hcju/isabelleps huggingface dataset:
dghm_comp_is_ft_dghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"
sup_continuousD: fixes F :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> 'c" assumes "sup_continuous F"   and "incseq M" shows "F (Sup (range M)) = (SUP i. F (M i))"
rel_state_def: fixes P :: "heap \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "(heap, 'g) state"   and g :: "'h Heap" shows "heap_mem_defs.rel_state P R f g \<equiv> \<forall>heap. P heap \<longrightarrow> (case run_state f heap of (v1, heap1) \<Rightarrow> case execute g heap of None \<Rightarrow> False | Some (v2, heap2) \<Rightarrow> R v1 v2 \<and> heap1 = heap2 \<and> P heap2)"
WTrts_hext_shext_mono: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and es :: "char list exp list"   and Ts :: "ty list"   and h' :: "heap"   and sh' :: "sheap" assumes "P,E,h,sh \<turnstile> es [:] Ts"   and "h \<unlhd> h'"   and "sh \<unlhd>\<^sub>s sh'"   and "\<not> sub_RIs es" shows "P,E,h',sh' \<turnstile> es [:] Ts"
cgenideal_eq_genideal: fixes i :: "int" assumes "i \<in> UNIV" shows "PIdl\<^bsub>\<Z>\<^esub> i = Idl\<^bsub>\<Z>\<^esub> {i}"
simps: fixes v :: "nat"   and t :: "nat"   and e :: "nat"   and bdd :: "bdd" shows "ifmi v t e bdd = (if t = e then (t, bdd) else let (r, pm) = pointermap_getmk (v, t, e) (dpm bdd) in (Suc (Suc r), dpm_update (const pm) bdd))"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(ext.Val x2 = ext.Val y2) = (x2 = y2)"
intros_9: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and a' :: "'a"   and n :: "'b SDG_node"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "a' \<in> get_return_edges a"   and "n = CFG_node (sourcenode a)"   and "n' = CFG_node (targetnode a')" shows "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n None (Some (p, True)) True n'"
finite_global_assms': fixes i :: "'var itself" shows "finite (global_assms' i)"
jt_def: fixes v :: "Value" shows "jt v = (case v of arrV T a \<Rightarrow> at2jt T)"
lc_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'e" shows "lc p = lookup p (lt p)"
wp3Assign: fixes x :: "char list"   and e :: "aexp"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "wp\<^sub>3\<^sub>' (x ::= e) Q = ((\<lambda>(ps, t). vars e \<union> {x} \<subseteq> dom ps \<and> Q (ps(x \<mapsto> paval e ps), t)) \<and>* $ 1)"
ccw_minus3: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real"   and x :: "real \<times> real" assumes "det3 a b c \<noteq> 0" shows "ccw (x - a) (x - b) (x - c) = ccw a b c"
set_swap: fixes i :: "nat"   and xs :: "'a list"   and j :: "nat" assumes "i < length xs"   and "j < length xs" shows "set (xs[i := xs ! j, j := xs ! i]) = set xs"
simps_6: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat" shows "(case Triple x21 x22 x23 of Prime x \<Rightarrow> f1 x | Triple x xa xb \<Rightarrow> f2 x xa xb) = f2 x21 x22 x23"
Fract: fixes a :: "int"   and b :: "int" assumes "q = Fract a b"   and "0 < b"   and "a \<noteq> 0"   and "coprime a b" shows "C"
select_convs_2: fixes carrier :: "'f set"   and more :: "'e" shows "partial_object.more \<lparr>carrier = carrier, \<dots> = more\<rparr> = more"
gt_sameI: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t = head s"   and "chksubs (>\<^sub>t) t s"   and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)" shows "gt_same t s"
swap_commute: fixes n :: "nat"   and m :: "nat"   and p :: "poly" shows "swap n m p = swap m n p"
RedT_induct: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ttas :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and P :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set \<Rightarrow> ('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list \<Rightarrow> ('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set \<Rightarrow> bool" assumes "s -\<triangleright>ttas\<rightarrow>* s'"   and "\<And>s. P s [] s"   and "\<And>s ttas s' t ta s''. \<lbrakk>s -\<triangleright>ttas\<rightarrow>* s'; P s ttas s'; s' -t\<triangleright>ta\<rightarrow> s''\<rbrakk> \<Longrightarrow> P s (ttas @ [(t, ta)]) s''" shows "P s ttas s'"
