hcju/isabelleps huggingface dataset:
path_not_Nil: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and g :: "'a"   and ns :: "'b list" assumes "graph_path \<alpha>e \<alpha>n invar inEdges'"   and "graph_path_base.path \<alpha>n invar inEdges' g ns" shows "ns \<noteq> []"
assms_5: shows "\<forall>k. \<not> j cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
sel_2: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "ltl (iterates f x) = iterates f (f x)"
vector_smult_component: fixes c :: "'b"   and y :: "('b, 'c) vec"   and i :: "'c" shows "(c *s y) $ i = c * y $ i"
FrameStatImpTrans: fixes F :: "'b frame"   and G :: "'b frame"   and H :: "'b frame" assumes "F \<hookrightarrow>\<^sub>F G"   and "G \<hookrightarrow>\<^sub>F H" shows "F \<hookrightarrow>\<^sub>F H"
orthogonal_complement_of_closure: fixes A :: "'a set" shows "orthogonal_complement A = orthogonal_complement (closure A)"
IH: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'a state" shows "\<exists>a. evaluate_decs True mn env s ds a"
simps_7: fixes P :: "pi"   and Q :: "pi" shows "depth (P \<oplus> Q) = max (depth P) (depth Q)"
distinct_27: fixes x4 :: "'b" shows "WakeUp x4 \<noteq> WokenUp"
strict_implies_order: fixes b :: "'b list"   and a :: "'b list" assumes "b <s a" shows "b \<le>s a"
max_perm_degree_eqI: fixes A :: "('c poly, 'b) sq_matrix"   and x :: "nat" assumes "\<And>p. p permutes UNIV \<Longrightarrow> (\<Sum>i\<in>UNIV. degree (to_fun A i (p i))) \<le> x"   and "\<exists>p. p permutes UNIV \<and> (\<Sum>i\<in>UNIV. degree (to_fun A i (p i))) = x" shows "max_perm_degree A = x"
pq: shows "P = p\<^sup>2 + 3 * q\<^sup>2"
bp_def: fixes P :: "'a set set" shows "bp P = (partition_on U P \<and> (\<forall>B\<in>P. W B \<le> real c))"
Loop: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and n :: "nat"   and b :: "val"   and s1 :: "state"   and c :: "stmt"   and s2 :: "state"   and l :: "label"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<midarrow>n\<rightarrow> s1"   and "if the_Bool b then G\<turnstile>s1 \<midarrow>c\<midarrow>n\<rightarrow> s2 \<and> G\<turnstile>abupd (absorb (Cont l)) s2 \<midarrow>l\<bullet> While(e) c\<midarrow>n\<rightarrow> s3 else s3 = s1" shows "G\<turnstile>Norm s0 \<midarrow>l\<bullet> While(e) c\<midarrow>n\<rightarrow> s3"
no_spoofing_algorithm_sound: fixes rs :: "'b common_primitive rule list"   and iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option" assumes "simple_ruleset rs"   and "\<forall>r\<in>set rs. normalized_nnf_match (get_match r)"   and "no_spoofing_iface iface ipassmt rs" shows "nospoof TYPE('c) iface ipassmt rs"
Loss_L6: fixes t :: "nat"   and a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and y :: "nat \<Rightarrow> aType list"   and i :: "nat \<Rightarrow> ECall_Info list" assumes "\<forall>t2\<le>t. a t2 = []"   and "Loss lose a i2 y i"   and "ts lose" shows "\<forall>t2\<le>t. y t2 = []"
take_induct: fixes P :: "C \<Rightarrow> bool"   and x :: "C" assumes "adm P"   and "\<And>n. P (C_take n\<cdot>x)" shows "P x"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Inv (t \<^bold>\<otimes> u) = Inv t \<^bold>\<otimes> Inv u"
is_subseq_of_trans: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and s' :: "nat \<Rightarrow> nat \<Rightarrow> int"   and s'' :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "is_subseq_of s s'"   and "is_subseq_of s' s''" shows "is_subseq_of s s''"
simps_1: fixes MI :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U"   and P :: "nat"   and vs :: "nat list" shows "FEval MI e (PAtom P vs) = (let IP = snd MI P in IP (map e vs))"
