hcju/isabelleps huggingface dataset:
6: shows "F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star> \<le> top * e * F"
wellFormedAppend: fixes subst :: "'d \<Rightarrow> name list \<Rightarrow> 'e list \<Rightarrow> 'd"   and \<sigma> :: "('f list \<times> 'g list) list"   and \<sigma>' :: "('f list \<times> 'g list) list" assumes "substType subst" shows "wellFormedSubst (\<sigma> @ \<sigma>') = (wellFormedSubst \<sigma> \<and> wellFormedSubst \<sigma>')"
eql: shows "range (flow0 (flow0 y t)) = range (flow0 y)"
list_all_append: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list" shows "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
arcosh_double: fixes x :: "real" assumes "1 \<le> x" shows "2 * arcosh x = arcosh (2 * x\<^sup>2 - 1)"
RangeRepl: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "HOLZF.Range (Lambda A f) = Repl A f"
1: shows "\<turnstile> \<not> SF(A)_v = (\<box>\<diamond>Enabled \<langle>A\<rangle>_v \<and> \<not> \<box>\<diamond>\<langle>A\<rangle>_v)"
set_list_of_vec: fixes v :: "'a vec" shows "set (list_of_vec v) = set\<^sub>v v"
qfmq: shows "qfree (simpfm (minusinf (fst (chooset p))))"
restrict_nonempty_product_less_eq: fixes ks :: "'c list"   and k :: "'c"   and ls :: "'c list"   and l :: "'c"   and ms :: "'c list"   and m :: "'c"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd"   and h :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<not> List.member ks k"   and "\<not> List.member ls l"   and "\<not> List.member ms m" shows "((k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms) \<preceq> (k # ks) \<langle>h\<rangle> (m # ms)) = ([k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m] \<preceq> [k] \<langle>h\<rangle> [m] \<and> [k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms \<preceq> [k] \<langle>h\<rangle> ms \<and> ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m] \<preceq> ks \<langle>h\<rangle> [m] \<and> ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms \<preceq> ks \<langle>h\<rangle> ms)"
unop_f32: fixes c :: "f32"   and fop :: "unop_f" shows "\<lparr>[$C ConstFloat32 c, $Unop_f T_f32 fop]\<rparr> \<leadsto> \<lparr>[$C ConstFloat32 (app_unop_f fop c)]\<rparr>"
cont_Pair: fixes lub :: "'a set \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'c" assumes "cont lub ord luba orda f"   and "cont lub ord lubb ordb g" shows "cont lub ord (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
as_a: shows "a \<in> at_least_as_good x carrier relation - as_good_as x carrier relation"
less_eq_flat_complete_lattice_def: fixes x :: "'a flat_complete_lattice"   and y :: "'a flat_complete_lattice" shows "x \<le> y \<equiv> case x of flat_complete_lattice.Value v1 \<Rightarrow> case y of flat_complete_lattice.Value v2 \<Rightarrow> v1 = v2 | flat_complete_lattice.Bot \<Rightarrow> False | flat_complete_lattice.Top \<Rightarrow> True | flat_complete_lattice.Bot \<Rightarrow> True | flat_complete_lattice.Top \<Rightarrow> y = flat_complete_lattice.Top"
h_times: fixes x :: "'a"   and y :: "'a" shows "h (x * y) = h x * h y"
vars_monom_single_cases: fixes v :: "nat"   and k :: "nat"   and a :: "'b" shows "vars (monom (Poly_Mapping.single v k) a) = (if k = 0 \<or> a = (0::'b) then {} else {v})"
weak_GB_is_strong_GB: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>f. f \<in> pmdl G \<Longrightarrow> (red G)\<^sup>*\<^sup>* f 0" shows "is_Groebner_basis G"
defAss'_def: fixes g :: "'a"   and m :: "'node"   and v :: "'var" shows "CFG2.defAss' g m v = (\<forall>ns. CFG2.path2 g (Entry2 g) ns m \<longrightarrow> (\<exists>n\<in>set ns. v \<in> defs2 g n))"
