hcju/isabelleps huggingface dataset:
rev_conv_fold: fixes xs :: "'b list" shows "rev xs = fold (#) xs \<epsilon>"
Alice_case: fixes \<phi> :: "complex Matrix.mat"   and q :: "complex Matrix.mat"   and p :: "real" assumes "state 1 \<phi>"   and "state 3 q"   and "List.member (alice_meas \<phi>) (p, q)" shows "alice_pos \<phi> q"
IH: fixes l :: "int" shows "(0 \<le> k AND l) = (0 \<le> k \<or> 0 \<le> l)"
obsf_distinguisher_def: fixes \<D> :: "('c, 'd) distinguisher" shows "obsf_distinguisher \<D> = map_gpv' (\<lambda>x. x = Some True) id option_of_exception (gpv_stop \<D>)"
zero_hom_0_def: fixes hom :: "'a \<Rightarrow> 'b" shows "zero_hom_0 hom \<equiv> zero_hom hom \<and> zero_hom_0_axioms hom"
Valid_def: fixes A :: "'b \<Rightarrow> bool" shows "\<turnstile> A \<equiv> \<forall>w. A w"
P2_S2_def: fixes \<A> :: "(bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> 'd spmf)"   and \<sigma>' :: "bool"   and z :: "'e"   and x\<sigma> :: "'grp"   and aux_out :: "('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f" shows "P2_S2 \<A> \<sigma>' z x\<sigma> aux_out = (let (\<A>1, \<A>2, \<A>3) = \<A>; ((h0, h1, a, b0, b1), s) = aux_out in sample_uniform (order \<G>) \<bind> (\<lambda>u0. sample_uniform (order \<G>) \<bind> (\<lambda>v0. sample_uniform (order \<G>) \<bind> (\<lambda>u1. sample_uniform (order \<G>) \<bind> (\<lambda>v1. let w0 = a [^] u0 \<otimes> \<^bold>g [^] v0; w1 = a [^] u1 \<otimes> \<^bold>g [^] v1; z0 = b0 [^] u0 \<otimes> h0 [^] v0 \<otimes> (if \<sigma>' then \<one> else x\<sigma>); z1 = (b1 \<otimes> inv \<^bold>g) [^] u1 \<otimes> h1 [^] v1 \<otimes> (if \<sigma>' then x\<sigma> else \<one>); e0 = (w0, z0); e1 = (w1, z1) in \<A>3 e0 e1 s)))))"
b_accs_as_closed: fixes \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d set" assumes "bacc_step \<delta> Q \<subseteq> Q" shows "b_accessible \<delta> \<subseteq> Q"
int_simps_2: shows "\<turnstile> \<not> #True = #False"
smult_0_right: fixes a :: "'c" shows "Polynomial.smult a 0 = 0"
rel_distinct_9: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x21 :: "'c"   and x22 :: "'a tm list"   and y4 :: "('b, 'd) fm" shows "\<not> rel_fm R1 R2 (\<^bold>\<ddagger> x21 x22) (\<^bold>\<forall> y4)"
sums_mult_D: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "(\<lambda>n. c * f n) sums a"   and "c \<noteq> (0::'a)" shows "f sums (a / c)"
not_is_NoMsg: fixes m :: "'b" shows "(\<not> is_NoMsg m) = is_Msg m"
a1: shows " M1 \<longrightarrow>\<^sub>1\<^sup>* M2"
abs_summable_on_nat_iff: fixes f :: "nat \<Rightarrow> 'd"   and A :: "nat set" shows "(f abs_summable_on A) = summable (\<lambda>n. if n \<in> A then norm (f n) else 0)"
inverse_nonpositive_iff_nonpositive: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse a) zero = less_eq a zero"
dgrad_sig_set_closed_Koszul_syz: fixes a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading dgrad"   and "a \<in> dgrad_sig_set' (length fs) dgrad"   and "b \<in> dgrad_sig_set' (length fs) dgrad" shows "rep_list a \<odot> b - rep_list b \<odot> a \<in> dgrad_sig_set' (length fs) dgrad"
proj_psi_le_one: shows "proj_psi \<le>\<^sub>L 1\<^sub>m N"
distinct_enum_upto: fixes b :: "'b word" shows "distinct [0 .e. b]"
ConE_1: fixes p :: "'a pfm"   and q :: "'a pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<and>\<^sub>! q)" shows "\<turnstile>\<^sub>! p"
1: shows "card (Rep_nat_int i \<union> Rep_nat_int j) = card (Rep_nat_int i) + card (Rep_nat_int j)"
aug_pm_aug_pm_minus: fixes a :: "'e"   and i :: "'e \<Rightarrow> 'e"   and z :: "'e" assumes "ipp_cond1 {a} i" shows "aug_pm_set z i {a} = aug_pm_set z i {\<^sub>i- a}"
tensors_from_net_Conv_copy_first: fixes m :: "real mat convnet"   and i :: "nat"   and nr :: "nat" assumes "valid_net' m"   and "i < nr"   and "0 < output_size' m" shows "tensors_from_net (Conv (copy_first_matrix nr (output_size' m)) m) $ i = tensors_from_net m $ 0"
