hcju/isabelleps huggingface dataset:
split3_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b"   and p :: "'c \<times> 'd \<times> 'e" shows "split3 f p \<equiv> case p of (a, b, c) \<Rightarrow> f a b c"
pos_assign_monom_list: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and m :: "('c \<times> nat) list" assumes "pos_assign \<alpha>" shows "(0::'d) \<le> eval_monom_list \<alpha> m"
normalize_idem: fixes a :: "'a"   and b :: "'a" shows "normalize (a \<^bold>* b) = a \<^bold>* b"
nat_euclid_pyth_triples: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a\<^sup>2 + b\<^sup>2 = c\<^sup>2"   and "coprime a b"   and "odd a" shows "\<exists>p q. a = p\<^sup>2 - q\<^sup>2 \<and> b = 2 * p * q \<and> c = p\<^sup>2 + q\<^sup>2 \<and> coprime p q"
card_bijections_domain_permutation_eq_1: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "card A = card B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_permutation A B) = 1"
distinct_39: fixes x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp"   and x61 :: "'d"   and x62 :: "'a \<Rightarrow> nat set"   and x63 :: "'a \<Rightarrow> 'b"   and x64 :: "('a, 'b, 'c, 'd) seqp" shows "x31 \<oplus> x32 \<noteq> {x61}groupcast(x62, x63) . x64"
subadditive_converges_bounded': fixes u :: "nat \<Rightarrow> real"   and N0 :: "nat" assumes "eventually_subadditive u N0"   and "bdd_below {u n / real n |n. N0 < n}" shows "(\<lambda>n. u n / real n) \<longlonglongrightarrow> Inf {u n / real n |n. N0 < n}"
NSDERIVD4: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and D :: "real"   and h :: "real star" assumes "NSDERIV f x :> D"   and "h \<in> Infinitesimal" shows "(*f* f) (hypreal_of_real x + h) - hypreal_of_real (f x) \<approx> hypreal_of_real D * h"
set_insert: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" obtains B :: "'b set" where "A = Set.insert x B" and "x \<notin> B"
int_pow_eq_id: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier (G\<lparr>carrier := generate G {a}\<rparr>)" shows "(x [^]\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> i = \<one>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub>) = (int (A.ord x) dvd i)"
simps: fixes \<Gamma> :: "'b formula set"   and a :: "'b formula" shows "((\<Gamma>::'b formula set) \<turnstile>\<^sub>H (a::'b formula)) = ((\<exists>F::'b formula. a = F \<and> F \<in> \<Gamma>) \<or> (\<exists>(F::'b formula) G::'b formula. a = G \<and> (\<Gamma> \<turnstile>\<^sub>H F) \<and> (\<Gamma> \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G)))"
D_mono': fixes e :: "('b, 'c, 'd) exp"   and A :: "'b set option"   and A' :: "'b set option" assumes "\<D> e A"   and "A \<sqsubseteq> A'" shows "\<D> e A'"
csorted_list_of_set_split: fixes P :: "'c list \<Rightarrow> bool"   and A :: "'c set" shows "P (csorted_list_of_set A) = ((\<forall>xs. ID ccompare \<noteq> None \<longrightarrow> finite A \<longrightarrow> A = set xs \<longrightarrow> distinct xs \<longrightarrow> linorder.sorted cless_eq xs \<longrightarrow> P xs) \<and> (ID ccompare = None \<or> infinite A \<longrightarrow> P undefined))"
effective_matrix_tensor_elements: fixes i :: "nat"   and M1 :: "'a list list"   and M2 :: "'a list list"   and j :: "nat" assumes "(i < row_length M1 * row_length M2 \<and> j < length M1 * length M2) \<and> mat (row_length M1) (length M1) M1 \<and> mat (row_length M2) (length M2) M2" shows "(M1 \<otimes> M2) ! j ! i = M1 ! (j div length M2) ! (i div row_length M2) * M2 ! (j mod length M2) ! (i mod row_length M2)"
remove1_append: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "remove1 x (xs @ ys) = (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
