hcju/isabelleps huggingface dataset:
makes_compatible_eval_sched: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms\<^sub>1 :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and cms\<^sub>2 :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b"   and mems :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list"   and sched :: "nat list"   and cms\<^sub>1' :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>1' :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.makes_compatible dma \<C>_vars eval (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "sifum_security_init.sound_mode_use dma \<C>_vars eval (cms\<^sub>1, mem\<^sub>1)"   and "sifum_security_init.sound_mode_use dma \<C>_vars eval (cms\<^sub>2, mem\<^sub>2)"   and "sifum_security_init.meval_sched eval sched (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>1', mem\<^sub>1')" shows "\<exists>cms\<^sub>2' mem\<^sub>2' mems'. sifum_security_init.sound_mode_use dma \<C>_vars eval (cms\<^sub>1', mem\<^sub>1') \<and> sifum_security_init.sound_mode_use dma \<C>_vars eval (cms\<^sub>2', mem\<^sub>2') \<and> map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> sifum_security_init.meval_sched eval sched (cms\<^sub>2, mem\<^sub>2) (cms\<^sub>2', mem\<^sub>2') \<and> sifum_security_init.makes_compatible dma \<C>_vars eval (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2', mem\<^sub>2') mems'"
residue_closed: fixes b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "b \<in> carrier Zp" shows "b k \<in> carrier (residue_ring (p ^ k))"
distinct_alist_of: fixes q :: "('c, 'd) pq" shows "distinct (alist_of q)"
weak_total_order_axioms_def: fixes L :: "('a, 'b) gorder_scheme" shows "weak_total_order_axioms L \<equiv> \<forall>x y. x \<in> carrier L \<longrightarrow> y \<in> carrier L \<longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> y \<or> y \<sqsubseteq>\<^bsub>L\<^esub> x"
strict_iff_order: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a <s b) = (a \<le>s b \<and> a \<noteq> b)"
diff_fun_submanifold: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'e"   and S :: "'c set" assumes "diff_fun k charts f"   and "open S" shows "diff_fun k (manifold.charts_submanifold charts S) f"
content_dvd: fixes p :: "'b poly" shows "[:content p:] dvd p"
vi: fixes x :: "real" assumes "x \<in> {0..1} - s" shows "(\<phi> has_real_derivative D x) (at x within {0..1})"
addr_of_array_inj: fixes a :: "'b array"   and a' :: "'b array" shows "(addr_of_array a = addr_of_array a') = (a = a')"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_on_right_moving (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> ml + mr = Suc (Suc m) \<and> (if lm1 = [] then l = Oc \<up> ml @ Bk # Bk # ires else l = Oc \<up> ml @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> (r = Oc \<up> mr @ [Bk] @ <lm2> @ Bk \<up> rn \<or> r = Oc \<up> mr \<and> lm2 = []))"
forward_terminating_iff3: fixes x :: "'a" assumes "path x" shows "forward_terminating x = (x \<le> x\<^sup>\<star> ; - (x ; 1))"
substBinp_preserves_good: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and binp :: "'h \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodBinp binp" shows "goodBinp (binp %%[X / x]_xs)"