restrict_spmf_eq_return_pmf_None: fixes p :: "'d spmf"   and A :: "'d set" shows "(p \<upharpoonleft> A = return_pmf None) = (set_spmf p \<inter> A = {})"
estep_Pr_def: fixes e :: "nat" shows "estep_Pr e \<equiv> if e2ls e = 0 then if e2rv e = 0 then prod_encode (e_cons (triple_encode (pdec122 (e2i e)) (e_tl (e2xs e)) 0) (e2stack e), 0) else prod_encode (e_cons (triple_encode (e2i e) (e2xs e) (singleton_encode (e2rv e - 1))) (e2tail e), 0) else if e2lenls e = Suc (e_hd (e2xs e)) then prod_encode (e2tail e, Suc (e_hd (e2ls e))) else if e2rv e = 0 then prod_encode (e_cons (triple_encode (pdec222 (e2i e)) (e_cons (e2lenls e - 1) (e_cons (e_hd (e2ls e)) (e_tl (e2xs e)))) 0) (e2stack e), 0) else prod_encode (e_cons (triple_encode (e2i e) (e2xs e) (e_cons (e2rv e - 1) (e2ls e))) (e2tail e), 0)"
assms_1: shows "\<forall>p. (ps c p \<noteq> None) = (ps d p \<noteq> None)"
CHECK_def: fixes \<Phi> :: "bool"   and e :: "'b" shows "CHECK \<Phi> e \<equiv> if \<Phi> then ERETURN () else THROW e"
suc_underS: fixes a :: "'a" assumes "a \<in> Field r" shows "a = suc (local.underS a)"
poly_map_poly_eval_poly: fixes p :: "'a poly" shows "poly (map_poly hom p) = eval_poly hom p"
pochhammer_Suc0: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a (Suc 0) = a"
hcomp_simps_2: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.hseq \<nu> \<mu>" shows "S.trg (\<nu> \<star>\<^sub>S \<mu>) = S.trg \<nu>"
is_finite_closest: fixes v :: "('a, 'b) IEEE.float \<Rightarrow> real"   and p :: "('a, 'b) IEEE.float \<Rightarrow> bool"   and x :: "real" shows "is_finite (closest v p {a. is_finite a} x)"
CQLeftSubst_def: shows "CQLeftSubst = \<lparr>SFunctions = Map.empty, SPredicates = \<lambda>p. if p = vid1 then Some (Geq ($f (Inr vid1) local.empty) (Differential (Const 0))) else None, SContexts = Map.empty, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
iMODb_iprev_if: fixes t :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" shows "iprev t [ r, mod m, c ] = (if t \<in> [ r, mod m, c ] \<and> r < t then t - m else t)"
simple_match_valid_def: fixes m :: "('b, 'c) simple_match_scheme" shows "simple_match_valid m \<equiv> ({p. simple_match_port (sports m) p} \<noteq> UNIV \<or> {p. simple_match_port (dports m) p} \<noteq> UNIV \<longrightarrow> proto m \<in> Proto ` {TCP, UDP, L4_Protocol.SCTP}) \<and> valid_prefix_fw (src m) \<and> valid_prefix_fw (dst m)"
SoundWhileAux: fixes n :: "nat"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "SoundWhileProp n G c B b A"
q3: shows "(q1 - q2) * y = r2 - r1"
gfp_in_Fix: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "gfp f \<in> Fix f"
m_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier H"   and "y \<in> carrier H" shows "x \<oplus>\<^bsub>H\<^esub> y \<in> carrier H"
exec_mbindFPurge_success: fixes ioprog :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and a :: "'f"   and \<sigma> :: "'e"   and b :: "'d"   and \<sigma>' :: "'e"   and S :: "'f list"   and M :: "'d list \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "ioprog a \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e (a # S) ioprog) M) = (\<sigma>' \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e S ioprog) (M (b # s)))"
shrK_in_used: fixes A :: "agent"   and evs :: "event list" shows "Key (shrK A) \<in> used evs"
sum_SDG_param_summary_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and a' :: "'edge"   and x :: "nat"   and ns :: "'node SDG_node list"   and x' :: "nat"   and n :: "'node SDG_node"   and n' :: "'node SDG_node"   and ins :: "'var list"   and outs :: "'var list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "a' \<in> get_return_edges a"   and "matched (Formal_in (targetnode a, x)) ns (Formal_out (sourcenode a', x'))"   and "n = Actual_in (sourcenode a, x)"   and "n' = Actual_out (targetnode a', x')"   and "(p, ins, outs) \<in> set procs"   and "x < length ins"   and "x' < length outs" shows "n s-p\<rightarrow>\<^bsub>sum\<^esub> n'"
