hcju/isabelleps huggingface dataset:
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>\<Delta>.map f : \<Delta>.map a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] \<Delta>.map b\<guillemotright>"
abelian_subgroupI3: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "additive_subgroup H G"   and "abelian_group G" shows "abelian_subgroup H G"
child_sc: shows "h \<turnstile> get_scdom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<rightarrow>\<^sub>r child_sc"
intra_proc_additional_edge: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>a''. valid_edge a'' \<and> sourcenode a'' = targetnode a \<and> targetnode a'' = sourcenode a' \<and> kind a'' = (\<lambda>cf. False)\<^sub>\<surd>"
upper_Interval: fixes x :: "'a \<times> 'a" assumes "fst x \<le> snd x" shows "upper (Interval x) = snd x"
bf_subset: fixes M :: "nat set"   and q :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and u :: "nat \<times> nat" shows "fst (bf M q u) \<subseteq> M \<and> snd (bf M q u) \<subseteq> M"
map_prs_aux: fixes R1 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs1 :: "'c \<Rightarrow> 'd"   and rep1 :: "'d \<Rightarrow> 'c"   and R2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs2 :: "'e \<Rightarrow> 'f"   and rep2 :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f"   and l :: "'d list" assumes "Quotient3 R1 abs1 rep1"   and "Quotient3 R2 abs2 rep2" shows "map abs2 (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"
limi_subset: fixes T :: "'a set"   and g :: "'b" shows "limi T g \<subseteq> T"
vsv_vrange_vempty: fixes A :: "V" assumes "\<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A) = 0" shows "r \<restriction>\<^sup>l\<^sub>\<circ> A = 0"
mset_le_decr_left2: fixes c :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" assumes "c + a \<subseteq># b" shows "a \<subseteq># b"
induct: fixes P :: "\<I> \<Rightarrow> event_data option list set list \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "\<I>"   and a1 :: "event_data option list set list"   and a2 :: "nat list" assumes "\<And>I ts. P I [] ts"   and "\<And>I v va. P I (v # va) []"   and "\<And>I v va t. P I (v # va) [t]"   and "\<And>I x xs t t' ts. P I xs (t' # ts) \<Longrightarrow> P I (x # xs) (t # t' # ts)" shows "P a0 a1 a2"
in_restrict_rel_iff: fixes x :: "'a"   and y :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a) set" shows "((x, y) \<in> restrict_rel A R) = ((x, y) \<in> R \<and> x \<in> A \<and> y \<in> A)"
assms: shows "(p, p') \<in> fully_unsorted_poly_rel"
tseyting_update_keep_subformula_sema: fixes fresh :: "'a set \<Rightarrow> 'a"   and G :: "'a formula"   and F :: "'a formula"   and \<A> :: "'a \<Rightarrow> bool" assumes "freshstuff_sema fresh"   and "G \<in> set (subformulae F)" shows "freshstuff_sema.tseytin_update fresh \<A> F \<Turnstile> G = \<A> \<Turnstile> G"
max2_among: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "max2 a b \<in> {a, b}"
rbt_delete: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and k :: "'c" assumes "rbt t" shows "rbt (RBT_Map.delete k t)"
freshFrame: fixes F :: "'d frame"   and C :: "'e" obtains A\<^sub>F :: "name list" and \<Psi>\<^sub>F :: "'d" where "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>" and "distinct A\<^sub>F" and "A\<^sub>F \<sharp>* C"
dual_less_eq_iff: fixes x :: "'d dual"   and y :: "'d dual" shows "(x \<le> y) = (undual y \<le> undual x)"
leg0_in_hom_2: shows "\<guillemotleft>f' : f' \<Rightarrow> f'\<guillemotright>"
