hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "'f \<Rightarrow>\<^sub>L 'e"   and xa :: "'d \<Rightarrow>\<^sub>L 'f" shows "blinfun_apply (x o\<^sub>L xa) = blinfun_apply x \<circ> blinfun_apply xa"
integral_eq_nn_integral: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x" shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M)"
path_component_eq: fixes y :: "'a"   and S :: "'a set"   and x :: "'a" assumes "y \<in> path_component_set S x" shows "path_component_set S y = path_component_set S x"
cball_subset_cball_iff: fixes a :: "'b"   and r :: "real"   and a' :: "'b"   and r' :: "real" shows "(cball a r \<subseteq> cball a' r') = (dist a a' + r \<le> r' \<or> r < 0)"
gSubstBinp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'b, 'j) Recursion.model_scheme"   and ys :: "'e"   and Y' :: "('c, 'd, 'e, 'h, 'g) qTerm set"   and Y :: "'i"   and y :: "'h"   and binp' :: "'a \<Rightarrow> ('c, 'd, 'e, 'h, 'g) qAbs set option"   and binp :: "'a \<Rightarrow> 'b option" shows "gSubstBinp MOD ys Y' Y y binp' binp \<equiv> lift2 (gSubstAbs MOD ys Y' Y y) binp' binp"
foldD_def: fixes D :: "'e set"   and f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and e :: "'e"   and A :: "'f set" shows "foldD D f e A = (THE x. (A, x) \<in> foldSetD D f e)"
invertibleE: fixes u :: "'a"   and thesis :: "bool" assumes "invertible u"   and "\<And>v. \<lbrakk>u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>; v \<in> G\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> G" shows "thesis"
in_homE: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "S.VV.in_hom f a b" obtains "S.VV.arr f" and "S.VV.dom f = a" and "S.VV.cod f = b"
valid_plan_valid_tail: fixes h :: "'c"   and as :: "'c list"   and PROB :: "'c set" assumes "h # as \<in> valid_plans PROB" shows "as \<in> valid_plans PROB"
rBangTauI: fixes S :: "('a, 'b, 'c) psi"   and S' :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi>' \<rhd> S \<parallel> S \<Longrightarrow>\<^sub>\<tau> S'"   and "guarded S" shows "\<exists>T. \<Psi>' \<rhd> !S \<Longrightarrow>\<^sub>\<tau> T \<and> (\<Psi>', T, S' \<parallel> !S) \<in> Rel'"
simps_2: fixes k :: "nat" shows "htuple (Suc k) = \<langle>0, htuple k\<rangle>"
le_cases: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" obtains "le xa x y"   | "le xa y x"
reindex_bij_witness: fixes S :: "'a set"   and i :: "'b \<Rightarrow> 'a"   and j :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"   and "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "prod g S = prod h T"
rtos_inj: fixes r :: "nat"   and s :: "nat" assumes "0 < r"   and "0 < s" shows "inj_on (rtos r s) {i. i \<le> r * s - Suc 0}"
fbb_fbd_demorgan: fixes R :: "'c \<Rightarrow> 'd set" shows "\<partial> \<circ> bb\<^sub>\<F> R = bd\<^sub>\<F> R \<circ> \<partial>"
subset_\<I>\<J>Suc: fixes k :: "nat"   and u :: "nat" shows "\<I> k \<subseteq> \<J> (Suc k) u"
binomial_strict_antimono: fixes k :: "nat"   and k' :: "nat"   and n :: "nat" assumes "k < k'"   and "n \<le> 2 * k"   and "k' \<le> n" shows "n choose k' < n choose k"
pth_c_1: fixes c :: "real"   and x :: "'b"   and d :: "real"   and y :: "'b" shows "c *\<^sub>R x + d *\<^sub>R y \<equiv> d *\<^sub>R y + c *\<^sub>R x"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'d \<Rightarrow> 'b"   and option :: "'d option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
BisT_matchC_TMC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>T d" shows "matchC_TMC BisT c d"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "tree \<Rightarrow> tree \<Rightarrow> 'b"   and tree :: "tree" shows "h (case tree of Leaf \<Rightarrow> f1 | Branching x xa \<Rightarrow> f2 x xa) = (case tree of Leaf \<Rightarrow> h f1 | Branching x1 x2 \<Rightarrow> h (f2 x1 x2))"
