D j \<gamma>) \<or> (\<exists>d \<alpha> ix. d < length D \<and> LeftDerivation a (take d D) \<alpha> \<and> LeftDerivationIntro \<alpha> (fst (D ! d)) (snd (D ! d)) ix (drop (Suc d) D) j \<gamma>)"
upd_surj: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "upd ` {..<n} = {..<n}"
squash_mult: fixes a :: "real"   and b :: "real" shows "squash (a * b) = squash a * squash b"
chain_map_minus: fixes p :: "nat"   and g :: "'d \<Rightarrow> 'c"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int" shows "chain_map p g (- c) = - chain_map p g c"
wf_cdecl_unique: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "wf_cdecl G (C, c)" shows "unique (cfields c) \<and> unique (methods c)"
preferred_alts_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "preferred_alts R x = {y. R x y}"
frechet_derivative_const: fixes c :: "'c"   and a :: "'b" shows "frechet_derivative (\<lambda>x. c) (at a) = (\<lambda>x. 0::'c)"
apply_operations_added_ids: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and f :: "'b \<Rightarrow> 'a set"   and x :: "'b" assumes "es prefix of j"   and "apply_operations es = Some f" shows "f x \<subseteq> set (added_ids es x)"
event_splitter_def: fixes e :: "char list \<times> 'a list" shows "event_splitter e = \<Union> (strategy ` {v. wf_tuple (MFOTL.nfv \<phi>) (fv \<phi>) v \<and> MFOTL.matches (map the v) \<phi> e})"
update_matchP_2: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and aux :: "(nat \<times> (mregex, event_data option list set) mapping) list"   and ne :: "nat"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and mrs :: "mregex list"   and rels :: "event_data option list set list"   and rel :: "event_data option list set"   and aux' :: "(nat \<times> (mregex, event_data option list set) mapping) list" assumes "wf_matchP_aux \<sigma> V n R I r aux ne"   and "safe_regex Past Strict r"   and "to_mregex r = (mr, \<phi>s)"   and "mrs = sorted_list_of_set (RPDs mr)"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne \<phi>)) \<phi>s rels"   and "(rel, aux') = update_matchP n I mr mrs rels (\<tau> \<sigma> ne) aux" shows "qtable n (fv_regex r) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne (formula.MatchP I r)) rel"
nat_mult_1: fixes n :: "nat" shows "1 * n = n"
simps_2: fixes n :: "nat"   and u :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and m :: "nat" shows "muh (Suc n) u m = upclose (u m (Suc n)) (muh n u m)"
continuous_on_topological: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
distinct_9: fixes x61 :: "nat"   and x62 :: "recf" shows "Z \<noteq> Mn x61 x62"
sup_aci_1: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> y = y \<oplus> x"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> 'b mat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "'b mat" assumes "\<And>i j A. (i < j \<Longrightarrow> P (Suc i) j (swap_cols_rows i j A)) \<Longrightarrow> P i j A" shows "P a0 a1 a2"
lcp_ruler: fixes r :: "'b list"   and w1 :: "'b list"   and w2 :: "'b list" assumes "r \<bowtie> w1"   and "r \<bowtie> w2"   and "\<not> w1 \<bowtie> w2" shows "r \<le>p w1 \<and>\<^sub>p w2"
env_restr_mono: fixes m1 :: "'c \<Rightarrow> 'd"   and m2 :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "m1 \<sqsubseteq> m2" shows "m1 f|` S \<sqsubseteq> m2 f|` S"
exhaust: fixes y :: "('a, 'b) Node" obtains x11 :: "'a" and x12 :: "'b" where "y = Tip x11 x12"   | x21 :: "'b" and x22 :: "('a, 'b) Node" and x23 :: "('a, 'b) Node" where "y = Node2 x21 x22 x23"   | x31 :: "'b" and x32 :: "('a, 'b) Node" and x33 :: "('a, 'b) Node" and x34 :: "('a, 'b) Node" where "y = Node3 x31 x32 x33 x34"