ZObisT_transT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>01T d"   and "s \<approx> t"   and "(c, s) \<rightarrow>t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>t t'" and "s' \<approx> t'"
submod: fixes K :: "('b, 'c) ring_scheme"   and W :: "'d set"   and V :: "('b, 'd, 'e) module_scheme" assumes "subspace K W V" shows "LinearCombinations.submodule K W V"
intro: fixes G :: "'c set"   and K :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "\<And>g k. \<lbrakk>g \<in> G; k \<in> K\<rbrakk> \<Longrightarrow> composition (composition (monoid.inverse G composition unit g) k) g \<in> K" shows "normal_subgroup_axioms K G composition unit"
frechet_derivative_at_real_eq_scaleR: fixes f :: "real \<Rightarrow> 'b"   and x :: "real"   and v :: "real" assumes "f differentiable at x"   and "NO_MATCH (1::'c) v" shows "frechet_derivative f (at x) v = v *\<^sub>R frechet_derivative f (at x) 1"
list_bind_cong: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'b list"   and g :: "'a \<Rightarrow> 'b list" assumes "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "List.bind xs f = List.bind ys g"
set_integrable_Un: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "set_integrable M A f"   and "set_integrable M B f"   and "A \<in> sets M"   and "B \<in> sets M" shows "set_integrable M (A \<union> B) f"
distinct_5: shows "ou_witness.b \<noteq> ou_witness.c"
some_gcd_0_1: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.is_gcd times (dvd.some_gcd times a zero) a zero"
subspace_UNIV: fixes p :: "'a" shows "subspace (tangent_space p)"
acyclic_regressively_finite: fixes x :: "'a" assumes "is_acyclic x" shows "regressively_finite x"
clop_Fix_range: fixes f :: "'b \<Rightarrow> 'b" assumes "clop f" shows "Fix f = range f"
w_values_WriteMemD: fixes ad :: "'a"   and al :: "addr_loc"   and v :: "'a val"   and obs :: "('a, 'b) obs_event action list"   and P :: "'c prog"   and vs0 :: "'a \<times> addr_loc \<Rightarrow> 'a val set" assumes "NormalAction (WriteMem ad al v) \<in> set obs" shows "v \<in> w_values P vs0 obs (ad, al)"
build_rule_index_s_def: fixes \<delta> :: "('a, 'b) ta_rule dlist" shows "build_rule_index_s \<delta> \<equiv> build_rule_index lhs \<delta>"
cwiseext_compat_cons: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "cwiseext gt ys xs" shows "cwiseext gt (x # ys) (x # xs)"
simps_2: fixes f :: "'b \<rightarrow> 'c \<rightarrow> 'b"   and z :: "'b"   and x :: "'c"   and xs :: "['c]" shows "foldl\<cdot>f\<cdot>z\<cdot>(x : xs) = foldl\<cdot>f\<cdot>(f\<cdot>z\<cdot>x)\<cdot>xs"
supp_Rep_formula: fixes x :: "('e, 'f, 'g) formula" shows "supp (Rep_formula x) = supp x"
eventually_nhds_x_imp_x: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "class.topological_space open"   and "eventually P (topological_space.nhds open x)" shows "P x"
restrict_relation_carrier: shows "restrict_relation (\<Union> (set As)) (of_weak_ranking As) = of_weak_ranking As"
dist_of_real: fixes x :: "real"   and y :: "real" shows "dist (of_real x) (of_real y) = dist x y"
markedElementsPrefixAreMarkedElementsTrail: fixes p :: "('a \<times> bool) list"   and M :: "('a \<times> bool) list"   and m :: "'a" assumes "isPrefix p M"   and "m \<in> set (markedElements p)" shows "m \<in> set (markedElements M)"
measure: shows "\<forall>\<sigma>. \<not> exec_stop \<sigma> \<and> cond \<sigma> \<and> P \<sigma> \<longrightarrow> M \<sigma> \<noteq> None \<and> f (snd (the (M \<sigma>))) < f \<sigma>"
smc_SemiCAT_Comp_app: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<^bsub>smc_SemiCAT \<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<^bsub>smc_SemiCAT \<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>A\<^bsub>smc_SemiCAT \<alpha>\<^esub> \<FF> = \<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>"
altinsertc: shows "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) c = C"
rtos_rs_1: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "rtos r s i < r * s"   and "\<not> Suc (rtos r s i) < r * s" shows "rtos r s i = r * s - Suc 0"
