hcju/isabelleps huggingface dataset:
case_sum_inject: fixes f1 :: "'d \<Rightarrow> 'f"   and f2 :: "'e \<Rightarrow> 'f"   and g1 :: "'d \<Rightarrow> 'f"   and g2 :: "'e \<Rightarrow> 'f" assumes "case_sum f1 f2 = case_sum g1 g2" obtains "f1 = g1" and "f2 = g2"
update_convs_8: fixes dests' :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> nat option"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "dests_update dests' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests' dests, pre = pre, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr>"
x14: shows "\<exists>x1. p x1 \<in> p ` fst ` f \<and> x1 \<in> fst ` f"
lexmin_mem: fixes C :: "(nat \<Rightarrow> 'a) set"   and i :: "nat" assumes "C \<subseteq> SEQ A"   and "C \<noteq> {}" shows "lexmin C i \<in> ith (eq_upto C (lexmin C) i) i"
simps_13: fixes C' :: "char list"   and V :: "'a"   and T :: "ty"   and e :: "'a exp" shows "not_init C' {V:T; e} = not_init C' e"
star_one_sup_below: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y\<^sup>\<star> * ((1::'a) \<squnion> z) \<le> x * (y \<squnion> z)\<^sup>\<star>"
simps_2: fixes k :: "'c"   and s :: "'c list"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "star_matrix' (k # s) g = (let r = [k]; a = r \<langle>g\<rangle> r; b = r \<langle>g\<rangle> s; c = s \<langle>g\<rangle> r; d = s \<langle>g\<rangle> s; as = r \<langle>star \<circ> a\<rangle> r; ds = star_matrix' s d; e = a \<oplus> b \<odot> ds \<odot> c; es = r \<langle>star \<circ> e\<rangle> r; f = d \<oplus> c \<odot> as \<odot> b; fs = star_matrix' s f in es \<oplus> as \<odot> b \<odot> fs \<oplus> ds \<odot> c \<odot> es \<oplus> fs)"
f': shows "(\<lambda>x. if x \<in> S then f x else (0::'b)) measurable_on UNIV"
check_eqv_def: shows "WS1S_Presburger_Equivalence.check_eqv \<equiv> WS1S_Presb.check_eqv"
erH_rHom_unique: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B" shows "\<exists>!g. g \<in> pHom R S X, A B Y \<and> (\<forall>x\<in>carrier S. h x = g x)"
incl_lsbis1: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and R1 :: "('d \<times> 'd) set"   and R2 :: "('e \<times> 'e) set" assumes "sbis B1 B2 s1 s2 R1 R2" shows "R1 \<subseteq> lsbis1 B1 B2 s1 s2"
Qmp_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" shows "Qm\<^bsub>G H,N\<^esub> = (\<lambda>X\<in>set_rcs G H. {z. \<exists>x\<in>X. \<exists>y\<in>N. y \<cdot>\<^bsub>G\<^esub> x = z})"
abs_distrib_div: fixes c :: "real"   and a :: "real"   and b :: "real" assumes "0 < c" shows "\<bar>a / c - b / c\<bar> = \<bar>a - b\<bar> / c"
no_passive: shows "FL.passive_subset (Liminf_llist (lmap lclss_of_state Sts)) = {}"
sp_fundefs_generalize: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef" assumes "sp_fundefs (Fubx_get F)"   and "Fubx_get F f = Some fd" shows "sp_fundefs (Fubx_get (Fubx_add F f (Subx.generalize_fundef fd)))"
image_comp: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'd"   and r :: "'e set" shows "f ` g ` r = (f \<circ> g) ` r"
AssertionStatEqClosed: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and p :: "(name \<times> name) list" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>'" shows "assertionAux.AssertionStatEq SImp (p \<bullet> \<Psi>) (p \<bullet> \<Psi>')"
subst_atm_mset_list_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AAA :: "'b multiset list"   and \<sigma> :: "'c" shows "substitution_ops.subst_atm_mset_list subst_atm AAA \<sigma> = map (\<lambda>AA. substitution_ops.subst_atm_mset subst_atm AA \<sigma>) AAA"
