hcju/isabelleps huggingface dataset:
obs_strong_postdominate: fixes n :: "'node"   and n' :: "'node"   and S :: "'node set" assumes "n \<in> obs n' (PDG_BS S)"   and "n \<noteq> (_Exit_)" shows "n strongly-postdominates n'"
ProcModifyReturnSameFaults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and ModifAbr :: "'e \<Rightarrow> 'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P call init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P call init p return c Q,A"
dterms_rel': shows "dterms_rel = (\<lambda>gq gp. (gq, gp) \<in> {((\<Gamma>, q), \<Gamma>', p). \<Gamma> = \<Gamma>' \<and> p \<leadsto>\<^bsub>\<Gamma>\<^esub> q})"
assms_2: shows "(\<lambda>n. to_Gromov_completion (t n)) \<longlonglongrightarrow> xi"
assms_3: shows "\<forall>\<^sub>F x in at_top. - l x \<le> u' x"
set_sel: fixes a :: "'d strand_label" assumes "a \<noteq> \<star>" shows "the_LabelN a \<in> set_strand_label a"
transitionI_2: fixes P :: "pi" shows "P \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P"
wlsAbs_Abs: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs' :: "'varSort"   and s :: "'sort"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "wlsAbs (xs', s) (Abs xs x X) = (isInBar (xs', s) \<and> xs = xs' \<and> wls s X)"
b_acc_subset_\<delta>_states: fixes \<delta> :: "('a, 'b) ta_rule set" shows "b_accessible \<delta> \<subseteq> \<delta>_states \<delta>"
simps_3: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int option" shows "paval (Plus a\<^sub>1 a\<^sub>2) s = paval a\<^sub>1 s + paval a\<^sub>2 s"
span_pr: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "span \<pp>\<^sub>1[a, b] \<pp>\<^sub>0[a, b]"
take_Suc_prefix: fixes n :: "nat"   and p :: "'b list" shows "prefix (take n p) (take (Suc n) p)"
distinct_56: fixes x13 :: "Syntax.term"   and x2 :: "Syntax.var" shows "Roll x13 \<noteq> Var x2"
cong_mult_self_left: 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 division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (times a b) zero a"
is_oc_projection_def: fixes n :: "nat"   and w :: "'a vec"   and S :: "'a vec set"   and v :: "'a vec" shows "gram_schmidt.is_oc_projection n w S v = (w \<in> carrier_vec n \<and> v - w \<in> LinearCombinations.module.span class_ring (module_vec TYPE('a) n) S \<and> (\<forall>u. u \<in> S \<longrightarrow> w \<bullet> u = (0::'a)))"
same_messages_if_no_occurrence: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and p :: "'a"   and q :: "'a"   and cid :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev \<noteq> p"   and "occurs_on ev \<noteq> q"   and "channel cid = Some (p, q)" shows "msgs c cid = msgs c' cid \<and> cs c cid = cs c' cid"
it_paths_Union: fixes t :: "'a rose_tree" shows "it_paths t \<subseteq> insert [] (\<Union>(i, t)\<in>set (enumerate 0 (children t)). (#) i ` it_paths t)"
gamma_deriv_at_within: fixes a :: "real"   and b :: "real"   and x :: "real"   and \<gamma> :: "real \<Rightarrow> 'a" assumes "a < b"   and "x \<in> {a..b}"   and "\<forall>x\<in>{a..b}. \<gamma> differentiable at x" shows "vector_derivative \<gamma> (at x within {a..b}) = vector_derivative \<gamma> (at x)"
mono: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<le> c"   and "b \<le> d" shows "min a b \<le> min c d"
closed_singleton: fixes "open" :: "'a set \<Rightarrow> bool"   and a :: "'a" assumes "class.t1_space open" shows "topological_space.closed open {a}"