i_in_Liminf_or_Red_F: fixes Ns :: "'f set llist"   and i :: "nat" assumes "chain (\<rhd>) Ns"   and "enat i < llength Ns" shows "lnth Ns i \<subseteq> Red_F (Liminf_llist Ns) \<union> Liminf_llist Ns"
adjoint_eqI: fixes F :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and G :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "\<And>x y. (F *\<^sub>V x) \<bullet>\<^sub>C y = x \<bullet>\<^sub>C (G *\<^sub>V y)" shows "F = G*"
comp_ide_self: fixes a :: "'a" assumes "L.ide a" shows "L.comp a a = a"
code: shows "code = [Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]"
comp_preserves_can: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "can f"   and "can g"   and "local.dom f = cod g" shows "can (f \<cdot> g)"
less_eq_dual_def: shows "(\<le>) \<equiv> map_fun undual (map_fun undual id) (\<lambda>x y. y \<le> x)"
prebicategory_with_homs_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" shows "prebicategory_with_homs V H \<a> src trg \<equiv> prebicategory V H \<a> \<and> weak_composition_with_homs V H src trg"
prems_2: fixes i :: "nat" assumes "i < length (x # xl)" shows "intT (tpOfV ((x # xl) ! i)) ((a # al) ! i)"
ordLeq_transitive: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "ordLeq3 r r'"   and "ordLeq3 r' r''" shows "ordLeq3 r r''"
dvdmE': fixes p :: "int poly"   and q :: "int poly" assumes "p dvdm q" obtains r :: "int poly" where "q =m p * Mp r"
terminates_mark_guards_to_terminates_Normal: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and c :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>mark_guards f c \<down> Normal s" shows "\<Gamma>\<turnstile>c \<down> Normal s"
iMOD_div_right_inj_on: fixes k :: "nat"   and m :: "nat"   and r :: "nat" assumes "0 < k"   and "k \<le> m" shows "inj_on (\<lambda>x. x div k) [ r, mod m ]"
disc_1: fixes x1 :: "'g" shows "is_Pure (Pure x1)"
eq: fixes A :: "'i \<Rightarrow> 'a set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)" shows "emeasure P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
error_msg_ne_SUCCES_3: fixes \<phi> :: "char list \<Rightarrow> nat"   and i :: "'a"   and m :: "char list" assumes "poly_embed \<phi>" shows "is_cfailed (error_msg i m)"
env_axioms: shows "env Fubx_empty Fubx_get Fubx_add Fubx_to_list"
simps_2: fixes t :: "'b"   and n :: "nat" shows "unlock_lock \<lfloor>(t, n)\<rfloor> = (case n of 0 \<Rightarrow> None | Suc n' \<Rightarrow> \<lfloor>(t, n')\<rfloor>)"
ntr_is_tr: fixes c :: "'a list multiset"   and ww :: "('b, 'c) label list list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "(c, ww, c') \<in> trcl (ntr fg)" shows "(c, foldl (@) [] ww, c') \<in> trcl (tr fg)"
freshAbs_psubstAbs_I: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "goodAbs A"   and "goodEnv rho"   and "rho zs z = None \<Longrightarrow> freshAbs zs z A"   and "\<And>ys y Y. rho ys y = Some Y \<Longrightarrow> freshAbs ys y A \<or> fresh zs z Y" shows "freshAbs zs z A $[rho]"
lemma2_6_1_set: fixes r :: "('a \<times> 'a) set"   and S :: "'a set"   and T :: "'a set" shows "r \<down>s (S \<union> T) = r \<down>s S \<union> r \<down>s T"
Equality: fixes a :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c (update\<^sub>s\<^sub>t \<S> (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
Mf: shows "f \<in> borel_measurable (Pi\<^sub>M (insert v V) N)"
XD_XH_gfp_F_1: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "fst XD_XH \<subseteq> fst (gfp_F ds)"
