hcju/isabelleps huggingface dataset:
eval_fds_log_deriv: fixes f :: "'b fds"   and s :: "'b" assumes "fds_nth f 1 \<noteq> (0::'b)"   and "abs_conv_abscissa f < ereal (s \<bullet> (1::'b))"   and "abs_conv_abscissa (fds_deriv f / f) < ereal (s \<bullet> (1::'b))"   and "eval_fds f s \<noteq> (0::'b)" shows "eval_fds (fds_deriv f / f) s = eval_fds (fds_deriv f) s / eval_fds f s"
adjoint_transpose_left_2: fixes u :: "'a"   and v :: "'a" assumes "ide u"   and "ide v"   and "src f = trg v"   and "src g = trg u" shows "trnl\<^sub>\<epsilon> u \<in> hom v (g \<star> u) \<rightarrow> hom (f \<star> v) u"
inject_2: fixes x2 :: "'d"   and y2 :: "'d" shows "(Leaf x2 = Leaf y2) = (x2 = y2)"
fBallI: fixes A :: "'c fset"   and P :: "'c \<Rightarrow> bool" assumes "\<And>x. x |\<in>| A \<Longrightarrow> P x" shows "fBall A P"
itrm_brackets_Nil: fixes x :: "nat itrm" shows "itrm_brackets [] x = Some x"
mkAutoSim_simps_1: fixes a :: "'a" shows "pInit (mkAutoSim a) = simInit a"
denseI: fixes T :: "'b set set"   and A :: "'b set" assumes "carrier\<^bsub>T\<^esub> \<subseteq> closure\<^bsub>T\<^esub> A" shows "A dense\<^bsub>T\<^esub>"
continuous_on_cases_local: fixes s :: "'b set"   and t :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "closedin (top_of_set (s \<union> t)) s"   and "closedin (top_of_set (s \<union> t)) t"   and "continuous_on s f"   and "continuous_on t g"   and "\<And>x. x \<in> s \<and> \<not> P x \<or> x \<in> t \<and> P x \<Longrightarrow> f x = g x" shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
Tsmult1_def: shows "(\<star>\<currency>) \<equiv> \<lambda>a T f. a \<sharp>\<star> T f"
IH: fixes i :: "nat" assumes "i < length T" shows "(T ! i, S ! i) \<in> timpl_closure' c"
case_unit_Unity: fixes f :: "'a"   and u :: "unit" shows "(case u of () \<Rightarrow> f) = f"
difind_difelem: fixes i :: "nat"   and xs :: "'a list"   and j :: "nat"   and a :: "'a" assumes "i < length xs"   and "distinct xs"   and "xs ! j = a"   and "j < length xs"   and "i \<noteq> j" shows "a \<noteq> xs ! i"
cinsert_not_cempty: fixes a :: "'c"   and A :: "'c cset" shows "cinsert a A \<noteq> {}\<^sub>c"
PBaP: shows "P * B * adjoint P = P * adjoint B * adjoint P"
CALL_lemma: fixes q :: "pname"   and pre :: "state"   and c :: "com" shows "(\<Union>p. {(\<lambda>z s. (z = s \<and> body p \<down> s) \<and> ((p, s), q, pre) \<in> termi_call_steps, CALL p, \<lambda>z. exec z (body p))}) \<turnstile>\<^sub>t {\<lambda>z s. (z = s \<and> body q \<down> pre) \<and> (\<exists>cs. ([body q], pre) \<rightarrow>\<^sup>* (c # cs, s))} c {\<lambda>z. exec z c}"
prems_2: shows "fdisjnt (fset_of_list \<Gamma>) (frees (\<Lambda> t))"
get_shadow_root_safe_def: shows "get_shadow_root_safe \<equiv> l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_shadow_root_safe get_shadow_root get_mode"
assms: shows "zs_y \<in> Joints2 ZS Y"
horizontal_chop_empty_left: shows "\<forall>v. \<exists>u. v=u\<parallel>v"
assert_bound_unsat: fixes s :: "('i, 'a) state"   and as :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom"   and s' :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "index_valid as s"   and "assert_bound a s = s'"   and "\<U> s'" shows "minimal_unsat_state_core s'"
orthogonal_matrix_norm: fixes A :: "((real, 'b) vec, 'b) vec"   and i :: "'b" assumes "orthogonal_matrix A" shows "norm (column i A) = 1"
kmin: fixes k' :: "nat" assumes "k' < k" shows "\<gamma> \<notin> elts (\<omega> \<up> ord_of_nat n * ord_of_nat k')"
assms_4: shows "inj_on f (VS\<^sub>1.span S)"
