hcju/isabelleps huggingface dataset:
ds2_unique: fixes M :: "('e, 'a, 'f) Module_scheme"   and M1 :: "('e, 'a, 'f) Module_scheme"   and M2 :: "('e, 'a, 'f) Module_scheme"   and m1 :: "'e"   and m1' :: "'e"   and m2 :: "'e"   and m2' :: "'e" assumes "R module M"   and "R module M1"   and "R module M2"   and "M1 \<Oplus>\<^bsub>R,M\<^esub> M2"   and "m1 \<in> carrier M1"   and "m1' \<in> carrier M1"   and "m2 \<in> carrier M2"   and "m2' \<in> carrier M2"   and "m1 \<plusminus>\<^bsub>M\<^esub> m2 = m1' \<plusminus>\<^bsub>M\<^esub> m2'" shows "m1 = m1' \<and> m2 = m2'"
op_single_inter_ivl_def: fixes a :: "'b set"   and fxs :: "'b set" shows "op_single_inter_ivl a fxs = (let isa = op_inter_ivl_coll (fxs ::: clw_rel lvivl_rel) (a ::: lvivl_rel) in if op_coll_is_empty isa then RETURN op_empty_coll else op_ivl_of_ivl_coll isa \<bind> (\<lambda>ivl. RETURN (mk_coll ((ivl ::: lvivl_rel) \<inter> a))))"
simps: fixes \<zeta> :: "('s, 'v) hd"   and ss :: "('s, 'v) tm list" shows "wt (apps (Hd \<zeta>) ss) = PSum ([wt0 \<zeta>, PNum (\<delta>\<^sub>h * (arity_sym\<^sub>h (min_ground_head \<zeta>) - of_nat (length ss)))] @ map2 (\<lambda>s i. PMult [coef_hd \<zeta> i, wt s]) ss [0..<length ss])"
wlp_while_exists: fixes WS :: "complex mat \<Rightarrow> complex mat"   and M :: "nat \<Rightarrow> complex mat"   and P :: "complex mat" assumes "\<And>P. is_quantum_predicate P \<Longrightarrow> is_quantum_predicate (WS P)"   and "\<And>P Q. \<lbrakk>is_quantum_predicate P; is_quantum_predicate Q; P \<le>\<^sub>L Q\<rbrakk> \<Longrightarrow> WS P \<le>\<^sub>L WS Q"   and "measurement d 2 M"   and "is_quantum_predicate P" shows "is_quantum_predicate (wlp_while (M 0) (M 1) WS P) \<and> (\<forall>n. wlp_while (M 0) (M 1) WS P \<le>\<^sub>L wlp_while_n (M 0) (M 1) WS n P) \<and> (\<forall>W'. (\<forall>n. W' \<le>\<^sub>L wlp_while_n (M 0) (M 1) WS n P) \<longrightarrow> W' \<le>\<^sub>L wlp_while (M 0) (M 1) WS P) \<and> limit_mat (\<lambda>n. wlp_while_n (M 0) (M 1) WS n P) (wlp_while (M 0) (M 1) WS P) d"
3: shows "finite {v $ <a> |v a. llength v = enat n}"
errMOD_iwlsFSbSwTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSb MOD" shows "iwlsFSbSwTR (errMOD MOD)"
pair_sym_digraph_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs G"   and "w \<notin> pverts G" shows "pair_sym_digraph (subdivide G e w)"
induct: fixes P :: "formula \<Rightarrow> bool"   and formula :: "formula" assumes "\<And>x1 x2. P (Pred x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P x2; P x3\<rbrakk> \<Longrightarrow> P (formula.Let x1 x2 x3)"   and "\<And>x1 x2. P (formula.Eq x1 x2)"   and "\<And>x1 x2. P (Less x1 x2)"   and "\<And>x1 x2. P (LessEq x1 x2)"   and "\<And>x. P x \<Longrightarrow> P (Neg x)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Or x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (And x1 x2)"   and "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> P xa) \<Longrightarrow> P (Ands x)"   and "\<And>x. P x \<Longrightarrow> P (Exists x)"   and "\<And>x1 x2 x3 x4 x5. P x5 \<Longrightarrow> P (Agg x1 x2 x3 x4 x5)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (Prev x1 x2)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (Next x1 x2)"   and "\<And>x1 x2 x3. \<lbrakk>P x1; P x3\<rbrakk> \<Longrightarrow> P (Since x1 x2 x3)"   and "\<And>x1 x2 x3. \<lbrakk>P x1; P x3\<rbrakk> \<Longrightarrow> P (Until x1 x2 x3)"   and "\<And>x1 x2. (\<And>x2a. x2a \<in> regex.atms x2 \<Longrightarrow> P x2a) \<Longrightarrow> P (MatchF x1 x2)"   and "\<And>x1 x2. (\<And>x2a. x2a \<in> regex.atms x2 \<Longrightarrow> P x2a) \<Longrightarrow> P (MatchP x1 x2)" shows "P formula"
