hcju/isabelleps huggingface dataset:
degenerate_lotteries_def: shows "local.degenerate_lotteries = {x \<in> local.\<P>. card (set_pmf x) = 1}"
set_less_eq_aux'_def: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubseteq>'' B) = (A \<sqsubseteq>' B \<or> A = {} \<and> B \<in> infinite_complement_partition)"
upd_cond_alt: fixes u :: "'v"   and v' :: "'v" shows "upd_cond (Q' Q \<pi> u) (\<pi>' Q \<pi> u) u v' = ((v', u) \<in> edges g \<and> v' \<notin> S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u)) \<and> enat (w {v', u}) < Q' Q \<pi> u v')"
sol: shows "solve_factored_ratfps' p fctrs = (b, es)"
assms_1: shows "((ps, a) # branch) !. v = Some (qs, b)"
div_lt_mult: fixes i :: "'a word"   and k :: "'a word"   and x :: "'a word" assumes "i < k div x"   and "0 < x" shows "i * x < k"
mem_existence_ivl_iv_defined_1: fixes t :: "real"   and x0 :: "'c" assumes "t \<in> existence_ivl0 x0" shows "0 \<in> UNIV"
ceovisidiffimi_ffie: shows "(e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>) O (f \<union> f\<inverse> \<union> e) \<subseteq> e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>"
Carmichael_number_imp_squarefree: fixes n :: "nat" assumes "Carmichael_number n" shows "squarefree n"
assms_2: shows "\<forall>x. x \<notin> A \<longrightarrow> f x = x"
map_2: fixes f :: "'e \<Rightarrow> 'd"   and x2 :: "'e" shows "map_option f (Some x2) = Some (f x2)"
pdom_res_minus: fixes A :: "'a set"   and f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" shows "A \<lhd>\<^sub>p (f - g) = A \<lhd>\<^sub>p f - g"
inj_map_strong: fixes x :: "'c \<Rightarrow> 'b"   and xa :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'd"   and fa :: "'b \<Rightarrow> 'd" assumes "\<And>z za. \<lbrakk>z \<in> range x; za \<in> range xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "f \<circ> x = fa \<circ> xa" shows "x = xa"
carrier_vs_is_self: fixes N :: "'c set" shows "carrier (vs N) = N"
less_le_trans: fixes x :: "'c word"   and y :: "'c word"   and z :: "'c word" assumes "x <s y"   and "y \<le>s z" shows "x <s z"
GF_advice_sync_lesseq: fixes n :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set"   and \<psi> :: "'b ltln" assumes "\<And>i. i \<le> n \<Longrightarrow> \<exists>j. suffix (i + j) w \<Turnstile>\<^sub>n af \<phi> (w [i \<rightarrow> j + i])[X]\<^sub>\<nu>"   and "\<exists>j. suffix (n + j) w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> j + n])[X]\<^sub>\<nu>" shows "\<exists>k\<ge>n. (\<forall>j\<le>n. suffix k w \<Turnstile>\<^sub>n af \<phi> (w [j \<rightarrow> k])[X]\<^sub>\<nu>) \<and> suffix k w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> k])[X]\<^sub>\<nu>"
assms_3: fixes k :: "nat" assumes "k < n" shows "trace (f k) \<le> trace (g k)"
mult_le_cancel_iff2: fixes z :: "'c"   and x :: "'c"   and y :: "'c" assumes "(0::'c) < z" shows "(z * x \<le> z * y) = (x \<le> y)"
strip_acom: fixes f :: "'b \<Rightarrow> 'a"   and c :: "'b acom" shows "strip (ACom.map_acom f c) = strip c"
poly_pos_between_less_leq: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a < x \<and> x \<le> b \<longrightarrow> 0 < poly p x) = (b \<le> a \<or> p \<noteq> 0 \<and> 0 < poly p b \<and> count_roots_between p a b = 0)"
infdist_proper_attained: fixes C :: "'c set"   and x :: "'c" assumes "proper C"   and "C \<noteq> {}" shows "\<exists>c\<in>C. infdist x C = dist x c"
make_map_def: fixes ks :: "'d fset"   and v :: "'e" shows "make_map ks v \<equiv> \<lambda>k. if k \<in> fset ks then Some v else None"
