hcju/isabelleps huggingface dataset:
exec_move_CondI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e1 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and e2 :: "'addr expr1" assumes "exec_move ci P t e1 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (if (e) e1 else e2) h (stk, loc, Suc (length (compE2 e) + pc), xcp) ta h' (stk', loc', Suc (length (compE2 e) + pc'), xcp')"
unit_eq_div2: 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 "(a = divide c b) = (times a b = c)"
prems_1: shows "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var x) = \<Gamma> (Var y)"
list_all2_rev: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
even_succ_div_two: 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 "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "divide (plus a one) (numeral.numeral one plus (num.Bit0 num.One)) = divide a (numeral.numeral one plus (num.Bit0 num.One))"
distinct_tl: fixes xs :: "'e list" assumes "distinct xs" shows "distinct (tl xs)"
T_eq: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type"   and T' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> t : T"   and "T = T'" shows "\<Gamma> \<turnstile> t : T'"
symmetric_bibd_axioms: shows "symmetric_bibd \<V> \<B> \<k> \<Lambda>"
from_nat_mono': fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "i \<le> j"   and "j < CARD('b)" shows "less_eq (mod_type.from_nat Abs i) (mod_type.from_nat Abs j)"
\<R>_bisim: shows "strong_low_bisim_mm \<R> \<and> (\<langle>c\<^sub>1, mds\<^sub>1, mem\<^sub>1\<rangle>, \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>) \<in> \<R>"
fact'_1: fixes q :: "int poly" assumes "q \<in> set qs" shows "irreducible q \<and> 0 < lead_coeff q \<and> degree q \<le> degree p \<and> degree q \<noteq> 0"
summable: shows "AE x in M. summable (\<lambda>i. norm (f i x))"
dependent_live_vars_same_pathsD: fixes V :: "'var"   and as :: "'edge list"   and n' :: "'node" assumes "(V, as, as) \<in> dependent_live_vars n'"   and "as \<noteq> [] \<longrightarrow> n' = last (targetnodes as)" shows "V \<in> Use n' \<and> (\<forall>as' a as''. as = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n')"
nth_rotate1_Suc: fixes n :: "nat"   and ls :: "'a list" assumes "Suc n < |ls|" shows "ls ! Suc n = rotate1 ls ! n"
filter_eq_Cons_iff: fixes P :: "'a \<Rightarrow> bool"   and ys :: "'a list"   and x :: "'a"   and xs :: "'a list" shows "(filter P ys = x # xs) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
weak_ranking_Union: shows "\<Union> (set (weak_ranking le)) = carrier"
map_eq_imp_length_eq: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list"   and g :: "'g \<Rightarrow> 'e"   and ys :: "'g list" assumes "map f xs = map g ys" shows "length xs = length ys"
wf_monitor: fixes i :: "nat"   and v :: "'b option list"   and \<pi> :: "'a prefix" assumes "(i, v) \<in> M \<pi>" shows "wf_tuple nfv fv v"
emsapss_encode_help2_def: fixes M' :: "bit list"   and emBits :: "nat" shows "emsapss_encode_help2 M' emBits = emsapss_encode_help3 (sha1 M') emBits"
aezfun_setspanI: fixes x :: "('b, 'a) aezfun"   and A :: "'a set" assumes "supp (aezfun x) \<subseteq> A" shows "x \<in> aezfun_setspan A"
base: fixes f :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "f a \<noteq> b" shows "f a \<in> segment f a b"
matroid_ax_3_alt': fixes X :: "Points set"   and y :: "Points"   and z :: "Points" shows "rk (X \<union> {y}) = rk (X \<union> {z}) \<longrightarrow> rk (X \<union> {z}) = rk X \<longrightarrow> rk X = rk (X \<union> {y, z})"
assms_3: shows "A *\<^sub>v (A\<^sup>H *\<^sub>v v) = 0\<^sub>v nr"
store_sops_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "nat set \<times> (nat \<Rightarrow> bool option)" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>\<Union> (fst ` store_sops is). i < t"