inject: fixes x2 :: "nat"   and y2 :: "nat" shows "(Suc x2 = Suc y2) = (x2 = y2)"
assms_2: shows "M \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<turnstile> t"
imp_igSubstBinpIPresIGWlsBinpSTR: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" assumes "igSubstAbsIPresIGWlsAbsSTR MOD" shows "igSubstBinpIPresIGWlsBinpSTR MOD"
Cauchy_theorem_homotopic_paths: fixes S :: "complex set"   and g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" assumes "homotopic_paths S g h"   and "open S"   and "f holomorphic_on S"   and "valid_path g"   and "valid_path h" shows "contour_integral g f = contour_integral h f"
axis_eq_axis: fixes i :: "'c"   and x :: "'b"   and j :: "'c"   and y :: "'b" shows "(axis i x = axis j y) = (x = y \<and> i = j \<or> x = (0::'b) \<and> y = (0::'b))"
simCases: fixes Q :: "pi"   and P :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "\<And>a y Q'. \<lbrakk>Q \<longmapsto>\<^sub>e a<\<nu>y> \<prec>\<^sub>e Q'; y \<sharp> P\<rbrakk> \<Longrightarrow> \<exists>P'. P \<longmapsto>\<^sub>e a<\<nu>y> \<prec>\<^sub>e P' \<and> (P', Q') \<in> Rel"   and "\<And>\<alpha> Q'. Q \<longmapsto>\<^sub>e \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<longmapsto>\<^sub>e \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" shows "P \<leadsto>\<^sub>e[Rel] Q"
ccField_ccBind: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and v :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ccField (CoCallAnalysis.ccBind ccExp v e\<cdot>(ae, G)) \<subseteq> fv e"
dist: fixes P :: "face \<Rightarrow> bool" shows "distinct (filter P (facesAt g v))"
Arg2pi_real: fixes z :: "complex" assumes "z \<in> \<real>" shows "Arg2pi z = (if 0 \<le> Re z then 0 else pi)"
accI_pos: fixes x :: "'b"   and y :: "'b"   and n :: "nat" assumes "0 < K2.p x y n" shows "(x, y) \<in> K2.acc"
ctr: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and w :: "'c llist"   and a :: "'b" shows "lscan f w a = (case w of <> \<Rightarrow> a | _ \<Rightarrow> a) % (case w of <> \<Rightarrow> <> | x % xs \<Rightarrow> lscan f xs (f x a))"
norm_minus_cancel: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (uminus x) = norm x"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "cring R" shows "UP_cring R"
rep_eq: fixes x :: "'a \<Rightarrow> 'c"   and xa :: "('b, 'c) uexpr" shows "\<lbrakk>x \<dagger> xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>xa\<rbrakk>\<^sub>e (x b))"
all_in_path_found': fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and path' :: "(ctx \<times> cld) list"   and path :: "(ctx \<times> cld) list"   and ctxcld :: "ctx \<times> cld" assumes "find_path_rec_f P ctx cl path' = Some path"   and "ctxcld \<in> set path" shows "ctxcld \<in> set path' \<or> (\<exists>ctx' fqn'. find_cld_f P ctx' fqn' = Some ctxcld)"
MkIde_Dom': fixes a :: "('obj, 'obj) concrete_category.arr" assumes "ide a" shows "MkIde (Dom a) = a"
variance_expectation_2: fixes X :: "'a \<Rightarrow> real" assumes "integrable M (\<lambda>x. (X x)\<^sup>2)"   and "random_variable borel X" shows "expectation (\<lambda>x. (X x - expectation X)\<^sup>2) = expectation (\<lambda>x. (X x)\<^sup>2) - (expectation X)\<^sup>2"
prefixToElementToPrefixElement: fixes p :: "('b \<times> bool) list"   and M :: "('b \<times> bool) list"   and e :: "'b" assumes "isPrefix p M"   and "e \<in> set (elements p)" shows "prefixToElement e M = prefixToElement e p"
td_c: shows "type_definition Rep_c Abs_c (Collect (Domainp C))"
