hcju/isabelleps huggingface dataset:
ewp_neg_and_omega: fixes x :: "'a" shows "(\<not> ewp x) = (x\<^sup>\<omega> = (0::'a))"
r_inv: fixes x :: "cltn2" assumes "x \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "x \<otimes>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = \<one>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"
seq_char: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "A_B.seq g f = (A_B.arr f \<and> A_B.arr g \<and> A_B.Dom g = A_B.Cod f)"
iso_assoc: fixes f :: "'d"   and g :: "'d"   and h :: "'d" 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 "D.iso \<a>\<^sub>D[f, g, h]"
cases: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b stream" assumes "class.order less_eq less"   and "order.sdescending less_eq a" obtains aa :: "'b" and b :: "'b" and w :: "'b stream" where "a = aa ## b ## w" and "less_eq b aa" and "order.sdescending less_eq (b ## w)"
power_gt1: fixes a :: "'a"   and n :: "nat" assumes "(1::'a) < a" shows "(1::'a) < a ^ Suc n"
set_take_auxlist: fixes x :: "nat \<times> event_data option list set \<times> event_data option list set" assumes "x \<in> set (take (length done) auxlist)" shows "check_before I nt x"
LTa: shows "LT1 ars r (s, [\<beta>_step]) (s, [\<alpha>_step]) \<gamma>1 \<gamma>2 \<gamma>3"
transitionI_1: fixes P :: "pi"   and P''' :: "pi"   and \<alpha> :: "freeRes"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'''"   and "P''' \<longmapsto> \<alpha> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P'"
assms: shows "sp F xs \<Sigma> = Ok \<Sigma>'"
extreme_points_of_convex_hull_eq: fixes S :: "'b set" assumes "compact S"   and "\<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S" shows "{x. x extreme_point_of convex hull S} = S"
measurable_measure_Diff: fixes A :: "'c set"   and M :: "'c measure"   and B :: "'c set" assumes "A \<in> fmeasurable M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M B"
linorder_on_linorder_class: shows "linorder_on UNIV {(x, y). x \<le> y}"
dirichlet_prod_altdef1: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "dirichlet_prod f g = (\<lambda>n. \<Sum>d | d dvd n. f (n div d) * g d)"
eq''': shows "sum (sum (\<lambda>m. \<Phi> m * unity_root d (int m))) (rcosets\<^bsub>G\<^esub> kernel G H f) = (\<Sum>b\<in>rcosets\<^bsub>G\<^esub> kernel G H f. of_nat (totient k div totient d) * (\<Phi> (the_elem (f ` b)) * unity_root d (int (the_elem (f ` b)))))"
true_cls_mset_empty: fixes I :: "'b set" shows "I \<Turnstile>m {#}"
count_empty: fixes a :: "'a" shows "count {#} a = 0"
HBL1_2_3_axioms: shows "HBL1_2_3 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc P"
le_fin: fixes y :: "enat" assumes "y \<in> A" shows "y \<le> enat n"
monom_eq_zero_iff: fixes a :: "nat \<Rightarrow>\<^sub>0 nat"   and b :: "'a" shows "(MPoly_Type.monom a b = 0) = (b = (0::'a))"
length_sorted_list_of_set: fixes A :: "'a word set" shows "length (signed.sorted_list_of_set A) = card A"
vlrestriction_vdomain: fixes r :: "V" assumes "vbrelation r" shows "r \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> r = r"
IH: assumes "eval (qe_eq_repeat_helper var A' L F) (xs @ x # \<Gamma>)" shows "\<exists>x. eval (list_conj (map (fm.Atom \<circ> Eq) A' @ map fm.Atom L @ F)) (xs @ x # \<Gamma>)"
