hcju/isabelleps huggingface dataset:
idem2: fixes r :: "'b \<times> 'c"   and r' :: "'b \<times> 'c"   and s :: "'a" assumes "fst r \<noteq> fst r'" shows "\<gamma> s r = \<gamma> ((s \<bullet> r) \<bullet> r') r"
Frct_code_post_1: fixes a :: "int" shows "Frct (0, a) = 0"
VarP_def: fixes x :: "tm" shows "VarP x \<equiv> OrdP x AND Zero IN x"
same_sign_cond_rationals_reals: fixes qs :: "rat poly list" assumes "0 < length (fst (factorize_polys qs))" shows "set (find_sgas (map real_of_rat_poly (fst (factorize_polys qs)))) = consistent_sign_vectors (fst (factorize_polys qs)) UNIV"
IH: shows "insert (QuoteP (Var i) (Var i')) (quote_all p Vs) \<turnstile> \<alpha> IMP PfP (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi)"
simps_1: fixes sort1 :: "('b list \<Rightarrow> 'b) \<Rightarrow> 'b list list \<Rightarrow> 'b list list"   and xss :: "'b list list" assumes "Radix_Sort sort1" shows "Radix_Sort.radix_sort sort1 0 xss = xss"
that: fixes x' :: "'b" assumes "x' \<in> s'"   and "(x, x') \<in> R" shows "thesis"
Infinitesimal_pi_divide_HNatInfinite: fixes N :: "nat star" assumes "N \<in> HNatInfinite" shows "hypreal_of_real pi / hypreal_of_hypnat N \<in> Infinitesimal"
atMost_def: fixes u :: "'b \<times> 'c" shows "prod.atMost u \<equiv> {x. less_eq_prod x u}"
ifc_split: fixes P :: "'a ifex \<Rightarrow> bool"   and v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "P (IFC v t e) = ((t = e \<longrightarrow> P t) \<and> (t \<noteq> e \<longrightarrow> P (IF v t e)))"
subst_apply_left_idemp: fixes \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and x :: "'c"   and t :: "('a, 'c) Term.term"   and s :: "('a, 'c) Term.term" assumes "\<sigma> x = t \<cdot> \<sigma>" shows "s \<cdot> subst x t \<cdot> \<sigma> = s \<cdot> \<sigma>"
verticesFrom_hd: fixes f :: "face"   and v :: "nat" shows "hd (verticesFrom f v) = v"
distributed_measurable: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> ennreal" assumes "distributed M N X f" shows "X \<in> M \<rightarrow>\<^sub>M N"
span_clauses_1: fixes a :: "'c"   and S :: "'c set" assumes "a \<in> S" shows "a \<in> ideal S"
intro: fixes p :: "int" assumes "p = int CARD('b)" shows "mod_ring_locale TYPE('b) p"
closure_approachableD: fixes x :: "'b"   and S :: "'b set"   and e :: "real" assumes "x \<in> closure S"   and "0 < e" shows "\<exists>y\<in>S. dist x y < e"
assms: shows "bal_up\<^sub>i (Up\<^sub>i (Node as a) x (Node bs b))"
good_map_is_surj: fixes h :: "'d \<times> 'e \<Rightarrow> 'f" assumes "good_map h" shows "surj h"
assms_2: shows "left_nesting f \<noteq> left_nesting g"
continuous_map_subset_aux1: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f" shows "\<forall>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S"
closed_subst: fixes e :: "db"   and k :: "nat"   and s :: "db" assumes "\<forall>v. freedb e v \<longrightarrow> v < k" shows "e<s/k> = e"
power2_diff: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "power.power one times (minus x y) 2 = minus (plus (power.power one times x 2) (power.power one times y 2)) (times (times (numeral.numeral one plus (num.Bit0 num.One)) x) y)"
compact_imp_complete: fixes s :: "'c set" assumes "compact s" shows "complete s"
simps_1: fixes x :: "'a" shows "invert (Pos x) = Neg x"
simps_28: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x31 :: "nat \<Rightarrow> nat option"   and x32 :: "nat" shows "(case Rerr x31 x32 of Rreq x xa xb xc xd xe xf \<Rightarrow> f1 x xa xb xc xd xe xf | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f3 x31 x32"