countable_lists: fixes A :: "'b set" assumes "countable A" shows "countable (lists A)"
x_le_xs: shows "\<forall>v\<in>set xs. x \<le> v"
IH: fixes A :: "V set" assumes "A \<subseteq> \<Union> \<A>"   and "\<A> \<noteq> {}"   and "\<alpha> \<le> ordertype A VWF"   and "small (\<Union> \<A>)" shows "\<exists>X\<in>\<A>. \<alpha> \<le> ordertype X VWF"
param_map_option: fixes A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'd) set" shows "(map_option, map_option) \<in> (A \<rightarrow> B) \<rightarrow> \<langle>A\<rangle>option_rel \<rightarrow> \<langle>B\<rangle>option_rel"
to_mtx_ith_2: fixes A :: "((real, 'a) vec, 'a) vec"   and i1 :: "'a"   and i2 :: "'a" shows "to_mtx A $$ i1 $ i2 = A $ i1 $ i2"
ex_match: fixes n :: "nat"   and ts :: "Formula.trm list"   and v :: "event_data option list" assumes "wf_tuple n (\<Union> (fv_trm ` set ts)) v"   and "\<forall>t\<in>set ts. (\<forall>x\<in>fv_trm t. x < n) \<and> (trm.is_Var t \<or> trm.is_Const t)" shows "\<exists>f. match ts (map (Formula.eval_trm (map the v)) ts) = Some f \<and> v = tabulate f 0 n"
RPred1_def: shows "\<R>\<^sub>0\<^sub>1 \<equiv> {f \<in> \<R>. \<forall>x. f x \<down>= 0 \<or> f x \<down>= 1}"
image_def: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "f ` A = rel_ext (\<lambda>y. \<exists>x\<in>A. y = f x)"
Max_ge: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq x (linorder.Max less_eq A)"
del_UnEdge_even: fixes v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "(v, w, v') \<in> G.E"   and "finite G.E" shows "(v \<in> odd_nodes_set (del_unEdge v w v' G)) = even (degree v G)"
f_2: shows "\<forall>(k, \<gamma>)\<in>one_chain1. if k = fst (f (k, \<gamma>)) then reparam \<gamma> (snd (f (k, \<gamma>))) else reparam \<gamma> (reversepath (snd (f (k, \<gamma>))))"
simps_2: fixes x :: "'b"   and hs :: "'b heap list" shows "size_hp (Hp x hs) = size_hps hs + 1"
simps: fixes a1 :: "int"   and a2 :: "int"   and a3 :: "intv'" shows "valid_intv' a1 a2 a3 = ((\<exists>l uu_. a1 = l \<and> a2 = uu_ \<and> a3 = Smaller' (- l)) \<or> (\<exists>l d u. a1 = l \<and> a2 = u \<and> a3 = Const' d \<and> - l \<le> d \<and> d \<le> u) \<or> (\<exists>l d u. a1 = l \<and> a2 = u \<and> a3 = Intv' d \<and> - l \<le> d \<and> d < u) \<or> (\<exists>uv_ u. a1 = uv_ \<and> a2 = u \<and> a3 = Greater' u))"
distinct_4: fixes list' :: "char list"   and var' :: "NominalSequents.var"   and form' :: "form"   and nat :: "nat"   and list :: "NominalSequents.var list" shows "list' \<nabla> [var'].form' \<noteq> At nat list"
InterD: fixes A :: "'a"   and C :: "'a set set"   and X :: "'a set" assumes "A \<in> \<Inter> C"   and "X \<in> C" shows "A \<in> X"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case EV_SIGNAL_PREP of EV_SIGNAL_PREP \<Rightarrow> f1 | EV_SIGNAL_FINISH \<Rightarrow> f2) = f1"
prems: fixes x :: "'a" assumes "x \<in> set (x__ # xs)" shows "\<exists>y. execute (f x) h = Some (y, h)"
fps_conv_radius_def: fixes f :: "'a fps" shows "fps_conv_radius f = conv_radius (($) f)"
abs_eq: fixes xa :: "'a ltl"   and x :: "'a set" shows "af_abs_opt.f_abs (Abs xa) x = Abs (af_letter_opt xa x)"
qfree_qe_FM\<^sub>1: fixes as :: "atom list" shows "qfree (qe_FM\<^sub>1 as)"
slist_bottom': fixes xs :: "[:'b:]" shows "(\<Lambda> [::]. \<bottom>)\<cdot>xs = \<bottom>"