elims_3: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a tree" assumes "\<not> bst_wrt x xa" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and l :: "'a tree" and a :: "'a" and r :: "'a tree" where "x = P" and "xa = \<langle>l, a, r\<rangle>" and "\<not> ((\<forall>x\<in>set_tree l. P x a) \<and> Ball (set_tree r) (P a) \<and> bst_wrt P l \<and> bst_wrt P r)"
assms: shows " Var x \<longrightarrow>\<^sub>1 M"
add3_self: fixes p :: "real \<times> real"   and q :: "real \<times> real" shows "ccw' 0 p (p + q) = ccw' 0 p q"
Inf_le_iff_less: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and x :: "'a" shows "(Inf (f ` A) \<le> x) = (\<forall>y>x. \<exists>i\<in>A. f i \<le> y)"
rel_interior_affine: fixes S :: "'a set" assumes "affine S" shows "rel_interior S = S"
a_gla2: fixes x :: "'a"   and y :: "'a" assumes "ar x \<le> ar y" shows "y \<cdot> ar x = (zero_class.zero::'a)"
infiter_induct: fixes c :: "'a::refinement_lattice"   and x :: "'a::refinement_lattice" assumes "(c::'a::refinement_lattice) ; (x::'a::refinement_lattice) \<sqsubseteq> x" shows "(c::'a::refinement_lattice)\<^sup>\<infinity> \<sqsubseteq> (x::'a::refinement_lattice)"
mat1_sum_eq: fixes s :: "'e"   and t :: "'e" shows "(\<Sum>k\<in>UNIV. mat (1::'d) $ s $ k * mat (1::'d) $ k $ t) = mat (1::'d) $ s $ t"
IFmap_cong: fixes x :: "'d IF1"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and y :: "'d IF2" shows "((\<forall>a\<in>IF1set x. f a = g a) \<longrightarrow> IF1map f x = IF1map g x) \<and> ((\<forall>a\<in>IF2set y. f a = g a) \<longrightarrow> IF2map f y = IF2map g y)"
frameResChainFresh''': fixes x :: "name"   and xvec :: "name list"   and F :: "'b frame" assumes "x \<sharp> xvec" shows "x \<sharp> \<lparr>\<nu>*xvec\<rparr>F = x \<sharp> F"
is_Var_def: fixes hd :: "('a, 'b) hd" shows "is_Var hd = (\<exists>x1. hd = Var x1)"
slg_succs_def: fixes E :: "('b \<times> 'a) set"   and v :: "'b" shows "slg_succs E v \<equiv> E `` {v}"
simps: fixes \<delta>\<^sub>M :: "'c ltl_prop_equiv_quotient \<Rightarrow> 'c set \<Rightarrow> 'c ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'c ltl \<Rightarrow> 'c ltl_prop_equiv_quotient"   and \<pi> :: "'c ltl \<Rightarrow> nat option"   and \<chi> :: "'c ltl" shows "ltl_to_rabin_base_def.Acc_inf \<delta>\<^sub>M q\<^sub>0\<^sub>M \<pi> \<chi> = \<Union> (embed_transition_snd ` \<Union> (\<upharpoonleft>\<^sub>\<chi> ` mojmir_to_rabin_def.succeed\<^sub>R \<delta>\<^sub>M (q\<^sub>0\<^sub>M (theG \<chi>)) (Collect (ltl_prop_entails_abs (dom \<pi>))) (the (\<pi> \<chi>))))"
abs_summable_on_Sigma_project2: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'd set"   and x :: "'c" assumes "f abs_summable_on Sigma A B"   and "x \<in> A" shows "(\<lambda>y. f (x, y)) abs_summable_on B x"
prod_n_ideal_contTr: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set" assumes "\<forall>l\<le>n. ideal R (J l)" shows "i\<Pi>\<^bsub>R,n\<^esub> J \<subseteq> \<Inter> {X. \<exists>k\<le>n. X = J k}"
reachable_if_obtainable_step: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat" assumes "obtainable s r e"   and "\<exists>l i t. (s', t) |\<in>| possible_steps e s r l i" shows "reachable s' e"
estep_Pr1: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((Pr n f g, xs, ls) # fs, rv)"   and "ls \<noteq> []"   and "length ls \<noteq> Suc (hd xs)"   and "rv \<down>"   and "recfn (length xs) (Pr n f g)" shows "estep_Pr (encode_config c) = encode_config (step c)"