antisymPD: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "antisymp r"   and "r a b"   and "r b a" shows "a = b"
homeomorphism_image2: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "homeomorphism S T f g" shows "g ` T = S"
bdd_below_UN: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'b set" assumes "class.lattice inf less_eq less sup"   and "finite I" shows "preorder.bdd_below less_eq (\<Union> (A ` I)) = (\<forall>i\<in>I. preorder.bdd_below less_eq (A i))"
red_dfs_def: fixes E :: "('a \<times> 'a) set"   and onstack :: "'a set"   and V :: "'a set"   and u :: "'a" shows "red_dfs E onstack V u \<equiv> REC\<^sub>T (\<lambda>D (V, u). let V = insert u V in FOREACH\<^sub>C (E `` {u}) (\<lambda>brk. brk = None) (\<lambda>t _. if t \<in> onstack then RETURN (red_init_witness u t) else RETURN None) None \<bind> (\<lambda>brk. case brk of None \<Rightarrow> FOREACH\<^sub>C (E `` {u}) (\<lambda>(V, brk). brk = None) (\<lambda>t (V, uu_). if t \<notin> V then D (V, t) \<bind> (\<lambda>(V, brk). RETURN (V, prep_wit_red u brk)) else RETURN (V, None)) (V, None) | Some a \<Rightarrow> RETURN (V, brk))) (V, u)"
simps_5: fixes x31 :: "comb"   and x32 :: "comb" shows "x31 \<bullet> x32 \<noteq> K"
assms: shows "hn_ctxt A x y \<or>\<^sub>A hn_ctxt A' x y \<Longrightarrow>\<^sub>t hn_ctxt Am x y"
subst_lsst_unlabel: fixes A :: "('f strand_label \<times> ('d, 'e) stateful_strand_step) list"   and \<delta> :: "'e \<Rightarrow> ('d, 'e) Term.term" shows "unlabel (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>) = unlabel A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>"
NSLIMSEQ_mult: fixes X :: "nat \<Rightarrow> 'a"   and a :: "'a"   and Y :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a"   and "Y \<longlonglongrightarrow>\<^sub>N\<^sub>S b" shows "(\<lambda>n. X n * Y n) \<longlonglongrightarrow>\<^sub>N\<^sub>S a * b"
le_left_mono: fixes x :: "'d"   and y :: "'d"   and a :: "'d" assumes "x \<le> y" shows "y \<le> a \<longrightarrow> x \<le> a"
G: fixes x :: "'d" assumes "x \<in> s" shows "norm (g x) \<le> G"
prv_psubst_scnj_imp: fixes F :: "'fmla set"   and txs :: "('trm \<times> 'var) list" assumes "F \<subseteq> fmla"   and "finite F"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)" shows "prv (imp (scnj ((\<lambda>\<phi>. psubst \<phi> txs) ` F)) (psubst (scnj F) txs))"
section_if_inj_1: fixes f :: "'s" assumes "arr f"   and "inj_on (Fun f) (Dom f)"   and "Dom f = {} \<longrightarrow> Cod f = {}" shows "section f"
res_ri3: fixes y :: "'a"   and x :: "'a" shows "y \<cdot> (y \<rightarrow> y \<cdot> (y \<rightarrow> x)) = y \<cdot> (y \<rightarrow> x)"
apex_def: shows "\<pi>.dom.apex \<equiv> C.dom \<pi>.dom.leg0"
cring_simprules_3: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "cring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<in> carrier R"
simps_127: 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 x6 :: "16 word" shows "(case VlanPriority x6 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) = f6 x6"
hyps_1: shows "\<Gamma> \<turnstile> a1 : t"
set_less_aux_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" shows "ord.set_less_aux less_eq A B = (finite A \<and> finite B \<and> (\<exists>y\<in>B - A. \<forall>z\<in>A - B \<union> (B - A). less_eq y z \<and> (less_eq z y \<longrightarrow> y = z)))"