esssup_pos_measure: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and z :: "'c" assumes "f \<in> borel_measurable M"   and "z < esssup M f" shows "0 < emeasure M {x \<in> space M. z < f x}"
is_natural_2: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "\<alpha> (VVV.cod f) \<cdot> H.HoHV f = \<alpha> f"
ordered_map_keys: fixes S :: "('a \<times> 'b) set" assumes "finite S"   and "is_map S" shows "map fst (ordered_map S) = sorted_list_of_set (fst ` S)"
jump_poly0_1: fixes p :: "real poly"   and x :: "real" shows "jump_poly 0 p x = 0"
synth_payload: fixes Y :: "msg set"   and X :: "msg set" assumes "Y \<inter> payload = {}" shows "synth (X \<union> Y) \<subseteq> synth X \<union> - payload"
final_state: shows "passive_subset (Liminf_llist (lmap snd Ns)) = {}"
condensation_condense2: fixes n :: "nat" shows "(\<Sum>k = 1..<2 ^ n. f (2 * 2 ^ Discrete.log k)) = (\<Sum>k<n. 2 ^ k * f (2 ^ Suc k))"
InitConf_CompFun_Ancestor: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and SA :: "('d, 'e, 'f) seqauto"   and T :: "'d" assumes "S \<in> HAStates A"   and "SA \<in> the (CompFun A S)"   and "T \<in> InitConf A"   and "T \<in> States SA" shows "S \<in> InitConf A"
iteratei_def: fixes list_it :: "'d \<Rightarrow> (('b \<times> 'c) list \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list) \<Rightarrow> ('b \<times> 'c) list \<Rightarrow> ('b \<times> 'c) list"   and S :: "'d" shows "poly_map_iteratei_defs.iteratei list_it S \<equiv> it_to_it (list_it S)"
xivp_2: shows "\<mu> (X t\<^sub>0) = \<nu> (X t\<^sub>0)"
eq_idx_for_lvar_aux: fixes x :: "nat"   and t :: "(nat \<times> linear_poly) list"   and i :: "nat" assumes "x \<in> lvars t" shows "let idx = eq_idx_for_lvar_aux t x i in i \<le> idx \<and> idx < i + length t \<and> lhs (t ! (idx - i)) = x"
operator_preconditions_hold: shows "are_all_operators_applicable (\<Phi>\<^sub>S\<inverse> \<Pi> \<A> k) ((\<Phi>\<inverse> \<Pi> \<A> t) ! k) \<and> are_all_operator_effects_consistent ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
dvd_mult_unit_iff: 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 b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times a (times c b) = dvd.dvd times a c"
beta: fixes v :: "exp"   and x :: "nat"   and e :: "exp" assumes "is_val v" shows "EApp (ELam x e) v \<longrightarrow> subst x v e"
is_le_rel_trans: fixes rel :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and x :: "'t"   and y :: "'t"   and z :: "'t" assumes "ord_term_lin.is_le_rel rel"   and "rel x y"   and "rel y z" shows "rel x z"
root_imp_deg_nonzero: fixes p :: "'a poly"   and x :: "'a" assumes "p \<noteq> 0"   and "poly p x = (0::'a)" shows "degree p \<noteq> 0"
independent_in_carrier: fixes K :: "'a list set"   and Us :: "'a list list" assumes "UP.independent K Us" shows "set Us \<subseteq> carrier (K [X])"
integrable_bound: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "integrable M f"   and "g \<in> borel_measurable M"   and "AE x in M. norm (g x) \<le> norm (f x)" shows "integrable M g"
iterate_sel_def: fixes it :: "('a option \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a option \<Rightarrow> 'a option) \<Rightarrow> 'a option \<Rightarrow> 'a option"   and f :: "'b \<Rightarrow> 'a option" shows "iterate_sel it f = it (\<lambda>\<sigma>. \<sigma> = None) (\<lambda>x \<sigma>. f x) None"
kmin_le: fixes f :: "nat \<Rightarrow> real" shows "\<forall>i\<in>kmin f PR. \<forall>r\<in>reduce f PR. f i \<le> r"
