hcju/isabelleps huggingface dataset:
induct: fixes x :: "msg"   and evs :: "event list"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> responses evs"   and "P END"   and "\<And>RA KAB B A NB. \<lbrakk>RA \<in> responses evs; P RA; Key KAB \<notin> used evs\<rbrakk> \<Longrightarrow> P \<lbrace>Crypt (shrK B) \<lbrace>Key KAB, Agent A, Nonce NB\<rbrace>, RA\<rbrace>" shows "P x"
singleton: fixes x :: "'a" shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>n {x} = x"
span_add1: fixes U :: "rat vec set"   and v :: "rat vec"   and w :: "rat vec" assumes "U \<subseteq> Rn"   and "v \<in> gs.span U"   and "w \<in> gs.span U" shows "v + w \<in> gs.span U"
subspace_span: fixes S :: "'a set" shows "csubspace (cspan S)"
is_REL_def: fixes thread :: "thread" shows "is_REL thread = (\<exists>x4. thread = REL x4)"
sr_u_pos: fixes u :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "u \<in> X"   and "r u = sr" shows "lt_vec 0 u"
isometry_quasi_isometry_on: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "1 0 -quasi_isometry_on X f"
lex_ord_pair_EqD: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option"   and k :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "comparator.oalist_inv_raw comp ys"   and "comparator.lex_ord_pair comp f xs ys = Some Eq"   and "k \<in> fst ` set xs \<union> fst ` set ys" shows "f k (comparator.lookup_pair comp xs k) (comparator.lookup_pair comp ys k) = Some Eq"
BT_lemma_2_22: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2"   and "3 \<le> i"   and "i \<le> k" shows "Polynomial.smult (subresultant_prs_locale.\<alpha> \<delta> f i ^ \<delta> (i - 1)) (subresultant (n i) (F (i - 2)) (F (i - 1))) = Polynomial.smult ((- 1) ^ ((\<delta> (i - 2) + \<delta> (i - 1)) * \<delta> (i - 1)) * f (i - 1) ^ (\<delta> (i - 2) + \<delta> (i - 1)) * f i ^ (\<delta> (i - 1) - 1) * \<beta> i ^ \<delta> (i - 1)) (F i)"
add_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "x + y \<in> A"
simps_2: fixes A :: "agent" shows "invKey (priK A) = pubK A"
Unfold1: fixes \<phi>' :: "'a ltl" shows "\<G> \<Turnstile>\<^sub>P \<phi>' = (\<G> \<inter> {\<psi>. G_depth \<psi> \<le> G_depth \<phi>'}) \<Turnstile>\<^sub>P \<phi>'"
incseq_sumI2: fixes A :: "'b set"   and f :: "'b \<Rightarrow> nat \<Rightarrow> 'c" assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)" shows "incseq (\<lambda>i. \<Sum>n\<in>A. f n i)"
lens_indep_left_ext: fixes y :: "'b \<Longrightarrow> 'c"   and z :: "'d \<Longrightarrow> 'c"   and x :: "'e \<Longrightarrow> 'b" assumes "y \<bowtie> z" shows "x ;\<^sub>L y \<bowtie> z"
ts: shows "\<lambda>s. 1 \<tturnstile> wp s (\<lambda>s. 1)"
abc_abc_neq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c"
e2frame_def: fixes e :: "nat" shows "e2frame e \<equiv> e_hd (e2stack e)"
map_cong0: fixes x :: "('h, 'g, 'i, 'j) F"   and f1 :: "'g \<Rightarrow> 'k"   and g1 :: "'g \<Rightarrow> 'k"   and f2 :: "'i \<Rightarrow> 'l"   and g2 :: "'i \<Rightarrow> 'l"   and f3 :: "'j \<Rightarrow> 'm"   and g3 :: "'j \<Rightarrow> 'm" assumes "\<And>z1. z1 \<in> Fset1 x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> F1set1 x \<Longrightarrow> f2 z2 = g2 z2"   and "\<And>z3. z3 \<in> F2set x \<Longrightarrow> f3 z3 = g3 z3" shows "Fmap f1 f2 f3 x = Fmap g1 g2 g3 x"
almost_full_on_Int: fixes P1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A1 :: "'b set"   and P2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A2 :: "'b set" assumes "almost_full_on P1 A1"   and "almost_full_on P2 A2" shows "almost_full_on (\<lambda>x y. P1 x y \<and> P2 x y) (A1 \<inter> A2)"