invC_prop: shows "G\<turnstile>invmode statM e\<rightarrow>invC\<preceq>statT"
preserved_def: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a" shows "preserved f h h' = (\<forall>x. h \<turnstile> f \<rightarrow>\<^sub>r x = h' \<turnstile> f \<rightarrow>\<^sub>r x)"
WLI_2: fixes r :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and p :: "'g process_name" assumes "r \<in> ghost_honorary_grey (s p)" shows "r \<in> WL p s"
summable_divide: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "summable f" shows "summable (\<lambda>n. f n / c)"
m_closed: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
b_rel_simp: fixes x :: "'a"   and y :: "'b"   and R :: "('a \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" shows "((x, y) \<in> b_rel R P) = ((x, y) \<in> R \<and> P y)"
aff: fixes X :: "'a set" assumes "X \<in> \<F>" shows "aff_dim X < aff_dim T"
cong_add_lcancel: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a"   and x :: "'a"   and y :: "'a"   and n :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo (plus a x) (plus a y) n = unique_euclidean_semiring.cong modulo x y n"
simps_2: fixes x :: "brick"   and y :: "brick list" shows "is_cup (x # y) = (if x = cup then is_cup y else False)"
assms_2: shows "\<forall>n'''>n'. n''' \<le> n'' \<longrightarrow> length sbc \<le> length (bc n''')"
asm: shows "\<forall>x\<in>vars\<^sub>l\<^sub>s L. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x"
rec: shows "scalar_prod (map (\<lambda>k. scalar_prod r (m ! k)) [0..<ncc]) cc = scalar_prod r (map (\<lambda>k. scalar_prod (row m k) cc) [0..<nr])"
Nf_def: shows "Nf = (\<lambda>N. if N = 0 then 0 else if N = 1 then N1 + 1 else max (N1 + 1) (Max {Ne (eps n) |n. n \<le> N}) + N)"
rel_converter_Grp: fixes f :: "'m \<Rightarrow> 'i"   and B :: "'j set"   and g :: "'j \<Rightarrow> 'n"   and C :: "'k set"   and h :: "'k \<Rightarrow> 'o"   and k :: "'p \<Rightarrow> 'l" shows "rel_converter (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp B g) (BNF_Def.Grp C h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse> = BNF_Def.Grp {conv. results_converter (\<I>_uniform (range f) UNIV) (\<I>_uniform UNIV (range k)) conv \<subseteq> B \<and> outs_converter (\<I>_uniform (range f) UNIV) (\<I>_uniform UNIV (range k)) conv \<subseteq> C} (map_converter f g h k)"
sup_right_zero: fixes x :: "'b" shows "top_homomorphism ((\<squnion>) (- x))"
aNDSubsetaux: fixes a :: "('b, 'c) Combinators list"   and b :: "('b, 'c) Combinators list" assumes "singleCombinators a"   and "set a \<subseteq> set b" shows "set (net_list_aux a) \<subseteq> set (net_list_aux b)"
fs_init: shows "set fs_init \<subseteq> carrier_vec n"
sel'I: fixes m :: "'s"   and P :: "'u \<times> 'v \<Rightarrow> bool" assumes "invar m"   and "\<forall>u v. \<alpha> m u = Some v \<longrightarrow> \<not> P (u, v)" shows "sel' m P = None"
R: shows "topspace (product_topology X I) = {} \<longrightarrow> topspace (X i) = {}"
intros_2: fixes v :: "'a" assumes "v \<in> V" shows "walk [v]"
OrdP_IN_SUCC_lemma: fixes x :: "tm"   and y :: "tm" shows "{OrdP x, y IN x} \<turnstile> SUCC y IN SUCC x"
path_connected_sphere: fixes a :: "'a"   and r :: "real" assumes "2 \<le> DIM('a)" shows "path_connected (sphere a r)"
simps_2: fixes t :: "'trm"   and t1 :: "'trm"   and x1 :: "'var"   and txs :: "('trm \<times> 'var) list" shows "rawpsubstT t ((t1, x1) # txs) = rawpsubstT (substT t t1 x1) txs"