split_ismeq: fixes a :: "nat"   and b :: "nat"   and X :: "bool" shows "(a \<le> b \<and> X) = (a < b \<and> X \<or> a = b \<and> X)"
ensure_testable_def: fixes f :: "'b" shows "ensure_testable f = f"
eval_fps_1: fixes z :: "'a" shows "eval_fps 1 z = (1::'a)"
assms_3: shows "f ` C \<noteq> f ` D"
inj_map_strong: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'c"   and fa :: "'a \<Rightarrow> 'c" assumes "\<And>z za. \<lbrakk>s2p (range x) z; s2p (range xa) za; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "f \<circ> x = fa \<circ> xa" shows "x = xa"
ActualBoxI: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<box>\<phi> in v]" shows "[\<^bold>\<A>(\<^bold>\<box>\<phi>) in v]"
simps: fixes a1 :: "s_context"   and a2 :: "t_context"   and a3 :: "e list"   and a4 :: "tf" shows "a1\<bullet>a2 \<turnstile> a3 : a4 = ((\<exists>\<C> b_es tf \<S>. a1 = \<S> \<and> a2 = \<C> \<and> a3 = $* b_es \<and> a4 = tf \<and> \<C> \<turnstile> b_es : tf) \<or> (\<exists>\<S> \<C> es t1s t2s e t3s. a1 = \<S> \<and> a2 = \<C> \<and> a3 = es @ [e] \<and> a4 = t1s _> t3s \<and> \<S>\<bullet>\<C> \<turnstile> es : t1s _> t2s \<and> \<S>\<bullet>\<C> \<turnstile> [e] : t2s _> t3s) \<or> (\<exists>\<S> \<C> es t1s t2s ts. a1 = \<S> \<and> a2 = \<C> \<and> a3 = es \<and> a4 = ts @ t1s _> ts @ t2s \<and> \<S>\<bullet>\<C> \<turnstile> es : t1s _> t2s) \<or> (\<exists>\<S> \<C> tf. a1 = \<S> \<and> a2 = \<C> \<and> a3 = [Trap] \<and> a4 = tf) \<or> (\<exists>\<S> ts i vs es n \<C>. a1 = \<S> \<and> a2 = \<C> \<and> a3 = [Local n i vs es] \<and> a4 = [] _> ts \<and> \<S>\<bullet>Some ts \<tturnstile>_ i vs;es : ts \<and> length ts = n) \<or> (\<exists>\<S> cl tf \<C>. a1 = \<S> \<and> a2 = \<C> \<and> a3 = [Callcl cl] \<and> a4 = tf \<and> cl_typing \<S> cl tf) \<or> (\<exists>\<S> \<C> e0s ts t2s es n. a1 = \<S> \<and> a2 = \<C> \<and> a3 = [Label n e0s es] \<and> a4 = [] _> t2s \<and> \<S>\<bullet>\<C> \<turnstile> e0s : ts _> t2s \<and> \<S>\<bullet>\<C>\<lparr>label := [ts] @ label \<C>\<rparr> \<turnstile> es : [] _> t2s \<and> length ts = n))"
dim_le_card': fixes s :: "('b, 'c) vec set" assumes "finite s" shows "vec.dim s \<le> card s"
assms_2: shows "b \<in> {real_of_float bl..real_of_float bu}"
E_def: shows "E = sum_list (ss @ [s, t]) `\<rightarrow> C0"
atLeastAtMost_singleton': fixes a :: "'a word"   and b :: "'a word" assumes "a = b" shows "signed.atLeastAtMost a b = {a}"
status_cases: fixes y :: "'a \<times> nat \<Rightarrow> nat list" assumes "y \<in> {\<sigma>. \<forall>f k. set (\<sigma> (f, k)) \<subseteq> {0..<k}}" obtains x :: "'a status" where "y = status x"
encrypt_def: fixes \<alpha> :: "'grp"   and msg :: "bool list" shows "encrypt \<alpha> msg = lift_spmf (sample_uniform (order \<G>)) \<bind> (\<lambda>y. hash (\<alpha> [^] y) \<bind> (\<lambda>h. Done (\<^bold>g [^] y, h [\<oplus>] msg)))"
set3_F_map: fixes l1 :: "'p \<Rightarrow> 'aa"   and l2 :: "'q \<Rightarrow> 'ab"   and l3 :: "'r \<Rightarrow> 'z" shows "set3_F \<circ> mapl_F l1 l2 l3 = (`) l3 \<circ> set3_F"
int_prod: fixes f :: "'b \<Rightarrow> nat"   and A :: "'b set" shows "int (prod f A) = (\<Prod>x\<in>A. int (f x))"
Y_pos: fixes y :: "(real, 'n) Finite_Cartesian_Product.vec"   and i :: "'n" assumes "y \<in> Y" shows "0 < y $h i"
