hcju/isabelleps huggingface dataset:
simps: fixes a :: "nat \<times> 'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" shows "(a \<in> his t n nid) = (a = (n, nid) \<and> \<parallel>nid\<parallel>\<^bsub>t n\<^esub> \<or> (\<exists>n' nid'. a = (SOME x. (\<exists>n. latestAct_cond nid' t n' n) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>nid' \<leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>nid'\<^esub>t n')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>nid'\<^esub>t n') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>nid'\<^esub>t n')))) \<and> (n', nid') \<in> his t n nid \<and> (\<exists>x. (\<exists>n. latestAct_cond nid' t n' n) \<and> \<parallel>snd x\<parallel>\<^bsub>t (fst x)\<^esub> \<and> fst x = \<langle>nid' \<leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> (prefix (bc (\<sigma>\<^bsub>nid'\<^esub>t n')) (bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x))) \<or> (\<exists>b. bc (\<sigma>\<^bsub>nid'\<^esub>t n') = bc (\<sigma>\<^bsub>snd x\<^esub>t (fst x)) @ [b] \<and> mining (\<sigma>\<^bsub>nid'\<^esub>t n'))))))"
le_supI: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and x :: "'b"   and b :: "'b" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a x"   and "less_eq b x" shows "less_eq (sup a b) x"
conjoin_def: fixes c :: "('b com option list \<times> 'b) list"   and clist :: "('b com option \<times> 'b) list list" shows "c \<propto> clist \<equiv> same_length c clist \<and> same_state c clist \<and> same_program c clist \<and> compat_label c clist"
abs_eq: fixes xa :: "'c set"   and x :: "'c set" assumes "eq_onp ord_class.filter xa xa"   and "eq_onp ord_class.filter x x" shows "Filters.filter.Abs_filter xa \<sqinter> Filters.filter.Abs_filter x = Filters.filter.Abs_filter (xa \<inter> x)"
var_not_pair: fixes x :: "'b"   and A :: "'b trm"   and B :: "'b trm" shows "Var x \<noteq> SimplyTyped.Pair A B"
uniform_alt_def_all: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size" shows "\<forall>bl\<in>#block_collection. int (card bl) = u_block_size"
append_Nil2: fixes xs :: "'c list" shows "xs @ [] = xs"
pnet_net_ips_net_tree_ips: fixes s :: "'a net_state"   and np :: "nat \<Rightarrow> ('a, 'b seq_action) automaton"   and p :: "net_tree"   and I :: "'b node_action \<Rightarrow> bool" assumes "s \<in> reachable (pnet np p) I" shows "net_ips s = net_tree_ips p"
spec2: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" assumes "\<forall>x y. P x y" shows "P x y"
exec_gpv_conv_inline1: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and gpv :: "('d, 'f, 'g) gpv"   and s :: "'e" shows "exec_gpv callee gpv s = map_spmf projl (inline1 (\<lambda>s c. lift_spmf (callee s c)) gpv s)"
is_interval_1: fixes s :: "real set" shows "is_interval s = (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> s)"
finprod_exp: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and k :: "int" assumes "A \<subseteq> carrier G"   and "f \<in> A \<rightarrow> carrier G" shows "finprod G f A [^] k = finprod G ((\<lambda>a. a [^] k) \<circ> f) A"
all_args_def: fixes f :: "'o \<Rightarrow> ('l, 'm, 'n) prog" shows "all_args f = (\<Union>arg. {f arg})"
map_replicate_trivial: fixes x :: "'d"   and i :: "nat" shows "map (\<lambda>i. x) [0..<i] = replicate i x"
ide_pbdom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "partial_magma.ide C (elementary_category_with_pullbacks.pbdom C prj0 f g)"
card_Q: shows "real (card Q) = \<pi> (real n) - \<pi> (real n powr \<alpha>)"
