hcju/isabelleps huggingface dataset:
map_prod_simp: fixes f :: "'e \<Rightarrow> 'b"   and g :: "'f \<Rightarrow> 'd"   and a :: "'e"   and b :: "'f" shows "map_prod f g (a, b) = (f a, g b)"
simps: fixes N :: "('a literal multiset \<times> nat) multiset"   and P :: "('a literal multiset \<times> nat) multiset"   and Q :: "('a literal multiset \<times> nat) multiset"   and n :: "nat" shows "wP_of_wstate (N, P, Q, n) = P"
NatTransMapsTo: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and X :: "'g" assumes "NatTransP \<eta>"   and "X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>" shows "\<eta> $$ X maps\<^bsub>NTCatCod \<eta>\<^esub> NTDom \<eta> @@ X to NTCod \<eta> @@ X"
get_root_node_same_no_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_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 child" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r None"
modRules2_def: shows "modRules2 \<equiv> {(xa, x). modRules2p xa x}"
prod_comp_eq: fixes f :: "'a \<Rightarrow> 'b \<times> 'c"   and g :: "'a \<Rightarrow> 'c \<times> 'b" assumes "f = prod.swap \<circ> g" shows "prod.swap \<circ> f = g"
noNeg: fixes m :: "'a common_primitive match_expr" assumes "normalized_nnf_match m" shows "\<not> has_disc_negated disc False (rewrite_neg m)"
ifbd_comp_pres: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and \<psi> :: "'e set \<Rightarrow> 'c set" assumes "Sup_pres \<phi>" shows "bd\<^sup>-\<^sub>\<F> (\<phi> \<circ> \<psi>) = bd\<^sup>-\<^sub>\<F> \<psi> \<circ>\<^sub>K bd\<^sup>-\<^sub>\<F> \<phi>"
IH_1: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "(X, Y) \<in> qSwapped" shows "qFresh zs z Y \<longrightarrow> qGood Y \<and> qFreshEnv zs z rho \<longrightarrow> qFresh zs z Y #[[rho]]"
normal_series_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list" shows "normal_series_axioms G \<GG> \<equiv> (\<GG> \<noteq> [] \<and> hd \<GG> = {\<one>\<^bsub>G\<^esub>}) \<and> last \<GG> = carrier G \<and> (\<forall>i. i + 1 < length \<GG> \<longrightarrow> \<GG> ! i \<lhd> G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>)"
wf_tuple_proj_idle: fixes n :: "nat"   and X :: "nat set"   and as :: "'b option list" assumes "wf_tuple n X as" shows "proj_tuple (join_mask n X) as = as"
Ord_induct: fixes k :: "V"   and P :: "V \<Rightarrow> bool" assumes "Ord k"   and "\<And>x. \<lbrakk>Ord x; \<And>y. y \<in> elts x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P k"
rep_eq: fixes x :: "p_point"   and xa :: "p_point"   and xb :: "p_point" shows "p_between x xa xb = poincare_between (Rep_p_point x) (Rep_p_point xa) (Rep_p_point xb)"
outer_regular_lborel_le: fixes B :: "'b set"   and e :: "real" assumes "B \<in> sets borel"   and "0 < e" obtains U :: "'b set" where "open U" and "B \<subseteq> U" and "emeasure lborel (U - B) \<le> ennreal e"
prod_list_def: shows "prod_list \<equiv> monoid_list.F (*) (1::'a)"
a_inv_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<ominus> x \<in> carrier G"
root_int: fixes p :: "nat"   and x :: "int" assumes "p \<noteq> 0 \<or> x \<noteq> 1" shows "set (root_int p x) = {y. y ^ p = x}"
neg_mask_mono_le: fixes x :: "'b word"   and y :: "'b word"   and n :: "nat" assumes "x \<le> y" shows "x AND NOT (mask n) \<le> y AND NOT (mask n)"
is_lub_def: fixes S :: "'a set"   and x :: "'a" shows "S <<| x = (S <| x \<and> (\<forall>u. S <| u \<longrightarrow> x \<sqsubseteq> u))"