guessed4_1: shows "zs2 = zs3 @- zs4"
elims: fixes x :: "hterm"   and y :: "fterm" assumes "fterm_of_hterm x = y" obtains p :: "char list" and ts :: "hterm list" where "x = HFun p ts" and "y = Fun p (map fterm_of_hterm ts)"
per_union_memI1: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "(x, y) \<in> R" shows "(x, y) \<in> per_union R a b"
main: fixes s :: "'a" shows "SN_on (S\<^sup>* O R O S\<^sup>*) {s} = SN_on (S\<^sup>* O R) {s}"
aUU: shows "adjoint U * U = 1\<^sub>m n"
simps_1: fixes a' :: "nat \<Rightarrow> int"   and b' :: "nat \<Rightarrow> int"   and B :: "int"   and N :: "nat" shows "get_c a' b' B N 0 = round (real_of_int (B * b' N) / real_of_int (a' N))"
of_rat_1_2: fixes x :: "rat" shows "real_of_1 (of_rat_1 x) = real_of_rat x"
dg_prod_Dom_app_in_Obj: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>" shows "dg_prod I \<AA>\<lparr>Dom\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>"
subspace_sum: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "module scale"   and "module.subspace scale A"   and "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" shows "sum f B \<in> A"
upper_max_interval: fixes x :: "'a interval"   and y :: "'a interval" shows "upper (max_interval x y) = max (upper x) (upper y)"
map_snd_zip_take: fixes xs :: "'e list"   and ys :: "'d list" shows "map snd (zip xs ys) = take (min (length xs) (length ys)) ys"
mirror_DD: fixes r :: "('a \<times> 'a) set"   and ars :: "('b \<times> 'a \<times> 'b) set"   and \<tau> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma> :: "'b \<times> ('a \<times> 'b) list"   and \<sigma>' :: "'b \<times> ('a \<times> 'b) list"   and \<tau>' :: "'b \<times> ('a \<times> 'b) list" assumes "trans r"   and "irrefl r"   and "DD ars r (\<tau>, \<sigma>, \<sigma>', \<tau>')" shows "DD ars r (\<sigma>, \<tau>, \<tau>', \<sigma>')"
cond_or_distr: fixes b :: "'a upred"   and P :: "'a upred"   and Q :: "'a upred"   and R :: "'a upred"   and S :: "'a upred" shows "(P \<or> Q) \<triangleleft> b \<triangleright> (R \<or> S) = (P \<triangleleft> b \<triangleright> R \<or> Q \<triangleleft> b \<triangleright> S)"
less_eq_st_rep_def: fixes ps1 :: "(char list \<times> 'a) list"   and ps2 :: "(char list \<times> 'a) list" shows "less_eq_st_rep ps1 ps2 = (\<forall>x\<in>set (map fst ps1) \<union> set (map fst ps2). fun_rep ps1 x \<le> fun_rep ps2 x)"
a_closed: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> y \<in> carrier G"
wf_lprod: fixes R :: "('a \<times> 'a) set" assumes "wf R" shows "wf (lprod R)"
semantics_put_imps: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and z :: "('b, 'c) form list"   and p :: "('b, 'c) form" shows "(e,f,g,z \<Turnstile> p) = eval e f g (put_imps p z)"
rbt_map_rel_finite: fixes Rk :: "('g \<times> 'h) set"   and eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and bhc :: "nat \<Rightarrow> 'g \<Rightarrow> nat"   and Rv :: "('i \<times> 'j) set" assumes "GEN_ALGO_tag (is_bounded_hashcode Rk eq bhc)" shows "finite_map_rel (\<langle>Rk, Rv\<rangle>ahm_rel bhc)"
is_int_rat_division: fixes y :: "int"   and x :: "int" assumes "y \<noteq> 0" shows "is_int_rat (rat_of_int x / rat_of_int y) = (x mod y = 0)"
ffun_app_minus: fixes x :: "'a"   and g :: "('a, 'b) ffun"   and f :: "('a, 'b) ffun" assumes "x \<notin> fdom g" shows "(f - g)(x)\<^sub>f = f(x)\<^sub>f"
mult_left: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)" shows "(\<lambda>x. h x * f x) \<in> O[F](\<lambda>x. h x * g x)"