conditionalize: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A" shows "prod g A = (\<Prod>a. if a \<in> A then g a else (1::'c))"
dvd_div_eq_mult: 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 a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "dvd.dvd times a b" shows "(divide b a = c) = (b = times c a)"
lowlink_more_cong: fixes s :: "('c, 'd) tarjan_state_scheme"   and s' :: "('c, 'd) tarjan_state_scheme" assumes "state.more s = state.more s'" shows "lowlink s = lowlink s'"
basis_left_null_space_def: fixes A :: "(('a, 'c) vec, 'b) vec" shows "basis_left_null_space A = {row i (P_Gauss_Jordan A) |i. rank A \<le> mod_type_class.to_nat i}"
11: shows "path s\<^sub>2 (target s\<^sub>1 q)"
simps_5: fixes c :: "('p, 's, 'm) configuration"   and i :: "nat"   and p :: "'p"   and q :: "'p"   and s :: "'s"   and s' :: "'s"   and msg :: "'m"   and c' :: "('p, 's, 'm) configuration" shows "c \<turnstile> Recv i p q s s' msg \<mapsto> c' = (can_occur (Recv i p q s s' msg) c \<and> (states c p = s \<and> states c' p = s') \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c i = Msg msg # msgs c' i \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>r. ps c r = ps c' r) \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> cs c i' = cs c' i') \<and> (if snd (cs c i) = Recording then cs c' i = (fst (cs c i) @ [msg], Recording) else cs c i = cs c' i))"
chain_boundary_add: fixes p :: "nat"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "chain_boundary p (a + b) = chain_boundary p a + chain_boundary p b"
count_greater_zero_iff: fixes M :: "'k multiset"   and x :: "'k" shows "(0 < count M x) = (x \<in># M)"
inf_concat_Suc: fixes n :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "\<exists>\<^sub>\<infinity>i. 0 < n i"   and "\<And>i. f i (n i) = f (Suc i) 0"   and "inf_concat n k = (i, j)"   and "inf_concat n (Suc k) = (i', j')" shows "f i' j' = f i (Suc j)"
poly_roots_prod_mset: fixes A :: "'a poly multiset" assumes "0 \<notin># A" shows "poly_roots (\<Prod>\<^sub># A) = \<Sum>\<^sub># (image_mset poly_roots A)"
hyps_2: shows "\<not> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
mk_linear_orders_decisive_on_total_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and r :: "'a list"   and C :: "'a set" assumes "f_range_on A f"   and "r \<in> mk_linear_orders f C A"   and "decisive_on A f" shows "total_on A (linord_of_list r)"
sym_mpoly_altdef: fixes A :: "nat set"   and k :: "nat" shows "sym_mpoly A k = (\<Sum>X | X \<subseteq> A \<and> card X = k. MPoly_Type.monom (monom_of_set X) (1::'b))"
emeasure_join: fixes M :: "'c measure measure"   and N :: "'c measure"   and A :: "'c set" assumes "sets M = sets (subprob_algebra N)"   and "A \<in> sets N" shows "emeasure (join M) A = \<integral>\<^sup>+ M'. emeasure M' A \<partial>M"
columnvector_row_def: fixes v :: "('d, 'f) vec"   and i :: "'f" shows "columnvector_row v i = (\<chi>j. v $ i)"
cone_axioms: shows "D'.cone a \<tau>o\<chi>.map"
zero_less_divide_1_iff: fixes a :: "'a" shows "((0::'a) < (1::'a) / a) = ((0::'a) < a)"
assms: shows "w \<Turnstile>\<^sub>n \<phi>"
that: fixes d' :: "'b" assumes "u(c := d') \<turnstile>\<^bsub>v,n\<^esub> M" shows "thesis"
lemma_2_10_26: fixes a :: "'a"   and b :: "'a" shows "a \<le> (a r\<rightarrow> b) l\<rightarrow> b"
lemma3: fixes r :: "dB"   and r' :: "dB"   and s :: "dB"   and s' :: "dB"   and x :: "nat" assumes "r \<rightarrow>\<^sub>s r'"   and "s \<rightarrow>\<^sub>s s'" shows "r[s/x] \<rightarrow>\<^sub>s r'[s'/x]"