EF: shows "is_sentence (\<delta> @ \<alpha>) \<and> is_sentence \<gamma> \<and> LeftDerivation (\<delta> @ \<alpha>) D \<gamma> \<and> i < length (\<delta> @ \<alpha>) \<and> j < length \<gamma> \<and> (\<delta> @ \<alpha>) ! i = \<gamma> ! j \<and> D = E @ derivation_shift F 0 (Suc j) \<and> LeftDerivation (take i (\<delta> @ \<alpha>)) E (take j \<gamma>) \<and> LeftDerivation (drop (Suc i) (\<delta> @ \<alpha>)) F (drop (Suc j) \<gamma>)"
pochhammer_Suc_prod_rev: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a (Suc n) = comm_monoid_mult.prod times one (\<lambda>i. plus a (semiring_1.of_nat one plus zero (n - i))) {0..n}"
tauLaw3CongSimRight: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<alpha> :: "'a prefix" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi> P Q R S. \<lbrakk>\<Psi> \<rhd> P \<sim> Q; (\<Psi>, Q, R) \<in> Rel; \<Psi> \<rhd> R \<sim> S\<rbrakk> \<Longrightarrow> (\<Psi>, P, S) \<in> Rel"   and "\<And>\<Psi>. (\<Psi>, P, \<tau>.Q) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>P \<leadsto>\<guillemotleft>Rel\<guillemotright> \<alpha>\<cdot>(\<tau>.Q)"
card_of_Field_ordIso: fixes r :: "('b \<times> 'b) set" assumes "Card_order r" shows "|Field r| =o r"
intros: fixes t :: "trm"   and k :: "stack" shows "t \<star> k \<rhd> k"
assms_1: shows "b = r\<lparr>a\<rparr>"
rep_eq: fixes x :: "nat"   and xa :: "float"   and xb :: "float" shows "real_of_float (float_divr x xa xb) = real_divr x (real_of_float xa) (real_of_float xb)"
simps_1: fixes f :: "nat \<Rightarrow> 'a" shows "is_prefix [] f = True"
ch2ch_fun: fixes S :: "nat \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'b" assumes "chain S" shows "chain (\<lambda>i. S i x)"
conj_wo_inv: fixes Q :: "'a set"   and P :: "'a set"   and g :: "'a" assumes "subgroup Q G"   and "subgroup P G"   and "g \<in> carrier G"   and "inv g <# (Q #> g) = P" shows "Q #> g = g <# P"
OR1: fixes evs1 :: "event list"   and NA :: "nat"   and A :: "agent"   and B :: "agent"   and M :: "nat" assumes "evs1 \<in> orb"   and "Nonce NA \<notin> used evs1" shows "Says A B \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce NA, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> # evs1 \<in> orb"
assms_2: shows "SubRel.subpath g v1 es v2 subs"
mu_const: fixes P :: "'b" shows "(\<mu> X \<bullet> P) = P"
arl_is_empty_def: shows "arl_is_empty \<equiv> \<lambda>(a, n). return (n = 0)"
concat_bit_of_zero_1: fixes n :: "nat"   and l :: "int" shows "concat_bit n 0 l = push_bit n l"
sublens_greatest: fixes X :: "'d \<Longrightarrow> 'e" assumes "vwb_lens X" shows "X \<subseteq>\<^sub>L 1\<^sub>L"
ZassenhausTr3: fixes H :: "'a set"   and H1 :: "'a set"   and K :: "'a set"   and K1 :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> H1 "   and "G \<guillemotright> K "   and "G \<guillemotright> K1 "   and "\<natural>H \<triangleright> H1"   and "\<natural>K \<triangleright> K1" shows "\<natural>H1 \<diamondop> H \<inter> K \<triangleright> H1 \<diamondop> H \<inter> K1"
max_var: fixes v :: "nat"   and f :: "nat formula" assumes "v \<in> atoms f" shows "v \<le> find_max (formula_vars f)"
lnth_Suc_LCons: fixes x :: "'c"   and xs :: "'c llist"   and n :: "nat" shows "LCons x xs $ Suc n = xs $ n"
i_aggregate_nth: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b"   and n :: "nat" shows "i_aggregate f k ag n = ag (f \<Up> (n * k) \<Down> k)"
openE: fixes m :: "'a set" assumes "m open" obtains "m \<in> T"
width_bound_V_le: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set" assumes "TreeDecomposition G T bag" shows "TreeDecomposition.width T bag \<le> card V\<^bsub>G\<^esub> - 1"