cbox_division_memE: fixes K :: "'a set"   and \<D> :: "'a set set"   and S :: "'a set" assumes "K \<in> \<D>"   and "\<D> division_of S" obtains a :: "'a" and b :: "'a" where "K = cbox a b" and "K \<noteq> {}" and "K \<subseteq> S"
elims: fixes x :: "nat"   and y :: "tm" assumes "ORD_OF x = y" obtains "x = 0" and "y = Zero"   | k :: "nat" where "x = Suc k" and "y = SUCC (ORD_OF k)"
perm_is_finite: fixes f :: "atom \<Rightarrow> atom" assumes "f \<in> perm" shows "finite {a. f a \<noteq> a}"
independent_insert: fixes a :: "('a, 'b) vec"   and S :: "('a, 'b) vec set" shows "vec.independent (insert a S) = (if a \<in> S then vec.independent S else vec.independent S \<and> a \<notin> vec.span S)"
cases: fixes a1 :: "'b"   and a2 :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a1, a2) \<in> r\<^sup>*" obtains a :: "'b" where "a1 = aa__" and "a2 = aa__"   | a :: "'b" and b :: "'b" and c :: "'b" where "a1 = a" and "a2 = c" and "(a, b) \<in> r\<^sup>*" and "(b, c) \<in> r"
unique: fixes q :: "'a list" assumes "q \<in> carrier (K [X])"   and "lead_coeff q = \<one>"   and "a_kernel (K [X]) R h = PIdl\<^bsub>K [X]\<^esub> q" shows "q = p"
rprodl_extend_state_oracle: fixes "oracle" :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf" shows "(rprodl ---> id ---> map_spmf (map_prod id lprodr)) \<dagger>\<dagger>oracle = \<dagger>oracle"
cSup_upper2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and X :: "'b set"   and y :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "less_eq y x"   and "preorder.bdd_above less_eq X" shows "less_eq y (Sup X)"
last_ConsR: fixes xs :: "'a list"   and x :: "'a" assumes "xs \<noteq> []" shows "last (x # xs) = last xs"
iIN_inext: fixes n :: "nat"   and t :: "nat"   and d :: "nat" assumes "n \<le> t"   and "t < n + d" shows "inext t [n\<dots>,d] = Suc t"
liftLM_comm_ctxt: fixes E :: "ctxt"   and n :: "nat"   and m :: "nat" shows "liftL_ctxt (liftM_ctxt E n) m = liftM_ctxt (liftL_ctxt E m) n"
xs: shows "xs = concat (map (roots_of_real_main \<circ> fst) pis)"
set_rel_empty_iff_2: fixes x :: "'a set"   and A :: "('a \<times> 'b) set" shows "((x, {}) \<in> \<langle>A\<rangle>set_rel) = (x = {})"
eq_fin_symp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a list"   and v :: "'a list" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* u v" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* v u"
le_less: fixes x :: "'a list"   and y :: "'a list" shows "prefix x y = (strict_prefix x y \<or> x = y)"
is_sublist_in1: fixes x :: "'a"   and y :: "'a"   and vs :: "'a list" assumes "is_sublist [x, y] vs" shows "y \<in> set vs"
validE_append: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list" assumes "\<turnstile> (\<Delta> @ \<Gamma>) ok" shows "\<turnstile> \<Gamma> ok"
cnvalid1_eq: fixes A :: "triple set"   and P :: "assn"   and c :: "stmt"   and Q :: "assn" shows "A |\<Turnstile> {(P, c, Q)} \<equiv> \<forall>n. |\<Turnstile>n: A \<longrightarrow> (\<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q t)"
List_case_def: fixes c :: "'c"   and d :: "('b, unit) Old_Datatype.node set \<Rightarrow> ('b, unit) Old_Datatype.node set \<Rightarrow> 'c" shows "List_case c d = Case (\<lambda>x. c) (Split d)"
Pos_Melem_subst_atm_subst_cls: 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 A :: "'b"   and C :: "'b literal multiset"   and \<sigma> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms"   and "Pos A \<in># C" shows "Pos (subst_atm A \<sigma>) \<in># substitution_ops.subst_cls subst_atm C \<sigma>"