strict_prefixE: fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix xs ys" obtains "prefix xs ys" and "xs \<noteq> ys"
variation_def: fixes x :: "real"   and y :: "real" shows "variation x y = (if 0 \<le> x * y then 0 else if x < y then 1 else - 1)"
mapCollect_def: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and m :: "'f \<Rightarrow> 'g option" shows "mapCollect f m = {f k v |k v. m k = Some v}"
s_mul_ext_mono: fixes ns :: "('a \<times> 'a) set"   and ns' :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and s' :: "('a \<times> 'a) set" assumes "ns \<subseteq> ns'"   and "s \<subseteq> s'" shows "s_mul_ext ns s \<subseteq> s_mul_ext ns' s'"
mp: shows "cf_pos_poly (poly_mult_rat r1 p2) \<noteq> 0"
hoare_while_mbt: fixes p :: "'b \<Rightarrow> 'c Assertion"   and b :: "'c Assertion"   and x :: "'c"   and q :: "'c Assertion"   and w :: "'b" assumes "\<forall>w r. (\<forall>v<w. p v \<le> r) \<longrightarrow> hoare (p w \<sqinter> b) x r"   and "\<forall>u. p u \<le> q" shows "hoare (p w) (While b do x) (q \<sqinter> - b)"
funs_term_Fun_subterm: fixes f :: "'c"   and t :: "('c, 'd) Term.term" assumes "f \<in> funs_term t" shows "\<exists>T. Fun f T \<sqsubseteq> t"
\<tau>Runs_table_into_\<tau>Runs: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g"   and stlsss :: "('h \<times> 'g, 'g option) tllist" assumes "\<tau>trsys.\<tau>Runs_table trsys \<tau>move s stlsss" shows "\<tau>trsys.\<tau>Runs trsys \<tau>move s (tmap fst id stlsss)"
distinct_1: fixes x1 :: "'a"   and x21 :: "'a"   and x22 :: "'b"   and x23 :: "'c" shows "Idle x1 \<noteq> State x21 x22 x23"
lfp_while: fixes f :: "'a set \<Rightarrow> 'a set"   and C :: "'a set" assumes "mono f"   and "\<And>X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C"   and "finite C" shows "lfp f = while (\<lambda>A. f A \<noteq> A) f {}"
Domain_Int_subset: fixes A :: "('a \<times> 'b) set"   and B :: "('a \<times> 'b) set" shows "Domain (A \<inter> B) \<subseteq> Domain A \<inter> Domain B"
V_subset_simp: fixes v :: "nat" assumes "v \<in> set V" shows "v \<in> \<V> g"
disc_nodes_document_ptr_h2: shows "h2 \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes_h3"
LNil_eq_lappend_iff: fixes xs :: "'a llist"   and ys :: "'a llist" shows "(LNil = lappend xs ys) = (xs = LNil \<and> ys = LNil)"
assms: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R - {\<zero>}"   and "b \<in> carrier R - {\<zero>}" shows "\<exists>q r. q \<in> carrier R \<and> r \<in> carrier R \<and> a = b \<otimes> q \<oplus> r \<and> (r = \<zero> \<or> \<phi> r < \<phi> b)"
Liminf_upto_llist_eq_Liminf_llist_ltake: fixes Xs :: "'b set llist"   and j :: "enat" shows "Liminf_upto_llist Xs j = Liminf_llist (ltake (eSuc j) Xs)"
z: shows "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p"
param_rev: fixes R :: "('g \<times> 'h) set" shows "(rev, rev) \<in> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel"
g: shows "\<forall>x\<in>S. (\<lambda>n. f n x) \<longlonglongrightarrow> g x"
one_or_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "or one a = plus a (zero_neq_one.of_bool one zero (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a))"
order_zmset_exists_foundation: fixes M :: "'a zmultiset"   and t :: "'a" assumes "0 < zcount M t" shows "\<exists>s\<le>t. 0 < zcount M s \<and> (\<forall>u. 0 < zcount M u \<longrightarrow> \<not> u < s)"
assms_1: shows "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (new_shadow_root_ptr, h')"