monom_1_dvd_iff': fixes n :: "nat"   and p :: "'a poly" shows "(Polynomial.monom (1::'a) n dvd p) = (\<forall>k<n. poly.coeff p k = (0::'a))"
sint_ucast_eq_uint: fixes x :: "'b word" assumes "\<not> is_down UCAST('b \<rightarrow> 'c)" shows "sint (UCAST('b \<rightarrow> 'c) x) = uint x"
inner_commute: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x y = inner y x"
s_init_foreach_def: fixes U0 :: "'c set" shows "s_init_foreach U0 \<equiv> FOREACH U0 (\<lambda>x (U0, ws). RETURN (insert x U0, ws(x \<mapsto> []))) ({}, Map.empty) \<bind> (\<lambda>(U0, ws). RETURN (None, U0, ws))"
eq_upred_refl: fixes x :: "('b, 'a) uexpr" shows "(x =\<^sub>u x) = true"
Domainp_Grp: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "Domainp (BNF_Def.Grp A f) = (\<lambda>x. x \<in> A)"
assms_5: shows "Smith_normal_form_mat (P * A * Q)"
image_ideal_eq_surj: fixes h :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "\<And>x y. h (x + y) = h x + h y"   and "\<And>x y. h (x * y) = h x * h y"   and "surj h" shows "h ` ideal B = ideal (h ` B)"
plus_adds_pp_right: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and s :: "'b"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "term_powerprod.adds_pp pair_of_term (s + t) v" shows "term_powerprod.adds_pp pair_of_term t v"
extract_def: shows "extract \<equiv> \<lambda>A ts. card (A \<inter> ts)"
connex_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "connex A less_eq \<equiv> \<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> less_eq x y \<or> less_eq y x"
nop_onto_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "nop_onto A n f \<equiv> vsv f \<and> nop_onto_axioms A n f"
mod_hom_axioms_def: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and N :: "('b, 'f, 'g) module_scheme"   and f :: "'d \<Rightarrow> 'f" shows "mod_hom_axioms R M N f \<equiv> f \<in> module_hom R M N"
conversion_inv: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" shows "((x, y) \<in> R\<^sup>\<leftrightarrow>\<^sup>*) = ((y, x) \<in> R\<^sup>\<leftrightarrow>\<^sup>*)"
assms_2: shows "\<guillemotleft>f : a' \<rightarrow> a\<guillemotright>"
S5: shows "c_fold (c_fst u # c_unfold k (c_snd u)) = c_pair (c_fst u) (c_snd u)"
join\<^sub>\<epsilon>: fixes s :: "'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 r' :: "'r" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"   and "s r' = None" shows "revision_step r s \<epsilon>"
map_id: fixes t :: "'a error_result" shows "map_error_result id t = t"
term_variants_Fun_2: fixes i :: "nat" assumes "i < length T" shows "term_variants_pred P (T ! i) (S ! i)"
neq: shows "local.xd (Suc i) i \<noteq> (0::'a)"
MtransC_StepC: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and cf'' :: "('test, 'atom) com \<times> 'state" assumes "cf \<rightarrow>*c cf'"   and "cf' \<rightarrow>c cf''" shows "cf \<rightarrow>*c cf''"
gc2_def: shows "gc2 = gcd pR2 pI2"
measurable_envelopeD_1: fixes M :: "'b measure"   and A :: "'b set"   and E :: "'b set" assumes "measurable_envelope M A E" shows "A \<subseteq> E"
map_component_term_of_pair: fixes f :: "nat \<Rightarrow> nat"   and t :: "'a"   and k :: "nat" shows "map_component f (term_of_pair (t, k)) = term_of_pair (t, f k)"
wtSB_def: fixes \<pi> :: "Preliminaries.var \<Rightarrow> ('fsym, 'tp) efsym trm" shows "GE.wtSB \<pi> \<equiv> \<forall>x. GE.wt (\<pi> x) \<and> GE.tpOf (\<pi> x) = GE.tpOfV x"
