hcju/isabelleps huggingface dataset:
wellformed_items_\<J>_k_u_if_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "LocalLexing.wellformed_items \<RR> Doc (LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc k 0)" shows "LocalLexing.wellformed_items \<RR> Doc (LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc k u)"
continuous_on_Times_swap: fixes Y :: "'b set"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd" assumes "continuous_on (Y \<times> X) (\<lambda>(y, x). f x y)" shows "continuous_on (X \<times> Y) (\<lambda>(x, y). f x y)"
mod_poly_def: fixes f :: "'a poly"   and g :: "'a poly" shows "f mod g = (if g = 0 then f else pseudo_mod (smult (((1::'a) / lead_coeff g) ^ (Suc (degree f) - degree g)) f) g)"
hom_one: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
assms_2: shows "\<forall>\<sigma>. \<not> strategy_winning_by_Odd \<sigma> p"
restrict_codomain: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and Z :: "'c set set" assumes "ChamberComplexMorphism X Y f"   and "ChamberComplex.ChamberSubcomplex Y Z"   and "order.greater_eq Z (f \<turnstile> X)" shows "ChamberComplexMorphism X Z f"
Int_assoc: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
compatible_induce_subgraph: fixes G :: "('c, 'd) pre_digraph"   and S :: "'c set" shows "compatible (G \<restriction> S) G"
solution_def: shows "solution = (\<lambda>(a, b). a\<^sup>2 = (1::'a) + of_nat D * b\<^sup>2)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" shows "\<guillemotleft>\<tau> f : F a \<rightarrow>\<^sub>B G b\<guillemotright>"
antisym: fixes b :: "'b"   and a :: "'b" assumes "ordb b a"   and "ordb a b" shows "a = b"
cut_less_Int_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t = I \<inter> {..<t}"
alphaAbs_qAbs_imp_alphaAbs_all_distinct_qAFresh: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d"   and xs' :: "'e"   and x' :: "'d"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "alphaAbs_all_distinct_qAFresh xs x X xs' x' X'"
intro: fixes G :: "('a, 'b) graph" assumes "fst ` edges G \<subseteq> nodes G"   and "snd ` snd ` edges G \<subseteq> nodes G" shows "valid_graph G"
sums_def_le: fixes f :: "nat \<Rightarrow> 'e"   and s :: "'e" shows "f sums s = (\<lambda>n. sum f {..n}) \<longlonglongrightarrow> s"
left_right_inverse_eq: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and h :: "'c \<Rightarrow> 'b" assumes "f \<circ> g = id"   and "g \<circ> h = id" shows "f = h"
path_split_3: fixes n :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as @ a # as'\<rightarrow>* n'" shows "targetnode a -as'\<rightarrow>* n'"
struct_subst_type_command: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta>1 :: "nat \<Rightarrow> Syntax.type"   and c :: "cmd"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and E :: "ctxt"   and U :: "Syntax.type"   and T1 :: "Syntax.type"   and \<alpha> :: "nat"   and \<beta> :: "nat" assumes "\<Gamma> , \<Delta>1 \<turnstile>\<^sub>C c"   and "\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : U \<Leftarrow> T1"   and "\<Delta>1 = \<Delta>\<langle>\<alpha>:T1\<rangle>" shows "\<Gamma> , \<Delta>\<langle>\<beta>:U\<rangle> \<turnstile>\<^sub>C c[\<alpha>=\<beta> liftM_ctxt E \<beta>]\<^sup>C"
dist_ge_\<mu>: fixes v :: "'a" assumes "v \<in> verts G"   and "enum v \<noteq> \<infinity>"   and "dist v \<noteq> - \<infinity>"   and "\<mu> c s s = ereal 0"   and "dist s = 0"   and "\<And>u. \<lbrakk>u \<in> verts G; u \<noteq> s; enum u \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> enum u \<noteq> enat 0" shows "\<mu> c s v \<le> dist v"
chine_assoc'_props_1: shows "\<guillemotleft>chine_assoc' : \<mu>\<nu>\<pi>.chine \<rightarrow>\<^sub>C \<mu>\<nu>_\<pi>.chine\<guillemotright>"
corresponding_union_find_empty: shows "corresponding_union_find (per_init V) {}"
MClkidle: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan" shows "\<turnstile> \<not> $Calling send p \<and> $(cst!p) = #clkA \<longrightarrow> \<not> MClkNext send rcv cst p"