parZero: fixes P :: "pi" shows "P \<parallel> \<zero> \<sim>\<^sub>e P"
rr_def: shows "rr = 1 + max \<bar>mx\<bar> \<bar>mn\<bar>"
simps_1: fixes q :: "('c monom \<times> 'd) list" shows "check_poly_ge [] q = list_all (\<lambda>(uu_, d). d \<le> (0::'d)) q"
coeff_smult: fixes a :: "'a"   and p :: "'a poly"   and n :: "nat" shows "coeff (smult a p) n = a * coeff p n"
set_integrable_abs: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "set_integrable M A f" shows "set_integrable M A (\<lambda>x. \<bar>f x\<bar>)"
is_bisimulation_def: fixes R :: "'state \<Rightarrow> 'state \<Rightarrow> bool" shows "S_transform.is_bisimulation R \<equiv> symp R \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<phi>. P \<turnstile>\<^sub>S \<phi> \<longrightarrow> Q \<turnstile>\<^sub>S \<phi>)) \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> P \<rightarrow>\<^sub>S \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. Q \<rightarrow>\<^sub>S \<langle>\<alpha>,Q'\<rangle> \<and> R P' Q')))"
that: fixes i :: "'b" assumes "i \<in> A" shows "coprime a (f i)"
simps_1: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "[] @- \<sigma> = \<sigma>"
assms_2: shows "ky = (k, the (fmlookup (fmmap f x) k))"
I: shows "(xs @ ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys X) \<in> failures P"
gen_iapath_def: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "G.gen_iapath V u p v \<equiv> u \<in> V \<and> v \<in> V \<and> G.apath u p v \<and> set (G.inner_verts p) \<inter> V = {} \<and> p \<noteq> []"
monom_coeff_def: fixes l :: "linear_poly" shows "monom_coeff l = coeff l (monom_var l)"
rtrancl_into_trancl1: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
x0_1: fixes x :: "nat" assumes "x0 \<le> x" shows "\<bar>g x\<bar> < \<epsilon>"
Object_subcls1: fixes P :: "'a prog"   and C :: "String.literal" shows "\<not> P \<turnstile> Object \<prec>\<^sup>1 C"
hom_ord_dvd_ord: fixes a :: "'a" assumes "a \<in> carrier G" shows "H.ord (h a) dvd G.ord a"
aff_dim_convex_hull: fixes S :: "'a set" shows "aff_dim (convex hull S) = aff_dim S"
on_curvep_iff_on_curve: fixes a :: "'a"   and b :: "'a"   and p :: "'a \<times> 'a \<times> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "in_carrierp p" shows "on_curvep a b p = on_curve a b (make_affine p)"
frontier_translation: fixes a :: "'c"   and s :: "'c set" shows "frontier ((+) a ` s) = (+) a ` frontier s"
adj_sub_geq1: fixes \<sigma> :: "('b, 'c, 'a) subst"   and x1 :: "('b, 'a) trm"   and x2 :: "('b, 'a) trm" shows "(\<Union>i\<in>SIGT x1. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x) \<subseteq> \<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF (Geq x1 x2)))"
simps_4: fixes f1 :: "'x"   and f2 :: "'y \<Rightarrow> 'x" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
assms: shows "pseudonaturally_equivalent V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G"
assms_4: fixes n :: "nat" shows "g n \<in> S"
assms_2: shows "P \<turnstile> \<sigma> -jvm\<rightarrow> \<sigma>'"
hyps: shows "normalize (prod_list P) = prod_list (map normalize P)"
map_of_zip_is_Some: fixes xs :: "'b list"   and ys :: "'c list"   and x :: "'b" assumes "length xs = length ys" shows "(x \<in> set xs) = (\<exists>y. map_of (zip xs ys) x = Some y)"
map_of_map_restrict: fixes f :: "'a \<Rightarrow> 'b"   and ks :: "'a list" shows "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
Lazy_llist_inverse: fixes xs :: "unit \<Rightarrow> ('a \<times> 'a llist) option" shows "force (Lazy_llist xs) = xs ()"
