hcju/isabelleps huggingface dataset:
pas_alt_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "D x y \<and> D (x \<oplus> y) z" shows "D y z \<and> D x (y \<oplus> z) \<and> x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
mig_set_def: fixes D :: "nat"   and X :: "'c set" shows "mig_set D X = op_ivl_rep_of_set (X ::: appr_rel) \<bind> (\<lambda>(i, s). let migc = mig_componentwise i s in ASSUME (D = DIM('c)) \<bind> (\<lambda>_. let norm_fas = [Norm (map floatarith.Var [0..<D])] ::: fas_rel; env = list_of_eucl ` ({migc..migc} ::: appr_rel) ::: appr_rell in approx_slp_appr norm_fas (slp_of_fas norm_fas) env \<bind> (\<lambda>n. Inf_spec (n ::: appr_rel) \<bind> (\<lambda>ni. RETURN (rnv_of_lv ni)))))"
complement_symmetric: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.bounded_lattice inf less_eq less sup bot top"   and "sup x y = top \<and> inf x y = bot" shows "sup y x = top \<and> inf y x = bot"
map_cong_pred: fixes x :: "'c pmf"   and ya :: "'c pmf"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "pred_pmf (\<lambda>z. f z = g z) ya" shows "map_pmf f x = map_pmf g ya"
pr_comp3_3': fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and k :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "pr_conv_3_to_1 f \<in> PrimRec1'"   and "pr_conv_3_to_1 g \<in> PrimRec1'"   and "pr_conv_3_to_1 h \<in> PrimRec1'"   and "pr_conv_3_to_1 k \<in> PrimRec1'" shows "pr_conv_3_to_1 (\<lambda>x y z. f (g x y z) (h x y z) (k x y z)) \<in> PrimRec1'"
absorb_iff2: fixes b :: "'b::semilattice_inf"   and a :: "'b::semilattice_inf" shows "((b::'b::semilattice_inf) \<le> (a::'b::semilattice_inf)) = (a \<sqinter> b = b)"
dist_bound: fixes f :: "'a \<Rightarrow>\<^sub>C 'b"   and g :: "'a \<Rightarrow>\<^sub>C 'b"   and b :: "real" assumes "\<And>x. dist (apply_bcontfun f x) (apply_bcontfun g x) \<le> b" shows "dist f g \<le> b"
ord_Zp_ultrametric_eq': fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "ord_Zp y < ord_Zp x" shows "ord_Zp (x \<ominus> y) = ord_Zp y"
order_trans: fixes xa :: "'o"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "le xa x y"   and "le xa y z" shows "le xa x z"
keysFor_parts_initState: fixes C :: "agent" shows "keysFor (parts (initState C)) = {}"
iterate_sound: fixes Ee :: "val list \<Rightarrow> val set"   and \<rho> :: "val list"   and v :: "val"   and \<sigma> :: "ty"   and \<tau> :: "ty"   and \<eta> :: "val set list" assumes "iterate Ee \<rho> v"   and "\<forall>v. v \<in> T (\<sigma> \<rightarrow> \<tau>) \<eta> \<longrightarrow> Ee (v # \<rho>) \<subseteq> T (\<sigma> \<rightarrow> \<tau>) \<eta>" shows "v \<in> T (\<sigma> \<rightarrow> \<tau>) \<eta>"
assms: shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
verit_sko_ex': fixes P :: "'b \<Rightarrow> bool"   and A :: "bool" assumes "P (SOME x. P x) = A" shows "(\<exists>x. P x) = A"
lemma2b: fixes p :: "bool list"   and q :: "bool list"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> (eps R)\<^sup>*" shows "(False # p, False # q) \<in> (eps (or L R))\<^sup>*"
Inf_greatest: fixes A :: "('c \<times> 'd) set"   and z :: "'c \<times> 'd" assumes "\<And>x. s2p A x \<Longrightarrow> less_eq_prod z x" shows "less_eq_prod z (Inf_prod A)"
snd_initmissing: fixes x :: "(nat \<Rightarrow> 'g option) \<times> 'a" shows "snd (initmissing x) = snd x"
eq: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and c :: "'a" shows "poly_eval a (extend_indets (monomial c t)) = poly_eval (\<lambda>x. a (Some x)) (monomial c t)"