axioms_1: fixes L :: "('a, 'b) gorder_scheme" assumes "partial_order L" shows "weak_partial_order L"
fresh_distinct: fixes S :: "'d set"   and \<Gamma> :: "('d \<times> 'e) list" assumes "atom ` S \<sharp>* \<Gamma>" shows "S \<inter> domA \<Gamma> = {}"
ctxt_subst_mem_TVarB: fixes X :: "tyvrs"   and T :: "ty"   and \<Gamma> :: "binding list"   and Y :: "tyvrs"   and U :: "ty" assumes "TVarB X T \<in> set \<Gamma>" shows "TVarB X (T[Y \<mapsto> U]\<^sub>\<tau>) \<in> set (\<Gamma>[Y \<mapsto> U]\<^sub>e)"
A: fixes n :: "nat" shows "A n \<subseteq> space M"
xfer_fair_run_aux: fixes as\<^sub>0 :: "'b"   and alstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and cs\<^sub>0 :: "'e"   and clstep :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and cs :: "nat \<Rightarrow> 'e" assumes "simulation as\<^sub>0 alstep cs\<^sub>0 clstep R"   and "df_system.is_fair_run cs\<^sub>0 clstep cs" shows "\<exists>as. (\<forall>i. R (as i) (cs i)) \<and> df_system.is_fair_run as\<^sub>0 alstep as"
prems: shows "Vagree \<nu> \<nu>' (FVDiff (Times t1 t2))"
distr_orD: fixes F :: "'d \<Rightarrow> 'e decision option"   and F2 :: "'d \<Rightarrow> 'e decision option"   and F1 :: "'d \<Rightarrow> 'e decision option"   and g :: "'h \<times> 'e \<Rightarrow> 'g"   and N :: "'i \<Rightarrow> 'h decision option"   and f :: "'f \<Rightarrow> 'i \<times> 'd" assumes "F = F1 \<Oplus> F2" shows "g o\<^sub>f (N \<Otimes>\<^sub>\<or>\<^sub>D F \<circ> f) = (g o\<^sub>f (N \<Otimes>\<^sub>\<or>\<^sub>D F1 \<circ> f)) \<Oplus> (g o\<^sub>f (N \<Otimes>\<^sub>\<or>\<^sub>D F2 \<circ> f))"
AboveS_def: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" shows "AboveS r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. b \<noteq> a \<and> (a, b) \<in> r}"
subdivision_bidir: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd" assumes "subdivision (G, rev_G) (H, rev_H)" shows "bidirected_digraph H rev_H"
invar2_ctd_finish: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" assumes "prim_invar2_ctd Q \<pi>"   and "Q = (\<lambda>_. \<infinity>)" shows "is_MST w rg (graph {r} {(u, v). \<pi> u = Some v})"
ndet_def: fixes P :: "'a process"   and Q :: "'a process" shows "P |-| Q \<equiv> Abs_process (F P \<union> F Q, D P \<union> D Q)"
tauTransitionChain: fixes P :: "pi"   and P' :: "pi" shows "P \<Longrightarrow>\<^sup>^\<tau> \<prec> P' = P \<Longrightarrow>\<^sub>\<tau> P'"
case_prod_beta: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'f"   and p :: "'g \<times> 'h" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
of_int_mod_ring_def: shows "of_int_mod_ring \<equiv> (id ---> Abs_mod_ring) (\<lambda>x. x mod int CARD('b))"
Un_Diff_Int: fixes A :: "'b set"   and B :: "'b set" shows "A - B \<union> A \<inter> B = A"
UP_root_rem_def: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" shows "UP_root_rem f a = monom P (T\<^bsub>a\<^esub> f 0) 0"
ComplI: fixes c :: "'b"   and A :: "'b set" assumes "c \<in> A \<Longrightarrow> False" shows "c \<in> - A"
conv_iso: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x\<^sup>T \<le> y\<^sup>T)"
hyps_2: fixes U :: "POPLmarkRecord.type"   and U' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> S' <: (\<forall><:U. U')" obtains S' :: "POPLmarkRecord.type" where "\<Gamma> \<turnstile> U <: S" and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'" and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'"
max_bot2: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.order_bot bot less_eq less" shows "ord.max less_eq x bot = x"