collapse_4: fixes prot_fun :: "('a, 'b, 'c) prot_fun" assumes "is_Abs prot_fun" shows "Abs (the_Abs prot_fun) = prot_fun"
push_action_bound: fixes p :: "pr_operation list"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts" shows "length (filter is_PUSH p) \<le> 22 * (card V)\<^sup>2 * card E"
degree_lagrange_basis_poly: fixes xs :: "'b list"   and xj :: "'b" shows "degree (lagrange_basis_poly xs xj) \<le> length (filter (\<lambda>x. x \<noteq> xj) xs)"
intros_4: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "ev\<^sub>B mem b"   and "no_await c\<^sub>1"   and "\<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto>\<^sup>+ \<langle>c\<^sub>2, mds', mem'\<rangle>"   and "is_final c\<^sub>2" shows "\<langle>Await b c\<^sub>1, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>2, mds', mem'\<rangle>"
simps_1: fixes i :: "nat"   and n :: "nat" shows "select [] i n = (if i \<le> 0 then selecthelp [] n else select [] (i - 1) (n - 1))"
Cons_4: shows "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"
DERIV_cos: fixes x :: "'b" shows "(cos has_field_derivative - sin x) (at x)"
fp64_of_float_def: shows "fp64_of_float \<equiv> map_fun Rep_float id (\<lambda>(s, e, f). word_cat s (word_cat e f))"
bijective_def: fixes R :: "('b \<times> 'c) set" shows "bijective R \<equiv> (\<forall>x y z. (x, y) \<in> R \<and> (x, z) \<in> R \<longrightarrow> y = z) \<and> (\<forall>x y z. (x, z) \<in> R \<and> (y, z) \<in> R \<longrightarrow> x = y)"
subs_Bot_G_entails: fixes sB :: "'g set"   and N :: "'g set" assumes "sB \<noteq> {}"   and "sB \<subseteq> Bot_G" shows "sB \<Turnstile>G N"
simps: fixes x :: "int poly \<times> rat \<times> rat"   and y :: "int poly \<times> rat \<times> rat" shows "mult_1 x y = (case (x, y) of ((p1, l1, r1), p2, l2, r2) \<Rightarrow> if 0 < r1 then if 0 < r2 then mult_1_pos x y else uminus_2 (mult_1_pos x (uminus_1 y)) else if 0 < r2 then uminus_2 (mult_1_pos (uminus_1 x) y) else mult_1_pos (uminus_1 x) (uminus_1 y))"
aux_split_1: shows "drop (length done) auxlist = hd_aux # tl_aux"
key_present_imp_eq_lookup_finds_value: fixes k :: "'a"   and al :: "('a \<times> 'b) list" assumes "k \<in> fst ` set al" shows "\<exists>v. lookup (\<lambda>x. x = k) al = Some v"
independent_vertices_finite: fixes S :: "'a set" assumes "independent_vertices E S" shows "finite S"
assms: shows "r \<cdot> u = u \<cdot> r"
ListReds2: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v :: "val" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>es',s',b'\<rangle>" shows "P \<turnstile> \<langle>Val v # es,s,b\<rangle> [\<rightarrow>]* \<langle>Val v # es',s',b'\<rangle>"
infty_relI_1: fixes R :: "('a \<times> 'b) set" shows "(infty.Infty, infty.Infty) \<in> \<langle>R\<rangle>infty_rel"
n_circ_1: fixes k :: "nat" shows "valid_path (n_circ k)"
aristo_conversion3: fixes A :: "'b set"   and B :: "'b set" assumes "A Q B"   and "B \<noteq> {}" shows "B I A"
qp_D3_post: shows "is_quantum_predicate (adjoint exM0 * P' * exM0 + adjoint exM1 * Q * exM1)"
wt_\<Phi>: shows "F.wtPB \<Phi>"
unit_simps_5: shows "S.trg (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f)) = S.src (S.UP f)"
fsafe_Prop: fixes args :: "'a \<Rightarrow> ('b, 'a) trm"   and p :: "'a" assumes "\<And>i. dsafe (args i)" shows "fsafe ($\<phi> p args)"
