hcju/isabelleps huggingface dataset:
triangle_right: shows "(f\<^sup>* \<star> \<epsilon>) \<cdot> \<a>[f\<^sup>*, f, f\<^sup>*] \<cdot> (\<eta> \<star> f\<^sup>*) = \<r>\<^sup>-\<^sup>1[f\<^sup>*] \<cdot> \<l>[f\<^sup>*]"
assms_1: shows "a \<notin> \<real>\<^sub>\<le>\<^sub>0 \<or> a = 0 \<and> 0 < Re b"
similar_matrix_wit_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and P :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and Q :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "similar_matrix_wit A B P Q = (P ** Q = mat (1::'b) \<and> Q ** P = mat (1::'b) \<and> A = P ** B ** Q)"
sums_summable: fixes f :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "f sums l" shows "summable f"
assigns_hoare_r: fixes p :: "'a upred"   and \<sigma> :: "'a \<Rightarrow> 'a"   and q :: "'a upred" assumes "`p \<Rightarrow> \<sigma> \<dagger> q`" shows "\<lbrace>p\<rbrace> \<langle>\<sigma>\<rangle>\<^sub>a \<lbrace>q\<rbrace>\<^sub>u"
prefI': fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" assumes "ys = xs \<cdot> zs" shows "xs \<le>p ys"
prob_space_rT: fixes x :: "'s" shows "prob_space (rT x)"
permutes_inv: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "p permutes S" shows "inv p permutes S"
absorb_iff2: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "semilattice_order f less_eq less" shows "less_eq b a = (f a b = b)"
npeSum2_sub_muly: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "y \<cdot>\<^sub>r \<Sigma>\<^sub>e R (\<lambda>i.  (n choose i) \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (n - i)\<^esup> \<cdot>\<^sub>r y^\<^bsup>R i\<^esup>)) n = \<Sigma>\<^sub>e R (\<lambda>i.  (n choose i) \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (n - i)\<^esup> \<cdot>\<^sub>r y^\<^bsup>R (i + 1)\<^esup>)) n"
sum_of_aezfun_is_aezfun: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> aezfun_set"   and "g \<in> aezfun_set" shows "f + g \<in> aezfun_set"
rel_refl: fixes Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b message_af" assumes "\<And>x. Ra x x" shows "rel_message_af Ra x x"
foldr_conv_foldl: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'a list"   and a :: "'b" assumes "comp_fun_commute f" shows "foldr f xs a = foldl (\<lambda>a b. f b a) a xs"
Moebius_function_eq_zero: fixes t :: "real"   and w :: "complex" shows "Moebius_function t w w = 0"
lunit_char_2: fixes f :: "'d" assumes "D.ide f" shows "D.L \<l>\<^sub>D[f] = (\<i>\<^sub>D[trg\<^sub>D f] \<star>\<^sub>D f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[trg\<^sub>D f, trg\<^sub>D f, f]"
of_nat_def: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero n = (plus one ^^ n) zero"
assms_1: fixes c :: "'a"   and var :: "Vars.var" shows "P1 c (Var var)"
finite_abs_reach: fixes \<phi> :: "'b ltl" shows "finite (af_abs_opt.abs_reach (Abs \<phi>))"
poly_root_constant: fixes c :: "'a"   and p :: "'a poly"   and k :: "'a" assumes "c \<noteq> (0::'a)" shows "(poly (p * [:c:]) k = (0::'a)) = (poly p k = (0::'a))"
veqpoll_omega_vcard: fixes n :: "V"   and A :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "n \<approx>\<^sub>\<circ> A" shows "vcard A = n"
not_finished: shows "\<forall>i<k. \<not> (CS (f i) = cs \<and> CSS (f i) = css)"
