hcju/isabelleps huggingface dataset:
Ide_in_Hom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Ide C t" shows "t \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = t \<and> monoidal_language.Cod C t = t}"
op_out2c: shows "open (outside (path_image c2 \<union> path_image c))"
ipresIGSwap_def: fixes h :: "'c \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'c, 'k, 'l) model_scheme"   and MOD' :: "('m, 'n, 'g, 'o, 'p, 'j, 'd, 'q, 'r) model_scheme" shows "ipresIGSwap h MOD MOD' \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> h (igSwap MOD zs z1 z2 X) = igSwap MOD' zs z1 z2 (h X)"
filter_id_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(List.filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
R: fixes ws :: "bool list" assumes "length ws = n" shows "P (bdd_lookup (Branch l r) ws)"
assms_4: shows "h \<turnstile> create_element document_ptr tag \<rightarrow>\<^sub>r result"
primepows_le_subset: fixes x :: "real"   and l :: "nat" assumes "0 < x"   and "0 < l" shows "{(p, i). prime p \<and> l \<le> i \<and> real (p ^ i) \<le> x} \<subseteq> {..nat \<lfloor>root l x\<rfloor>} \<times> {..nat \<lfloor>log 2 x\<rfloor>}"
assms: shows "max_Var_floatarith fa \<le> length xs"
finite_set_avoid: fixes S :: "'b set"   and a :: "'b" assumes "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
list_succ_altdef: fixes xs :: "'b list"   and x :: "'b" shows "list_succ xs x = (let n = index xs x in if n + 1 = length xs then xs ! 0 else if n + 1 < length xs then xs ! (n + 1) else x)"
valid_Exit_path_descending_path: 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 n :: "'b"   and as :: "'a list" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as Exit" obtains as' :: "'a list" where "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as' Exit" and "set (CFG.sourcenodes sourcenode as') \<subseteq> set (CFG.sourcenodes sourcenode as)" and "\<forall>a'\<in>set as'. intra_kind (kind a') \<or> (\<exists>Q f p. kind a' = Q\<hookleftarrow>\<^bsub>p\<^esub>f)"
triple_valid_Suc: fixes n :: "nat"   and t :: "'b triple" assumes "|=Suc n:t" shows "|=n:t"
le_less: fixes A :: "'b set set"   and f :: "'b \<Rightarrow> real" assumes "sigma_algebra A"   and "\<forall>a. {w. f w \<le> a} \<in> A" shows "\<forall>a. {w. f w < a} \<in> A"
intros: fixes P :: "pi"   and Q :: "pi" assumes "P \<leadsto>\<^sup>^<weakBisim> Q"   and "Q \<approx> P" shows "P \<approx> Q"
conjug_nemp_iff: fixes u :: "'b list"   and v :: "'b list" assumes "u \<sim> v" shows "(u = \<epsilon>) = (v = \<epsilon>)"
fags_bpp_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e" shows "fags_bpp f z i a = (\<lambda>x\<in>fags_carr f z i a. \<lambda>y\<in>fags_carr f z i a. ((THE n. x = n\<Odot>a\<^bsub>f,i,z\<^esub>) + (THE m. y = m\<Odot>a\<^bsub>f,i,z\<^esub>))\<Odot>a\<^bsub>f,i,z\<^esub>)"
pnormal_length: fixes p :: "'a list" assumes "pnormal p" shows "0 < length p"
\<psi>_def: shows "\<psi> \<equiv> THE \<psi>. \<guillemotleft>\<psi> : \<I>\<^sub>D \<rightarrow>\<^sub>D F \<I>\<^sub>C\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<iota>\<^sub>D = \<iota>\<^sub>1 \<cdot>\<^sub>D (\<psi> \<otimes>\<^sub>D \<psi>)"