assms_4: shows "Xs \<sharp>* T[xvec::=Tvec]"
measure_exclude: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> events"   and "B \<in> events"   and "prob A = prob (space M)"   and "A \<inter> B = {}" shows "prob B = 0"
inv_into_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "inv_into A f = (\<lambda>x. SOME y. y \<in> A \<and> f y = x)"
cases: fixes core1 :: "('a, 'b, 'c, 'd, 'e, 'f) core"   and core2 :: "('g, 'b, 'c, 'd, 'e, 'f) core"   and E :: "'b set"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'g spmf"   and a1 :: "'a spmf"   and a2 :: "'g spmf" assumes "trace_core_closure core1 core2 E IA IU p q a1 a2" obtains tr :: "('b + 'c \<times> 'e + 'd \<times> 'f) list" where "a1 = trace_core_aux core1 p tr" and "a2 = trace_core_aux core2 q tr" and "set tr \<subseteq> E <+> IA \<times> UNIV <+> IU \<times> UNIV"
someI2_bex: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<exists>a\<in>A. P a"   and "\<And>x. x \<in> A \<and> P x \<Longrightarrow> Q x" shows "Q (SOME x. x \<in> A \<and> P x)"
smap_is_snil_conv_2: fixes f :: "'b \<rightarrow> 'a"   and xs :: "[:'b:]" shows "([::] = smap\<cdot>f\<cdot>xs) = (xs = [::])"
unique_norm: fixes f :: "'c local.term set" assumes "ARR f" shows "\<exists>!t. \<forall>u. u \<in> f \<longrightarrow> \<^bold>\<parallel>u\<^bold>\<parallel> = t"
gb_sig_pprod_def: fixes to :: "(('a, 'b) pp \<times> nat) nat_term_order"   and rword_strict :: "(('a, 'b) pp \<times> nat) nat_term_order \<Rightarrow> (('a, 'b) pp \<times> nat) \<times> (('a, 'b) pp \<Rightarrow>\<^sub>0 'c) \<Rightarrow> (('a, 'b) pp \<times> nat) \<times> (('a, 'b) pp \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and fs :: "(('a, 'b) pp \<Rightarrow>\<^sub>0 'c) list" shows "gb_sig_pprod to rword_strict fs \<equiv> gb_sig_pprod' to (rword_strict to) (map (change_ord (proj_ord to)) fs)"
Deduct2_with_False_axioms: shows "Deduct2_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv"
frontier_inside_subset: fixes S :: "'a set" assumes "closed S" shows "frontier (inside S) \<subseteq> S"
divisors_nat_1: fixes n :: "nat" assumes "n \<noteq> 0" shows "set (divisors_nat n) = {p. p dvd n}"
gL_def: fixes x :: "'a"   and y :: "'a" shows "gL x y = (x \<preceq>\<^sub>L y \<and> y \<preceq>\<^sub>L x)"
m_inv_closed: fixes H :: "'f set"   and G :: "('f, 'g) monoid_scheme"   and x :: "'f" assumes "subgroup H G"   and "x \<in> H" shows "inv\<^bsub>G\<^esub> x \<in> H"
simps_1: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "'a list"   and y1 :: "'a"   and y2 :: "'b"   and y3 :: "'a list" shows "(x1 \<rightarrow> x2 x3 = y1 \<rightarrow> y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
norm_eq_zero: 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 x = 0) = (x = zero)"
log_mult_eq: fixes A :: "real"   and B :: "real" shows "log b (A * B) = (if 0 < A * B then log b \<bar>A\<bar> + log b \<bar>B\<bar> else log b 0)"
HBL4: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla" assumes "{\<phi>1, \<phi>2} \<subseteq> fmla"   and "Fvars \<phi>1 = {}"   and "Fvars \<phi>2 = {}" shows "prv (imp (cnj (PP \<langle>\<phi>1\<rangle>) (PP \<langle>\<phi>2\<rangle>)) (PP \<langle>cnj \<phi>1 \<phi>2\<rangle>))"
group_add_ow_def: fixes U :: "'a set"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a" shows "group_add_ow U minus plus zero uminus \<equiv> (minus_ow U minus \<and> uminus_ow U uminus) \<and> monoid_add_ow U plus zero \<and> group_add_ow_axioms U minus plus zero uminus"