silent_moves_slice_intra_path: fixes S :: "'node SDG_node set"   and m :: "'node"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and m' :: "'node"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,slice_kind S \<turnstile> (m # ms,s) =as\<Rightarrow>\<^sub>\<tau> (m' # ms',s')"   and "\<forall>mx\<in>set ms. \<exists>mx'. call_of_return_node mx mx' \<and> mx' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "\<forall>a\<in>set as. intra_kind (kind a)"
szip\<^sub>E_def: fixes a :: "'a"   and b :: "'b" shows "szip\<^sub>E a b \<equiv> \<lambda>(\<omega>1, \<omega>2). szip (K1.force_enabled a \<omega>1) (K2.force_enabled b \<omega>2)"
cases: fixes r :: "'c monoid" obtains carrier :: "'c set" and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" and one :: "'c" where "r = \<lparr>carrier = carrier, monoid.mult = mult, one = one\<rparr>"
unique_row_col_2: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and nr1 :: "nat"   and nc1 :: "nat"   and M :: "'c list list"   and nr2 :: "nat"   and nc2 :: "nat" assumes "plus_mult id f zer g inver"   and "mat nr1 nc1 M"   and "mat nr2 nc2 M"   and "M \<noteq> []" shows "nc1 = nc2"
to_nat_suc: fixes x :: "'a" assumes "mod_type_class.to_nat x + 1 < CARD('a)" shows "mod_type_class.to_nat (x + (1::'a)) = mod_type_class.to_nat x + 1"
m: shows "real (m div 2) \<le> real n / (2 * (2 * real d + 1)) + 1 / 2"
assms_3: shows "\<forall>x\<in>S. (x, L) \<in> r"
prems_1: shows "set \<sigma> \<subseteq> {x, y}"
dbl_simps_3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus one = numeral.numeral one plus (num.Bit0 num.One)"
subformulas\<^sub>\<mu>_semantics: fixes \<phi> :: "'a ltln" shows "subformulas\<^sub>\<mu> \<phi> = {\<psi> \<in> subfrmlsn \<phi>. \<exists>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi> = \<psi>\<^sub>1 U\<^sub>n \<psi>\<^sub>2 \<or> \<psi> = \<psi>\<^sub>1 M\<^sub>n \<psi>\<^sub>2}"
cong_diff: fixes b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a"   and e :: "'a" assumes "[b = c] (mod a)"   and "[d = e] (mod a)" shows "[b - d = c - e] (mod a)"
satisfies_bounded_conjuncts: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b \<phi>) = (\<forall>\<psi>\<in>conjuncts \<phi>. \<AA> \<Turnstile>\<^sub>b \<psi>)"
has_vector_derivative_eq_has_derivative_blinfun: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and U :: "real set" shows "(f has_vector_derivative f') (at x within U) = (f has_derivative blinfun_apply (blinfun_scaleR_left f')) (at x within U)"
assms_1: shows "Y \<in> space M \<rightarrow> space N"
alw_ev_shift: fixes P :: "'a stream \<Rightarrow> bool"   and u :: "'a list"   and v :: "'a stream" shows "alw (ev P) (u @- v) = alw (ev P) v"
irreflexive_complement_reflexive: fixes x :: "'a" shows "(x \<le> - top) = (top \<le> - x)"
sup_left_divisibility: fixes y :: "'a"   and x :: "'a" shows "(y \<le> x) = (\<exists>z. x \<sqinter> z = y)"
dg_prod_3_ObjI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and c :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "digraph \<alpha> \<CC>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "c \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "[a, b, c]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Obj\<rparr>"
Spec: fixes s :: "'a"   and t :: "'a"   and r :: "('a \<times> 'a) set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option" assumes "(s, t) \<in> r" shows "\<Gamma>\<turnstile> (Spec r, Normal s) \<rightarrow> (Skip, Normal t)"
assms_5: shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr \<in> set c"
unit_mult_right_cancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "(b * a = c * a) = (b = c)"