rep_eq: fixes x :: "'a fset fset" shows "fset (ffUnion x) = \<Union> (fset ` fset x)"
iMOD_cut_ge: fixes r :: "nat"   and m :: "nat"   and t :: "nat" shows "[ r, mod m ] \<down>\<ge> t = (if t \<le> r then [ r, mod m ] else if m = 0 then {} else [ t + m - Suc ((t - Suc r) mod m), mod m ])"
edom_cHeap: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "edom (cHeap \<Delta> e\<cdot>a) = edom (Aheap \<Delta> e\<cdot>a)"
preserves_arr: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.arr (VoV.map f)"
abs_conv_abscissa_MInf_iff: fixes f :: "'a fds" shows "(abs_conv_abscissa f = - \<infinity>) = All (fds_abs_converges f)"
translation_Compl: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and t :: "'a set" assumes "class.ab_group_add plus zero minus uminus" shows "plus a ` (- t) = - plus a ` t"
measure_eqI_PiM_infinite: fixes P :: "('b \<Rightarrow> 'c) measure"   and I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and Q :: "('b \<Rightarrow> 'c) measure" assumes "sets P = sets (Pi\<^sub>M I M)"   and "sets Q = sets (Pi\<^sub>M I M)"   and "\<And>A J. \<lbrakk>finite J; J \<subseteq> I; \<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)\<rbrakk> \<Longrightarrow> emeasure P (prod_emb I M J (Pi\<^sub>E J A)) = emeasure Q (prod_emb I M J (Pi\<^sub>E J A))"   and "finite_measure P" shows "P = Q"
complement_2: fixes x :: "'a"   and y :: "'a" shows "- x = - (- (- x \<squnion> - y) \<squnion> - (- x \<squnion> - - y))"
mult_of_nat_commute: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "nat"   and y :: "'b" assumes "class.semiring_1 one times plus zero" shows "times (semiring_1.of_nat one plus zero x) y = times y (semiring_1.of_nat one plus zero x)"
roots_def: fixes p :: "'a list list" shows "UP.roots p = Abs_multiset (UP.alg_mult p)"
refl_onD1: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "refl_on A r"   and "x \<succeq>[r] y" shows "x \<in> A"
succ_rtran_def: fixes G :: "'b list_graph"   and v :: "'b" shows "FiniteListGraph.succ_rtran G v = rtrancl_list_impl (edgesL G) [v]"
\<sigma>'_1: shows "Unifier \<sigma>' (s \<cdot> \<I>) (t \<cdot> \<I>)"
from_to_enum: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "nat" assumes "class.enum enum enum_all enum_ex"   and "x \<le> enum.fromEnum enum (enum.maxBound enum)" shows "enum.fromEnum enum (enum.toEnum enum x) = x"
FG_advice_b1: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and Y :: "'b ltln set" assumes "\<F>\<G> \<phi> w \<subseteq> Y"   and "w \<Turnstile>\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<Sigma>\<^sub>1"
set_product: fixes xs :: "'b list"   and ys :: "'c list" shows "set (List.product xs ys) = set xs \<times> set ys"
assoc'_in_hom_2: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<guillemotleft>\<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] : D.dom f \<star>\<^sub>D D.dom g \<star>\<^sub>D D.dom h \<Rightarrow>\<^sub>D (D.cod f \<star>\<^sub>D D.cod g) \<star>\<^sub>D D.cod h\<guillemotright>"
simps_9: shows "subcomp sA9 = {sA91, sA92, sA93}"
disjoint_alt_simp1: fixes A :: "'a set"   and B :: "'a set" shows "(A - B = A) = (A \<inter> B = {})"
iMin_Un: fixes A :: "'a set"   and B :: "'a set" assumes "A \<noteq> {}"   and "B \<noteq> {}" shows "iMin (A \<union> B) = min (iMin A) (iMin B)"
ith_def: fixes B :: "('c \<Rightarrow> 'b) set"   and i :: "'c" shows "ith B i = {f i |f. f \<in> B}"
I_intP_Fext_imp: fixes p :: "'psym"   and al :: "univ list" assumes "polC p = Fext"   and "\<not> intP p al" shows "\<not> I_intP p al"