5: shows "(v \<squnion> e\<^sup>T * top)\<^sup>T * (t \<squnion> e) \<le> (v \<squnion> e\<^sup>T * top)\<^sup>T"
mod_emp: fixes h :: "heap \<times> nat set" shows "(h \<Turnstile> emp) = (snd h = {})"
integer_of_uint32_signed_def: fixes n :: "uint32" shows "integer_of_uint32_signed n = (if bit n 31 then undefined integer_of_uint32 n else integer_of_uint32 n)"
self_le_power: fixes a :: "'a"   and n :: "nat" assumes "(1::'a) \<le> a"   and "0 < n" shows "a \<le> a ^ n"
insertI: fixes A :: "'b set"   and a :: "'b" assumes "finite (A::'b set)" shows "finite ((a::'b) \<triangleright> (A::'b set))"
i_shrink_eq_Msg_iUntil_finish_event_conv: fixes k :: "nat"   and m :: "'a message_af"   and event :: "nat \<Rightarrow> bool"   and t0 :: "nat"   and t :: "nat"   and s :: "nat \<Rightarrow> 'a message_af" assumes "1 < k"   and "m \<noteq> NoMsg"   and "\<And>t. event t = (t mod k = k - 1)"   and "t0 = t * k" shows "((s \<div> k) t = m) = ((\<not> event t1. t1 \<U> t2 [0\<dots>] \<oplus> t0. event t2 \<and> s t2 = m) \<or> (\<not> event t1. t1 \<U> t2 [0\<dots>] \<oplus> t0. \<not> event t2 \<and> s t2 = m \<and> (\<circle> t' t2 [0\<dots>]. s t3 = NoMsg. t3 \<U> t4 [0\<dots>] \<oplus> t'. event t4 \<and> s t4 = NoMsg)))"
inducts: fixes x :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set"   and P :: "(('a, 'b, 'c) prot_fun, 'd) Term.term \<Rightarrow> bool" assumes "x \<in> timpl_closure t TI"   and "P t"   and "\<And>u a b s. \<lbrakk>u \<in> timpl_closure t TI; P u; (a, b) \<in> TI; term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) u s\<rbrakk> \<Longrightarrow> P s" shows "P x"
simps_66: fixes v :: "int"   and va :: "MIR.num" shows "isrlfm (NDvd v va) = (isatom (NDvd v va) \<and> bound0 (NDvd v va))"
rel_gpv''_colossless_gpvD2: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and gpv :: "('g, 'i, 'k) gpv"   and gpv' :: "('h, 'j, 'l) gpv"   and \<I>' :: "('j, 'l) \<I>"   and \<I> :: "('i, 'k) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "colossless_gpv \<I>' gpv'"   and "rel_\<I> C R \<I> \<I>'" shows "colossless_gpv \<I> gpv"
rep_eq: fixes x :: "uint16" shows "Rep_uint (uint_of_uint16 x) = ucast (Rep_uint16 x)"
ScopeExt: fixes y :: "name"   and R :: "pi"   and z :: "name"   and S :: "pi" assumes "y \<sharp> R" shows "(<\<nu>y><\<nu>z>(R \<parallel> S), <\<nu>z>(R \<parallel> <\<nu>y>S)) \<in> Rel"
restrict_cong: fixes I :: "'d set"   and J :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "I = J"   and "\<And>i. i \<in> J =simp=> f i = g i" shows "restrict f I = restrict g J"
assms_2: fixes x :: "nat" assumes "P (x + k)" shows "Q (x + k)"
homeomorphic_map_def: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "homeomorphic_map X Y f \<equiv> quotient_map X Y f \<and> inj_on f (topspace X)"
assms: fixes v :: "(real, 'n) vec"   and i :: "'n" assumes "v \<in> A" shows "v $ i \<in> \<int>"
fim: shows "f ` C \<subseteq> S \<times> T"
2: fixes v :: "'a stream" assumes "pred_stream (Rangep R) v" shows "stream_all2 R (smap (\<lambda>y. SOME x. R x y) v) v"
interval_of: fixes x :: "float" shows "real_of_float x \<in>\<^sub>r interval_of x"
vdiff_vsingleton_vinsert: fixes a :: "V"   and A :: "V" shows "set {a} -\<^sub>\<circ> vinsert a A = 0"