to_subset: fixes T :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "T \<subseteq> carrier D"   and "ord_isom D (Iod D T) f" shows "\<forall>a. a \<in> carrier D \<longrightarrow> a \<preceq> f a"
sorted_wrt_dist_last_take_mono: fixes p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and n :: "nat" assumes "sorted_wrt_dist p ps"   and "n \<le> length ps"   and "0 < n" shows "dist (last (take n ps)) p \<le> dist (last ps) p"
seq_distrib_left_def: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice" shows "seq_distrib_left (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) \<equiv> sequential seq nil \<and> seq_distrib_left_axioms seq"
tmap_eq_TCons_conv: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and xs :: "('g, 'h) tllist"   and y :: "'e"   and ys :: "('e, 'f) tllist" shows "(tmap f g xs = TCons y ys) = (\<exists>z zs. xs = TCons z zs \<and> f z = y \<and> tmap f g zs = ys)"
steps: fixes i :: "nat" shows "(f i, f (Suc i)) \<in> SN_rel_ext_step (A\<^sup>* O (P \<inter> M') O A\<^sup>*) (A\<^sup>* O ((P \<union> Pw) \<inter> M') O A\<^sup>*) (A\<^sup>* O ((P \<union> R) \<inter> M') O A\<^sup>*) (A\<^sup>*) (ty i)"
atLeastAtMost_def: fixes l :: "'a word"   and u :: "'a word" shows "signed.atLeastAtMost l u \<equiv> signed.atLeast l \<inter> signed.atMost u"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and bop :: "bop" shows "P (case bop of Eq \<Rightarrow> f1 | Add \<Rightarrow> f2) = (\<not> (bop = Eq \<and> \<not> P f1 \<or> bop = Add \<and> \<not> P f2))"
osum_def: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" shows "r +o r' = map_prod Inl Inl ` r \<union> map_prod Inr Inr ` r' \<union> {(Inl a, Inr a') |a a'. a \<in> Field r \<and> a' \<in> Field r'}"
unique_extends_subst: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and t :: "('b, 'a) Term.term" assumes "extends_subst \<sigma> \<tau>"   and "extends_subst \<sigma> \<rho>"   and "dom \<tau> = vars_term t \<union> dom \<sigma>"   and "dom \<rho> = vars_term t \<union> dom \<sigma>"   and "subst_atm_abbrev t (subst_of_map Var \<rho>) = subst_atm_abbrev t (subst_of_map Var \<tau>)" shows "\<rho> = \<tau>"
b: shows "n x \<oplus> n z = n y \<oplus> n z"
measurable_comb_seq: fixes i :: "nat"   and M :: "'b measure" shows "(\<lambda>(\<omega>, \<omega>'). comb_seq i \<omega> \<omega>') \<in> Pi\<^sub>M UNIV (\<lambda>i. M) \<Otimes>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M) \<rightarrow>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M)"
Image_Un: fixes R :: "('e \<times> 'd) set"   and A :: "'e set"   and B :: "'e set" shows "R `` (A \<union> B) = R `` A \<union> R `` B"
finite_params_extend: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat" assumes "infinite (\<Inter>p\<in>S. - params p)" shows "infinite (\<Inter>p\<in>extend S C f n. - params p)"
subspace_refl: fixes V :: "'c set" assumes "vectorspace V" shows "Subspace.subspace V V"
"2_1": shows "\<forall>x\<in>set T1. stateful_strand_step.is_Send x"
ss0: shows "0 \<le> s - real_of_int \<lfloor>s\<rfloor>"
power_dvd_iff_le_multiplicity: fixes x :: "'a"   and p :: "'a"   and n :: "nat" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p" shows "(p ^ n dvd x) = (n \<le> multiplicity p x)"
not_one_le_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
af\<^sub>G_simps_3: fixes \<phi> :: "'a ltl"   and w :: "'a set list" shows "af\<^sub>G (G \<phi>) w = G \<phi>"