homeomorphism_of_subsets: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and S' :: "'a set"   and T'' :: "'b set"   and T' :: "'b set" assumes "homeomorphism S T f g"   and "S' \<subseteq> S"   and "T'' \<subseteq> T"   and "f ` S' = T'" shows "homeomorphism S' T' f g"
closed_cspan_range_ket: shows "closure (cspan (range ket)) = UNIV"
assms_1: shows "x \<in> rvars_eq (pivot_eq e v)"
triangle_in_hom_2: shows "\<guillemotleft>(g' \<star> \<epsilon>') \<cdot> \<a>[g', f, g'] \<cdot> (\<eta>' \<star> g') : trg g' \<star> g' \<Rightarrow> g' \<star> src g'\<guillemotright>"
rep_eq: fixes x :: "'a\<^sup>+"   and xa :: "'a\<^sup>+" shows "Rep_pos (x * xa) = Rep_pos x * Rep_pos xa"
matrix_scalar_vector_ac: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and k :: "'a"   and v :: "('a, 'c) Finite_Cartesian_Product.vec" shows "A *v (k *s v) = k *k A *v v"
union_filter_mset_complement: fixes P :: "'k \<Rightarrow> bool"   and Q :: "'k \<Rightarrow> bool"   and M :: "'k multiset" assumes "\<forall>x. P x = (\<not> Q x)" shows "filter_mset P M + filter_mset Q M = M"
component_in_hom: fixes j :: "'j" assumes "J.arr j" shows "\<guillemotleft>\<chi> j : S.unity \<rightarrow> D (J.cod j)\<guillemotright>"
assm: shows "left (space ts v c) \<le> right (ext v)"
rmlab_def: fixes k :: "'a"   and A :: "'a \<Rightarrow> 'b set" shows "rmlab k A = (\<lambda>x. if x = k then UNIV else A x)"
fset_of_list_empty: fixes l :: "'b list" shows "(fset_of_list l = {||}) = (l = [])"
0: shows "prv (imp \<phi>G (neg (PP \<langle>\<phi>G\<rangle>)))"
compress_interfaces_None_2: fixes \<beta> :: "'c common_primitive \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and ifces :: "iface negation_type list"   and \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('c, 'd) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>"   and "compress_interfaces ifces = None" shows "\<not> matches (\<beta>, \<alpha>) (alist_and (NegPos_map OIface ifces)) a p"
S_transform_observable_transition_iff: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and \<alpha> :: "'c"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.observable_transition (nominal_ts.S_transition satisfies transition) (S_action.Act \<tau>) P (S_action.Act \<alpha>) P' = weak_nominal_ts.observable_transition transition \<tau> P \<alpha> P'"
BRelRel': fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> bangRel Rel" shows "(P, Q) \<in> bangRel Rel'"
subst_rel_lift_comp: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and \<rho> :: "'b \<Rightarrow> 'b" shows "\<lceil>\<sigma>\<rceil>\<^sub>s \<circ> \<lceil>\<rho>\<rceil>\<^sub>s = \<lceil>\<sigma> \<circ> \<rho>\<rceil>\<^sub>s"
qfree_lift_nnf_qe: fixes aneg :: "'c \<Rightarrow> 'c fm"   and anormal :: "'c \<Rightarrow> bool"   and I\<^sub>a :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and depends\<^sub>0 :: "'c \<Rightarrow> bool"   and decr :: "'c \<Rightarrow> 'c"   and qe :: "'c fm \<Rightarrow> 'c fm"   and \<phi> :: "'c fm" assumes "ATOM aneg anormal I\<^sub>a depends\<^sub>0 decr"   and "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)" shows "qfree (ATOM.lift_nnf_qe aneg qe \<phi>)"
arr_char: fixes f :: "'d" shows "D'.arr f = (D.arr f \<and> Obj (src\<^sub>D f) \<and> Obj (trg\<^sub>D f))"
iprev_le_iMin: fixes n :: "nat"   and I :: "nat set" assumes "n \<le> iMin I" shows "iprev n I = n"
jb_imp_uppert: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" assumes "??.Jordan_Normal_Form_Existence.jb A i j" shows "??.Jordan_Normal_Form_Existence.uppert A i j"