inC_\<psi>: fixes b :: "'c"   and a :: "'c"   and x :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "C.ide b"   and "C.ide a"   and "x \<in> inC ` C.hom b a" shows "inC (HomC.\<psi> (b, a) x) = x"
curry_forall_eq: fixes P :: "('c \<times> 'd \<Rightarrow> 'e) \<Rightarrow> bool" shows "(\<forall>f. P f) = (\<forall>f. P (\<lambda>(x, y). f x y))"
cpI1: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and P :: "('a state \<times> 'a state \<Rightarrow> 'd) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'b" assumes "\<forall>X \<tau>. f X \<tau> = f (\<lambda>_. X \<tau>) \<tau>"   and "cp P" shows "cp (\<lambda>X. f (P X))"
Un_upper2: fixes B :: "'e set"   and A :: "'e set" shows "B \<subseteq> A \<union> B"
fg: fixes x :: "'a"   and y :: "'b" assumes "x \<in> set xs1"   and "y \<in> set ys1" shows "f x y = g x y"
Phase1or2Write_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" shows "Phase1or2Write s s' p d = (phase s p \<in> {1, 2} \<and> disk s' = (disk s)(d := (disk s d)(p := dblock s p)) \<and> disksWritten s' = (disksWritten s)(p := disksWritten s p \<union> {d}) \<and> inpt s' = inpt s \<and> outpt s' = outpt s \<and> phase s' = phase s \<and> dblock s' = dblock s \<and> blocksRead s' = blocksRead s)"
cont_nec_fact2_4: fixes v :: "i" shows "[\<^bold>\<not>WeaklyContingent L in v]"
assms_1: shows "cbox c d \<subseteq> box a b"
eval_models: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a"   and \<phi> :: "('a, 'p) Kform"   and s :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t"   and "subjective a \<phi>"   and "s \<in> toSet (snd ec)" shows "eval envVal envObs ec \<phi> = spr_repMC (toSet (fst ec)), s \<Turnstile> \<phi>"
bound0_qf: fixes p :: "fm" assumes "bound0 p" shows "qfree p"
idom_potent_nonzero: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "x \<noteq> \<zero>" shows "x^\<^bsup>R n\<^esup> \<noteq> \<zero>"
assms_4: shows "isLastAssertedLiteral (opposite l) (oppositeLiteralList c) (elements M)"
mgp_vertices3: fixes g :: "graph"   and f :: "face" assumes "minGraphProps g"   and "f \<in> \<F> g" shows "3 \<le> |vertices f|"
nsqn_invalidate_other: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option" assumes "dip \<in> kD rt"   and "dip \<notin> dom dests" shows "nsqn (invalidate rt dests) dip = nsqn rt dip"
Not_eqvt: fixes p :: "perm"   and x :: "('a, 'b, 'c) formula" shows "p \<bullet> Formula.Not x = Formula.Not (p \<bullet> x)"
STL6: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>F \<and> \<diamond>\<box>G \<longrightarrow> \<diamond>\<box>(F \<and> G)"
tminus_sound: fixes P :: "'a \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "sound (\<lambda>s. P s \<ominus> c)"
insert_Diff_if: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "insert x A - B = (if x \<in> B then A - B else insert x (A - B))"
induct: fixes P :: "('b, 'c) Combinators list \<Rightarrow> bool"   and a0 :: "('b, 'c) Combinators list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x v va. P (v # va) \<Longrightarrow> P (x # v # va)" shows "P a0"
infinite_disjoint_family_imp_infinite_UNION: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c set" assumes "infinite A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> {}"   and "disjoint_family_on f A" shows "infinite (\<Union> (f ` A))"
Diag_TensorE_4: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "Diag u"
less_le_not_le: fixes x :: "'a list"   and y :: "'a list" shows "strict_border x y = (border x y \<and> \<not> border y x)"
inv_approx_preserv2: fixes S :: "'a set"   and X :: "'a set"   and s :: "nat"   and x :: "'a" assumes "inv_approx S X s"   and "x \<in> V - X" shows "inv_approx (insert x S) (X \<union> neighbors x \<union> {x}) (max (card (neighbors x \<union> {x} - X)) s)"
bij_lift: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw ((`) f) (Pow A) (Pow B)"
f_f'_def_2: shows "f' \<equiv> \<lambda>z. \<Sum>n. diffs ((gchoose) a) n * z ^ n"