cases: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and "structure" :: "'f set"   and sat :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('g \<times> bool \<times> 'g) set"   and Ord :: "('h \<times> 'h) set"   and \<sigma> :: "'g \<times> 'f \<Rightarrow> 'h"   and pointsTo :: "'i \<Rightarrow> ('e, 'd, 'i) ctree"   and a :: "('e, 'd, 'i) ctree" assumes "Infinite_Soundness eff rules structure sat \<delta> Ord \<sigma>"   and "\<forall>l l'. pointsTo l \<noteq> Link l'"   and "Infinite_Soundness.cwf eff rules pointsTo a" obtains l :: "'i" where "a = Link l" and "Infinite_Soundness.cwf eff rules pointsTo (pointsTo l)"   | r :: "'d" and s :: "'e" and cts :: "('e, 'd, 'i) ctree fset" where "a = cNode (s, r) cts" and "r \<in> i.R rules" and "eff r s (Infinite_Soundness.seqOf pointsTo |`| cts)" and "\<And>ct'. ct' |\<in>| cts \<Longrightarrow> Infinite_Soundness.cwf eff rules pointsTo ct'"
fps_exp_eq_fps_const_iff: fixes c :: "'b"   and c' :: "'b" shows "(fps_exp c = fps_const c') = (c = (0::'b) \<and> c' = (1::'b))"
reset_sets_all_equiv: fixes u :: "'c \<Rightarrow> real"   and u' :: "'c \<Rightarrow> real"   and r :: "'c list"   and d :: "nat"   and x :: "'c" assumes "u \<in> V"   and "u' \<in> [[r\<rightarrow>real d]u]\<^sub>\<R>"   and "x \<in> set r"   and "set r \<subseteq> \<X>"   and "d \<le> k x" shows "u' x = real d"
rank_le: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and v :: "nat \<times> 'a"   and u :: "nat \<times> 'a" assumes "finite (nodes A)"   and "w \<notin> language A"   and "v \<in> gunodes A w"   and "u \<in> gusuccessors A w v" shows "rank A w u \<le> rank A w v"
normalized: fixes X :: "'b set"   and v :: "'b \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'b"   and k :: "'b \<Rightarrow> nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "Beta_Regions' X v n not_in_X" shows "Beta_Regions'.normalized X k v n not_in_X M \<equiv> (\<forall>i j. 0 < i \<and> i \<le> n \<and> 0 < j \<and> j \<le> n \<and> M i j \<noteq> \<infinity> \<longrightarrow> Lt (real_of_int (- int ((k \<circ> Beta_Regions'.v' X v n not_in_X) j))) \<le> M i j \<and> M i j \<le> Le (real ((k \<circ> Beta_Regions'.v' X v n not_in_X) i))) \<and> (\<forall>i\<le>n. 0 < i \<longrightarrow> (M i 0 \<le> Le (real ((k \<circ> Beta_Regions'.v' X v n not_in_X) i)) \<or> M i 0 = \<infinity>) \<and> Lt (real_of_int (- int ((k \<circ> Beta_Regions'.v' X v n not_in_X) i))) \<le> M 0 i)"
single: fixes y :: "'a set" assumes "y \<in> insert x ` Pow F"   and "card y = 1" shows "y = {x}"
elims: fixes x :: "complex_interval"   and xa :: "complex_interval"   and y :: "complex_interval" assumes "x + xa = y" obtains rx :: "real interval" and ix :: "real interval" and ry :: "real interval" and iy :: "real interval" where "x = Complex_Interval rx ix" and "xa = Complex_Interval ry iy" and "y = Complex_Interval (rx + ry) (ix + iy)"
sorted_distinct_set_unique: fixes xs :: "'d Word.word list"   and ys :: "'d Word.word list" assumes "signed.sorted xs"   and "distinct xs"   and "signed.sorted ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
unification_def: shows "unification \<equiv> \<forall>\<sigma> L. finite L \<longrightarrow> unifier\<^sub>l\<^sub>s \<sigma> L \<longrightarrow> (\<exists>\<theta>. mgu\<^sub>l\<^sub>s \<theta> L)"
post_i: shows "postrecording_event t (Max {i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}})"
bounded_by_imp_bounded: fixes b :: "real"   and P :: "'a \<Rightarrow> real" assumes "bounded_by b P" shows "Expectations.bounded P"
div_hiding_lub: fixes A :: "'b set"   and Y :: "nat \<Rightarrow> 'b process" assumes "finite A"   and "chain Y" shows "D (\<Squnion>i. Y i \ A) \<subseteq> D ((\<Squnion>i. Y i) \ A)"