assms_6: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp" assumes "P scrut"   and "P e1"   and "P e2" shows "P (scrut ? e1 : e2)"
ruleSet: fixes R1 :: "(sequent list \<times> sequent) set"   and R2 :: "(sequent list \<times> sequent) set"   and R3 :: "(sequent list \<times> sequent) set"   and R :: "(sequent list \<times> sequent) set"   and Ps :: "sequent list"   and C :: "sequent" assumes "R1 \<subseteq> upRules"   and "R2 \<subseteq> nprovRules"   and "R3 \<subseteq> provRules"   and "R = Ax \<union> R1 \<union> R2 \<union> R3"   and "(Ps, C) \<in>  R*" shows "\<exists>S r. extendRule S r = (Ps, C) \<and> (r \<in> R1 \<or> r \<in> R2 \<or> r \<in> R3 \<or> r \<in> Ax)"
mat_hom_one: fixes n :: "nat" shows "of_int_hom.mat_hom (1\<^sub>m n) = 1\<^sub>m n"
poly_eval: shows "\<forall>y. poly [:c, b, a:] y = a * y\<^sup>2 + b * y + c"
step: fixes gpv :: "('a, 'out, 'in) gpv"   and gpv' :: "('a', 'out, 'in) gpv" assumes "X gpv gpv'" shows "rel_spmf (eq_\<I>_generat A \<I> (\<lambda>gpv gpv'. X gpv gpv' \<or> eq_\<I>_gpv A \<I> gpv gpv' \<or> (\<exists>gpv'' gpv''' B f g. gpv = gpv'' \<bind> f \<and> gpv' = gpv''' \<bind> g \<and> eq_\<I>_gpv B \<I> gpv'' gpv''' \<and> rel_fun B X f g))) (the_gpv gpv) (the_gpv gpv')"
tl_nempty_length2: fixes s :: "'a list" assumes "s \<noteq> []"   and "tl s \<noteq> []" shows "Suc 0 < length s"
Abs_SkewBinomialHeap_inverse: fixes y :: "unit + ('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" assumes "y \<in> {q. BsSkewBinomialHeapStruc.bs_invar q}" shows "Rep_SkewBinomialHeap (Abs_SkewBinomialHeap y) = y"
fail_bind: fixes f :: "'a \<Rightarrow> 'm" shows "bind fail f = fail"
v0_neq_v1: shows "v0 \<noteq> v1"
ker_diff_carrier: fixes K :: "'a set"   and x :: "'a" assumes "subring K R" shows "a_kernel (K [X]) R (\<lambda>p. local.eval p x) \<noteq> carrier (K [X])"
mult_of_nat_commute: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and x :: "nat"   and y :: "'q" 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)"
count_mset_0_iff: fixes xs :: "'a list"   and x :: "'a" shows "(count (mset xs) x = 0) = (x \<notin> set xs)"
diff_zero_eq: fixes n :: "nat"   and m :: "nat" assumes "n = 0" shows "m = m - n"
assms_3: fixes B :: "'a set" assumes "B \<in> \<T>" shows "open B"
induct: fixes P :: "'c \<Rightarrow> (('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> bool"   and a0 :: "'c"   and a1 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x l a b c uu_ r. \<lbrakk>\<lbrakk>cmp x a = LT; l \<noteq> \<langle>\<rangle> \<and> color l = Black\<rbrakk> \<Longrightarrow> P x l; \<lbrakk>cmp x a = LT; \<not> (l \<noteq> \<langle>\<rangle> \<and> color l = Black)\<rbrakk> \<Longrightarrow> P x l; \<lbrakk>cmp x a = GT; r \<noteq> \<langle>\<rangle> \<and> color r = Black\<rbrakk> \<Longrightarrow> P x r; \<lbrakk>cmp x a = GT; \<not> (r \<noteq> \<langle>\<rangle> \<and> color r = Black)\<rbrakk> \<Longrightarrow> P x r\<rbrakk> \<Longrightarrow> P x \<langle>l, ((a, b), c, uu_), r\<rangle>" shows "P a0 a1"
not_reach_out: fixes u :: "'a"   and v :: "'a" assumes "u \<in> S"   and "v \<notin> S" shows "\<not> u \<rightarrow>\<^sup>* v \<or> \<not> v \<rightarrow>\<^sup>* u"
