hcju/isabelleps huggingface dataset:
map_le_iff_add: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "(f \<subseteq>\<^sub>m g) = (\<exists>h. dom f \<inter> dom h = {} \<and> f ++ h = g)"
fundfoldpairs_fundchamber_image: fixes X :: "'b set set"   and C0 :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "ThinChamberComplexManyFoldings X C0"   and "(f, g) \<in> ThinChamberComplexManyFoldings.fundfoldpairs X C0" shows "Abs_permutation (OpposedThinChamberComplexFoldings.induced_automorphism X f g) `\<rightarrow> C0 = g ` C0"
plus_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(f + g) x = f x + g x"
mat1_mult_right: fixes nr :: "nat"   and nc :: "nat"   and m :: "'a list list" assumes "mat nr nc m" shows "mat_mult nr m (mat1 nc) = m"
M1leq: shows "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m d"
add_stalk_def: shows "stk1.add_stalk \<equiv> stk1.add_rel"
at_prm_fresh2: fixes pi :: "('a \<times> 'a) list"   and c :: "'a" assumes "at TYPE('a)"   and "\<forall>(a, b)\<in>set pi. c \<noteq> a \<and> c \<noteq> b" shows "pi \<bullet> c = c"
moebius_pt_moebius_translation: fixes v :: "complex"   and z :: "complex" shows "moebius_pt (moebius_translation v) (of_complex z) = of_complex (z + v)"
strict_trans2: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a < b"   and "b \<le> c" shows "a < c"
\<phi>1s': shows "\<phi>1s' = remdups \<phi>1s"
l2wi_intersect: fixes l :: "('b word \<times> 'b word) list" shows "wordinterval_to_set (l2wi_intersect l) = (\<Inter>(i, j)\<in>set l. {i..j})"
card_range_greater_zero: fixes f :: "'d \<Rightarrow> 'c" assumes "finite (range f)" shows "0 < card (range f)"
wp_UN_subset: fixes act :: "('b \<times> 'b) set"   and A :: "'c \<Rightarrow> 'b set"   and I :: "'c set" shows "(\<Union>i\<in>I. wp act (A i)) \<subseteq> wp act (\<Union> (A ` I))"
low_proc_0_s\<^sub>L\<^sub>0: shows "low_proc_0\<^sub>2 s\<^sub>L\<^sub>0"
simps_2: fixes v :: "nat \<times> (nat \<times> 'a) list"   and va :: "(nat \<times> (nat \<times> 'a) list) list" shows "add_spmat (v # va) [] = v # va"
rel_symp: fixes R1 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "symp R1"   and "symp R2" shows "symp (rel_sum R1 R2)"
is_monoidal_category: shows "monoidal_category (\<cdot>\<^sub>S) T\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C"
simps_1: fixes A :: "'b set" assumes "finite A" shows "esize A = enat (card A)"
equal_deg_sum': fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P"   and "g \<in> carrier P"   and "deg R g < n"   and "deg R f = n" shows "deg R (f \<oplus>\<^bsub>P\<^esub> g) = n"
eqvt_def: fixes Rel :: "('a \<times> 'a) set" shows "eqvt Rel \<equiv> \<forall>x perma. x \<in> Rel \<longrightarrow> perma \<bullet> x \<in> Rel"
le_ceiling_iff: fixes z :: "int"   and x :: "'a" shows "(z \<le> \<lceil>x\<rceil>) = (of_int z - (1::'a) < x)"
distr_uminus_ring_count_space: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_density M (count_space UNIV) f" shows "has_density (distr M (count_space UNIV) uminus) (count_space UNIV) (\<lambda>x. f (- x))"
cnj_cos: fixes z :: "complex" shows "cnj (cos z) = cos (cnj z)"
concI: fixes u :: "'a list"   and A :: "'a list set"   and v :: "'a list"   and B :: "'a list set" assumes "u \<in> A"   and "v \<in> B" shows "u @ v \<in> A \<cdot> B"
prefs_wf'_1: fixes i :: "'agent" assumes "i \<in> agents'" shows "total_preorder_on alts' (R i)"
