hcju/isabelleps huggingface dataset:
call_option_def: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b"   and G :: "nat \<Rightarrow> bool stream measure"   and T :: "nat"   and K :: "real" assumes "CRR_market p M geom_proc u d init Mkt r risk_free_asset stk G" shows "CRR_market.call_option Mkt stk T K = (\<lambda>w. max (prices Mkt stk T w - K) 0)"
are_operators_interfering_symmetric: fixes op\<^sub>1 :: "('b, 'c) strips_operator_scheme"   and op\<^sub>2 :: "('b, 'd) strips_operator_scheme" assumes "are_operators_interfering op\<^sub>1 op\<^sub>2" shows "are_operators_interfering op\<^sub>2 op\<^sub>1"
ccNeighbors_disjoint_empty: fixes x :: "Vars.var"   and G :: "CoCalls" shows "(ccNeighbors x G = {}) = (x \<notin> ccField G)"
R_def: shows "R \<equiv> measures [\<lambda>(n, cds). n, \<lambda>(n, y). length y]"
productive_imp_INTERP: fixes M :: "'a literal multiset set"   and N :: "'a literal multiset set"   and C :: "'a literal multiset" assumes "gr.production M N C \<noteq> {}" shows "gr.INTERP M N \<TTurnstile> C"
acquire_all_def: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and ln :: "'a \<Rightarrow>f nat" shows "acquire_all ls t ln \<equiv> (\<lambda>(l, la). acquire_locks l t la) \<circ>$ ($ls, ln$)"
Id_O_R: fixes R :: "('a \<times> 'b) set" shows "Id ; R = R"
inv2: shows "opnet onp p\<^sub>2 \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U (net_tree_ips p\<^sub>2) \<rightarrow>) globala (\<lambda>a. case a of (\<sigma>, aa, \<sigma>') \<Rightarrow> castmsg (I \<sigma>) aa \<and> (aa \<noteq> \<tau> \<and> (\<forall>i d. aa \<noteq> i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p\<^sub>2. S (\<sigma> i) (\<sigma>' i))) \<and> (aa = \<tau> \<or> (\<exists>i d. aa = i:deliver(d)) \<longrightarrow> (\<forall>i\<in>net_tree_ips p\<^sub>2. U (\<sigma> i) (\<sigma>' i)) \<and> (\<forall>i. i \<notin> net_tree_ips p\<^sub>2 \<longrightarrow> \<sigma>' i = \<sigma> i)))"
has_bottomI: fixes z :: "'a"   and P :: "'a set" assumes "z \<in> P"   and "\<And>x. x \<in> P \<Longrightarrow> x \<succeq> z" shows "has_bottom P"
set_def: fixes r :: "'d ref"   and x :: "'d" shows "Ref.set r x = refs_update (\<lambda>h. h(TYPEREP('d) := (h TYPEREP('d))(addr_of_ref r := to_nat x)))"
greatest_noteq_card: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) \<noteq> nrows A"
monic_factor: fixes p :: "'b poly"   and q :: "'b poly" assumes "monic (p * q)"   and "monic p" shows "monic q"
ccpo: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lb :: "'b set \<Rightarrow> 'b" assumes "partial_function_definitions ord lb" shows "class.ccpo lb ord (mk_less ord)"
squarefree_factorization_degree: fixes p :: "'a poly"   and c :: "'a"   and fs :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, fs)" shows "degree p = (\<Sum>(f, c)\<leftarrow>fs. (c + 1) * degree f)"
composable_arrs_def: fixes \<CC> :: "V" shows "composable_arrs \<CC> = ZFC_in_HOL.set {[g, f]\<^sub>\<circ> |g f. \<exists>a b c. g : b \<mapsto>\<^bsub>\<CC>\<^esub> c \<and> f : a \<mapsto>\<^bsub>\<CC>\<^esub> b}"
upto_empty: fixes j :: "int"   and i :: "int" assumes "j < i" shows "[i..j] = []"
wf\<^sub>s\<^sub>t\<^sub>s_wf\<^sub>s\<^sub>t\<^sub>s': fixes \<S> :: "('fun, 'var) strand_step list set" shows "wf\<^sub>s\<^sub>t\<^sub>s \<S> = wf\<^sub>s\<^sub>t\<^sub>s' \<S> []"
a1: shows "pathfinish (p +++ linepath a b) = pathstart (p +++ linepath a b)"
