hcju/isabelleps huggingface dataset:
ltlc_aps_list'_distinct: fixes l :: "'a list"   and \<phi> :: "'a ltlc" assumes "distinct l" shows "distinct (ltlc_aps_list' \<phi> l)"
sup_assoc: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
le_diff_imp_le: fixes i :: "nat"   and j :: "nat"   and m :: "nat" assumes "i \<le> j - m" shows "i \<le> j"
eventually_not_iff: fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "ultrafilter F" shows "(\<forall>\<^sub>F x in F. \<not> P x) = (\<not> eventually P F)"
indep_in_subset: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set"   and X :: "'b set"   and Y :: "'b set" assumes "indep_system carrier indep"   and "\<E> \<subseteq> carrier"   and "indep_system.indep_in indep \<E> X"   and "Y \<subseteq> X" shows "indep_system.indep_in indep \<E> Y"
Inf_from_def: fixes N :: "'f set" shows "Inf_from N = {\<iota> \<in> Inf_F. set (prems_of \<iota>) \<subseteq> N}"
simps_2: fixes r :: "'a simple_rule"   and rs :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme" shows "simple_fw_alt (r # rs) p = (if simple_matches (match_sel r) p then case action_sel r of Accept \<Rightarrow> Decision FinalAllow | Drop \<Rightarrow> Decision FinalDeny else simple_fw_alt rs p)"
\<iota>_def: shows "\<iota>\<^sub>F\<^sub>M\<^sub>C \<equiv> \<l>[\<I>]"
assms_2: shows "\<forall>x xs. P xs \<longrightarrow> P (x # xs)"
tendsto_unique: fixes "open" :: "'a set \<Rightarrow> bool"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.t2_space open"   and "F \<noteq> \<bottom>"   and "filterlim f (topological_space.nhds open a) F"   and "filterlim f (topological_space.nhds open b) F" shows "a = b"
set_ConsD: fixes y :: "'e"   and x :: "'e"   and xs :: "'e list" assumes "y \<in> set (Cons x xs)" shows "y = x \<or> y \<in> set xs"
codomains_char': fixes f :: "Arr" shows "codomains f = (if f = Null then {} else {Cod f})"
simps_2: fixes t' :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.ins t' (t # bq) = (if BinomialHeapStruc.rank t' < BinomialHeapStruc.rank t then t' # t # bq else if BinomialHeapStruc.rank t < BinomialHeapStruc.rank t' then t # BinomialHeapStruc.ins t' bq else BinomialHeapStruc.ins (BinomialHeapStruc.link t' t) bq)"
Cons_in_lists_iff: fixes x :: "'d"   and xs :: "'d list"   and A :: "'d set" shows "(x # xs \<in> lists A) = (x \<in> A \<and> xs \<in> lists A)"
simps_2: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "black \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = black"
eq_elems_eq_val: fixes x :: "'e"   and y :: "'e"   and f :: "'e \<Rightarrow> 'f" assumes "x = y" shows "f x = f y"
holdfm: fixes m :: "nat" shows "deriv (f ^^ m) holomorphic_on s"
assms_1: fixes q :: "'a \<Rightarrow>\<^sub>0 'b" assumes "q * f \<in> ideal (set fs)" shows "q \<in> ideal (set fs)"
rk2_increment_rk2_fas1: fixes cx :: "floatarith list"   and vs :: "real list"   and x0 :: "floatarith list"   and h :: "floatarith"   and s1' :: "real"   and s2 :: "floatarith"   and rkp :: "floatarith" assumes "einterpret cx vs = flow0 (einterpret x0 vs) (interpret_floatarith h vs * s1')"   and "length x0 = DIM('c)"   and "length cx = DIM('c)"   and "DIM('c) \<le> length vs"   and "einterpret x0 vs \<in> Csafe"   and "einterpret cx vs \<in> Csafe"   and "einterpret x0 vs + (interpret_floatarith s2 vs * interpret_floatarith h vs * interpret_floatarith rkp vs) *\<^sub>R ode (einterpret x0 vs) \<in> Csafe" shows "einterpret (take DIM('c) (rk2_fas rkp x0 h cx s2)) vs = discrete_evolution (rk2_increment (interpret_floatarith rkp vs) (\<lambda>_. ode)) (interpret_floatarith h vs) 0 (einterpret x0 vs) + (heun_remainder1 (flow0 (einterpret x0 vs)) ode_na ode_d_na ode_d2_na 0 (interpret_floatarith h vs) s1' - heun_remainder2 (interpret_floatarith rkp vs) (flow0 (einterpret x0 vs)) ode_na ode_d2_na 0 (interpret_floatarith h vs) (interpret_floatarith s2 vs))"
