hcju/isabelleps huggingface dataset:
parent_child_rel_parent_in_heap: fixes parent :: "(_) object_ptr"   and child_ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "(parent, child_ptr) \<in> parent_child_rel h" shows "parent |\<in>| object_ptr_kinds h"
map_fun_def: fixes f :: "'b \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd"   and h :: "'e \<Rightarrow> 'f" shows "map_fun f g h = g \<circ> h \<circ> f"
order_linear: fixes a :: "'a" shows "Polynomial.order a [:- a, 1::'a:] = Suc 0"
take_Suc_j: shows "take (Suc j) ((Array.get h\<^sub>2 a)[i' := key]) = take i' (Array.get h\<^sub>2 a) @ key # take (j - i') (drop (Suc i') (Array.get h\<^sub>2 a))"
m_lcancel: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "domain R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "(a \<otimes>\<^bsub>R\<^esub> b = a \<otimes>\<^bsub>R\<^esub> c) = (b = c)"
intro: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'e set"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" assumes "finite_dimensional_vector_space s1 B1"   and "vector_space s2" shows "finite_dimensional_vector_space_pair_1 s1 B1 s2"
disjoint_zero_sym: fixes x :: "'a" shows "(0::'a) ## x"
exp_conj_comm: fixes a :: "'b \<Rightarrow> real"   and b :: "'b \<Rightarrow> real" shows "a && b = b && a"
map_simp: fixes f :: "'j" assumes "J.arr f" shows "A.map f = a"
simps: fixes cl :: "(String.literal \<times> String.literal) set"   and uu :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" shows "subclass (cl, uu) = cl"
comp_ide_self: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "local.comp a a = a"
divide_cancel_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a / c = b / c) = (c = (0::'a) \<or> a = b)"
count_pre_surj: fixes B :: "'c set"   and w :: "'d"   and Y :: "'d \<Rightarrow> 'c" assumes "countable B"   and "w \<in> Y -` B" shows "\<exists>i. w \<in> countable_preimages B Y i"
empty_neutral_2: fixes x :: "'f multiset" shows "x + {#} = x"
return_bind: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'b) set \<times> bool" shows "return x >>= f = f x"
indexle_trans: fixes k'' :: "nat"   and u'' :: "nat"   and k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" assumes "indexle k'' u'' k' u'"   and "indexle k' u' k u" shows "indexle k'' u'' k u"
insert_inc1: fixes A :: "'h set"   and a :: "'h" shows "A \<subseteq> insert a A"
rbt_comp_union_with_key_def: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt" shows "local.rbt_comp_union_with_key f t1 t2 = (case RBT_Impl.compare_height t1 t1 t2 t2 of compare.LT \<Rightarrow> RBT_Impl.fold (local.rbt_comp_insert_with_key (\<lambda>k v w. f k w v)) t1 t2 | compare.GT \<Rightarrow> RBT_Impl.fold (local.rbt_comp_insert_with_key f) t2 t1 | compare.EQ \<Rightarrow> rbtreeify (local.comp_sunion_with f (RBT_Impl.entries t1) (RBT_Impl.entries t2)))"
nn_integral_measure_pmf: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> ennreal" shows "integral\<^sup>N (measure_pmf p) f = \<integral>\<^sup>+ x. ennreal (pmf p x) * f x \<partial>count_space UNIV"
member_insert: fixes x :: "'b"   and xs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "DList_Set.member (DList_Set.insert x xs) = (DList_Set.member xs)(x := True)"
memb_equiv_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and t :: "'b local.term"   and u :: "'b local.term" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "t \<in> f"   and "u \<in> f" shows "monoidal_language.Diagonalize C t = monoidal_language.Diagonalize C u"
true_conj_zero: fixes P :: "'b \<Rightarrow> bool" shows "(P \<and> false) = false"
that_2: fixes x :: "'a"   and xsa :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "xa = x # xsa"   and "xb = ys"   and "xc = x # zs"   and "append2 xsa ys zs" shows "thesis"
poly_1: fixes x :: "'b" shows "poly 1 x = (1::'b)"
hyps_1: shows "n_nodes (reflect t1) = n_nodes t1"
DynPDG_ddep_edge: fixes n :: "'node"   and V :: "'var"   and n' :: "'node"   and as :: "'edge list" assumes "n influences V in n' via as" shows "n -{V}as\<rightarrow>\<^sub>d\<^sub>d n'"