star_of_1_eq: fixes y :: "'a" shows "(1 = star_of y) = ((1::'a) = y)"
the_elem_is_Proc: fixes cnt :: "d" assumes "isProc cnt" shows "the_elem |cnt| \<in> |cnt|"
disc_iff_1: fixes x :: "('i, 'g, 'h) Resumption.resumption"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "Resumption.resumption.is_Done (x \<bind> f) = (Resumption.resumption.is_Done x \<and> (Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x)))))"
lex_pp_antisym: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" assumes "lex_pp s t"   and "lex_pp t s" shows "s = t"
subgroup_imp_finite_comm_group: fixes H :: "'a set" assumes "subgroup H G" shows "finite_comm_group (G\<lparr>carrier := H\<rparr>)"
has_as_pullback_def: fixes f0 :: "nat"   and f1 :: "nat"   and p0 :: "nat"   and p1 :: "nat" shows "J'J.A.has_as_pullback f0 f1 p0 p1 \<equiv> J'J.A.cospan f0 f1 \<and> limit_cone J.comp J' (cospan_diagram.map J' f0 f1) (J'J.A.dom p0) (cospan_diagram.mkCone J' f0 f1 p0 p1)"
subst_bv_bind_fv: fixes t :: "Core.term"   and v :: "variable"   and T :: "typ" assumes "is_closed t" shows "subst_bv (Fv v T) (bind_fv (v, T) t) = t"
step: fixes CC' :: "'a literal multiset multiset"   and D :: "'a literal multiset"   and E :: "'a literal multiset"   and CC :: "'a literal multiset set" assumes "Infer CC' D E \<in> \<Gamma>"   and "\<And>C'. C' \<in># CC' \<Longrightarrow> C' \<in> saturate CC"   and "D \<in> saturate CC" shows "E \<in> saturate CC"
steps_preserve_SN_on_relto: fixes a :: "'a"   and b :: "'a"   and R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "(a, b) \<in> (R \<union> S)\<^sup>*"   and "SN_on (S\<^sup>* O R O S\<^sup>*) {a}" shows "SN_on (S\<^sup>* O R O S\<^sup>*) {b}"
that_2: assumes "r = C\<^sup>\<infinity>" shows "thesis"
assms: fixes p :: "'a" assumes "p \<in># A" shows "prime p"
distr_cong: fixes M :: "'d measure"   and K :: "'d measure"   and N :: "'e measure"   and L :: "'e measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "M = K"   and "sets N = sets L"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = g x" shows "distr M N f = distr K L g"
igOpIPresIGWlsSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igOpIPresIGWlsSTR MOD \<equiv> \<forall>delta inp binp. igWls MOD (stOf delta) (igOp MOD delta inp binp) = (igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp)"
normalize_positive_src_ports_nnf: fixes m :: "'b common_primitive match_expr"   and m' :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc_negated is_Src_Ports False m"   and "m' \<in> set (normalize_positive_src_ports m)" shows "normalized_nnf_match m'"
HInfinite_FreeUltrafilterNat: fixes X :: "nat \<Rightarrow> 'b"   and u :: "real" assumes "star_n X \<in> HInfinite" shows "\<forall>\<^sub>F n in \<U>. u < norm (X n)"
fmdrop_fset_insert: fixes x :: "'c"   and S :: "'c fset"   and m :: "('c, 'd) fmap" shows "fmdrop_fset (finsert x S) m = fmdrop x (fmdrop_fset S m)"
Substable_Cond_\<o>_def: shows "Substable_Cond \<equiv> \<lambda>\<phi> \<psi>. \<forall>v. [\<phi> \<^bold>\<equiv> \<psi> in v]"
psi_even_def: fixes n :: "nat" shows "psi_even n = sum mangoldt_even {1..n}"
in_own_orbit: fixes a :: "'a" shows "a \<in> Orb a"
continuous_map_id: fixes X :: "'b topology" shows "continuous_map X X id"
simps_1: fixes a :: "predicate"   and vs :: "'ent list" shows "wf_fmla_atom (Atom (predAtm a vs)) = wf_pred_atom (a, vs)"
simps_4: fixes a :: "'b"   and b :: "'b" shows "(dRETURN a \<le> dRETURN b) = (a = b)"