pdivides_imp_degree_le: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<noteq> []"   and "p pdivides\<^bsub>K [X]\<^esub> q" shows "degree p \<le> degree q"
has_integralD: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and a :: "'a"   and b :: "'a"   and e :: "real" assumes "(f has_integral y) (cbox a b)"   and "0 < e" obtains \<gamma> :: "'a \<Rightarrow> 'a set" where "gauge \<gamma>" and "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - y) < e"
g_value_image_filter_def: fixes f :: "'k \<Rightarrow> 'v1 \<Rightarrow> 'v2 option"   and m1 :: "'s1" shows "g_value_image_filter f m1 \<equiv> m1.iterate m1 (\<lambda>(k, v) \<sigma>. case f k v of None \<Rightarrow> \<sigma> | Some v' \<Rightarrow> m2.update_dj k v' \<sigma>) (m2.empty ())"
sum_le_included: fixes s :: "'b set"   and t :: "'c set"   and g :: "'c \<Rightarrow> 'd multiset"   and i :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'd multiset" assumes "finite s"   and "finite t"   and "\<forall>y\<in>t. {#} \<subseteq># g y"   and "\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> f x \<subseteq># g y" shows "comm_monoid_add.sum (+) {#} f s \<subseteq># comm_monoid_add.sum (+) {#} g t"
representation_basis: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set"   and b :: "'c" assumes "module scale"   and "\<not> module.dependent scale basis"   and "b \<in> basis" shows "module.representation scale basis b = (\<lambda>v. if v = b then 1::'b else (0::'b))"
\<delta>_states_remap: fixes f :: "'c \<Rightarrow> 'a"   and \<delta> :: "('c, 'b) ta_rule set" shows "\<delta>_states (remap_rule f ` \<delta>) = f ` \<delta>_states \<delta>"
assms: shows "atom ` (edom \<rho> - S) \<sharp>* e"
scopeExtSum: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<oplus> Q) \<sim>\<^sub>l P \<oplus> <\<nu>x>Q"
gen_qe_eval': fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (gen_qe var L F) (xs @ \<Gamma>))"
list_relI_2: fixes x :: "'a"   and x' :: "'b"   and R :: "('a \<times> 'b) set"   and l :: "'a list"   and l' :: "'b list" assumes "(x, x') \<in> R"   and "(l, l') \<in> \<langle>R\<rangle>list_rel" shows "(x # l, x' # l') \<in> \<langle>R\<rangle>list_rel"
fmupd_idem: fixes a :: "'a"   and x :: "'b"   and y :: "'b"   and m :: "('a, 'b) fmap" shows "fmupd a x (fmupd a y m) = fmupd a x m"
pgalleryI_gallery: fixes X :: "'b set set"   and xs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimpchain X xs"   and "distinct xs" shows "SimplicialComplex.pmaxsimpchain X xs"
comp_fun_commute_axioms: shows "comp_fun_commute (good_entry x e \<rho> \<circ> good_prod)"
take_takefill: fixes m :: "nat"   and n :: "nat"   and fill :: "'a"   and w :: "'a list" assumes "m \<le> n" shows "take m (takefill fill n w) = takefill fill m w"
Int_ac_3: fixes A :: "'h set"   and B :: "'h set" shows "A \<inter> B = B \<inter> A"
adm_edges_def: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" shows "adm_edges f l \<equiv> {(u, v). (u, v) \<in> cfE_of f \<and> l u = l v + 1}"
ordered_term_axioms: shows "ordered_term pair_of_term term_of_pair (\<preceq>) (\<prec>) (\<preceq>\<^sub>t) (\<prec>\<^sub>t)"
assms: shows "\<exists>x. [x\<^sub>v*A]=c"
Cons: fixes x :: "'a"   and xs' :: "'a list" assumes "xs = x # xs'"   and "R x y"   and "list_all2 R xs' ys'" shows "phi"
Inf_nat_def1: fixes K :: "nat set" assumes "K \<noteq> {}" shows "Inf K \<in> K"
inducedHOMTr0: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and a :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "a \<in> carrier F" shows "(f\<dieresis>\<^bsub>F,G\<^esub>) (gker\<^bsub>F,G\<^esub> f \<bullet>\<^bsub>F\<^esub> a) = f a"
PROD_E: fixes v :: "val"   and t1 :: "pdf_type"   and t2 :: "pdf_type" assumes "val_type v = PRODUCT t1 t2" obtains a :: "val" and b :: "val" where "val_type a = t1" and "val_type b = t2" and "v = <|a, b|>"