elims: fixes x :: "Formula.formula"   and y :: "Formula.formula" assumes "remove_neg x = y" obtains \<phi> :: "Formula.formula" where "x = formula.Neg \<phi>" and "y = \<phi>"   | v :: "String.literal" and va :: "Formula.trm list" where "x = formula.Pred v__ va__" and "y = formula.Pred v__ va__"   | v :: "String.literal" and va :: "Formula.formula" and vb :: "Formula.formula" where "x = formula.Let vb__ vaa__ vba__" and "y = formula.Let vb__ vaa__ vba__"   | v :: "Formula.trm" and va :: "Formula.trm" where "x = formula.Eq vc__ vab__" and "y = formula.Eq vc__ vab__"   | v :: "Formula.trm" and va :: "Formula.trm" where "x = formula.Less vd__ vac__" and "y = formula.Less vd__ vac__"   | v :: "Formula.trm" and va :: "Formula.trm" where "x = formula.LessEq ve__ vad__" and "y = formula.LessEq ve__ vad__"   | v :: "Formula.formula" and va :: "Formula.formula" where "x = formula.Or vf__ vae__" and "y = formula.Or vf__ vae__"   | v :: "Formula.formula" and va :: "Formula.formula" where "x = formula.And vg__ vaf__" and "y = formula.And vg__ vaf__"   | v :: "Formula.formula list" where "x = formula.Ands vh__" and "y = formula.Ands vh__"   | v :: "Formula.formula" where "x = formula.Exists vi__" and "y = formula.Exists vi__"   | v :: "nat" and va :: "Formula.agg_type \<times> event_data" and vb :: "nat" and vc :: "Formula.trm" and vd :: "Formula.formula" where "x = formula.Agg vj__ vag__ vbb__ vc vd" and "y = formula.Agg vj__ vag__ vbb__ vc vd"   | v :: "\<I>" and va :: "Formula.formula" where "x = formula.Prev vk__ vah__" and "y = formula.Prev vk__ vah__"   | v :: "\<I>" and va :: "Formula.formula" where "x = formula.Next vl__ vai__" and "y = formula.Next vl__ vai__"   | v :: "Formula.formula" and va :: "\<I>" and vb :: "Formula.formula" where "x = formula.Since vm__ vaj__ vbc__" and "y = formula.Since vm__ vaj__ vbc__"   | v :: "Formula.formula" and va :: "\<I>" and vb :: "Formula.formula" where "x = formula.Until vn__ vak__ vb" and "y = formula.Until vn__ vak__ vb"   | v :: "\<I>" and va :: "Formula.formula Regex.regex" where "x = formula.MatchF vo__ val__" and "y = formula.MatchF vo__ val__"   | v :: "\<I>" and va :: "Formula.formula Regex.regex" where "x = formula.MatchP v va" and "y = formula.MatchP v va"
eq_dfa_def: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "eq_dfa n ks l = (let (is, js) = dioph_dfs n ks l in (map (\<lambda>j. make_bdd (\<lambda>xs. if eval_dioph ks xs mod 2 = j mod 2 then the (is ! int_encode ((j - eval_dioph ks xs) div 2)) else length js) n []) js @ [Leaf (length js)], map (\<lambda>j. j = 0) js @ [False]))"
iu_2: shows "root n (real_of_1 (p, l, r)) \<le> real_of_rat (local.initial_upper_bound r)"
eval_elim_cases_28: fixes G :: "prog"   and s :: "st"   and accC :: "qtname"   and statDeclC :: "qtname"   and stat :: "bool"   and e :: "expr"   and fn :: "vname"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In2 ({accC,statDeclC,stat}e..fn)\<succ>\<rightarrow> (v, s')" obtains s1 :: "state" and a :: "val" and s2 :: "state" and va :: "vvar" and s2' :: "state" where "v = In2 va" and "s' = check_field_access G accC statDeclC fn stat a s2'" and "G\<turnstile>Norm s \<midarrow>Init statDeclC\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>e-\<succ>a\<rightarrow> s2" and "(va, s2') = fvar statDeclC stat fn a s2"
mrw_change_prefix: fixes E' :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and r :: "nat"   and w :: "nat"   and n :: "enat" assumes "E' \<in> \<E>"   and "P,E \<turnstile> r \<leadsto>mrw w"   and "thread_start_actions_ok E'"   and "ltake n E [\<approx>] ltake n E'"   and "enat r < n"   and "enat w < n" shows "P,E' \<turnstile> r \<leadsto>mrw w"
bdd_aboveI2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and M :: "'c" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "preorder.bdd_above less_eq (f ` A)"
max_rank_of_def: fixes \<Sigma> :: "'a set set"   and \<psi> :: "'a ltl" shows "\<A>.max_rank_of \<Sigma> \<psi> \<equiv> semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<psi>))"