hyperUn_assoc: fixes A :: "char list set option"   and B :: "char list set option"   and C :: "char list set option" shows "A \<squnion> B \<squnion> C = A \<squnion> (B \<squnion> C)"
ereal_add_right_mono: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<le> b" shows "a + c \<le> b + c"
valid_Cn_equiv: fixes p :: "'a"   and q :: "'a" shows "\<tturnstile> (p .\<longleftrightarrow>. q) = (Cn {p} = Cn {q})"
arrI: fixes f1 :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f2 :: "'a" assumes "A_B.arr f1"   and "A.arr f2" shows "A_BxA.arr (f1, f2)"
star_rtc2: fixes x :: "'a list set" shows "(p_one \<union> p_prod x x \<subseteq> x) = (x = p_star x)"
atLeastAtMost_def: fixes l :: "'b word"   and u :: "'b word" shows "signed.atLeastAtMost l u \<equiv> signed.atLeast l \<inter> signed.atMost u"
Lxy_index_mono: fixes x :: "'a"   and S :: "'a set"   and y :: "'a"   and xs :: "'a list" assumes "x \<in> S"   and "y \<in> S"   and "index xs x < index xs y"   and "index xs y < length xs"   and "x \<noteq> y" shows "index (Lxy xs S) x < index (Lxy xs S) y"
P_t_const: fixes t :: "'a"   and name :: "name" assumes "P t (const name)" shows "t = const name"
triangle_num_less_iff: fixes m :: "nat"   and n :: "nat" shows "(triangle_num m < triangle_num n) = (m < n)"
max_input_def: fixes g :: "vname aexp" shows "AExp.max_input g = (let inputs = enumerate_aexp_inputs g in if inputs = {} then None else Some (Max inputs))"
c_intersect_is_pr: shows "c_intersect \<in> PrimRec2"
dependent_zero: fixes A :: "'b set" assumes "(0::'b) \<in> A" shows "cdependent A"
sum_split_even_odd: fixes f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and n :: "nat" shows "(\<Sum>i<2 * n. if even i then f i else g i) = (\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1))"
proj_psi_lR_mult_proj_psi_lR: shows "proj_psi_l R * proj_psi_l R = proj_psi_l R"
C'sees: shows "P \<turnstile> C' sees_methods Mm'"
ot\<omega>: fixes K :: "nat set"   and j :: "nat"   and i :: "nat" assumes "K \<in> \<K> j i"   and "i < j" shows "ordertype ((\<lambda>k. list_of (b k (j, i))) ` {Max (insert j K)<..}) (lenlex less_than) = \<omega>"
Nil: fixes P :: "(char list list \<times> char list exp) prog"   and s :: "heap \<times> sfields \<times> sheap" shows "P \<turnstile> \<langle>[],s\<rangle> [\<Rightarrow>] \<langle>[],s\<rangle>"
size_nat_multiset_eq: fixes n :: "nat"   and N :: "nat multiset" assumes "number_partition n N" shows "size N = sum (count N) {..n}"
SUP_le_iff: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set"   and u :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Sup::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set))) (u::'b::type) = (\<forall>i::'c::type\<in>A. less_eq (f i) u)"
supp_some: fixes x :: "'b" shows "supp (Some x) = supp x"
simps: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" shows "kbo'.kbo s t = (if vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> weight t \<le> weight s then if weight t < weight s then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> x = y | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> if pr_strict' (f, length ss) (g, length ts) then (True, True) else if pr_weak' (f, length ss) (g, length ts) then lex_ext_unbounded kbo'.kbo ss ts else (False, False) else (False, False))"
cart_basis'_i: fixes i :: "'b" shows "cart_basis' $ i = axis i (1::'a)"