bounded_real: fixes S :: "real set" shows "bounded S = (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"
append_is_Nil_conv: fixes xs :: "'e list"   and ys :: "'e list" shows "(xs @ ys = []) = (xs = [] \<and> ys = [])"
New: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option" assumes "new_Addr h = \<lfloor>a\<rfloor>"   and "P \<turnstile> C has_fields FDTs"   and "h' = h(a \<mapsto> (C, init_fields FDTs))" shows "P \<turnstile> \<langle>new C,(h, l)\<rangle> \<Rightarrow> \<langle>addr a,(h', l)\<rangle>"
G_def: shows "G \<equiv> \<lambda>x. if x = 4 then pi else if x = 0 then - pi else F x"
r_distr: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z * (x + y) = z * x + z * y"
pcompose_sum: fixes f :: "'b \<Rightarrow> 'a poly"   and A :: "'b set"   and p :: "'a poly" shows "sum f A \<circ>\<^sub>p p = (\<Sum>i\<in>A. f i \<circ>\<^sub>p p)"
less_PInf_Ex_of_nat: fixes x :: "ereal" shows "(x \<noteq> \<infinity>) = (\<exists>n. x < ereal (real n))"
degree_le: fixes n :: "nat"   and p :: "'a poly" assumes "\<forall>i>n. coeff p i = (0::'a)" shows "degree p \<le> n"
im_sheaf_morphisms_def: fixes U :: "'a set"   and V :: "'a set" shows "idim.im_sheaf_morphisms U V \<equiv> \<rho> (identity S \<^sup>\<inverse> S U) (identity S \<^sup>\<inverse> S V)"
ChamberSubcomplex_def: fixes X :: "'b set set"   and Y :: "'b set set" assumes "ChamberComplex X" shows "ChamberComplex.ChamberSubcomplex X Y \<equiv> order.greater_eq X Y \<and> ChamberComplex Y \<and> (\<forall>C. SimplicialComplex.maxsimp Y C \<longrightarrow> SimplicialComplex.maxsimp X C)"
list_all_iff: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g list" shows "list_all P x = Ball (set x) P"
finitesumbase_sub_carrier: fixes f :: "'f \<Rightarrow> 'a set"   and I :: "'f set" assumes "f \<in> I \<rightarrow> {X. submodule R M X}" shows "finitesum_base M I f \<subseteq> carrier M"
LINEAR_inject: fixes b :: "'g"   and r :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and b' :: "'g"   and r' :: "('f + 'g + 'h, 'i) Old_Datatype.node set" shows "(LINEAR b r = LINEAR b' r') = (b = b' \<and> r = r')"
set: fixes x :: "'x" shows "set_node_ptr (node_ptr.Ext x) = {x}"
nonempty_K: shows "{h. h \<le> k \<and> P h} \<noteq> {}"
allj_exn: shows "\<forall>j\<in>{0..<m}. \<exists>n. P j n"
Units: fixes m :: "'a list list" assumes "mat nr nc m" shows "\<exists>m'. mat nr nc m' \<and> mat_plus m' m = mat0 nr nc \<and> mat_plus m m' = mat0 nr nc"
exec_gpv_bind: fixes callee :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and c :: "('e, 'c, 'd) gpv"   and f :: "'e \<Rightarrow> ('a, 'c, 'd) gpv"   and s :: "'b" shows "exec_gpv callee (c \<bind> f) s = exec_gpv callee c s \<bind> (\<lambda>xa. case xa of (x, s') \<Rightarrow> exec_gpv callee (f x) s')"
derivation_useful: fixes X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "derivation X"   and "x \<in> X"   and "y \<in> X"   and "x \<sqsubset> y" shows "f x \<sqsubseteq> y"
older_seniors_finite: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "finite (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n)"