HSem_eq: fixes ESem :: "'f \<Rightarrow> ('d \<Rightarrow> 'e) \<rightarrow> 'e"   and \<Gamma> :: "('d \<times> 'f) list"   and \<rho> :: "'d \<Rightarrow> 'e" shows "has_ESem.HSem ESem \<Gamma>\<cdot>\<rho> = \<rho> ++\<^bsub>domA \<Gamma>\<^esub> evalHeap \<Gamma> (\<lambda>e. ESem e\<cdot>(has_ESem.HSem ESem \<Gamma>\<cdot>\<rho>))"
simps: fixes uu :: "'b tree"   and x :: "'b"   and uv :: "'b tree" shows "get_min \<langle>uu, x, uv\<rangle> = x"
simple_path_subpath: fixes g :: "real \<Rightarrow> 'a"   and u :: "real"   and v :: "real" assumes "simple_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "u \<noteq> v" shows "simple_path (subpath u v g)"
arev_eq: fixes a :: "'a \<times> 'a" assumes "a \<notin> arcs (with_proj G)" shows "edge_rev M a = a"
canonical_retraction_def: fixes A :: "'a set set"   and C :: "'a set" shows "canonical_retraction A C = (\<lambda>a. restrict1 (the_apartment_iso (supapartment (supchamber a) C) A) (\<Union> X) a)"
trans_source_target_relation_impl_fully_abstract: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel"   and "SRel = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel)"   and "TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel)"   and "trans Rel" shows "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"
"1_1": shows "x \<in> {0..<2 * pi}"
isOmax_def: fixes R :: "('b \<times> 'b) set set"   and r :: "('b \<times> 'b) set" shows "isOmax R r \<equiv> r \<in> R \<and> (\<forall>r'\<in>R. r' \<le>o r)"
INF_cong: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. s2p B x \<Longrightarrow> C x = D x" shows "Inf (\<P> C A) = Inf (\<P> D B)"
axioms_4: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'" shows "group_homomorphism \<eta> R addition zero R' addition' zero'"
prems: fixes p :: "'a poly" assumes "p \<in> set (p__ # ps)" shows "p \<noteq> 0"
cases: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T3 :: "ty" assumes "is_lub_sup P T1 T2 T3" obtains "SemiType.sup P T1 T2 = OK T3"
simps_3: fixes P :: "'b prog"   and ad :: "'a"   and al :: "addr_loc"   and v :: "'a val" shows "action_loc_aux P (NormalAction (WriteMem ad al v)) = {(ad, al)}"
\<O>': shows "\<O>' = acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>"
pth': shows "pm_pth (dpm s') n = (v, x1a, x2a)"
i: shows "i < dim_row (A $$ (0, 0) \<cdot>\<^sub>m B)"
prime_dvd_power_nat: fixes p :: "nat"   and x :: "nat"   and n :: "nat" assumes "normalization_semidom_class.prime p"   and "p dvd x ^ n" shows "p dvd x"
VA': shows "Vagree \<nu> \<omega> (\<Union>x\<in>{x. Inl (Inr x) \<in> SIGF \<phi>}. FVT (\<sigma> x))"
d_a_export: fixes x :: "'a"   and y :: "'a" shows "d (ad x \<cdot> y) = ad x \<cdot> d y"
sublist_append_back: fixes l1 :: "'b list"   and l2 :: "'b list" shows "subseq l1 (l2 @ l1)"
le_neq_trans: fixes a :: "'b list"   and b :: "'b list" assumes "a \<le>p b"   and "a \<noteq> b" shows "a <p b"
intro: fixes succ :: "'c list \<Rightarrow> 'c list"   and rel :: "('c \<times> 'c) set" assumes "\<And>as. set (succ as) = {b. \<exists>a\<in>set as. (a, b) \<in> rel}" shows "set_access_succ_axioms succ rel"
Sum_any_when_equal: fixes f :: "'c \<Rightarrow> 'a"   and b :: "'c" shows "(\<Sum>a. f a when a = b) = f b"
\<psi>_inverts_coext: fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "arrow_from_functor.is_coext (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (G x) (\<epsilon>.map x) y (\<psi> x g) g"