assms: shows "set \<phi>s \<subseteq> fmla"
coincidence_game: fixes \<omega> :: "variable \<Rightarrow> real"   and \<omega>' :: "variable \<Rightarrow> real"   and V :: "variable set"   and \<alpha> :: "game"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set" assumes "Vagree \<omega> \<omega>' V"   and "FVG \<alpha> \<subseteq> V" shows "(\<omega> \<in> game_sem I \<alpha> (restrictto X V)) = (\<omega>' \<in> game_sem I \<alpha> (restrictto X V))"
length_nSend: fixes activation :: "nat \<Rightarrow> nat list"   and nReturn :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and i :: "'b"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nStore :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "'b \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "'b \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat" assumes "BusInterface activation (nReturn i) recv (nStore i) (nSend i) (nGet i)"   and "\<forall>t. length (nReturn i t) \<le> Suc 0" shows "length (nSend i t) \<le> Suc 0"
plus_\<I>_mono: fixes \<I>1 :: "('a, 'b) \<I>"   and \<I>1' :: "('a, 'b) \<I>"   and \<I>2 :: "('c, 'd) \<I>"   and \<I>2' :: "('c, 'd) \<I>" assumes "\<I>1 \<le> \<I>1'"   and "\<I>2 \<le> \<I>2'" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<le> \<I>1' \<oplus>\<^sub>\<I> \<I>2'"
trace_callee_eq_coinduct: fixes S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf"   and A :: "'c set"   and callee1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and callee2 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf" assumes "S p q"   and "\<And>p q a. \<lbrakk>S p q; a \<in> A\<rbrakk> \<Longrightarrow> bind_spmf p (\<lambda>s. map_spmf fst (callee1 s a)) = bind_spmf q (\<lambda>s. map_spmf fst (callee2 s a))"   and "\<And>p q a res res' b s'' s'. \<lbrakk>S p q; a \<in> A; res \<in> set_spmf p; (b, s'') \<in> set_spmf (callee1 res a); res' \<in> set_spmf q; (b, s') \<in> set_spmf (callee2 res' a)\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (bind_spmf p (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (bind_spmf q (\<lambda>s. callee2 s a)) b)" shows "trace_callee_eq callee1 callee2 A p q"
is_offending_flows_def: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and f :: "('c \<times> 'c) set"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "SecurityInvariant_withOffendingFlows.is_offending_flows sinvar f G nP \<equiv> \<not> sinvar G nP \<and> sinvar (FiniteGraph.delete_edges G f) nP"
h1: shows "\<forall>m\<in>monomials (p * q). lookup m var \<le> MPoly_Type.degree p var + MPoly_Type.degree q var"
ub_ln_lb_ln_bounds': fixes x :: "float"   and prec :: "nat" assumes "1 \<le> x" shows "real_of_float (the (lb_ln prec x)) \<le> ln (real_of_float x) \<and> ln (real_of_float x) \<le> real_of_float (the (ub_ln prec x))"
z: shows "z \<in> inside (path_image c1 \<union> path_image c)"
cases: fixes a1 :: "'d set"   and a2 :: "'d \<Rightarrow> nat"   and a3 :: "'d \<Rightarrow> Regions.intv"   and a4 :: "('d \<times> 'd) set" assumes "Regions.valid_region a1 a2 a3 a4" obtains X\<^sub>0 :: "'d set" and X :: "'d set" and I :: "'d \<Rightarrow> Regions.intv" and r :: "('d \<times> 'd) set" and k :: "'d \<Rightarrow> nat" where "a1 = X" and "a2 = k" and "a3 = I" and "a4 = r" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d}" and "refl_on X\<^sub>0 r" and "trans r" and "total_on X\<^sub>0 r" and "\<forall>x\<in>X. Regions.valid_intv (k x) (I x)"