D: shows "Crypt (SesK SK) (PubKey C) \<notin> parts (used s)"
mk_disjoint_insert: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "\<exists>B. A = Set.insert a B \<and> a \<notin> B"
mds\<^sub>A_of_def: fixes mds\<^sub>C :: "Mode \<Rightarrow> 'Var set" shows "gen_refine.mds\<^sub>A_of mds\<^sub>C \<equiv> \<lambda>m. inv id ` (range id \<inter> mds\<^sub>C m)"
uminus_aform_def: fixes X :: "'b \<times> 'c pdevs" shows "uminus_aform X = (- fst X, uminus_pdevs (snd X))"
connected_component_of_subset: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and T :: "'a set" assumes "connected_component S x y"   and "S \<subseteq> T" shows "connected_component T x y"
bheight_joinL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invh l"   and "invh r"   and "bheight l \<le> bheight r" shows "bheight (joinL l x r) = bheight r"
e_type_comp_conc: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and es :: "e list"   and t1s :: "t list"   and t2s :: "t list"   and es' :: "e list"   and t3s :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> es : t1s _> t2s"   and "\<S>\<bullet>\<C> \<turnstile> es' : t2s _> t3s" shows "\<S>\<bullet>\<C> \<turnstile> es @ es' : t1s _> t3s"
has_fps_expansion_cmult_left: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps"   and c :: "'a" assumes "f has_fps_expansion F" shows "(\<lambda>x. c * f x) has_fps_expansion fps_const c * F"
arr_cod: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.arr (VoV.cod f)"
case_eq_if: fixes f :: "('e \<Rightarrow> 'f) \<Rightarrow> 'd"   and envT :: "('e, 'f) envT" shows "(case envT of EnvT x \<Rightarrow> f x) = f (run_env envT)"
rank_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "rank (Finite_Cartesian_Product.transpose A) = rank A"
in_hhomI: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "B.arr \<mu>"   and "src\<^sub>B \<mu> = a"   and "trg\<^sub>B \<mu> = b" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>B b\<guillemotright>"
iso_dghm_ObjMap_v11: shows "v11 (\<FF>\<lparr>ObjMap\<rparr>)"
uses_in_\<alpha>n: fixes v :: "'d"   and g :: "'a"   and n :: "'b" assumes "v \<in> oldUses g n" shows "n \<in> set (\<alpha>n g)"
\<eta>_map_simp: fixes y :: "'d" assumes "D.ide y" shows "\<eta>.map y = \<phi> y (F y)"
select_convs_1: fixes lens_get :: "'b \<Rightarrow> 'c"   and lens_put :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and more :: "'d" shows "get\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_get"
axiom_predcong_def: fixes i :: "String.literal"   and l :: "tm list"   and l' :: "tm list" shows "axiom_predcong i l l' \<equiv> if equal_length l l' then Thm (foldr Imp (zip_eq l l') (Imp (Atom (Rl i l)) (Atom (Rl i l')))) else Thm Truth"
reachable_configs_induct: fixes xa :: "nat"   and xb :: "integer"   and bp :: "instr array"   and c\<^sub>s' :: "nat"   and c\<^sub>s'' :: "integer"   and P :: "nat \<Rightarrow> integer \<Rightarrow> bool" assumes "(xa, xb) \<in> reachable_configs bp (c\<^sub>s', c\<^sub>s'')"   and "P c\<^sub>s' c\<^sub>s''"   and "\<And>a b aa ba. \<lbrakk>(a, b) \<in> reachable_configs bp (c\<^sub>s', c\<^sub>s''); P a b; (aa, ba) \<in> set (nexts bp (a, b))\<rbrakk> \<Longrightarrow> P aa ba" shows "P xa xb"
simps_1: fixes x1 :: "'f"   and x2 :: "'g"   and y1 :: "'f"   and y2 :: "'g" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
restrict_idemp: fixes x :: "('e, 'f) labeled_graph" shows "restrict (restrict x) = restrict x"
