hcju/isabelleps huggingface dataset:
integral_component_nonneg: fixes k :: "'a"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "k \<in> Basis"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x \<bullet> k" shows "0 \<le> integral S f \<bullet> k"
coll_zero: fixes z :: "real \<times> real" shows "coll 0 z 0"
usubst_taboos_mon: fixes U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<alpha> :: "game" shows "U \<subseteq> fst (usubstappp \<sigma> U \<alpha>)"
compact_uniformly_continuous: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on S f"   and "compact S" shows "uniformly_continuous_on S f"
summable_divide: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "summable (\<lambda>n. f n / c)"
simple_proto_conjunct_correct: fixes p1 :: "protocol"   and pkt :: "8 word"   and p2 :: "protocol" shows "(match_proto p1 pkt \<and> match_proto p2 pkt) = (case simple_proto_conjunct p1 p2 of None \<Rightarrow> False | Some proto \<Rightarrow> match_proto proto pkt)"
qAFreshAll_imp_qFreshAll: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and A :: "('f, 'g, 'd, 'e, 'h) qAbs" shows "(qAFresh xs x X \<longrightarrow> qFresh xs x X) \<and> (qAFreshAbs xs x A \<longrightarrow> qFreshAbs xs x A)"
g_closed: shows "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
not_is_word_\<SS>: shows "\<not> is_word [\<SS>]"
hyps: shows "card (deg_le_sect X d) = d + card X choose card X"
cod_char: fixes Arr :: "'a set"   and f :: "(bool, 'a list) concrete_category.arr" shows "partial_magma.cod (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) f = (if partial_magma.arr (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) f then if f = quiver.Zero then quiver.Zero else quiver.One else partial_magma.null (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)))"
lunit_inv: fixes a :: "'c" assumes "C.ide a" shows "C.inverse_arrows \<l>\<^sub>C[a] (C.lunit' a)"
complex_neq_0: fixes z :: "complex" shows "(z \<noteq> 0) = (0 < (Re z)\<^sup>2 + (Im z)\<^sup>2)"
dom_update: fixes k :: "'c"   and v :: "'d"   and al :: "('c \<times> 'd) list" shows "fst ` set (update k v al) = {k} \<union> fst ` set al"
Class_eq: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E(\<eta>)" shows "Class a = Class b"
not_HLD: fixes X :: "'a set" shows "(\<lambda>xs. \<not> HLD X xs) = HLD (- X)"
sup_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "sup x y = (sup (fst x) (fst y), sup (snd x) (snd y))"
crepresentation_from_representation: fixes B :: "'d set"   and b :: "'d"   and \<psi> :: "'d" assumes "cindependent B"   and "b \<in> B"   and "finite B" shows "crepresentation B \<psi> b = complex_of_real (representation (B \<union> (*\<^sub>C) \<i> ` B) \<psi> b) + \<i> *\<^sub>C complex_of_real (representation (B \<union> (*\<^sub>C) \<i> ` B) \<psi> (\<i> *\<^sub>C b))"
s2: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq g (Abs \<tau> (g $ Bv 0))"
adjust_nonzero: fixes us :: "complex vec list"   and w :: "complex vec" assumes "set us \<subseteq> carrier_vec d"   and "distinct us"   and "w \<in> carrier_vec d"   and "w \<notin> local.span (set us)" shows "adjuster d w us + w \<noteq> 0\<^sub>v d"
swap: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and B :: "'e set"   and A :: "'d set" shows "F (\<lambda>i. F (g i) B) A = F (\<lambda>j. F (\<lambda>i. g i j) A) B"
hcomp_in_vhom': fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b"   and f :: "'b"   and h :: "'b"   and g :: "'b"   and k :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.arr V (H \<nu> \<mu>)"   and "partial_magma.dom V \<mu> = f"   and "partial_magma.dom V \<nu> = h"   and "partial_magma.cod V \<mu> = g"   and "partial_magma.cod V \<nu> = k"   and "partial_magma.in_hom V \<mu> f g"   and "partial_magma.in_hom V \<nu> h k"   and "src h = trg f" shows "partial_magma.in_hom V (H \<nu> \<mu>) (H h f) (H k g)"
LLL_inv_wD_3: fixes fs :: "int vec list" assumes "LLL_invariant_weak fs" shows "set fs \<subseteq> carrier_vec n"