that: fixes ns :: "'node SDG_node list" assumes "matched n ns n'"   and "n \<in> set ns"   and "get_proc (parent_node (last ns)) = p" shows "thesis"
thr_init_fin_list_state: fixes s :: "status"   and \<sigma> :: "('d, 'e) locks \<times> (('e \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option) \<times> 'f) \<times> ('e \<Rightarrow> 'g wait_set_status option) \<times> 'e set"   and t :: "'e" shows "thr (init_fin_lift_state s \<sigma>) t = map_option (\<lambda>(x, ln). ((s, x), ln)) (thr \<sigma> t)"
dims2_def: shows "st.dims2 = nths [2, 2] (- {0})"
\<omega>_sum_in_tpD: fixes \<gamma> :: "V"   and k :: "nat" assumes "\<gamma> \<in> elts (\<omega> \<up> \<alpha>s ! k)"   and "k < length \<alpha>s" shows "\<omega>_sum (take k \<alpha>s) + \<gamma> \<in> elts (tp D)"
sum_bounded_above_strict: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and K :: "'d" assumes "\<And>i. i \<in> A \<Longrightarrow> f i < K"   and "0 < card A" shows "sum f A < of_nat (card A) * K"
non_strict_ending: fixes t :: "nat \<Rightarrow> 'b"   and R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "\<forall>i. (t i, t (Suc i)) \<in> R \<union> S"   and "R O S \<subseteq> S"   and "SN_on S {t 0}" shows "\<exists>j. \<forall>i\<ge>j. (t i, t (Suc i)) \<in> R - S"
Ker_rw: shows "carrier (V.vs kerT) = {\<zero>\<^bsub>V\<^esub>}"
pp_a_neg: fixes Q :: "'b \<Rightarrow> bool" shows "pp_a \<lceil>Q\<rceil> = \<lceil>- Q\<rceil>"
prv_cnj_com_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (cnj \<phi> \<chi>) (cnj \<chi> \<phi>))"
ce_rel_lm_6: fixes r :: "(nat \<times> nat) set" assumes "r \<in> ce_rels" shows "ce_rel_to_set r \<in> ce_sets"
psiCasesFreshChain_1: fixes xvec :: "name list"   and Cs :: "('g \<times> ('e, 'f, 'g) psi) list" shows "xvec \<sharp>* psiCases Cs = xvec \<sharp>* Cs"
pseudo_proj_True_snth: fixes n :: "nat"   and x :: "bool stream"   and w :: "bool stream"   and i :: "nat" assumes "pseudo_proj_True n x = pseudo_proj_True n w"   and "Suc i \<le> n" shows "x !! i = w !! i"
plus_preal_def: fixes x :: "preal"   and y :: "preal" shows "x + y \<equiv> preal_min x y"
p_def: shows "p = (\<lambda>x. \<psi> j (x + 1))"
simps: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "basis_reduction_main (upw, i, fs) = (if i < m \<and> LLL_invariant upw i fs then basis_reduction_main (basis_reduction_step upw i fs) else fs)"
FAddSA_dom_emptyset: fixes S :: "'d"   and A :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and SA :: "('d, 'e, 'f) seqauto"   and S' :: "'d" assumes "S \<in> dom A"   and "S \<notin> States SA"   and "S' \<in> States SA" shows "(A [f+] (S, SA)) S' = Some {}"
edge'_simps_1: fixes x :: "'v vertex ??.MFMC_Flow_Attainability.flow_attainability.node" shows "??.local.\<Psi>.edge' ??.MFMC_Flow_Attainability.flow_attainability.node.SOURCE x = (x = ??.MFMC_Flow_Attainability.flow_attainability.node.Inner (source \<Psi>))"
rep_eq: fixes x :: "'f"   and xa :: "'f cset" shows "cin x xa = (x \<in> rcset xa)"
multi_member_split: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "\<exists>A. M = add_mset x A"
env_restr_below_itself: fixes S :: "'a set"   and m :: "'a \<Rightarrow> 'b" shows "m f|` S \<sqsubseteq> m"
orthogonalD: fixes vs :: "'a vec list"   and i :: "nat"   and j :: "nat" assumes "orthogonal vs"   and "i < length vs"   and "j < length vs" shows "(vs ! i \<bullet> vs ! j = (0::'a)) = (i \<noteq> j)"