cases: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'a + 'c"   and a2 :: "'b + 'd" assumes "rel_sum R1 R2 a1 a2" obtains a :: "'a" and c :: "'b" where "a1 = Inl a" and "a2 = Inl c" and "R1 a c"   | b :: "'c" and d :: "'d" where "a1 = Inr b" and "a2 = Inr d" and "R2 b d"
assms_4: fixes x :: "'a" assumes "x \<in> S" shows "norm (f x) \<le> g x"
refl_onD: fixes A :: "'d set"   and r :: "('d \<times> 'd) set"   and a :: "'d" assumes "refl_on A r"   and "a \<in> A" shows "(a, a) \<in> r"
fs_int'_def: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and fs :: "int vec list" shows "fs_int' n m fs_init fs \<equiv> LLL.LLL_invariant_weak n m fs_init fs"
rk_triple_to_rk_couple: fixes A :: "Points"   and B :: "Points"   and C :: "Points" assumes "rk {A, B, C} = 3" shows "rk {A, B} = 2"
simps_13: shows "cospan_shape.comp cospan_shape.BB cospan_shape.BT = cospan_shape.Null"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> s" shows "s\<lparr>a\<rparr> = b"
is_group: shows "Group.group \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"
assms_2: shows "[e * d = 1] (mod P - 1)"
coprime_poly_x_mult_y_poly_lift: fixes p :: "'a poly"   and q :: "'a poly" assumes "0 < degree p"   and "0 < degree q"   and "poly p (0::'a) \<noteq> (0::'a) \<or> poly q (0::'a) \<noteq> (0::'a)" shows "comm_monoid_mult_class.coprime (poly_x_mult_y p) (poly_lift q)"
sets_PiM_cong: fixes I :: "'c set"   and J :: "'c set"   and M :: "'c \<Rightarrow> 'd measure"   and N :: "'c \<Rightarrow> 'd measure" assumes "I = J"   and "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (Pi\<^sub>M I M) = sets (Pi\<^sub>M J N)"
simps_1: fixes f :: "int \<Rightarrow> int \<Rightarrow> bool"   and a :: "int"   and b :: "int" shows "MaybeBoolInt f (Some (Num a)) (Some (Num b)) = (if f a b then trilean.true else trilean.false)"
inf_SUP: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and 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 a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and B :: "'d set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf a (Sup (f ` B)) = Sup ((\<lambda>b. inf a (f b)) ` B)"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_singleton: fixes t :: "('b, 'a) Term.term"   and s :: "('c, 'a) Term.term" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s [(t, s)] = fv t \<union> fv s"
Cond_\<tau>execI3: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e'' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and e' :: "'addr expr1" assumes "\<tau>exec_move ci P t e'' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (if (e) e' else e'') h (stk, loc, Suc (Suc (length (compE2 e) + length (compE2 e') + pc)), xcp) (stk', loc', Suc (Suc (length (compE2 e) + length (compE2 e') + pc')), xcp')"
len_comp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and a :: "nat"   and c :: "nat"   and xs :: "nat list"   and b :: "nat"   and ys :: "nat list" assumes "class.linordered_ab_monoid_add plus zero less_eq less" shows "linordered_ab_monoid_add.len plus m a c (xs @ b # ys) = plus (linordered_ab_monoid_add.len plus m a b xs) (linordered_ab_monoid_add.len plus m b c ys)"
weakTransitionI_2: fixes P :: "pi" shows "P \<Longrightarrow>\<^sup>^\<tau> \<prec> P"
count_const_lt_components_def: fixes hs :: "(('a \<Rightarrow>\<^sub>0 'd) \<times> 'e) list" shows "punit.count_const_lt_components hs = length (remdups (filter (\<lambda>x. x \<noteq> None) (map (punit.const_lt_component \<circ> fst) hs)))"