fixp_above_Field: fixes a :: "'a" assumes "a \<in> Field leq" shows "fixp_above a \<in> Field leq"
star_sq_le_rd: fixes x :: "'a" shows "star_sq x \<le> star_rd x"
rtranclpD: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "R\<^sup>*\<^sup>* a b" shows "a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
AddSA_InitConf: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates A = {}"   and "S \<in> InitConf A" shows "InitConf (A [++] (S, SA)) = insert (InitState SA) (InitConf A)"
sorted_mono_map: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "sorted xs"   and "order_class.mono_on f (set xs)" shows "sorted (map f xs)"
split_max_induct: fixes P :: "('b \<times> bal) tree \<Rightarrow> bool"   and a0 :: "('b \<times> bal) tree" assumes "\<And>l a ba r. (r \<noteq> \<langle>\<rangle> \<Longrightarrow> P r) \<Longrightarrow> P \<langle>l, (a, ba), r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
tabR: shows "table (args_n args) (args_R args) rel2"
H: shows "\<Delta> @ VarB x U # \<Gamma> \<turnstile> t : T"
act_ref: fixes s :: "'a v_state_scheme"   and v :: "val"   and S :: "process set" assumes "s \<in> Vinv1" shows "process_mru (votes s) ++ const_map (v_state.next_round s, v) S = process_mru ((votes s)(v_state.next_round s := const_map v S))"
R_linear: fixes y :: "real"   and x :: "real" assumes "\<not> y \<le> x" shows "x \<le> y"
times_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<cdot> B = {u \<cdot> v |u v. u \<in> A \<and> v \<in> B}"
distinct_287: fixes x14 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Ln x14 \<noteq> x151 ^\<^sub>e x152"
mem_zmset_frontier: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'d"   and M :: "'d zmultiset" assumes "dataflow_topology summary results_in" shows "(x \<in>#\<^sub>z zmset_of (mset_set (set_antichain (frontier M)))) = (x \<in>\<^sub>A frontier M)"
assms_1: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
intros_1: fixes u :: "'a \<Rightarrow> real"   and x :: "'a"   and d :: "nat" assumes "u x = real d" shows "Regions.intv_elem x u (Regions.intv.Const d)"
set_iterator_genord_I: fixes l0 :: "'d list"   and S :: "'d set"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" assumes "distinct l0"   and "S = set l0"   and "sorted_wrt R l0"   and "iti = foldli l0" shows "set_iterator_genord iti S R"
simps_1: fixes ok :: "nat"   and f :: "'b list"   and f' :: "'b list"   and r :: "'b list"   and r' :: "'b list" shows "invalidate (Rev ok f f' r r') = Rev (ok - 1) f f' r r'"
assms_2: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and m :: "nat"   and h :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and g :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd" assumes "dickson_grading d"   and "set gs \<union> set bs \<union> set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "h \<in> set hs"   and "g \<in> set gs \<union> set bs \<union> set hs"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "h \<noteq> g"   and "fst h \<noteq> 0"   and "fst g \<noteq> 0"   and "\<And>a b. \<lbrakk>(a, b) \<in>\<^sub>p set (ap gs bs ps hs data); fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)"   and "\<And>a b. \<lbrakk>a \<in> set gs \<union> set bs; b \<in> set gs \<union> set bs; fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)" shows "crit_pair_cbelow_on d m (fst ` B) (fst h) (fst g)"
exists_least_lemma: fixes P :: "nat \<Rightarrow> bool" assumes "\<not> P 0"   and "\<exists>n. P n" shows "\<exists>n. \<not> P n \<and> P (Suc n)"