match_term_list_sound: fixes tus :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term option" assumes "match_term_list tus \<sigma> = Some \<tau>" shows "extends_subst \<sigma> \<tau> \<and> dom \<tau> = (\<Union>(t, u)\<in>set tus. vars_term t) \<union> dom \<sigma> \<and> (\<forall>(t, u)\<in>set tus. subst_atm_abbrev t (subst_of_map Var \<tau>) = u)"
product_run_finite_snd: fixes \<delta>\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and \<delta>\<^sub>2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>1 :: "'a"   and q\<^sub>2 :: "'b"   and w :: "nat \<Rightarrow> 'c" assumes "finite (range (run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w))" shows "finite (range (run\<^sub>t \<delta>\<^sub>2 q\<^sub>2 w))"
subspace_kernel: shows "module.subspace (*\<^sub>R) {x. f x = (0::'b)}"
distinct_1: fixes x11 :: "subject"   and x12 :: "name \<Rightarrow> pi noption"   and x21 :: "freeRes"   and x22 :: "pi" shows "BoundR_Rep x11 x12 \<noteq> FreeR_Rep x21 x22"
ProcNoRec1: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P :: "'d \<Rightarrow> 'f set"   and p :: "'e"   and Q :: "'d \<Rightarrow> 'f set"   and A :: "'d \<Rightarrow> 'f set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P Z) the (\<Gamma> p) (Q Z),(A Z)"   and "p \<in> dom \<Gamma>" shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P Z) Call p (Q Z),(A Z)"
simps: fixes a :: "('a, 'b, 'c) Tree\<^sub>\<alpha>" shows "hereditarily_fs a = ((\<exists>tset\<^sub>\<alpha>. a = Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<and> finite (supp tset\<^sub>\<alpha>) \<and> (\<forall>x. x \<in> set_bset tset\<^sub>\<alpha> \<longrightarrow> hereditarily_fs x)) \<or> (\<exists>t\<^sub>\<alpha>. a = Not\<^sub>\<alpha> t\<^sub>\<alpha> \<and> hereditarily_fs t\<^sub>\<alpha>) \<or> (\<exists>\<phi>. a = Pred\<^sub>\<alpha> \<phi>) \<or> (\<exists>t\<^sub>\<alpha> \<alpha>. a = Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> \<and> hereditarily_fs t\<^sub>\<alpha>))"
integrable_subinterval: fixes integrable :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool"   and integral :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and a' :: "real"   and b' :: "real" assumes "akra_bazzi_integral integrable integral"   and "integrable f a b"   and "a \<le> a'"   and "b' \<le> b" shows "integrable f a' b'"
sum_two_elements: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'a" assumes "class.comm_monoid_add plus zero"   and "a \<noteq> b" shows "comm_monoid_add.sum plus zero f {a, b} = plus (f a) (f b)"
map_filter_map_of: fixes P :: "'c \<Rightarrow> bool"   and m :: "('c \<times> 'd) list" shows "map_filter P (map_of m) = map_of (filter (\<lambda>(k, uu_). P k) m)"
rep_eq: fixes x :: "'c \<Rightarrow> 'e \<Rightarrow> 'd"   and xa :: "('c, 'e) fmap" shows "($$) (fmmap_keys x xa) = (\<lambda>a. map_option (x a) (xa $$ a))"
m_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "submonoid H G"   and "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"
step: shows "LeftDerivationIntrosAt (\<alpha> @ \<delta>) D L (Suc index)"
ccomp_correct: fixes c :: "com"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "ccomp c \<Turnstile> cfs\<box>" shows "cpred c (cfs ! 0) (cfs ! (length cfs - 1))"
summable_suminf_not_top: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "(\<Sum>i. ennreal (f i)) \<noteq> \<top>" shows "summable f"
zip_append1: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'c list" shows "zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
SV_inv3_inductive_2: shows "{SV_inv3} TS.trans sv_TS {> SV_inv3}"