netmap_none_not_in_net_ips: fixes ns :: "'d net_state"   and i :: "nat" assumes "netmap ns i = None" shows "i \<notin> net_ips ns"
globi_agree_def: fixes gs :: "'a list"   and n :: "nat"   and g :: "'a" shows "globi_agree gs n g = (n < length gs \<and> gs ! n = g)"
evtEA: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n (\<diamond>\<^sub>b\<gamma>)" shows "\<exists>n'\<ge>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub> \<and> (\<forall>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub>. n'' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n'\<^esub> \<longrightarrow> eval c t t' n'' \<gamma>)) \<or> \<not> (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> eval c t t' n' \<gamma>"
valid_reusables_under_some_conditions: fixes w :: "nat \<times> nat"   and as :: "'b list"   and t :: "'b tree" assumes "0 < fst w"   and "well_valued0 as t"   and "well_shaped t"   and "l t < fst w"   and "r t \<le> snd w" shows "\<forall>t'\<in>set (reusables t w). valid as t'"
N: fixes n :: "nat" assumes "N \<le> n" shows "eNorm (\<LL> p M) (f (r0 n)) < \<infinity>"
assms_5: shows "\<exists>w. minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T"
left_commute: fixes b :: "'k"   and a :: "'k"   and c :: "'k" shows "b + (a + c) = a + (b + c)"
Null_notin_List: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and ps :: "ref list" assumes "List p h ps" shows "Null \<notin> set ps"
assms_1: shows "\<forall>p q. dfa_is_node M q \<and> p < q \<longrightarrow> tr_lookup T q p = (\<exists>n<m. dist_nodes M n v p q)"
wp1_is_pre: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com" assumes "finite (support Q)" shows "\<turnstile>\<^sub>1 {wp\<^sub>1 c Q} c { \<lambda>s. \<down>\<^sub>t (c, s) \<Down> Q}"
Ghost: fixes A :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and L :: "nat set"   and R :: "nat set"   and \<O>s :: "nat set list"   and i :: "nat"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool" assumes "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = {}"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> (\<O>s ! j \<union> dom (\<R>s ! j)) = {}" shows "\<O>s,\<R>s,i\<turnstile> (Ghost A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
dbm_add_not_inf: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<noteq> \<infinity>"   and "b \<noteq> \<infinity>" shows "dbm_add a b \<noteq> \<infinity>"
l_zero: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<zero> \<oplus> x = x"
Subset_sf: fixes t :: "tm"   and u :: "tm" shows "Sigma_fm (t SUBS u)"
IH: fixes a :: "int" assumes "0 \<le> a"   and "a < 2 ^ p + 5"   and "2 < p" shows "gen_lucas_lehmer_sequence'' p a n \<in> {0..<2 ^ p + 5}"
transpose_mat: fixes n :: "'a" shows "Finite_Cartesian_Product.transpose (mat n) = mat n"
ide_composite: shows "ide ((\<mu> \<star> \<nu>) \<star> \<pi>)"
cnj_Gamma: fixes z :: "complex" shows "cnj (Gamma z) = Gamma (cnj z)"
intervalE: fixes a :: "'e"   and x :: "'e"   and b :: "'e" assumes "a \<le> x \<and> x \<le> b" obtains "x \<in> {a..b}"