cases: fixes x :: "('a, 'b) S_action" obtains \<alpha> :: "'a" where "x = S_action.Act \<alpha>"   | uu_ :: "'b" where "x = S_action.Pred uu_"
\<o>_\<iota>: fixes x :: "hf" assumes "x \<in> UNIV" shows "\<o> (UP x) = x"
no_tick_sub: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and a :: "clock" assumes "dilating f sub r" shows "(\<nexists>n\<^sub>0. f n\<^sub>0 = n) \<longrightarrow> \<not> hamlet (Rep_run r n a)"
Le_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Le C D \<or> B A Le C D \<or> A B Le D C \<or> B A Le D C" shows "A B Le C D"
sum_mat_mono_gt: fixes sd :: "nat"   and n :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "sd \<le> n"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "mat_gt (\<succ>) sd A B" shows "sum_mat A \<succ> sum_mat B"
basis_row_space: fixes A :: "(('a, 'b) vec, 'c) vec" shows "vec.independent (basis_row_space A) \<and> vec.span (basis_row_space A) = row_space A"
state_io_relI: fixes m :: "('c, 'b) state"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a s s'. run_state m s = (a, s') \<Longrightarrow> P s s'" shows "state_io_rel P m"
homeomorphic_compactness: fixes s :: "'c set"   and t :: "'d set" assumes "s homeomorphic t" shows "compact s = compact t"
lower_times: fixes A :: "'a interval"   and B :: "'a interval" shows "lower (A * B) = Min {lower A * lower B, lower A * upper B, upper A * lower B, upper A * upper B}"
VPG_MST_G: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "VP_G G"   and "MST_valid" shows "G_valid G"
summable_pre_sin: fixes x :: "real" shows "summable (\<lambda>n. mssnth sin_series_stream n * x ^ n)"
P\<^sub>1_def: shows "P\<^sub>1 \<equiv> ts_process {[], [Some a], [Some a, None]}"
int_pow_consistent: fixes H :: "'a set"   and x :: "'a"   and n :: "int" assumes "subgroup H G"   and "x \<in> H" shows "x [^] n = x [^]\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> n"
primefact_contains: fixes ps :: "nat list"   and n :: "nat"   and p :: "nat" assumes "primefact ps n"   and "prime p"   and "p dvd n" shows "p \<in> set ps"
transformation_by_components_axioms: shows "transformation_by_components (\<cdot>) (\<cdot>) L local.map lunit"
proper_interval_Compl_set_aux_fusion_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and g1 :: "('b, 'c) generator"   and g2 :: "('b, 'd) generator"   and ao :: "'b option"   and s1 :: "'c"   and s2 :: "'d" shows "proper_intrvl.proper_interval_Compl_set_aux_fusion less proper_interval g1 g2 ao s1 s2 = proper_intrvl.proper_interval_Compl_set_aux less proper_interval ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
inner_prod_eq: fixes i :: "'b" shows "(\<lambda>v. v \<bullet> axis i 1) = (\<lambda>v. v $ i)"
kernel_from_trivial_group: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h"   and "trivial_group G" shows "kernel G H h = carrier G"
All_None_def: fixes Ts :: "('b ann_com option \<times> 'b set) list" shows "All_None Ts \<equiv> \<forall>(c, q)\<in>set Ts. c = None"
fps_integral_def: fixes a :: "'a fps"   and a0 :: "'a" shows "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else inverse (of_nat n) * a $ (n - 1))"
span_mono: fixes A :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a"   and B :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set" assumes "A \<subseteq> cotangent_space p"   and "B \<subseteq> cotangent_space p"   and "A \<subseteq> B" shows "span A \<subseteq> span B"