pdg_dghm_proj_is_dghm: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i : dg_prod I \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA> i"
access_alloc: fixes t :: "New"   and l :: "Location"   and s :: "Store" assumes "isNewArr t \<longrightarrow> l \<noteq> arr_len (new s t)" shows "s\<langle>t\<rangle>@@l = s@@l"
l_cancel_one': fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier H'"   and "a \<in> carrier H'" shows "(x = x \<otimes>\<^bsub>H'\<^esub> a) = (a = \<one>\<^bsub>H'\<^esub>)"
simps_4: fixes uy :: "Formula.trm"   and uz :: "Formula.trm" shows "Formula.future_bounded (formula.Less uy uz) = True"
up_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" shows "up M \<equiv> \<lambda>i j. if 0 < i then if j = 0 then \<infinity> else min (dbm_add (M i 0) (M 0 j)) (M i j) else M i j"
A'0: fixes i :: "nat" assumes "i < n - 1" shows "A' $$ (Suc i, 0) = (0::'a)"
assms_3: shows "is_parallel_solution_for_problem \<Pi> \<pi>"
Y_def: shows "Y = {(if x $ n = (0::'a) then 1::'a else (1::'a) / x $ n) \<cdot>\<^sub>v x |x. x \<in> X}"
elims_2: fixes x :: "'b list"   and xa :: "'b list set" assumes " x \<in>B xa " obtains b :: "'b list" and G :: "'b list set" where "x = b" and "xa = G" and "b \<in> G \<and> (\<forall>us. us \<in> lists G\<^sub>+ \<and> concat us = b \<longrightarrow> \<^bold>|us\<^bold>| = 1)"
equiv_restrict: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and B :: "'a set" assumes "equiv A r" shows "equiv (A \<inter> B) (Restr r B)"
tc_def: fixes x :: "'a" shows "tc x = 1' \<cdot> - x"
Float_num_5: shows "real_of_float (Float 1 (- 2)) = 1 / 4"
assms: shows "sources \<mu> \<inter> sources \<nu> \<noteq> {}"
span_subsetI: fixes ws :: "'a vec set"   and us :: "'a vec set" assumes "ws \<subseteq> carrier_vec n"   and "us \<subseteq> local.span ws" shows "local.span us \<subseteq> local.span ws"
J_rad_mem: fixes x :: "'a" assumes "x \<in> J_rad R" shows "x \<in> carrier R"
map_poly_comm_ring_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "map_poly_comm_ring_hom hom \<equiv> comm_ring_hom hom"
ide_char: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "ide f = (Dom f = Cod f \<and> Fun f = (\<lambda>x\<^bold>\<in>Dom f. x))"
\<R>\<^sub>1_weak_bisim: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" shows "weak_bisim (\<R>\<^sub>1 \<Gamma>' \<S>' P') (\<R> \<Gamma>' \<S>' P')"
has_as_limit_iff_terminal: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c"   and a :: "'c" assumes "empty_diagram J C D" shows "(\<exists>\<chi>. limit_cone J C D a \<chi>) = category.terminal C a"
distinct_11: fixes x3 :: "'b formula" shows "\<bottom> \<noteq> \<^bold>\<not> x3"
D_3: shows "[(l, u)] \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta> = [(l, s')]"
simps_10: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = (\<forall>i. suffix i \<xi> \<Turnstile>\<^sub>n \<phi> \<or> (\<exists>j\<le>i. suffix j \<xi> \<Turnstile>\<^sub>n \<psi>))"
pderiv_minus: fixes p :: "'a poly" shows "pderiv (- p) = - pderiv p"
l_set_disconnected_nodes_get_ancestors_axioms: shows "l_set_disconnected_nodes_get_ancestors set_disconnected_nodes_locs get_ancestors_locs"
sym_sum_mset_inter_sets_count: fixes blv :: "'a set"   and p :: "'a" assumes "blv \<in># \<B>"   and "p \<in> blv" shows "int (count (\<Sum>bl\<in>#remove1_mset blv \<B>. mset_set (bl \<inter> blv)) p) = \<Lambda> * (\<v> - 1) div (\<k> - 1) - 1"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'b" assumes "Network c s t"   and "RGraph_axioms c s t cf" shows "RGraph c s t cf"