fA_rel_def: shows "fA_rel = {(in_sync, hp_Idle, fA, fM, []) |fA fM in_sync. \<not> in_sync \<longrightarrow> fA = fM} \<union> {(in_sync, hp_IdleInit, fA, \<not> fA, []) |fA in_sync. True} \<union> {(in_sync, hp_InitMark, fA, \<not> fA, [mw_fA (\<not> fA)]) |fA in_sync. in_sync} \<union> {(in_sync, hp_InitMark, fA, fM, []) |fA fM in_sync. \<not> in_sync \<longrightarrow> fA \<noteq> fM} \<union> {(in_sync, hp_Mark, fA, fA, []) |fA in_sync. True} \<union> {(in_sync, hp_IdleMarkSweep, fA, fA, []) |fA in_sync. True}"
App_mono: fixes t :: "CL"   and t' :: "CL"   and u :: "CL"   and u' :: "CL" assumes "(t, t') \<in> red\<^sup>*"   and "(u, u') \<in> red\<^sup>*" shows "(` t u, ` t' u') \<in> red\<^sup>*"
assms: shows "\<exists>a. atom a \<sharp> (h, h a)"
lunit_char_2: fixes f :: "'a" assumes "S.ide f" shows "L \<l>[f] = (\<iota> \<star> f) \<cdot>\<^sub>S local.inv \<a>[b, b, f]"
not_inits_def': fixes C :: "char list"   and es :: "'a exp list" assumes "not_inits C es" shows "\<forall>e\<in>set es. not_init C e"
dagger_unfoldr: fixes x :: "'b \<times> 'c" shows "less_eq_prod (sup_prod one_prod (times_prod' (prod.qstar x) x)) (prod.qstar x)"
lm016: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Y :: "'c set" shows "P || X \<subseteq> P || (X \<union> Y) \<and> P outside X \<subseteq> P outside (X \<inter> Y)"
assms_1: shows "P (nsqn\<^sub>r (dsn, dsk, val, hops, nhip, pre))"
ev_le_vec: shows "le_vec (cmod \<alpha> *s map_vector cmod v) (A *v map_vector cmod v)"
interleave_induct: fixes x3 :: "'a list"   and x1 :: "'a list"   and x2 :: "'a list"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "x3 \<in> x1 \<otimes> x2"   and "P [] [] []"   and "\<And>xs ys zs x. \<lbrakk>zs \<in> xs \<otimes> ys; P xs ys zs\<rbrakk> \<Longrightarrow> P (x # xs) ys (x # zs)"   and "\<And>xs ys zs x. \<lbrakk>zs \<in> xs \<otimes> ys; P xs ys zs\<rbrakk> \<Longrightarrow> P xs (x # ys) (x # zs)" shows "P x1 x2 x3"
prj_simps_6: shows "C.cod \<mu>\<nu>.prj\<^sub>0 = \<nu>.apex"
ALA: shows "Domainp ALA = (\<lambda>x. x \<in> U)"
awalk_ends: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and p :: "'c list"   and v :: "'b"   and u' :: "'b"   and v' :: "'b" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "pre_digraph.awalk G u' p v'" shows "p \<noteq> [] \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
nths_Cons: fixes x :: "'c"   and l :: "'c list"   and A :: "nat set" shows "nths (x # l) A = (if 0 \<in> A then [x] else []) @ nths l {j. Suc j \<in> A}"
iso_counit: fixes a :: "'c" assumes "C.obj a" shows "C.iso (\<epsilon>'.counit a)"
prems: fixes x :: "'a \<times> 'b" assumes "x \<in># {#a#} \<times># add_mset x__ B" shows "P x = Q (snd x)"
simps_8: shows "OUTfromCh data8 = {data13}"
tensor_vec_eval: fixes i :: "nat"   and dims :: "nat list"   and vars :: "nat set"   and v1 :: "'a vec"   and v2 :: "'a vec" assumes "i < state_sig.d dims" shows "partial_state.tensor_vec dims vars v1 v2 $ i = v1 $ partial_state.encode1 dims vars i * v2 $ partial_state.encode2 dims vars i"
sin_monotone_2pi: fixes y :: "real"   and x :: "real" assumes "- (pi / 2) \<le> y"   and "y < x"   and "x \<le> pi / 2" shows "sin y < sin x"
hp_step_rel_def: shows "hp_step_rel = {True} \<times> ({(ht_NOOP, hp, hp) |hp. hp \<in> {hp_Idle, hp_IdleInit, hp_InitMark, hp_Mark}} \<union> {(ht_GetRoots, hp_IdleMarkSweep, hp_IdleMarkSweep), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}) \<union> {False} \<times> {(ht_NOOP, hp_Idle, hp_IdleMarkSweep), (ht_NOOP, hp_IdleInit, hp_Idle), (ht_NOOP, hp_InitMark, hp_IdleInit), (ht_NOOP, hp_Mark, hp_InitMark), (ht_GetRoots, hp_IdleMarkSweep, hp_Mark), (ht_GetWork, hp_IdleMarkSweep, hp_IdleMarkSweep)}"
