hcju/isabelleps huggingface dataset:
continuous_on_blinfun_componentwise: fixes s :: "'f set"   and f :: "'f \<Rightarrow> 'e \<Rightarrow>\<^sub>L 'g" assumes "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on s (\<lambda>x. blinfun_apply (f x) i)" shows "continuous_on s f"
supp_list_append: fixes xs :: "'e list"   and ys :: "'e list" shows "supp (xs @ ys) = supp xs \<union> supp ys"
enum_0_bot: fixes a :: "nat \<Rightarrow> nat" assumes "a \<in> s" shows "(a = enum 0) = (\<forall>a'\<in>s. a \<le> a')"
pair_term: fixes p :: "'a \<times> 'k" shows "pair_of_term (term_of_pair p) = p"
coinduct: fixes X :: "'a stream \<Rightarrow> 'a treeFI \<Rightarrow> bool"   and x1 :: "'a stream"   and x2 :: "'a treeFI" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>as tr tr'. x1 = as \<and> x2 = tr \<and> shd as = lab tr \<and> tr' \<in> set (sub tr) \<and> (X (stl as) tr' \<or> properPath (stl as) tr')" shows "properPath x1 x2"
card_length_sum_list_rec: fixes m :: "nat"   and N :: "nat" assumes "1 \<le> m" shows "card {l. length l = m \<and> sum_list l = N} = card {l. length l = m - 1 \<and> sum_list l = N} + card {l. length l = m \<and> sum_list l + 1 = N}"
wf_abc_mopup_measure: shows "wf abc_mopup_measure"
anga_sym: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.QCongAAcute Bet Cong a"   and "a A B C" shows "a C B A"
ANR_imp_absolute_neighbourhood_retract: fixes S :: "'b set"   and S' :: "'c set"   and U :: "'c set" assumes "ANR S"   and "S homeomorphic S'"   and "closedin (top_of_set U) S'" obtains V :: "'c set" where "openin (top_of_set U) V" and "S' retract_of V"
case_eq_if: fixes f1 :: "'h"   and f2 :: "'i \<Rightarrow> 'h"   and exception :: "'i exception" shows "(case exception of Fault \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = (if exception = Fault then f1 else f2 (ok exception))"
objE: fixes a :: "'a arrow_of_spans_data" assumes "obj a" obtains "V.ide a" and "src a = a" and "trg a = a"
leftderives1_\<alpha>_\<alpha>': shows "LeftDerives1 \<alpha> i r \<alpha>'"
logical_weak_head_closure: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and s' :: "trm"   and t' :: "trm" assumes "\<Gamma> \<turnstile> s is t : T"   and "s' \<leadsto> s"   and "t' \<leadsto> t" shows "\<Gamma> \<turnstile> s' is t' : T"
fst_half_segments: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "half_segments_of_aform X \<noteq> []" shows "fst (half_segments_of_aform X ! 0) = lowest_vertex (fst X, nlex_pdevs (snd X))"
poly_complex_to_real: fixes p :: "int poly"   and x :: "real" shows "(poly (complex_of_int_poly p) (complex_of_real x) = 0) = (poly (real_of_int_poly p) x = 0)"
CompDiag_Can_Inv: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t"   and "monoidal_language.Diag C t" shows "monoidal_language.CompDiag C t (monoidal_language.Inv C t) = monoidal_language.Cod C t"
msetext_dersh_imp_mult_rel: fixes ys :: "'b list"   and A :: "'b set"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "ys \<in> lists A"   and "xs \<in> lists A"   and "msetext_dersh gt ys xs" shows "(mset xs, mset ys) \<in> mult {(x, y). x \<in> A \<and> y \<in> A \<and> gt y x}"
cycle_of_permutation: fixes p :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "permutation p" shows "cycle (support p a)"
vsv_vrange_vsubset: fixes A :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_inverse \<Longrightarrow> x\<inverse>\<^sub>\<rat> \<in>\<^sub>\<circ> A" shows "\<R>\<^sub>\<circ> vrat_inverse \<subseteq>\<^sub>\<circ> A"