apath_ends: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "apath u p v"   and "apath u' p v'" shows "p \<noteq> [] \<and> u \<noteq> v \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
50: shows "sset (gtrace r v) \<subseteq> graph A w k"
conj_commute: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice"   and b :: "'b::refinement_lattice" assumes "conjunction (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (a::'b::refinement_lattice) (b::'b::refinement_lattice) = conj b a"
frequently_const_iff: fixes P :: "bool"   and F :: "'b filter" shows "(\<exists>\<^sub>F x in F. P) = (P \<and> F \<noteq> bot)"
cases: fixes x :: "'a ms" obtains xs :: "('a \<times> real) msllist" and f :: "real \<Rightarrow> real" where "x = MS xs f"
X: shows "\<forall>x\<ge>X. (\<forall>h\<in>set hs'. \<bar>h x\<bar> \<le> hb * x / ln x powr (1 + e)) \<and> (\<forall>b\<in>set bs. C * x \<le> b * x - hb * x / ln x powr (1 + e)) \<and> (\<forall>b\<in>set bs. akra_bazzi_asymptotics b hb e p x) \<and> (\<forall>b>x. \<exists>c. \<forall>x\<in>{x..b}. g' x \<le> c) \<and> (\<forall>u\<in>{C * x..x}. g' u \<le> c2 * g' x) \<and> 0 < f2' (nat \<lfloor>x\<rfloor>) \<and> 0 \<le> g' x"
fresh_subset: fixes B :: "'b set"   and x :: "atom"   and A :: "'b set" assumes "finite B"   and "x \<sharp> B"   and "A \<subseteq> B" shows "x \<sharp> A"
col_line_eq_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points" assumes "A \<noteq> B"   and "A \<noteq> C"   and "col A B C" shows "line A B = line A C"
purgeIdle_append: fixes sl1 :: "('St, 'U, 'C) state list"   and sl2 :: "('St, 'U, 'C) state list" shows "purgeIdle (sl1 @ sl2) = purgeIdle sl1 @ purgeIdle sl2"
count_gt_imp_in_mset: fixes n :: "nat"   and M :: "'b multiset"   and x :: "'b" assumes "n < count M x" shows "x \<in># M"
singleton_multisets: fixes x :: "'b"   and A :: "'b set" shows "({#x#} \<in> multisets A) = (x \<in> A)"
wtTE_intT: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and x :: "Preliminaries.var" assumes "wtE \<xi>" shows "intT (tpOfV x) (\<xi> x)"
leq2: shows "matrix_sum d (\<lambda>k. tensor_P (1\<^sub>m N) (proj_k k)) R \<le>\<^sub>L 1\<^sub>m d"
rel_mono_strong: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c stream"   and y :: "'d stream"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "stream_all2 R x y"   and "\<And>z yb. \<lbrakk>z \<in> sset x; yb \<in> sset y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "stream_all2 Ra x y"
mtf2_forward_effect1': fixes q :: "'a"   and xs :: "'a list"   and x :: "'a"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q < index xs x \<and> index xs x < length xs" shows "index (mtf2 n q xs) x = index xs x \<and> index xs q < index (mtf2 n q xs) x \<and> index (mtf2 n q xs) x < length xs"
is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) node_ptr" shows "is_element_ptr_kind ptr = (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> True)"
parent_r_child: fixes l :: "int"   and n :: "int" shows "parent l (r_child l n) = n"
comp_inv_arr: fixes f :: "'c"   and g :: "'c" assumes "C.inverse_arrows f g" shows "g \<cdot>\<^sub>C f = C.dom f"
memo_int_def: fixes low :: "int"   and up :: "int"   and f :: "int \<Rightarrow> 'c" shows "memo_int low up f \<equiv> let m = Mapping.tabulate [low..up] f in (\<lambda>x. if low \<le> x \<and> x \<le> up then the (Mapping.lookup m x) else f x)"
fcompress_eq: fixes M :: "'a fset" shows "fcompress M = M"