bound: shows "interaction_bounded_by' (Not \<circ> isl) \<A> q"
merge_apply_3: fixes J :: "'d set"   and I :: "'d set"   and i :: "'d"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "J \<inter> I = {}"   and "i \<in> I" shows "merge I J (x, y) i = x i"
cone_axioms: shows "Da.cone (A_B.Map x a) (at a \<chi>)"
isomorphic_reduced_homology_by_contractible: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "contractible_space (subtopology X S)"   and "topspace X \<inter> S \<noteq> {}" shows "reduced_homology_group p X \<cong> relative_homology_group p X S"
invar_delete: fixes s :: "'b tree"   and x :: "'b" assumes "splay.invar s" shows "splay.invar (delete x s)"
diff_ge_0_iff_ge: fixes plus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and zero :: "'i"   and minus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and uminus :: "'i \<Rightarrow> 'i"   and less_eq :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and less :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'i"   and b :: "'i" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (minus a b) = less_eq b a"
str_not_num: fixes s :: "char list"   and x1 :: "int" shows "EFSM.Str s \<noteq> Num x1"
elims_1: fixes x :: "'edge list"   and xa :: "'edge list"   and y :: "bool" assumes "same_level_path_aux x xa = y" obtains cs :: "'edge list" where "x = csb__" and "xa = []" and "y"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> same_level_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> False | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> same_level_path_aux cs' as | _ \<Rightarrow> same_level_path_aux cs as)"
subset_iff_less_eq_V: fixes B :: "V set"   and A :: "V set" assumes "small B" shows "(A \<subseteq> B) = (set A \<subseteq>\<^sub>\<circ> set B \<and> small A)"
replace_primitive_matchexpr_unchanged_if_not_has_disc: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and replace_f :: "'c negation_type \<Rightarrow> 'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc disc m" shows "replace_primitive_matchexpr (disc, sel) replace_f m = m"
a_3: shows "x \<sqsubseteq>\<^bsub>\<X>\<^esub> y"
af_homomorphism: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "Applicative.ap (pure f) (pure x) = pure (f x)"
prob_eq_0_AE: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s" assumes "AE x in T s. \<not> P x" shows "T.prob s {x \<in> space (T s). P x} = 0"
Domainp_pred_fun_eq: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "left_unique T" shows "Domainp (T ===> S) = pred_fun (Domainp T) (Domainp S)"
Sigma_Un_distrib1: fixes I :: "'c set"   and J :: "'c set"   and C :: "'c \<Rightarrow> 'd set" shows "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and altc :: "'c cset \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_altc return bind altc" shows "monad_altc_axioms bind altc"
select_convs_1: fixes edge :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more :: "'b" shows "edge \<lparr>edge = edge, \<dots> = more\<rparr> = edge"
image_inv_f_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj f" shows "Hilbert_Choice.inv f ` f ` A = A"
conjugate_id: fixes a :: "'b" shows "conjugate (conjugate a) = a"
ordered_positive_resolvent_def: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set" shows "ordered_positive_resolvent P1 P2 C = (\<exists>L. C = P1 - {L} \<union> (P2 - {complement L}) \<and> (negative_part P1 = {} \<and> strictly_maximal_literal P1 L) \<and> strictly_maximal_literal (negative_part P2) (complement L))"