induct: fixes P :: "(char list \<Rightarrow> 'c rule list option) \<Rightarrow> 'c rule list \<Rightarrow> bool"   and a0 :: "char list \<Rightarrow> 'c rule list option"   and a1 :: "'c rule list" assumes "\<And>uu_. P uu_ []"   and "\<And>\<Gamma> m chain rs. (\<And>x2. \<lbrakk>\<Gamma> chain = Some x2; \<not> \<not> terminal_chain x2\<rbrakk> \<Longrightarrow> P \<Gamma> rs) \<Longrightarrow> P \<Gamma> (Rule m (Goto chain) # rs)"   and "\<And>\<Gamma> v rs. P \<Gamma> rs \<Longrightarrow> P \<Gamma> (Rule v Accept # rs)"   and "\<And>\<Gamma> v rs. P \<Gamma> rs \<Longrightarrow> P \<Gamma> (Rule v Drop # rs)"   and "\<And>\<Gamma> v rs. P \<Gamma> rs \<Longrightarrow> P \<Gamma> (Rule v Log # rs)"   and "\<And>\<Gamma> v rs. P \<Gamma> rs \<Longrightarrow> P \<Gamma> (Rule v Reject # rs)"   and "\<And>\<Gamma> v vb rs. P \<Gamma> rs \<Longrightarrow> P \<Gamma> (Rule v (Call vb) # rs)"   and "\<And>\<Gamma> v rs. P \<Gamma> rs \<Longrightarrow> P \<Gamma> (Rule v Return # rs)"   and "\<And>\<Gamma> v rs. P \<Gamma> rs \<Longrightarrow> P \<Gamma> (Rule v Empty # rs)"   and "\<And>\<Gamma> v rs. P \<Gamma> rs \<Longrightarrow> P \<Gamma> (Rule v Unknown # rs)" shows "P a0 a1"
loopfix: fixes \<alpha> :: "game"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set" shows "loopfp\<tau> \<alpha> I X = game_sem I \<alpha>** X"
select_object_def: fixes mt :: "'e"   and incl :: "'e \<Rightarrow> 'f \<Rightarrow> 'e"   and smash :: "'e \<Rightarrow> 'd"   and deref :: "'g \<Rightarrow> 'f"   and l :: "'g list" shows "select_object mt incl smash deref l = smash (foldl incl mt (map deref l))"
Status_select: fixes ST :: "('d, 'e, 'f) status" shows "Status (HA ST) (Conf ST) (Events ST) (Value ST)"
Basis_imp: fixes k :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "real" assumes "k \<in> Basis" shows "content (cbox a b) = content (cbox a b \<inter> {x. x \<bullet> k \<le> c}) + content (cbox a b \<inter> {x. c \<le> x \<bullet> k})"
cases: fixes a1 :: "('a, 'b) sequent"   and a2 :: "nat"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "(a1, a2) \<in> derivable R" obtains C :: "('a, 'b) sequent" where "a1 = C" and "a2 = 0" and "([], C) \<in> R"   | r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent" and m :: "nat" where "a1 = snd r" and "a2 = m + 1" and "r \<in> R" and "fst r \<noteq> []" and "\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R"
l_one: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<zero> \<oplus> x = x"
wf\<^sub>s\<^sub>t\<^sub>s'_update\<^sub>s\<^sub>t_snd: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" assumes "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"   and "send\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "wf\<^sub>s\<^sub>t\<^sub>s' (update\<^sub>s\<^sub>t \<S> (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)) (\<A> @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)])"
lift_eq: shows "lift (\<lambda>d lm p \<alpha>. snd (up' d lm p \<alpha>)) dm lm d \<alpha> p = (\<Sum>p'\<in>lgrid (base {d} p) {d} lm. if p' \<in> grid p {d} - {p} then \<alpha> p' * l2_\<phi> (p' ! d) (p ! d) else 0)"
is_while_algo: fixes WA :: "'b det_while_algo" assumes "det_while_algo WA" shows "while_algo (det_wa_wa WA)"
not_col_permutation_5: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "\<not> Tarski_neutral_dimensionless.Col Bet A C B"
cltn2_rep_in: fixes B :: "cltn2" shows "cltn2_rep B \<in> Rep_cltn2 B"
ss: shows "{a. \<exists>b\<in>elts A. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r} \<subseteq> (\<lambda>x. x\<lparr>0\<^sub>\<nat>\<rparr>) ` elts r"