axioms_6: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" assumes "inverse_transformations A B F G \<tau> \<tau>'" shows "natural_transformation A B G F \<tau>'"
ih: assumes "\<forall>i<length xs. a \<le> xs ! i" shows "of_nat (length xs) * a \<le> sum_list xs"
add_less_imp_less_left: fixes c :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" assumes "c + a \<subset># c + b" shows "a \<subset># b"
of_nat_unat: fixes w :: "'c word" shows "of_nat (unat w) = unsigned w"
transitive_closurep_trans'_4: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "r a b"   and "r\<^sup>+\<^sup>+ b c" shows "r\<^sup>+\<^sup>+ a c"
fst_eqD: fixes x :: "'g"   and y :: "'h"   and a :: "'g" assumes "fst (x, y) = a" shows "x = a"
semiring_normalization_rules_3: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "'b"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "plus m (times a m) = times (plus a one) m"
GreatestI_ex_nat: fixes P :: "nat \<Rightarrow> bool"   and b :: "nat" assumes "\<exists>k. P k"   and "\<And>y. P y \<Longrightarrow> y \<le> b" shows "P (Greatest P)"
prime_factors_multiplicity: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and n :: "'c" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize n) = {p. normalization_semidom.prime zero times one normalize p \<and> 0 < factorial_semiring.multiplicity times one p n}"
list_all2_nthD2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length ys" shows "P (xs ! p) (ys ! p)"
set_nebset_inverse: fixes x :: "'d set!['e]" shows "Abs_nebset (set_nebset x) = x"
prefixToElementAppend: fixes e :: "'b"   and M1 :: "('b \<times> bool) list"   and M2 :: "('b \<times> bool) list" shows "prefixToElement e (M1 @ M2) = (if e \<in> set (elements M1) then prefixToElement e M1 else M1 @ prefixToElement e M2)"
rv_mult: fixes v :: "(real, 'n) Finite_Cartesian_Product.vec" shows "A *v (\<chi>i. complex_of_real (v $h i)) = (\<chi>i. complex_of_real ((local.B *v v) $h i))"
red_indI1: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "f \<in> F"   and "f \<noteq> 0"   and "p \<noteq> 0"   and "lt f adds\<^sub>t lt p" shows "red F p (p - monom_mult (lc p / lc f) (lp p - lp f) f)"
\<I>_full_sel_2: fixes x :: "'d" shows "responses_\<I> \<I>_full x = UNIV"
fresh_substEnv_updEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "freshEnv ys y rho" shows "rho &[Y / y]_ys = rho [y \<leftarrow> Y]_ys"
dd_a: fixes x :: "'a" shows "domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> x = a (a x)"
bij_add: fixes y :: "'a" shows "bij (\<lambda>x. x + y)"
SeqWhile: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and P :: "'b set"   and b :: "'b set"   and a :: "('b, 'a, 'c) ann"   and c :: "('b, 'a, 'c) com"   and A :: "'b set"   and r :: "'b set"   and i :: "'b set" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> (P \<inter> b) a c P, A" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P (AnnWhile r i a) (While b c) (P \<inter> - b), A"
member_rec_1: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" shows "List.member (x # xs) y = (x = y \<or> List.member xs y)"
Id_on_def: fixes A :: "'c set" shows "Id_on A = (\<Union>x\<in>A. {(x, x)})"
n_of_wstate_weighted_RP_increasing: fixes St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat" assumes "St \<leadsto>\<^sub>w St'" shows "n_of_wstate St \<le> n_of_wstate St'"
Image_mono: fixes r' :: "('d \<times> 'e) set"   and r :: "('d \<times> 'e) set"   and A' :: "'d set"   and A :: "'d set" assumes "r' \<subseteq> r"   and "A' \<subseteq> A" shows "r' `` A' \<subseteq> r `` A"