min_of_mono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and m :: "'a"   and n :: "'a" assumes "class.linorder less_eq less"   and "order.mono less_eq f" shows "min (f m) (f n) = f (ord.min less_eq m n)"
strong_remove1: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and t :: "('a, 'b) BinomialHeapStruc.BinomialTree" assumes "BinomialHeapStruc.invar bq" shows "BinomialHeapStruc.invar (remove1 t bq)"
get_root_node_impl: shows "get_root_node = local.a_get_root_node"
intro: fixes f :: "nat \<Rightarrow> 'a"   and f_prime_power :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f_prime :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and g_prime_power :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and g_prime :: "nat \<Rightarrow> 'a" assumes "multiplicative_function' f f_prime_power f_prime"   and "multiplicative_function' g g_prime_power g_prime" shows "multiplicative_dirichlet_prod' f g f_prime_power g_prime_power f_prime g_prime"
HK_1: shows "(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p (subtopology X S)\<^esub> y) \<in> Group.iso (subgroup_generated (homology_group p (subtopology X S)) H \<times>\<times> subgroup_generated (homology_group p (subtopology X S)) K) (homology_group p (subtopology X S))"
thetaSeq_def: shows "thetaSeq \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. c1 \<approx>s d1 \<and> c2 \<approx>s d2}"
start_eq: fixes Ia :: "'s pmf"   and t :: "'s"   and s :: "'s" shows "\<P>(\<omega> in T' Ia. shd (stl \<omega>) = t \<and> shd \<omega> = s) = \<P>(\<omega> in T' Ia. shd \<omega> = s) * pmf (K s) t"
assms_4: shows "(\<lambda>w. f w * (w - z)) \<midarrow>z\<rightarrow> c"
validI: fixes Q :: "'d set \<Rightarrow> 'x set \<Rightarrow> bool"   and ds :: "'d set"   and P :: "'d set \<Rightarrow> 'x set \<Rightarrow> bool" assumes "Q ds {}"   and "\<And>fp. \<lbrakk>P ds fp; Q ds fp\<rbrakk> \<Longrightarrow> Q ds (cop_F ds fp)" shows "valid ds P Q"
identity_functor_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "identity_functor C \<equiv> category C"
sat_fvi_cong: fixes \<phi> :: "'a MFOTL.formula"   and v :: "'a list"   and v' :: "'a list"   and \<sigma> :: "(char list \<times> 'a list) trace"   and i :: "nat" assumes "\<forall>x\<in>fv \<phi>. v ! x = v' ! x" shows "MFOTL.sat \<sigma> v i \<phi> = MFOTL.sat \<sigma> v' i \<phi>"
A_B_C_D_3: fixes n :: "nat" shows "C n = (\<Sum>(a, i)\<in>as - UNIV \<times> {0..<n}. (\<Prod>(b, j)\<in>as - UNIV \<times> {0..<n} - {(a, i)}. b) * smult (of_nat (Suc i - n)) (pderiv a))"
has_subprob_density_measurable: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "f \<in> borel_measurable N"
occur_in_subterm: fixes u :: "'b trm"   and t :: "'b trm"   and s :: "'b trm" assumes "occurs_in u t"   and "occurs_in t s" shows "occurs_in u s"
simps_35: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Stop \<noteq> x31@[x32]"
lagrange_interpolation_poly_def: fixes xs_ys :: "('b \<times> 'b) list" shows "lagrange_interpolation_poly xs_ys \<equiv> let xs = map fst xs_ys in \<Sum>(xj, yj)\<leftarrow>xs_ys. smult yj (lagrange_basis_poly xs xj)"