exI_realizer: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'b"   and x :: "'c" assumes "P y x" shows "P (snd (x, y)) (fst (x, y))"
\<Delta>_def: fixes f :: "'d \<Rightarrow> 'c"   and k :: "int" shows "\<Delta> f k = f (of_int (k + 1)) - f (of_int k)"
prod_zero_def: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" shows "prod_zero I A = (\<lambda>x\<in>I. \<zero>\<^bsub>A x\<^esub>)"
S: fixes h :: "'a" assumes "h \<in> carrier G" shows "inv g \<otimes> h \<otimes> g \<in> carrier G"
valid_head_and_tail_valid_plan: fixes h :: "'d"   and PROB :: "'d set"   and as :: "'d list" assumes "h \<in> PROB"   and "as \<in> valid_plans PROB" shows "h # as \<in> valid_plans PROB"
BigOr_is_not_cnf': fixes ls :: "'b formula list" assumes "\<forall>l\<in>set ls. is_lit_plus l" shows "is_cnf \<^bold>\<Or>ls"
negligible_lowdim: fixes S :: "'a set" assumes "dim S < DIM('a)" shows "negligible S"
add_monom: fixes a :: "'c"   and n :: "nat"   and b :: "'c" shows "monom a n + monom b n = monom (a + b) n"
iFROM_mult: fixes n :: "nat"   and k :: "nat" shows "[n\<dots>] \<otimes> k = [ n * k, mod k ]"
f_bit_sur: fixes a :: "'a"   and b :: "'b"   and c :: "'c" assumes "f' a = f b c" shows "\<exists>c'. c = f' c'"
cases: fixes a :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set" assumes "a \<in> fake bad IK chan" obtains M :: "chan" where "a = Ma__" and "Ma__ \<in> chan"   | M :: "msg" and c :: "tag" and A :: "agent" and B :: "agent" where "a = Chan c A B M" and "M \<in> IK" and "c = insec \<or> c = confid \<or> A \<in> bad \<or> B \<in> bad"
index_ends: fixes h :: "real \<Rightarrow> complex" assumes "\<forall>t\<in>{0<..<1}. Re (h t) \<noteq> Re z"   and "valid_path h" shows "cindex_pathE h z = jumpF_pathstart h z - jumpF_pathfinish h z"
ideD_1: fixes a :: "'a" assumes "hhom\<^sub>C.ide a" shows "hhom\<^sub>C.arr a"
path_Entry_target_1: 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 n :: "'b"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as Entry" shows "n = Entry"
antisym_less_nmultiset: fixes X :: "'a nmultiset"   and Y :: "'a nmultiset" assumes "X < Y"   and "Y < X" shows "False"
rcliff_1: shows "(s, t) \<in> (E a)\<inverse>"
isometryD: fixes U :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" assumes "isometry U" shows "U* o\<^sub>C\<^sub>L U = id_cblinfun"
iuf_1: shows "a1 \<in> fst ` set (routing_port_ranges tbl wordinterval_UNIV) \<and> b1 = wordinterval_Union (map snd (filter (\<lambda>x. a1 = fst x) (routing_port_ranges tbl wordinterval_UNIV)))"
empty: fixes i :: "'i" assumes "i \<in> I"   and "G i = {}" shows "\<mu> i = 0"
pdigraph_base_axioms: shows "pdigraph_base \<alpha> (ZFC_in_HOL.set {j}) (\<lambda>i. \<CC>)"
authKeys_simp: fixes K :: "nat"   and A :: "agent"   and K' :: "nat"   and Peer :: "agent"   and Ta :: "nat"   and evs :: "event list" assumes "K \<in> authKeys (Says Kas A (Crypt (shrK A) \<lbrace>Key K', Agent Peer, Number Ta, Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key K', Number Ta\<rbrace>\<rbrace>) # evs)" shows "K = K' \<or> K \<in> authKeys evs"
hd_SubobjsR: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list" assumes "Subobjs\<^sub>R P C Cs" shows "\<exists>Cs'. Cs = C # Cs'"
mk_nat_vecs_bound: fixes n :: "nat" shows "\<forall>xs\<in>set (mk_nat_vecs n). \<forall>x\<in>set xs. x < 2"
bot_unique: fixes a :: "'a" shows "(a \<le> (zero_class.zero::'a)) = (a = (zero_class.zero::'a))"
Ints_0: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_1 one times plus zero minus uminus" shows "zero \<in> ring_1.Ints one plus zero minus"
ennreal_minus_eq_top: fixes a :: "ennreal"   and b :: "ennreal" shows "(a - b = top) = (a = top)"
Lim_linear: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter"   and h :: "'b \<Rightarrow> 'c" assumes "(f \<longlongrightarrow> l) F"   and "linear h" shows "((\<lambda>x. h (f x)) \<longlongrightarrow> h l) F"