zag_t_def: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "zag_t x y = (if snd x * snd y = 0 then if 0 < fst x * snd y + snd x * fst y then (0, 1) else if fst x * snd y + snd x * fst y = 0 then (fst x * fst y, 0) else (0, - 1) else if 0 < snd x * snd y then (0, 1) else (0, - 1))"
list_of_eucl_eucl_of_list: fixes xs :: "real list" assumes "length xs = DIM('b)" shows "list_of_eucl (eucl_of_list xs) = xs"
vpairs_def: fixes r :: "V" shows "vpairs r = set {x. x \<in>\<^sub>\<circ> r \<and> (\<exists>a b. x = \<langle>a, b\<rangle>)}"
i_expand_if: fixes f :: "nat \<Rightarrow> 'b message_af"   and k :: "nat" shows "f \<odot> k = (if k = 0 then \<lambda>n. NoMsg else (\<lambda>n. if n mod k = 0 then f (n div k) else NoMsg))"
elim_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto>[Rel] Q"   and "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<longmapsto> a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
r_one: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x"
scalar_prod_Cauchy: fixes u :: "'a vec"   and n :: "nat"   and v :: "'a vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n" shows "(u \<bullet> v)\<^sup>2 \<le> \<parallel>u\<parallel>\<^sup>2 * \<parallel>v\<parallel>\<^sup>2"
trms\<^sub>s\<^sub>s\<^sub>t_cons: fixes a :: "('d, 'e) stateful_strand_step"   and A :: "('d, 'e) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t (a # A) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> trms\<^sub>s\<^sub>s\<^sub>t A"
sort_of_Rep: fixes a :: "'a" shows "sort_of (Rep a) = s"
d_def: shows "d = rec_list (Generative_Probabilistic_Value.Pause x (\<lambda>y'. Generative_Probabilistic_Value.Done (y = y'))) (\<lambda>(x, y) xs rec. Generative_Probabilistic_Value.Pause x (\<lambda>input. if input = y then rec else Generative_Probabilistic_Value.Done False))"
simps: fixes n :: "('form, 'rule) graph_node" shows "terminal_node n = (outPorts n = {||})"
wiring: shows "wiring \<I>2 \<I>3 conv w"
cring_simprules_13: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
coeff_closed: fixes p :: "nat \<Rightarrow> 'c"   and R :: "('c, 'd) ring_scheme"   and n :: "nat" assumes "p \<in> carrier (UP R)" shows "coeff (UP R) p n \<in> carrier R"
mrNnt: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators" assumes "applied_rule_rev C x p = \<lfloor>a\<rfloor>" shows "p \<noteq> []"
slim_is_slim: shows "pair_pre_digraph.is_slim sG.slim (verts3 (with_proj slim))"
sel_1: fixes x21 :: "'c"   and x22 :: "'c list" shows "lead_coeff (x21 # x22) = x21"
PdXXX_range: fixes d :: "'b" shows "Field (Pd' d) \<subseteq> {x. Xd x = d}"
shadow_root: shows "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
powhr_hlog_cancel: fixes a :: "real star"   and x :: "real star" assumes "0 < a"   and "a \<noteq> 1"   and "0 < x" shows "a powhr hlog a x = x"
closed_Collect_eq: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "closed {x. f x = g x}"
urel32_mult: fixes x :: "uint32"   and y :: "int"   and x' :: "uint32"   and y' :: "int" assumes "urel32 x y"   and "urel32 x' y'" shows "urel32 (mult_p32 pp x x') (mult_p p y y')"
T2_def: shows "T2 \<equiv> unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_selects T))"
echelon_foldl_condition6: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and k :: "nat"   and A :: "(('d, 'f) vec, 'e) vec"   and mc :: "'e" assumes "is_bezout_ext bezout"   and "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'e) \<le> mc"   and "A $ mc $ mod_type_class.from_nat k \<noteq> (0::'d)" shows "\<exists>m. \<not> is_zero_row_upt_k m (Suc k) (bezout_iterate (interchange_rows A ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'e)) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'d) \<and> (GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'e) \<le> n)) (nrows A - Suc 0) ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'e)) (mod_type_class.from_nat k) bezout)"