that: fixes i :: "nat" assumes "\<And>\<chi> j. \<chi> \<in> \<G> \<Longrightarrow> suffix i w \<Turnstile> \<chi> = suffix (i + j) w \<Turnstile> \<chi>" shows "thesis"
QR_decomposition_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "QR_decomposition A = (let Q = divide_by_norm (Gram_Schmidt_matrix A) in (Q, Finite_Cartesian_Product.transpose Q ** A))"
that: fixes C' :: "'a set set" assumes "C' \<subseteq> C"   and "finite C'"   and "s \<subseteq> \<Union> C'" shows "thesis"
rel_compp: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_pmf (R OO S) = rel_pmf R OO rel_pmf S"
zero_power: fixes n :: "nat" assumes "0 < n" shows "(0::'a) ^ n = (0::'a)"
IF1map_simps: fixes f :: "'e \<Rightarrow> 'd"   and v :: "('e, 'e IF1, 'e IF2) F1" shows "IF1map f (ctor1 v) = ctor1 (F1map f (IF1map f) (IF2map f) v)"
Diff_partition: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "A \<union> (B - A) = B"
intros_19: fixes h :: "heap"   and v :: "val"   and T\<^sub>1 :: "ty"   and P :: "(char list list \<times> char list exp) prog"   and T :: "ty"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<^sub>1\<rfloor>"   and "P \<turnstile> T\<^sub>1 \<le> T"   and "P,E(V \<mapsto> T),h,sh \<turnstile> e\<^sub>2 :' T\<^sub>2" shows "P,E,h,sh \<turnstile> {V:T; V:=Val v;; e\<^sub>2} :' T\<^sub>2"
induced_arrowI_1: fixes \<chi>' :: "'j \<Rightarrow> 'c"   and a' :: "'c" assumes "\<chi>' \<in> D.cones a'" shows "\<guillemotleft>induced_arrow a' \<chi>' : a' \<rightarrow> a\<guillemotright>"
fun_vars_list_def: fixes f :: "nat \<Rightarrow> 'a" shows "fun_vars_list f = sorted_list_of_set {v. f v \<noteq> (0::'a)}"
order_iff: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a = max a b)"
fixpt_True: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list" assumes "fst (iter M T)" shows "fixpt M T = fixpt M (snd (iter M T))"
set_eq_subset: fixes A :: "'e set"   and B :: "'e set" shows "(A = B) = (A \<subseteq> B \<and> B \<subseteq> A)"
valid_fold_combine: fixes w :: "nat \<times> nat"   and ts :: "'a tree list"   and h :: "'a tree"   and ts' :: "'a tree list"   and z :: "'a tree" assumes "0 < fst w"   and "ts = h # ts'"   and "Ball (set ts) local.valid"   and "adjacent (fst w, l h - 1) ts'"   and "local.valid z"   and "z \<noteq> Leaf"   and "l z = (case ts' of [] \<Rightarrow> fst w | t\<^sub>1 # ts'' \<Rightarrow> Suc (r t\<^sub>1))"   and "r z = snd w" shows "local.valid (fold combine ts' z) \<and> l (fold combine ts' z) = fst w \<and> r (fold combine ts' z) = snd w"
preserves_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "trg\<^sub>D (G \<mu>) = G.map\<^sub>0 (trg\<^sub>C \<mu>)"
FunctorCompPreserved: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and f :: "'f" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "F ## f \<in> mor\<^bsub>CatCod F\<^esub>"
a_ptr_kinds_M_def: shows "object_ptr_kinds_M = Heap_Error_Monad.bind get_heap (\<lambda>h. return (sorted_list_of_set (fset (object_ptr_kinds h))))"
induct: fixes P :: "('b state \<Rightarrow> exp0 \<Rightarrow> 'b state \<times> (v, v) result) \<Rightarrow> 'b state \<Rightarrow> exp0 list \<Rightarrow> bool"   and a0 :: "'b state \<Rightarrow> exp0 \<Rightarrow> 'b state \<times> (v, v) result"   and a1 :: "'b state"   and a2 :: "exp0 list" assumes "\<And>eval s. P eval s []"   and "\<And>eval s e es. (\<And>x y x1. \<lbrakk>(x, y) = fix_clock s (eval s e); y = Rval x1\<rbrakk> \<Longrightarrow> P eval x es) \<Longrightarrow> P eval s (e # es)" shows "P a0 a1 a2"