inheritedInsert: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and Gamma :: "'b"   and A :: "(nat \<times> 'b) set"   and n :: "nat" assumes "inherited subs P"   and "\<not> terminal subs Gamma"   and "P A" shows "P (insert (n, Gamma) A)"
vc_rpost_imp_spec: fixes V :: "'v set"   and r :: "bool" assumes "rpost {} ({}, src) (V, r)" shows "r = ((src, tgt) \<in> E\<^sup>*)"
size_4: fixes x1 :: "nat" shows "size (Var x1) = 0"
assms_3: shows "I \<subseteq> {..<brn (cl ! pickFT cl)}"
emeasure_eq_measure: fixes A :: "'a set" shows "emeasure (density M (\<lambda>x. ennreal (D x))) A = ennreal (N.prob A)"
all_info: shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set [[1], [0], [- 1]]"
dotprod_le_take: fixes a :: "nat list"   and b :: "nat list"   and k :: "nat" assumes "length a = length b"   and "k \<le> length a" shows "take k a \<bullet> take k b \<le> a \<bullet> b"
quot_of_fract_add_aux: fixes x :: "'b \<times> 'b"   and y :: "'b \<times> 'b" assumes "snd x \<noteq> (0::'b)"   and "snd y \<noteq> (0::'b)" shows "(fst x * snd y + fst y * snd x) * (snd (normalize_quot x) * snd (normalize_quot y)) = snd x * snd y * (fst (normalize_quot x) * snd (normalize_quot y) + snd (normalize_quot x) * fst (normalize_quot y))"
prems: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a"   and y :: "'a" assumes "xs__ @ ys__ = xs @ ys"   and "x \<in> set xs"   and "y \<in> set ys" shows "x \<le> y"
Rats_eq_int_div_int: shows "\<rat> = {real_of_int i / real_of_int j |i j. j \<noteq> 0}"
b: shows "[A,e,B]\<^bsub>T\<^esub> = [A',e',B']\<^bsub>T\<^esub>"
atoms_of_cnf_alt: fixes c :: "'a literal set set" shows "atoms_of_cnf c = \<Union> ((`) atoms_of_lit ` c)"
a_get_tag_name_locs_def: fixes element_ptr :: "(_) element_ptr" shows "a_get_tag_name_locs element_ptr \<equiv> {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name)}"
hyps_4: shows "bn \<alpha> \<sharp>* \<Psi>"
h1a: shows "\<forall>x. (\<Sum>i = 0..<n + 1. f i / B ^ i) * x = (\<Sum>i = 0..<n + 1. f i / B ^ i * x)"
m_comm: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> carrier (R Quot I)"   and "y \<in> carrier (R Quot I)" shows "x \<oplus>\<^bsub>R Quot I\<^esub> y = y \<oplus>\<^bsub>R Quot I\<^esub> x"
path_length_eq: fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and nn :: "nat"   and nn' :: "nat" assumes "path_length P ctx cl nn"   and "path_length P ctx cl nn'" shows "nn = nn'"
qAbs_preserves_alpha: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "X #= X'"   and "FixVars.qGood X \<or> FixVars.qGood X'" shows "qAbs xs x X $= qAbs xs x X'"
mk_eqcl'_nth': fixes xs :: "nat option list"   and l :: "nat"   and T :: "bool list list"   and j :: "nat"   and i :: "nat"   and j' :: "nat" assumes "\<And>x k. \<lbrakk>x \<in> set xs; x = Some k\<rbrakk> \<Longrightarrow> k < l"   and "\<And>i'. \<lbrakk>i' < length xs; \<not> tr_lookup T (i' + j) i\<rbrakk> \<Longrightarrow> xs ! i' = None"   and "i < j"   and "j' < length xs" shows "(mk_eqcl' xs i j l T ! j' = Some l) = (\<not> tr_lookup T (j' + j) i)"
sublist_valid_plan__alt: fixes as1 :: "'d list"   and PROB :: "'d set"   and as2 :: "'d list" assumes "as1 \<in> valid_plans PROB"   and "subseq as2 as1" shows "as2 \<in> valid_plans PROB"
f: shows "C1.iso (fst f) \<and> C2.iso (snd f)"