subsets_int: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "finite A"   and "B \<subseteq> A"   and "C \<subseteq> A"   and "card A < card B + card C" shows "B \<inter> C \<noteq> {}"
map_cond_spmf_fst: fixes f :: "'b \<Rightarrow> 'a"   and p :: "('c \<times> 'b) spmf"   and x :: "'c" shows "map_spmf f (cond_spmf_fst p x) = cond_spmf_fst (map_spmf (apsnd f) p) x"
add_eq_0_iff: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "(plus a b = zero) = (b = uminus a)"
\<omega>_of_Cantor_subset: fixes \<alpha>s :: "'a list"   and ms :: "nat list" shows "list.set (\<omega>_of_Cantor \<alpha>s ms) \<subseteq> list.set \<alpha>s"
fact_Suc: 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_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero (Suc n) = times (semiring_1.of_nat one plus zero (Suc n)) (semiring_char_0.fact one plus zero n)"
cscc_outer_invar_def: shows "cscc_outer_invar \<equiv> \<lambda>it (l, D). cscc_outer_invar_loc G it l D"
exe0_def: shows "exe0 \<equiv> map_fun eval\<o> make\<o> id"
onorm_scaleR_left: fixes r :: "'a \<Rightarrow> real"   and f :: "'b" assumes "bounded_linear r" shows "onorm (\<lambda>x. r x *\<^sub>R f) = onorm r * norm f"
SeqAbstFormP_imp_OrdP: fixes v :: "tm"   and u :: "tm"   and x :: "tm"   and x' :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqAbstFormP v u x x' s k} \<turnstile> OrdP k"
Gromov_extension_isometry_2: fixes f :: "'b \<Rightarrow> 'c" assumes "isometry f" shows "continuous_on UNIV (Gromov_extension f)"
simps_1: fixes e :: "name list" shows "trans_tm e Zero = DBZero"
hsq: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) sq = SQ"
le_sup_iff: fixes z :: "'c"   and x :: "'c"   and y :: "'c" shows "(z \<le> x \<sqinter> y) = (z \<le> x \<and> z \<le> y)"
lookup_pp_of_exps: fixes a :: "nat"   and b :: "nat"   and i :: "nat" shows "lookup_pp (fst (rep_nat_term (Term_Order.of_exps a b i))) = (\<lambda>x. if x = 0 then a else if x = 1 then b else 0)"
CIInterpretation: fixes T :: "(ZF, ZF) Axioms" assumes "ZFAxioms T" shows "Interpretation (CI T)"
difg_def: shows "difg \<equiv> \<lambda>m t. diff m t - ((\<Sum>p<n - m. diff (m + p) 0 / fact p * t ^ p) + B * (t ^ (n - m) / fact (n - m)))"
select_convs_1: fixes carrier :: "'f set"   and more :: "'g" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
inv_fun_plus: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a" assumes "Abstract_Linear_Poly.inv f1"   and "Abstract_Linear_Poly.inv f2" shows "Abstract_Linear_Poly.inv (fun_plus f1 f2)"
subs: shows "\<Gamma> \<subseteq> \<Gamma>'"
Image_singleton_iff: fixes b :: "'a"   and r :: "('b \<times> 'a) set"   and a :: "'b" shows "s2p (r `` \<eta> a) b = s2p r (a, b)"
tranclp_less_basic_type: fixes \<tau> :: "'c basic_type"   and \<sigma> :: "'c basic_type" shows "((\<tau>, \<sigma>) \<in> {(\<tau>, \<sigma>). \<tau> \<sqsubset>\<^sub>B \<sigma>}\<^sup>+) = (\<tau> < \<sigma>)"
simps_3: fixes x :: "'a"   and y :: "'a" shows "Some x \<nabla> Some y = Some (x \<nabla> y)"
simps_185: fixes x9 :: "Term.var" shows "Acc x9 \<noteq> Super"
igFreshIGVar_def: fixes MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igFreshIGVar MOD \<equiv> \<forall>ys y xs x. ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igFresh MOD ys y (igVar MOD xs x)"
frameImpEAux: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and F :: "'d frame"   and \<Phi> :: "'e"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.FrameImp SImp F \<Phi>"   and "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"   and "A\<^sub>F \<sharp>* \<Phi>" shows "SImp \<Psi>\<^sub>F \<Phi>"