HEndPhase0_some: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv1 s" shows "(SOME b. b \<in> allBlocksRead s p \<and> (\<forall>t\<in>allBlocksRead s p. bal t \<le> bal b)) \<in> allBlocksRead s p \<and> (\<forall>t\<in>allBlocksRead s p. bal t \<le> bal (SOME b. b \<in> allBlocksRead s p \<and> (\<forall>t\<in>allBlocksRead s p. bal t \<le> bal b)))"
type_definition_eint: shows "type_definition Rep_eint Abs_eint UNIV"
reachable_target_elim: fixes q :: "'a"   and execute :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and enabled :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a" assumes "q \<in> transition_system.reachable execute enabled p" obtains r :: "'b list" where "transition_system.path execute enabled r p" and "q = fold execute r p"
imageE: fixes b :: "'f"   and f :: "'g \<Rightarrow> 'f"   and A :: "'g set" assumes "b \<in> f ` A" obtains x :: "'g" where "b = f x" and "x \<in> A"
map_graph_in: fixes G :: "('b, 'c) labeled_graph"   and a :: "'c"   and b :: "'c"   and e :: "'b allegorical_term"   and f :: "'c \<Rightarrow> 'd" assumes "graph G"   and "(a, b) \<in> :G:\<lbrakk>e\<rbrakk>" shows "(f a, f b) \<in> :map_graph_fn G f:\<lbrakk>e\<rbrakk>"
vsubset_vintersection_right: fixes A :: "V"   and B :: "V"   and C :: "V" shows "(A \<subseteq>\<^sub>\<circ> B \<inter>\<^sub>\<circ> C) = (A \<subseteq>\<^sub>\<circ> B \<and> A \<subseteq>\<^sub>\<circ> C)"
KBasic2_8: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<diamond>(\<phi> \<^bold>& \<psi>) \<^bold>\<rightarrow> \<^bold>\<diamond>\<phi> \<^bold>& \<^bold>\<diamond>\<psi> in v]"
run_reachable: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'c"   and m :: "'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" assumes "graph_defs.is_run (b_graph.to_gbg_ext (degeneralize_ext ma) m) r" shows "range r \<subseteq> (g_E (b_graph.to_gbg_ext (degeneralize_ext ma) m))\<^sup>* `` g_V0 (b_graph.to_gbg_ext (degeneralize_ext ma) m)"
interior_limit_point: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> interior S" shows "x islimpt S"
simps_2: shows "left \<langle>\<rangle> = \<langle>\<rangle>"
iT_Plus_neg_iprev: fixes n :: "nat"   and k :: "nat"   and I :: "nat set" shows "iprev (n - k) (I \<oplus>- k) = iprev n (I \<down>\<ge> k) - k"
trans_MapOps_def: shows "trans_MapOps \<equiv> \<lparr>MapOps.empty = empty_trie, lookup = trans_MapOps_lookup, update = trans_MapOps_update\<rparr>"
projection_def: fixes M :: "'b set" shows "projection M \<equiv> SOME \<pi>. is_projection_on \<pi> M"
f2: shows "Deliver (i, Create i e2) \<notin> set pre"
hd_Cons_tl: fixes xs :: "'b list" assumes "xs \<noteq> \<epsilon>" shows "hd xs # tl xs = xs"
index_row_2: fixes A :: "'c mat"   and i :: "nat" shows "dim_vec (row A i) = dim_col A"
homij: shows "(\<lambda>(x, y). i x \<otimes>\<^bsub>G\<^esub> j y) \<in> hom (A \<times>\<times> B) G"
uniformity_float_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist (real_of_float x) (real_of_float y) < e})"
assms_1: shows "\<exists>i\<ge>n. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"
op_igba_sys_prod_def: shows "op_igba_sys_prod \<equiv> igba_sys_prod_precond.prod"
the_elem_image_unique: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "A \<noteq> {}"   and "\<And>y. y \<in> A \<Longrightarrow> f y = f x" shows "the_elem (f ` A) = f x"