simps_2: fixes l :: "'b tree"   and uu :: "'b"   and r :: "'b tree" shows "\<Phi> \<langle>l, uu, r\<rangle> = \<Phi> l + \<Phi> r + int (rh l r)"
seqeq_imp_sim: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "s = t" shows "s \<approx> t"
size_in_args: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "s \<in> set (args t)" shows "size s < size t"
SHOmsgVectors_def: fixes A :: "('f, 'h, 'g, 'i) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'f"   and cfg :: "'f \<Rightarrow> 'h"   and HOp :: "'f set"   and SHOp :: "'f set" shows "SHOmsgVectors A r p cfg HOp SHOp \<equiv> {\<mu>. (\<forall>q. (q \<in> HOp) = (\<mu> q \<noteq> None)) \<and> (\<forall>q. q \<in> SHOp \<inter> HOp \<longrightarrow> \<mu> q = Some (sendMsg A r q p (cfg q)))}"
arg_in_args: fixes s :: "('a, 'b) tm" assumes "is_App s" shows "arg s \<in> set (args s)"
path_append: fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and path' :: "(ctx \<times> cld) list"   and path :: "(ctx \<times> cld) list" assumes "find_path_rec_f P ctx cl path' = Some path" shows "\<exists>path''. path = path' @ path''"
one_closed: shows "\<zero>\<^bsub>M\<^esub> \<in> carrier M"
SeqFormP_Neg: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and k1 :: "tm"   and x :: "tm" assumes "atom s \<sharp> (k, s1, k1, x)"   and "atom k \<sharp> (s1, k1, x)" shows "{SeqFormP s1 k1 x} \<turnstile> SyntaxN.Ex k (SyntaxN.Ex s (SeqFormP (Var s) (Var k) (Q_Neg x)))"
SeqAbstFormP_graph_aux_def: shows "SeqAbstFormP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqAbstFormP_aux l s k v sli sl sl' m n smi sm sm' sni sn sn' i x x'. x1 = (v, i, x, x', s, k) \<and> x2 = LstSeqP s k (HPair i (HPair x x')) AND All2 l (SUCC k) (SyntaxN.Ex sli (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sli) (HPair (Var sl) (Var sl'))) IN s AND (AbstAtomicP v (Var sli) (Var sl) (Var sl') OR OrdP (Var sli) AND SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex smi (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sni (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var smi) (HPair (Var sm) (Var sm'))) IN s AND HPair (Var n) (HPair (Var sni) (HPair (Var sn) (Var sn'))) IN s AND (Var sli EQ Var smi AND Var sli EQ Var sni AND Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR Var sli EQ Var smi AND Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR SUCC (Var sli) EQ Var smi AND Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))))) \<and> atom l \<sharp> (s, k, v, sli, sl, sl', m, n, smi, sm, sm', sni, sn, sn') \<and> atom sli \<sharp> (s, v, sl, sl', m, n, smi, sm, sm', sni, sn, sn') \<and> atom sl \<sharp> (s, v, sl', m, n, smi, sm, sm', sni, sn, sn') \<and> atom sl' \<sharp> (s, v, m, n, smi, sm, sm', sni, sn, sn') \<and> atom m \<sharp> (s, n, smi, sm, sm', sni, sn, sn') \<and> atom n \<sharp> (s, smi, sm, sm', sni, sn, sn') \<and> atom smi \<sharp> (s, sm, sm', sni, sn, sn') \<and> atom sm \<sharp> (s, sm', sni, sn, sn') \<and> atom sm' \<sharp> (s, sni, sn, sn') \<and> atom sni \<sharp> (s, sn, sn') \<and> atom sn \<sharp> (s, sn') \<and> atom sn' \<sharp> s)"
WT_plus_oracle_iff: fixes \<I>l :: "('h, 'j) \<I>"   and \<I>r :: "('i, 'k) \<I>"   and left :: "'l \<Rightarrow> 'h \<Rightarrow> ('j \<times> 'l) spmf"   and right :: "'l \<Rightarrow> 'i \<Rightarrow> ('k \<times> 'l) spmf"   and s :: "'l" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd> = (\<I>l \<turnstile>c left s \<surd> \<and> \<I>r \<turnstile>c right s \<surd>)"