fast_select_correct: fixes k :: "nat"   and xs :: "'b list" assumes "k < length xs" shows "fast_select k xs = select k xs"
T: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], x # h) v = length v - 2"
refine: fixes n :: "nat"   and ai :: "real list \<times> real list"   and a :: "'b set"   and bi :: "real list \<times> real list"   and b :: "'b set" assumes "DIM_precond TYPE('b) n"   and "(ai, a) \<in> lvivl_rel"   and "(bi, b) \<in> lvivl_rel" shows "(nres_of (disjoint_sets_impl n ai bi), disjoint_sets $ a $ b) \<in> \<langle>bool_rel\<rangle>nres_rel"
s0g_domI: fixes x :: "'a s0g_state" assumes "Domain (az x \<union> lkr x) \<subseteq> Domain (kn x)" shows "x \<in> s0g_dom"
minus_zero: shows "\<ominus>\<^bsub>R Quot I\<^esub> \<zero>\<^bsub>R Quot I\<^esub> = \<zero>\<^bsub>R Quot I\<^esub>"
oth_class_taut_5_c: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<chi> \<^bold>\<rightarrow> (\<phi> \<^bold>\<rightarrow> \<chi>)) in v]"
iff_sem: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and I :: "('b, 'c, 'a) interp"   and A :: "('b, 'c, 'a) formula"   and B :: "('b, 'c, 'a) formula" shows "(\<nu> \<in> fml_sem I (A \<leftrightarrow> B)) = ((\<nu> \<in> fml_sem I A) = (\<nu> \<in> fml_sem I B))"
fps_lr_inverse_gp_ring1'_1: shows "fps_left_inverse (Abs_fps (\<lambda>n. 1::'a)) (1::'a) = 1 - fps_X"
basis_def: fixes X :: "'a set" shows "basis X = (indep X \<and> (\<forall>x\<in>carrier - X. \<not> indep (insert x X)))"
basis_def: fixes A :: "'c set" shows "V.basis A = (V.module.lin_indpt A \<and> V.module.gen_set A \<and> A \<subseteq> carrier V)"
v2: shows "g k \<cdot>\<^sub>v gs ! k \<in> carrier_vec n"
try_spmf_def: fixes p :: "'h spmf"   and q :: "'h spmf" shows "TRY p ELSE q = p \<bind> (\<lambda>x. case x of None \<Rightarrow> q | Some y \<Rightarrow> return_spmf y)"
tkbo_coefs_axioms: shows "tkbo_coefs ground_heads_var (>\<^sub>s) \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"
assms: shows "\<forall>x\<in>set xs. is_measured_node x"
hext_refl: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "h \<unlhd> h"
Rep_cltn2: fixes x :: "cltn2" shows "Rep_cltn2 x \<in> Collect invertible // invertible_proportionality"
simps_2: fixes uu :: "'a" shows "- extB.Val uu = extB.Bot"
subscene_refl: fixes X :: "'a scene" shows "X \<subseteq>\<^sub>S X"
prod_cases3: fixes y :: "'a \<times> 'b \<times> 'd" obtains a :: "'a" and b :: "'b" and c :: "'d" where "y = (a, b, c)"
wf_iff_acyclic_if_finite: fixes r :: "('a \<times> 'a) set" assumes "finite r" shows "wf r = acyclic r"
bex_Infinitesimal_iff: fixes x :: "'c star"   and z :: "'c star" shows "(\<exists>y\<in>Infinitesimal. x - z = y) = (x \<approx> z)"
lappend_lnull1: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lnull xs" shows "lappend xs ys = ys"
collapse: fixes exetheory :: "exetheory" shows "ExeTheory (exesig exetheory) (exeaxioms_of exetheory) = exetheory"
dom_eqI: fixes a :: "J.Arr"   and f :: "J.Arr" assumes "J.ide a"   and "J.seq f a" shows "J.dom f = a"
Cond_sound: fixes pre :: "'b set"   and rely :: "('b \<times> 'b) set"   and P1 :: "'b com"   and b :: "'b set"   and guar :: "('b \<times> 'b) set"   and post :: "'b set"   and P2 :: "'b com" assumes "stable pre rely"   and "\<Turnstile> P1 sat [pre \<inter> b, rely, guar, post]"   and "\<Turnstile> P2 sat [pre \<inter> - b, rely, guar, post]"   and "\<forall>s. (s, s) \<in> guar" shows "\<Turnstile> Cond b P1 P2 sat [pre, rely, guar, post]"