rel_reflp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "reflp R" shows "reflp (fmrel R)"
mult_less_cancel_right_disj: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a c) (times b c) = (less zero c \<and> less a b \<or> less c zero \<and> less b a)"
sp: shows "splay a \<langle>ls, x, rs\<rangle> = \<langle>l, e, r\<rangle>"
assms: fixes i :: "nat" assumes "i \<in> set X0" shows "i < length vs"
prod_div_sub: fixes A :: "nat set"   and B :: "nat set"   and f :: "nat \<Rightarrow> complex" assumes "finite A"   and "B \<subseteq> A"   and "\<forall>b\<in>B. f b \<noteq> 0" shows "prod f (A - B) = prod f A / prod f B"
inductive_on_accessible_on_conv: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "inductive_on P A = (\<forall>x\<in>A. accessible_on P A x)"
lists_eq_set: fixes A :: "'e set" shows "lists A = {xs. set xs \<subseteq> A}"
stable_rank_tower: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and j :: "nat"   and i :: "nat"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "j \<le> i"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some j"   and "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w y n = Some i"   and "semi_mojmir.stable_rank \<Sigma> \<delta> q\<^sub>0 w y i" shows "semi_mojmir.stable_rank \<Sigma> \<delta> q\<^sub>0 w x j"
sel_5: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "occurs_on (RecvMarker x51 x52 x53) = x52"
fset_rev_mp: fixes c :: "'x"   and A :: "'x fset"   and B :: "'x fset" assumes "c |\<in>| A"   and "A |\<subseteq>| B" shows "c |\<in>| B"
M3_def: shows "M3 \<equiv> PM \<and> M3_axioms"
inv_o_cancel: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f" shows "inv f \<circ> f = id"
le_add_diff_inverse2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus (minus a b) b = a"
self_join1: fixes xs :: "'b option list"   and ys :: "'b option list"   and zs :: "'b option list" assumes "join1 (xs, ys) \<noteq> Some xs" shows "join1 (zs, ys) \<noteq> Some xs"
semiring_Gcd: shows "class.semiring_Gcd (1::'a) (0::'a) (*) normalization_euclidean_semiring_class.gcd normalization_euclidean_semiring_class.lcm normalization_euclidean_semiring_class.Gcd normalization_euclidean_semiring_class.Lcm (div) (+) (-) unit_factor normalize"
s_cf_expr: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "pol_coeff S (s_cf R S X p) \<and> p = polyn_expr R X (fst (s_cf R S X p)) (s_cf R S X p) \<and> snd (s_cf R S X p) (fst (s_cf R S X p)) \<noteq> \<zero>\<^bsub>S\<^esub>"
clinear_def: fixes f :: "'b \<Rightarrow> 'c" shows "clinear f \<equiv> Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) f"
DDDC: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans f" shows "CatDom (NTDom f) = CatDom (NTCod f)"
Skip_ir_valid: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c' :: "com" assumes "\<And>t t'. Q t t' \<Longrightarrow> \<exists>s'. P t s' \<and> (c', s') \<Rightarrow> t'" shows "ir_valid P SKIP c' Q"
in_lex_prod_alt: fixes x :: "'e \<times> 'f"   and y :: "'e \<times> 'f"   and r :: "('e \<times> 'e) set"   and s :: "('f \<times> 'f) set" shows "((x, y) \<in> r <*lex*> s) = ((fst x, fst y) \<in> r \<or> fst x = fst y \<and> (snd x, snd y) \<in> s)"
NUM_closed_subseteq: fixes U :: "(nat \<Rightarrow> nat option) set"   and V :: "(nat \<Rightarrow> nat option) set" assumes "U \<in> NUM"   and "V \<subseteq> U" shows "V \<in> NUM"
subrelI: fixes r :: "('e \<times> 'f) set"   and s :: "('e \<times> 'f) set" assumes "\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s" shows "r \<subseteq> s"
c1: shows "{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y} \<inter> {(x, y). swaps ss init ! s = x \<and> swaps ss init ! Suc s = y \<or> swaps ss init ! s = y \<and> swaps ss init ! Suc s = x} = {(x, y). x < y \<and> swaps ss init ! s = x \<and> swaps ss init ! Suc s = y} \<union> {(x, y). x < y \<and> swaps ss init ! s = y \<and> swaps ss init ! Suc s = x}"