set_rel_sv: fixes R :: "('c \<times> 'd) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>set_rel)"
analyzed_def: fixes M :: "('fun, 'var) Term.term set" shows "analyzed M \<equiv> \<forall>t. (M \<turnstile> t) = (M \<turnstile>\<^sub>c t)"
ga_automaton_def: shows "ga_automaton \<equiv> \<lparr>g_V = UNIV, g_E = ga_step, g_V0 = {pid_init_gc}, sa_L = pid_interp_gc\<rparr>"
\<psi>_2: shows "Fvars \<psi> = {xx}"
aux'_eq: shows "aux' = add_new_table_msaux args rel2 aux0"
Y_fnz: fixes y :: "'a" assumes "y \<in> Y" shows "blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} y)) (f (poincare_map {x \<in> S. s x = 0} y)) \<noteq> 0"
oth_class_taut_5_d: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<equiv> \<psi> \<^bold>\<equiv> (\<^bold>\<not>\<phi> \<^bold>\<equiv> \<^bold>\<not>\<psi>) in v]"
i_append_i_take_i_drop_id: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Down> n \<frown> f \<Up> n = f"
vreal_impl_vdisjnt: shows "vreal_impl \<inter>\<^sub>\<circ> \<omega> = []\<^sub>\<circ>"
paths_many_calls_subset: fixes t :: "'b ttree"   and x :: "'b" shows "paths t \<subseteq> paths (many_calls x \<otimes>\<otimes> without x t)"
list_eq_iff_nth_eq: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs = ys) = (length xs = length ys \<and> (\<forall>i<length xs. xs ! i = ys ! i))"
part_fn_subset: fixes r :: "nat"   and s :: "nat"   and YY :: "'a set"   and f :: "'a set \<Rightarrow> nat"   and Y :: "'a set" assumes "part_fn r s YY f"   and "Y \<subseteq> YY" shows "part_fn r s Y f"
sum_character: fixes G :: "('a, 'b) monoid_scheme"   and \<chi> :: "'a \<Rightarrow> complex" assumes "character G \<chi>" shows "sum \<chi> (carrier G) = (if \<chi> = principal_char G then of_nat (order G) else 0)"
stardiff_1: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and m :: "nat"   and ps :: "char list \<Rightarrow> int option"   and n :: "nat" assumes "(P \<and>* $ m) (ps, n)" shows "P (ps, n - m)"
M: shows "\<forall>t\<in>M. \<exists>s \<delta>. s \<in> N \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = s \<cdot> \<delta>"
inf_continuous_inf: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "inf_continuous f"   and "inf_continuous g" shows "inf_continuous (\<lambda>x. inf (f x) (g x))"
inject_3: fixes x1 :: "name"   and x2 :: "lam"   and y1 :: "name"   and y2 :: "lam" shows "(LAM [x1].x2 = LAM [y1].y2) = ([x1].x2 = [y1].y2)"
distinct_13: fixes x21 :: "(char list \<times> tid_or_exn) option"   and x22 :: "v list"   and x41 :: "v sem_env"   and x42 :: "(char list \<times> char list \<times> exp0) list"   and x43 :: "char list" shows "Conv x21 x22 \<noteq> Recclosure x41 x42 x43"
lift_poly_syz_def: fixes n :: "nat"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and i :: "nat" shows "lift_poly_syz n b i = Abs_poly_mapping (\<lambda>x. if pair_of_term x = (0::'a, i) then 1::'c else if n \<le> component_of_term x then lookup b (map_component (\<lambda>k. k - n) x) else (0::'c))"
less_eq_aexp_def: fixes e1 :: "'b aexp"   and e2 :: "'b aexp" shows "e1 \<le> e2 \<equiv> e1 < e2 \<or> e1 = e2"
unstream_append_trans_Inr: fixes g :: "('e, 'f) generator"   and h :: "('e, 'g) generator"   and sh :: "'g"   and sh' :: "'g" shows "unstream (append_trans g h sh) (Inr sh') = unstream h sh'"
xor_compl_right: fixes x :: "'c"   and y :: "'c" shows "x XOR NOT y = NOT (x XOR y)"
disj_imp_eq_proj_exec: fixes a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "fmdom' (snd a) \<inter> vs = {}" shows "fmrestrict_set vs s = fmrestrict_set vs (state_succ s a)"