disjoint_family_elem_disjnt: fixes A :: "'a set"   and C :: "'b set"   and B :: "'a \<Rightarrow> 'b set" assumes "infinite A"   and "finite C"   and "disjoint_family_on B A" obtains x :: "'a" where "x \<in> A" and "disjnt C (B x)"
pdevs_val_trunc_pdevs: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and X :: "'a pdevs" shows "pdevs_val e (trunc_pdevs p X) = pdevs_val e X + pdevs_val e (trunc_err_pdevs p X)"
resTau: fixes x :: "name"   and P :: "pi" shows "<\<nu>x>\<tau>.P \<sim> \<tau>.(<\<nu>x>P)"
empty: fixes g :: "'b \<Rightarrow> 'a measure" shows "sup_measure.F g {} = bot"
env_restr_env_delete_other: fixes x :: "'a"   and S :: "'a set"   and m :: "'a \<Rightarrow> 'b" assumes "x \<notin> S" shows "env_delete x m f|` S = m f|` S"
R23s_defs_2: fixes implem :: "chan \<Rightarrow> msg"   and t :: "l3_state" assumes "sklvl3 implem" shows "sklvl3.med23s implem t \<equiv> \<lparr>ik = ik t \<inter> payload, secret = secret t, progress = progress t, signalsInit = signalsInit t, signalsResp = signalsResp t, signalsInit2 = signalsInit2 t, signalsResp2 = signalsResp2 t, chan = semivalid_implem.abs implem (ik t), bad = l3_state.bad t\<rparr>"
nb: fixes v :: "'a"   and a :: "'b" assumes "(v, a) \<in> set (effect_of op)" shows "set (filter ((\<noteq>) a) (the (range_of \<Psi> v))) = {a' \<in> \<R>\<^sub>+ \<Psi> v. a' \<noteq> a}"
simps_2: fixes t :: "'a trm"   and s :: "'a trm" shows "mset_lit (Neg (Eq t s)) = {#t, t, s, s#}"
KRP_subst_fm: fixes i :: "name"   and \<beta> :: "fm"   and j :: "name" shows "{KRP \<guillemotleft>Var i\<guillemotright> \<guillemotleft>\<beta>\<guillemotright> (Var j)} \<turnstile> Var j EQ \<guillemotleft>\<beta>(i::=\<guillemotleft>\<beta>\<guillemotright>)\<guillemotright>"
rt_fresh_as_sym: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<approx>\<^bsub>dip\<^esub> rt2" shows "rt2 \<approx>\<^bsub>dip\<^esub> rt1"
fun_upd_other: fixes z :: "'c"   and x :: "'c"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "z \<noteq> x" shows "(f(x := y)) z = f z"
i_cut_defs_2: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t \<equiv> {x \<in> I. x < t}"
sel_9: fixes x1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x2 :: "('b, 'a) state \<Rightarrow> (nat, 'b \<times> 'a) mapping"   and x3 :: "('b, 'a) state \<Rightarrow> (nat, 'b \<times> 'a) mapping"   and x4 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'a option"   and x5 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'a option"   and x6 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'b"   and x7 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'b"   and x8 :: "((nat, 'b \<times> 'a) mapping \<Rightarrow> (nat, 'b \<times> 'a) mapping) \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and x9 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a atom"   and x10 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.LE (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x9"
restrict_ext: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "restrict f A = restrict g A"
etran_in_comm: fixes P :: "'b com option"   and t :: "'b"   and xs :: "('b com option \<times> 'b) list"   and guar :: "('b \<times> 'b) set"   and post :: "'b set"   and s :: "'b" assumes "(P, t) # xs \<in> comm (guar, post)" shows "(P, s) # (P, t) # xs \<in> comm (guar, post)"
SUP_eq_top_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Sup (f ` A) = top) = (\<forall>x. less x top \<longrightarrow> (\<exists>i\<in>A. less x (f i)))"