nth_append_length_plus: fixes xs :: "'b list"   and ys :: "'b list"   and n :: "nat" shows "(xs @ ys) ! (length xs + n) = ys ! n"
less_upper_bintrunc_id: fixes n :: "int"   and b :: "nat" assumes "n < 2 ^ b"   and "0 \<le> n" shows "take_bit b n = n"
converse_r_into_equivclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "r y x" shows "equivclp r x y"
big_step_pres_approx: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int option" assumes "(c, s) \<Rightarrow> s'"   and "approx t s" shows "approx (defs c t) s'"
isOK_bind: fixes m :: "'a + 'c"   and f :: "'c \<Rightarrow> 'a + 'b" shows "isOK (m \<bind> f) = (isOK m \<and> isOK (f (projr m)))"
canI_associator_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "horizontal_homs.obj V src a" shows "\<a> a a a = bicategory.can V H \<a> \<i> src trg (bicategorical_language.Hcomp (bicategorical_language.Prim\<^sub>0 a) (bicategorical_language.Hcomp (bicategorical_language.Prim\<^sub>0 a) (bicategorical_language.Prim\<^sub>0 a))) (bicategorical_language.Hcomp (bicategorical_language.Hcomp (bicategorical_language.Prim\<^sub>0 a) (bicategorical_language.Prim\<^sub>0 a)) (bicategorical_language.Prim\<^sub>0 a))"
comp_inv_arr: fixes f :: "'b"   and g :: "'b" assumes "inv.B.inverse_arrows f g" shows "B g f = inv.B.dom f"
left_commute: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "abel_semigroup f" shows "f b (f a c) = f a (f b c)"
simps_4: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
continuous_openin_preimage: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and U :: "'c set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "openin (top_of_set T) U" shows "openin (top_of_set S) (S \<inter> f -` U)"
abs_eq: fixes xa :: "'a \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'b" assumes "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'b)}) xa xa"   and "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'b)}) x x" shows "Abs_poly_mapping xa + Abs_poly_mapping x = Abs_poly_mapping (\<lambda>k. xa k + x k)"
strips_state_to_state_inverse_is_iv: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'd option"   and v :: "'c"   and a :: "'d" assumes "is_valid_problem_sas_plus \<Psi>"   and "dom s \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "s v = Some a"   and "a \<in> \<R>\<^sub>+ \<Psi> v" shows "(\<phi>\<^sub>S\<inverse> \<Psi> \<phi>\<^sub>S \<Psi> s) v = Some a"
cball_divide_subset: fixes d :: "real"   and x :: "'b"   and e :: "real" assumes "1 \<le> d" shows "cball x (e / d) \<subseteq> cball x e"
asym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubset> y"   and "y \<sqsubset> x"   and "x \<in> P"   and "y \<in> P" shows "False"
not_simple_match_ip: fixes s :: "'a word"   and aa :: "simple_action" assumes "\<not> simple_match_ip (dst m) d" shows "(runFw s d c (SimpleRule m aa # rs) = Decision FinalAllow) = (runFw s d c rs = Decision FinalAllow)"
distinct_55: fixes x71 :: "poly"   and x72 :: "nat"   and x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly" shows "Pw x71 x72 \<noteq> CN x81 x82 x83"
degrees_celcius_def: fixes x :: "'b" shows "x\<degree>C = (x *\<^sub>Q kelvin) + ((27315::'b) / (10::'b)\<^sup>2 *\<^sub>Q kelvin)"
smult_one: fixes c :: "'b" shows "Polynomial.smult c 1 = [:c:]"
with_proj_simps_2: fixes G :: "'a pair_pre_digraph" shows "arcs (with_proj G) = parcs G"
inj_on_map_of_zip: fixes xs :: "'a list"   and ys :: "'b list" assumes "length xs = length ys"   and "distinct ys" shows "inj_on (map_of (zip xs ys)) (set xs)"
key_order_of_nat_term_order_inv_def: fixes to :: "'a nat_term_order" shows "key_order_of_nat_term_order_inv to = Abs_key_order (nat_term_compare_inv to)"
isomorphic_implies_hpar_4: fixes f :: "'b"   and f' :: "'b" assumes "f \<cong>\<^sub>D f'" shows "trg\<^sub>D f = trg\<^sub>D f'"
nths_vars2': shows "nths ps2_P.dims0 (- ps2_P.vars1') = ps2_P.dims2"