pairwiseD: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "pairwise R S"   and "x \<in> S"   and "y \<in> S"   and "x \<noteq> y" shows "R x y"
assms_5: shows "A' B' C' D' E' F' SumA G H I"
lcount_def: fixes n :: "nat" shows "lcount n = card {l. length l = n \<and> valid l}"
lzip_lmap2: fixes xs :: "'b llist"   and f :: "'d \<Rightarrow> 'c"   and ys :: "'d llist" shows "lzip xs (lmap f ys) = lmap (\<lambda>(x, y). (x, f y)) (lzip xs ys)"
U_nonneg: fixes x :: "'s"   and y :: "'s" shows "0 \<le> U x y"
sgn_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less a zero" shows "sgn a = uminus one"
open_ball: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and d :: "real" assumes "class.metric_space dist uniformity open" shows "open {y. dist x y < d}"
vsum_is_minimal: fixes W1 :: "'c set"   and W2 :: "'c set"   and W :: "'c set" assumes "subspace K W1 V"   and "subspace K W2 V"   and "subspace K W V" shows "(subspace_sum W1 W2 \<subseteq> W) = (W1 \<subseteq> W \<and> W2 \<subseteq> W)"
elims: fixes x :: "('c, 'd) SkewBinomialTree list"   and y :: "('c \<times> 'd) multiset" assumes "queue_to_multiset x = y" obtains "x = []" and "y = {#}"   | t :: "('c, 'd) SkewBinomialTree" and q :: "('c, 'd) SkewBinomialTree list" where "x = t # q" and "y = tree_to_multiset t + queue_to_multiset q"
step_Fault: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and f :: "'c" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Fault f" shows "s' = Fault f"
soundI: fixes P :: "'b \<Rightarrow> real" assumes "Expectations.bounded P"   and "nneg P" shows "sound P"
hom_int_pow: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and n :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) neg (x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n) = hom_induced (int n) (nsphere n) (equator n) (nsphere n) (equator n) neg x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n"
sub_commutative: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - y = - y \<squnion> - x"
cmp_simps_5: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "src f = trg g" shows "S.cod (S.cmp\<^sub>U\<^sub>P (f, g)) = S.UP (f \<star> g)"
inorder_paint: fixes c :: "color"   and t :: "('b \<times> color) tree" shows "Tree2.inorder (paint c t) = Tree2.inorder t"
times_sat_def: fixes x :: "'a sat"   and y :: "'a sat" shows "x * y = Abs_sat' (nat_of x * nat_of y)"
cSup_eq_non_empty: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and 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 X :: "'b set"   and a :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x a"   and "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> less_eq x y) \<Longrightarrow> less_eq a y" shows "Sup X = a"
supp_rel_tNot: fixes t :: "('e, 'f, 'g) Tree" shows "supp_rel (=\<^sub>\<alpha>) (tNot t) = supp_rel (=\<^sub>\<alpha>) t"
PO_m1_inv3r_init: shows "reach m1 \<subseteq> m1_inv3r_init"
nat_set_cset_collapse_def: shows "nat_set_cset_collapse = inv bit_seq_of_nat_set \<circ> seq_inj \<circ> cset2seq \<circ> (`\<^sub>c) bit_seq_of_nat_set"