induct: fixes x1 :: "expr list"   and x2 :: "val list"   and P :: "expr list \<Rightarrow> val list \<Rightarrow> bool" assumes "map_val x1 x2"   and "P [] []"   and "\<And>xs ys y. \<lbrakk>map_val xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (Val y # xs) (y # ys)" shows "P x1 x2"
fimage_eq_to_f: fixes f1 :: "'b \<Rightarrow> 'a"   and S1 :: "'b fset"   and f2 :: "'c \<Rightarrow> 'a"   and S2 :: "'c fset" assumes "f1 |`| S1 = f2 |`| S2" obtains f :: "'c \<Rightarrow> 'b" where "\<And>x. x |\<in>| S2 \<Longrightarrow> f x |\<in>| S1 \<and> f1 (f x) = f2 x"
converged_cd_diverge_cs: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and j :: "nat"   and j' :: "nat"   and l :: "nat"   and m :: "nat"   and m' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"   and "j < l"   and "\<nexists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"   and "l < m"   and "cs\<^bsup>\<pi>\<^esup> m = cs\<^bsup>\<pi>'\<^esup> m'" obtains k :: "nat" and k' :: "nat" where "j \<le> k" and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'" and "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" and "\<pi> (Suc k) \<noteq> \<pi>' (Suc k')"
r_4: shows "\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0"
sunflower_subset: fixes F :: "'b set set"   and G :: "'b set set" assumes "F \<subseteq> G"   and "sunflower G" shows "sunflower F"
match_list_matches: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and ms :: "'a match_expr list"   and a :: "action"   and p :: "'b" shows "match_list \<gamma> ms a p = (\<exists>m\<in>set ms. matches \<gamma> m a p)"
prodag_aGroup: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "aGroup (a\<Pi>\<^bsub>I\<^esub> A)"
gbijec_def: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" shows "gbij\<^bsub>F,G\<^esub> f = (gsurj\<^bsub>F,G\<^esub> f \<and> ginj\<^bsub>F,G\<^esub> f)"
arr_char: fixes f :: "('obj, 'obj) concrete_category.arr" shows "arr f = (f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f \<in> Obj \<and> Map f \<in> (if Dom f = Cod f then {Dom f} else {}))"
eq_acom_iff_strip_annos: fixes C1 :: "'b acom"   and C2 :: "'b acom" shows "(C1 = C2) = (strip C1 = strip C2 \<and> annos C1 = annos C2)"
transp_less: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "transp less"
monomial_inj: fixes s :: "'a"   and c :: "'b"   and t :: "'a"   and d :: "'b" shows "(monomial c s = monomial d t) = (c = (0::'b) \<and> d = (0::'b) \<or> c = d \<and> s = t)"
degree_gcd1: fixes a :: "'a poly"   and b :: "'a poly" assumes "a \<noteq> 0" shows "degree (gcd a b) \<le> degree a"
simps_1: fixes P :: "'a" shows "pre \<lbrace> P \<rbrace> = P"
eventually_ae_filter_negligible: fixes P :: "'a \<Rightarrow> bool" shows "almost_everywhere lebesgue P = (\<exists>N. negligible N \<and> {x. \<not> P x} \<subseteq> N)"
surjective: fixes r :: "'a BlockProc_scheme" shows "r = \<lparr>block = block r, proc = proc r, \<dots> = BlockProc.more r\<rparr>"
MsgHasMinTime: shows "\<forall>msg. enabled (last exec) msg \<and> (\<exists>p. isReceiverOf p msg) \<longrightarrow> (\<exists>n<length exec. enabled (execConf n) msg \<and> (\<forall>n'\<ge>n. n' < length trace \<longrightarrow> msg \<noteq> execMsg n') \<and> (\<forall>n0. n0 < length exec \<and> enabled (execConf n0) msg \<and> (\<forall>n'\<ge>n0. n' < length trace \<longrightarrow> msg \<noteq> execMsg n') \<longrightarrow> n \<le> n0))"
list_rel_simp_2: fixes l :: "'b list"   and R :: "('b \<times> 'c) set" shows "((l, []) \<in> \<langle>R\<rangle>list_rel) = (l = [])"
pt3_rev: fixes pi1 :: "('b \<times> 'b) list"   and pi2 :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and " pi1 \<triangleq> pi2 " shows "rev pi1 \<bullet> x = rev pi2 \<bullet> x"
dom_cod: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.dom (CC.cod f) = CC.cod f"