terminates_call: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and init :: "'d \<Rightarrow> 'd"   and s :: "'d"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile>bdy \<down> Normal (init s)"   and "\<forall>t. \<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Normal t \<longrightarrow> \<Gamma>\<turnstile>c s t \<down> Normal (return s t)" shows "\<Gamma>\<turnstile>call init p return c \<down> Normal s"
padic_val_a_inv: fixes p :: "int"   and a :: "nat \<Rightarrow> int" assumes "prime p"   and "a \<in> carrier (padic_int p)" shows "padic_val p a = padic_val p (\<ominus>\<^bsub>padic_int p\<^esub> a)"
simps_1: fixes I :: "interp"   and a :: "char" shows "game_sem I (Game a) = Games I a"
choose: fixes p :: "assertion"   and c1 :: "command"   and q :: "assertion"   and c2 :: "command" assumes "prov_triple (p, c1, q)"   and "prov_triple (p, c2, q)" shows "prov_triple (p, Choose c1 c2, q)"
ring_distrib2: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(y \<plusminus>\<^bsub>R\<^esub> z) \<cdot>\<^sub>r\<^bsub>R\<^esub> x = y \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<plusminus>\<^bsub>R\<^esub> z \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
cycle_free_diag_intro: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs" shows "cycle_free M n"
walk_middle_edge: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list"   and v :: "'a"   and w :: "'a"   and ys :: "'a list" assumes "Digraph G"   and "Digraph.walk G (xs @ v # w # ys)" shows "v \<rightarrow>\<^bsub>G\<^esub> w"
sym_trancl: fixes r :: "('e \<times> 'e) set" assumes "sym r" shows "sym (r\<^sup>+)"
fresh_prodD_1: fixes a :: "'a"   and x :: "'b"   and y :: "'c" assumes "a \<sharp> (x, y)" shows "a \<sharp> x"
antisymp_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "antisymp_on P A = (\<forall>a\<in>A. \<forall>b\<in>A. P a b \<and> P b a \<longrightarrow> a = b)"
integer_of_uint8_signed_def: fixes n :: "uint8" shows "integer_of_uint8_signed n = (if n !! 7 then undefined integer_of_uint8 n else integer_of_uint8 n)"
sorted_insortb: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list"   and x :: "('b, 'c) Combinators" assumes "FWNormalisationCore.sorted xs l"   and "all_in_list (x # xs) l"   and "distinct (x # xs)"   and "singleCombinators [x]"   and "singleCombinators xs" shows "FWNormalisationCore.sorted (FWNormalisationCore.insort x xs l) l"
sel_1: fixes R :: "('f, 'g, 'h) Resumption.resumption set" assumes "\<forall>r\<in>R. Resumption.resumption.is_Done r" shows "Resumption.resumption.result (resumption_lub R) = flat_lub None (Resumption.resumption.result ` R)"
polynomial_function_iff_Basis_inner: fixes f :: "'c \<Rightarrow> 'd" shows "polynomial_function f = (\<forall>b\<in>Basis. real_polynomial_function (\<lambda>x. f x \<bullet> b))"
isCont_cdf: fixes x :: "real" shows "isCont (cdf M) x = (prob {x} = 0)"
factorize_root_free_def: fixes p :: "rat poly" shows "factorize_root_free p = (if degree p = 0 then (coeff p 0, []) else factorize_root_free_main p (roots_of_rat_poly p) [])"
inj_map_def: fixes \<pi> :: "'c \<Rightarrow> 'd option" shows "inj_map \<pi> = (\<forall>x y. \<pi> x = \<pi> y \<and> \<pi> x \<noteq> None \<longrightarrow> x = y)"
\<Phi>_inv: fixes yx :: "'d \<times> 'c" assumes "DopxC.ide yx" shows "S.inverse_arrows (\<Phi> yx) (\<Psi>o yx)"
assms_2: shows "basis_in \<E> B\<^sub>2"
start_prog_Start_fields: fixes C :: "char list"   and M :: "char list"   and P :: "jvm_prog"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and F :: "char list" assumes "class_add P (start_class C M) \<turnstile> Start has_fields FDTs" shows "map_of FDTs (F, Start) = None"