path_weight_conv_path: fixes s :: "'lbl"   and l1 :: "'vtx"   and l2 :: "'vtx" assumes "s \<in>\<^sub>A path_weight l1 l2" shows "\<exists>xs. path l1 l2 xs \<and> s = sum_weights (map (\<lambda>(s, l, t). l) xs) \<and> (\<forall>ys. path l1 l2 ys \<longrightarrow> \<not> sum_weights (map (\<lambda>(s, l, t). l) ys) < sum_weights (map (\<lambda>(s, l, t). l) xs))"
pair_prod_smooth_pw_smooth: fixes f :: "real \<Rightarrow> real"   and s :: "real set"   and g :: "real \<Rightarrow> real" assumes "f C1_differentiable_on s"   and "g piecewise_C1_differentiable_on s" shows "(\<lambda>x. (f x, g x)) piecewise_C1_differentiable_on s"
assms_5: shows "p = q \<oplus>\<^bsub>P\<^esub> monom P a n"
LI_preserves_tfr_single: fixes S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list"   and \<theta>' :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(S, \<theta>) \<leadsto> (S', \<theta>')"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t S)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)"   and "list_all tfr\<^sub>s\<^sub>t\<^sub>p S" shows "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t S') \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S')"
strict_coboundedI1: fixes c :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" assumes "c \<prec> a" shows "c \<prec> ordered_powerprod_lin.max a b"
Out: fixes out :: "'a"   and \<I>' :: "('a, 'b) \<I>"   and conv :: "('d, 'c, 'a, 'b) converter"   and a :: "'d"   and \<I> :: "('d, 'c) \<I>" assumes "out \<in> outs_gpv \<I>' (run_converter conv a)"   and "a \<in> outs_\<I> \<I>" shows "out \<in> outs_converter \<I> \<I>' conv"
True: fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F bf_True"
length_stirling_row: fixes n :: "nat" shows "length (stirling_row n) = Suc n"
ENR_convex_closed: fixes S :: "'a set" assumes "closed S"   and "convex S" shows "ENR S"
jonsson1c: shows "(\<forall>x y. x \<rhd> y = (x \<rhd> (1::'a)) \<cdot> y) = (\<forall>x y. y \<lhd> x = (1::'a) \<lhd> (x \<lhd> y))"
cases: fixes x :: "com" obtains "x = SKIP"   | xa :: "char list" and a :: "aexp" where "x = xa ::= a"   | c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "x = c\<^sub>1__;; c\<^sub>2__"   | b :: "bexp" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "x = IF b__ THEN c\<^sub>1 ELSE c\<^sub>2"   | b :: "bexp" and c :: "com" where "x = WHILE b DO c"
trg_inv: fixes \<mu> :: "'a" assumes "local.iso \<mu>" shows "trg (local.inv \<mu>) = trg \<mu>"
assms_1: shows "is_sig_red sing_reg top_tail F p"
Trie_inverse: fixes y :: "('c, 'd) Trie.trie" assumes "y \<in> {t. invar_trie t}" shows "impl_of (Trie2.trie.Trie y) = y"
eval\<^sub>G_respectfulness_2: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl"   and S :: "'b ltl set" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "eval\<^sub>G S \<phi> \<equiv>\<^sub>P eval\<^sub>G S \<psi>"
list_comb: fixes f :: "'a"   and xs :: "'a list"   and S :: "name fset" shows "closed_except (list_comb f xs) S = (closed_except f S \<and> list_all (\<lambda>t. closed_except t S) xs)"
hyps: shows "p \<noteq> [] \<longrightarrow> Cp (list2policyR p) p = list2policy (map Cp p) p"
F_aprog_def: shows "F_aprog \<equiv> let (aprog, rs_pos, a_md) = rec_ci rec_F in aprog [+] dummy_abc (Suc (Suc 0))"
P3: shows "A B Perp P A \<and> Col A B T \<and> Bet Q T P"
d_OUT_Edge: fixes f :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" assumes "flow \<Delta>'' f" shows "d_OUT f (Edge x y) = f (Edge x y, Vertex y)"
len': shows "length (map of_int_hom.vec_hom fs) = m"