in_non_special_solutions: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> set (non_special_solutions a b)" shows "(x, y) \<in> local.Solutions"
A: shows "\<guillemotleft>\<nu> \<cdot> \<r>[r] \<cdot> (r \<star> \<epsilon>) \<cdot> \<a>[r, f, f\<^sup>*] \<cdot> (\<rho> \<star> f\<^sup>*) : g \<star> f\<^sup>* \<Rightarrow> g \<star> w\<guillemotright>"
es_def_2: shows "Lfilled i l (vs @ [$Br (i + Suc k)]) LK"
L_init: fixes \<sigma> :: "nat \<Rightarrow> 'c list"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and init :: "'d rexp \<Rightarrow> 'e"   and delta :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and final :: "'e \<Rightarrow> bool"   and wf_state :: "'e \<Rightarrow> bool"   and post :: "'e \<Rightarrow> 'e"   and L :: "'e \<Rightarrow> 'c list set"   and n :: "nat"   and r :: "'d rexp" assumes "rexp_DA \<sigma> wf_atom project lookup init delta final wf_state post L n"   and "alphabet.wf wf_atom n r" shows "L (init r) = project.lang (set \<circ> \<sigma>) project lookup n r"
finite_dom_comp: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'c \<Rightarrow> 'a" assumes "finite (dom f)"   and "inj_on g (dom (f \<circ> g))" shows "finite (dom (f \<circ> g))"
intro: fixes xs :: "V" assumes "\<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>" shows "vfsequence_axioms xs"
face_cycle_succ_vu: shows "GM.face_cycle_succ (rev_G uv) = proj_arcs_H (H.face_cycle_succ (rev_H uw))"
cod_comp: fixes g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" assumes "J.seq g f" shows "J.cod (g \<cdot>\<^sub>J f) = J.cod g"
lin_independent_imp_independent_take: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list"   and n :: "nat" assumes "Module smult M"   and "set ms \<subseteq> M"   and "R_scalar_mult.R_lin_independent UNIV smult ms" shows "R_scalar_mult.R_lin_independent UNIV smult (take n ms)"
singleton_example_1: fixes S :: "'a set set" assumes "\<forall>x\<in>S. \<forall>y\<in>S. x \<subseteq> y" shows "\<exists>z. S \<subseteq> {z}"
comp_timpl_closure_def: fixes FP :: "(('b, 'c, 'd) prot_fun, 'e) Term.term set"   and TI :: "('d set \<times> 'd set) set" shows "comp_timpl_closure FP TI \<equiv> let f = \<lambda>X. FP \<union> (\<Union>x\<in>X. \<Union>(a, b)\<in>TI. set \<langle>a --\<guillemotright> b\<rangle>\<langle>x\<rangle>) in while (\<lambda>X. f X \<noteq> X) f {}"
monomial_set_is_GB: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_monomial_set G" shows "is_Groebner_basis G"
timpl_closure'_timpls_rtrancl_subset: fixes c :: "('g set \<times> 'g set) set" shows "timpl_closure' (c\<^sup>*) \<subseteq> timpl_closure' c"
power_less_imp_less_base: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat"   and b :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less (power.power one times a n) (power.power one times b n)"   and "less_eq zero b" shows "less a b"
ucast_le_maskI: fixes a :: "'b word"   and n :: "nat" assumes "a \<le> mask n" shows "UCAST('b \<rightarrow> 'c) a \<le> mask n"
unit_vectors_is_basis: fixes n :: "nat" shows "vectorspace.basis cpx_rng (module_cpx_vec n) (unit_vectors n)"
is_digit_char_intro: fixes c :: "char" shows "is_digit (char_to_digit c) = is_digit_char c"