simps_3: fixes v :: "'a" shows "drop_half [v] = [v]"
synth_def: shows "synth \<equiv> \<lambda>H. {x. synthp (\<lambda>x. x \<in> H) x}"
mirror_elem_Min: fixes I :: "nat set" assumes "finite I"   and "I \<noteq> {}" shows "mirror_elem (iMin I) I = Max I"
sds_lowering_sd_efficient_axioms: shows "sds_lowering_sd_efficient agents alts sds agents' alts'"
bind_sample1: fixes p :: "'p pmf"   and f :: "'p \<Rightarrow> 'm"   and g :: "'a \<times> 's \<Rightarrow> 'm" shows "bind (sample p f) g = sample p (\<lambda>x. bind (f x) g)"
poly_mult_closed: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p1 :: "'c list"   and p2 :: "'c list" assumes "ring R"   and "subring K R"   and "polynomial\<^bsub>R\<^esub> K p1"   and "polynomial\<^bsub>R\<^esub> K p2" shows "polynomial\<^bsub>R\<^esub> K (ring.poly_mult R p1 p2)"
dim_orthogonal_sum: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0" shows "dim (A \<union> B) = dim A + dim B"
closed_contains_Sup: fixes S :: "real set" assumes "S \<noteq> {}"   and "bdd_above S"   and "closed S" shows "Sup S \<in> S"
constrI: fixes E :: "nat \<Rightarrow> 'a upred"   and F :: "'a upred \<Rightarrow> 'a upred" assumes "chain E"   and "\<And>X n. (F X \<and> E (n + 1)) = (F (X \<and> E n) \<and> E (n + 1))" shows "constr F E"
lq_s: shows "set (Lxy qs {x, y}) \<subseteq> {x, y}"
mset_points_distinct: shows "distinct_mset (mset_set \<V>)"
res_to_fun_monic_monom: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat"   and n :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a k = b k" shows "(monom (UP Zp) \<one> n \<bullet> a) k = (monom (UP Zp) \<one> n \<bullet> b) k"
bisimScopeExtChainSym: fixes xvec :: "name list"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* Q" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<parallel> Q \<sim> (\<lparr>\<nu>*xvec\<rparr>P) \<parallel> Q"
noWhile_transC: fixes c :: "('a, 'b) com"   and tval :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and aval :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c"   and c' :: "('a, 'b) com"   and s' :: "'c" assumes "noWhile c"   and "PL.transC tval aval (c, s) (c', s')" shows "noWhile c'"
prefix_append_subsequence: fixes i :: "nat"   and j :: "nat"   and w :: "nat \<Rightarrow> 'a" assumes "i \<le> j" shows "prefix i w @ w [i \<rightarrow> j] = prefix j w"
sup_ge1: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x (sup x y)"
Ana: fixes t :: "('fun, 'var) Term.term"   and M :: "('fun, 'var) Term.term set"   and K :: "('fun, 'var) Term.term list"   and T :: "('fun, 'var) Term.term list"   and k :: "('fun, 'var) Term.term" assumes "t \<in> SMP M"   and "Ana t = (K, T)"   and "k \<in> set K" shows "k \<in> SMP M"
set_mrbst_union: fixes t' :: "'a tree"   and t1 :: "'a tree"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_union t1 t2)"   and "bst t1"   and "bst t2" shows "set_tree t' = set_tree t1 \<union> set_tree t2"
a_star_subst1: fixes M :: "trm"   and M' :: "trm"   and y :: "name"   and c :: "coname"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>a* M'" shows "M{y:=<c>.P} \<longrightarrow>\<^sub>a* M'{y:=<c>.P}"
7: shows "d \<le> top * e\<^sup>T * (w \<sqinter> - d)\<^sup>T\<^sup>\<star>"
Gromov_completion_dist_comparison_1: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "ereal (dist x y) \<le> esqrt (extended_Gromov_distance x y)"