wf_ipath_epath: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and t :: "('e \<times> 'd) tree"   and steps :: "('e \<times> 'd) stream" assumes "RuleSystem_Defs.wf eff rules t"   and "ipath t steps" shows "RuleSystem_Defs.epath eff rules steps"
zero_ow_def: fixes U :: "'a set"   and zero :: "'a" shows "zero_ow U zero \<equiv> neutral_ow U zero"
prems: shows "s\<^sub>0 = (h\<^sub>0, l\<^sub>0(V := None), sh\<^sub>0)"
equivalence_pseudofunctor_axioms: shows "equivalence_pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
span_on_def: fixes b :: "'a set" shows "subspace.span b = {\<Sum>a\<in>t. r a *\<^sub>R a |t r. finite t \<and> t \<subseteq> b}"
Rep_hyp2_cltn2: fixes J :: "cltn2"   and p :: "hyp2" assumes "is_K2_isometry J" shows "Rep_hyp2 (hyp2_cltn2 p J) = apply_cltn2 (Rep_hyp2 p) J"
Diff_eq_empty_iff: fixes A :: "'c set"   and B :: "'c set" shows "(A - B = {}) = (A \<subseteq> B)"
SD_preorderD_3: fixes q :: "'a pmf"   and p :: "'a pmf"   and x :: "'a" assumes "SD le q p"   and "x \<in> carrier" shows "measure_pmf.prob q (preferred_alts le x) \<le> measure_pmf.prob p (preferred_alts le x)"
pre: shows "wf_mbuf2' \<sigma> P V j n R \<phi> \<psi> buf"
memory_write_privilege: fixes r :: "'b sparc_state option"   and asi :: "word8"   and addr :: "word32"   and byte_mask :: "word4"   and data :: "word32"   and s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "r = memory_write asi addr byte_mask data s \<and> r = Some s' \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
rt_fresher_trans_rt_strictly_fresher': 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"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubset>\<^bsub>dip\<^esub> rt3" shows "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt3"
Seq: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C\<^sub>2 :: "com"   and R :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} C\<^sub>1 { Q}"   and "\<turnstile>\<^sub>3\<^sub>a {Q} C\<^sub>2 { R}" shows "\<turnstile>\<^sub>3\<^sub>a {P} C\<^sub>1;; C\<^sub>2 { R}"
luckyFind_eval': fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (luckyFind' var L F) (xs @ \<Gamma>))"
true_clss_mono_strong: fixes DD :: "'a literal multiset set"   and CC :: "'a literal multiset set"   and I :: "'a set" assumes "\<forall>D\<in>DD. \<exists>C\<in>CC. C \<subseteq># D"   and "I \<Turnstile>s CC" shows "I \<Turnstile>s DD"
strong_law_of_large_numbers_iid: fixes X :: "nat \<Rightarrow> (nat \<Rightarrow> bool) \<Rightarrow> real" assumes "indep_vars (\<lambda>_. borel) X UNIV"   and "\<And>i. distr M borel (X i) = distr M borel (X 0)"   and "integrable M (X 0)" shows "AE x in M. (\<lambda>n. (\<Sum>i<n. X i x) / real n) \<longlonglongrightarrow> expectation (X 0)"
trcl_rev_uncons: fixes c :: "'c"   and w :: "'d list"   and e :: "'d"   and c' :: "'c"   and T :: "('c \<times> 'd \<times> 'c) set" assumes "(c, w @ [e], c') \<in> trcl T" shows "\<exists>ch. (c, w, ch) \<in> trcl T \<and> (ch, e, c') \<in> T"
step_abs: fixes s :: "'C"   and s' :: "'C" assumes "s \<in> invar_addc"   and "s \<in> condc"   and "\<alpha> s \<in> wa_invar WAA"   and "(s, s') \<in> stepc" shows "(\<alpha> s, \<alpha> s') \<in> wa_step WAA"
p1_def: fixes p :: "'sz"   and x :: "'sz" shows "p1 p x = $\<phi> p (singleton (Var x))"
map_filter_def: fixes f :: "'d \<Rightarrow> 'c option"   and xs :: "'d list" shows "List.map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
ALL_LIST_def: fixes A :: "bool list" shows "ALL_LIST A \<equiv> \<forall>x\<in>set A. x"
integrable_on_indicator: fixes S :: "'a set"   and T :: "'a set" shows "(indicat_real S integrable_on T) = (S \<inter> T \<in> lmeasurable)"
Mojmir_reject: shows "(\<not> accept\<^sub>M) = (\<exists>\<^sub>\<infinity>x. \<not> token_succeeds x)"