simps_2: fixes n :: "nat" shows "local.wf n Full = True"
Sec_def: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and R :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and S :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" shows "Sec A R S \<Phi> s t = (A s t \<and> (\<forall>r. R s r \<longrightarrow> \<Phi> (t, r)) \<and> (\<forall>r. \<Phi> (r, s) \<longrightarrow> S r t))"
root_def: fixes tree :: "'b tree" shows "root tree = (case tree of Node x1 x2 x3 \<Rightarrow> x1)"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "'a"   and f6 :: "'a"   and f7 :: "'a"   and f8 :: "'a"   and f9 :: "'a"   and f10 :: "'a"   and f11 :: "'a"   and f12 :: "'a"   and f13 :: "'a"   and f14 :: "'a"   and f15 :: "'a"   and f16 :: "'a"   and f17 :: "'a"   and bop :: "bop" shows "P (case bop of Eq \<Rightarrow> f1 | NotEq \<Rightarrow> f2 | LessThan \<Rightarrow> f3 | LessOrEqual \<Rightarrow> f4 | GreaterThan \<Rightarrow> f5 | GreaterOrEqual \<Rightarrow> f6 | Add \<Rightarrow> f7 | Subtract \<Rightarrow> f8 | Mult \<Rightarrow> f9 | Div \<Rightarrow> f10 | Mod \<Rightarrow> f11 | BinAnd \<Rightarrow> f12 | BinOr \<Rightarrow> f13 | BinXor \<Rightarrow> f14 | ShiftLeft \<Rightarrow> f15 | ShiftRightZeros \<Rightarrow> f16 | ShiftRightSigned \<Rightarrow> f17) = (\<not> (bop = Eq \<and> \<not> P f1 \<or> bop = NotEq \<and> \<not> P f2 \<or> bop = LessThan \<and> \<not> P f3 \<or> bop = LessOrEqual \<and> \<not> P f4 \<or> bop = GreaterThan \<and> \<not> P f5 \<or> bop = GreaterOrEqual \<and> \<not> P f6 \<or> bop = Add \<and> \<not> P f7 \<or> bop = Subtract \<and> \<not> P f8 \<or> bop = Mult \<and> \<not> P f9 \<or> bop = Div \<and> \<not> P f10 \<or> bop = Mod \<and> \<not> P f11 \<or> bop = BinAnd \<and> \<not> P f12 \<or> bop = BinOr \<and> \<not> P f13 \<or> bop = BinXor \<and> \<not> P f14 \<or> bop = ShiftLeft \<and> \<not> P f15 \<or> bop = ShiftRightZeros \<and> \<not> P f16 \<or> bop = ShiftRightSigned \<and> \<not> P f17))"
points_index_singleton: fixes b :: "'b set"   and ps :: "'b set" shows "({#b#} index ps = 1) = (ps \<subseteq> b)"
a2: shows "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)"
ccloses_def: fixes \<theta>_c :: "(coname \<times> name \<times> trm) list"   and \<Delta> :: "(coname \<times> ty) list" shows "\<theta>_c ccloses \<Delta> \<equiv> \<forall>a B. (a, B) \<in> set \<Delta> \<longrightarrow> (\<exists>x P. \<theta>_c cmaps a to Some (x, P) \<and> x:P \<in> \<parallel>(B)\<parallel>)"
YMapImage: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and X :: "'d"   and \<eta> :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) NatTrans" assumes "LSCategory C"   and "Ftor F : Op C \<longrightarrow> SET"   and "X \<in> obj\<^bsub>C\<^esub>"   and "NT \<eta> : YFtor C @@ X \<Longrightarrow> F" shows "YMap C X \<eta> |\<in>| (F @@ X)"
galleryD_chamber: fixes xs :: "'b set list"   and x :: "'b set" assumes "codomain.gallery xs"   and "x \<in> set xs" shows "codomain.chamber x"
matchCases: fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual"   and F :: "name \<Rightarrow> name \<Rightarrow> bool" assumes "[a\<frown>b]P \<longmapsto> Rs"   and "P \<longmapsto> Rs \<Longrightarrow> F a a" shows "F a b"
IH: assumes "no_Emptys (h3 # hs)" shows "\<Phi>_hps (pass\<^sub>1 (h3 # hs)) - \<Phi>_hps (h3 # hs) \<le> sum_ub (h3 # hs)"
bounded_by_mono_alt: fixes b :: "real"   and Q :: "'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "bounded_by b Q"   and "P \<tturnstile> Q" shows "bounded_by b P"