Not_ExPre: fixes a :: "'a" assumes "a \<in> carrier D" shows "\<not> ExPre (Iod D {a}) a"
lspace_mono: fixes f :: "'b \<Rightarrow> real"   and q :: "ennreal"   and S :: "'b set"   and p :: "ennreal" assumes "f \<in> lspace (lebesgue_on S) q"   and "S \<in> lmeasurable"   and "0 < p"   and "p \<le> q"   and "q < \<infinity>" shows "f \<in> lspace (lebesgue_on S) p"
setcompr_eq_image: fixes f :: "'d \<Rightarrow> 'c"   and P :: "'d \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
simps_4: fixes g :: "safety"   and r :: "'a regex"   and s :: "'a regex" shows "local.safe_regex Futu g (Times r s) = ((g = Lax \<or> fv_regex fv r \<subseteq> fv_regex fv s) \<and> local.safe_regex Futu g s \<and> local.safe_regex Futu Lax r)"
QgrpUnit_2: fixes G :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme"   and H :: "'e set" assumes "Group G"   and "Ugp E"   and "G \<triangleright> H"   and "carrier G = H" shows "(G / H) \<cong> E"
of_int_vec_def: fixes v :: "(int, 'b) vec" shows "of_int_vec v = (\<chi>i. of_int (v $ i))"
splitFace_edges_incr: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and f :: "face"   and vs :: "nat list"   and f\<^sub>1 :: "face"   and f\<^sub>2 :: "face"   and g' :: "graph" assumes "pre_splitFace g ram1 ram2 f vs"   and "(f\<^sub>1, f\<^sub>2, g') = splitFace g ram1 ram2 f vs" shows "\<E> g \<subseteq> \<E> g'"
assms_1: shows "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1"
P_LCons: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c" assumes "vm_path G P v0" shows "P = LCons v0 (ltl P)"
WbisT_If_cross: fixes c1 :: "(test, atom) com"   and c2 :: "(test, atom) com"   and tst :: "test" assumes "c1 \<approx>wT c2"   and "c1 \<approx>wT c1"   and "c2 \<approx>wT c2" shows "if tst then c1 else c2 \<approx>wT if tst then c1 else c2"
assms_2: shows "\<forall>var\<in>tvars t. v1 var = v2 var"
fba_union: fixes S :: "'a set set"   and T :: "'a set set" assumes "S \<in> fba"   and "T \<in> fba" shows "S \<union> T \<in> fba"
JF1map_simps: fixes u :: "'e \<Rightarrow> 'd"   and v :: "'e JF1" shows "dtor1 (JF1map u v) = F1map u (JF1map u) (JF2map u) (dtor1 v)"
blin: shows "b = lincomb (ls p) (a ` Is p)"
pfp_least: fixes f :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x0 :: "'b"   and x :: "'b" assumes "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y"   and "f p \<sqsubseteq> p"   and "x0 \<sqsubseteq> p"   and "pfp f x0 = Some x" shows "x \<sqsubseteq> p"
4: shows "greachable A w V u \<subseteq> greachable A w V v"
length_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'c list" shows "length (permute_list f xs) = length xs"
assms: shows "\<forall>e f g. eval e f g p \<longrightarrow> eval e f g q"
ef_echelon_form_of_det: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'c) vec" assumes "is_bezout_ext bezout" shows "echelon_form (snd (echelon_form_of_det A bezout))"
eq1: shows "inner_prod (alpha_l l \<cdot>\<^sub>v \<alpha>) (psi_l l) = alpha_l l * alpha_l l"