assms_3: fixes P'' :: "ccs" assumes "P \<longmapsto>\<tau> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "Prop P'"
f2: shows "y \<cdot> y\<^sup>\<star> + y\<^sup>\<star> = y\<^sup>\<star>"
rel_Grp: fixes A1 :: "'d set"   and f1 :: "'d \<Rightarrow> 'f"   and A2 :: "'e set"   and f2 :: "'e \<Rightarrow> 'g" shows "rel_optionT' (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. phantom_optionT x \<subseteq> A1 \<and> set_optionT x \<subseteq> A2} (map_optionT' f1 f2)"
basis_dec: fixes p :: "'b list"   and G :: "'b list set"   and s :: "'b list" assumes "p \<in> \<langle>G\<rangle>"   and "s \<in> \<langle>G\<rangle>"   and "p \<cdot> s \<in> \<BB> G" shows "p = \<epsilon> \<or> s = \<epsilon>"
Inf_between_q_def: fixes q :: "'q" shows "Inf_between_q q = inference_system.Inf_between (Inf_q q)"
da_elim_cases_6: fixes Env :: "env"   and B :: "lname set"   and l :: "jump"   and c :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (l\<bullet> c)\<guillemotright> A" obtains C :: "assigned" and la :: "label" where "l = Break la" and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "nrm A = nrm C \<inter> brk C la" and "brk A = rmlab la (brk C)"
nres_of_nress_alt_def: shows "nres_of_nress \<equiv> \<lambda>P. rec_list (RETURN []) (\<lambda>x xs xsa. x \<bind> (\<lambda>r. xsa \<bind> (\<lambda>rs. RETURN (r # rs))))"
nub_body'_def: shows "nub_body' \<equiv> \<Lambda> f r. case a2c\<cdot>r of lnil \<Rightarrow> lnil | x :@ xs \<Rightarrow> x :@ f\<cdot>(c2a\<cdot>(lfilter\<cdot>(neg oo Abs_cfun ((=\<^sub>B) x))\<cdot>xs))"
init_fin_simulation_silent1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and t :: "'b"   and sxm1 :: "(status \<times> 'a) \<times> 'c"   and sxm2 :: "(status \<times> 'g) \<times> 'h"   and sxm1' :: "(status \<times> 'a) \<times> 'c" assumes "FWdelay_bisimulation_diverge final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWbisimulation_base.init_fin_bisim final1 final2 bisim t sxm1 sxm2"   and "\<tau>trsys.silent_move (multithreaded_base.init_fin final1 r1 t) (\<tau>multithreaded.init_fin_\<tau>move \<tau>move1) sxm1 sxm1'" shows "\<exists>sxm2'. (\<tau>trsys.silent_move (multithreaded_base.init_fin final2 r2 t) (\<tau>multithreaded.init_fin_\<tau>move \<tau>move2))\<^sup>*\<^sup>* sxm2 sxm2' \<and> FWbisimulation_base.init_fin_bisim final1 final2 bisim t sxm1' sxm2'"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('d \<Rightarrow> ('b, 'c, 'd) env option) \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Val x1 of Val x \<Rightarrow> f1 x | Env x xa \<Rightarrow> f2 x xa) = f1 x1"
is_auto_reduced_def: fixes B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "punit.is_auto_reduced B \<equiv> \<forall>b\<in>B. \<not> punit.is_red (B - {b}) b"
assms_2: fixes cfg :: "'a \<Rightarrow> 'b" assumes "CHOinitConfig A cfg (coords 0)" shows "P cfg"
prv_nprv2I: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<psi> :: "'fmla" assumes "prv (imp \<phi>1 (imp \<phi>2 \<psi>))"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2} \<psi>"
convex_linear_image: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "linear f"   and "convex S" shows "convex (f ` S)"