returnS_def: shows "returnS = (\<Lambda> x. lcons\<cdot>x\<cdot>lnil)"
left_closed: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E(\<eta>)" shows "a \<in> M"
scale_left_diff_distrib: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'d"   and b :: "'d"   and x :: "'e" assumes "module scale" shows "scale (a - b) x = scale a x - scale b x"
conj_one_left: fixes x :: "'a" shows "\<zero> \<^bold>\<squnion> x = x"
orderE: fixes b :: "'c \<times> 'c \<Rightarrow> 'd"   and a :: "'c \<times> 'c \<Rightarrow> 'd" assumes "b \<preceq> a" obtains "a = a \<oplus> b"
mr_cos_sprod_assoc: fixes H :: "'a set"   and a :: "'b"   and b :: "'b"   and X :: "'a set" assumes "submodule R M H"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "X \<in> set_mr_cos M H" shows "mr_cos_sprod M H (a \<cdot>\<^sub>r\<^bsub>R\<^esub> b) X = mr_cos_sprod M H a (mr_cos_sprod M H b X)"
ct_suffixI: fixes ys :: "ct list"   and as :: "ct list"   and zs :: "ct list"   and xs :: "ct list" assumes "ys = as @ zs"   and "ct_list_eq zs xs" shows "ct_suffix xs ys"
rtranclD: fixes a :: "'b"   and b :: "'b"   and R :: "('b \<times> 'b) set" assumes "(a, b) \<in> R\<^sup>*" shows "a = b \<or> a \<noteq> b \<and> (a, b) \<in> R\<^sup>+"
zero_neq_one: shows "(0::'a) \<noteq> (1::'a)"
rtrancl_trans: fixes x :: "'x"   and y :: "'x"   and r :: "('x \<times> 'x) set"   and z :: "'x" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
distinct_length_2_or_more: fixes a :: "'b"   and b :: "'b"   and xs :: "'b list" shows "distinct (a # b # xs) = (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
append_take_drop_id: fixes n :: "nat"   and xs :: "'b list" shows "take n xs @ drop n xs = xs"
float_distinct_4: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_nan a \<and> is_zero a)"
hd_in_butlast: fixes xs :: "'b list" assumes "1 < length xs" shows "hd xs \<in> set (butlast xs)"
C_index: fixes vs :: "'d list"   and xs :: "'d list"   and M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<in> C M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs" shows "Suc (length xs) = i"
iMOD_subset_imp_iMODb_subset: fixes r' :: "nat"   and m' :: "nat"   and r :: "nat"   and m :: "nat"   and c' :: "nat"   and c :: "nat" assumes "[ r', mod m' ] \<subseteq> [ r, mod m ]"   and "r' + m' * c' \<le> r + m * c" shows "[ r', mod m', c' ] \<subseteq> [ r, mod m, c ]"
sint_neg_hom: fixes w :: "32 Word.word" assumes "uint w \<noteq> 2 ^ LENGTH(31)" shows "sint (- w) = - sint w"
aext_lit: fixes v :: "'b"   and a :: "'d \<Longrightarrow> 'c" shows "\<guillemotleft>v\<guillemotright> \<oplus>\<^sub>p a = \<guillemotleft>v\<guillemotright>"
non_atomic_Enc: fixes x :: "msg"   and y :: "msg" shows "composed (Enc x y)"
faces: fixes X :: "'b set set"   and x :: "'b set"   and y :: "'b set" assumes "ChamberComplex X"   and "x \<in> X"   and "order.greater_eq x y" shows "y \<in> X"
red_setE: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F p q" obtains f :: "'t \<Rightarrow>\<^sub>0 'c" and t :: "'a" where "f \<in> F" and "red_single p q f t"
assms: shows "x \<in> \<guillemotleft>\<nat>\<guillemotright>"
cfields_cbody: fixes c :: "class" shows "cfields (cbody c) = cfields c"