wtE_pickE: fixes xl :: "Preliminaries.var list"   and al :: "'univ list" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> intT (tpOfV (xl ! i)) (al ! i)" shows "wtE (pickE xl al)"
f1: assumes "x \<noteq> y"   and "atom y \<sharp> T"   and "atom x \<sharp> (y \<leftrightarrow> x) \<bullet> T" shows "atom x \<sharp> f x T"
replicate_0: fixes x :: "'b" shows "x\<^bsup>0\<^esup> = []"
append1_eq_conv: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list"   and y :: "'a" shows "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
scaleR_fun_beta: fixes r :: "real"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(r *\<^sub>R f) x = r *\<^sub>R f x"
IK_nf: fixes f :: "'a set \<Rightarrow> 'a set" shows "IK f = IK_nf f"
le_nat_iff': fixes x :: "nat"   and y :: "int" shows "(x \<le> nat y) = (x = 0 \<and> y \<le> 0 \<or> int x \<le> y)"
msg_ok_Newpkt: fixes \<sigma> :: "nat \<Rightarrow> state"   and data :: "nat"   and dst :: "nat" shows "msg_ok \<sigma> (Newpkt data dst)"
Un_assoc: fixes A :: "'f set"   and B :: "'f set"   and C :: "'f set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
has_as_product_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and J :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'b"   and a :: "'b" assumes "category C" shows "category.has_as_product C J D a \<equiv> \<exists>\<pi>. product_cone J C D a \<pi>"
Lcm_def: fixes A :: "'a set" shows "normalization_euclidean_semiring_class.Lcm A = (if \<exists>l. l \<noteq> (0::'a) \<and> (\<forall>a\<in>A. a dvd l) then Let (SOME l. l \<noteq> (0::'a) \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = (LEAST n. \<exists>l. l \<noteq> (0::'a) \<and> (\<forall>a\<in>A. a dvd l) \<and> euclidean_size l = n)) normalize else (0::'a))"
elims_2: fixes x :: "nat"   and xa :: "cell list \<times> cell list" assumes "inv_loop1_exit x xa" obtains n :: "nat" and l :: "cell list" and r :: "cell list" where "x = n" and "xa = (l, r)" and "0 < n \<and> (l, r) = ([], Bk # Oc # Bk \<up> n @ Oc \<up> n)"
preserves_runit_2: fixes f :: "'c" assumes "C.ide f" shows "H \<r>\<^sub>C\<^sup>-\<^sup>1[f] = \<Phi>\<^sub>H (f, src\<^sub>C f) \<cdot>\<^sub>D (H f \<star>\<^sub>D H.unit (src\<^sub>C f)) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[H f]"
wf_implies_regular: fixes R :: "(ZF \<times> ZF) set" assumes "wf R" shows "regular R"
FOREACH_body_def: fixes f :: "'m \<Rightarrow> 'n \<Rightarrow> 'o nres" shows "FOREACH_body f \<equiv> \<lambda>(xs, \<sigma>). let x = hd xs in f x \<sigma> \<bind> (\<lambda>\<sigma>'. RETURN (tl xs, \<sigma>'))"
dims2_def: shows "ps2_P.dims2 = nths dims vars2"
allocationsUniverseCharacterization: fixes a :: "('c \<times> 'd set) set" shows "(a \<in> allocationsUniverse) = (a \<in> allAllocations'' (Domain a) (\<Union> (Range a)))"
cases: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'d bset"   and a2 :: "'e bset" assumes "rel_bset R a1 a2" obtains x :: "('d \<times> 'e) bset" where "a1 = map_bset fst x" and "a2 = map_bset snd x" and "set_bset x \<subseteq> {(x, y). R x y}"
Red_I_def: fixes N :: "('f \<times> 'l) set" shows "std.Red_I N = (\<Inter>q\<in>Q. Red_I_\<G>_L_q q N)"
closed_orbitE: fixes x :: "'a" assumes "closed_orbit x" obtains T :: "real" where "0 < T" and "\<And>t. flow0 x (t + T) = flow0 x t"
atLeast_eq_UNIV_iff: fixes x :: "'a" shows "({x..} = UNIV) = (x = (0::'a))"