condensation_edges_def: fixes g :: "'g"   and P :: "'val set" shows "condensation_edges g P \<equiv> (\<lambda>(x, y). (scc_of (induced_phi_graph g P) x, scc_of (induced_phi_graph g P) y)) ` induced_phi_graph g P - Id"
1: shows "2 ^ card (fmdom' s) < card (state_set (state_list s as))"
keys_monom_mult_subset: fixes c :: "'b"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (punit.monom_mult c t p) \<subseteq> punit.splus t ` keys p"
assms_2: shows "g ` {- 1..1} \<subseteq> cbox (- 1) 1"
tyvrs_vrs_prm_simp: fixes pi :: "(vrs \<times> vrs) list"   and a :: "binding" shows "tyvrs_of (pi \<bullet> a) = tyvrs_of a"
OAlist_tc_lookup_single: fixes k :: "'c"   and v :: "'b"   and k' :: "'c" shows "OAlist_tc_lookup (oalist_tc_of_list [(k, v)]) k' = (if k = k' then v else (0::'b))"
homotopic_with_product_topology: fixes I :: "'b set"   and p :: "'b \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and X :: "'b \<Rightarrow> 'c topology"   and Y :: "'b \<Rightarrow> 'd topology"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and q :: "(('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'd) \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> homotopic_with (p i) (X i) (Y i) (f i) (g i)"   and "\<And>h. (\<And>i. i \<in> I \<Longrightarrow> p i (h i)) \<Longrightarrow> q (\<lambda>x. \<lambda>i\<in>I. h i (x i))" shows "homotopic_with q (product_topology X I) (product_topology Y I) (\<lambda>z. \<lambda>i\<in>I. f i (z i)) (\<lambda>z. \<lambda>i\<in>I. g i (z i))"
Collect_conj_eq: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
map_SSup_prop: fixes F :: "'a downset \<Rightarrow> 'b downset" shows "ds \<circ> map_SSup F = F \<circ> ds"
alpha_def_3: shows "val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> \<alpha>)"
otherI: fixes I :: "'d set"   and \<sigma>' :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>i. i \<in> I \<Longrightarrow> \<sigma>' i = \<sigma> i"   and "\<And>j. j \<notin> I \<Longrightarrow> P (\<sigma> j) (\<sigma>' j)" shows "other P I \<sigma> \<sigma>'"
has_fields_fun: fixes P :: "'b prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and FDTs' :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C has_fields FDTs'" shows "FDTs' = FDTs"
simps: fixes st :: "nat"   and l :: "cell list"   and r :: "cell list" shows "trpl_code (st, l, r) = trpl (bl2wc l) st (bl2wc r)"
inext_mono2: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> I"   and "\<exists>x\<in>I. n < x" shows "n < inext n I"
vec_carrier: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "Matrix.vec n f \<in> carrier_vec n"
downset_directed_downset_var: fixes X :: "'a set" shows "directed (\<Down> X) = directed X"
"assume": fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_term (sig \<Theta>) A"   and "\<turnstile>\<^sub>\<tau> A : propT"   and "A \<in> \<Gamma>" shows "\<Theta>,\<Gamma> \<turnstile> A"
sethi_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (sethi_instr instr s1)) \<and> t2 = snd (fst (sethi_instr instr s2))" shows "low_equal t1 t2"
sup_left_idem: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup x y) = sup x y"
my_wf_induct_triple': fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set"   and r3 :: "('c \<times> 'c) set"   and P :: "'a \<times> 'b \<times> 'c \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'b"   and x3 :: "'c" assumes "wf (r1 <*lex*> r2 <*lex*> r3)"   and "\<And>x1 x2 x3. (\<And>y1 y2 y3. ((y1, y2, y3), x1, x2, x3) \<in> r1 <*lex*> r2 <*lex*> r3 \<longrightarrow> P (y1, y2, y3)) \<Longrightarrow> P (x1, x2, x3)" shows "P (x1, x2, x3)"
