hcju/isabelleps huggingface dataset:
distinct_11: fixes x1 :: "('d, 'e) Term.term"   and x71 :: "'e list"   and x72 :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list"   and x73 :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" shows "send\<langle>x1\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
L2_set_le_sum_abs: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set" shows "L2_set f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
shift_additivity: fixes r :: "real_int"   and x :: "real"   and y :: "real" shows " shift r x + y =  shift  shift r x y"
type_definition_resumption: shows "type_definition Rep_resumption Abs_resumption resumption"
chine_in_hom: shows "\<guillemotleft>chine : dom.apex \<rightarrow> cod.apex\<guillemotright>"
eq_bernoulli_pmfI: fixes p :: "bool pmf"   and x :: "real" assumes "pmf p True = x" shows "p = bernoulli_pmf x"
stone_phi_inf_dense: fixes x :: "'c"   and y :: "'c" shows "stone_phi (Abs_regular (- x)) \<sqinter> up_filter (Abs_dense (y \<squnion> - y)) \<le> up_filter (Abs_dense (y \<squnion> - y \<squnion> x))"
finite_nat_iff_bounded2: fixes S :: "nat set" shows "finite S = (\<exists>k. \<forall>n\<in>S. n < k)"
diff_right_commute: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a c) b = minus (minus a b) c"
Cd_Above_dX: fixes X :: "'a set"   and d :: "'b" assumes "PdXXX.dX X d \<subseteq> Field (Pd' d)" shows "PdXXX.Cd d X = Above (Pd' d) (PdXXX.dX X d) \<inter> X"
mult_zero_right: fixes times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e" assumes "class.mult_zero times zero" shows "times a zero = zero"
bot_least: 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" shows "less_eq bot a"
unsat_limit_iff: fixes Ns :: "('a literal multiset \<times> label) set llist" assumes "chain (\<rhd>L) Ns"   and "chain FL.entails_\<G> Ns" shows "FL.entails_\<G> (Liminf_llist Ns) ({{#}} \<times> UNIV) = FL.entails_\<G> (lhd Ns) ({{#}} \<times> UNIV)"
simps: fixes b :: "nat list"   and ys :: "nat list"   and a :: "nat"   and x :: "nat"   and xs :: "nat list"   and s :: "nat" shows "c1_incs b ys a x (xs, s) = (let t = s + a * x in if cond1 b ys (x # xs) t then (x # xs, t) # c1_incs b ys a (Suc x) (xs, s) else [])"
bisim1FAss1: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and F :: "String.literal"   and D :: "String.literal"   and e2 :: "'addr expr1" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e\<bullet>F{D} := e2,h \<turnstile> (e'\<bullet>F{D} := e2, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
list_nth_append0: fixes i :: "nat"   and x :: "'d list"   and z :: "'d list" assumes "i < length x" shows "x ! i = (x @ z) ! i"
fps_power_zeroth: fixes a :: "'a fps"   and n :: "nat" shows "fps_nth (a ^ n) 0 = fps_nth a 0 ^ n"
add_mset_commute: fixes x :: "'d"   and y :: "'d"   and M :: "'d multiset" shows "add_mset x (add_mset y M) = add_mset y (add_mset x M)"
linear_injective_left_inverse: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f"   and "inj f" shows "\<exists>g. linear g \<and> g \<circ> f = id"
Inv2b_def: fixes s :: "state" shows "Inv2b s = (\<forall>p d. Inv2b_inner s p d)"
fdia_export_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "ad y \<cdot>  |x\<rangle> z =  |ad y \<cdot> x\<rangle> z"
le_fract: fixes b :: "'a"   and d :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<noteq> (0::'a)"   and "d \<noteq> (0::'a)" shows "(Fract a b \<le> Fract c d) = (a * d * (b * d) \<le> c * b * (b * d))"
comp_min_basis_def: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "comp_min_basis xs = filter_min (\<lambda>x y. lt x adds\<^sub>t lt y) (filter (\<lambda>x. x \<noteq> 0) xs)"
suminf_eq_lim: fixes f :: "nat \<Rightarrow> 'b" shows "suminf f = lim (\<lambda>n. sum f {..<n})"