pt: fixes n :: "nat" assumes "0 < n" shows "PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n (replicate n 0) = fold (\<lambda>i P. P|||RPHIL i) [1..<n] LPHIL0"
mcont_monoD: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "mcont luba orda lubb ordb f"   and "orda x y" shows "ordb (f x) (f y)"
that: fixes df :: "real" assumes "(f has_real_derivative df) (at x within s)" shows "thesis"
RelStay: fixes P :: "pi"   and Q :: "pi" assumes "(P \<parallel> !P, Q) \<in> bisim O bangRel weakBisim O bisim" shows "(!P, Q) \<in> bisim O bangRel weakBisim O bisim"
lossless: shows "lossless_gpv (\<I>_full \<oplus>\<^sub>\<I> \<I>_full) \<A>"
eq_iff: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "bijection f" shows "(f a = f b) = (a = b)"
i_Exec_Stream_Acc_LocalState_take: fixes k :: "nat"   and n :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" assumes "0 < k" shows "i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c \<Down> n = f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun (input \<Down> n) c"
span_eq_iff: fixes s :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a" assumes "s \<subseteq> cotangent_space p" shows "(span s = s) = subspace s"
conj_disj_distrib2: fixes y :: "'c"   and z :: "'c"   and x :: "'c" shows "(y OR z) AND x = y AND x OR z AND x"
step_Mn_correct: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "valid ((Mn n f, xs, []) # rest)"   and "\<And>xs rest. valid ((f, xs, []) # rest) \<Longrightarrow> correct ((f, xs, []) # rest, None)" shows "correct ((Mn n f, xs, []) # rest, None)"
is_root_merge: fixes h1 :: "'b heap"   and h2 :: "'b heap" assumes "is_root' h1"   and "is_root' h2" shows "is_root' (merge' h1 h2)"
select_convs_1: fixes orderA :: "('a, 'b) gorder_scheme"   and orderB :: "('c, 'd) gorder_scheme"   and lower :: "'a \<Rightarrow> 'c"   and upper :: "'c \<Rightarrow> 'a"   and more :: "'e" shows "\<X>\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = orderA"
Sem_no_zero_height_derivsAux: fixes c :: "OBJ" shows "\<forall>s t.  s , c \<rightarrow>\<^sub>0  t  \<longrightarrow> False"
ghom_unit_unit: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "f \<one>\<^bsub>F\<^esub> = \<one>\<^bsub>G\<^esub>"
cos_plus_pi: fixes z :: "'b" shows "cos (z + of_real pi) = - cos z"
inf_Inf: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "inf_prod x y = Inf_prod (insert x (\<eta> y))"
pasting_lemma_exists: fixes X :: "'c topology"   and T :: "'d \<Rightarrow> 'c set"   and I :: "'d set"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'e" assumes "topspace X \<subseteq> \<Union> (T ` I)"   and "\<And>i. i \<in> I \<Longrightarrow> openin X (T i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"   and "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x" obtains g :: "'c \<Rightarrow> 'e" where "continuous_map X Y g" and "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
prems_1: shows "x < count_weights True (Pool m1 m2)"
swap_msgs_Trans_Trans: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isTrans ev"   and "isTrans ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
subadditiveD: fixes u :: "nat \<Rightarrow> real"   and m :: "nat"   and n :: "nat" assumes "subadditive u" shows "u (m + n) \<le> u m + u n"
4: fixes y :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and x :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "(y, x) \<in>\<^sub>p snd ` set ps" shows "x \<in> set (p # q # hs @ map (fst \<circ> snd) ps @ map (snd \<circ> snd) ps)"
collapse_DList_set_def: fixes dxs :: "'b set_dlist" shows "collapse_DList_set dxs = set (list_of_dlist dxs)"
cobounded1: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<subseteq># a \<union># b"
belowD: fixes d :: "'a \<rightarrow> 'a"   and f :: "'a \<rightarrow> 'a"   and x :: "'a" assumes "deflation d"   and "f \<sqsubseteq> d"   and "f\<cdot>x = x" shows "d\<cdot>x = x"
simps_1: fixes s :: "'b" shows "tLast (tInit s) = s"