1: shows "\<forall>PROB x. P PROB \<and> finite PROB \<longrightarrow> x \<in> MPLS_charles PROB \<longrightarrow> x < f PROB"
c: shows "Cn A \<div> .\<not> \<phi> \<subseteq> Cn A \<div> .\<not> (\<phi> .\<and>. \<psi>)"
bind_nondetT_def: shows "Monad_Overloading.bind = nondetM_base.bind_nondet Monad_Overloading.bind (mmerge Monad_Overloading.return Monad_Overloading.bind)"
that: fixes k :: "'b" assumes "k \<in> keys p"   and "c = lookup p k" shows "thesis"
SLC_app: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and x :: "'v"   and e :: "('r, 'l, 'v) expr"   and v :: "('r, 'l, 'v) val"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Apply (VE (Lambda x e)) (VE v)])"   and "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [subst (VE v) x e]))"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
f_aggregate_append_mod: fixes xs :: "'b list"   and k :: "nat"   and ys :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" assumes "length xs mod k = 0" shows "f_aggregate (xs @ ys) k ag = f_aggregate xs k ag @ f_aggregate ys k ag"
mult_poly_raw_def: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset" shows "mult_poly_raw p q = (\<Sum>y\<in>#p. mult_poly_by_monom y q)"
sortnMTQ: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "p \<noteq> []" shows "qsort p l \<noteq> []"
af_ltl_continuation: fixes w :: "'b set list"   and w' :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "(w \<frown> w') \<Turnstile> \<phi> = w' \<Turnstile> af \<phi> w"
disjointed_def: fixes A :: "nat \<Rightarrow> 'c set"   and n :: "nat" shows "disjointed A n = A n - \<Union> (A ` {0..<n})"
min_top2: fixes x :: "'d upred" shows "ord.min (\<le>) x true = x"
D: shows "\<forall>A\<in>P\<^sub>2 \<union> wrap B\<^sub>2. \<forall>B\<in>P\<^sub>2 \<union> wrap B\<^sub>2. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
assms: shows "eval cid t t' n (\<gamma> \<or>\<^sup>b \<gamma>')"
continuous_on_fst: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. fst (f x))"
imod_iadd_self2: fixes m :: "enat"   and n :: "nat" shows "(m + enat n) mod enat n = m mod enat n"
pred_inject_1: fixes P :: "'b \<Rightarrow> bool" shows "pred_option P None"
axioms_2: fixes R :: "('b, 'c) ring_scheme" assumes "cring R" shows "Group.comm_monoid R"
not_outside_2: fixes x :: "'alt pmf"   and y :: "'alt pmf" assumes "Pareto (SD \<circ> R) x y" shows "y \<in> lotteries_on alts"
seql_def: fixes i :: "'d"   and P :: "'e \<times> 'f \<Rightarrow> bool" shows "seql i P \<equiv> \<lambda>(\<sigma>, p). P (\<sigma> i, p)"
ideal_condition1: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "I \<subseteq> carrier R"   and "I \<noteq> {}"   and "\<forall>x\<in>I. \<forall>y\<in>I. x \<plusminus>\<^bsub>R\<^esub> y \<in> I"   and "\<forall>r\<in>carrier R. \<forall>x\<in>I. r \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<in> I" shows "ideal R I"
assms_4: shows "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"
lexord_transI: fixes r :: "('b \<times> 'b) set" assumes "trans r" shows "trans (lexord r)"
has_derivative_exp_scaleRl: fixes f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and t :: "real"   and T :: "real set"   and A :: "'b" assumes "D f \<mapsto> f' at t within T" shows "D (\<lambda>t. exp (f t *\<^sub>R A)) \<mapsto> (\<lambda>h. f' h *\<^sub>R (exp (f t *\<^sub>R A) * A)) at t within T"