simps_2: fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist"   and x :: "'b"   and xs :: "'b llist" shows "lrev_body1\<cdot>r\<cdot>(x :@ xs) = list2H\<cdot>(wrapH\<cdot>r\<cdot>xs :++ x :@ lnil)"
SourcesA72_L1: shows "Sources level1 sA72 = {sA6}"
d_delta_lnexp_cf4_nonpos: fixes x :: "real" shows "diff_delta_lnexp_cf4 x \<le> 0"
final_iff: fixes e :: "('a, 'b, 'c) exp" shows "final e = ((\<exists>v. e = Val v) \<or> (\<exists>a. e = Throw a))"
scaling_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "scaling t \<equiv> \<forall>P c x. sound P \<and> 0 \<le> c \<longrightarrow> c * t P x = t (\<lambda>x. c * P x) x"
decode_unfold: fixes e :: "nat"   and M :: "hmultiset" shows "local.decode e M = (\<Sum>m\<in>#hmsetmset M. base ^ local.decode 0 m) div base ^ e"
sup_least: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> x" shows "y + z \<le> x"
NoActivityNoMessageLoss: fixes cfg1 :: "('p, 'v, 's) configuration"   and Q :: "'p set"   and cfg2 :: "('p, 'v, 's) configuration"   and p :: "'p"   and m' :: "('p, 'v) message" assumes "qReachable cfg1 Q cfg2"   and "p \<notin> Q"   and "isReceiverOf p m'" shows "msgs cfg1 m' \<le> msgs cfg2 m'"
Ord_pred: fixes x :: "hf" assumes "Ord x" shows "Ord (pred x)"
mult_nonneg_nonpos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq b zero" shows "less_eq (times a b) zero"
scalar_prod_right_zero: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v \<bullet> 0\<^sub>v n = (0::'a)"
transferNonTauFrame: fixes \<Psi>\<^sub>F :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and A\<^sub>F :: "name list"   and A\<^sub>G :: "name list"   and \<Psi>\<^sub>G :: "'b" assumes "\<Psi>\<^sub>F \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "distinct (bn \<alpha>)"   and "\<langle>A\<^sub>F, \<Psi>\<^sub>F \<otimes> \<Psi>\<^sub>P\<rangle> \<hookrightarrow>\<^sub>F \<langle>A\<^sub>G, \<Psi>\<^sub>G \<otimes> \<Psi>\<^sub>P\<rangle>"   and "A\<^sub>F \<sharp>* P"   and "A\<^sub>G \<sharp>* P"   and "A\<^sub>F \<sharp>* subject \<alpha>"   and "A\<^sub>G \<sharp>* subject \<alpha>"   and "A\<^sub>P \<sharp>* A\<^sub>F"   and "A\<^sub>P \<sharp>* A\<^sub>G"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>F"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>G"   and "\<alpha> \<noteq> \<tau>" shows "\<Psi>\<^sub>G \<rhd> P \<longmapsto> \<alpha> \<prec> P'"
more_arith_simps_8: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> - b = - (a \<cdot> b)"
simps_1: fixes t1 :: "('c \<times> nat) tree"   and b :: "'c"   and lvb :: "nat"   and t2 :: "('c \<times> nat) tree"   and a :: "'c"   and lva :: "nat"   and t3 :: "('c \<times> nat) tree" shows "skew \<langle>\<langle>t1, (b, lvb), t2\<rangle>, (a, lva), t3\<rangle> = (if lva = lvb then \<langle>t1, (b, lvb), \<langle>t2, (a, lva), t3\<rangle>\<rangle> else \<langle>\<langle>t1, (b, lvb), t2\<rangle>, (a, lva), t3\<rangle>)"
Eq_RESUMPTION_implies_ntrunc_equality: fixes r :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and r' :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and k :: "nat" assumes "Eq_RESUMPTION r r'" shows "ntrunc k r = ntrunc k r'"
vwalk_to_vpath_hd: fixes xs :: "'a list" shows "hd (vwalk_to_vpath xs) = hd xs"
FL_valid_Tree\<^sub>\<alpha>_Not\<^sub>\<alpha>: fixes P :: "'state"   and t\<^sub>\<alpha> :: "('b, 'pred, 'act, 'effect) Tree\<^sub>\<alpha>" shows "FL_valid_Tree\<^sub>\<alpha> P (Not\<^sub>\<alpha> t\<^sub>\<alpha>) = (\<not> FL_valid_Tree\<^sub>\<alpha> P t\<^sub>\<alpha>)"