fls_base_factor_to_fps_right_inverse: fixes f :: "'b fls"   and y :: "'b" shows "fls_base_factor_to_fps (fls_right_inverse f y) = fps_right_inverse (fls_base_factor_to_fps f) y"
set_child_nodes_def: shows "Core_DOM_Functions.set_child_nodes \<equiv> l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes"
inext_nth_Max_fix: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" assumes "finite I"   and "I \<noteq> {}"   and "I \<rightarrow> a = Max I"   and "a \<le> b" shows "I \<rightarrow> b = Max I"
vars\<^sub>0_correct: fixes p :: "prog" assumes "vars p = vars\<^sub>0" shows "io_vars p"
csppa_Formal_in_notin_slice: 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 Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and x :: "nat"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and m :: "'b"   and i :: "nat"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "x < length fs"   and "Formal_in (m, x + i) \<notin> S" shows "SDG.csppa m S i fs ! x = Map.empty"
in_rem_implicit_pres_\<delta>: fixes op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "op \<in> set (ast_problem.ast\<delta> prob)" shows "rem_implicit_pres op \<in> set (ast_problem.ast\<delta> (rem_implicit_pres_ops prob))"
is_spanning_tree_def: fixes G :: "'b ugraph"   and T :: "'b ugraph" shows "is_spanning_tree G T \<equiv> tree T \<and> nodes T = nodes G \<and> edges T \<subseteq> edges G"
not_eq_order_implies_strict: fixes a :: "'a Word.word"   and b :: "'a Word.word" assumes "a \<noteq> b"   and "b \<le>s a" shows "b <s a"
R_in: fixes A :: "'a set" assumes "A \<in> {{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}" shows "(SOME y. y \<in> A) \<in> A"
pseudonatural_equivalence_def: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" shows "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> pseudonatural_transformation V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<and> pseudonatural_equivalence_axioms V\<^sub>C src\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<tau>\<^sub>0"
valid_weight_repr_def: fixes l :: "(('a \<times> 'a) \<times> 'b) list" shows "valid_weight_repr l \<equiv> distinct (map (uedge \<circ> fst) l)"
hyps_7: fixes x :: "real" assumes "i < x"   and "x \<le> b" shows "f2 x = f x"
Lambert_W'_times_exp_self: fixes x :: "real" assumes "x \<in> {- exp (- 1)..<0}" shows "Lambert_W' x * exp (Lambert_W' x) = x"
int_set_bits_unfold_BIT: shows "set_bits f = of_bool (f 0) + 2 * set_bits (f \<circ> Suc)"
assms_1: shows "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>"
inject_1: fixes x1 :: "rat"   and y1 :: "rat" shows "(Rational x1 = Rational y1) = (x1 = y1)"
word_ao_absorbs_7: fixes x :: "'a word"   and y :: "'a word" shows "(x OR y) AND x = x"
induces_hom_adjunction: shows "hom_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) replete_setcat.comp \<phi>\<psi>.\<phi>C \<phi>\<psi>.\<phi>D F G \<phi>\<psi>.\<Phi> \<phi>\<psi>.\<Psi>"
rewrite_negated_primitives_normalized_preserves_unrelated_helper: fixes disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and disc2 :: "'b common_primitive \<Rightarrow> bool"   and m :: "'b common_primitive match_expr"   and sel2 :: "'b common_primitive \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> bool"   and a :: "'b common_primitive match_expr" assumes "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> disc2 (C a)"   and "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated disc False m"   and "normalized_nnf_match m"   and "normalized_n_primitive (disc2, sel2) f m"   and "a \<in> set (normalize_match (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one m))" shows "normalized_n_primitive (disc2, sel2) f a"
scaleR_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR 1 x = x"
equals0D: fixes A :: "'b set"   and a :: "'b" assumes "A = {}" shows "a \<notin> A"