in_minimal_antichain: fixes x :: "'b"   and A :: "'b set" shows "(x \<in> minimal_antichain A) = (x \<in> A \<and> \<not> (\<exists>y\<in>A. y < x))"
d_less: shows "d < return_time {x \<in> S. s x = 0} x"
hoare_valids_def: fixes G :: "'b triple set"   and ts :: "'b triple set" shows "G||=ts = (\<forall>n. ||=n:G \<longrightarrow> ||=n:ts)"
slice_complete: fixes xs :: "'c list" shows "Misc.slice 0 (length xs) xs = xs"
simps_1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "list_assn P [] [] = emp"
is_interval_existence_ivl: fixes x0a :: "'a"   and t0 :: "real"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" shows "is_interval (Y.mvar.existence_ivl x0a t0 x0)"
inv_BijGroup: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "f \<in> Bij S" shows "inv\<^bsub>BijGroup S\<^esub> f = restrict (inv_into S f) S"
char_to_digit_simps_10: shows "char_to_digit CHR ''9'' = 9"
divide_inverse: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a b = times a (inverse b)"
map_poly_hom_monom: fixes aa :: "'a"   and a :: "'a poly"   and i :: "nat" shows "map_poly (\<lambda>p. poly p aa) (monom a i) = monom (poly a aa) i"
commutativity1: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set"   and L :: "'b set set"   and T :: "'b set"   and G :: "'b program" assumes "F \<in> A \<longmapsto> B"   and "lattice L"   and "B \<in> L"   and "T \<in> L"   and "F \<in> stable T"   and "\<And>X. X \<in> L \<Longrightarrow> G \<in> X - B co X"   and "commutes F T B L" shows "F \<squnion> G \<in> T \<inter> A \<longmapsto> B"
obs: shows " |if p then x else y] q = d p \<cdot>  |if p then x else y] q + ad p \<cdot>  |if p then x else y] q"
theSubjectFreshChain: fixes \<alpha> :: "'a action"   and xvec :: "name list" assumes "\<alpha> \<noteq> \<tau>" shows "xvec \<sharp>* the (subject \<alpha>) = xvec \<sharp>* subject \<alpha>"
sums_mono_reindex: fixes g :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "strict_mono g"   and "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0::'b)" shows "(\<lambda>n. f (g n)) sums c = f sums c"
abc_ac_neq: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "a \<noteq> c"
unique_connecting_path_set_2: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "w \<in> set (v \<leadsto>\<^bsub>G\<^esub> w)"
ThinChamberComplexManyFoldingsI: fixes X :: "'b set set"   and C0 :: "'b set" assumes "ThinChamberComplex X"   and "SimplicialComplex.maxsimp X C0"   and "\<And>C D. \<lbrakk>SimplicialComplex.maxsimp X C; SimplicialComplex.maxsimp X D; C \<sim> D; C \<noteq> D\<rbrakk> \<Longrightarrow> \<exists>f g. OpposedThinChamberComplexFoldings X f g C \<and> D = g ` C" shows "ThinChamberComplexManyFoldings X C0"
hyps_2: fixes U :: "POPLmarkRecord.type"   and U' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> S' <: U \<rightarrow> U'" obtains S' :: "POPLmarkRecord.type" where "\<Gamma> \<turnstile> U <: S" and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'" and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'"
norm_powr_real_powr: fixes w :: "complex"   and z :: "complex" assumes "w \<in> \<real>"   and "0 \<le> Re w" shows "cmod (w powr z) = Re w powr Re z"
zero_cap_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> cf.E" shows "cf (u, v) = (0::'capacity)"
to_ns_unsat: fixes I :: "'i set"   and cs :: "('i \<times> constraint) list" assumes "minimal_unsat_core_ns I (set (to_ns cs))" shows "minimal_unsat_core I cs"
add_sup_inf_distribs_3: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "sup a b + c = sup (a + c) (b + c)"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_sub a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "is_App t" and "fun t \<ge>\<^sub>t s \<or> arg t \<ge>\<^sub>t s"