A_updates_sec: shows "\<forall>x. dma mem\<^sub>1 [\<parallel>\<^sub>1 A] x = Low \<and> (x \<notin> mds AsmNoReadOrWrite \<or> x \<in> \<C>) \<longrightarrow> mem\<^sub>1 [\<parallel>\<^sub>1 A] x = mem\<^sub>2 [\<parallel>\<^sub>2 A] x"
induct: fixes P :: "int list \<Rightarrow> bool"   and a0 :: "int list" assumes "\<And>xs. (\<And>x xa y. \<lbrakk>xs \<noteq> []; x = take2 xs; (xa, y) = x\<rbrakk> \<Longrightarrow> P y) \<Longrightarrow> P xs" shows "P a0"
mwb_lens_weak: fixes x :: "'a \<Longrightarrow> 'b" assumes "mwb_lens x" shows "weak_lens x"
g_max_impl: shows "map_max \<alpha> invar g_max"
iMin_remove: fixes n :: "'a"   and I :: "'a set" assumes "n \<in> I" shows "iMin I = (if I - {n} = {} then n else min n (iMin (I - {n})))"
octo_of_nat_def: fixes r :: "nat" shows "octo_of_nat r = real r *\<^sub>R 1"
compositional_rsp3: fixes R1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and R3 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and R4 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R5 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and R6 :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "(R1 ===> R2 ===> R3) C C"   and "(R4 ===> R5 ===> R6) C C" shows "(R1 OOO R4 ===> R2 OOO R5 ===> R3 OOO R6) C C"
chain_singleton: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.ccpo Sup less_eq less" shows "Complete_Partial_Order.chain less_eq {x}"
p_max_eq: shows "p_max = s0 - v\<^sup>2 / a / 2"
S3: shows "nat_to_pr (sch_to_nat t) = sch_to_pr (nat_to_sch (sch_to_nat t))"
Subject_reduction: 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"
J2_def: shows "J2 = linepath (flow0 x t2) (flow0 x t1)"
convex_polychain_Cons2: fixes x :: "(real \<times> real) \<times> real \<times> real"   and y :: "(real \<times> real) \<times> real \<times> real"   and zs :: "((real \<times> real) \<times> real \<times> real) list" shows "convex_polychain (x # y # zs) = (snd x = fst y \<and> 0 < det3 (fst x) (fst y) (snd y) \<and> convex_polychain (y # zs))"
vec_inj: fixes x :: "'a"   and y :: "'a" shows "(vec x = vec y) = (x = y)"
not_on_path_ball: fixes g :: "real \<Rightarrow> 'a"   and z :: "'a" assumes "path g"   and "z \<notin> path_image g" shows "\<exists>e>0. ball z e \<inter> path_image g = {}"
cc_inst_true: shows "I \<Turnstile>m mset CAs \<cdot>cm \<sigma> \<cdot>cm \<eta>"
avl_ins_case: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t" shows "case ins x t of Same t' \<Rightarrow> avl t' \<and> height t' = height t | Diff t' \<Rightarrow> avl t' \<and> height t' = height t + 1 \<and> (\<forall>l a r. t' = \<langle>l, (a, Bal), r\<rangle> \<longrightarrow> a = x \<and> l = \<langle>\<rangle> \<and> r = \<langle>\<rangle>)"
ring_hom: shows "(\<lambda>x. the_elem (f ` x)) \<circ> (+>) I \<in> ring_hom R (image_ring f R)"
oprod_Well_order: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Well_order r"   and "Well_order r'" shows "Well_order (r *o r')"
assms_2: shows "v \<in> fv\<^sub>s\<^sub>t S"
smod_int_0_mod: fixes x :: "int" shows "0 smod x = 0"
list_emb_Nil: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c list" shows "list_emb P [] ys"
singleton_rev_conv: fixes x :: "'f"   and xs :: "'f list" shows "([x] = rev xs) = (xs = [x])"
simps_1: fixes P :: "'c" shows "post \<lbrace> P \<rbrace> = P"
times_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "f * g = (\<lambda>x. f x * g x)"
int_subst: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and T :: "'fsym trm" shows "int \<xi> (subst \<rho> T) = int (int \<xi> \<circ> \<rho>) T"