OUTfromChCorrect_def: fixes x :: "chanID" shows "OUTfromChCorrect x \<equiv> OUTfromCh x \<noteq> {} \<longrightarrow> (\<exists>Z. x \<in> OUT Z \<and> (\<forall>y\<in>OUTfromCh x. y \<in> IN Z))"
orthogonal_spanningset_subspace: fixes S :: "'a set" assumes "subspace S" obtains B :: "'a set" where "B \<subseteq> S" and "pairwise orthogonal B" and "span B = S"
Field_card_of: fixes A :: "'c set" shows "Field |A| = A"
constVal_Boolean: fixes e :: "expr"   and c :: "val"   and Env :: "env" assumes "constVal e = Some c"   and "Env\<turnstile>e\<Colon>-PrimT Boolean" shows "typeof empty_dt c = Some (PrimT Boolean)"
is_rewritable_def: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and u :: "'t" shows "is_rewritable bs p u = (\<exists>b\<in>set bs. b \<noteq> 0 \<and> lt b adds\<^sub>t u \<and> rword_strict (spp_of p) (spp_of b))"
assms_3: shows "p\<^sub>0 \<noteq> p\<^sub>1"
fps_unit_factor_mult_unit_factor_2: fixes f :: "'a fps"   and g :: "'a fps" shows "unit_factor (unit_factor f * g) = unit_factor (f * g)"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>x r. P \<langle>\<langle>\<rangle>, x, r\<rangle>"   and "\<And>v va vb x r. P \<langle>v, va, vb\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb\<rangle>, x, r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
simps_2: fixes literal :: "Literal"   and clause :: "Literal list"   and formula :: "Literal list list" shows "literal el clause # formula = (literal el clause \<or> literal el formula)"
union_fold_mset_add_mset: fixes A :: "'a multiset"   and B :: "'a multiset" shows "A + B = fold_mset add_mset A B"
filter_nb_cont: fixes cond :: "'c \<Rightarrow> bool"   and M :: "'c nres set" shows "filter_nb cond (Sup M) = Sup {filter_nb cond m |m. m \<in> M}"
finite_psupset_wf: fixes S :: "'a set" assumes "finite S" shows "wf (finite_psupset S)"
\<rho>': shows "\<rho> \<circ> Suc \<in> space (state_measure V' \<Gamma>)"
alc0: shows "\<bar>coeff p n\<bar> \<noteq> 0"
wf_subclass_loc_def: fixes cs :: "(String.literal \<times> String.literal) set" shows "wf_subclass_loc cs \<equiv> wf_subclass cs"
assms: fixes x :: "'a"   and y :: "'b" assumes "fmlookup m x = Some y" shows "P x y"
irreducible_imp_irreducible_mult: fixes a :: "'a" assumes "a \<in> carrier R"   and "irreducible R a" shows "irreducible (Ring_Divisibility.mult_of R) a"
combine_eq_linorder: fixes D1 :: "'a set"   and cmp1 :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and D2 :: "'a set"   and cmp2 :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and D :: "'a set" assumes "eq_linorder_on D1 cmp1"   and "eq_linorder_on D2 cmp2"   and "D = D1 \<union> D2" shows "eq_linorder_on D (cmp_combine D1 cmp1 D2 cmp2)"
path_via_reverse: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and D :: "char list"   and Cs :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> Path C to D via Cs "   and "wf_prog wf_md P" shows "\<forall>Cs'. P \<turnstile> Path D to C via Cs'  \<longrightarrow> Cs = [C] \<and> Cs' = [C] \<and> C = D"
ip_cidr_set_change_base: fixes j :: "'a word"   and i :: "'a word"   and r :: "nat" assumes "j \<in> ip_cidr_set i r" shows "ip_cidr_set j r = ip_cidr_set i r"
not_in_env_smaller: fixes x :: "char list"   and e :: "'b environment"   and a :: "char list"   and X :: "'b" assumes "x \<notin> env_dom (e\<lparr>a:X\<rparr>)"   and "x \<noteq> a"   and "ok (e\<lparr>a:X\<rparr>)" shows "x \<notin> env_dom e"