simps_11: fixes \<phi> :: "'b formula"   and I :: "\<I>"   and \<psi> :: "'b formula" shows "safe_formula (Until \<phi> I \<psi>) = (fv \<phi> \<subseteq> fv \<psi> \<and> (safe_formula \<phi> \<or> (case \<phi> of Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False)) \<and> safe_formula \<psi>)"
red_stat_red_is_stat_red: shows "reducedly_statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F = statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
Abs_image: shows "Abs_bit0 ` {0..<2 * int CARD('a)} = UNIV"
sel_2: fixes x21 :: "('c, 'd) tm"   and x22 :: "('c, 'd) tm" shows "head (App x21 x22) = head0 x21"
r_neg: fixes x :: "'a set" assumes "x \<in> carrier (R Quot I)" shows "x \<oplus>\<^bsub>R Quot I\<^esub> \<ominus>\<^bsub>R Quot I\<^esub> x = \<zero>\<^bsub>R Quot I\<^esub>"
rcos_assoc_lcos: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier H'"   and "K \<subseteq> carrier H'"   and "x \<in> carrier H'" shows "H #>\<^bsub>H'\<^esub> x <#>\<^bsub>H'\<^esub> K = H <#>\<^bsub>H'\<^esub> (x <#\<^bsub>H'\<^esub> K)"
bounded_linear_via_derivative: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'c"   and f' :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and y :: "'b"   and x :: "'b" assumes "\<And>i. ((\<lambda>x. blinfun_apply (f x) i) has_derivative (\<lambda>x. f' y x i)) (at y)" shows "bounded_linear (f' y x)"
is_factorization_of_roots: fixes a :: "'a"   and fctrs :: "('a \<times> nat) list"   and p :: "'a poly" assumes "is_factorization_of (a, fctrs) p"   and "p \<noteq> 0" shows "set (map fst fctrs) = {x. poly p x = (0::'a)}"
in_Sup_multiset_iff: fixes A :: "'k multiset set"   and x :: "'k" assumes "subset_mset.bdd_above A" shows "(x \<in># Sup A) = (\<exists>X\<in>A. x \<in># X)"
not_empty_hd_fst: fixes l :: "'a list"   and a :: "'a" assumes "l \<noteq> []" shows "hd (l @ [a]) = l ! 0"
functor_axioms: shows "functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a')"
uminus_apply: fixes A :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "(- A) x = - A x"
msubst2_def: fixes p :: "fm"   and c :: "poly"   and t :: "tm" shows "msubst2 p c t = Parametric_Ferrante_Rackoff.disj (Parametric_Ferrante_Rackoff.conj (lt (CP (~\<^sub>p c))) (simpfm (msubstpos p c t))) (Parametric_Ferrante_Rackoff.conj (lt (CP c)) (simpfm (msubstneg p c t)))"
bounded_sesquilinear_axioms: shows "bounded_sesquilinear (**)"
0: shows "(x \<oplus>\<^bsub>F\<^esub> y \<oplus>\<^bsub>F\<^esub> z) a = x a \<oplus> y a \<oplus> z a"
\<tau>move2_AAccD1: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and a :: "'addr expr1"   and i :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (a\<lfloor>i\<rceil>) pc xcp"   and "pc < length (compE2 a)" shows "\<tau>move2 P h stk a pc xcp"
exect_eq_abs_execute: fixes \<pi> :: "char list"   and s :: "nat \<Rightarrow> nat option"   and name :: "char list"   and preconds :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat" assumes "enabled \<pi> s"   and "lookup_operator \<pi> = Some (name, preconds, effs, layer)"   and "is_standard_operator' (name, preconds, effs, layer)" shows "execute \<pi> s = s \<then>\<^sub>+ (abs_ast_operator \<circ> the \<circ> lookup_operator) \<pi>"
mult_imp_less_div_pos: fixes y :: "'c"   and z :: "'c"   and x :: "'c" assumes "(0::'c) < y"   and "z * y < x" shows "z < x / y"