wlsFSb_recAll_termFSbMorph: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) Recursion.model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.wlsFSb Delta MOD" shows "FixSyn.termFSbMorph Delta (FixSyn.rec Delta MOD) (FixSyn.recAbs Delta MOD) MOD"
higher_differentiable_on_open_Un: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and T :: "'b set" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on T f n"   and "open S"   and "open T" shows "higher_differentiable_on (S \<union> T) f n"
conj_xor_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y XOR z) AND x = y AND x XOR z AND x"
hyps: assumes "k < min (nrows A) (ncols A)" shows "isDiagonal (diagonal_to_Smith_aux A [0..<k] bezout)"
awalk_verts_in_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "u \<in> pverts (subdivide G e w)"   and "set p \<subseteq> parcs (subdivide G e w)"   and "v \<in> set (pawalk_verts u p)" shows "v \<in> pverts (subdivide G e w)"
vbrelation_vconverse: shows "vbrelation (vreal_uminus\<inverse>\<^sub>\<circ>)"
assms: shows "is_parallel_solution_for_problem \<Psi> \<psi>"
vars_tableau_vars_constraints: fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat" shows "rvars (Tableau (preprocess' cs start)) \<subseteq> vars_constraints (map snd cs)"
closures_universal: fixes \<phi> :: "'a \<Rightarrow> \<o>" assumes "\<And>x. [[\<phi> x]]" shows "[[\<^bold>\<forall>x. \<phi> x]]"
ex_post_efficient': fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and x :: "'alt"   and y :: "'alt" assumes "is_pref_profile R"   and "x \<prec>[Pareto R] y" shows "pmf (sds R) x = 0"
INF_ereal_minus_right: fixes I :: "'a set"   and c :: "ereal"   and f :: "'a \<Rightarrow> ereal" assumes "I \<noteq> {}"   and "\<bar>c\<bar> \<noteq> \<infinity>" shows "(INF i\<in>I. c - f i) = c - Sup (f ` I)"
x_ivp: shows "X \<in> Sols f U S t\<^sub>0 s"
simps_1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and lt :: "('a, 'b) rbt"   and z :: "'a"   and v :: "'b"   and rt :: "('a, 'b) rbt"   and y :: "'a"   and s :: "'b"   and b :: "('a, 'b) rbt" shows "ord.rbt_del_from_left less x (Branch B lt z v rt) y s b = balance_left (ord.rbt_del less x (Branch B lt z v rt)) y s b"
akra_bazzi_light_aux2: fixes a :: "real"   and c :: "real"   and C\<^sub>1 :: "real"   and n\<^sub>0 :: "nat"   and f :: "nat \<Rightarrow> real"   and b :: "real"   and d :: "real"   and C\<^sub>2 :: "real"   and C\<^sub>3 :: "real"   and C\<^sub>4 :: "real"   and n :: "nat" assumes "0 < a"   and "0 < c"   and "a + c < 1"   and "0 \<le> C\<^sub>1"   and "\<forall>n>n\<^sub>0. f n = f (nat \<lceil>a * real n + b\<rceil>) + f (nat \<lceil>c * real n + d\<rceil>) + C\<^sub>1 * real n + C\<^sub>2"   and "(max 0 b + max 0 d + 2) / (1 - a - c) < real n\<^sub>0"   and "C\<^sub>1 / (1 - a - c) \<le> C\<^sub>3"   and "(C\<^sub>1 * real n\<^sub>0 + C\<^sub>2 + C\<^sub>4) / ((1 - a - c) * real n\<^sub>0 - max 0 b - max 0 d - 2) \<le> C\<^sub>3"   and "\<forall>n\<le>n\<^sub>0. f n \<le> C\<^sub>4" shows "f n \<le> C\<^sub>3 * real n + C\<^sub>4"
derf: fixes x :: "'a" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
isomorphic_symmetric: fixes f :: "'c"   and g :: "'c" assumes "f \<cong>\<^sub>C g" shows "g \<cong>\<^sub>C f"
quot_setD: fixes a :: "'a"   and A :: "'a set"   and x :: "'a" assumes "a \<in> A \<div> x" shows "x * a \<in> A"
is_run_def: fixes r :: "nat \<Rightarrow> 'b" shows "b.is_run r \<equiv> r 0 \<in> b.V0 \<and> ipath b.E r"
state_in_successor_proj_in_state_in_successor: fixes s' :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and vs :: "'a set"   and s :: "('a, 'b) fmap" assumes "s' \<in> state_successors (prob_proj PROB vs) s" shows "fmrestrict_set vs s' \<in> state_successors (prob_proj PROB vs) (fmrestrict_set vs s)"
ord_inj_restrict_isom: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and T :: "'a set" assumes "Order E"   and "ord_inj D E f"   and "T \<subseteq> carrier D" shows "ord_isom (Iod D T) (Iod E (f ` T)) (restrict f T)"