simps_2: fixes nodesL :: "'a list"   and edgesL :: "('a \<times> 'a) list"   and more :: "'b" shows "edgesL \<lparr>nodesL = nodesL, edgesL = edgesL, \<dots> = more\<rparr> = edgesL"
hyp_c1: shows "\<not> AlternativeSmallStep.inf \<Gamma> [c1] [] (Normal s)"
theI: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "P (THE x. P x)"
recurrent_subset_def: fixes A :: "'a set" shows "recurrent_subset A = (\<Union>n\<in>{1..}. A \<inter> (T ^^ n) -` A)"
L9: fixes js :: "nat list"   and k :: "nat"   and i :: "nat" assumes "\<And>f h. associative f \<Longrightarrow> foldl1 f (map h js) = foldl1 f (map h [0..<k + 1])"   and "i \<le> k" shows "foldl1 f1 (map (h1 k i) js) = One"
vdwE: fixes k :: "nat"   and l :: "nat"   and n :: "nat"   and a :: "int"   and b :: "int"   and col :: "int \<Rightarrow> nat" assumes "vdw k l n"   and "a + int n \<le> b + 1"   and "col \<in> {a..b} \<rightarrow> {..<k}" obtains j :: "nat" and start :: "int" and step :: "nat" where "j < k" and "0 < step" and "is_arith_prog_on l start step a b" and "arith_prog start step ` {..<l} \<subseteq> col -` {j} \<inter> {a..b}"
div_by_1: fixes a :: "'a" shows "a div (1::'a) = a"
pell_mul_cnj_left: fixes z :: "'a \<times> 'a" assumes "solution z" shows "pell_mul (pell_cnj z) z = (1::'a, 0::'a)"
simps_18: fixes x2 :: "'b" shows "vars_hd (Sym x2) = {}"
bind_assoc: fixes x :: "'m"   and f :: "'a option \<Rightarrow> 'm"   and g :: "'a option \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
linearI: fixes f :: "'c \<Rightarrow> 'd" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>R b) = r *\<^sub>R f b" shows "linear f"
lim: shows "((\<lambda>x. f (x - a) /\<^sub>R norm (x - a)) \<longlongrightarrow> (0::'b)) (at a within S)"
succ_chain_swap_rl_exists: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat"   and x :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i < j"   and "j < length l"   and "x \<in> rvars (\<T> (l ! i))"   and "x \<in> lvars (\<T> (l ! j))" shows "\<exists>k\<ge>i. k < j \<and> k + 1 < length l \<and> x \<in> rvars (\<T> (l ! k)) \<and> x \<in> lvars (\<T> (l ! (k + 1)))"
sum_permutations_inverse: fixes f :: "('c \<Rightarrow> 'c) \<Rightarrow> 'b"   and S :: "'c set" shows "sum f {p. p permutes S} = (\<Sum>p | p permutes S. f (inv p))"
WT_converter_map_converter: fixes f :: "'m \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'n"   and \<I> :: "('m, 'n) \<I>"   and f' :: "'k \<Rightarrow> 'o"   and g' :: "'p \<Rightarrow> 'l"   and \<I>' :: "('o, 'p) \<I>"   and conv :: "('i, 'j, 'k, 'l) converter" assumes "map_\<I> (inv f) (inv g) \<I>, map_\<I> f' g' \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "inj f"   and "surj g" shows "\<I>, \<I>' \<turnstile>\<^sub>C map_converter f g f' g' conv \<surd>"
cases: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "'g"   and a2 :: "('g \<times> 'h \<times> 'g) llist" assumes "trsys.inf_step_table trsys a1 a2" obtains s :: "'g" and s' :: "'g" and stls :: "('g \<times> 'h \<times> 'g) llist" and tl :: "'h" where "a1 = s" and "a2 = LCons (s, tl, s') stls" and "trsys s tl s'" and "trsys.inf_step_table trsys s' stls"
nat_pow_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "[n] \<cdot> x \<in> carrier R"