entails_trans: fixes Bot :: "'b set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and N1 :: "'b set"   and N2 :: "'b set"   and N3 :: "'b set" assumes "consequence_relation Bot entails"   and "entails N1 N2"   and "entails N2 N3" shows "entails N1 N3"
VPow_vsingleton: fixes a :: "V" shows "VPow (set {a}) = set {0, set {a}}"
snth_in: fixes s :: "'c stream"   and X :: "'c set"   and n :: "nat" assumes "s \<in> streams X" shows "s !! n \<in> X"
add_comm: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "sup_prod x y = sup_prod y x"
freshBoundOutputTransition: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "c \<sharp> P"   and "c \<noteq> x" shows "c \<sharp> P'"
fps_nth_power_0: fixes a :: "'a fps"   and n :: "nat" shows "a ^ n $ 0 = (a $ 0) ^ n"
wb_lens_def: fixes x :: "'e \<Longrightarrow> 'f" shows "wb_lens x \<equiv> weak_lens x \<and> wb_lens_axioms x"
Xd1'_Xd2_stable: shows "BSI.stable_on UNIV {X5.Xd1', X5.Xd2}"
var_dist_subst_Var: fixes vs :: "nat list"   and s :: "dB"   and i :: "nat"   and j :: "nat" shows "var_dist vs s[Var i/j] = var_dist (map (\<lambda>v. vsubst v i j) vs) (s[Var i/j])"
left_unique_rel_resource: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "left_total A"   and "left_unique B" shows "left_unique (rel_resource A B)"
norm_axis_1: fixes m :: "'b" shows "\<parallel>\<e> m\<parallel> = 1"
asIMOD_igSubstBinp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and ys :: "'varSort"   and Y :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and y :: "'var"   and binp :: "'c \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs) option" assumes "sWlsVal SEM val" shows "lift (\<lambda>A. A val) (igSubstBinp (asIMOD SEM) ys Y y binp) = lift (\<lambda>A. A (val (y := Y val)_ys)) binp"
invsL_def: shows "invsL = (\<lambda>s. gc.invsL s \<and> (\<forall>x. mut_m.invsL x s))"
ENC_FOr: fixes n :: "nat"   and \<phi> :: "'a formula"   and \<psi> :: "'a formula" assumes "wf_formula n (FOr \<phi> \<psi>)" shows "lang n (ENC n (FOV (FOr \<phi> \<psi>))) \<subseteq> lang n (ENC n (FOV \<phi>)) \<inter> lang n (ENC n (FOV \<psi>))"
SDG_cdep_edge_CFG_node: 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 n :: "'b SDG_node"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n None None n'" shows "\<exists>m. n = CFG_node m"
Cons: shows "(gval (foldr gAnd G (Bc True)) s = true) = (\<forall>g\<in>set G. gval g s = true)"
orthogonal_complementI: fixes M :: "'b set"   and y :: "'b" assumes "\<And>x. x \<in> M \<Longrightarrow> is_orthogonal y x" shows "y \<in> orthogonal_complement M"
\<nu>_simps_5: shows "cod \<nu> = g \<star> w"
firstOccurrence_def: fixes msg :: "('p, 'v) message"   and n :: "nat" shows "firstOccurrence msg n \<equiv> (\<exists>p. isReceiverOf p msg) \<and> enabled (last exec) msg \<and> n < length exec \<and> enabled (execConf n) msg \<and> (\<forall>n'\<ge>n. n' < length trace \<longrightarrow> msg \<noteq> execMsg n') \<and> (n \<noteq> 0 \<longrightarrow> \<not> enabled (execConf (n - 1)) msg \<or> msg = execMsg (n - 1))"
power2_commute: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "power.power one times (minus x y) 2 = power.power one times (minus y x) 2"
tsubstT_ith: fixes i :: "nat"   and Ts :: "typ list"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" assumes "i < length Ts" shows "map (\<lambda>T. tsubstT T \<rho>) Ts ! i = tsubstT (Ts ! i) \<rho>"
ap_cfD1: fixes f :: "'a itrm"   and x :: "'a itrm" assumes "f \<diamondop> x \<in> CF" shows "f \<in> CF"
fro_1: shows "frontier (inside (path_image p)) = path_image p"