closed_scaling: fixes S :: "'b set"   and c :: "real" assumes "closed S" shows "closed ((*\<^sub>R) c ` S)"
match_rel: fixes t :: "pterm"   and u :: "sterm"   and p :: "Term_Class.term" assumes "t = sterm_to_pterm u" shows "rel_option srelated.P_env (match p t) (match p u)"
regular_weak_composition_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "regular_weak_composition V H \<equiv> weak_composition V H \<and> regular_weak_composition_axioms V H"
0: shows "\<Union> S = {a..<x} \<union> {x<..b}"
horner_eval_interval: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a interval"   and x :: "'a"   and I :: "'a interval" assumes "\<And>i. i < n \<Longrightarrow> f i \<in>\<^sub>i g i"   and "x \<in>\<^sub>i I" shows "horner_eval f x n \<in>\<^sub>i horner_eval g I n"
map_of_zip_nth: fixes xs :: "'c list"   and ys :: "'d list"   and i :: "nat" assumes "length xs = length ys"   and "distinct xs"   and "i < length ys" shows "map_of (zip xs ys) (xs ! i) = Some (ys ! i)"
neg_mask_le_high_bits: fixes n :: "nat"   and w :: "'b word" shows "(NOT (mask n) \<le> w) = (\<forall>i\<in>{n..<size w}. w !! i)"
lnths_cong: fixes xs :: "'b llist"   and ys :: "'b llist"   and A :: "nat set"   and B :: "nat set" assumes "xs = ys"   and "\<And>n. enat n < llength ys \<Longrightarrow> (n \<in> A) = (n \<in> B)" shows "lselect A xs = lselect B ys"
IH: shows "\<exists>p s. xs = p @ s \<and> insert_body xs e = p @ e # s"
simps_1: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "upd_lock l t Lock = lock_lock l t"
exponential_order_add: fixes M :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a" assumes "exponential_order M a f"   and "exponential_order M a g" shows "exponential_order (2 * M) a (\<lambda>x. f x + g x)"
Abs_uexpr_cases: fixes x :: "('d, 'c) uexpr" obtains y :: "'c \<Rightarrow> 'd" where "x = mk\<^sub>e y" and "y \<in> UNIV"
nopaid: fixes s :: "'b list \<times> 'b list"   and q :: "'b" shows "snd (fst (TS_step_d s q)) = []"
pos_less_pos_iff: fixes A :: "'a"   and B :: "'a" shows "(Pos A < Pos B) = (A < B)"
affine_dependent_iff_dependent2: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S" shows "affine_dependent S = dependent ((+) (- a) ` (S - {a}))"
SLC_new: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and v :: "('r, 'l, 'v) val"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and l :: "'l"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])"   and "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (Loc l)]))"   and "l \<notin> LID\<^sub>G s\<^sub>1"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'"   and "reachable s\<^sub>1"   and "infinite UNIV" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
T_on'_append: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and s :: "'state \<times> 'b"   and xs :: "'request list"   and ys :: "'request list" shows "T_on' A s (xs @ ys) = T_on' A s xs + T_on' A (config' A s xs) ys"
42: fixes x :: "'a"   and y :: "'a" shows "(- x \<squnion> y) \<sqinter> x = y \<sqinter> x"
search_enum_nonempty_neq0: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and i :: "nat"   and y\<^sub>0 :: "nat" assumes "total_cons s"   and "s b \<down>= i"   and "\<phi> i y\<^sub>0 \<down>"   and "\<nexists>y. \<phi> i y \<down> \<and> e_length b \<le> y" shows "\<nexists>x. eval (total_cons.r_search_enum s) [x, b] \<down>= 0"
is_sup_def: fixes x :: "int"   and y :: "int"   and s :: "int" shows "int.is_sup x y s = (x \<le> s \<and> y \<le> s \<and> (\<forall>z. x \<le> z \<and> y \<le> z \<longrightarrow> s \<le> z))"