Inf_ne_def: fixes X :: "'b set" shows "Inf_ne X = FOLD_bin_ne inf X"
leg_simps_4: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "span_in_category C S" shows "partial_magma.dom C (Leg1 S) = span_in_category.apex C S"
less_imp_le: fixes x :: "'b Word.word"   and y :: "'b Word.word" assumes "x <s y" shows "x \<le>s y"
topological_basis: fixes "open" :: "'d set \<Rightarrow> bool"   and B :: "'d set set" assumes "class.topological_space open" shows "topological_space.topological_basis open B = (\<forall>x. open x = (\<exists>B'\<subseteq>B. \<Union> B' = x))"
fds_nth_fds': fixes f :: "nat \<Rightarrow> 'a" assumes "f 0 = (0::'a)" shows "fds_nth (fds f) = f"
bdd_below_Ici: fixes a :: "'a multiset" shows "subset_mset.bdd_below (subset_mset.atLeast a)"
left_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "f a (f b c) = f (f a b) c"
gr_implies_not_zero: fixes m :: "'e multiset"   and n :: "'e multiset" assumes "m \<subset># n" shows "n \<noteq> {#}"
elem_natUnion: fixes x :: "'b"   and f :: "nat \<Rightarrow> 'b set" assumes "\<forall>n. x \<in> f n" shows "x \<in> natUnion f"
sat_eq: fixes c :: "('c, 'd) conf" shows "Conf.sat c = (states c \<noteq> {})"
seq_comp_secure_aux_1_case_3: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and Q :: "'d option process"   and R :: "'d option process"   and ws :: "'d option list"   and ys' :: "'d option list"   and Y :: "'d option set"   and xs :: "'d option list"   and y :: "'d option"   and ys :: "'d option list" assumes "secure_termination I D"   and "ref_union_closed Q"   and "sequential Q"   and "secure Q I D"   and "secure R I D"   and "ws \<in> sentences Q"   and "(ys', Y) \<in> failures R"   and "ws @ ys' = xs @ y # ys" shows "(xs @ ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys Y) \<in> seq_comp_failures Q R"
cases: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and a1 :: "('a, 'c) com \<times> 'b"   and a2 :: "'b" assumes "PL.transT tval aval a1 a2" obtains atm :: "'c" and s :: "'b" where "a1 = (Atm atm, s__)" and "a2 = aval atm s__"   | tst :: "'a" and s :: "'b" and c :: "('a, 'c) com" where "a1 = (while tst do c, s)" and "a2 = s" and "\<not> tval tst s"
dist_eq_tr_4: shows "\<bottom> \<noteq> TT"
runit_simps_5: fixes f :: "'a" assumes "C.ide f" shows "C.cod \<r>\<^sub>C[f] = f"
simps_1: fixes x1 :: "bool"   and y1 :: "bool" shows "(Det x1 = Det y1) = (x1 = y1)"
assms: fixes P :: "'a \<Rightarrow> ('b \<times> 'b) set" assumes "Pcond A Is P" shows "complete A (scf P)"
factorize_int_poly_generic_def: fixes f :: "int poly" shows "local.factorize_int_poly_generic f = (case x_split f of (n, g) \<Rightarrow> if g = 0 then (0, []) else case local.factorize_int_last_nz_poly g of (a, fs) \<Rightarrow> if n = 0 then (a, fs) else (a, (monom 1 1, n - 1) # fs))"
bottom_closed: shows "\<bottom>\<^bsub>fpl L f\<^esub> \<in> carrier (fpl L f)"
eint_ile: fixes n :: "eint"   and m :: "int" assumes "n \<le> eint m" shows "\<exists>k. n = eint k"
absorb2: fixes b :: "'b::semilattice_inf"   and a :: "'b::semilattice_inf" assumes "(b::'b::semilattice_inf) \<le> (a::'b::semilattice_inf)" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) = b"
ring_hom_ring_axioms: shows "ring_hom_ring R (S Quot P) ((+>\<^bsub>S\<^esub>) P \<circ> h)"