measure_pmf_eq_density: fixes p :: "'c pmf" shows "measure_pmf p = density (count_space UNIV) (\<lambda>x. ennreal (pmf p x))"
showsp_list_def: fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and p :: "nat" shows "showsp_list s p = pshowsp_list p \<circ> map (s 0)"
a_owner_document_valid_def: fixes h :: "(_) heap" shows "local.CD.a_owner_document_valid h = (\<forall>node_ptr\<in>fset (node_ptr_kinds h). (\<exists>document_ptr. document_ptr |\<in>| document_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r) \<or> (\<exists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r))"
logical_relation_def: fixes R :: "('b \<Rightarrow> ValD) set" shows "logical_relation R \<equiv> (\<forall>fs\<in>R. \<forall>xs\<in>R. (\<lambda>j. appF\<cdot>(fs j)\<cdot>(xs j)) \<in> R) \<and> (\<forall>fs\<in>R. \<forall>xs\<in>R. (\<lambda>j. strictify\<cdot>(appF\<cdot>(fs j))\<cdot>(xs j)) \<in> R) \<and> (\<forall>fs. (\<forall>xs\<in>R. (\<lambda>j. fs j\<cdot>(xs j)) \<in> R) \<longrightarrow> (\<lambda>j. ValF\<cdot>(fs j)) \<in> R) \<and> (\<forall>fs. (\<forall>xs\<in>R. (\<lambda>j. strictify\<cdot>(fs j)\<cdot>(xs j)) \<in> R) \<longrightarrow> (\<lambda>j. ValF\<cdot>(strictify\<cdot>(fs j))) \<in> R) \<and> (\<forall>xs\<in>R. (\<lambda>j. fixD\<cdot>(xs j)) \<in> R) \<and> (\<forall>cs\<in>R. \<forall>ts\<in>R. \<forall>es\<in>R. (\<lambda>j. cond\<cdot>(cs j)\<cdot>(ts j)\<cdot>(es j)) \<in> R) \<and> (\<forall>xs\<in>R. (\<lambda>j. succ\<cdot>(xs j)) \<in> R) \<and> (\<forall>xs\<in>R. (\<lambda>j. pred\<cdot>(xs j)) \<in> R) \<and> (\<forall>xs\<in>R. (\<lambda>j. isZero\<cdot>(xs j)) \<in> R)"
of_int_poly_P: fixes q :: "int poly" assumes "q \<in> P" shows "Ring_Hom_Poly.of_int_poly q = Polynomial.smult (of_int (Polynomial.lead_coeff q)) (\<Prod>x\<in>Roots q. [:- x, 1:])"
plain_floatarith_approx: fixes f :: "floatarith"   and p :: "nat" assumes "plain_floatarith 0 f" shows "interpret_floatarith f [] \<in>\<^sub>r the (approx p f [])"
f_last_message_hold_idem: fixes xs :: "'b message_af list"   and k :: "nat" shows "xs \<longmapsto> k \<longmapsto> k = xs \<longmapsto> k"
\<nu>_simps_5: shows "cod \<nu> = g' \<star> w"
measurable_emeasure: fixes N :: "'c measure"   and A :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space lborel"   and "{x \<in> space (N \<Otimes>\<^sub>M lborel). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M lborel)" shows "(\<lambda>x. emeasure lborel (A x)) \<in> borel_measurable N"
env_restr_eqI: fixes S :: "'c set"   and m\<^sub>1 :: "'c \<Rightarrow> 'd"   and m\<^sub>2 :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> S \<Longrightarrow> m\<^sub>1 x = m\<^sub>2 x" shows "m\<^sub>1 f|` S = m\<^sub>2 f|` S"
bilinear_l_add1: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m11 :: "'g"   and m12 :: "'g"   and m2 :: "'i"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m11 \<in> carrier M1"   and "m12 \<in> carrier M1"   and "m2 \<in> carrier M2"   and "bilinear_map f R M1 M2 N" shows "f (m11 \<plusminus>\<^bsub>M1\<^esub> m12, m2) \<plusminus>\<^bsub>N\<^esub> -\<^sub>a\<^bsub>N\<^esub> (f (m11, m2) \<plusminus>\<^bsub>N\<^esub> f (m12, m2)) = \<zero>\<^bsub>N\<^esub>"
c1: shows "\<langle>v|v\<rangle> \<noteq> 0"
aboveS_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b" shows "Order_Relation.aboveS r a \<equiv> {b. b \<noteq> a \<and> (a, b) \<in> r}"
union: fixes A :: "'a word set"   and B :: "'a word set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "signed.Min (A \<union> B) = signed.min (signed.Min A) (signed.Min B)"