bind_return_pmf: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b pmf" shows "return_pmf x \<bind> f = f x"
MakeNT_def: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "MakeNT \<eta> \<equiv> \<lparr>NTDom = NTDom \<eta>, NTCod = NTCod \<eta>, NatTransMap = restrict (($$) \<eta>) (obj\<^bsub>NTCatDom \<eta>\<^esub>)\<rparr>"
commute: fixes a :: "'a"   and b :: "'a" shows "a * b = b * a"
filter_mset_image_mset: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" shows "filter_mset P (image_mset f A) = image_mset f {#x \<in># A. P (f x)#}"
trans: fixes s1 :: "'a"   and s2 :: "'a"   and s3 :: "'a" assumes "s1 \<preceq> s2"   and "s2 \<preceq> s3" shows "s1 \<preceq> s3"
euclid_ext_i: shows "(poly_rel ===> poly_rel ===> rel_prod (rel_prod poly_rel poly_rel) poly_rel) (poly.euclid_ext_aux_i (arith_ops_record.one (poly_ops ops)) (arith_ops_record.zero (poly_ops ops)) (arith_ops_record.zero (poly_ops ops)) (arith_ops_record.one (poly_ops ops))) (euclid_ext_aux 1 0 0 1)"
BIND_D: fixes name :: "char list"   and inp :: "nat"   and P :: "bool" assumes "Case_Labeling.BIND name inp P" shows "P"
snoc: shows "(oid \<in> set (inserted_ids ops)) = (\<exists>ref val. (oid, Insert ref val) \<in> set ops)"
mod_diff_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus (modulo a c) (modulo b c)) c = modulo (minus a b) c"
surj_pair: fixes p :: "'a \<times> 'c" shows "\<exists>x y. p = (x, y)"
cones_map_mapsto: fixes f :: "'c" assumes "C.arr f" shows "cones_map f \<in> extensional (cones (C.cod f)) \<inter> (cones (C.cod f) \<rightarrow> cones (C.dom f))"
carrier_Quot: shows "carrier (R Quot I 0) = (+>) (I 0) ` carrier R"
polyn_addTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "pol_coeff S (n, f) \<and> pol_coeff S (n, g) \<longrightarrow> polyn_expr R X n (n, f) \<plusminus> polyn_expr R X n (n, g) = \<Sigma>\<^sub>e R (\<lambda>j. (f j \<plusminus>\<^bsub>S\<^esub> g j) \<cdot>\<^sub>r X^\<^bsup>R j\<^esup>) n"
content_box_cases: fixes a :: "'a"   and b :: "'a" shows "content (box a b) = (if \<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i then \<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i else 0)"
I: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier"   and "x \<noteq> y" shows "\<exists>u\<in>nhds x. \<exists>v\<in>nhds y. u \<inter> v = {}"
scaleR_eq_0_iff: fixes a :: "real"   and x :: "'a" shows "(a *\<^sub>R x = (0::'a)) = (a = 0 \<or> x = (0::'a))"
simps_1: fixes \<sigma> :: "'e \<Rightarrow> ('c monom \<times> 'd) list" shows "poly_subst \<sigma> [] = zero_poly"
distinct_n_lists: fixes xs :: "'a list"   and n :: "nat" assumes "distinct xs" shows "distinct (List.n_lists n xs)"
adds_plus_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a adds b" shows "a adds b + c"
red_preserves_hconf: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and E :: "String.literal \<Rightarrow> ty option"   and T :: "ty" assumes "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>"   and "P,E,hp s \<turnstile> e : T"   and "hconf (hp s)" shows "hconf (hp s')"
in_Ext_RTrans_implies_Elem_Ext_ZF_hull: fixes R :: "(ZF \<times> ZF) set"   and x :: "ZF"   and s :: "ZF" assumes "set_like R"   and "x \<in> Ext (R\<^sup>+) s" shows "Elem x (Ext_ZF_hull R s)"