nonvar_term_has_composed_shallow_term: fixes t :: "('b, 'a) Term.term" assumes "\<nexists>x. t = Var x" shows "\<exists>f T. Fun f T \<sqsubseteq> t \<and> (\<forall>s\<in>set T. (\<exists>c. s = Fun c []) \<or> (\<exists>x. s = Var x))"
A: shows "F \<l>\<^sub>C[g] = (D.inv (unit (trg\<^sub>C g)) \<star>\<^sub>D F g) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C g, g))"
assoc_simps_3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "cod (assoc a b c) = local.prod a (local.prod b c)"
continuous_map_subtopology_fst: fixes X :: "'b topology"   and Y :: "'c topology"   and Z :: "('b \<times> 'c) set" shows "continuous_map (subtopology (prod_topology X Y) Z) X fst"
nsqn\<^sub>r_zero: fixes dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" shows "nsqn\<^sub>r (0, dsk, flag, hops, nhip) = 0"
bvars_wellformed_transaction_unfold_1: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" assumes "wellformed_transaction T" shows "bvars_transaction T = bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_checks T)"
leqc: fixes c :: "QDelta le_constraint" assumes "c \<in> cs" shows "\<exists>p const. c = Le_Constraint Leq_Rel p const"
pair_finite_arcs: shows "finite (parcs (list_digraph G_list))"
assms_3: shows "\<bar>valof a - valof b\<bar> < threshold TYPE(('e, 'f) IEEE.float)"
span_span: fixes A :: "'d set" shows "cspan (cspan A) = cspan A"
add_mono_thms_linordered_field_3: fixes i :: "'d"   and j :: "'d"   and k :: "'d"   and l :: "'d" assumes "i < j \<and> k \<le> l" shows "i + k < j + l"
one_not_le_zero_ereal: shows "\<not> 1 \<le> 0"
Always_ConstrainsD: fixes F :: "'c program"   and INV :: "'c set"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> Always INV"   and "F \<in> A Co A'" shows "F \<in> A Co INV \<inter> A'"
simps_2: fixes x :: "gauss_int"   and y :: "gauss_int" shows "ImZ (x * y) = ReZ x * ImZ y + ImZ x * ReZ y"
defs'_def: fixes g :: "'a"   and n :: "'node" shows "i.defs' g n \<equiv> (\<lambda>v. (v, n, SimpleDef)) ` defs2 g n"
mult_implies_one_step: fixes r :: "('b \<times> 'b) set"   and M :: "'b multiset"   and N :: "'b multiset" assumes "trans r"   and "(M, N) \<in> mult r" shows "\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r)"
pred_numeral_def: fixes k :: "num" shows "pred_numeral k = numeral k - 1"
closed_segment_iv_subset_domain: fixes t :: "real" assumes "t \<in> cball t_max et" shows "{t_max--t} \<subseteq> cball t_max et"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_stop (as, lm) (s, l, r) ires = (\<exists>rn. l = Bk # Bk # ires \<and> r = <lm> @ Bk \<up> rn)"
mem_Aleph_succ: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "\<aleph>\<alpha> \<in> elts (\<aleph>ZFC_in_HOL.succ \<alpha>)"
agree: fixes e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and \<pi>A :: "Syntax.term list"   and i :: "nat"   and \<pi>' :: "Syntax.term list"   and eV' :: "Syntax.term" assumes "{$$} \<turnstile> e, eP, eV : \<tau>"   and "\<lless>\<pi>A, eV\<ggreater> V\<rightarrow>i \<lless>\<pi>', eV'\<ggreater>" shows "\<exists>e' eP' \<pi> j \<pi>0 s s'. \<lless>[], e\<ggreater> I\<rightarrow>i \<lless>[], e'\<ggreater> \<and> \<lless>[], eP\<ggreater> P\<rightarrow>i \<lless>\<pi>, eP'\<ggreater> \<and> \<pi>A = \<pi> @ \<pi>' \<and> {$$} \<turnstile> e', eP', eV' : \<tau> \<or> j \<le> i \<and> \<lless>[], eP\<ggreater> P\<rightarrow>j \<lless>\<pi>0 @ [s], eP'\<ggreater> \<and> \<pi>A = \<pi>0 @ [s'] @ \<pi>' \<and> s \<noteq> s' \<and> hash s = hash s'"