A_def: shows "A = (\<lambda>i. of_nat (i choose nat zn) * fact (nat zn) * (deriv ^^ (i - nat zn)) g z)"
Id_compose: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "f \<in> A \<rightarrow> B"   and "f \<in> extensional A" shows "compose A (\<lambda>y\<in>B. y) f = f"
ucl_def: shows "ucl = map getGMUserCom (tl (purgeIdle sl))"
ImpP: fixes H :: "('a, 'b) fm set"   and p :: "('a, 'b) fm"   and q :: "('a, 'b) fm" assumes "Hintikka H"   and "(p \<^bold>\<longrightarrow> q) \<in> H" shows "(\<^bold>\<not> p) \<in> H \<or> q \<in> H"
induct: fixes x1 :: "cmd"   and x2 :: "action + brk_ctd"   and x3 :: "cmd"   and P :: "cmd \<Rightarrow> action + brk_ctd \<Rightarrow> cmd \<Rightarrow> bool" assumes "cfg x1 x2 x3"   and "\<And>c x e. P (Assign c x e) (Inl (AAssign c x e)) Skip"   and "\<And>c x e. P (Assign_local c x e) (Inl (AAssign_local c x e)) Skip"   and "\<And>c x e. P (Assign_global c x e) (Inl (AAssign_global c x e)) Skip"   and "\<And>e. P (Test e) (Inl (ATest e)) Skip"   and "\<And>c2. P (Seq Skip c2) (Inl ASkip) c2"   and "\<And>c1 e c2. \<lbrakk>cfg c1 e Skip; P c1 e Skip\<rbrakk> \<Longrightarrow> P (Seq c1 c2) e c2"   and "\<And>c1 e c1' c2. \<lbrakk>cfg c1 e c1'; P c1 e c1'; c1' \<noteq> Skip\<rbrakk> \<Longrightarrow> P (Seq c1 c2) e (Seq c1' c2)"   and "\<And>c1 e c1' c2. \<lbrakk>cfg c1 e c1'; P c1 e c1'\<rbrakk> \<Longrightarrow> P (Or c1 c2) e c1'"   and "\<And>c2 e c2' c1. \<lbrakk>cfg c2 e c2'; P c2 e c2'\<rbrakk> \<Longrightarrow> P (Or c1 c2) e c2'"   and "P Break (Inr AIBreak) Invalid"   and "P Continue (Inr AIContinue) Invalid"   and "\<And>c. P (Iterate Skip c) (Inl ASkip) (Iterate c c)"   and "\<And>c e cb. \<lbrakk>cfg c (Inl e) Skip; P c (Inl e) Skip\<rbrakk> \<Longrightarrow> P (Iterate c cb) (Inl e) (Iterate cb cb)"   and "\<And>c e c' cb. \<lbrakk>cfg c (Inl e) c'; P c (Inl e) c'; c' \<noteq> Skip\<rbrakk> \<Longrightarrow> P (Iterate c cb) (Inl e) (Iterate c' cb)"   and "\<And>c uu_ cb. \<lbrakk>cfg c (Inr AIBreak) uu_; P c (Inr AIBreak) uu_\<rbrakk> \<Longrightarrow> P (Iterate c cb) (Inl ASkip) Skip"   and "\<And>c uv_ cb. \<lbrakk>cfg c (Inr AIContinue) uv_; P c (Inr AIContinue) uv_\<rbrakk> \<Longrightarrow> P (Iterate c cb) (Inl ASkip) (Iterate cb cb)" shows "P x1 x2 x3"
drop_bit_drop_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit m (drop_bit n a) = drop_bit (m + n) a"
lookup_eq: fixes env :: "('f, 'g, 'h) env"   and xs :: "'h list" shows "lookup env xs = (case xs of [] \<Rightarrow> Some env | x # xs \<Rightarrow> case env of Val x \<Rightarrow> Map.empty x | Env b es \<Rightarrow> case es x of None \<Rightarrow> None | Some e \<Rightarrow> lookup e xs)"
diff_cancel: fixes a :: "'b" shows "a - a = (0::'b)"
Subject_reductions: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "P,E,s \<turnstile> e : T \<surd>" shows "\<exists>T'. P,E,s' \<turnstile> e' : T' \<surd> \<and> P \<turnstile> T' \<le> T"
one_over_poly_closed: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "one_over_poly p \<in> carrier P"
polyfun_single: fixes i :: "nat"   and N :: "nat set" assumes "i \<in> N" shows "polyfun N (\<lambda>x. x i)"
wf_trm_subterms: fixes t :: "('fun, 'a) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subterms t)"