cases: fixes x :: "('b \<times> nat) tree" obtains "x = \<langle>\<rangle>"   | uu_ :: "('b \<times> nat) tree" and uv_ :: "'b" and lv :: "nat" and uw_ :: "('b \<times> nat) tree" where "x = \<langle>uu_, (uv_, lv), uw_\<rangle>"
satisfiable_antimono: fixes CC :: "'b literal multiset set"   and DD :: "'b literal multiset set" assumes "CC \<subseteq> DD"   and "satisfiable DD" shows "satisfiable CC"
simps_1: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "Unf (\<phi> U\<^sub>n \<psi>) = (\<phi> U\<^sub>n \<psi> and\<^sub>n Unf \<phi>) or\<^sub>n Unf \<psi>"
nxt_def: fixes \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "\<circle>\<^sub>b\<gamma> \<equiv> \<lambda>t n. \<gamma> t (Suc n)"
null_fun_def: shows "null \<equiv> \<lambda>x. null"
lexordp_irreflexive: fixes xs :: "'a list" assumes "\<forall>x. \<not> x < x" shows "\<not> lexordp xs xs"
chine_simps_3: shows "C.cod \<nu>'.chine = \<nu>'.cod.apex"
JF1col_0: fixes x :: "'d JF1" shows "JF1col 0 x = {}"
N: fixes p :: "nat" assumes "N < p" shows "\<exists>m>p. u m < u p"
red2_simps_3: fixes u :: "'a local.term"   and v :: "'a local.term" assumes "Ide u"   and "Nml u"   and "Ide v"   and "Nml v"   and "Src u = Trg v" shows "Dom (u \<^bold>\<Down> v) = u \<^bold>\<star> v"
euler_maclaurin_strong_nat': fixes b :: "nat" assumes "1 \<le> b" shows "(\<Sum>k = 1..b. 1 / (real k)\<^sup>2) = - 1 / real b + pi\<^sup>2 / 6 + (1 / 2) *\<^sub>R (1 / (real b)\<^sup>2) + (\<Sum>i = 1..N. (bernoulli (2 * i) / fact (2 * i)) *\<^sub>R fs (2 * i - 1) (real b)) - EM_remainder (2 * N + 1) (fs (2 * N + 1)) (int b)"
Joints_map_split_aform: fixes x :: "(real \<times> real pdevs) list"   and i :: "nat" shows "Joints x \<subseteq> Joints (map (\<lambda>a. fst (split_aform a i)) x) \<union> Joints (map (\<lambda>b. snd (split_aform b i)) x)"
sas_plus_parallel_plan_to_strips_parallel_plan_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list list" shows "\<phi>\<^sub>P \<Psi> \<psi> \<equiv> map (map (sasp_op_to_strips \<Psi>)) \<psi>"
conjugate_strict: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "conjugate f \<bottom> = \<bottom>"
list_digraph_ext_def: fixes ext :: "'d"   and G_list :: "'c list \<times> ('c \<times> 'c) list" shows "list_digraph_ext ext G_list \<equiv> \<lparr>pverts = set (fst G_list), parcs = set (snd G_list), \<dots> = ext\<rparr>"
rw0: shows "(\<Sum>i<n. four_block_mat A B C D $$ (i, n) * Determinant.cofactor (four_block_mat A B C D) i n) = (0::'a)"
sum_over_product_left: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "O y z" shows "P (x \<oplus> y \<otimes> z) ((x \<oplus> y) \<otimes> (x \<oplus> z))"
lex_pm_lin: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "lex_pm s t \<or> lex_pm t s"
less_imp_le: fixes x :: "'c multiset"   and y :: "'c multiset" assumes "x \<subset># y" shows "x \<subseteq># y"
has_bottomI: fixes z :: "'b"   and P :: "'b set" assumes "z \<in> P"   and "\<And>x. x \<in> P \<Longrightarrow> order.greater_eq x z" shows "order.has_bottom P"
nprv_clear2_2: fixes \<phi>1 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>2 :: "'fmla" assumes "nprv {\<phi>1} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2} \<psi>"
bot': fixes f :: "'b \<Rightarrow> 'c" shows "O[bot](f) = UNIV"
valid_W_inv_sys_ghg_empty_iff: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state" assumes "valid_W_inv s" shows "sys_ghost_honorary_grey s = {}"
distinct_ParamDefs: fixes a :: "'edge" assumes "valid_edge a" shows "distinct (ParamDefs (targetnode a))"