part_def: fixes X :: "'b set" shows "part X = (K \<circ> I = I)"
simps: fixes x :: "'b" shows "the_af (Msg x) = x"
isLub_def: fixes S :: "'b set"   and po :: "'b potype" shows "isLub S po = (\<lambda>L. L \<in> pset po \<and> (\<forall>y\<in>S. (y, L) \<in> order po) \<and> (\<forall>z\<in>pset po. (\<forall>y\<in>S. (y, z) \<in> order po) \<longrightarrow> (L, z) \<in> order po))"
dB: fixes d :: "real"   and t :: "real"   and x :: "'a" assumes "0 < d"   and "d < dB"   and "t \<in> cball t0 d"   and "x \<in> cball x0 d" shows "norm (f t x) \<le> B"
openI: fixes S :: "'b set" assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" shows "open S"
measurable_Pair: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and M1 :: "'c measure"   and g :: "'b \<Rightarrow> 'd"   and M2 :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M M1"   and "g \<in> M \<rightarrow>\<^sub>M M2" shows "(\<lambda>x. (f x, g x)) \<in> M \<rightarrow>\<^sub>M M1 \<Otimes>\<^sub>M M2"
completenessForUNSAT: fixes F0 :: "Literal list list"   and decisionVars :: "nat set"   and state0 :: "State"   and state :: "State" assumes "vars F0 \<subseteq> decisionVars"   and "\<not> satisfiable F0"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "isFinalState state F0 decisionVars" shows "getConflictFlag state = True \<and> getC state = []"
ltln_expand_WeakUntil: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = \<xi> \<Turnstile>\<^sub>n \<psi> or\<^sub>n (\<phi> and\<^sub>n X\<^sub>n (\<phi> W\<^sub>n \<psi>))"
set_l_eq: shows "set (g_to_list m) = map_to_set (\<alpha> m)"
cong_axioms: shows "BNF_Corec.cong rel eval retr"
ipath_mkTree_Saturated: fixes s :: "'state"   and rs :: "'rule stream"   and steps :: "('state \<times> 'rule) stream" assumes "s \<in> S"   and "fair rs"   and "ipath (mkTree rs s) steps" shows "Saturated steps"
expands_to_lift: fixes f :: "'a \<Rightarrow> 'b"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list"   and c :: "real \<Rightarrow> real"   and C :: "'a" assumes "is_lifting f basis basis'"   and "(c expands_to C) basis" shows "(c expands_to f C) basis'"
comp_assoc: fixes h :: "'a"   and g :: "'a"   and f :: "'a" shows "(h \<cdot>\<^sub>S g) \<cdot>\<^sub>S f = h \<cdot>\<^sub>S g \<cdot>\<^sub>S f"
list_update_nonempty: fixes xs :: "'c list"   and k :: "nat"   and x :: "'c" shows "(xs[k := x] = []) = (xs = [])"
cong: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set" shows "Reachability_Problem.F_sup K S S1 S2 \<equiv> Reachability_Problem.F_sup K S S1 S2"
prime_factorization_times_prime: fixes x :: "'a"   and p :: "'a" assumes "x \<noteq> (0::'a)"   and "prime p" shows "prime_factorization (p * x) = {#p#} + prime_factorization x"
assms_3: shows "\<forall>var\<in>V. var < start"
mat_def: fixes k :: "'c" shows "mat k = (\<chi>i j. if i = j then k else (0::'c))"
ln_bounds_2: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x"   and "x < 1" shows "ln (x + 1) \<le> (\<Sum>i = 0..<2 * n + 1. (- 1) ^ i * (1 / real (i + 1)) * x ^ Suc i)"
fixedaction_Empty: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and ms :: "'c match_expr list" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m Empty) ms) Undecided = Undecided"
projection_idempotent: fixes l :: "'b list"   and X :: "'b set" shows "l \<upharpoonleft> X = l \<upharpoonleft> X \<upharpoonleft> X"
wt_ge_\<delta>: 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 s :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym" shows "\<delta> \<le> kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s"
Sretr_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "Sretr theta \<equiv> {(c, d). matchC_C theta c d \<and> matchT_T c d}"
assume_SE_def: fixes P :: "'d \<Rightarrow> bool" shows "assume\<^sub>S\<^sub>E P = (\<lambda>\<sigma>. if \<exists>\<sigma>. P \<sigma> then Some ((), SOME \<sigma>. P \<sigma>) else None)"