meld_def: fixes q1 :: "('c, 'd) SkewBinomialHeap"   and q2 :: "('c, 'd) SkewBinomialHeap" shows "meld q1 q2 \<equiv> Abs_SkewBinomialHeap (BsSkewBinomialHeapStruc.bs_meld (Rep_SkewBinomialHeap q1) (Rep_SkewBinomialHeap q2))"
dg_length: shows "vcard \<DD> = 4\<^sub>\<nat>"
mult_le_cancel2_gr0: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "(m * k \<le> n * k) = (m \<le> n)"
scale_eq_0_iff: fixes a :: "'b"   and x :: "'b" shows "(a * x = (0::'b)) = (a = (0::'b) \<or> x = (0::'b))"
split_asm: fixes P :: "'k \<Rightarrow> bool"   and f1 :: "'k"   and f2 :: "'l \<Rightarrow> 'k"   and option :: "'l option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"
l6_7: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "P Out A B"   and "P Out B C" shows "P Out A C"
the_inverseD_2: fixes g :: "'a" assumes "g \<in> G" shows "binop g (\<ii> g) = e"
simps: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and x :: "complex"   and y :: "complex" shows "(a, b, c, d) *\<^sub>m\<^sub>v (x, y) = (x * a + y * b, x * c + y * d)"
imp_igSubstBinpIPresIGWlsBinp: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" assumes "igSubstAbsIPresIGWlsAbs MOD" shows "igSubstBinpIPresIGWlsBinp MOD"
weakenStatImpI: fixes Q :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'"   and "insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi>"   and "(\<Psi>, P, Q') \<in> Rel" shows "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<Rel> Q"
continuous_at_imp_continuous_at_within: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd"   and s :: "'c set" assumes "isCont f x" shows "continuous (at x within s) f"
af_congruent_axioms_def: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool" shows "af_congruent_axioms eq \<equiv> \<forall>\<phi> \<psi> \<nu>. eq \<phi> \<psi> \<longrightarrow> eq (af_letter \<phi> \<nu>) (af_letter \<psi> \<nu>)"
wf_mdecl_def: fixes G :: "prog"   and C :: "qtname" shows "wf_mdecl G C = (\<lambda>(sig, m). wf_mhead G (pid C) sig (mhead m) \<and> unique (lcls (mbody m)) \<and> (\<forall>(vn, T)\<in>set (lcls (mbody m)). is_acc_type G (pid C) T) \<and> (\<forall>pn\<in>set (pars m). table_of (lcls (mbody m)) pn = None) \<and> jumpNestingOkS {Ret} (stmt (mbody m)) \<and> is_class G C \<and> \<lparr>prg = G, cls = C, lcl = callee_lcl C sig m\<rparr>\<turnstile>stmt (mbody m)\<Colon>\<surd> \<and> (\<exists>A. \<lparr>prg = G, cls = C, lcl = callee_lcl C sig m\<rparr>\<turnstile> parameters m \<guillemotright>\<langle>stmt (mbody m)\<rangle>\<guillemotright> A \<and> Result \<in> nrm A))"
fw_act_def: shows "first_world \<equiv> \<lambda>sigma. (st1 sigma, st2 sigma)"
gcd_eq_0_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(gcd a b = zero) = (a = zero \<and> b = zero)"
x: shows "x \<equiv> the_inv_into (Pow (w `\<rightarrow> C0)) ((`) \<psi>) y"
tan_total: fixes y :: "real" shows "\<exists>!x. - (pi / 2) < x \<and> x < pi / 2 \<and> tan x = y"
pick_def: fixes A :: "'d quot" shows "pick A = (SOME a. A = \<lfloor>a\<rfloor>)"
capacity_const: shows "\<forall>e. (0::'capacity) \<le> f' e \<and> f' e \<le> cf e"
assms_3: shows "z ^ h = \<chi> (a [^] h)"
map_G_id: fixes x :: "('j, 'k, 'l, 'm, 'n, 'o, 'p) G" shows "map_G id id id id id id x = x"
return_writerT_def: shows "Monad_Overloading.return = return_writer Monad_Overloading.return"
simps_3: fixes v :: "String.literal"   and va :: "formula"   and vb :: "formula" shows "get_and_list (formula.Let v va vb) = [formula.Let v va vb]"
substBinp_def: fixes xs :: "'varSort"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and x :: "'var" shows "substBinp xs X x \<equiv> psubstBinp (idEnv [x \<leftarrow> X]_xs)"
distinct_243: fixes x151 :: "floatarith"   and x152 :: "nat" shows "Pi \<noteq> x151 ^\<^sub>e x152"
dual_assertion_prop: fixes x :: "'a" assumes "x \<in> assertion" shows "x ^ o * \<bottom> \<squnion> (1::'a) = x ^ o"