dom_char: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" shows "A2_B.dom f = (if A2_B.arr f then A2_B.MkIde (A2_B.Dom f) else A2_B.null)"
pos_zcount_after_summary: fixes M :: "'t zmultiset"   and t :: "'t"   and s :: "'sum"   and S :: "'sum antichain" assumes "\<And>t. 0 \<le> zcount M t"   and "0 < zcount M t"   and "s \<in>\<^sub>A S" shows "0 < zcount (after_summary M S) (results_in t s)"
merge_list_cong: fixes x :: "'c list"   and y :: "'c list"   and m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and m' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option" assumes "\<And>a b. \<lbrakk>a \<in> set x; b \<in> set y\<rbrakk> \<Longrightarrow> m a b = m' a b" shows "merge_list m x y = merge_list m' x y"
simps: fixes a1 :: "'a"   and a2 :: "'a multiset"   and a3 :: "'b el_step"   and a4 :: "'a"   and a5 :: "'a multiset"   and S :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" shows "(((a1, a2), a3, a4, a5) \<in> gtrp S) = ((\<exists>s c e s' c'. a1 = s \<and> a2 = c \<and> a3 = LOC e \<and> a4 = s' \<and> a5 = c' \<and> ((s, c), e, s', c') \<in> S) \<or> (\<exists>s sl c e s' c'. a1 = sl \<and> a2 = add_mset s c \<and> a3 = ENV e \<and> a4 = sl \<and> a5 = add_mset s' c' \<and> ((s, add_mset sl c), e, s', add_mset sl c') \<in> S))"
card_V_est_E: shows "card V \<le> 2 * card E"
odd_two_times_div_two_succ: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "plus (times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (numeral.numeral one plus (num.Bit0 num.One)))) one = a"
strong_induct: fixes P :: "'b \<Rightarrow> freeRes \<Rightarrow> bool"   and z :: "'b"   and freeRes :: "freeRes" assumes "\<And>name1 name2 z. P z (name1<name2>)"   and "\<And>name1 name2 z. P z (name1[name2])"   and "\<And>z. P z (\<tau>)" shows "P z freeRes"
pmf_r: fixes x :: "'alt" shows "pmf r x = pmf p x + \<epsilon> * (pmf q x - pmf p x)"
elims_2: fixes x :: "real_min_top"   and xa :: "real_min_top" assumes "x \<le> xa" obtains uu_ :: "real_min_top" where "xa__ = uu_" and "xa = PInfty"   | x :: "real" and y :: "real" where "xa__ = real_min_top.R x" and "xa = real_min_top.R y" and "x \<le> y"
completely_multiplicative_fds_deriv': fixes f :: "'b fds" assumes "completely_multiplicative_function (fds_nth f)" shows "fds_deriv f = - fds (\<lambda>n. fds_nth f n * mangoldt n) * f"
directed_prop: fixes X :: "'a set" assumes "X \<noteq> {}"   and "\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. x \<le> z \<and> y \<le> z" shows "directed X"
en_def: shows "en \<equiv> \<lambda>F. {x. case x of \<omega>\<nu> \<omega> \<Rightarrow> False | \<alpha>\<nu> y \<Rightarrow> make\<Pi>\<^sub>1 F \<in> y}"
joinableE: fixes x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'b) set" assumes "(x, y) \<in> joinable R" obtains z :: "'b" where "(x, z) \<in> R" and "(y, z) \<in> R"
vector_derivative_of_real_left: fixes f :: "real \<Rightarrow> real"   and x :: "real" assumes "f differentiable at x" shows "vector_derivative (\<lambda>x. of_real (f x)) (at x) = of_real (deriv f x)"
bij_betw_imp_surj_on: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "f ` A = B"
Branch_def: fixes c :: "'h"   and rs :: "'i \<Rightarrow> ('f, 'g, 'h, 'i) resumption" shows "Branch c rs = Abs_resumption (BRANCH c (\<lambda>d. Rep_resumption (rs d)))"
conversep_noteq: shows "(\<noteq>)\<inverse>\<inverse> = (\<noteq>)"