IH: assumes "\<forall>i. var \<notin> vars (f i)" shows "var \<notin> vars (sum f {0..<n})"
rep_eq: fixes x :: "'a vec" shows "list_of_vec x = (case Rep_vec x of (n, v) \<Rightarrow> map v [0..<n])"
h_2: shows "A *\<^sub>v h \<le> b"
sinh_real_at_bot: shows "filterlim sinh at_bot at_bot"
simps_1: fixes i :: "int"   and ks :: "int list"   and xs :: "int list" shows "I\<^sub>Z (Le i ks) xs = (i \<le> \<langle>ks,xs\<rangle>)"
insert_seq_rem_last: fixes start :: "'a option"   and xs :: "('a \<times> 'a option) list"   and x :: "'a \<times> 'a option" assumes "insert_seq start (xs @ [x])"   and "xs \<noteq> []" shows "insert_seq start xs"
sum_zero_eq_neg: fixes x :: "'a"   and y :: "'a" assumes "True"   and "True"   and "x + y = (0::'a)" shows "x = - y"
assms_4: fixes s :: "'a set" assumes "s \<in> set ss" shows "(0::'a) \<in> s"
multiset_inter_def: fixes A :: "'b multiset"   and B :: "'b multiset" shows "A \<inter># B = A - (A - B)"
mor_UNIV: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f1 :: "'d \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'h" shows "mor UNIV UNIV s1 s2 UNIV UNIV s1' s2' f1 f2 = (F1map id f1 f2 \<circ> s1 = s1' \<circ> f1 \<and> F2map id f1 f2 \<circ> s2 = s2' \<circ> f2)"
awalk_verts_append: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and q :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.awalk G u (p @ q) v" shows "pre_digraph.awalk_verts G u (p @ q) = pre_digraph.awalk_verts G u p @ tl (pre_digraph.awalk_verts G (last (pre_digraph.awalk_verts G u p)) q)"
new_cond: shows "all_unshared (x # sb) \<inter> dom \<S>' - dom \<S> = {}"
measurable_on_restrict: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "f measurable_on UNIV"   and "S \<in> sets lebesgue" shows "(\<lambda>x. if x \<in> S then f x else (0::'d)) measurable_on UNIV"
supchamberD_2: fixes v :: "'a" assumes "v \<in> \<Union> X" shows "v \<in> supchamber v"
Goal: fixes xa :: "name"   and Pa :: "pi" assumes "a<xa>.Pa \<longmapsto> \<alpha> \<prec> P'"   and "xa \<sharp> \<alpha>"   and "xa \<sharp> P'"   and "xa \<noteq> a"   and "\<And>u. Prop (a<u>) (Pa[xa::=u])" shows "Prop \<alpha> P'"
compatible_l_def: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('a \<times> 'a) set" shows "compatible_l R1 R2 \<equiv> R1 O R2 \<subseteq> R2"
vertex_2_char_def: shows "vertex_2 \<equiv> CHR ''B''"
l2_inv1I: fixes x :: "l2_state" assumes "\<And>A B. can_signal x A B \<Longrightarrow> A \<notin> bad x \<and> B \<notin> bad x" shows "x \<in> l2_inv1"
fst_snd_f_le_nrows: shows "fst (snd f) \<le> nrows (snd (snd f))"
updS_Rmi: fixes s :: "bdd"   and x :: "nat \<times> nat \<times> nat"   and r :: "nat" shows "Rmi (updS s x r) = Rmi s"
preserves_cod: fixes f :: "J.Arr" assumes "J.arr f" shows "C.cod (E.map f) = E.map (J.cod f)"
set_sel_1: fixes a :: "'c list" assumes "a \<noteq> []" shows "hd a \<in> set a"
map_lookup: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow>\<^sub>0 'b"   and x :: "'c" assumes "g (0::'b) = (0::'a)" shows "lookup (Poly_Mapping.map g f) x = g (lookup f x)"