Disjoint_Eats_D1: fixes x :: "tm"   and y :: "tm"   and z :: "tm" shows "{Disjoint (Eats x y) z} \<turnstile> Disjoint x z"
prems_1: shows "M, g \<Turnstile>\<^sub>B (ps, a)"
fg: fixes x :: "'a" assumes "x \<in> s \<inter> t" shows "f x \<le> g x"
ext: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.comp g f \<noteq> B.VVV.null" shows "B.VVV.seq g f"
fminus_fsubset_conv: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" shows "(A |-| B |\<subseteq>| C) = (A |\<subseteq>| B |\<union>| C)"
t: shows "t \<notin> real_of_ereal ` {a<..<b}"
\<epsilon>_less: shows "\<epsilon> < weight \<Gamma> a - d_OUT f a"
nvnv: shows "normal_vector W \<bullet> normal_vector W \<noteq> (0::'a)"
AssignE: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and ct :: "com \<times> (char list \<Rightarrow> int)" assumes "(x ::= a, s) \<rightarrow> ct" obtains "ct = (SKIP, s(x := aval a s))"
cut4: fixes A :: "fmla"   and B :: "fmla"   and C :: "fmla"   and D :: "fmla"   and E :: "fmla"   and H :: "fmla set" assumes "{A, B, C, D} \<turnstile> E"   and "H \<turnstile> A"   and "H \<turnstile> B"   and "H \<turnstile> C"   and "H \<turnstile> D" shows "H \<turnstile> E"
\<Sigma>_comp_of_int: fixes f :: "'d \<Rightarrow> 'c" shows "\<Sigma> (f \<circ> of_int) = \<Sigma> f"
real_sqrt_less_iff: fixes x :: "real"   and y :: "real" shows "(sqrt x < sqrt y) = (x < y)"
assertLiteralEffect_2: fixes state :: "State"   and l :: "Literal"   and d :: "bool" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)" shows "getF (assertLiteral l d state) = getF state"
all_total_extend: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program" assumes "Extend h"   and "all_total F" shows "all_total (extend h F)"
left_mult_zero_mat': fixes A :: "'a mat"   and n :: "nat"   and nr :: "nat" assumes "dim_row A = n" shows "0\<^sub>m nr n * A = 0\<^sub>m nr (dim_col A)"
perzeta_int: fixes q :: "real" assumes "q \<in> \<int>" shows "perzeta q = zeta"
StepP: shows "cfg \<turnstile> m \<mapsto> cfg1"
publicKey_neq_privateKey: fixes c :: "keymode"   and A' :: "agent"   and b :: "keymode"   and A :: "agent" shows "publicKey c A' \<noteq> privateKey b A"
in_gamma_join_UpI: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and test_num' :: "int \<Rightarrow> 'b \<Rightarrow> bool"   and filter_plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b"   and filter_less' :: "bool \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and s :: "char list \<Rightarrow> int"   and S1 :: "'b Abs_State.st option"   and S2 :: "'b Abs_State.st option" assumes "Abs_Int1 num' plus' test_num' filter_plus' filter_less' \<gamma>"   and "s \<in> \<gamma>_option (\<gamma>_st \<gamma>) S1 \<or> s \<in> \<gamma>_option (\<gamma>_st \<gamma>) S2" shows "s \<in> \<gamma>_option (\<gamma>_st \<gamma>) (S1 \<squnion> S2)"
ttl_tSup: fixes Y :: "('d, 'b) tllist set" assumes "Complete_Partial_Order.chain local.tllist_ord Y"   and "\<exists>xs\<in>Y. \<not> is_TNil xs" shows "ttl (local.tSup Y) = local.tSup (ttl ` (Y \<inter> {xs. \<not> is_TNil xs}))"
assms_5: shows "finite {v. g v \<noteq> (0::'a)}"
less_le: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "ko.lt ko x y = (ko.le ko x y \<and> x \<noteq> y)"