ExecSteps: shows "\<forall>i<length exec' - 1. exec' ! i \<turnstile> trace' ! i \<mapsto> exec' ! (i + 1)"
strict_trans2: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "strict_prefix b a"   and "Sublist.prefix c b" shows "strict_prefix c a"
vle_def: fixes x :: "V"   and y :: "V" shows "x \<unlhd> y \<equiv> \<exists>z. x + z = y"
left_neutral: fixes a :: "'a" shows "1 \<cdot> a = a"
mat_of_rows_def: fixes n :: "nat"   and rs :: "'a vec list" shows "mat_of_rows n rs = mat (length rs) n (\<lambda>(i, j). rs ! i $ j)"
symp_combine_per: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "symp R" shows "symp (combine_per P R)"
Cons_tl_drop_update: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i < length xs" shows "x # tl (drop i xs) = drop i (xs[i := x])"
poly_mult_var'_2: fixes p :: "'a list list" assumes "set p \<subseteq> carrier (K [X])" shows "UP.poly_mult p X\<^bsub>K [X]\<^esub> = UP.normalize (p @ [\<zero>\<^bsub>K [X]\<^esub>])"
cases: fixes x :: "'b acom" obtains P :: "'b" where "x = SKIP {P__}"   | xa :: "char list" and e :: "aexp" and P :: "'b" where "x = xa ::= e {Pa__}"   | c1 :: "'b acom" and c2 :: "'b acom" where "x = c1__;; c2__"   | b :: "bexp" and c1 :: "'b acom" and c2 :: "'b acom" and P :: "'b" where "x = IF b__ THEN c1 ELSE c2 {Pb__}"   | Inv :: "'b" and b :: "bexp" and c :: "'b acom" and P :: "'b" where "x = {Inv} WHILE b DO c {P}"
t2: shows "\<forall>y. clinear (\<lambda>x. h x y)"
exposed_face_of_parallel: fixes T :: "'b set"   and S :: "'b set" shows "(T exposed_face_of S) = (T face_of S \<and> (\<exists>a b. S \<subseteq> {x. a \<bullet> x \<le> b} \<and> T = S \<inter> {x. a \<bullet> x = b} \<and> (T \<noteq> {} \<longrightarrow> T \<noteq> S \<longrightarrow> a \<noteq> (0::'b)) \<and> (T \<noteq> S \<longrightarrow> (\<forall>w\<in>affine hull S. w + a \<in> affine hull S))))"
transpos_surjec: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "surj_to (transpos i j) {i. i \<le> n} {i. i \<le> n}"
crit_def: shows "crit = (\<lambda>t f. f \<in> pmdl F \<and> f \<in> dgrad_p_set d m \<and> f \<noteq> 0 \<and> t = lt f)"
Sup_action: fixes S :: "('c, 'd) action set" shows "Sup S \<equiv> if S = {} then bot else action_of (\<lambda>A. A \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` S})"
upd_set'_aux4: fixes k :: "'b" assumes "k \<notin> fst ` A" shows "upd_set' Mapping.empty d f {b. (k, b) \<in> A} = Mapping.empty"
simps_3: fixes v :: "trm" shows "Timeso (Aterm v) undeft = undeft"
simps: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" shows "ord.lexordp less a1 a2 = ((\<exists>y ys. a1 = [] \<and> a2 = y # ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> less x y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> less x y \<and> \<not> less y x \<and> ord.lexordp less xs ys))"
mr_coset_def: fixes a :: "'e"   and M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set" shows "mr_coset a M H = a \<uplus>\<^bsub>M\<^esub> H"
distrib_imp1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.lattice inf less_eq less sup"   and "\<And>x y z. inf x (sup y z) = sup (inf x y) (inf x z)" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
nabla_separation: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star>" shows "\<nabla>(x + y) = \<nabla>x +  |x\<^sup>\<star>\<rangle> \<nabla>y"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (F\<lparr>ArrVal\<rparr>)" shows "F\<lparr>ArrVal\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (F\<lparr>ArrVal\<rparr>)"