antilinear_from_conjugate_space: shows "antilinear from_conjugate_space"
sound_system_def: fixes \<Sigma> :: "ground_action set"   and M\<^sub>0 :: "object atom formula set"   and s\<^sub>0 :: "predicate \<times> object list \<Rightarrow> bool"   and f :: "ground_action \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option" shows "sound_system \<Sigma> M\<^sub>0 s\<^sub>0 f = ((\<forall>fmla\<in>close_world M\<^sub>0. close_eq s\<^sub>0 \<Turnstile> fmla) \<and> wm_basic M\<^sub>0 \<and> (\<forall>\<alpha>\<in>\<Sigma>. sound_opr \<alpha> (f \<alpha>)))"
abc_list_crsp_simp3: fixes lma :: "nat list"   and lmb :: "nat list"   and m :: "nat"   and n :: "nat" assumes "abc_list_crsp lma lmb"   and "\<not> m < length lma"   and "m < length lmb" shows "abc_list_crsp (lma @ 0 \<up> (m - length lma) @ [n]) (lmb[m := n])"
simps_7: fixes M :: "'b trm"   and N :: "'b trm"   and M' :: "'b trm"   and N' :: "'b trm" shows "unify (M \<cdot> N) (M' \<cdot> N') = (case unify M M' of None \<Rightarrow> None | Some \<theta> \<Rightarrow> case unify (N \<lhd> \<theta>) (N' \<lhd> \<theta>) of None \<Rightarrow> None | Some \<sigma> \<Rightarrow> Some (\<theta> \<lozenge> \<sigma>))"
max_bot2: fixes x :: "'a list" shows "suffix_order.max x [] = x"
vs_def: shows "vs \<equiv> sorted_list_of_set \<circ> FV \<phi>"
sum_upto_moebius_mu_integrable: fixes a :: "real"   and b :: "real" assumes "1 \<le> a" shows "(\<lambda>t. M t / t) integrable_on {a..b}"
admissible_dom_prog: fixes P :: "'x \<Rightarrow> 'y \<Rightarrow> 'y \<Rightarrow> 'z \<Rightarrow> bool" shows "dom_prog.admissible (\<lambda>f. \<forall>x h h' r. h \<turnstile> f x \<rightarrow>\<^sub>r r \<longrightarrow> h \<turnstile> f x \<rightarrow>\<^sub>h h' \<longrightarrow> P x h h' r)"
cfun_arg_cong: fixes x :: "'c"   and y :: "'c"   and f :: "'c \<rightarrow> 'd" assumes "x = y" shows "f\<cdot>x = f\<cdot>y"
irreflexive_transitive_orientation_asymmetric: fixes x :: "'a"   and y :: "'a" assumes "irreflexive_inf x"   and "transitive y"   and "y \<squnion> y\<^sup>T = x" shows "asymmetric y"
order_iff: fixes b :: "'a word"   and a :: "'a word" shows "(b \<le>s a) = (a = signed.max a b)"
rassume_def: fixes c :: "'b upred" shows "?[c] = II \<triangleleft> c \<triangleright>\<^sub>r false"
is_ground_clss_union: fixes CC :: "'a literal multiset set"   and DD :: "'a literal multiset set" shows "is_ground_clss (CC \<union> DD) = (is_ground_clss CC \<and> is_ground_clss DD)"
dom_eqI: fixes a :: "'a"   and f :: "'a" assumes "Left_a.ide a"   and "Left_a.seq f a" shows "Left_a.dom f = a"
vcard_finite_set: fixes A :: "V set" assumes "finite A" shows "vcard (set A) = card A\<^sub>\<nat>"
F_deriv: fixes x :: "real" assumes "x \<in> {real_of_int a..} - Y" shows "(F has_vector_derivative f x) (at x)"
fold_mset_fun_left_comm: fixes x :: "'a"   and s :: "'b"   and M :: "'a multiset" shows "f x (fold_mset f s M) = fold_mset f (f x s) M"
Assign_is_CSP: fixes x :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and e :: "'d \<Rightarrow> 'e" shows "is_CSP_process (R (true \<turnstile> \<lambda>(S, S'). tr S' = tr S \<and> \<not> wait S' \<and> alpha_rp.more S' = update x (\<lambda>_. e (alpha_rp.more S)) (alpha_rp.more S)))"
hom_components_def: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "hom_components p = hom_component p ` deg_pm ` keys p"
prime_multiplicity_gt_zero_iff: fixes p :: "'a"   and x :: "'a" assumes "prime_elem p"   and "x \<noteq> (0::'a)" shows "(0 < multiplicity p x) = (p dvd x)"
can_select_def: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b set" shows "can_select P A = (\<exists>!x. x \<in> A \<and> P x)"