elims: fixes x :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and y :: "(String.literal \<times> String.literal) set" assumes "subclass x = y" obtains cl :: "(String.literal \<times> String.literal) set" and uu_ :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" where "x = (cl, uu_)" and "y = cl"
symmetry_preserves_midpoint: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Z :: "'a"   and A :: "'a"   and D :: "'a"   and B :: "'a"   and E :: "'a"   and C :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong Z A D"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong Z B E"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong Z C F"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B A C" shows "Tarski_neutral_dimensionless.Midpoint Bet Cong E D F"
negate_negate: fixes P :: "'a \<Rightarrow> bool" shows "\<N> (\<N> P) = P"
simps_3: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'b" shows "lcl \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = lcl"
ListRed2: fixes E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and v :: "val" assumes "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>" shows "P,E \<turnstile> \<langle>Val v # es,s\<rangle> [\<rightarrow>] \<langle>Val v # es',s'\<rangle>"
dvd_0_right: fixes a :: "'a" shows "a dvd (0::'a)"
assms_1: shows "s' \<in> pmdl.syzygy_module (set bs)"
Rep_cfun_inverse: fixes x :: "'d \<rightarrow> 'e" shows "Abs_cfun (Rep_cfun x) = x"
assms_1: shows "Case_Labeling.CTXT IC CT OC (ovalidNF P f Q)"
append_eq_appendI: fixes xs :: "'c list"   and xs1 :: "'c list"   and zs :: "'c list"   and ys :: "'c list"   and us :: "'c list" assumes "xs @ xs1 = zs"   and "ys = xs1 @ us" shows "xs @ ys = zs @ us"
ring_hom_def: fixes R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" shows "ring_hom R S = {h \<in> carrier R \<rightarrow> carrier S. (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and> h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
map_iterator_dom_filter_def: fixes P :: "'e \<times> 'f \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "map_iterator_dom_filter P it \<equiv> set_iterator_image_filter (\<lambda>xy. if P xy then Some (fst xy) else None) it"
cases_scheme: fixes r :: "('c, 'd) tarjan_state_scheme" obtains counter :: "nat" and discovered :: "'c \<Rightarrow> nat option" and finished :: "'c \<Rightarrow> nat option" and pending :: "('c \<times> 'c) set" and stack :: "'c list" and tree_edges :: "('c \<times> 'c) set" and back_edges :: "('c \<times> 'c) set" and cross_edges :: "('c \<times> 'c) set" and sccs :: "'c set set" and lowlink :: "'c \<Rightarrow> nat option" and tj_stack :: "'c list" and more :: "'d" where "r = \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr>"
cka_2: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> \<cdot> (x\<^sup>\<star> \<cdot> y)\<^sup>\<star> = x\<^sup>\<star> \<cdot> y\<^sup>\<star>"
61422: fixes x :: "'a"   and y :: "'a" shows "- (- - x \<sqinter> y) = - (- - y \<sqinter> x)"
assms_1: shows "n \<in>\<^sub>\<circ> \<omega>"
less_eq_matrix_def: fixes f :: "'b \<times> 'b \<Rightarrow> 'c"   and g :: "'b \<times> 'b \<Rightarrow> 'c" shows "(f \<preceq> g) = (\<forall>e. f e \<le> g e)"
maxdeg_def: fixes A :: "(('e \<Rightarrow>\<^sub>0 'd) \<Rightarrow>\<^sub>0 'f) set" shows "maxdeg A = Max (poly_deg ` A)"
simps_1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "bexp"   and c :: "com" shows "get_back P b c 0 = P"
sel_1: fixes x1 :: "'r"   and x2 :: "'s" shows "prod.fst (x1, x2) = x1"
pdom_upd: fixes f :: "('a, 'b) pfun"   and k :: "'a"   and v :: "'b" shows "pdom (f(k \<mapsto> v)\<^sub>p) = insert k (pdom f)"