inject_3: fixes x31 :: "'b"   and x32 :: "('a, 'b) form list"   and y31 :: "'b"   and y32 :: "('a, 'b) form list" shows "(Modal x31 x32 = Modal y31 y32) = (x31 = y31 \<and> x32 = y32)"
assms: fixes x :: "'a" shows "B *\<^sub>V ket x = (case \<pi> x of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i)"
red: shows "s -t\<triangleright>ta\<rightarrow> s'"
invar_empty_2: shows "SkewBinomialHeapStruc.tail_invar []"
distinct_270: fixes x151 :: "floatarith"   and x152 :: "nat"   and x12a :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Exp x12a"
mset_zero_iff_right: fixes x :: "'k list" shows "({#} = mset x) = (x = [])"
assign_load_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and y :: "Eg1.var"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = (x \<leftarrow> aexp.Load y) ;; c\<^sub>A_tail"   and "c\<^sub>C = (Eg2_var\<^sub>C_of_Eg1 x \<leftarrow> aexp\<^sub>C.Load (Eg2_var\<^sub>C_of_Eg1 y)) ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoWrite"   and "y \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> \<C>"   and "y \<notin> \<C>"   and "\<forall>mds\<^sub>A' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
nat_power_1_right: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.nat_power one times plus zero nat_power" shows "nat_power n one = semiring_1.of_nat one plus zero n"
hoare_total_correct_def: fixes dims :: "nat list"   and P :: "complex mat"   and S :: "com"   and Q :: "complex mat" shows "state_sig.hoare_total_correct dims P S Q = (\<forall>\<rho>\<in>state_sig.density_states dims. trace (P * \<rho>) \<le> trace (Q * state_sig.denote dims S \<rho>))"
trace_simps_7: shows "schedule \<equiv> filter_act \<circ> snd"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state return bind get put"   and "monad_altc return bind altc"   and "monad_state_altc_axioms get put altc" shows "monad_state_altc return bind get put altc"
assms_3: fixes \<phi> :: "'pred" assumes "\<alpha>\<^sub>S = S_action.Pred \<phi>"   and "P' = P"   and "P \<turnstile> \<phi>" shows "R"
pgallery_last_eq_W_image: fixes Cs :: "'a set list"   and C :: "'a set" assumes "pgallery (C0 # Cs @ [C])" shows "\<exists>w\<in>\<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. C = w `\<rightarrow> C0"
nstut_and: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "NSTUTINV F"   and "NSTUTINV G" shows "NSTUTINV (F \<and> G)"
omit_del: fixes p :: "('c, 'd) fm"   and ps :: "('c, 'd) fm list"   and xs :: "nat set" assumes "p \<in>. ps"   and "p \<notin> set (omit xs ps)" shows "\<exists>v. ps !. v = Some p \<and> v \<in> xs"
assms_3: shows "(i, Append i e1) \<in> set (node_deliver_messages xs)"
"apply": fixes x :: "'b perm" shows "apply x \<in> {f. bij f \<and> finite {a. f a \<noteq> a}}"
mm: fixes c\<^sub>1 :: "'Com\<^sub>A"   and mdsa :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem\<^sub>1 :: "'Var\<^sub>A \<Rightarrow> 'Val"   and c\<^sub>2 :: "'Com\<^sub>A"   and mem\<^sub>2 :: "'Var\<^sub>A \<Rightarrow> 'Val" assumes "(\<langle>c\<^sub>1, mdsa, mem\<^sub>1\<rangle>\<^sub>A, \<langle>c\<^sub>2, mdsa, mem\<^sub>2\<rangle>\<^sub>A) \<in> \<R>\<^sub>A" shows "abs.low_mds_eq mdsa mem\<^sub>1 mem\<^sub>2"
k_ins_def: fixes a :: "'a"   and ec :: "'s odlist"   and A :: "(('s, 'aAct list) trie, ('s, ('obs, 's odlist) mapping) trie) AlgState" shows "SPRsingle.KBP.k_ins a ec A \<equiv> \<lparr>aActs = SPRsingle.KBP.actsUpdate a ec A, aTrans = foldr (SPRsingle.KBP.transUpdate a ec) (local.spr_simTrans a ec) (aTrans A)\<rparr>"
of_real_poly_map_plus: fixes p :: "real poly"   and q :: "real poly" shows "map_poly of_real (p + q) = map_poly of_real p + map_poly of_real q"
\<epsilon>_map_simp: fixes x :: "'c" assumes "C.ide x" shows "transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) FG.map \<epsilon>o x = \<psi> x (G x)"
space_in_measure_of: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "\<Omega> \<in> sets (measure_of \<Omega> M \<mu>)"