load_tmps_distinct_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "load_tmps_distinct ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> distinct_load_tmps is"
segmentTr: fixes T :: "'a set" assumes "T \<subseteq> carrier D"   and "\<forall>b\<in>T. \<forall>x. x \<prec> b \<and> x \<in> carrier D \<longrightarrow> x \<in> T" shows "T = carrier D \<or> (\<exists>a. a \<in> carrier D \<and> T = segment D a)"
left_right_inverse_power: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x + y = (0::'a)" shows "x +^ n + y +^ n = (0::'a)"
dist_degree_factorize_main_i: fixes F :: "'i list"   and f :: "'a mod_ring poly"   and G :: "'i list"   and g :: "'a mod_ring poly"   and Res :: "(nat \<times> 'i list) list"   and res :: "(nat \<times> 'a mod_ring poly) list"   and d :: "nat" assumes "poly_rel F f"   and "poly_rel G g"   and "list_all2 (rel_prod (=) poly_rel) Res res" shows "list_all2 (rel_prod (=) poly_rel) (dist_degree_factorize_main_i p ff_ops (arith_ops_record.zero ff_ops) (arith_ops_record.one ff_ops) (degree_i F) F G d Res) (dist_degree_factorize_main f g d res)"
catch_option_def: fixes return :: "'d option \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d option \<Rightarrow> 'e) \<Rightarrow> 'e"   and m :: "('d, 'e) optionT"   and h :: "('d, 'e) optionT" shows "catch_option return bind m h = OptionT (bind (run_option m) (\<lambda>x. if x = None then run_option h else return x))"
ms_1: shows "g \<turnstile> n-ms\<rightarrow>m'"
extendAssoc: fixes S :: "('a, 'b) sequent"   and R :: "('a, 'b) sequent"   and c :: "('a, 'b) sequent" shows "extend S (extend R c) = extend (extend S R) c"
hhom_is_subcategory: fixes a :: "'a"   and b :: "'a" shows "subcategory (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>)"
inv_allD: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "local.inv_all p A"   and "i < n"   and "j < n" shows "p A i j"
mon_compose_rev: fixes f :: "'g \<Rightarrow> 'h"   and G :: "('g, 'i) monoid_scheme"   and H :: "('h, 'j) monoid_scheme"   and g :: "'h \<Rightarrow> 'k"   and K :: "('k, 'l) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H K"   and "g \<circ> f \<in> mon G K" shows "f \<in> mon G H"
robbins_is_huntington: shows "class.huntington_algebra uminus (\<sqinter>) (\<squnion>) \<bottom> \<top>"
measurable_ident: fixes M :: "'b measure" shows "id \<in> M \<rightarrow>\<^sub>M M"
stalk_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and I :: "'d set set"   and x :: "'d" shows "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x \<equiv> direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I \<and> stalk_axioms S is_open I x"
X: shows "X = prod_emb I M J (Pi\<^sub>E J E)"
rfn_compseries_iM: fixes G :: "('e, 'f) Group_scheme"   and r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and h :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "0 < r"   and "0 < s"   and "compseries G r f"   and "h \<in> wcsr_rfns G r f s" shows "card (h ` {i. i \<le> s * r}) = r + 1"
inangle3123: 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"   and "C \<noteq> B" shows "Tarski_neutral_dimensionless.InAngle Bet C A B C"
ipurge_ref_aux_cons: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and x :: "'e"   and xs :: "'e list"   and X :: "'e set" shows "ipurge_ref_aux I D U (x # xs) X = ipurge_ref_aux I D (sinks_aux I D U [x]) xs X"