mpoly_induct: fixes P :: "'a mpoly \<Rightarrow> bool"   and p :: "'a mpoly" assumes "\<And>m a. P (MPoly_Type.monom m a)"   and "\<And>p1 p2 m a. \<lbrakk>P p1; P p2; p2 = MPoly_Type.monom m a; m \<notin> keys (mapping_of p1)\<rbrakk> \<Longrightarrow> P (p1 + p2)" shows "P p"
iTILL_iMOD_neq: fixes m :: "nat"   and n :: "nat"   and r :: "nat" assumes "0 < m" shows "[\<dots>n] \<noteq> [ r, mod m ]"
le_sup_iff: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<le> x \<sqinter> y) = (z \<le> x \<and> z \<le> y)"
subtopology_superset: fixes U :: "'a topology"   and V :: "'a set" assumes "topspace U \<subseteq> V" shows "subtopology U V = U"
equivp_reflp2: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "equivp R" shows "reflp R"
cempty_iff: fixes c :: "'f" shows "cin c cempty = False"
range_oVeblen_subset: fixes x :: "ordinal"   and y :: "ordinal" assumes "x \<le> y" shows "range (oVeblen F y) \<subseteq> range (oVeblen F x)"
Sum_def: fixes x2 :: "ccs"   and x1 :: "ccs" shows "x2 \<oplus> x1 \<equiv> Abs_ccs (Sum_Rep (Rep_ccs x2) (Rep_ccs x1))"
split_option_all: fixes P :: "\<langle>'a\<rangle>\<^sub>\<bottom> \<Rightarrow> bool" shows "(\<forall>x. P x) = (P \<bottom> \<and> (\<forall>x. P \<lfloor>x\<rfloor>))"
extended_max: fixes F :: "'b formula"   and S :: "'b formula set" shows "F \<in> extended S \<or> \<^bold>\<not> F \<in> extended S"
simps_2: fixes s :: "'a \<Rightarrow> 'b option"   and op :: "('a, 'b) sas_plus_operator"   and ops :: "('a, 'b) sas_plus_operator list" shows "execute_serial_plan_sas_plus s (op # ops) = (if is_operator_applicable_in s op then execute_serial_plan_sas_plus (s \<then>\<^sub>+ op) ops else s)"
arcs_ends_mono: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "subgraph H G" shows "arcs_ends H \<subseteq> arcs_ends G"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "category A"   and "category B"   and "functor B A F"   and "functor A B G"   and "inverse_functors_axioms A B F G" shows "inverse_functors A B F G"
bigo_mult: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "O(f) * O(g) \<subseteq> O(f * g)"
rtn: shows "the (rt dip) = (dsn\<^sub>n, dsk\<^sub>n, f\<^sub>n, hops\<^sub>n, nhip\<^sub>n)"
le_approx_implies_le_ref: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq> Q" shows "P \<le> Q"
change: fixes j :: "nat" assumes "j < degree f - J + (degree g - J)" shows "((if j < degree f - J then j + (degree g - J) else j - (degree f - J)) < degree g - J) = (\<not> j < degree f - J)"
bell00_def: shows "|\<beta>\<^sub>0\<^sub>0\<rangle> \<equiv> complex_of_real (1 / sqrt 2) \<cdot>\<^sub>m |vec 4 (\<lambda>i. if i = 0 \<or> i = 3 then 1 else 0)\<rangle>"
CC_2: shows "qReachable cJ (Proc - {pJ}) cc"
match_iface_eqI: fixes x :: "iface"   and y :: "char list" assumes "x = Iface y" shows "match_iface x y"
Ipoly_snd_tmf_polys: fixes z :: "float interval list"   and x :: "real interval" shows "set_of (horner_eval (real_interval \<circ> centered \<circ> (!) z) x (length z)) \<subseteq> set_of (Ipoly [x] (map_poly real_interval (snd (tmf_polys z))))"
Arr_iff_cod_in_codomain: fixes f :: "'a" shows "left (trg \<mu>) f = (cod f \<in> Left.codomains f)"
subset_code_1: fixes xs :: "'h list"   and B :: "'h set" shows "(set xs \<subseteq> B) = (\<forall>x\<in>set xs. x \<in> B)"
hyps: fixes ys :: "'a list" shows "set ys \<subseteq> set (local.filter_min_aux xs ys)"
ord_trans: assumes "ord s t"   and "ord t u" shows "ord s u"
nub_body_nub_body'_eq: shows "unwrap oo nub_body oo wrap = nub_body'"
nodesI_1: fixes u :: "'b"   and v :: "'b"   and g :: "'b ugraph" assumes "(u, v) \<in> edges g" shows "u \<in> nodes g"
lc_higher: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<prec>\<^sub>t lt p" shows "lc (higher p v) = lc p"
distinct_10: fixes tyvrs' :: "tyvrs"   and ty1' :: "ty"   and ty2' :: "ty" shows "(\<forall>tyvrs'<:ty2'. ty1') \<noteq> Top"