iarray_to_vec_def: fixes A :: "'c iarray" shows "iarray_to_vec A = (\<chi>i. A !! mod_type_class.to_nat i)"
subprob_space_axioms: shows "subprob_space (M \<bind> f)"
count_roots_above_def: fixes p :: "real poly"   and a :: "real" shows "count_roots_above p a = (if p = 0 then 0 else let ps = sturm_squarefree p in sign_changes ps a - sign_changes_inf ps)"
x: shows "(x solves_ode f) {t0--t1} X"
rquot_well_formed_aut: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat" assumes "wf_dfa M n" shows "wf_dfa (rquot M n) n"
algebraic_power_iff: fixes n :: "nat"   and x :: "'a" assumes "0 < n" shows "algebraic (x ^ n) = algebraic x"
i_cut_Int_conv_2: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t = I \<inter> {..<t}"
RID_skip_r: fixes x :: "'b \<Longrightarrow> 'c" assumes "vwb_lens x" shows "RID x II = II"
list_allI: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "list_all P xs"
MASTER_BOUND''_def: fixes a :: "real"   and x :: "nat" shows "MASTER_BOUND'' a x = real x powr a"
typeof_lit_typeof: fixes v :: "val"   and T :: "ty"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "typeof v = \<lfloor>T\<rfloor>" shows "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>"
special_points_on_y_axis'_1: shows "on_circline y_axis 0\<^sub>h"
L3_5v_lemma1: fixes S :: "'a com"   and s :: "'a"   and T :: "'a com"   and t :: "'a"   and n :: "nat" assumes "(S, s) -Pn\<rightarrow> (T, t)"   and "S = \<Omega>" shows "\<nexists>Rs. T = Parallel Rs \<and> All_None Rs"
lcoset_refl: fixes A :: "'a set"   and a :: "'a" assumes "(0::'a) \<in> A" shows "a \<in> a +o A"
relative_ending: fixes t :: "nat \<Rightarrow> 'a"   and R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and X :: "'a set" assumes "\<forall>i. (t i, t (Suc i)) \<in> R \<union> S"   and "t 0 \<in> X"   and "SN_on (S\<^sup>* O R O S\<^sup>*) X" shows "\<exists>j. \<forall>i\<ge>j. (t i, t (Suc i)) \<in> S - R"
assms_7: shows "has_greatest_fixpoint (g \<circ> h)"
mset_drop_upto: fixes a :: "nat"   and N :: "'a list" shows "mset (drop a N) = (!) N `# mset_set {a..<length N}"
inverse2: fixes f :: "'b \<Rightarrow> 'a set" shows "U (C f) = f"
assms: fixes a :: "'a"   and b :: "'b"   and c :: "'c"   and d :: "'d" assumes "ab = (a, b)"   and "cd = (c, d)" shows "f a b c d \<le> ESPEC \<Phi> \<Psi>"
gcd_neg1_int: fixes x :: "int"   and y :: "int" shows "gcd (- x) y = gcd x y"
abs_eq_iff: fixes x :: "'d + 'd"   and y :: "'d + 'd" shows "(abs_opt x = abs_opt y) = ignore_Inl x y"
inject_3: fixes x2 :: "trm"   and x1 :: "trm"   and y2 :: "trm"   and y1 :: "trm" shows "(App x2 x1 = App y2 y1) = (x2 = y2 \<and> x1 = y1)"
impl_mp1: fixes P :: "'b upred"   and Q :: "'b upred" shows "(P \<and> (P \<Rightarrow> Q)) = (P \<and> Q)"
zip_map_map: fixes f :: "'d \<Rightarrow> 'b"   and xs :: "'d list"   and g :: "'e \<Rightarrow> 'c"   and ys :: "'e list" shows "zip (map f xs) (map g ys) = map2 (\<lambda>x y. (f x, g y)) xs ys"
cong_scalar_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b"   and d :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (times b d) (times c d) a"
GC_tautology_step: fixes A :: "'a"   and C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" assumes "Neg A \<in># C"   and "Pos A \<in># C" shows "lclss_of_state (N \<union> {C}, P, Q) \<leadsto>GC lclss_of_state (N, P, Q)"
that: shows "X \<subseteq> {n. real n \<le> x}"
rts_def: shows "rts = {x. \<exists>q ri. (q, ri) \<in> orig \<and> poly (real_of_int_poly q) x = 0}"
assms: shows "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong>"