iterate_length: fixes m :: "nat"   and n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "length (iterate m n f x) = n - m"
Red: shows "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
state_successors_def: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" shows "state_successors PROB s \<equiv> state_succ s ` PROB - {s}"
is_GB_cofactor_boundI: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and b :: "nat" assumes "punit.is_Groebner_basis G"   and "ideal G = ideal F"   and "\<Union> (indets ` G) \<subseteq> \<Union> (indets ` F)"   and "\<And>g. g \<in> G \<Longrightarrow> \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f\<in>F'. poly_deg (q f * f) \<le> b)" shows "is_GB_cofactor_bound F b"
zip_map_fst_snd: fixes zs :: "('d \<times> 'e) list" shows "map fst zs || map snd zs = zs"
tendsto_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and "open" :: "'e set \<Rightarrow> bool"   and F :: "'f filter"   and f :: "'f \<Rightarrow> 'e"   and x :: "'e"   and g :: "'f \<Rightarrow> 'e"   and y :: "'e" assumes "class.linorder_topology less_eq less open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open x) F"   and "filterlim g (topological_space.nhds open y) F"   and "\<forall>\<^sub>F x in F. less_eq (g x) (f x)" shows "less_eq y x"
M_def: shows "M \<equiv> uniform_measure lborel {a..b}"
enat_the_enat_minus_le: fixes x :: "enat"   and y :: "enat" shows "enat (the_enat (x - y)) \<le> x"
share_unshared_in: fixes a :: "nat"   and sb :: "'b memref list"   and S :: "nat \<Rightarrow> bool option" assumes "a \<in> dom (share sb S)" shows "a \<in> dom (share sb Map.empty) \<or> a \<in> dom S \<and> a \<notin> all_unshared sb"
F1: shows "\<forall>x\<^sub>2 x\<^sub>1. x\<^sub>1 \<subseteq> x\<^sub>1 \<union> x\<^sub>2"
distinct_degree_factorization: fixes f :: "'a mod_ring poly"   and facts :: "(nat \<times> 'a mod_ring poly) list" assumes "distinct_degree_factorization f = facts"   and "square_free f"   and "monic f" shows "f = prod_list (map snd facts) \<and> (\<forall>i f. (i, f) \<in> set facts \<longrightarrow> factors_of_same_degree i f)"
cmod_le: fixes z :: "complex" shows "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
Frame_R: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and F :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3 {P} C { Q}"   and "Frame P' P F" shows "\<turnstile>\<^sub>3 {P'} C { Q \<and>* F}"
get_pending_def: fixes s :: "('v, 'es) state_scheme" shows "get_pending s \<equiv> let u = hd (stack s); Vs = pending s `` {u} in if Vs = {} then RETURN (u, None, s) else RES Vs \<bind> (\<lambda>v. let s = s\<lparr>pending := pending s - {(u, v)}\<rparr> in RETURN (u, Some v, s))"
agent_abs_def: fixes a :: "'a"   and t :: "'s Trace" shows "agent_abs a t \<equiv> rel_ext (\<lambda>uu_. \<exists>t'. uu_ = tLast t' \<and> t' \<in> Clock.jkbpC \<and> clock_jview a t' = clock_jview a t)"
GPicard4: fixes k :: "real"   and f :: "complex \<Rightarrow> complex"   and B :: "real" assumes "0 < k"   and "f holomorphic_on ball 0 k - {0}"   and "\<And>e. \<lbrakk>0 < e; e < k\<rbrakk> \<Longrightarrow> \<exists>d>0. d < e \<and> (\<forall>z\<in>sphere 0 d. cmod (f z) \<le> B)" obtains \<epsilon> :: "real" where "0 < \<epsilon>" and "\<epsilon> < k" and "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> cmod (f z) \<le> B"
Perp2_def: fixes P :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" shows "P Perp2 A B C D \<equiv> \<exists>X Y. Col P X Y \<and> X Y Perp A B \<and> X Y Perp C D"
cobounded2: fixes b :: "'a"   and a :: "'a" shows "b \<le> a \<squnion> b"
rbt_map_entry_color_of: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and t :: "('a, 'b) rbt" shows "color_of (ord.rbt_map_entry less k f t) = color_of t"
app_invimageI1: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> A)" shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A"