runit_hcomp_gen: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<r>\<^sub>D[f \<star>\<^sub>D g] \<star>\<^sub>D h = ((f \<star>\<^sub>D \<r>\<^sub>D[g]) \<star>\<^sub>D h) \<cdot>\<^sub>D (\<a>\<^sub>D[f, g, src\<^sub>D g] \<star>\<^sub>D h)"
ratfps_nth_aux_correct: fixes p :: "'b poly"   and n :: "nat" shows "ratfps_nth_aux p n = natfun_inverse (fps_of_poly p) n"
comp_par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: fixes public :: "'a \<Rightarrow> bool"   and arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('a, 'b \<times> nat) Term.term list \<times> 'c"   and \<Gamma> :: "('a, 'b \<times> nat) Term.term \<Rightarrow> ('d, 'e) Term.term"   and pair_fun :: "'a"   and A :: "('f strand_label \<times> ('a, 'b \<times> nat) stateful_strand_step) list"   and M :: "'f \<Rightarrow> ('a, 'b \<times> nat) Term.term list"   and C :: "('a, 'b \<times> nat) Term.term list" shows "comp_par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t public arity Ana \<Gamma> pair_fun A M C \<equiv> let L = remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A)); MP0 = \<lambda>B. remdups (trms_list\<^sub>s\<^sub>s\<^sub>t B @ map (pair' pair_fun) (setops_list\<^sub>s\<^sub>s\<^sub>t B)); pr = \<lambda>l. MP0 (proj_unl l A) in 1 < length L \<and> list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) (MP0 (unlabel A)) \<and> list_all (wf\<^sub>t\<^sub>r\<^sub>m' arity) C \<and> has_all_wt_instances_of \<Gamma> (subterms\<^sub>s\<^sub>e\<^sub>t (set C)) (set C) \<and> is_TComp_var_instance_closed \<Gamma> C \<and> (\<forall>i\<in>set L. \<forall>j\<in>set L. i \<noteq> j \<longrightarrow> comp_GSMP_disjoint public arity Ana \<Gamma> (pr i) (pr j) (M i) (M j) C) \<and> (\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. i \<noteq> j \<longrightarrow> (let s = pair' pair_fun p; t = pair' pair_fun q in mgu s (t \<cdot> var_rename (max_var_set (fv s))) = None))"
simple_conn_matches_simple_match_any: fixes c :: "parts_connection" shows "simple_conn_matches simple_match_any c"
induct: fixes P :: "nat \<Rightarrow> C \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "C" assumes "\<And>c. P c (C [])"   and "\<And>c ns. P c (C (C [] # ns))"   and "\<And>c n ns ms. P c (C (funC (C (n # ns)) (Suc (Suc c)) @ ms)) \<Longrightarrow> P c (C (C (n # ns) # ms))" shows "P a0 a1"
almost_diff_const_sound: shows "sound ([], [], [Equals (Differential (Const 0)) (Const 0)])"
0: shows "rawpsubstT (pls r1 r2) (zip (map Var us) (map snd txs)) = pls (rawpsubstT r1 (zip (map Var us) (map snd txs))) (rawpsubstT r2 (zip (map Var us) (map snd txs)))"
P_k_hits_Q: shows "Bex (set P_k) hitting_Q_or_new_last"
simps_2: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a option" shows "Some x \<bind> f = f x"
left_neutral: fixes a :: "'a" shows "(0::'a) XOR a = a"
lt_f: shows "lt (monom_mult (lc p / lc f) (lp p - lp f) f) = lt p"
assms_3: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "plossless_gpv \<I>' (callee s x)"
star_n_def: fixes X :: "nat \<Rightarrow> 'c" shows "star_n X = Abs_star (starrel `` {X})"
fing: shows "finite (dom ((\<lambda>a. Some (y a)) |` {a. y a \<noteq> b}))"
inext_le_mono: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a \<le> b"   and "a \<in> I"   and "b \<in> I" shows "inext a I \<le> inext b I"
S: shows "g C1_differentiable_on {0..1} - S"
geq_def: shows "geq \<equiv> \<lambda>A i. \<forall>k. i < k \<and> k < length A \<longrightarrow> pivot \<le> A ! k"