case_Neg_in_iff: fixes x :: "Formula.formula"   and y :: "Formula.formula" shows "(x \<in> (case y of formula.Neg \<phi>' \<Rightarrow> {\<phi>'} | _ \<Rightarrow> {})) = (y = formula.Neg x)"
rt_comb: fixes t\<^sub>1 :: "'a"   and u\<^sub>1 :: "'a"   and t\<^sub>2 :: "'a"   and u\<^sub>2 :: "'a" assumes "R\<^sup>*\<^sup>* t\<^sub>1 u\<^sub>1"   and "R\<^sup>*\<^sup>* t\<^sub>2 u\<^sub>2" shows "R\<^sup>*\<^sup>* (app t\<^sub>1 t\<^sub>2) (app u\<^sub>1 u\<^sub>2)"
baEA: fixes n :: "nat"   and c :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'ND"   and \<phi> :: "'ND \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n [\<phi>]\<^sub>b" shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>)"
NS_Says_staticSecret: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list"   and m :: "msg"   and C :: "agent" assumes "Says A B X \<in> set evs"   and "m \<in> staticSecret C"   and "m \<in> parts {X}"   and "evs \<in> ns_public" shows "A = Spy"
ltl_prop_equiv_true: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>P true\<^sub>n = {} \<Turnstile>\<^sub>P \<phi>"
mset_poly_rel_def: shows "mset_poly_rel = {(a, b). b = polynomial_of_mset a}"
lookup_minus: fixes f :: "'r \<Rightarrow>\<^sub>0 'q"   and g :: "'r \<Rightarrow>\<^sub>0 'q"   and k :: "'r" shows "lookup (f - g) k = lookup f k - lookup g k"
zero_less_one: fixes one :: "'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "class.zero_less_one one zero less_eq less" shows "less zero one"
typeof_unboxed_inversion_2: fixes x :: "('a, 'b, 'c) unboxed" assumes "typeof x = Some Ubx1" shows "\<exists>n. x = OpUbx1 n"
select_convs_2: fixes Chn :: "'b"   and Dom :: "'b span_data"   and Cod :: "'b span_data"   and more :: "'c" shows "Dom \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> = Dom"
select_convs_2: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
normalized_append: fixes xs :: "('c, 'd) bintree option list"   and ys :: "('c, 'd) bintree option list" assumes "normalized xs"   and "normalized ys" shows "normalized (xs @ ys)"
trms\<^sub>s\<^sub>s\<^sub>t_subst_cons: fixes a :: "('c, 'd) stateful_strand_step"   and A :: "('c, 'd) stateful_strand_step list"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t (a # A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) \<union> trms\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
free_strict_monoidal_category_axioms: shows "free_strict_monoidal_category (\<cdot>\<^sub>C)"
one_pratt1_def: shows "1 \<equiv> [0]\<^sub>1"
prod_emb_Pi: fixes X :: "'d \<Rightarrow> 'e set"   and J :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and K :: "'d set" assumes "X \<in> (\<Pi> j\<in>J. sets (M j))"   and "J \<subseteq> K" shows "prod_emb K M J (Pi\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))"
simps_10: fixes pn :: "'c" shows "ctermsl (call(pn)) = {call(pn)}"
e2tail_def: fixes e :: "nat" shows "e2tail e \<equiv> e_tl (e2stack e)"
equivalence_functor_F: shows "equivalence_functor hom.comp hom'.comp (\<lambda>\<mu>. e \<star> \<mu> \<star> src f)"
eq_TT_1: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and xs :: "'a"   and ys :: "'a" assumes "class.Eq_def eq" shows "(TT = eq\<cdot>xs\<cdot>ys) = (xs \<noteq> \<bottom> \<and> ys \<noteq> \<bottom> \<and> xs = ys)"
vifintersection_vsubset_lower: fixes i :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V" assumes "i \<in>\<^sub>\<circ> I" shows "\<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I) \<subseteq>\<^sub>\<circ> f i"