mult_inv_same_comm: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and m :: "nat"   and n :: "nat" assumes "Group.monoid G"   and "a \<in> Units G" shows "inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> m) \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> n) = inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> n) \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> m)"
stutter_extend_en_def: fixes en :: "'b \<Rightarrow> 'd set" shows "stutter_extend_en en \<equiv> \<lambda>s. let as = en s in if as = {} then {None} else Some ` as"
a: shows "fic (M[x\<turnstile>n>y]) c"
dvd_smult_int: fixes c :: "int"   and q :: "int poly"   and p :: "int poly" assumes "c \<noteq> 0"   and "q dvd smult c p" shows "primitive_part q dvd p"
results_in_zero: fixes t :: "'t" shows "results_in t (0::'sum) = t"
map_eq_Cons_D: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and y :: "'d"   and ys :: "'d list" assumes "map f xs = y # ys" shows "\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys"
init_obj_pred_conv: 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 C :: "char list" shows "set_of_pred (init_obj_pred P C) = Collect (init_obj' P C)"
comp_rassocl\<^sub>C: fixes conv1 :: "('h, 'k, 'n, 'q) converter"   and conv2 :: "('i, 'l, 'o, 'r) converter"   and conv3 :: "('j, 'm, 'p, 's) converter" shows "(conv1 |\<^sub>= conv2 |\<^sub>= conv3) \<odot> rassocl\<^sub>C = rassocl\<^sub>C \<odot> ((conv1 |\<^sub>= conv2) |\<^sub>= conv3)"
refl_onI: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "r \<subseteq> A \<times> A"   and "\<And>x. x \<in> A \<Longrightarrow> (x, x) \<in> r" shows "refl_on A r"
fps_exp_unique_ODE: fixes a :: "'a fps"   and c :: "'a" shows "(fps_deriv a = fps_const c * a) = (a = fps_const (fps_nth a 0) * fps_exp c)"
runit_def: fixes a :: "'a" shows "\<r>[a] \<equiv> THE f. \<guillemotleft>f : a \<otimes> \<I> \<rightarrow> a\<guillemotright> \<and> f \<otimes> \<I> = (a \<otimes> \<iota>) \<cdot> \<a>[a, \<I>, \<I>]"
hyps_3: assumes "U \<subseteq> X" shows "(\<Sum>i\<in>U. f i * i) \<in> X"
monom_eq_1: shows "monom (1::'a) 0 = 1"
trans: fixes b :: "'t"   and a :: "'t"   and c :: "'t" assumes "b \<preceq>\<^sub>t a"   and "c \<preceq>\<^sub>t b" shows "c \<preceq>\<^sub>t a"
keys_empty_iff_map_empty: fixes m :: "('c, 'd) mapping" shows "(Mapping.keys m = {}) = (m = Mapping.empty)"
tendsto_offset_zero_iff: fixes a :: "'b"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and L :: "'c" assumes "NO_MATCH (0::'a) a"   and "a \<in> S"   and "open S" shows "(f \<longlongrightarrow> L) (at a within S) = (\<lambda>h. f (a + h)) \<midarrow>0::'b\<rightarrow> L"
minus_octo_def: shows "(-) \<equiv> \<lambda>uu uua. corec_octo (\<lambda>(x, y). Ree x - Ree y) (\<lambda>(x, y). Im1 x - Im1 y) (\<lambda>(x, y). Im2 x - Im2 y) (\<lambda>(x, y). Im3 x - Im3 y) (\<lambda>(x, y). Im4 x - Im4 y) (\<lambda>(x, y). Im5 x - Im5 y) (\<lambda>(x, y). Im6 x - Im6 y) (\<lambda>(x, y). Im7 x - Im7 y) (uu, uua)"
UP_def: fixes R :: "('c, 'd) ring_scheme" shows "UP R = \<lparr>carrier = up R, mult = \<lambda>p\<in>up R. \<lambda>q\<in>up R. \<lambda>n. \<Oplus>\<^bsub>R\<^esub>i\<in>{..n}. p i \<otimes>\<^bsub>R\<^esub> q (n - i), one = \<lambda>i. if i = 0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>, zero = \<lambda>i. \<zero>\<^bsub>R\<^esub>, add = \<lambda>p\<in>up R. \<lambda>q\<in>up R. \<lambda>i. p i \<oplus>\<^bsub>R\<^esub> q i, smult = \<lambda>a\<in>carrier R. \<lambda>p\<in>up R. \<lambda>i. a \<otimes>\<^bsub>R\<^esub> p i, monom = \<lambda>a\<in>carrier R. \<lambda>n i. if i = n then a else \<zero>\<^bsub>R\<^esub>, coeff = \<lambda>p\<in>up R. p\<rparr>"
linkage_quotient_webD_1: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and h :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "web_flow (quotient_web \<Gamma> f) g"   and "linkage (quotient_web \<Gamma> f) g"   and "trimming \<Gamma> f h" shows "web_flow \<Gamma> (plus_current h g)"