openin_subbase: fixes B :: "'c set \<Rightarrow> bool"   and U :: "'c set"   and S :: "'c set" shows "openin (topology (arbitrary union_of (finite intersection_of B relative_to U))) S = (arbitrary union_of (finite intersection_of B relative_to U)) S"
that: fixes V :: "'b set" assumes "open V"   and "S' retract_of V" shows "thesis"
Mp_smult_1: fixes a :: "int"   and f :: "int poly" shows "pl.eq_m (smult (pl.M a) f) (smult a f)"
assms_2: fixes n :: "nat" shows "f n \<in> space\<^sub>N (\<LL> \<infinity> M)"
insert_3: shows "dom f' = dom (f(l \<mapsto> t))"
1: shows "component_of_term (lt g) = component_of_term (lt h)"
max_fun_diff_max2: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> 'a" assumes "s.max_fun_diff f g = s.max_fun_diff g h \<longrightarrow> f (s.max_fun_diff f g) \<noteq> h (s.max_fun_diff g h)"   and "f \<noteq> g"   and "g \<noteq> h"   and "f \<noteq> h"   and "f \<in> FINFUNC"   and "g \<in> FINFUNC"   and "h \<in> FINFUNC" shows "s.max_fun_diff f h = s.max2 (s.max_fun_diff f g) (s.max_fun_diff g h)"
preserves_reflects_arr: fixes f :: "'a" shows "Adj.D.arr (G f) = Adj.C.arr f"
select_convs_1: fixes precondition_of :: "'c list"   and add_effects_of :: "'c list"   and delete_effects_of :: "'c list"   and more :: "'d" shows "strips_operator.precondition_of \<lparr>strips_operator.precondition_of = precondition_of, add_effects_of = add_effects_of, delete_effects_of = delete_effects_of, \<dots> = more\<rparr> = precondition_of"
simps: fixes s1 :: "('a, 'b, 'c) SLin_state"   and s2 :: "('a, 'b, 'c) SLin_state" shows "P25 (s1, s2) = (\<forall>ivs. ivs \<in> initSets s2 \<and> initialized s2 \<and> dstate s2 \<preceq> \<Sqinter> ivs \<longrightarrow> (\<exists>rs'\<in>pendingSeqs (f (s1, s2)). \<Sqinter> ivs = dstate s2 \<star> rs'))"
Unf\<^sub>G_abs_def: shows "\<up>Unf\<^sub>G \<equiv> map_fun Rep Abs Unf\<^sub>G"
bf: shows "b \<in>\<^sub>\<circ> f (1\<^sub>\<nat>)"
Collect_disjoint_atLeast: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "(Collect P \<inter> {x..} = {}) = (\<forall>y\<ge>x. \<not> P y)"
in_angle_trans: fixes C :: "'p"   and A :: "'p"   and B :: "'p"   and D :: "'p"   and E :: "'p" assumes "C InAngle A B D"   and "D InAngle A B E" shows "C InAngle A B E"
irreducible_linear_field_poly: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "irreducible [:a, b:]"
simps_1: fixes R :: "(ZF \<times> ZF) set"   and s :: "ZF" shows "Ext_ZF_n R s 0 = Ext_ZF R s"
add_ac_1: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> b \<oplus> c = a \<oplus> (b \<oplus> c)"
simps_3: fixes va :: "dag"   and vb :: "ref"   and vc :: "dag"   and v :: "ref"   and r :: "dag" shows "reduced (Node (Node va vb vc) v r) = (Node va vb vc \<noteq> r \<and> reduced (Node va vb vc) \<and> reduced r)"
glb_common_set: fixes ss :: "'a set"   and s0 :: "'a"   and rset :: "('b \<times> 'c) set" assumes "finite ss"   and "ss \<noteq> {}"   and "\<And>s. s \<in> ss \<Longrightarrow> \<exists>rs. s = s0 \<star> rs \<and> set rs \<subseteq> rset" shows "\<exists>rs. \<Sqinter> ss = s0 \<star> rs \<and> set rs \<subseteq> rset"
odd_mirror_bird_stern_brocot: shows "odd_mirror bird = stern_brocot_recurse"
chainp_imp_rtranclp: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "\<forall>i. P (S i) (S (Suc i))"   and "i \<le> j" shows "P\<^sup>*\<^sup>* (S i) (S j)"
ind_mor_btw_stalks_axioms_def: fixes X :: "'c set"   and x :: "'c" shows "ind_mor_btw_stalks_axioms X x \<equiv> x \<in> X"