integral_holo_aux: fixes a :: "real"   and b :: "real"   and A :: "complex set" assumes "0 < a"   and "a \<le> b"   and "b < 1" shows "(\<lambda>z. complex_interval_lebesgue_integral lborel (ereal a) (ereal b) (f z)) holomorphic_on A"
domain_eq: shows "Domainp (pcr_status (=)) = (\<lambda>\<sigma>. \<forall>f k. set (\<sigma> (f, k)) \<subseteq> {0..<k})"
simps_6: fixes \<A> :: "'b \<Rightarrow> bool"   and F :: "'b formula"   and G :: "'b formula" shows "\<A> \<Turnstile> F \<^bold>\<rightarrow> G = (\<A> \<Turnstile> F \<longrightarrow> \<A> \<Turnstile> G)"
cases: fixes a1 :: "('c \<times> 'd) option"   and a2 :: "('c \<times> 'd) option"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool" assumes "(a1, a2) \<in> whileLoop_results C B" obtains r :: "'c" and s :: "'d" where "a1 = Some (r__, s__)" and "a2 = Some (r__, s__)" and "\<not> C r__ s__"   | r :: "'c" and s :: "'d" where "a1 = Some (ra__, sa__)" and "a2 = None" and "C ra__ sa__" and "snd (B ra__ sa__)"   | r :: "'c" and s :: "'d" and r' :: "'c" and s' :: "'d" and z :: "('c \<times> 'd) option" where "a1 = Some (r, s)" and "a2 = z" and "C r s" and "(r', s') \<in> fst (B r s)" and "(Some (r', s'), z) \<in> whileLoop_results C B"
euclidean_size_poly_def: fixes p :: "'a poly" shows "euclidean_size p = (if p = 0 then 0 else 2 ^ degree p)"
personal_quorums_def: fixes quorum_of :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" shows "personal_quorums quorum_of \<equiv> \<forall>Q p p'. quorum_of p Q \<longrightarrow> p' \<in> Q \<longrightarrow> quorum_of p' Q"
domain_axioms_def: fixes R :: "('e, 'f) ring_scheme" shows "domain_axioms R \<equiv> \<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<forall>a b. a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub> \<longrightarrow> a \<in> carrier R \<longrightarrow> b \<in> carrier R \<longrightarrow> a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>)"
reindex: fixes B :: "nat set" shows "(\<Sum>x\<in>B. f (prod_decode x)) = sum f (prod_decode ` B)"
pt_\<delta>: fixes x :: "'a" assumes "x \<in> S - U" shows "\<delta> \<le> p x"
basis_ex: shows "\<exists>vs. set vs \<subseteq> V \<and> V = RSpan vs \<and> R_lin_independent vs"
var_app_type_eq: fixes e :: "nat \<Rightarrow> LambdaType.type"   and i :: "nat"   and ts :: "dB list"   and T :: "LambdaType.type"   and U :: "LambdaType.type" assumes "e \<turnstile> Var i \<degree>\<degree> ts : T"   and "e \<turnstile> Var i \<degree>\<degree> ts : U" shows "T = U"
Cons: fixes ys :: "'a list"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>y\<in>set ys. le x y"   and "linorder_list0.sortedP le ys" shows "linorder_list0.sortedP le (x # ys)"
nat_grSuc0_conv: fixes n :: "nat" shows "(Suc 0 < n) = (n \<noteq> 0 \<and> n \<noteq> Suc 0)"
LSQPS_def: shows "LSQPS = {p \<in> ps\<^sub>L. (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<in> LSQ}"
finite_at_most_singleton: fixes F :: "'b set" assumes "\<And>x y. \<lbrakk>x \<in> F; y \<in> F\<rbrakk> \<Longrightarrow> x = y" shows "finite F \<and> card F \<le> 1"
TESL_interp_unfold_stepwise_implies_not: fixes master :: "clock"   and slave :: "clock" shows "\<lbrakk> master implies not slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Inter> {Y. \<exists>n. Y = \<lbrakk> master implies not slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup>}"
simps_3: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and xs :: "'g list" shows "successively P (x # y # xs) = (P x y \<and> successively P (y # xs))"