defs_4: fixes r :: "'a args_scheme" shows "args.truncate r \<equiv> \<lparr>args_ivl = args_ivl r, args_n = args_n r, args_L = args_L r, args_R = args_R r, args_pos = args_pos r\<rparr>"
comm_monoid_def: fixes G :: "('c, 'd) monoid_scheme" shows "Group.comm_monoid G \<equiv> Group.monoid G \<and> Group.comm_monoid_axioms G"
l_one: fixes x :: "int" assumes "x \<in> carrier R" shows "\<zero> \<oplus> x = x"
braun_del_lo: fixes t :: "'b tree" assumes "braun t" shows "braun (del_lo t)"
star_inductr': fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z \<squnion> y \<cdot> x \<le> y" shows "z \<cdot> quantale_class.star x \<le> y"
nth_Cons': fixes x :: "'d"   and xs :: "'d list"   and n :: "nat" shows "(x # xs) ! n = (if n = 0 then x else xs ! (n - 1))"
simps_4: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "fv\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t = \<Union> (fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r ` set F) - set X"
chine_eq_apex: shows "\<mu>\<nu>\<pi>.chine = \<mu>\<nu>\<pi>.apex"
wt_jvm_progD: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "jvm_prog" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "\<exists>wt. wf_prog wt P"
derived_set_of_finite: fixes X :: "'b topology"   and S :: "'b set" assumes "Hausdorff_space X"   and "finite S" shows "X derived_set_of S = {}"
wt_subst: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and T :: "'fsym trm" assumes "wtSB \<pi>"   and "wt T" shows "wt (subst \<pi> T)"
Com_def: fixes X :: "char list set" shows "Com X = {c. \<forall>S\<in>set (annos c). domo S \<subseteq> X}"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop1 n (l, r) = (inv_loop1_loop n (l, r) \<or> inv_loop1_exit n (l, r))"
elims_2: fixes x :: "('b \<times> nat) tree"   and xa :: "('b \<times> nat) tree" assumes "balanced1 x xa" obtains t1 :: "('b \<times> nat) tree" and t2 :: "('b \<times> nat) tree" where "x = t1" and "xa = t2" and "10 * (size_wbt t2 + 1) \<le> 25 * (size_wbt t1 + 1)"
d_def: shows "d = (SOME d. \<forall>la. la \<in> snd ` set n_as \<longrightarrow> cmod la = 1 \<longrightarrow> d la \<noteq> 0 \<and> la ^ d la = 1)"
finprod_closed: fixes f :: "'e \<Rightarrow> 'a"   and A :: "'e set" assumes "f \<in> A \<rightarrow> carrier G" shows "finprod G f A \<in> carrier G"
map_upd_triv: fixes t :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and x :: "'c" assumes "t k = Some x" shows "t(k \<mapsto> x) = t"
outstanding_refs_takeWhile: fixes P :: "'b memref \<Rightarrow> bool"   and P' :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" shows "outstanding_refs P (takeWhile P' sb) \<subseteq> outstanding_refs P sb"
mset_merge: fixes t1 :: "('b \<times> nat) tree"   and t2 :: "('b \<times> nat) tree" shows "mset_tree (merge t1 t2) = mset_tree t1 + mset_tree t2"
bigger_prime: fixes n :: "nat" shows "\<exists>p. normalization_semidom_class.prime p \<and> n < p"
ide_to_hf_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "ide_to_hf a = HF (DOWN ` local.set a)"
zero_order_3: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and n :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
mapCommute: fixes S :: "('a, 'b) sequent"   and R :: "('a, 'b) sequent"   and c :: "('a, 'b) sequent list" shows "map (extend S) (map (extend R) c) = map (extend R) (map (extend S) c)"
cong_sym: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo c b a"