R_lin_independentD_all_scalars_nth: fixes rs :: "'r list"   and ms :: "'m list"   and k :: "nat" assumes "set rs \<subseteq> R"   and "set ms \<subseteq> M"   and "R_lin_independent ms"   and "rs \<bullet>\<cdot> ms = (0::'m)"   and "k < min (length rs) (length ms)" shows "rs ! k = (0::'r)"
eq_newpkt_def: fixes m :: "msg" shows "eq_newpkt m \<equiv> not_Pkt m"
right_adjoints_compose: fixes g :: "'b"   and g' :: "'b" assumes "D.is_right_adjoint g"   and "D.is_right_adjoint g'"   and "src\<^sub>D g' = trg\<^sub>D g" shows "D.is_right_adjoint (g' \<star>\<^sub>D g)"
nth_last_Suc_n: fixes ls :: "'a list"   and n :: "nat" assumes "distinct ls"   and "n < |ls|"   and "last ls = ls ! n" shows "Suc n = |ls|"
has_vderiv_on_cong: fixes S :: "real set"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a"   and T :: "real set" assumes "\<And>x. x \<in> S \<Longrightarrow> f x = g x"   and "\<And>x. x \<in> S \<Longrightarrow> f' x = g' x"   and "S = T" shows "(f has_vderiv_on f') S = (g has_vderiv_on g') T"
simps: fixes Enc :: "'a \<Rightarrow> 'b"   and a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc" shows "encoding.indRelLp Enc a1 a2 = (\<exists>S. a1 = TargetTerm (Enc S) \<and> a2 = SourceTerm S)"
assms_2: shows "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
Rep_filter_inverse: fixes x :: "'b Filters.filter" shows "Filters.filter.Abs_filter (Filters.filter.Rep_filter x) = x"
le_bot: fixes bot :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g" assumes "class.order_bot bot less_eq less"   and "less_eq a bot" shows "a = bot"
CLAMP_: fixes L :: "nat filter \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> (nat \<Rightarrow> real) set"   and L' :: "'b filter \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) set"   and Lr :: "nat filter \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> (nat \<Rightarrow> real) set"   and f :: "nat \<Rightarrow> real" assumes "landau_symbol L L' Lr" shows "L sequentially f \<equiv> L sequentially (CLAMP f)"
fair_imp: fixes t :: "'s"   and t' :: "'s"   and \<omega> :: "'s stream" assumes "fair t t' \<omega>"   and "alw (ev (HLD {t})) \<omega>" shows "alw (ev (HLD {t'})) \<omega>"
span_scale: fixes x :: "'d"   and S :: "'d set"   and c :: "real" assumes "x \<in> span S" shows "c *\<^sub>R x \<in> span S"
ada: shows "open_segment (a - complex_of_real d) a \<subseteq> inside (path_image p)"
stable_onI: fixes ds :: "D2 set"   and X :: "X5 set" assumes "BSI.individually_rational_on ds X"   and "BSI.stable_no_blocking_on ds X" shows "BSI.stable_on ds X"
emeasure_space_le_1: fixes p :: "'a spmf" shows "emeasure (measure_spmf p) (space (measure_spmf p)) \<le> 1"
mult_exp_exp: fixes x :: "'e"   and y :: "'e" shows "exp x * exp y = exp (x + y)"
poly_cond_def: fixes p :: "int poly" shows "poly_cond p = (0 < lead_coeff p \<and> irreducible p)"
removeAll_map: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and a :: "'a list" assumes "\<forall>x y. x \<noteq> y \<longrightarrow> f x \<noteq> f y" shows "removeAll (f x) (map f a) = map f (removeAll x a)"
mat_of_cols_carrier_3: fixes n :: "nat"   and vs :: "'c vec list" shows "dim_col (mat_of_cols n vs) = length vs"
collapse_1: fixes unboxed :: "('a, 'b, 'c) unboxed" assumes "is_dyn_operand unboxed" shows "OpDyn (un_OpDyn unboxed) = unboxed"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "comm_ring_isom hom"   and "inj_idom_hom hom" shows "idom_isom hom"
uses'_def: fixes g :: "'g"   and n :: "'node" shows "uses' g n \<equiv> lookupDef g n ` uses g n"
Can_rep_can: fixes f :: "'c local.term set" assumes "can f" shows "Can (rep f)"
simps_8: fixes vb :: "('a, 'b) Combinators"   and vc :: "('a, 'b) Combinators" shows "separate [vb \<oplus> vc] = [vb \<oplus> vc]"
\<A>_1: shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> Sec"
beta_cfun: fixes f :: "'g \<Rightarrow> 'h"   and u :: "'g" assumes "cont f" shows "(\<Lambda> x. f x)\<cdot>u = f u"