Dom_memb: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "t \<in> f" shows "monoidal_language.Dom C t = monoidal_language.Dom C (free_monoidal_category.rep C f)"
of_type_def: fixes oT :: "PDDL_STRIPS_Semantics.type"   and T :: "PDDL_STRIPS_Semantics.type" shows "of_type oT T \<equiv> set (primitives oT) \<subseteq> subtype_rel\<^sup>* `` set (primitives T)"
smallstepI_ps_eq: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> I\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<pi> = \<pi>'"
simps_73: fixes x3 :: "'a" shows "LID\<^sub>V (Loc x3) = {x3}"
ring_simprules_23: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y"
comm_neutral: fixes a :: "'c" shows "a AND - (1::'c) = a"
param_opt_lowest_tops_lem: fixes i :: "'d ifex"   and t :: "'d ifex"   and e :: "'d ifex" assumes "param_opt i t e = None" shows "\<exists>y. lowest_tops [i, t, e] = Some y"
lucas_lehmer_hom_cong: fixes x :: "int \<times> int"   and y :: "int \<times> int"   and m :: "nat" assumes "[fst x = fst y] (mod int m)"   and "[snd x = snd y] (mod int m)" shows "lucas_lehmer_hom m x = lucas_lehmer_hom m y"
prefix_diff: fixes l :: "'c list"   and t :: "'c list" shows "l @ t - l = t"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_left_moving (s, l, r) lm n ires = (\<exists>lnl lnr rn m. m = Suc (lm ! n) \<and> (l = Bk \<up> lnr @ Oc \<up> m @ Bk \<up> lnl @ Bk # Bk # ires \<and> r = Bk \<up> rn \<or> l = Oc \<up> (m - 1) @ Bk \<up> lnl @ Bk # Bk # ires \<and> r = Oc # Bk \<up> rn))"
inv_map_def: fixes \<pi> :: "'d \<Rightarrow> 'c option" shows "inv_map \<pi> = (\<lambda>y. if Some y \<in> range \<pi> then Some (inv \<pi> (Some y)) else None)"
has_vector_derivative_real_power_aux: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "real"   and c :: "'a" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < x" shows "LIM y at x. scaleR (inverse (norm (y - x))) (minus (minus (real_power y c) (real_power x c)) (scaleR (y - x) (times c (real_power x (minus c one))))) :> Inf (principal ` {S. open S \<and> zero \<in> S})"
mmonitorable_def: fixes \<phi> :: "'b MFOTL.formula" shows "mmonitorable \<phi> = (safe_formula \<phi> \<and> MFOTL.future_reach \<phi> \<noteq> \<infinity>)"
converse_rtranclpE2: fixes r :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"   and xa :: "'a"   and xb :: "'b"   and za :: "'a"   and zb :: "'b" assumes "r\<^sup>*\<^sup>* (xa, xb) (za, zb)" obtains "(xa, xb) = (za, zb)"   | a :: "'a" and b :: "'b" where "r (xa, xb) (a, b)" and "r\<^sup>*\<^sup>* (a, b) (za, zb)"
iT_Plus_icard: fixes I :: "nat set"   and k :: "nat" shows "icard (I \<oplus> k) = icard I"
Ratreal_code: shows "Ratreal = real_of_u \<circ> mau_of_rat"
bmi: shows "mat_rel R (berlekamp_mat_i p ff_ops f) (berlekamp_mat f')"
simps_3: fixes \<sigma> :: "(char list \<times> 'b list) Trace.trace"   and v :: "'b list"   and i :: "nat"   and \<phi> :: "'b formula" shows "sat \<sigma> v i (Neg \<phi>) = (\<not> sat \<sigma> v i \<phi>)"
simps_9: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Abs a) vs = \<bar>interpret_floatarith a vs\<bar>"
extend_Stable: fixes F :: "'a program"   and A :: "'a set" shows "(extend h F \<in> Stable (extend_set h A)) = (F \<in> Stable A)"
gr_implies_not_zero: fixes m :: "'a"   and n :: "'a" assumes "m < n" shows "n \<noteq> (0::'a)"
extensional_def: fixes A :: "'d set" shows "extensional A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = undefined}"