gr_implies_not_zero: fixes plus :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and zero :: "'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and m :: "'b::type"   and n :: "'b::type" assumes "class.canonically_ordered_monoid_add (plus::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (zero::'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (m::'b::type) (n::'b::type)" shows "(n::'b::type) \<noteq> (zero::'b::type)"
is_Groebner_basis_def: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "punit.is_Groebner_basis F \<equiv> relation.is_ChurchRosser (punit.red F)"
square_rtrancl_reflcl_commute: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "square R S S\<^sup>*\<^sup>* R\<^sup>=\<^sup>=" shows "commute R\<^sup>*\<^sup>* S\<^sup>*\<^sup>*"
cases: fixes x :: "('b \<times> color) tree \<times> 'b \<times> ('b \<times> color) tree" obtains t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (t1__, a__, R t2__ b__ (R t3__ c__ t4))"   | t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" where "x = (t1a__, aa__, R (R t2a__ ba__ t3a__) ca__ \<langle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (t1b__, ab__, R (R t2 b t3) c (B v__ vc__ vb__))"   | t1 :: "('b \<times> color) tree" and a :: "'b" where "x = (t1c__, ac__, \<langle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" where "x = (t1d__, ad__, B va__ vca__ vba__)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and va :: "'b \<times> color" where "x = (t1e__, ae__, \<langle>\<langle>\<rangle>, vaa__, \<langle>\<rangle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and vb :: "('b \<times> color) tree" and ve :: "'b" and vd :: "('b \<times> color) tree" and va :: "'b \<times> color" where "x = (t1f__, af__, \<langle>B vbb__ ve__ vd, vab__, \<langle>\<rangle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and v :: "('b \<times> color) tree" and vb :: "'b" where "x = (t1g__, ag__, B vf__ vbc__ \<langle>\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and va :: "'b \<times> color" and vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" where "x = (t1h__, ah__, \<langle>\<langle>\<rangle>, vac__, B vcb__ vfa__ vea__\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and vb :: "('b \<times> color) tree" and vh :: "'b" and vg :: "('b \<times> color) tree" and va :: "'b \<times> color" and vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" where "x = (t1i__, ai__, \<langle>B vbd__ vh vg, va, B vcc__ vfb__ veb__\<rangle>)"   | t1 :: "('b \<times> color) tree" and a :: "'b" and v :: "('b \<times> color) tree" and vb :: "'b" and vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" where "x = (t1, a, B v vb (B vc vf ve))"
split_along_ivls_def: fixes M :: "nat"   and X :: "'b set"   and IS :: "'b set" shows "split_along_ivls M X IS = sets_of_coll IS \<bind> (\<lambda>IS. FORWEAK IS (RETURN {}) (sctnbounds_of_ivl M) (\<lambda>sctns sctns'. RETURN (sctns' \<union> sctns)) \<bind> (\<lambda>sctns. FOREACH\<^bsup>\<lambda>_ R. R = X\<^esup> sctns split_ivls_at_halfspace X))"
pp_star_contr: fixes X :: "'a ppath set"   and Y :: "'a ppath set" shows "pp_prod (pp_star X) Y = (\<Union>n. pp_prod (ppath_dioid.power X n) Y)"
assms_1: shows "Derives1 u i r (v1 @ [N] @ v2)"
geq2: shows "r'\<^sub>2 \<le> r2"
extractableI: fixes x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg"   and I :: "msg set"   and K :: "msg set" assumes "implem (Chan x A B M) \<in> I"   and "x = insec \<or> x = auth \<or> (x = confid \<or> x = secure) \<and> (A, B) \<in> broken K" shows "implem (Chan x A B M) \<in> extractable K I"
a: shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) in v]"
clm_subs: shows "restrict v (clm ts) c = restrict v (res ts) c"
Tasked_Proof_Graph_axioms: shows "Tasked_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions vertices nodeOf edges vidx inst"
seqI': fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and b :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and c :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] c\<guillemotright>" shows "J_C.seq g f"