sel: fixes r :: "('f, 'd, 'e) gpv"   and f :: "'f \<Rightarrow> ('b, 'd, 'e) gpv" shows "the_gpv (r \<bind> f) = map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. r \<bind> f)))) (the_gpv r \<bind> case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x))) (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))"
simps_7: fixes r :: "'b rexp"   and c :: "'b"   and v :: "'b val"   and vs :: "'b val list" shows "injval (Star r) c (Seq v (Stars vs)) = Stars (injval r c v # vs)"
necessitation_1: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "\<And>s. P s" shows "(\<box>P) \<sigma>"
foldl_spmf_chain: fixes fff :: "'t \<Rightarrow> 'u \<Rightarrow> ('s list \<times> 't) spmf"   and s_event :: "'t"   and ievents :: "'u list"   and ggg :: "'v \<Rightarrow> 's \<Rightarrow> 'v spmf"   and s_core :: "'v"   and f :: "'v \<Rightarrow> 't \<Rightarrow> 'r" shows "foldl_spmf (\<lambda>(oevents, s_event) event. map_spmf (map_prod ((@) oevents) id) (fff s_event event)) (return_spmf ([], s_event)) ievents \<bind> (\<lambda>(oevents, s_event'). foldl_spmf ggg (return_spmf s_core) oevents \<bind> (\<lambda>s_core'. return_spmf (f s_core' s_event'))) = foldl_spmf (\<lambda>(s_event, s_core) event. fff s_event event \<bind> (\<lambda>(oevents, s_event'). map_spmf (Pair s_event') (foldl_spmf ggg (return_spmf s_core) oevents))) (return_spmf (s_event, s_core)) ievents \<bind> (\<lambda>(s_event', s_core'). return_spmf (f s_core' s_event'))"
discounted_value_def: fixes r :: "real"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "discounted_value r X = (\<lambda>n w. discount_factor r n w * X n w)"
ewu: shows "awalk s p' u \<and> awalk u [e] v"
subgroup_subgroup_generated: fixes G :: "('f, 'g) monoid_scheme"   and S :: "'f set" assumes "Group.group G" shows "subgroup (carrier (subgroup_generated G S)) G"
bheight_baliL: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "bheight l = bheight r" shows "bheight (baliL l a r) = Suc (bheight l)"
recurrence_series_infinite: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> ennreal" assumes "conservative M T"   and "f \<in> borel_measurable M" shows "AE x in M. 0 < f x \<longrightarrow> (\<Sum>n. f ((T ^^ n) x)) = \<infinity>"
bind_lunit: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b option" shows "\<lfloor>x\<rfloor> \<bind> f = f x"
encode_problem_parallel_complete_iv_b: 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))\<inverse>, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<^sup>+} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (delete_effects_of op)}}})" shows "clause_semantics \<A> C"
simTrans: shows "\<forall>a ec t. t \<in> jkbpC \<and> simAbs ec = sim_equiv_class a t \<longrightarrow> simAbs ` set (simTrans a ec) = {sim_equiv_class a (t' \<leadsto> s) |t' s. t' \<leadsto> s \<in> jkbpC \<and> jview a t' = jview a t}"
bangLeftSC: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "!P \<leadsto>[Rel] P \<parallel> !P"
T_im: fixes v :: "'c" assumes "v \<in> carrier V" shows "T v \<in> carrier W"
equivalence_map_is_ide: fixes f :: "'b" assumes "D.equivalence_map f" shows "D.ide f"
var_dist_append1: fixes vs :: "nat list"   and v :: "nat"   and t :: "dB" shows "var_dist (vs @ [v]) t = var_dist vs t \<degree> Var v"
subst_atm_mset_list_nth: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and i :: "nat"   and AAs :: "'b multiset list"   and \<eta> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms"   and "i < length AAs" shows "substitution_ops.subst_atm_mset_list subst_atm AAs \<eta> ! i = substitution_ops.subst_atm_mset subst_atm (AAs ! i) \<eta>"
P: shows "P \<equiv> P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>1,Up\<^sub>1\<rbrakk>"
assms_2: shows "Fubx_get F2 f = Some fd"
kleene_fixed_point_theorem: fixes psi :: "recf"   and k :: "nat" assumes "recfn 2 psi" shows "\<exists>n\<ge>k. \<forall>x. eval r_phi [n, x] = eval psi [n, x]"