has_integral_0_cbox_imp_0: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "continuous_on (cbox a b) f"   and "\<And>x. x \<in> box a b \<Longrightarrow> 0 \<le> f x"   and "(f has_integral 0) (cbox a b)"   and "box a b \<noteq> {}"   and "x \<in> cbox a b" shows "f x = 0"
assms_1: fixes c :: "command" shows "wr_com c = {}"
lt_def: shows "lt = (\<Lambda> x y. case compare\<cdot>x\<cdot>y of LT \<Rightarrow> TT | EQ \<Rightarrow> FF | GT \<Rightarrow> FF)"
execn_Skip': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and t :: "('e, 'f) xstate"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>Skip,t\<rangle> =n\<Rightarrow> t"
indep_sets_def: fixes F :: "'c \<Rightarrow> 'a set set"   and I :: "'c set" shows "indep_sets F I = ((\<forall>i\<in>I. F i \<subseteq> events) \<and> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> (\<forall>A\<in>Pi J F. prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j)))))"
cexpr_subst_val_def: fixes e :: "cexpr"   and v :: "val" shows "cexpr_subst_val e v \<equiv> cexpr_subst_val_aux 0 e v"
alw_holds2: fixes P :: "'a \<Rightarrow> bool"   and ss :: "'a stream" shows "alw (holds P) ss = (P (shd ss) \<and> alw (holds P) (stl ss))"
lt_map_scale: fixes c :: "'b"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "c \<noteq> (0::'b)" shows "i.lpp (c \<cdot> p) = i.lpp p"
HARoot_ran_CompFun: fixes HA :: "('a, 'b, 'c) hierauto" shows "HARoot HA \<notin> \<Union> (ran (CompFun HA))"
basis_enum_of_vec_span: fixes Y :: "complex vec set" assumes "length canonical_basis = dA"   and "Y \<subseteq> carrier_vec dA" shows "basis_enum_of_vec ` local.span Y = cspan (basis_enum_of_vec ` Y)"
eqpoll_lift: fixes x :: "V"   and y :: "V" shows "elts (lift x y) \<approx> elts y"
univ: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "type_definition Rep Abs A" shows "UNIV = Abs ` A"
expand: fixes resumption :: "('d, 'e, 'f) resumption"   and resumption' :: "('d, 'e, 'f) resumption" assumes "is_Done resumption = is_Done resumption'"   and "\<lbrakk>is_Done resumption; is_Done resumption'\<rbrakk> \<Longrightarrow> result resumption = result resumption'"   and "\<lbrakk>\<not> is_Done resumption; \<not> is_Done resumption'\<rbrakk> \<Longrightarrow> output resumption = output resumption' \<and> resume resumption = resume resumption'" shows "resumption = resumption'"
ts_tuple_rel': shows "ts_tuple_rel (set auxlist) = {tas \<in> ts_tuple_rel (set (linearize data_prev') \<union> set (linearize data_in')). valid_tuple tuple_since tas}"
simps_3: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and a :: "'c"   and ts :: "'b FOL_Fitting.term list" shows "eval e f g (Pred a ts) = g a (evalts e f ts)"
usubst_taboos: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game" assumes "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg" shows "U \<union> BVG (the (snd (usubstappp \<sigma> U \<alpha>))) \<subseteq> fst (usubstappp \<sigma> U \<alpha>)"
canonical_tINT: fixes t :: "lam" assumes "[] \<turnstile> t : tINT"   and "val t" shows "\<exists>n. t = NUM n"
select_convs_1: fixes CatDom :: "('d, 'e, 'f) Category_scheme"   and CatCod :: "('g, 'h, 'i) Category_scheme"   and MapM :: "'e \<Rightarrow> 'h"   and more :: "'j" shows "CatDom \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = CatDom"
l_get_ancestors_axioms: shows "l_get_ancestors get_ancestors_si"
assms_2: shows "ccw'.sortedP 0 (P # Q # Ps)"
finsum_vec_def: fixes ty :: "'b itself"   and n :: "nat" shows "finsum_vec ty n = finprod (monoid_vec ty n)"
fw_mono': fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "fw m n k i j i j \<le> m i j"