getRel_dom_1: fixes G :: "('b, 'c) labeled_graph"   and a :: "'c"   and b :: "'c"   and l :: "'b" assumes "graph G"   and "(a, b) \<in> getRel l G" shows "a \<in> vertices G"
prv_imp_implies_isTrue: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<chi> = {}"   and "bprv (imp \<phi> \<chi>)"   and "isTrue \<phi>" shows "isTrue \<chi>"
IH_2: assumes "sorted (map fst kvs')"   and "distinct (map fst kvs')" shows "rbt_sorted (fst (rbtreeify_g n kvs'))"
H_cond: fixes p :: "'a"   and r :: "'a"   and x :: "'a"   and q :: "'a"   and y :: "'a" assumes "H (tp \<cdot> tr) x q"   and "H (tp \<cdot> at r) y q" shows "H p (if r then x else y fi) q"
intros_2: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "ListMem x xs" shows "ListMem x (y # xs)"
list_rel_comp_list_wset_rel: fixes R :: "('a \<times> 'b) set"   and S :: "('b \<times> 'c) set" assumes "single_valued R" shows "\<langle>R\<rangle>list_rel O \<langle>S\<rangle>list_wset_rel = \<langle>R O S\<rangle>list_wset_rel"
simps_33: fixes x101 :: "Term.var"   and x102 :: "expr"   and x1 :: "qtname" shows "x101:=x102 \<noteq> NewC x1"
alphaDistinct: fixes \<alpha> :: "'a action"   and P :: "('a, 'b, 'c) psi"   and \<beta> :: "'a action"   and Q :: "('a, 'b, 'c) psi" assumes "\<alpha> \<prec> P = \<beta> \<prec> Q"   and "distinct (bn \<alpha>)"   and "\<And>x. x mem bn \<alpha> \<Longrightarrow> x \<in> supp (object \<alpha>)"   and "bn \<alpha> \<sharp>* bn \<beta>"   and "bn \<alpha> \<sharp>* object \<beta>"   and "bn \<alpha> \<sharp>* Q" shows "distinct (bn \<beta>)"
eq_id_iff: fixes f :: "'a \<Rightarrow> 'a" shows "(\<forall>x. f x = x) = (f = Fun.id)"
set_iterator_map_rev_linord_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<times> 'e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "('c \<times> 'e) set" assumes "class.linorder less_eq less" shows "linorder.set_iterator_map_rev_linord less_eq it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). less_eq k' k)"
gf_app: assumes "1\<^sub>\<nat> \<in>\<^sub>\<circ> 2\<^sub>\<nat>" shows "gf\<lparr>1\<^sub>\<nat>\<rparr> : ab\<lparr>1\<^sub>\<nat>\<rparr> \<mapsto>\<^bsub>if 1\<^sub>\<nat> = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>\<^esub> cd\<lparr>1\<^sub>\<nat>\<rparr>"
rbt_lookup_in_tree: fixes t :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c" assumes "rbt_sorted t" shows "(rbt_lookup t k = Some v) = ((k, v) \<in> set (entries t))"
invertibleI: fixes M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and u :: "'b"   and v :: "'b" assumes "Group_Theory.monoid M composition unit"   and "composition u v = unit"   and "composition v u = unit"   and "u \<in> M"   and "v \<in> M" shows "monoid.invertible M composition unit u"
F1: shows "1 \<le> \<parallel>F\<parallel>\<^sup>2"
ab_semigroup_add_ow_Ball_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "ab_semigroup_add_ow S pls = (semigroup_add_ow S pls \<and> (\<forall>a\<in>S. \<forall>b\<in>S. pls a b = pls b a))"
nth_prime_lower_bound: fixes n :: "nat" assumes "0 < n" shows "139 / 443 * (real n * ln (real n)) \<le> real (nth_prime (n - 1))"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "\<And>i. P (C i)"   and "\<And>n c t. P t \<Longrightarrow> P (CN n c t)"   and "\<And>c t s. P s \<Longrightarrow> P (CF c t s)"   and "\<And>v. P (Bound v)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Floor v)" shows "P a0"
mono_mult1: fixes r :: "('k \<times> 'k) set"   and r' :: "('k \<times> 'k) set" assumes "r \<subseteq> r'" shows "mult1 r \<subseteq> mult1 r'"