Lebesgue_number_lemma: fixes S :: "'c set"   and \<C> :: "'c set set" assumes "compact S"   and "\<C> \<noteq> {}"   and "S \<subseteq> \<Union> \<C>"   and "\<And>B. B \<in> \<C> \<Longrightarrow> open B" obtains \<delta> :: "real" where "0 < \<delta>" and "\<And>T. \<lbrakk>T \<subseteq> S; diameter T < \<delta>\<rbrakk> \<Longrightarrow> \<exists>B\<in>\<C>. T \<subseteq> B"
rel_reflp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "reflp R" shows "reflp (llist_all2 R)"
listsE: fixes x :: "'a"   and l :: "'a list"   and A :: "'a set" assumes "x # l \<in> lists A" obtains "x \<in> A" and "l \<in> lists A"
lambda_skolem_euclidean: fixes P :: "'a \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> bool" shows "(\<forall>i\<in>Basis. \<exists>f'. P i f') = (\<exists>f'. \<forall>i\<in>Basis. P i (\<lambda>x. f' x \<bullet> i))"
compet_def: fixes A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and c :: "real"   and S :: "'state set" shows "compet A c S = (\<forall>s\<in>S. \<exists>b\<ge>0. \<forall>rs. wf s rs \<longrightarrow> real (T_on A s rs) \<le> c * real (T_opt s rs) + b)"
f: shows "bij_betw ((#) 0) {l. length l = m - 1 \<and> sum_list l = N} {l. length l = m \<and> sum_list l = N \<and> hd l = 0}"
distrib_lattice_ow_def: fixes U :: "'c set"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ls :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "distrib_lattice_ow U inf le ls sup \<equiv> lattice_ow U inf le ls sup \<and> distrib_lattice_ow_axioms U inf sup"
vfind_match_elem_1: fixes cs :: "(Term_Class.term \<times> sterm) list"   and t :: "value"   and env :: "(name, value) fmap"   and pat :: "Term_Class.term"   and rhs :: "sterm" assumes "vfind_match cs t = Some (env, pat, rhs)" shows "(pat, rhs) \<in> set cs"
bound_L1_Lp_1: fixes M :: "'b measure"   and p :: "real"   and f :: "'b \<Rightarrow> real" assumes "prob_space M"   and "1 \<le> p"   and "f \<in> borel_measurable M"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)" shows "integrable M f"
K2_communicating: fixes c :: "'s"   and x :: "'s option" assumes "c \<in> C" shows "((Some c, x) \<in> K2.communicating) = (\<exists>c'\<in>C. x = Some c')"
a_parent_child_rel_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_parent_child_rel get_child_nodes h = {(parent, child). parent |\<in>| object_ptr_kinds h \<and> child \<in> 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 ` set |h \<turnstile> get_child_nodes parent|\<^sub>r}"
awp_def: fixes F :: "'b program"   and B :: "'b set" shows "awp F B \<equiv> \<Inter>act\<in>Acts F. wp act B"
vpathI: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph" assumes "vwalk p G"   and "distinct p" shows "vpath p G"
subdg_objD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
intfAPP_def: fixes f :: "interface \<Rightarrow> 'a"   and x :: "interface" shows "\<langle>x\<rangle>\<^sub>if \<equiv> f x"
nth_equal_first_eq: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" assumes "x \<notin> set xs"   and "n \<le> length xs" shows "((x # xs) ! n = x) = (n = 0)"
one_dim_iso_adjoint: shows "one_dim_iso\<^sup>\<dagger> = one_dim_iso"
insert_remove: fixes A :: "'q set"   and x :: "'q" assumes "finite A" shows "card (insert x A) = Suc (card (A - {x}))"
simps_6: fixes unop :: "unop"   and e :: "expr" shows "constVal (UnOp unop e) = (case constVal e of None \<Rightarrow> None | Some v \<Rightarrow> Some (eval_unop unop v))"
gConsIndif_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gConsIndif MOD \<equiv> gOpIndif MOD \<and> gAbsIndif MOD"
nth_subdegree_zero_iff: fixes f :: "'a fps" shows "(fps_nth f (subdegree f) = (0::'a)) = (f = 0)"
inf_in_circline_set: fixes H :: "circline" shows "(\<infinity>\<^sub>h \<in> circline_set H) = is_line H"