l1_step1_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent" shows "l1_step1 Ra A B \<equiv> {(s, s'). Ra \<notin> dom (progress s) \<and> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xgnx})\<rparr>}"
ind_in_set_bound: fixes M :: "nat"   and N :: "nat"   and A :: "nat set" assumes "M \<le> N" shows "ind_in_set A N \<notin> ind_in_set A ` ({0..<M} \<inter> A)"
beukers_key_inequality: fixes b :: "nat"   and a :: "int"   and n :: "nat" assumes "0 < b"   and "Re (zeta 3) = real_of_int a / real b" shows "1 \<le> real (2 * b) * Re (zeta 3) * real (Lcm {1..n} ^ 3) / 27 ^ n"
Times_mset_single_single: fixes a :: "'a"   and b :: "'b" shows "{#a#} \<times># {#b#} = {#(a, b)#}"
leg1_simps_4: shows "trg\<^sub>D (F g) = trg\<^sub>D (F r)"
norm_scaleR: fixes a :: "real"   and x :: "'b" shows "norm (a *\<^sub>R x) = \<bar>a\<bar> * norm x"
op_norm_set_proptys_2: fixes A :: "(('a, 'b) vec, 'c) vec" shows "bdd_above {\<parallel>A *v x\<parallel> |x. \<parallel>x\<parallel> = 1}"
lunit'_in_vhom: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.trg f = b" shows "Span.in_hom (Span.lunit' f) f (Span.hcomp b f)"
SIGT_plus1: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<sigma> :: "('b, 'c, 'a, 'd) subst_scheme"   and t1 :: "('b, 'e) trm"   and t2 :: "('b, 'e) trm" assumes "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT (Plus t1 t2). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)" shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT t1. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"
prj_simps_1: fixes h :: "'a"   and k :: "'a" assumes "ide h"   and "is_left_adjoint k"   and "trg h = trg k" shows "trg (prj\<^sub>0 h k) = src h"
p\<^sub>0_simps: shows "trg r\<^sub>0s\<^sub>1.p\<^sub>0 = src s\<^sub>1"
stalk_axioms: shows "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str {U. is_open U \<and> x \<in> U} x"
assms_2: shows "\<lbrakk>M; S\<rbrakk>\<^sub>c \<delta>"
cSup_eq: fixes X :: "'b set"   and c :: "'b" assumes "\<forall>x\<in>X. x \<le> c"   and "\<exists>x\<in>X. c \<le> x" shows "Sup X = c"
inv: fixes h :: "'a"   and H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "h \<in> H" shows "m_inv G h \<in> generate G H"
lsl_almost_lax: fixes x :: "'b"   and y :: "'b" shows "x \<cdot> \<nu> y \<le> \<nu> y"
of_int_sint: fixes a :: "'a Word.word" shows "word_of_int (sint a) = a"
hvcon': shows "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
inj_on_respects_domain_and_range_permutation: fixes A :: "'c set"   and B :: "'d set" shows "(\<lambda>f. inj_on f A) respects domain_and_range_permutation A B"
simps_9: fixes i :: "int"   and a :: "Cooper.num" shows "bound0 (Dvd i a) = numbound0 a"
obj_fields_marked_def: shows "obj_fields_marked = (\<lambda>s. \<forall>x. x \<in> (- gc_field_set) s \<longrightarrow> obj_at_field_on_heap (\<lambda>r. marked r s) (gc_tmp_ref s) x s)"
in_idem: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<in>| A" shows "min x (fMin A) = fMin A"
infinite: fixes A :: "'a set" assumes "infinite A" shows "mset_set A = {#}"
insert_insert': fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool"   and i :: "'c option" shows "Ordered_List.insert xs e i = insert' xs e i"
continuous_within: fixes x :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous (at x within s) f = (f \<longlongrightarrow> f x) (at x within s)"
simulation_this: shows "simulation (R\<inverse>) (stutter_extend B) (stutter_extend A)"
Irr_1: shows "Irr K x \<in> carrier (K [X])"
"case": fixes f :: "'b list \<Rightarrow> 'a"   and x :: "'b list" shows "(case IArray x of IArray x \<Rightarrow> f x) = f x"
extf_compat_append_left: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and f :: "'b"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   and ys :: "('b, 'a) tm list"   and xs :: "('b, 'a) tm list"   and zs :: "('b, 'a) tm list" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf"   and "extf f gt ys xs" shows "extf f gt (zs @ ys) (zs @ xs)"