app_vconverseE: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>" obtains a :: "V" and b :: "V" where "x = \<langle>b, a\<rangle>" and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
module_ow_axioms: shows "module_ow U\<^sub>M\<^sub>_\<^sub>2 (+) (0::'c) (-) uminus (*s\<^sub>2)"
simps_1: fixes uu :: "nat"   and i :: "int"   and \<beta> :: "nat \<Rightarrow> 'a option"   and ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" shows "\<aA> (C uu i) \<beta> ve = {}"
simps: fixes f :: "'a \<Rightarrow> 'a"   and p :: "'a \<Rightarrow> bool"   and n :: "nat"   and xs :: "'a list" shows "filter_list_length f p n xs = (let ys = filter p xs in if length ys = n then ys else filter_list_length f p n (map f ys))"
mult_cancellation: fixes a :: "V"   and x :: "V"   and y :: "V" assumes "a \<noteq> 0" shows "(a * x = a * y) = (x = y)"
path_path_weight: fixes l1 :: "'loc"   and l2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list" assumes "path l1 l2 xs" shows "\<exists>s. s \<in>\<^sub>A path_summary l1 l2 \<and> s \<le> sum_weights (map (\<lambda>(s, l, t). l) xs)"
sums_cong: fixes f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd"   and c :: "'d" assumes "\<And>n. f n = g n" shows "f sums c = g sums c"
intersecting_sets_def: fixes X :: "'b set"   and Z :: "'b set" shows "intersecting_sets X Z = sets_of_coll Z \<bind> (\<lambda>ZS. sets_of_coll X \<bind> (\<lambda>XS. FORWEAK XS (RETURN (op_empty_coll, op_empty_coll)) (\<lambda>X. FORWEAK ZS (RETURN True) (disjoint_sets X) (\<lambda>a b. RETURN (a \<and> b)) \<bind> (\<lambda>d. RETURN (if d then (op_empty_coll, mk_coll X) else (mk_coll X, op_empty_coll)))) (\<lambda>(R, S) (R', S'). RETURN (R' \<union> R, S' \<union> S))))"
product_eq: fixes x :: "i"   and y :: "i" shows "x \<otimes> y = (THE z. \<forall>v. P v z = (P v x \<and> P v y))"
match_fresh: fixes p :: "pat"   and u :: "trm"   and \<theta> :: "(name \<times> trm) list" assumes "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>"   and "supp p \<sharp>* u" shows "supp (map fst \<theta>) \<sharp>* map snd \<theta>"
resl_comp1: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y \<leftarrow> y) \<cdot> y = x \<cdot> y"
l3: fixes x :: "int set" assumes "x \<in> local.S2" shows "finite x"
extend_set_eq_Collect: fixes P :: "'a \<Rightarrow> bool" shows "extend_set h {s. P s} = {s. P (f s)}"
locally_reflects_ide: fixes f :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> a\<guillemotright>"   and "ide (cod \<iota> \<otimes> f)" shows "ide f"
correctCompositionKS_exprChannel_k_P: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and key :: "Keys"   and ch :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "kKS key \<notin> LocalSecrets PQ"   and "ch \<in> ins P"   and "exprChannel ch (kE key)"   and "kKS key \<notin> specKeysSecrets PQ"   and "correctCompositionIn PQ" shows "ch \<in> ins PQ \<and> exprChannel ch (kE key)"
Var_Idem: fixes v :: "'b"   and t :: "'b trm" assumes "\<not> Var v \<prec> t" shows "Idem [(v, t)]"
timpl_closure_Fun_inv'_1: fixes g :: "('a, 'b, 'c) prot_fun"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and f :: "('a, 'b, 'c) prot_fun"   and T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and TI :: "('c set \<times> 'c set) set" assumes "Fun g S \<in> timpl_closure (Fun f T) TI" shows "length S = length T"
counit_in_vhom: shows "\<guillemotleft>\<psi> : F f \<star>\<^sub>D g' \<Rightarrow>\<^sub>D src\<^sub>D g'\<guillemotright>"
parts_initState: fixes C :: "agent" shows "parts (initState C) = initState C"