order_iff: fixes a :: "'e"   and b :: "'e" shows "(a \<le> b) = (a = inf a b)"
append_eq_appendI: fixes xs :: "'d list"   and xs1 :: "'d list"   and zs :: "'d list"   and ys :: "'d list"   and us :: "'d list" assumes "xs @ xs1 = zs"   and "ys = xs1 @ us" shows "xs @ ys = zs @ us"
nonempty_interval: fixes a :: "'c"   and b :: "'c" shows "nonempty {a..b} = (a \<le> b)"
ReZ_numeral: fixes n :: "num" shows "ReZ (numeral n) = numeral n"
SUCC_Subset_Ord: fixes H :: "fm set"   and k' :: "tm"   and k :: "tm" assumes "H \<turnstile> k' IN k"   and "H \<turnstile> OrdP k" shows "H \<turnstile> SUCC k' SUBS k"
fract_expand_2: shows "1 = Fract (1::'b) (1::'b)"
list_app_one_length: fixes l :: "'a list"   and n :: "nat"   and s :: "'a" assumes "length l = n" shows "(l @ [s]) ! n = s"
acyclic_irrefl: fixes r :: "('a \<times> 'a) set" shows "acyclic r = irrefl (r\<^sup>+)"
init_def: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "f \<triangleright> n \<equiv> Partial_Recursive.list_encode (prefix f n)"
ipPartitioningAddSubset1: fixes ts :: "'b set set"   and a :: "'b set" assumes "disjoint ts" shows "ipPartition (insert a ts) (addSubsetSet a ts)"
determI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "Transformers.additive t"   and "maximal t" shows "determ t"
Longest_common_prefix_prefix: fixes xs :: "'e list"   and L :: "'e list set" assumes "xs \<in> L" shows "prefix (Longest_common_prefix L) xs"
list_of_mset_exi: fixes m :: "'b multiset" shows "\<exists>l. m = mset l"
crit_pair_cbelow_on_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.crit_pair_cbelow_on d m F p q = cbelow_on (punit.dgrad_p_set d m) punit.ord_strict_p (monomial (1::'c) (fst (lcs (punit.lp p) (punit.lp q), local.punit.component_of_term (punit.lt p)))) (\<lambda>a b. punit.red F a b \<or> punit.red F b a) (fst (punit.crit_pair p q)) (snd (punit.crit_pair p q))"
holf2: shows "f holomorphic_on {z \<in> S. k < d \<bullet> z}"
negligible_plus: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real" assumes "Negligible.negligible f"   and "Negligible.negligible g" shows "Negligible.negligible (\<lambda>\<eta>. f \<eta> + g \<eta>)"
lsc_def: fixes f :: "'b \<Rightarrow> 'c" shows "lsc f = (\<forall>x. lsc_at x f)"
cones_map_mapsto: fixes f :: "'c" assumes "C.arr f" shows "\<chi>.D.cones_map f \<in> extensional (\<chi>.D.cones (C.cod f)) \<inter> (\<chi>.D.cones (C.cod f) \<rightarrow> \<chi>.D.cones (C.dom f))"
Compl_Int: fixes A :: "'a set"   and B :: "'a set" shows "- (A \<inter> B) = - A \<union> - B"
normalize_quot_aux_2: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "b = snd (normalize_quot (a, b)) * (gcd a b * unit_factor b)"
llength_least_None: fixes r :: "'b llist"   and A :: "'b set" assumes "r \<in> A\<^sup>\<star>" shows "LList2.llength r = (LEAST i. r !! i = None)"
a2: shows "Esigma (a * r * c) = Esigma a * (r + 1) * (c + 1)"
canonical_nat: fixes n :: "nat"   and v :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "VNat n \<in> E v \<rho>"   and "isval v" shows "v = ENat n"
spec\<^sub>4_def: fixes p :: "prog" shows "spec\<^sub>4 p \<equiv> vars p = vars\<^sub>0 \<and> (\<exists>s\<^sub>H. body_split p s\<^sub>L\<^sub>0 s\<^sub>H \<and> wfs \<Gamma>\<^sub>0 s\<^sub>H \<and> high_proc\<^sub>4 s\<^sub>H \<and> high_proc_no_low_output_change s\<^sub>H)"
assms_2: shows "mat_det A2 \<noteq> 0"