subst_term_eqI: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term"   and \<tau> :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "\<And>t. t \<cdot> \<sigma> = t \<cdot> \<tau>" shows "\<sigma> = \<tau>"
HNatInfinite_of_hypnat_gt_zero: fixes N :: "nat star" assumes "N \<in> HNatInfinite" shows "0 < of_hypnat N"
smallomega_def: fixes F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "\<omega>[F](g) = {f. \<forall>c>0. \<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)}"
intro: fixes Xd :: "'e \<Rightarrow> 'f"   and Ch :: "'d \<Rightarrow> 'e set \<Rightarrow> 'e set" assumes "\<forall>h. Contracts.bilateral_substitutes_on Xd UNIV (Ch h)" shows "ContractsWithBilateralSubstitutes_axioms Xd Ch"
seq_comp_secure_aux_1_case_2: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and P :: "'d option process"   and Q :: "'d option process"   and xs :: "'d option list"   and y :: "'d option"   and ys :: "'d option list"   and X :: "'d option set"   and Y :: "'d option set" assumes "secure_termination I D"   and "sequential P"   and "secure P I D"   and "secure Q I D"   and "xs @ y # ys \<in> sentences P"   and "(xs @ y # ys, X) \<in> failures P"   and "([], Y) \<in> failures Q" shows "(xs @ ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys (insert None X \<inter> Y)) \<in> seq_comp_failures P Q"
swap_inj_on: fixes A :: "('a \<times> 'b) set" shows "inj_on (\<lambda>(i, j). (j, i)) A"
space_subset_in_sets: fixes J :: "'d set set"   and I :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure" assumes "J \<subseteq> I"   and "\<And>j. j \<in> J \<Longrightarrow> finite j" shows "space (Pi\<^sub>F J M) \<in> sets (Pi\<^sub>F I M)"
not_None_eq: fixes x :: "'c option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) atm" shows "GE.satL \<xi> (Pos a) = GE.satA \<xi> a"
qtable_mem_restr_UNIV: fixes n :: "nat"   and A :: "nat set"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set" shows "qtable n A (mem_restr UNIV) Q X = wf_table n A Q X"
sel_2: fixes x1 :: "real interval"   and x2 :: "real interval" shows "Im_interval (Complex_Interval x1 x2) = x2"
beta_reduction_left_sndE: fixes P :: "'a trm"   and M' :: "'a trm" assumes "Snd P \<rightarrow>\<beta> M'" shows "(\<exists>P'. P \<rightarrow>\<beta> P' \<and> M' = Snd P') \<or> (\<exists>A B. P = SimplyTyped.Pair A B \<and> M' = B)"
inter_add_right1: fixes x :: "'b"   and N :: "'b multiset"   and M :: "'b multiset" assumes "x \<notin># N" shows "N \<inter># add_mset x M = N \<inter># M"
arr_char: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "arr f = (f = Zero \<or> f = One \<or> f \<in> fromArr ` Arr)"
map_cong0: fixes x :: "'e set['f]"   and f :: "'e \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'g" assumes "\<And>z. z \<in> set_bset x \<Longrightarrow> f z = g z" shows "map_bset f x = map_bset g x"
update_arg_wf_gives_cases: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "update_arg_wf r" shows "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)"
csol_unique: fixes t1 :: "real"   and x0 :: "'a"   and t2 :: "real"   and t :: "real" assumes "t1 \<in> existence_ivl t0 x0"   and "t2 \<in> existence_ivl t0 x0"   and "t \<in> {t0--t1}"   and "t \<in> {t0--t2}" shows "csol t0 x0 t1 t = csol t0 x0 t2 t"
reach_cong: fixes r :: "state"   and r' :: "state" assumes "r \<in> reach"   and "r = r'" shows "r' \<in> reach"
assms_2: shows "ln (real (Suc n)) \<in> {l<..<u}"
vec_nth_eucl_of_list_eq: fixes M :: "real list"   and i :: "'a" assumes "length M = CARD('a)" shows "eucl_of_list M $ i = M ! index Basis_list (axis i 1)"
strict_prefix_simps_1: fixes xs :: "'a list" shows "strict_prefix xs [] = False"