TP_prim_Sound: fixes \<Sigma> :: "((Class \<times> Method) \<times> nat) list"   and p :: "Prim"   and n :: "nat"   and l :: "int"   and code :: "(int \<times> Instr) list"   and code1 :: "(int \<times> Instr) list"   and l1 :: "int"   and C :: "Class"   and m :: "Method"   and T :: "(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"   and MI :: "(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"   and Anno :: "(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)) list"   and 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"   and nn :: "nat"   and k :: "nat" assumes "(\<Sigma>, p, n) \<in> TP_prim"   and "Sig_good \<Sigma>"   and "(l, code, p, code1, l1) \<in> compilePrim"   and "MST\<down>(C, m) = Some (T, MI, Anno)"   and "Segment C m l l1 code1"   and "derivAssum G C m l1 (Cachera nn)"   and "k = n + nn" shows "deriv G C m l (Cachera k)"
lookup_except_eq_zeroI: fixes t :: "'b"   and S :: "'b set"   and p :: "'b \<Rightarrow>\<^sub>0 'c" assumes "t \<in> S" shows "lookup (except p S) t = (0::'c)"
assoc_inv: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "inverse_arrows (C.assoc a b c) (C.assoc' a b c)"
strict_order_iff: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subset># b) = (a = a \<inter># b \<and> a \<noteq> b)"
simps_7: fixes f1 :: "'a"   and f2 :: "(nat \<Rightarrow> InfTree \<times> 'a) \<Rightarrow> 'a"   and x2 :: "nat \<Rightarrow> InfTree" shows "rec_InfTree f1 f2 (InfTree.Node x2) = f2 ((\<lambda>InfTree. (InfTree, rec_InfTree f1 f2 InfTree)) \<circ> x2)"
simps_1: fixes q :: "('c monom \<times> 'd) list" shows "check_poly_eq [] q = (q = [])"
simps_1: fixes x21 :: "'b"   and x22 :: "('a, 'b) Expression"   and y21 :: "'b"   and y22 :: "('a, 'b) Expression" shows "(x21 E@ x22 = y21 E@ y22) = (x21 = y21 \<and> x22 = y22)"
lt_lower_eq_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" shows "(punit.lt (punit.lower p v) = punit.lt p) = (punit.lt p = punit.min_term \<or> punit.lt p \<prec> v)"
inf_mono: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "a \<le> c"   and "b \<le> d" shows "a \<sqinter> b \<le> c \<sqinter> d"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> synth H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>agt. a = Agent agt) \<or> (\<exists>n. a = Number n) \<or> (\<exists>X. a = Hash X \<and> X \<in> synth H) \<or> (\<exists>X Y. a = \<lbrace>X, Y\<rbrace> \<and> X \<in> synth H \<and> Y \<in> synth H) \<or> (\<exists>X K. a = Crypt K X \<and> X \<in> synth H \<and> Key K \<in> H))"
Nfunc_iNJTr: fixes g :: "nat \<Rightarrow> 'e"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "inj_on g {i. i \<le> n}"   and "i \<le> n"   and "j \<le> n"   and "i < j" shows "g i \<noteq> g j"
ord0_less_def: fixes x :: "ord0"   and y :: "ord0" shows "(x < y) = ((x, y) \<in> ord0_leq O ord0_prec\<^sup>+)"
space_eq: fixes p :: "real"   and n :: "nat" assumes "edge_space p" shows "space (edge_space.P n p) = Pow (edge_space.S_edges n)"
braun_add_hi: fixes t :: "'b tree"   and x :: "'b" assumes "braun t" shows "braun (update1 (Suc (size t)) x t)"
tensor_from_vec_def: fixes d :: "nat list"   and v :: "'b list" shows "tensor_from_vec d v = Abs_tensor (d, v)"
assms_1: shows "Ea \<subseteq> Pow (space A)"
points_in_chain: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and Q :: "'a set" assumes "[f[x .. y ..  z]Q]" shows "x \<in> Q \<and> y \<in> Q \<and> z \<in> Q"
simps_6: fixes f1 :: "nat \<Rightarrow> tf \<Rightarrow> t list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f2 :: "tf \<Rightarrow> host \<Rightarrow> 'a"   and x21 :: "tf"   and x22 :: "host" shows "(case Func_host x21 x22 of Func_native x xa xb xc \<Rightarrow> f1 x xa xb xc | Func_host x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