NTCatDom_def: fixes \<eta> :: "('g, 'j, 'h, 'k, 'i, 'l, 'm) NatTrans_scheme" shows "NTCatDom \<eta> \<equiv> CatDom (NTDom \<eta>)"
pgallery_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "pgallery (x # xs)" shows "pgallery xs"
assms_6: shows "rk {A, B, C, A', B', C', P} = 3"
isos_compose: fixes f :: "'b"   and f' :: "'b" assumes "D.iso f"   and "D.iso f'"   and "D.seq f' f" shows "D.iso (f' \<cdot>\<^sub>D f)"
fun_cong: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i"   and x :: "'h" assumes "f = g" shows "f x = g x"
unit_circle_set_def: shows "unit_circle_set = circline_set unit_circle"
convol_def: fixes f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)"
imult_ile_mono2: fixes i :: "enat"   and j :: "enat"   and k :: "enat" assumes "i \<le> j" shows "k * i \<le> k * j"
pp_isotone_sup: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "less_eq (uminus (uminus x)) (uminus (uminus (sup x y)))"
count_roots_between: fixes a :: "real"   and b :: "real" assumes "p' \<noteq> 0"   and "a \<le> b" shows "sign_changes (sturm_squarefree p) a - sign_changes (sturm_squarefree p) b = card {x. a < x \<and> x \<le> b \<and> poly p' x = 0}"
IH: assumes "k' \<le> k"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n"   and "j'' \<le> j"   and "i'' \<le> i" shows "fw m n k i j i' j' \<le> fw m n k' i'' j'' i' j'"
iT_Mult_inext2_if: fixes n :: "nat"   and I :: "nat set"   and k :: "nat" shows "inext n (I \<otimes> k) = (if n mod k = 0 then inext (n div k) I * k else n)"
find_first_some: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set xs" obtains i :: "nat" where "find_first x xs = Some i" and "i < length xs"
dg_SemiCAT_components_1: fixes \<alpha> :: "V" shows "dg_SemiCAT \<alpha>\<lparr>Obj\<rparr> = ZFC_in_HOL.set {\<CC>. semicategory \<alpha> \<CC>}"
rel_simps_26: fixes n :: "num" shows "(numeral n \<le> (1::'a)) = (n \<le> num.One)"
run_shift_elim: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a list"   and s :: "'a stream"   and p :: "'b" assumes "transition_system.run execute enabled (r @- s) p" obtains "transition_system.path execute enabled r p" and "transition_system.run execute enabled s (fold execute r p)"
dual_weak_order: shows "weak_partial_order (inv_gorder (inv_gorder L))"
simps_3: fixes x1 :: "'b"   and l1 :: "'b list"   and x2 :: "'b"   and l2 :: "'b list" shows "inter_sorted_rev (x1 # l1) (x2 # l2) = (if x2 < x1 then inter_sorted_rev l1 (x2 # l2) else if x1 = x2 then x1 # inter_sorted_rev l1 l2 else inter_sorted_rev (x1 # l1) l2)"
i_in_sup: shows "\<iota> \<in> Sup_llist (lmap Red_I Ns)"
size_replicate_mset: fixes n :: "nat"   and M :: "'k" shows "size (replicate_mset n M) = n"
subt: shows "path_image \<gamma> \<subseteq> interior T"
assms_3: shows "\<Sigma> \<in> br_invar \<delta>"
n_div_2_cases: fixes n :: "nat" obtains "n = 0"   | "n div 2 < n" and "0 < n"
simps_10: fixes x2 :: "val"   and x1 :: "nat" shows "Val x2 \<noteq> Var x1"
bs_env_dom: fixes \<rho> :: "(nat \<times> bval) list"   and \<rho>' :: "(nat \<times> exp) list" assumes "bs_env \<rho> \<rho>'" shows "set (map fst \<rho>) = sdom \<rho>'"
ccspan_leqI: fixes M :: "'a set"   and S :: "'a ccsubspace" assumes "M \<subseteq> space_as_set S" shows "ccspan M \<le> S"
fps_div_fps_X_power_nth': fixes f :: "'a fps"   and n :: "nat"   and k :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "fps_nth (f / fps_X ^ n) k = fps_nth f (k + n)"
fps_of_poly_smult: fixes c :: "'a"   and p :: "'a poly" shows "fps_of_poly (smult c p) = fps_const c * fps_of_poly p"