cod_mkArr: fixes A :: "'t set"   and B :: "'t set"   and F :: "'t \<Rightarrow> 't" assumes "S'.arr (S'.mkArr A B F)" shows "S'.cod (S'.mkArr A B F) = S'.mkIde B"
linorder_linear: fixes x :: "'g"   and y :: "'g" shows "x \<le> y \<or> y \<le> x"
assoc: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" shows "(a @ b) @ c = a @ b @ c"
subterms_inclusion_def: fixes E1 :: "'a trm set"   and E2 :: "'a trm set" shows "subterms_inclusion E1 E2 = (\<forall>x1\<in>E1. \<exists>x2\<in>E2. occurs_in x1 x2)"
f8: fixes A :: "'a set"   and Aa :: "'a set" shows "- A = - (A - Aa) - (A - - Aa)"
eval_monom: fixes b :: "'a"   and a :: "'a"   and n :: "nat" assumes "b \<in> carrier R"   and "a \<in> carrier R" shows "local.eval (local.monom b n) a = b \<otimes> a [^] n"
heap_insert: fixes q :: "'a tree"   and x :: "'a" assumes "heap q" shows "heap (insert x q)"
INF_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
Chebyshev_sum_upper: fixes n :: "nat"   and a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a" assumes "\<And>i j. \<lbrakk>i \<le> j; j < n\<rbrakk> \<Longrightarrow> a i \<le> a j"   and "\<And>i j. \<lbrakk>i \<le> j; j < n\<rbrakk> \<Longrightarrow> b j \<le> b i" shows "of_nat n * (\<Sum>k = 0..<n. a k * b k) \<le> sum a {0..<n} * sum b {0..<n}"
llist_ccpo: shows "class.ccpo lSup (\<sqsubseteq>) (mk_less (\<sqsubseteq>))"
distf\<^sub>2: shows "distinct (vertices f\<^sub>2)"
D_def: fixes n :: "nat"   and m :: "nat"   and fs :: "int vec list" shows "LLL.D n m fs = nat (prod (LLL.d n fs) {..<m})"
solvable_iff_trivial_derived_seq: shows "solvable G = (\<exists>n. (derived G ^^ n) (carrier G) = {\<one>})"
set_union_minus_same1: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B - B = A - B"
pathsCard_paths_nxt: fixes f :: "Vars.var ttree"   and x :: "Vars.var" shows "pathsCard (paths (nxt f x)) \<sqsubseteq> record_call x\<cdot>(pathsCard (paths f))"
isFDERIV_map_Var: fixes ds :: "nat list"   and N :: "nat"   and xs :: "nat list"   and vs :: "real list" assumes "length ds = N"   and "length xs = N" shows "isFDERIV N xs (map Var ds) vs"
decomp_snd_exists: fixes t :: "('fun, 'var) Term.term" shows "\<exists>D. decomp t = send\<langle>t\<rangle>\<^sub>s\<^sub>t # D"
unifiers_def: fixes E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" shows "unifiers E = {\<sigma>. \<forall>p\<in>E. Unifier \<sigma> (fst p) (snd p)}"
cont_compose: fixes c :: "'g \<Rightarrow> 'h"   and f :: "'i \<Rightarrow> 'g" assumes "cont c"   and "cont f" shows "cont (\<lambda>x. c (f x))"
sqnf_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "rt \<noteq> update rt dip (dsn, dsk, flg, hops, sip, {})" shows "sqnf (update rt dip (dsn, dsk, flg, hops, sip, {})) dip = dsk"
cis_neq_zero: fixes a :: "real" shows "cis a \<noteq> 0"
observable2_equiv_observable: fixes ob :: "'a \<Rightarrow> 'b"   and P :: "'a set" shows "observable2 ob P = observable ob P"
sorted_list_of_set_unique: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and l :: "'b list" assumes "class.linorder less_eq less"   and "finite A" shows "(linorder.strict_sorted less l \<and> set l = A \<and> length l = card A) = (linorder.sorted_list_of_set less_eq A = l)"
hr_comp_def: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> assn"   and R2 :: "('e \<times> 'g) set"   and a :: "'g"   and c :: "'f" shows "hr_comp R1 R2 a c \<equiv> \<exists>\<^sub>Ab. R1 b c * \<up> ((b, a) \<in> R2)"
disc_25: fixes x51 :: "Core.term"   and x52 :: "Core.term" shows "is_App (x51 $ x52)"
assms_2: assumes "\<phi> \<pi>l"   and "alw \<phi> (map stl \<pi>l)" shows "\<chi>"