lookup_pair_eq_valueI: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k :: "'a"   and v :: "'c" assumes "oalist_inv_raw x xs"   and "(k, v) \<in> set xs" shows "lookup_pair x xs k = v"
FreeGroup_diff_closed: fixes x :: "'b freeword"   and S :: "'b set"   and y :: "'b freeword" assumes "x \<in> FreeGroup S"   and "y \<in> FreeGroup S" shows "x - y \<in> FreeGroup S"
termFSwSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'b, 'c, 'd) model_scheme" shows "termFSwSbImorph h hA MOD \<equiv> termFSwImorph h hA MOD \<and> ipresSubstAll h hA MOD"
foldr_cong_plus: fixes s :: "'b set"   and f' :: "'a \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and g' :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and lst :: "'b list"   and a :: "'a"   and b :: "'a" assumes "\<And>a b c. \<lbrakk>b \<in> s; f' a = f b (f' c)\<rbrakk> \<Longrightarrow> g' a = g b (g' c)"   and "\<And>a b. f' a = f' b \<Longrightarrow> a = b"   and "\<And>a b c. f' a = f b c \<Longrightarrow> \<exists>c'. c = f' c'"   and "set lst \<subseteq> s"   and "f' a = foldr f lst (f' b)" shows "g' a = foldr g lst (g' b)"
that: fixes tr1 :: "'c list"   and tr2 :: "'c list" assumes "path M (w1 || tr1) q"   and "length w1 = length tr1"   and "path M (w2 || tr2) (target (w1 || tr1) q)"   and "length w2 = length tr2" shows "thesis"
subset_insert: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A" shows "(A \<subseteq> insert x B) = (A \<subseteq> B)"
Derivation_terminals_stay: fixes u :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and v :: "symbol list"   and t :: "symbol" assumes "Derivation u D v"   and "t \<in> set u"   and "is_terminal t" shows "t \<in> set v"
compl_le_compl_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "(\<^bold>- x \<sqsubseteq> \<^bold>- y) = (y \<sqsubseteq> x)"
ShiftP_Mem1: fixes f :: "tm"   and k :: "tm"   and del :: "tm"   and g :: "tm"   and a :: "tm"   and b :: "tm"   and a' :: "tm" shows "{ShiftP f k del g, HPair a b IN f, HaddP del a a', a IN k} \<turnstile> HPair a' b IN g"
subcls1_def: shows "subcls1 \<equiv> \<lambda>P. {(xa, x). subcls1p P xa x}"
countable_subset: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B"   and "countable B" shows "countable A"
simps_1: fixes orderA :: "('d, 'e) gorder_scheme"   and orderB :: "('f, 'g) gorder_scheme"   and lower :: "'d \<Rightarrow> 'f"   and upper :: "'f \<Rightarrow> 'd"   and more :: "'h" shows "\<X>\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = orderA"
comp_apply: fixes f :: "'ah \<Rightarrow> 'ag"   and g :: "'ai \<Rightarrow> 'ah"   and x :: "'ai" shows "(f \<circ> g) x = f (g x)"
coarse_run_locally_similar: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f" assumes "fg_run A rho HOs SHOs coords" shows "locally_similar (state \<circ> rho) (coarse_run rho)"
C_B_zcong_prod: shows "[prod id C = prod id B] (mod int p)"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and Y :: "'d list" shows "injectionsAlg (x # xs) Y = concat (map (\<lambda>R. map (\<lambda>y. R \<union> {(x, y)}) (filter (\<lambda>y. y \<notin> Range R) Y)) (injectionsAlg xs Y))"
abs_o_rep: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs :: "'g \<Rightarrow> 'h"   and Rep :: "'h \<Rightarrow> 'g" assumes "Quotient3 R Abs Rep" shows "Abs \<circ> Rep = id"
affine_hull_translation: fixes a :: "'a"   and S :: "'a set" shows "affine hull (+) a ` S = (+) a ` (affine hull S)"