\<rho>: shows "\<rho> \<in> space (state_measure (Suc ` V') (case_nat t \<Gamma>))"
simps: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM" shows "M1 \<preceq> M2 = (L M1 \<subseteq> L M2)"
mat_O_def: shows "mat_O = mat N N (\<lambda>(i, j). if i = j then if f i then - 1 else 1 else 0)"
j_stricter_bound: shows "bounded m r\<^sub>0 j"
mat_delete_carrier: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat m n" shows "mat_delete A i j \<in> carrier_mat (m - 1) (n - 1)"
pochhammer_rec: fixes a :: "'b"   and n :: "nat" shows "pochhammer a (Suc n) = a * pochhammer (a + (1::'b)) n"
LIMSEQ_const_iff: fixes k :: "'a"   and l :: "'a" shows "(\<lambda>n. k) \<longlonglongrightarrow> l = (k = l)"
exists_indexed_eval_monomial: fixes P :: "'d multiset \<Rightarrow> 'a"   and Qs :: "('d multiset \<Rightarrow> 'a) list"   and n :: "'d multiset"   and i :: "'d" assumes "carrier_coeff P"   and "list_all carrier_coeff Qs"   and "P n \<noteq> \<zero>"   and "list_all (\<lambda>Q. index_free Q i) Qs" obtains m :: "'d multiset" where "count m i = length Qs + count n i" and "indexed_eval (P # Qs) i m \<noteq> \<zero>"
combine_is_incomplete: fixes \<V> :: "'b set"   and \<B> :: "'b set multiset"   and \<k> :: "int"   and \<V>' :: "'b set"   and \<B>' :: "'b set multiset" assumes "two_incomplete_designs \<V> \<B> \<k> \<V>' \<B>'" shows "incomplete_design (\<V> \<union> \<V>') (\<B> + \<B>') \<k>"
mult2_alt_def: fixes b :: "bool"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "mult2_alt b ns s = {(X, Y). \<exists>X1 X2 Y1 Y2. X = X1 + X2 \<and> Y = Y1 + Y2 \<and> (X1, Y1) \<in> multpw ns \<and> (b \<or> Y2 \<noteq> {#}) \<and> (\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s))}"
assms: fixes i :: "'m"   and j :: "'n" shows "\<bar>A $ i $ j\<bar> \<le> B"
limit_nonemptyE: fixes x :: "nat \<Rightarrow> 'b" assumes "finite (range x)" obtains xa :: "'b" where "xa \<in> limit x"
Pi_iff: fixes f :: "'e \<Rightarrow> 'f"   and I :: "'e set"   and X :: "'e \<Rightarrow> 'f set" shows "(f \<in> Pi I X) = (\<forall>i\<in>I. f i \<in> X i)"
list_types_agree_imp_e_typing: fixes ts :: "t list"   and vs :: "v list"   and \<S> :: "s_context"   and \<C> :: "t_context" assumes "list_all2 types_agree ts vs" shows "\<S>\<bullet>\<C> \<turnstile> $$* vs : [] _> ts"
in_arcs_int_arcs: fixes G :: "('d, 'c) pre_digraph"   and u :: "'d" shows "in_arcs G u \<inter> arcs G = in_arcs G u"
ipathI: fixes r :: "nat \<Rightarrow> 'b"   and E :: "('b \<times> 'b) set" assumes "\<And>i. (r i, r (Suc i)) \<in> E" shows "ipath E r"
avg_def: fixes s :: "(nat, unit) HashMap.hashmap" shows "hs_ctx.avg s \<equiv> case hs_ctx.avg_aux s of (c, s) \<Rightarrow> s div c"
toList_inject: fixes x :: "'c odlist"   and y :: "'c odlist" shows "(toList x = toList y) = (x = y)"
leq_zero: fixes r :: "('d \<times> 'd) set"   and x :: "'d" assumes "wo_rel r"   and "Field r \<noteq> {}" shows "((x, zero r) \<in> r) = (x = zero r)"
theta_induct: fixes f :: "ValD \<rightarrow> ValD"   and f' :: "(('o ValK \<rightarrow> 'o) \<rightarrow> 'o) \<rightarrow> ('o ValK \<rightarrow> 'o) \<rightarrow> 'o"   and P :: "(ValD \<rightarrow> ValD) \<Rightarrow> ((('o ValK \<rightarrow> 'o) \<rightarrow> 'o) \<rightarrow> ('o ValK \<rightarrow> 'o) \<rightarrow> 'o) \<Rightarrow> bool" assumes "\<theta>: f \<mapsto> f'"   and "(\<And>x x'. \<eta>: x \<mapsto> x' \<Longrightarrow> \<eta>: f\<cdot>x \<mapsto> f'\<cdot>x') \<Longrightarrow> P f f'" shows "P f f'"