fixp_above_unfold: fixes lub :: "'a set \<Rightarrow> 'a"   and leq :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "bourbaki_witt_fixpoint lub leq f"   and "a \<in> Field leq" shows "bourbaki_witt_fixpoint.fixp_above lub leq f a = f (bourbaki_witt_fixpoint.fixp_above lub leq f a)"
pow_generator_mod: fixes k :: "nat" shows "\<^bold>g [^] (k mod order G) = \<^bold>g [^] k"
split: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('h, 'g) tllist \<Rightarrow> 'f"   and tllist :: "('h, 'g) tllist" shows "P (case tllist of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. tllist = TNil x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. tllist = TCons x21 x22 \<longrightarrow> P (f2 x21 x22)))"
stat_def: fixes C :: "'s set" shows "stat C = point_measure UNIV (\<lambda>x. indicator C x / U' x x)"
sel_2: fixes x21 :: "nat"   and x22 :: "'d"   and x23 :: "'d"   and x24 :: "'e"   and x25 :: "'e"   and x26 :: "'f" shows "occurs_on (Send x21 x22 x23 x24 x25 x26) = x22"
FDOM_state_succ_subset: fixes s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" shows "fmdom' (state_succ s a) \<subseteq> fmdom' s \<union> fmdom' (snd a)"
one_floatarith_def: shows "1 = floatarith.Num 1"
mtx_row_def: fixes sign_list :: "rat list list"   and index_list :: "nat list" shows "mtx_row sign_list index_list \<equiv> map (z index_list) sign_list"
nonzero_of_rat_divide: fixes b :: "rat"   and a :: "rat" assumes "b \<noteq> 0" shows "of_rat (a / b) = of_rat a / of_rat b"
VDMComp: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and A1 :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c2 :: "IMP"   and A2 :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "G \<rhd> c1 : A1"   and "G \<rhd> c2 : A2" shows "G \<rhd> Comp c1 c2 : (\<lambda>s t. \<exists>r. A1 s r \<and> A2 r t)"
comp_arr_ide: fixes a :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" assumes "ide a"   and "seq f a" shows "f \<cdot> a = f"
inf_sup_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> z = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
univalent_def: fixes R :: "('b \<times> 'c) set" shows "univalent R = (\<forall>x y z. (x, y) \<in> R \<and> (x, z) \<in> R \<longrightarrow> z = y)"
slengthE: fixes xs :: "[:'r:]" assumes "xs \<noteq> \<bottom>" obtains n :: "int" where "slength\<cdot>xs = MkI\<cdot>n" and "0 \<le> n"
create_element_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and document_ptr :: "(_) document_ptr"   and tag :: "char list" assumes "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs type_wf create_element known_ptr" shows "create_element document_ptr tag = Heap_Error_Monad.bind new_element (\<lambda>new_element_ptr. Heap_Error_Monad.bind (set_tag_name new_element_ptr tag) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes document_ptr (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr # disc_nodes)) (\<lambda>_. return new_element_ptr))))"
image_insert: fixes f :: "'f \<Rightarrow> 'e"   and a :: "'f"   and B :: "'f set" shows "f ` insert a B = insert (f a) (f ` B)"
tr_valid_preserve_s: fixes c :: "'n list multiset"   and e :: "('p, 'ba) label"   and c' :: "'n list multiset" assumes "(c, e, c') \<in> tr fg"   and "valid fg c" shows "valid fg c'"
fps_unit_factor_mult_fps_X_1: fixes f :: "'a fps" shows "unit_factor (fps_X * f) = unit_factor f"
seqSubsCons: fixes P :: "('a, 'b, 'c) psi"   and xvec :: "name list"   and Tvec :: "'a list"   and \<sigma> :: "(name list \<times> 'a list) list" shows "P[<((xvec, Tvec) # \<sigma>)>] = P[xvec::=Tvec][<\<sigma>>]"