extend_act_inverse: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and act :: "('c \<times> 'c) set" assumes "Extend h" shows "project_act h (extend_act h act) = act"
Abs_Ainteg_inject: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "x \<in> Ainteg"   and "y \<in> Ainteg" shows "(Abs_Ainteg x = Abs_Ainteg y) = (x = y)"
cone_def: fixes hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set" shows "cone hU = (*) (fst hU) ` P[snd hU]"
K6_imp_Gets: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and T3 :: "nat"   and evs :: "event list" assumes "Says B A (Crypt servK (Number T3)) \<in> set evs"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "\<exists>Ts X. Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>, X\<rbrace> \<in> set evs"
assms_1: fixes i :: "nat" assumes "p i \<noteq> 0" shows "1 \<le> i \<and> i \<le> n"
integral_cong_AE: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and g :: "'d \<Rightarrow> 'e" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L M g"
wf_prog_def: fixes G :: "prog" shows "wf_prog G = (let is = ifaces G; cs = classes G in ObjectC \<in> set cs \<and> (\<forall>m\<in>set Object_mdecls. accmodi m \<noteq> Package) \<and> (\<forall>xn. SXcptC xn \<in> set cs) \<and> (\<forall>i\<in>set is. wf_idecl G i) \<and> unique is \<and> (\<forall>c\<in>set cs. wf_cdecl G c) \<and> unique cs)"
\<pi>_def: shows "\<pi> = map (\<lambda>s. (s, L s)) sl @- sconst (toSink (last sl), L (toSink (last sl)))"
elims: fixes x :: "('o, 'a) arr"   and y :: "'o" assumes "Dom x = y" obtains A :: "'o" and uu_ :: "'o" and uv_ :: "'a" where "x = MkArr A uu_ uv_" and "y = A"   | "x = Null" and "y = undefined"
x_1: shows "x \<in> cbox (- 1) 1"
simps: fixes \<alpha> :: "'e \<Rightarrow> 'e"   and \<beta> :: "'d \<Rightarrow> 'd"   and s :: "'e \<Rightarrow> (('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('d \<Rightarrow> ('e, 'd, 'f) val option) \<times> ('e, 'd, 'f) expr) option"   and r :: "'e" shows "\<R>\<^sub>G \<alpha> \<beta> s r = s (inv \<alpha> r) \<bind> (\<lambda>ls. Some (\<R>\<^sub>L \<alpha> \<beta> ls))"
Fence: fixes \<O>s :: "nat set list"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and i :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" shows "\<O>s,\<R>s,i\<turnstile> (Fence # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
fv_delete_subset: fixes v :: "'b"   and \<Gamma> :: "('b \<times> 'c) list" shows "fv (delete v \<Gamma>) \<subseteq> fv \<Gamma>"
assms: fixes i :: "'d" assumes "i \<in> S" shows "borel_adapt_stoch_proc F (A i)"
x_div_x: fixes x :: "nat" assumes "0 < x" shows "x div x = 1"
pure_the_pure: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> assn" assumes "is_pure R" shows "pure (the_pure R) = R"
elims_2: fixes x :: "ty" assumes "is_Array x" obtains T :: "ty" where "x = T\<lfloor>\<rceil>"