heapToAssn_asToHeap: fixes as :: "assn" shows "heapToAssn (asToHeap as) = as"
islimpt_subset: fixes x :: "'c"   and S :: "'c set"   and T :: "'c set" assumes "x islimpt S"   and "S \<subseteq> T" shows "x islimpt T"
state_indep_prop: shows "\<forall>s\<in>P. R"
inv_inv: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> (inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x) = x"
prod_A: fixes n :: "nat" shows "(\<Prod>i = 0..<n. A i ^ Suc i) = (\<Prod>(a, i)\<in>as \<inter> UNIV \<times> {0..<n}. a ^ Suc i)"
liveness_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "liveness P = (\<forall>\<alpha>. \<exists>\<sigma>. P (\<alpha> @- \<sigma>))"
assms_4: fixes x :: "'a" assumes "x \<in> set xs - set ys" shows "f x = 0"
integral_measure_pmf: fixes A :: "'e set"   and M :: "'e pmf"   and f :: "'e \<Rightarrow> 'f" assumes "finite A"   and "\<And>a. \<lbrakk>a \<in> set_pmf M; f a \<noteq> (0::'f)\<rbrakk> \<Longrightarrow> a \<in> A" shows "measure_pmf.expectation M f = (\<Sum>a\<in>A. pmf M a *\<^sub>R f a)"
equivalent_objects_reflexive: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "horizontal_homs.obj V src a" shows "bicategory.equivalent_objects V H \<a> \<i> src trg a a"
vpeq_subj_obj_def: fixes u :: "partition_id_t"   and s :: "state_t"   and t :: "state_t" shows "vpeq_subj_obj u s t \<equiv> \<forall>ob m p1. (Policy.sp_spec_subj_obj u ob m \<longrightarrow> sp_impl_subj_obj s u ob m = sp_impl_subj_obj t u ob m) \<and> (Policy.sp_spec_subj_obj p1 ob PROVIDE \<and> (Policy.sp_spec_subj_obj u ob READ \<or> Policy.sp_spec_subj_obj u ob WRITE) \<longrightarrow> sp_impl_subj_obj s p1 ob PROVIDE = sp_impl_subj_obj t p1 ob PROVIDE)"
assms: shows "dip \<in> vD (invalidate rt1 dests)"
fimage_fsubsetI: fixes A :: "'b fset"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c fset" assumes "\<And>x. x |\<in>| A \<Longrightarrow> f x |\<in>| B" shows "f |`| A |\<subseteq>| B"
add_divide_distrib: 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"   and c :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
cong_4321: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Cong D C B A"
indI: fixes x :: "'a"   and y :: "'a"   and S :: "'a set" assumes "(x, y) \<in> r"   and "x \<in> S"   and "y \<in> S" shows "(x, y) \<in> induced S r"
mute_preserves_dimensions_1: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "mute mf q k l (r, A) = (r', A')" shows "dim_row A' = dim_row A"
cont: fixes s :: "complex" assumes "s \<in> \<nat>" shows "isCont (perzeta q) s"
assms_2: assumes "\<forall>a\<in>A. \<exists>b\<in>B. a .= b"   and "\<forall>b\<in>B. \<exists>a\<in>A. b .= a" shows "P"
transpose_row_def: fixes A :: "(('b, 'd) vec, 'c) vec"   and i :: "'d" shows "transpose_row A i = (\<chi>j. A $ j $ i)"
consistent_map_lists: fixes xs1 :: "('a \<times> 'b) list"   and xs2 :: "('a \<times> 'b) list"   and x :: "'a" assumes "distinct (map fst (xs1 @ xs2))"   and "x \<in> dom (map_of xs1) \<inter> dom (map_of xs2)" shows "map_of xs1 x = map_of xs2 x"
cases: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'a" assumes "r\<^sup>*\<^sup>* a1 a2" obtains a :: "'a" where "a1 = a__" and "a2 = a__"   | a :: "'a" and b :: "'a" and c :: "'a" where "a1 = a" and "a2 = c" and "r\<^sup>*\<^sup>* a b" and "r b c"
dummies_fv_subset: fixes S :: "stack_elem list" shows "dummies S \<subseteq> fv S"