baseE: fixes v :: "state \<Rightarrow> 'b"   and c :: "'b" assumes "basevars v" obtains x :: "state" where "v x = c"
ideD_1: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.ide a" shows "S.arr a"
loc_single: fixes a :: "'a el_step" shows "loc [a] = (case a of LOC e \<Rightarrow> [a] | ENV e \<Rightarrow> [])"
rev_subspaceD: fixes x :: "'b"   and U :: "'b set"   and V :: "'b set" assumes "x \<in> U"   and "Subspace.subspace U V" shows "x \<in> V"
not_eventually: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" shows "(\<not> eventually P F) = (\<exists>\<^sub>F x in F. \<not> P x)"
inc: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and n :: "nat" assumes "matrix_seq dim f" shows "f n \<le>\<^sub>L f (Suc n)"
nn_integral_cong_AE: fixes u :: "'a \<Rightarrow> ennreal"   and v :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "AE x in M. u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
Gramian_matrix_def: fixes n :: "nat"   and G :: "'a vec list"   and k :: "nat" shows "gram_schmidt.Gramian_matrix n G k = (let M = mat k n (\<lambda>(i, j). G ! i $ j) in M * M\<^sup>T)"
1: shows "M.sumlist (map (\<lambda>i. g i \<cdot>\<^sub>v gs ! i) [0..<k]) \<bullet> (h k \<cdot>\<^sub>v gs ! k) = (0::'a)"
smult_mat_def: fixes a :: "'a"   and A :: "'a mat" shows "a \<cdot>\<^sub>m A \<equiv> map_mat ((*) a) A"
SeqStTermP_imp_LstSeqP: fixes v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqStTermP v i t u s k} \<turnstile> LstSeqP s k (HPair t u)"
open_preimage_identity: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and B :: "'c set" assumes "topological_space S is_open"   and "is_open B" shows "identity S \<^sup>\<inverse> S B = B"
WSE_11: shows "Ball (set (tl ms\<^sub>1)) return_node"
f1: shows "low_equal (snd (fst (get_curr_win () s1))) (snd (fst (get_curr_win () s2)))"
ord_strict_p_recI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p = lt q"   and "lc p = lc q"   and "tail p \<prec>\<^sub>p tail q" shows "p \<prec>\<^sub>p q"
fvi_plus: fixes x :: "nat"   and b :: "nat"   and c :: "nat"   and \<phi> :: "formula" shows "(x \<in> fvi (b + c) \<phi>) = (x + c \<in> fvi b \<phi>)"
insert_Diff: fixes a :: "'h"   and A :: "'h set" assumes "a \<in> A" shows "insert a (A - {a}) = A"
in_below: fixes a :: "alt"   and b :: "alt"   and L :: "(alt \<times> alt) set"   and x :: "alt"   and y :: "alt" assumes "a \<noteq> b"   and "L \<in> Lin" shows "(x <\<^bsub>below L a b\<^esub> y) = (x \<noteq> y \<and> (if y = a then x <\<^bsub>L\<^esub> b else if x = a then y = b \<or> b <\<^bsub>L\<^esub> y else x <\<^bsub>L\<^esub> y))"
eq_iff_14: fixes "term" :: "Syntax.term"   and terma :: "Syntax.term" shows "(Unroll term = Unroll terma) = (term = terma)"
disjoint_elem: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<inter> B = {}"   and "x \<in> A" shows "x \<notin> B"
opt_bst_simps_2: fixes i :: "int"   and j :: "int"   and w :: "int \<Rightarrow> int \<Rightarrow> nat" assumes "i \<le> j" shows "Optimal_BST.opt_bst w i j = argmin (wpl.wpl w i j) (map (\<lambda>k. \<langle>Optimal_BST.opt_bst w i (k - 1), k, Optimal_BST.opt_bst w (k + 1) j\<rangle>) [i..j])"
sep_conj_commute: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P Q = sep_algebra.sep_conj plus sep_disj Q P"
simps_1: fixes e :: "'env" shows "add_list e [] = e"
g: shows "dom (map_of as\<^sub>2) = set (delete_effects_of op)"
id3: fixes d2 :: "nat"   and g2 :: "'a"   and h2 :: "'a" shows "(if d2 = 1 then g2 * h2 else if even d2 then - g2 * h2 ^ d2 else g2 * h2 ^ d2) = (- (1::'a)) ^ (d2 + 1) * g2 * h2 ^ d2"