truncate_aform_def: fixes p :: "nat"   and x :: "'a \<times> 'a pdevs" shows "truncate_aform p x = (eucl_truncate_down p (fst x), trunc_pdevs p (snd x))"
get_root_node_no_parent: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None" shows "h \<turnstile> get_root_node (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) \<rightarrow>\<^sub>r 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"
disc_rfr_proc_nonrandom: fixes r :: "real"   and n :: "nat"   and F :: "'d \<Rightarrow> 'c measure" shows "disc_rfr_proc r n \<in> borel_measurable (F (0::'d))"
compact_nest: fixes F :: "'b \<Rightarrow> 'c set" assumes "\<And>n. compact (F n)"   and "\<And>n. F n \<noteq> {}"   and "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m" shows "\<Inter> (range F) \<noteq> {}"
fresh_fact': fixes a :: "name"   and t2 :: "lam"   and t1 :: "lam" assumes "a \<sharp> t2" shows "a \<sharp> t1[a::=t2]"
bisim_bis: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set" assumes "Bisim.mono Retr" shows "bisim Retr (bis Retr)"
permute_minus_cancel_2: fixes p :: "perm"   and x :: "'a" shows "- p \<bullet> p \<bullet> x = x"
1: shows "\<forall>m<n. is_sum4sq_nat m"
lemma_1_i: fixes s :: "('c, 'd) fmap"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" assumes "s \<in> valid_states PROB"   and "a \<in> PROB" shows "state_succ s a \<in> valid_states PROB"
ide_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "partial_magma C" shows "partial_magma.ide C a \<equiv> C a a \<noteq> partial_magma.null C \<and> (\<forall>f. (C f a \<noteq> partial_magma.null C \<longrightarrow> C f a = f) \<and> (C a f \<noteq> partial_magma.null C \<longrightarrow> C a f = f))"
a_mult_idem: fixes x :: "('a \<times> 'a) set" shows "rel_ad x ; rel_ad x = rel_ad x"
Pd'_tax_linear: fixes d :: "'b" shows "Linear_order (Pd'_tax d)"
except_eq_zero_iff: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set" shows "(except p S = 0) = (keys p \<subseteq> S)"
filter_shuffles_disjoint2_2: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "set xs \<inter> set ys = {}"   and "zs \<in> shuffles xs ys" shows "filter (\<lambda>x. x \<notin> set ys) zs = xs"
lab: fixes accC :: "qtname"   and L :: "lenv"   and c :: "stmt"   and s0 :: "st"   and C :: "assigned"   and s1 :: "state"   and l :: "jump" assumes "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd (Norm s0))) \<guillemotright>In1r c\<guillemotright> C"   and "P L accC (Norm s0) (In1r c) \<diamondsuit> s1" shows "P L accC (Norm s0) (In1r (l\<bullet> c)) \<diamondsuit> (abupd (absorb l) s1)"
inv: shows "wp_inv G body \<guillemotleft> I \<guillemotright>"
IH: fixes S :: "(char list \<Rightarrow> int) set"   and S' :: "(char list \<Rightarrow> 'av) option"   and c' :: "(char list \<Rightarrow> 'av) option acom" assumes "S \<subseteq> \<gamma>\<^sub>o S'"   and "c1 \<le> \<gamma>\<^sub>c c'" shows "step S c1 \<le> \<gamma>\<^sub>c (step' S' c')"
noFaultStuck_Call_body': fixes p :: "'d"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e"   and F :: "'f set" assumes "p \<in> dom \<Gamma>" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) = \<Gamma>\<turnstile> \<langle>the (\<Gamma> p),Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)"
ih: assumes "set_mset Ys \<subseteq> A"   and "set_mset (mset xs) \<subseteq> A" shows "\<exists>Y X. Y \<noteq> {#} \<and> Y \<subseteq># Ys \<and> mset xs = Ys - Y + X \<and> (\<forall>x. x \<in># X \<longrightarrow> (\<exists>y. y \<in># Y \<and> gt y x))"