limit_single_step: fixes x :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and X :: "'a set" assumes "x \<in> f X" shows "x \<in> limit f X"
assms_2: shows "set ps = ps\<^sub>L \<union> ps\<^sub>R"
sig_red_singleD1: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "rep_list f \<noteq> 0"
lhr_fps: fixes cs :: "'b list"   and m :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "cs \<noteq> []"   and "\<And>n. m \<le> n \<Longrightarrow> (\<Sum>k\<le>length cs - 1. cs ! k * f (n + k)) = (0::'b)"   and "last cs \<noteq> (0::'b)" shows "Abs_fps f = fps_of_poly (lhr_fps_numerator m cs f) / fps_of_poly (lr_fps_denominator cs)"
dagger_subdist_var: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<dagger> + y\<^sup>\<dagger> \<le> (x + y)\<^sup>\<dagger>"
value_mI_gen1: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)" shows "\<forall>j\<le>n. (\<nu>\<^bsub>K P j\<^esub>) (mIg\<^bsub> K n P I\<^esub>) = LI K (\<nu>\<^bsub>K P j\<^esub>) I"
finite: fixes s :: "('a, unit) RBT.rbt" assumes "rs.invar s" shows "finite (rs.\<alpha> s)"
psi_in_P2: shows "\<psi> \<in> \<P>\<^sup>2"
A4: shows "A4 \<in> carrier_mat 1 m2"
L_4cases_def: fixes x :: "nat"   and y :: "nat" shows "L_4cases x y = verund [seq [question (Atom x), Atom y, Atom y], seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y], seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x], seq [Atom x, Atom x]]"
Lp_D_1: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> space\<^sub>N (\<LL> (ennreal p) M)" shows "f \<in> borel_measurable M"
app_vfieldI1: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<union>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r"
sup_bot_left: fixes x :: "'d multiset" shows "{#} \<union># x = x"
simps_12: fixes sh :: "sheap"   and C :: "char list"   and M :: "char list"   and es :: "'a exp list" shows "iconf sh (C\<bullet>\<^sub>sM(es)) = iconfs sh es"
matrix_vector_all_zero_from_index2: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and k :: "'c" shows "(\<forall>m>i. A $ m $ k = (0::'b)) = vector_all_zero_from_index (mod_type_class.to_nat i + 1, vec_to_iarray (column k A))"
DisjunctiveD: fixes x :: "'c"   and Sup_b :: "'d set \<Rightarrow> 'd"   and Sup_c :: "'e set \<Rightarrow> 'e"   and times_abc :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and X :: "'d set" assumes "x \<in> Disjunctive.Disjunctive Sup_b Sup_c times_abc" shows "times_abc x (Sup_b X) = Sup_c (times_abc x ` X)"
simps_1: fixes n :: "nat" shows "take n \<epsilon> = \<epsilon>"
fps_nth_Abs_fps: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "Abs_fps f $ n = f n"
square_free_factorizationD_1: fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "local.square_free_factorization p (c, bs)" shows "p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
sbintrunc_mod2p: fixes n :: "nat"   and w :: "int" shows "signed_take_bit n w = (w + 2 ^ n) mod 2 ^ Suc n - 2 ^ n"
simps_2: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" shows "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
simps_4: fixes \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.future_reach (formula.Or \<phi> \<psi>) = max (MFOTL.future_reach \<phi>) (MFOTL.future_reach \<psi>)"
eq_iff: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "pseudo_ordered_set A less_eq"   and "x \<in> A"   and "y \<in> A" shows "(x = y) = (less_eq x y \<and> less_eq y x)"
read_only_share: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "read_only (share sb \<S>) \<subseteq> read_only \<S> \<union> \<O> \<union> all_acquired sb"
compatible_iff_seq: fixes g :: "'a"   and f :: "'a" shows "(C.domains g \<inter> C.codomains f \<noteq> {}) = C.seq g f"
Suc: shows "poly_mult (poly_mult (local.monom a n) p) q = poly_mult (local.monom a n) (poly_mult p q)"