set_child_nodes_def: shows "set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = invoke CD.a_set_child_nodes_tups"
rep_eq: fixes x :: "'b"   and xa :: "'c" shows "lookup (monomial xa x) = (\<lambda>k'. xa when x = k')"
M'_def: shows "M' \<equiv> \<lambda>i' j'. if i' = i \<and> j' = j then Le d else if i' = j \<and> j' = i then Le (- d) else M i' j'"
one_eq_prime_power_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "(one = power.power one times p n) = (n = 0)"
abs_eq: fixes xa :: "'a \<Rightarrow> real"   and x :: "'a \<Rightarrow> real" shows "Abs_euclidean_space xa + Abs_euclidean_space x = Abs_euclidean_space (\<lambda>xb. xa xb + x xb)"
2: shows "(s - k * l * s + b1 * k) * d' = b"
plusinf_ex: fixes p :: "fm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list" assumes "islin p"   and "Ifm vs (x # bs) (plusinf p)" shows "\<exists>x. Ifm vs (x # bs) p"
s_def: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat" assumes "distributed_system channel" shows "distributed_system.s channel trans send recv c t i = (THE c'. distributed_system.trace channel trans send recv c (take i t) c')"
foldli_append: fixes xs1 :: "'e list"   and xs2 :: "'e list"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d" shows "foldli (xs1 @ xs2) c f \<sigma> = foldli xs2 c f (foldli xs1 c f \<sigma>)"
that: fixes W :: "nat \<Rightarrow> 'a set" assumes "carrier = \<Union> (range W)"   and "\<And>i. open (W i)"   and "\<And>i. compact (closure (W i))"   and "\<And>i. closure (W i) \<subseteq> carrier"   and "locally_finite_on carrier UNIV W" shows "thesis"
set_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "set (filter P xs) = {x. x \<in>. xs \<and> P x}"
bounded_linear_axioms: shows "bounded_linear (if x \<in> T then g' x else (\<lambda>x. 0::'b))"
inject_2: fixes x21 :: "'h"   and x22 :: "'i"   and y21 :: "'h"   and y22 :: "'i" shows "(IO x21 x22 = IO y21 y22) = (x21 = y21 \<and> x22 = y22)"
replace_lt_adds_stable_is_red: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and f :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.is_red F f"   and "q \<noteq> 0"   and "local.punit.adds_term (punit.lt q) (punit.lt p)" shows "punit.is_red (insert q (F - {p})) f"
spec_of_waiting: shows "\<forall>s a. rwaiting s (rcurrent s) a \<longrightarrow> rstep s a = s"
planning_dimacs_sound_code: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and h :: "nat"   and dimacs_M :: "int list" assumes "ast_problem.well_formed prob"   and "\<forall>\<pi>\<in>set (ast_problem.ast\<delta> prob). is_standard_operator' \<pi>" shows "let cnf_formula = SASP_to_DIMACS h prob; decoded_plan = decode_DIMACS_model dimacs_M h prob in dimacs_model dimacs_M cnf_formula \<longrightarrow> ast_problem.valid_plan prob decoded_plan"
mod_eq_imp_diff_dvd: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "a mod m = b mod m" shows "m dvd b - a"
assms_3: shows "mod_heap = heap(l := heap (r - 1))"
strict_implies_order: fixes a :: "'a list"   and b :: "'a list" assumes "strict_prefix a b" shows "prefix a b"
obj_UP\<^sub>0: fixes a :: "'a" assumes "B.obj a" shows "obj (UP\<^sub>0 a)"
le_oFix1: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" shows "F a \<le> oFix F a"