rep_eq: fixes x :: "'b fset"   and xa :: "'b fset" shows "fdisjnt x xa = disjnt (fset x) (fset xa)"
cSup_le: fixes S :: "'a set"   and b :: "'a" assumes "S \<noteq> {}"   and "S *<= b" shows "Sup S \<le> b"
pow_ex_2: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<forall>x'. pow_cond t n x' \<longrightarrow> length (bc (\<sigma>\<^bsub>MBC t n\<^esub>t n)) \<le> x'"
inject: fixes x21 :: "'a"   and x22 :: "'a msllist"   and y21 :: "'a"   and y22 :: "'a msllist" shows "(MSLCons x21 x22 = MSLCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
card_fset_def: shows "card_fset \<equiv> (rep_fset ---> id) card_list"
project_Always_D: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set" assumes "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> Always A" shows "extend h F \<squnion> G \<in> Always (extend_set h A)"
dense_le_bounded: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.dense_linorder less_eq less"   and "less x y"   and "\<And>w. \<lbrakk>less x w; less w y\<rbrakk> \<Longrightarrow> less_eq w z" shows "less_eq y z"
NB_Crypt_imp_Server_msg: fixes B :: "agent"   and A :: "agent"   and NB :: "msg"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, NB\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad"   and "evs \<in> woolam" shows "Says Server B (Crypt (shrK B) \<lbrace>Agent A, NB\<rbrace>) \<in> set evs"
finite_unsigned_Hahn_decomposition: fixes M :: "'b measure"   and N :: "'b measure" assumes "finite_measure M"   and "finite_measure N"   and "sets N = sets M" shows "\<exists>Y\<in>sets M. (\<forall>X\<in>sets M. X \<subseteq> Y \<longrightarrow> emeasure N X \<le> emeasure M X) \<and> (\<forall>X\<in>sets M. X \<inter> Y = {} \<longrightarrow> emeasure M X \<le> emeasure N X)"
prob_space_bind': fixes A :: "'b measure"   and M :: "'b measure"   and B :: "'b \<Rightarrow> 'c measure"   and N :: "'c measure" assumes "A \<in> space (prob_algebra M)"   and "B \<in> M \<rightarrow>\<^sub>M prob_algebra N" shows "prob_space (A \<bind> B)"
HomFtorMapContra_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and g :: "'e"   and X :: "'d" shows "HomC\<^bsub>C\<^esub>[g,X] \<equiv> ZFfun (Hom\<^bsub>C\<^esub> cod\<^bsub>C\<^esub> g X) (Hom\<^bsub>C\<^esub> dom\<^bsub>C\<^esub> g X) (\<lambda>f. m2z\<^bsub>C\<^esub>g ;;\<^bsub>C\<^esub> (z2m\<^bsub>C\<^esub>f))"
chine_eq_apex: shows "SPN_r_SPN_s.chine = SPN_r_SPN_s.apex"
remove_spec: fixes h :: "'b \<Rightarrow> int"   and t :: "'b Tree"   and e :: "'b" shows "sortedTree h t \<longrightarrow> sortedTree h (remove h e t) \<and> setOf (remove h e t) = setOf t - eqs h e"
fv1: fixes x :: "real" assumes "x \<in> {v..1}" shows "f x = f 1"
oalist_abstract_axioms: shows "oalist_abstract (\<lambda>x. x) list_of_oalist OAlist"
cut_le_not_in_imp: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<notin> I" shows "x \<notin> I \<down>\<le> t"
borel_measurable_iff_le: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>a. {w \<in> space M. f w \<le> a} \<in> sets M)"
simps: fixes P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and A :: "'at" shows "resolvent_upon P1 P2 A = P1 - {Pos A} \<union> (P2 - {Neg A})"
fold_apply_guards: fixes G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "(gval (fold gAnd G (Bc True)) s = true) = apply_guards G s"
compact_continuous_image: fixes s :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "continuous_on s f"   and "compact s" shows "compact (f ` s)"