simps_1: fixes v :: "nat" shows "opposite (Pos v) = Neg v"
incl_Rel_components_1: fixes A :: "V"   and B :: "V" shows "incl_Rel A B\<lparr>ArrVal\<rparr> = vid_on A"
bernoulli_Not: shows "map_pmf Not (bernoulli_pmf (1 / 2)) = bernoulli_pmf (1 / 2)"
fusion: fixes \<phi> :: "i \<Rightarrow> bool" assumes "\<exists>x. \<phi> x" shows "\<exists>z. \<forall>y. O y z = (\<exists>x. \<phi> x \<and> O y x)"
simps_23: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b" shows "AExp_Lexorder.less_aexp_aux (Times e1 e2) (V v) = False"
rquot_memb_def: shows "rquot_memb = (\<lambda>x l. l ! x)"
config_dist: shows "\<forall>n. \<forall>xa\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take n qs)). distinct (fst xa)"
assms_4: shows "deriv f \<xi> \<noteq> 0"
vimage_algebra_sigma: fixes X :: "'b set set"   and \<Omega>' :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and \<Omega> :: "'c set" assumes "X \<subseteq> Pow \<Omega>'"   and "f \<in> \<Omega> \<rightarrow> \<Omega>'" shows "vimage_algebra \<Omega> f (sigma \<Omega>' X) = sigma \<Omega> {f -` A \<inter> \<Omega> |A. A \<in> X}"
redT_updTs_upd: fixes ts :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option"   and t :: "'c"   and xw :: "'a \<times> 'b \<Rightarrow>f nat"   and tas :: "('c, 'a, 'd) new_thread_action list"   and xw' :: "'a \<times> 'b \<Rightarrow>f nat" assumes "ts t = \<lfloor>xw\<rfloor>"   and "thread_oks ts tas" shows "redT_updTs ts tas(t \<mapsto> xw') = redT_updTs (ts(t \<mapsto> xw')) tas"
restrict_count_space: fixes B :: "'d set"   and A :: "'d set" shows "restrict_space (count_space B) A = count_space (A \<inter> B)"
distinct_1: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" shows "\<langle>\<rangle> \<noteq> \<langle>x21, x22, x23\<rangle>"
s_def: shows "s \<equiv> smap g nats ||| smap P nats"
l2_skr_def: fixes R :: "fid_t"   and K :: "msg" shows "l2_skr R K \<equiv> {(s, s'). R \<noteq> test \<and> R \<notin> partners \<and> run_ended (progress s R) \<and> guessed_frame R xsk = Some K \<and> s' = s\<lparr>ik := {K} \<union> ik s\<rparr>}"
compact_imp_seq_compact: fixes U :: "'b set" assumes "compact U" shows "seq_compact U"
Contra': fixes A :: "fm"   and H :: "fm set" assumes "insert A H \<turnstile> Neg A" shows "H \<turnstile> Neg A"
lm071: fixes a :: "('c \<times> 'd set) set" shows "Domain (pseudoAllocation a) \<subseteq> Domain a"
mem_Collect_eq: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "(a \<in> rel_ext P) = P a"
CConcEnd2: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "((int \<times> int) set set, int) Combinators"   and xs :: "((int \<times> int) set set, int) Combinators list" assumes "x \<in> dom (C a)" shows "C (list2FWpolicy (xs @ [a])) x = C a x"
the_elem_hom: shows "(\<lambda>X. the_elem (h ` X)) \<in> ring_hom (R Quot a_kernel R S h) S"
unit_factor_0: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a" assumes "class.semidom_divide_unit_factor divide plus minus zero times one unit_factor" shows "unit_factor zero = zero"
\<chi>2: shows "\<chi>2 \<in> fmla"
NotNotD: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<not>\<^bold>\<not>\<phi> in v]" shows "[\<phi> in v]"
nextX_nextX: fixes x :: "vbl"   and y :: "vbl" shows "(nextX x = nextX y) = (x = y)"
init_class_obj_inited: fixes G :: "prog"   and C :: "qtname"   and s1 :: "st"   and s2 :: "st" assumes "init_class_obj G C s1\<le>|s2" shows "inited C (globs s2)"
lfp_exp_lemma2: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "mono_trans t"   and "t R \<tturnstile> R"   and "sound R" shows "t (lfp_exp t) \<tturnstile> lfp_exp t"