list_singleton_induct: fixes P :: "'c list \<Rightarrow> bool"   and xs :: "'c list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P xs"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_loop_start_on_rightmost m lm (l, r) = (\<exists>rn mr. rev l @ r = Oc \<up> Suc m @ Bk # Bk # <lm> @ Bk \<up> rn \<and> l \<noteq> [] \<and> r = Oc \<up> mr @ Bk \<up> rn)"
eventually_filtermap: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and F :: "'d filter" shows "eventually P (filtermap f F) = (\<forall>\<^sub>F x in F. P (f x))"
ilist_i_take_eq_conv: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "(f = g) = (\<forall>n. f \<Down> n = g \<Down> n)"
Axp_Ax_eq: shows "Axp = (\<lambda>x xa. (x, xa) \<in> Ax)"
val_def: fixes tree :: "'b tree" shows "val tree = (case tree of Leaf \<Rightarrow> None | Node x21 x22 x23 x24 x25 \<Rightarrow> x23)"
Bex_True: fixes P :: "'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt" shows "RBT_Impl.fold (\<lambda>k v s. s \<or> P k) t True = True"
wp_under_wlp_DC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "wp_under_wlp a"   and "wp_under_wlp b" shows "wp_under_wlp (a \<Sqinter> b)"
eqv_def: fixes \<phi> :: "WS1S_Nameful.ws1s"   and \<psi> :: "WS1S_Nameful.ws1s" shows "eqv \<phi> \<psi> = (let fs = List.union (fos \<phi>) (fos \<psi>); ss = List.union (sos \<phi>) (sos \<psi>) in check_eqv (Abs_idx (length fs, length ss)) (\<epsilon> fs ss \<phi>) (\<epsilon> fs ss \<psi>))"
obj_def': fixes a :: "'d" shows "D.obj a = (D.arr a \<and> trg\<^sub>D a = a)"
f_expand_length: fixes xs :: "'b message_af list"   and k :: "nat" shows "length (xs \<odot> k) = length xs * k"
sub_num_simps_6: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num"   and l :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (num.Bit0 k) (num.Bit0 l) = neg_numeral.dbl plus (neg_numeral.sub minus plus one k l)"
cpq: fixes p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and q :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "(p, q) \<in>\<^sub>p set sps"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` (set gs \<union> set (ab gs bs hs data'))) (fst p) (fst q)"
le_implies_top: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<le> x \<leadsto> y"
integral_abs_bound: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" shows "\<bar>integral\<^sup>L M f\<bar> \<le> LINT x|M. \<bar>f x\<bar>"
comp_ide_arr: fixes b :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" assumes "J.ide b"   and "J.seq b f" shows "b \<cdot>\<^sub>J f = f"
Bno: fixes x :: "complex" assumes "x \<in> (\<lambda>w. w - z) ` path_image \<gamma>" shows "cmod x \<le> B"
assms_2: shows "(b, c) \<in> rtrancl_on F r"
f_limit_0: shows "f \<midarrow>0\<rightarrow> 0"
Cond: fixes Env :: "env"   and c :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and B :: "lname set"   and C :: "assigned"   and E1 :: "assigned"   and E2 :: "assigned"   and A :: "assigned" assumes "\<not> Env\<turnstile>c ? e1 : e2\<Colon>-PrimT Boolean"   and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C"   and "Env\<turnstile> B \<union> assigns_if True c \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1"   and "Env\<turnstile> B \<union> assigns_if False c \<guillemotright>\<langle>e2\<rangle>\<guillemotright> E2"   and "nrm A = nrm E1 \<inter> nrm E2"   and "brk A = (\<lambda>l. UNIV)" shows "Env\<turnstile> B \<guillemotright>\<langle>c ? e1 : e2\<rangle>\<guillemotright> A"
mann3: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z = - (- (- (- (- x \<squnion> y) \<squnion> x \<squnion> y \<squnion> y) \<squnion> - (- x \<squnion> y) \<squnion> z) \<squnion> - (y \<squnion> z))"
hyps_2: shows "\<Gamma>\<turnstile>Language.normalize c\<^sub>1_ \<down> Normal s_"
unfold_lesub_err: fixes e1 :: "'b err"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e2 :: "'b err" shows "(e1 \<sqsubseteq>\<^bsub>le r\<^esub> e2) = le r e1 e2"
ResRel: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "(P, Q) \<in> Rel" shows "(<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"