eventually_frequently_simps_8: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and F :: "'e filter" shows "(\<forall>\<^sub>F x in F. P x \<and> Q x) = (eventually P F \<and> eventually Q F)"
candidate_values_def: fixes z :: "'a trm"   and CC :: "'a eclause"   and C' :: "'a literal set"   and C :: "'a literal set"   and s :: "'a trm"   and L :: "'a literal"   and L' :: "'a literal"   and \<sigma> :: "('a \<times> 'a trm) list"   and t' :: "'a trm"   and s' :: "'a trm"   and t :: "'a trm"   and S :: "'a eclause set" shows "candidate_values z CC C' C s L L' \<sigma> t' s' t S = (CC \<in> S \<and> t \<notin> subst_set (trms_ecl CC) \<sigma> \<and> trms_irreducible CC \<sigma> S t \<and> C' = cl_ecl CC \<and> (s, t) \<in> trm_ord \<and> ((s, t) \<in> trm_ord \<longrightarrow> z = trm_rep s S) \<and> orient_lit_inst L' t' s' pos \<sigma> \<and> sel C' = {} \<and> L' \<in> C' \<and> maximal_literal L C \<and> L = subst_lit L' \<sigma> \<and> C = subst_cl C' \<sigma> \<and> ground_clause C \<and> t = t' \<lhd> \<sigma> \<and> s = s' \<lhd> \<sigma> \<and> finite C' \<and> smaller_lits_are_false t C S \<and> maximal_literal_is_unique t s C' L' S \<sigma>)"
unify_computes_MGU: fixes M :: "'b trm"   and N :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "unify M N = Some \<sigma>" shows "MGU \<sigma> M N"
length_restrict: fixes A :: "nat set"   and v :: "'a option list" shows "length (restrict A v) = length v"
indRelRSTPO_trans: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "trans (indRelRSTPO SRel TRel)"
sub2: fixes C :: "'a set"   and x :: "'a" assumes "C \<in> \<F> \<and> x \<in> C" shows "\<exists>D. D \<in> \<F>' \<and> x \<in> D \<and> D \<subseteq> C"
less_imp_inext_le: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a < b"   and "a \<in> I"   and "b \<in> I" shows "inext a I \<le> b"
neg_cycle_empty: fixes n :: "nat"   and v :: "'d \<Rightarrow> nat"   and xs :: "nat list"   and i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'e DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "set xs \<subseteq> {0..n}"   and "i \<le> n"   and "len M i i xs < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
connectedin_closedin: fixes X :: "'c topology"   and S :: "'c set" shows "connectedin X S = (S \<subseteq> topspace X \<and> (\<nexists>E1 E2. closedin X E1 \<and> closedin X E2 \<and> S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 \<inter> S = {} \<and> E1 \<inter> S \<noteq> {} \<and> E2 \<inter> S \<noteq> {}))"
chine_def: shows "chine \<equiv> SOME w. is_induced_map w"
function_add_def: fixes S :: "'a set"   and R :: "('b, 'c) ring_scheme"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "function_add S R f g = (\<lambda>x\<in>S. f x \<oplus>\<^bsub>R\<^esub> g x)"
\<beta>_def: fixes n :: "nat"   and fs :: "'a vec list"   and l :: "nat" shows "gram_schmidt.\<beta> n fs l = gram_schmidt.Gramian_determinant n fs (Suc l) / gram_schmidt.Gramian_determinant n fs l"
seq_tail1_1: fixes s :: "'a"   and x :: "'b \<times> 'a"   and xs :: "('b \<times> 'a) list"   and lars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, x # xs) \<in> seq lars" shows "(snd x, xs) \<in> seq lars"
sign_changes_cong': fixes ps :: "real poly list"   and x :: "real"   and y :: "real" assumes "\<forall>p\<in>set ps. sgn (poly p x) = sgn (poly p y)" shows "sign_changes ps x = sign_changes ps y"
MkArr_in_hom: fixes A :: "'a"   and B :: "'a"   and f :: "'a set" assumes "A \<in> Collect obj"   and "B \<in> Collect obj"   and "f \<in> Maps.Hom A B" shows "Maps.in_hom (Maps.MkArr A B f) (Maps.MkIde A) (Maps.MkIde B)"