canonical_factorization: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and V :: "'a \<Rightarrow> 'b"   and t :: "'a local.term" assumes "evaluation_map C D T \<alpha> \<iota> V"   and "monoidal_language.Arr C t" shows "(D (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.red C (monoidal_language.Cod C t))) (evaluation_map.eval D T \<alpha> \<iota> V t) = D (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.Diagonalize C t)) (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.red C (monoidal_language.Dom C t)))) = (evaluation_map.eval D T \<alpha> \<iota> V t = D (category.inv D (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.red C (monoidal_language.Cod C t)))) (D (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.Diagonalize C t)) (evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.red C (monoidal_language.Dom C t)))))"
induct: fixes P :: "'a list \<Rightarrow> 'a vec \<Rightarrow> 'a vec list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a vec"   and a2 :: "'a vec list" assumes "\<And>wits w. P wits w []"   and "\<And>wits w u us. (\<And>x. x = w \<bullet> u / \<parallel>u\<parallel>\<^sup>2 \<Longrightarrow> P (x # wits) w us) \<Longrightarrow> P wits w (u # us)" shows "P a0 a1 a2"
4: shows "length M2 \<le> j \<longrightarrow> (i < row_length M * row_length M2 \<and> j - length M2 < length M * length M2) \<and> mat (row_length M) (length M) M \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> (M \<otimes> M2) ! (j - length M2) ! i = M ! ((j - length M2) div length M2) ! (i div row_length M2) * M2 ! ((j - length M2) mod length M2) ! (i mod row_length M2)"
partn_lst_less: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and \<alpha> :: "V list"   and n :: "nat"   and \<alpha>' :: "V list" assumes "Partitions.partn_lst r B \<alpha> n"   and "length \<alpha>' = length \<alpha>"   and "list.set \<alpha>' \<subseteq> ON"   and "\<And>i. i < length \<alpha> \<Longrightarrow> \<alpha>' ! i \<le> \<alpha> ! i"   and "wf r"   and "trans r"   and "total_on B r"   and "small B" shows "Partitions.partn_lst r B \<alpha>' n"
next_tame0_subset_plane: fixes p :: "nat"   and g :: "graph" shows "set (next_tame0\<^bsub>p\<^esub> g) \<subseteq> set (next_plane\<^bsub>p\<^esub> g)"
assms_5: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
lookup_union1: fixes k :: "'c"   and ys :: "'c \<rightharpoonup>\<^sub>f 'd"   and xs :: "'c \<rightharpoonup>\<^sub>f 'd" assumes "k |\<in>| fmdom ys" shows "lookup (xs ++\<^sub>f ys) k = lookup ys k"
sign_simps_27: fixes an :: "real"   and x :: "'b"   and y :: "'b" shows "an *\<^sub>R (x - y) = an *\<^sub>R x - an *\<^sub>R y"
intros_3: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> parts H" shows "Y \<in> parts H"
h2: shows "\<forall>t. length (nReturn i t) \<le> Suc 0"
wf\<^sub>t\<^sub>r\<^sub>m_code: fixes t :: "('fun, 'a) Term.term" shows "wf\<^sub>t\<^sub>r\<^sub>m t = wf\<^sub>t\<^sub>r\<^sub>m' arity t"
assms_1: shows "is_tangle_diagram (w2 * ws2)"
lifelock_free_def: fixes P :: "'b process" shows "lifelock_free P \<equiv> CHAOS UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
le_imp_less_or_eq: fixes x :: "'a"   and y :: "'a" assumes "cmp.le x y" shows "cmp.lt x y \<or> x = y"
mkInv_def: fixes MI :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "mkInv MI = (\<lambda>s0 s t. s = mkState s0 \<longrightarrow> MI s0 t)"
extra_wt_App: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym" shows "kbo_std.extra_wt ground_heads_var \<delta> arity_sym wt_sym \<rho> (App s t) = kbo_std.extra_wt ground_heads_var \<delta> arity_sym wt_sym \<rho> s + kbo_std.extra_wt ground_heads_var \<delta> arity_sym wt_sym \<rho> t"
assms: shows "\<aa> \<in>\<^sub>\<circ> Vset \<alpha>"
when_neq_zero: fixes zero :: "'b"   and a :: "'b"   and P :: "bool" shows "(zero.when zero a P \<noteq> zero) = (P \<and> a \<noteq> zero)"
axiom11_F: fixes x :: "('l, 'm, 'n, 'k) F" shows "|F'set2 x| \<le>o F'bd"
assms: shows "\<forall>x\<in>A. comm_monoid_mult_class.coprime (g a) (g x)"
rangeI: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "f x \<in> range f"
list_length_hint3: fixes l :: "'a list" assumes "length l = Suc 0" shows "l \<noteq> []"