image_empty: fixes f :: "'b \<Rightarrow> 'a" shows "bot_pres (\<P> f)"
hfunction_insf: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "hfunction s" shows "hfunction (insf s k y)"
lm5: fixes x :: "nat"   and key :: "nat"   and y :: "nat" assumes "c_assoc_have_key (pr_gr x) key = 0"   and "x \<le> y" shows "c_assoc_have_key (pr_gr y) key = 0"
bool_of_specF: fixes e :: "'c exp" assumes "bool_of e = \<lfloor>False\<rfloor>" shows "e = false"
rel_FGf_def: fixes L1 :: "'p \<Rightarrow> 'z \<Rightarrow> bool"   and L2 :: "'q \<Rightarrow> 'aa \<Rightarrow> bool"   and Co1 :: "'s \<Rightarrow> 'ab \<Rightarrow> bool"   and Co2 :: "'t \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra1 :: "'v \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'ae \<Rightarrow> bool" shows "rel_FGf L1 L2 Co1 Co2 Contra1 Contra2 = rel_F L1 L2 (=) Co1 Co2 (=) Contra1 Contra2 (=)"
le_imp_less_or_equiv: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x < y \<or> x \<approx> y"
distr_restrict: fixes J :: "'a set"   and K :: "'a set" assumes "J \<subseteq> K"   and "finite K" shows "Pi\<^sub>M J (\<lambda>i. M) = distr (Pi\<^sub>M K (\<lambda>i. M)) (Pi\<^sub>M J (\<lambda>i. M)) (\<lambda>f. restrict f J)"
distinct_5: fixes x1 :: "nat"   and x41 :: "trm"   and x42 :: "trm" shows "Var x1 \<noteq> x41 \<bullet> x42"
ccSup_empty: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
\<A>ss_OR_def: fixes X :: "'g \<times> 'a"   and C1 :: "('j \<times> 'd) \<times> 'e \<times> ('e \<times> 'k) \<times> 'e \<times> 'f"   and C2 :: "('j \<times> 'd) \<times> 'e \<times> ('e \<times> 'k) \<times> 'e \<times> 'f" shows "\<A>ss_OR X C1 C2 = (if fst (fst (snd (snd C1))) \<noteq> fst (fst (snd (snd C2))) then \<A>ss0 (fst X) (fst (fst C1), fst (fst (snd (snd C1))), snd (fst (snd (snd C1)))) (fst (fst C2), fst (fst (snd (snd C2))), snd (fst (snd (snd C2)))) \<bind> (\<lambda>w0. return_spmf (Inl w0)) else \<A>ss1 (snd X) (snd (fst C1), fst (snd (snd (snd C1))), snd (snd (snd (snd C1)))) (snd (fst C2), fst (snd (snd (snd C2))), snd (snd (snd (snd C2)))) \<bind> (\<lambda>w1. return_spmf (Inr w1)))"
Rep_List_inject: fixes x :: "'b SList.list"   and y :: "'b SList.list" shows "(Rep_List x = Rep_List y) = (x = y)"
sums_cong: fixes f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c"   and c :: "'c" assumes "\<And>n. f n = g n" shows "f sums c = g sums c"
Dynkin_system_axioms: shows "Dynkin_system (space M) {D \<in> events. prob (X \<inter> D) = prob X * prob D}"
simps_2: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and as :: "'b negation_type list" shows "NegPos_map f (Pos a # as) = Pos (f a) # NegPos_map f as"
minf_neq_0: shows "- \<infinity> \<noteq> 0"
exhaust: fixes y :: "TP" obtains "y = low"   | "y = high"
ltake_lappend1: fixes n :: "enat"   and xs :: "'c llist"   and ys :: "'c llist" assumes "n \<le> llength xs" shows "ltake n (lappend xs ys) = ltake n xs"
f1: shows "sys_reg s1 CCR AND 1 \<noteq> 0 \<and> sys_reg s2 CCR AND 1 \<noteq> 0"
d_delta_lnexp_cf5: fixes x :: "real" assumes "0 < numer_cf5 x"   and "0 < numer_cf5 (- x)" shows "((\<lambda>x. ln (exp_cf5 x) - x) has_real_derivative diff_delta_lnexp_cf5 x) (at x)"