intruder_synth_induct: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'c) Term.term set"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'c) Term.term"   and P :: "(('fun, 'atom, 'sets) prot_fun, 'c) Term.term set \<Rightarrow> (('fun, 'atom, 'sets) prot_fun, 'c) Term.term \<Rightarrow> bool" assumes "M \<turnstile>\<^sub>c t"   and "\<And>t. t \<in> M \<Longrightarrow> P M t"   and "\<And>T f. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t; \<And>t. t \<in> set T \<Longrightarrow> P M t\<rbrakk> \<Longrightarrow> P M (Fun f T)" shows "P M t"
simps_129: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x8 :: "32 prefix_match" shows "(case IPv4Dst x8 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f8 x8"
vars_Const: fixes c :: "'b" shows "vars (Const c) = {}"
bind_returns_heap_E2: fixes h :: "'ap"   and f :: "('ap, 'aq, 'as) prog"   and g :: "'as \<Rightarrow> ('ap, 'aq, 'ar) prog"   and h' :: "'ap" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h'"   and "pure f h" obtains x :: "'as" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
le_inf_range: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2" shows "sset w\<^sub>1 \<subseteq> sset w\<^sub>2"
lemma2: fixes K :: "nat"   and H :: "msg set"   and X :: "msg" assumes "Key (invKey K) \<in> analz H" shows "insert (Crypt K X) (analz (insert X H)) \<subseteq> analz (insert (Crypt K X) H)"
isolated_verts_da: shows "H.isolated_verts = G.isolated_verts"
ln_add_one_self_le_self: fixes x :: "real" assumes "0 \<le> x" shows "ln (1 + x) \<le> x"
is_pole_transform: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and g :: "'b \<Rightarrow> 'c"   and b :: "'b" assumes "is_pole f a"   and "\<forall>\<^sub>F x in at a. f x = g x"   and "a = b" shows "is_pole g b"
equivu_imp_le: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "equiv_utrans t u" shows "le_utrans t u"
weakly_preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.isomorphic (F (src\<^sub>C \<mu>)) (src\<^sub>D (F \<mu>))"
conf: shows "s0\<Colon>\<preceq>(G, L)"
c_tl_is_pr: shows "c_tl \<in> PrimRec1"
drop_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "n < m" shows "drop m (xs[n := x]) = drop m xs"
\<phi>_simps_5: fixes \<mu> :: "'a" assumes "\<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>" shows "cod (d\<^sub>0e\<^sub>0.\<phi> \<mu>) = e\<^sub>1 \<star> (d\<^sub>1 \<star> cod \<mu> \<star> e\<^sub>0) \<star> d\<^sub>0"
complex_double: fixes a :: "real"   and b :: "real" shows "Complex a b * 2 = Complex (2 * a) (2 * b)"
chine_hcomp_props_2: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "C.commutative_square (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>)) (Chn \<nu> \<cdot> \<p>\<^sub>1[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)]) (Chn \<mu> \<cdot> \<p>\<^sub>0[Leg0 (Dom \<nu>), Leg1 (Dom \<mu>)])"
assms_4: shows "Sup (f ` X) \<le> Sup (g ` X)"
polyhedral_cone_def: fixes n :: "nat"   and A :: "'b mat" shows "gram_schmidt.polyhedral_cone n A = {x \<in> carrier_vec n. A *\<^sub>v x \<le> 0\<^sub>v (dim_row A)}"
lm087: fixes f :: "'c \<Rightarrow> 'd" shows "toFunction (Graph f) = f"
dim_vec_of_basis_enum': fixes v :: "'a" shows "dim_vec (vec_of_basis_enum v) = length canonical_basis"
sup_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> y = x"
dem_def: fixes \<eta> :: "nat" shows "MAC.dem \<eta> \<equiv> CNV (stateless_callee (\<lambda>_. Pause (Inr ()) (\<lambda>am'. case am' of Inl aa \<Rightarrow> Done None | Inr None \<Rightarrow> Done None | Inr (Some (a, m)) \<Rightarrow> Pause (Inl m) Done \<bind> (\<lambda>r. lift_spmf (mac \<eta> (projl r) m) \<bind> (\<lambda>a'. Done (if a' = a then Some m else None)))))) ()"