seqll_def: fixes i :: "'a"   and P :: "('b \<times> 'c) \<times> 'd \<times> 'b \<times> 'c \<Rightarrow> bool" shows "seqll i P \<equiv> \<lambda>((\<sigma>, p), a, \<sigma>', p'). P ((\<sigma> i, p), a, \<sigma>' i, p')"
CallNone: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and ps :: "char list exp list"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b Ts T m D. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchMethodError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
contains_considers': fixes xs :: "(nat \<times> form) list" assumes "infinite (deriv s)" shows "\<forall>n y ys. snd (f n) = xs @ y # ys \<longrightarrow> (\<exists>m zs'. snd (f (n + m)) = y # zs')"
set_intros_2: fixes y :: "'g"   and x22 :: "'g list"   and x21 :: "'g" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
Inf_union_distrib: fixes A :: "'a set"   and B :: "'a set" shows "\<Sqinter> (A \<union> B) = \<Sqinter> A \<sqinter> \<Sqinter> B"
vector_scalar_mult_def: fixes c :: "'d"   and x :: "('d, 'e) vec" shows "c *s x = (\<chi>i. c * x $ i)"
newton_coefficients_int_def: fixes xs :: "int list"   and fs :: "int list" shows "newton_coefficients_int xs fs = map_option (map hd) (newton_coefficients_main_int xs (rev fs) (rev xs))"
mon_conv_mon: fixes x :: "nat \<Rightarrow> real"   and y :: "real"   and i :: "nat"   and m :: "nat" assumes "x\<up>y" shows "x i \<le> x (m + i)"
ENF_offending_imp_not_P: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('v \<times> 'v) set"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a"   and e1 :: "'v"   and e2 :: "'v" assumes "sinvar_all_edges_normal_form P"   and "F \<in> set_offending_flows G nP"   and "(e1, e2) \<in> F" shows "\<not> P (nP e1) (nP e2)"
star_denest: fixes x :: "'a list set"   and y :: "'a list set" shows "p_star (x \<union> y) = p_star (p_prod (p_star x) (p_star y))"
PAR_IMAGE_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd nres"   and X :: "'c set" shows "PAR_IMAGE P f X = list_spec X \<bind> (\<lambda>xs. nres_of_nress (\<lambda>(x, y). P x y) (Parallel.map (\<lambda>x. f x \<bind> (\<lambda>y. RETURN (x, y))) xs) \<bind> (\<lambda>fxs. RETURN (set fxs)))"
casts': shows "P \<turnstile> Class (last Cs') # Ts Casts Ref (a, Cs') # vs to Ref (a, Cs') # vs' "
assms_1: shows "lossless_gpv (\<I>_full \<oplus>\<^sub>\<I> \<I>_full) \<A>"
binary_chinese_remainder_int: fixes m1 :: "int"   and m2 :: "int"   and u1 :: "int"   and u2 :: "int" assumes "coprime m1 m2" shows "\<exists>x. [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
e_3: fixes s :: "'a" assumes "s \<in> cball x e" shows "f (s, u s) = (0::'c)"
fail_writer_def: fixes fail :: "'i" shows "fail_writer fail = WriterT fail"
catch_get: fixes f :: "'s \<Rightarrow> 'm"   and m :: "'m" shows "catch (get f) m = get (\<lambda>s. catch (f s) m)"
atLeastAtMost_def: fixes l :: "'a"   and u :: "'a" shows "{l..u} \<equiv> {l..} \<inter> {..u}"
cyclic_on_inI: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and s :: "'c" assumes "cyclic_on f S"   and "s \<in> S" shows "f s \<in> S"
assms_2: shows "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'b)) = (1::'b)"
\<L>\<^sub>P_split: fixes a :: "symbol list"   and b :: "symbol list" assumes "a @ b \<in> \<L>\<^sub>P" shows "a \<in> \<L>\<^sub>P"
p_sc: fixes U :: "'v set" assumes "U \<in> set p" shows "U \<times> U \<subseteq> (Restr lvE U)\<^sup>*"
SIGT_times1: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<sigma> :: "('b, 'c, 'a, 'd) subst_scheme"   and t1 :: "('b, 'e) trm"   and t2 :: "('b, 'e) trm" assumes "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT (Times t1 t2). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT t1. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"
reaches1_reaches_iff2: fixes x :: "'a"   and y :: "'a" shows "G'.reaches1 x y = (\<exists>z. G'.reaches x z \<and> E' z y)"