fresh_star_eqvt_1: fixes pi :: "(name \<times> name) list"   and a :: "name set"   and x :: "'a" shows "pi \<bullet> a \<sharp>* x = (pi \<bullet> a) \<sharp>* (pi \<bullet> x)"
is_type_def: 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 T :: "ty" shows "is_type P T \<equiv> case T of Class C \<Rightarrow> is_class P C | _ \<Rightarrow> True"
ide_char': fixes a :: "(bool, bool list) concrete_category.arr" shows "ide a = (arr a \<and> (local.dom a = a \<or> cod a = a))"
alt_consistency_def: fixes C :: "('a, 'b) form set set" shows "alt_consistency C = (\<forall>S. S \<in> C \<longrightarrow> (\<forall>p ts. \<not> (Pred p ts \<in> S \<and> Neg (Pred p ts) \<in> S)) \<and> FF \<notin> S \<and> Neg TT \<notin> S \<and> (\<forall>Z. Neg (Neg Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. And A B \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. Neg (Or A B) \<in> S \<longrightarrow> S \<union> {Neg A, Neg B} \<in> C) \<and> (\<forall>A B. Or A B \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (And A B) \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {Neg B} \<in> C) \<and> (\<forall>A B. Impl A B \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (Impl A B) \<in> S \<longrightarrow> S \<union> {A, Neg B} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Forall P \<in> S \<longrightarrow> S \<union> {P[t/0]} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Neg (Exists P) \<in> S \<longrightarrow> S \<union> {Neg (P[t/0])} \<in> C) \<and> (\<forall>P x. (\<forall>a\<in>S. new x a) \<longrightarrow> Exists P \<in> S \<longrightarrow> S \<union> {P[App x []/0]} \<in> C) \<and> (\<forall>P x. (\<forall>a\<in>S. new x a) \<longrightarrow> Neg (Forall P) \<in> S \<longrightarrow> S \<union> {Neg (P[App x []/0])} \<in> C))"
search_converg_2: fixes b :: "nat"   and i :: "nat" assumes "s b \<down>= i"   and "search b \<down>" shows "e_length b \<le> the (search b)"
0: shows "\<forall>(x, K)\<in>{(f xK, snd xK) |xK. xK \<in> p}. x \<notin> s"
fsubset_refl: fixes A :: "'c fset" shows "A |\<subseteq>| A"
lm011: fixes N :: "'d set"   and G :: "'c set" shows "possible_allocations_rel' G N \<subseteq> injectionsUniverse"
valuate_add: fixes p1 :: "linear_poly"   and p2 :: "linear_poly"   and v :: "nat \<Rightarrow> 'b" shows "p1 + p2 \<lbrace> v \<rbrace> = p1 \<lbrace> v \<rbrace> + p2 \<lbrace> v \<rbrace>"
long_division_zero_1: fixes K :: "'a list set"   and q :: "'a list list" assumes "subfield K (K [X])"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "UP.pdiv [] q = []"
inf_dual2: fixes x :: "'a"   and y :: "'a" shows "\<partial> (x \<sqinter> y) = \<partial> x \<squnion> \<partial> y"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) A.map F \<phi>'.map"
enum_le_p: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j < n" shows "enum i j \<le> p"
ivl_def: shows "ivl = rec_nat (f 0 + 1, f 0 + 2) (\<lambda>n x. (i (fst x) (snd x) (f n), j__ (fst x) (snd x) (f n)))"
associated_iff_same_ideal: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<sim> b = (PIdl a = PIdl b)"
admissibleI: fixes ord :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and lub :: "'e set \<Rightarrow> 'e" assumes "\<And>A. \<lbrakk>Complete_Partial_Order.chain ord A; A \<noteq> {}; \<forall>x\<in>A. P x\<rbrakk> \<Longrightarrow> P (lub A)" shows "ccpo.admissible lub ord P"
simps_2: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real" shows "landau_dominating_chain F [f] = ((\<lambda>x. 1) \<in> o[F](f))"