sorted1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq [x] = True"
cases: fixes a :: "'b formula" assumes "a \<in> AX0" obtains F :: "'b formula" and G :: "'b formula" where "a = Fa__ \<^bold>\<rightarrow> G__ \<^bold>\<rightarrow> Fa__"   | F :: "'b formula" and G :: "'b formula" and H :: "'b formula" where "a = (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H"
in_possible_steps: fixes s :: "nat"   and s' :: "nat"   and t :: "transition"   and e :: "((nat \<times> nat) \<times> transition) fset"   and l :: "String.literal"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "(((s, s'), t) |\<in>| e \<and> Label t = l \<and> can_take_transition t i r) = ((s', t) |\<in>| possible_steps e s r l i)"
left_total_conversep: fixes A :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "left_total A\<inverse>\<inverse> = right_total A"
l8_21: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B" shows "\<exists>P T. Tarski_neutral_dimensionless.Perp Bet Cong A B P A \<and> Tarski_neutral_dimensionless.Col Bet A B T \<and> Bet C T P"
winning_moves_remain_winning_Even: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a" assumes "even (length p)"   and "strategy_winning_by_Even \<sigma> p" shows "strategy_winning_by_Even \<sigma> (augment_list \<sigma> p)"
iprev_def: fixes n :: "nat"   and I :: "nat set" shows "iprev n I \<equiv> if n \<in> I \<and> I \<down>< n \<noteq> {} then Max (I \<down>< n) else n"
rep_eq: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "('a, 'b) RBT.rbt"   and xb :: "('a, 'b) RBT.rbt" shows "impl_of (combine_with_key x xa xb) = rbt_union_with_key x (impl_of xa) (impl_of xb)"
assms_4: shows "e2 \<in> in_arcs G v"
dim_matrices: shows "P \<in> carrier_mat m m \<and> S \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
induct: fixes P :: "nat \<Rightarrow> floatarith \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "floatarith" assumes "\<And>x a b. \<lbrakk>P x a; P x b\<rbrakk> \<Longrightarrow> P x (Add a b)"   and "\<And>x a b. \<lbrakk>P x b; P x a\<rbrakk> \<Longrightarrow> P x (Mult a b)"   and "\<And>x a. P x a \<Longrightarrow> P x (Minus a)"   and "\<And>x a. P x a \<Longrightarrow> P x (Inverse a)"   and "\<And>x a. P x a \<Longrightarrow> P x (Cos a)"   and "\<And>x a. P x a \<Longrightarrow> P x (Arctan a)"   and "\<And>x a b. P x (Min a b)"   and "\<And>x a b. P x (Max a b)"   and "\<And>x a. P x (Abs a)"   and "\<And>x. P x Pi"   and "\<And>x a. P x a \<Longrightarrow> P x (Sqrt a)"   and "\<And>x a. P x a \<Longrightarrow> P x (Exp a)"   and "\<And>x a b. \<lbrakk>P x b; P x a\<rbrakk> \<Longrightarrow> P x (Powr a b)"   and "\<And>x a. P x a \<Longrightarrow> P x (Ln a)"   and "\<And>x a. P x (a ^\<^sub>e 0)"   and "\<And>x a n. P x a \<Longrightarrow> P x (a ^\<^sub>e Suc n)"   and "\<And>x a. P x (Floor a)"   and "\<And>x f. P x (Num f)"   and "\<And>x n. P x (Var n)" shows "P a0 a1"
pderivs_lang_subsetI: fixes A :: "'a list set"   and r :: "'a rexp"   and C :: "'a rexp set" assumes "\<And>s. s \<in> A \<Longrightarrow> pderivs s r \<subseteq> C" shows "pderivs_lang A r \<subseteq> C"
sincos_total_2pi_le: fixes x :: "real"   and y :: "real" assumes "x\<^sup>2 + y\<^sup>2 = 1" shows "\<exists>t\<ge>0. t \<le> 2 * pi \<and> x = cos t \<and> y = sin t"