submodule_axioms_def: fixes H :: "'e set"   and R :: "('f, 'g) ring_scheme"   and M :: "('f, 'e, 'h) module_scheme" shows "submodule_axioms H R M \<equiv> \<forall>a x. a \<in> carrier R \<longrightarrow> x \<in> H \<longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<in> H"
sorted_inorder_list_separators: fixes ts :: "('b btree \<times> 'b) list" assumes "sorted_less (inorder_list ts)" shows "sorted_less (separators ts)"
process_prover_2: fixes y :: "('b event list \<times> 'b event set) set \<times> 'b event list set" assumes "y \<in> {p. is_process p}" shows "Rep_process (Abs_process y) = y"
"1_3": shows "u \<circ> f = v \<circ> g"
simps_3: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option"   and x :: "'a"   and xs :: "'a list" shows "proper_intrvl.set_less_aux_Compl less proper_interval ao (x # xs) [] = (proper_interval ao (Some x) \<or> proper_intrvl.set_less_aux_Compl less proper_interval (Some x) xs [])"
grey_protects_whiteD: fixes g :: "'a"   and w :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "(g grey_protects_white w) s" shows "grey g s \<and> (g = w \<or> white w s)"
preserves_arr: fixes f :: "'c" assumes "Cop.arr f" shows "S.arr (F f)"
vars_Var: fixes x :: "nat" shows "vars (Var x) = {x}"
assms: shows "\<Psi> \<rhd> P \<approx>\<^sub>w Q"
ZObisT_matchT_T: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01T t" shows "matchT_T s t"
g_pair_def: fixes c_ls :: "nat"   and key :: "nat" shows "g_pair c_ls key = (let n = c_fst key; x = c_snd key; m = c_snd n; m1 = c_fst m; m2 = c_snd m in if c_assoc_have_key c_ls (c_pair m1 x) = 0 then let y1 = c_assoc_value c_ls (c_pair m1 x) in if c_assoc_have_key c_ls (c_pair m2 x) = 0 then let y2 = c_assoc_value c_ls (c_pair m2 x) in c_cons (c_pair key (c_pair y1 y2)) c_ls else c_ls else c_ls)"
H_rcoset_reps_def: shows "H_rcoset_reps \<equiv> (0::'g) # nonzero_H_rcoset_reps"
zip_replicate_mapr: fixes xs :: "'a list"   and n :: "nat"   and x :: "'b" assumes "length xs = n" shows "zip xs (replicate n x) = map (\<lambda>y. (y, x)) xs"
ones: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool" assumes "\<forall>n'\<ge>n. f n'" shows "wf_set_bits_int f"
X_def: shows "X \<equiv> \<Union> (indets ` set fs)"
nonzero_of_real_inverse: fixes x :: "real" assumes "x \<noteq> 0" shows "of_real (inverse x) = inverse (of_real x)"
sound_bounded: fixes P :: "'a \<Rightarrow> real" assumes "sound P" shows "Expectations.bounded P"
is_in_insort: fixes y :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and x :: "('b, 'c) Combinators"   and l :: "'b set list" assumes "y \<in> set xs" shows "y \<in> set (FWNormalisationCore.insort x xs l)"
unfold: shows "eval_body' = (\<Lambda> r v s f. Fixrec.run\<cdot>(match_Val\<cdot>v\<cdot>(\<Lambda> n. Fixrec.succeed\<cdot>(s\<cdot>n)) +++ match_Add\<cdot>v\<cdot>(\<Lambda> x y. Fixrec.succeed\<cdot>(case wrapC\<cdot>r\<cdot>x of Nothing \<Rightarrow> f | Just\<cdot>n \<Rightarrow> case wrapC\<cdot>r\<cdot>y of Nothing \<Rightarrow> f | Just\<cdot>m \<Rightarrow> s\<cdot>(n + m))) +++ match_Throw\<cdot>v\<cdot>(Fixrec.succeed\<cdot>f) +++ match_Catch\<cdot>v\<cdot>(\<Lambda> x y. Fixrec.succeed\<cdot>(case wrapC\<cdot>r\<cdot>x of Nothing \<Rightarrow> case wrapC\<cdot>r\<cdot>y of Nothing \<Rightarrow> f | Just\<cdot>n \<Rightarrow> s\<cdot>n | Just\<cdot>n \<Rightarrow> s\<cdot>n))))"
bell10_index_4: shows "|\<beta>\<^sub>1\<^sub>0\<rangle> $$ (3, 0) = complex_of_real (- 1 / sqrt 2)"