encode_problem_parallel_complete_iv_a: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and C :: "sat_plan_variable literal set" assumes "is_parallel_solution_for_problem \<Pi> \<pi>"   and "\<forall>k op. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<longrightarrow> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"   and "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)"   and "\<forall>v l. length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<le> l \<and> l < length \<pi> + 1 \<longrightarrow> \<A> (State l (index (\<Pi>\<^sub>\<V>) v)) = \<A> (State (length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1) (index (\<Pi>\<^sub>\<V>) v))"   and "C \<in> \<Union> (\<Union>(k, v)\<in>{0..<length \<pi>} \<times> set (\<Pi>\<^sub>\<V>). {{{(State k (index (\<Pi>\<^sub>\<V>) v))\<^sup>+, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<inverse>} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (add_effects_of op)}}})" shows "clause_semantics \<A> C"
simps_13: fixes v :: "fm"   and va :: "fm" shows "disjuncts (And v va) = [And v va]"
iMOD_iFROM_eq_conv: fixes n :: "nat"   and r :: "nat"   and m :: "nat" shows "([n\<dots>] = [ r, mod m ]) = (n = r \<and> m = Suc 0)"
is_valid_problem_strips_operator_variable_sets_1: fixes \<Pi> :: "'b strips_problem"   and op :: "'b strips_operator" assumes "is_valid_problem_strips \<Pi>"   and "op \<in> set (\<Pi>\<^sub>\<O>)" shows "set (precondition_of op) \<subseteq> set (\<Pi>\<^sub>\<V>)"
subset_closed_def: fixes C :: "'b set set" shows "subset_closed C \<equiv> \<forall>S\<in>C. \<forall>s\<subseteq>S. s \<in> C"
alt_put: fixes s :: "'s"   and m :: "'m"   and m' :: "'m" shows "alt (put s m) (put s m') = put s (alt m m')"
range_TM_tm_const: fixes e :: "nat \<Rightarrow> real"   and c :: "float" shows "range_tm e (tm_const c) = interval_of (real_of_float c)"
zcf_eq_zero_unique: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "g \<in> carrier P \<and> f = X_poly R \<otimes>\<^bsub>P\<^esub> g"   and "h \<in> carrier P \<and> f = X_poly R \<otimes>\<^bsub>P\<^esub> h" shows "h = g"
simps: fixes l :: "name"   and s :: "tm"   and k :: "tm"   and t :: "tm"   and sl :: "name"   and m :: "name"   and n :: "name"   and sm :: "name"   and sn :: "name" assumes "atom l \<sharp> (s, k, t, sl, m, n, sm, sn)"   and "atom sl \<sharp> (s, k, t, m, n, sm, sn)"   and "atom m \<sharp> (s, k, t, n, sm, sn)"   and "atom n \<sharp> (s, k, t, sm, sn)"   and "atom sm \<sharp> (s, k, t, sn)"   and "atom sn \<sharp> (s, k, t)" shows "SeqFormP s k t = LstSeqP s k t AND All2 n (SUCC k) (SyntaxN.Ex sn (HPair (Var n) (Var sn) IN s AND (AtomicP (Var sn) OR SyntaxN.Ex m (SyntaxN.Ex l (SyntaxN.Ex sm (SyntaxN.Ex sl (Var m IN Var n AND Var l IN Var n AND HPair (Var m) (Var sm) IN s AND HPair (Var l) (Var sl) IN s AND MakeFormP (Var sn) (Var sm) (Var sl))))))))"
mult_iarray_def: fixes A :: "'b iarray"   and q :: "'b" shows "mult_iarray A q = IArray.of_fun (\<lambda>n. q * A !! n) (IArray.length A)"
prod_belowI: fixes p :: "'a \<times> 'b"   and q :: "'a \<times> 'b" assumes "fst p \<sqsubseteq> fst q"   and "snd p \<sqsubseteq> snd q" shows "p \<sqsubseteq> q"
that: assumes "getFrom e \<in> set (\<alpha>n g)"   and "getTo e \<in> set (\<alpha>n g)" shows "thesis"
step: fixes x :: "'a node \<times> 'a node set" assumes "(expand_init < name (fst x) \<and> (\<forall>nm\<in>incoming (fst x). nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` snd x) \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)) \<and> (\<forall>nd\<in>snd x. expand_init < name nd \<and> (\<forall>nm\<in>incoming nd. nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` snd x))" shows "f x \<le> SPEC (\<lambda>x'. \<forall>nd\<in>snd x'. expand_init < name nd \<and> (\<forall>nm\<in>incoming nd. nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` snd x'))"