reflpE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "reflp r" obtains "r x x"
bot_eq_sup_iff: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(bot = sup x y) = (x = bot \<and> y = bot)"
not_neutral_contains_not_neutral: fixes g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "aggregation.sum_0 g A \<noteq> zero" obtains a :: "'d" where "a \<in> A" and "g a \<noteq> zero"
sinvar_monoI: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" assumes "SecurityInvariant_preliminaries sinvar" shows "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"
strict_mono_add: fixes k :: "'a" shows "strict_mono (\<lambda>n. n + k)"
suminf_mult2: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "summable f" shows "suminf f * c = (\<Sum>n. f n * c)"
add_def: fixes a :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" shows "ell_field_class.add a p\<^sub>1 p\<^sub>2 = (case p\<^sub>1 of Infinity \<Rightarrow> p\<^sub>2 | Point x\<^sub>1 y\<^sub>1 \<Rightarrow> case p\<^sub>2 of Infinity \<Rightarrow> p\<^sub>1 | Point x\<^sub>2 y\<^sub>2 \<Rightarrow> if x\<^sub>1 = x\<^sub>2 then if y\<^sub>1 = - y\<^sub>2 then Infinity else let l = ((3::'b) * x\<^sub>1\<^sup>2 + a) / ((2::'b) * y\<^sub>1); x\<^sub>3 = l\<^sup>2 - (2::'b) * x\<^sub>1 in Point x\<^sub>3 (- y\<^sub>1 - l * (x\<^sub>3 - x\<^sub>1)) else let l = (y\<^sub>2 - y\<^sub>1) / (x\<^sub>2 - x\<^sub>1); x\<^sub>3 = l\<^sup>2 - x\<^sub>1 - x\<^sub>2 in Point x\<^sub>3 (- y\<^sub>1 - l * (x\<^sub>3 - x\<^sub>1)))"
assms_1: shows "\<forall>C. strip (f C) = strip C"
continuous_on_closed_invariant: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
Image_subset: fixes r :: "('c \<times> 'd) set"   and A :: "'c set"   and B :: "'d set"   and C :: "'c set" assumes "r \<subseteq> A \<times> B" shows "r `` C \<subseteq> B"
right: shows "\<forall>v. O v w \<longrightarrow> O v x \<or> O v y"
eligible_lit_exists: fixes C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "finite (cl_ecl C)"   and "cl_ecl C \<noteq> {}"   and "ground_clause (subst_cl (cl_ecl C) \<sigma>)" shows "\<exists>L. eligible_literal L C \<sigma> \<and> L \<in> cl_ecl C"
static_to_dynamic_accessible_from_aux: fixes G :: "prog"   and accC :: "qtname"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m of C accessible_from accC"   and "wf_prog G" shows "G \<turnstile> m in C dyn_accessible_from accC"
call_indirect_None: fixes s :: "s"   and i :: "nat"   and c :: "i32"   and cl :: "cl"   and j :: "nat"   and vs :: "v list" assumes "stab s i (nat_of_int c) = Some cl \<and> stypes s i j \<noteq> cl_type cl \<or> stab s i (nat_of_int c) = None" shows "\<lparr>s;vs;[$C ConstInt32 c, $Call_indirect j]\<rparr> \<leadsto>_ i \<lparr>s;vs;[Trap]\<rparr>"
1: shows "{brnL cl n..<brnL cl n + brn (cl ! n)} = (+) (brnL cl n) ` {..<brn (cl ! n)}"
assms_2: shows "lt q \<prec>\<^sub>t lt p"
length_strip_while_le: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "length (strip_while P xs) \<le> length xs"
2: shows "bf_between_arcs a b H d \<or> bf_between_arcs a c H d \<and> bf_between_arcs c b H d \<longrightarrow> bf_between_arcs a b H (d \<squnion> c)"
discI_1: fixes "term" :: "('a, 'b) Term.term"   and x1 :: "'b" assumes "term = Var x1" shows "is_Var term"
dvd_def: fixes b :: "'a"   and a :: "'a" shows "(b dvd a) = (\<exists>k. a = b * k)"
rel_funD: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i"   and x :: "'f"   and y :: "'g" assumes "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"