is_minimal_basisI: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>p. p \<in> B \<Longrightarrow> p \<noteq> 0"   and "\<And>p q. \<lbrakk>p \<in> B; q \<in> B; p \<noteq> q\<rbrakk> \<Longrightarrow> \<not> lt p adds\<^sub>t lt q" shows "is_minimal_basis B"
fresh_finfun_update: fixes a :: "atom"   and f :: "('a, 'b) finfun"   and x :: "'a"   and y :: "'b" assumes "a \<sharp> f"   and "a \<sharp> x"   and "a \<sharp> y" shows "a \<sharp> finfun_update f x y"
inj_on_impl_inj_on_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and X :: "'a set set" assumes "inj_on f A"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq> A" shows "inj_on ((`) f) X"
returns_toI: fixes t :: "real"   and x :: "'a"   and P :: "'a set" assumes "0 < t"   and "t \<in> existence_ivl0 x"   and "flow0 x t \<in> P"   and "\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> P"   and "closed P" shows "returns_to P x"
span_is_subset: fixes S :: "'a vec set"   and N :: "'a vec set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule class_ring N V" shows "local.span S \<subseteq> N"
zip_append: fixes xs :: "'g list"   and us :: "'h list"   and ys :: "'g list"   and vs :: "'h list" assumes "length xs = length us" shows "zip (xs @ ys) (us @ vs) = zip xs us @ zip ys vs"
terminal_lappendt_lfinite: fixes xs :: "'c llist"   and ys :: "('c, 'd) tllist" assumes "lfinite xs" shows "terminal (lappendt xs ys) = terminal ys"
singleton_conv: fixes a :: "'g" shows "{x. x = a} = {a}"
hyp: fixes n :: "enat" assumes "\<And>m. m < n \<Longrightarrow> P m" shows "P n"
fieldm_def: fixes n :: "vname"   and f :: "qtname \<times> Decl.field" shows "fieldm n f = (declclass f, fdecl (n, fld f))"
relcompp_bot1: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "\<bottom> OO R = \<bottom>"
dy_fake_msg_mono_chan: fixes G :: "chan set"   and H :: "chan set"   and bad :: "agent set"   and I :: "msg set" assumes "G \<subseteq> H" shows "dy_fake_msg bad I G \<subseteq> dy_fake_msg bad I H"
simps_1: fixes x :: "'a" shows "replicate 0 x = []"
elims_3: fixes x :: "'b nres"   and xa :: "'b nres" assumes "\<not> x \<le> xa" obtains a :: "'b set" and b :: "'b set" where "x = RES a" and "xa = RES b" and "\<not> a \<subseteq> b"   | uv_ :: "'b set" where "x = FAILi" and "xa = RES uv_"
tab\<^sub>1_simps_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.ide V r" shows "partial_magma.ide V (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg r)"
findnth_correct_pre: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and n :: "nat"   and f :: "nat \<Rightarrow> (nat \<times> cell list \<times> cell list) \<times> nat"   and P :: "(nat \<times> cell list \<times> cell list) \<times> nat \<Rightarrow> bool"   and Q :: "(nat \<times> cell list \<times> cell list) \<times> nat \<Rightarrow> bool" assumes "ly = layout_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "0 < n"   and "f = (\<lambda>stp. (steps0 (Suc 0, l, r) (findnth n) stp, n))"   and "P = (\<lambda>((s, l, r), n). s = Suc (2 * n))"   and "Q = (\<lambda>((s, l, r), n). findnth_inv ly n (as, lm) (s, l, r) ires)" shows "\<exists>stp. P (f stp) \<and> Q (f stp)"
fF: shows "finite {f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<n}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
semiring_normalization_rules_23: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "plus (plus a b) c = plus (plus a c) b"
comb: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "f = g"   and "x = y" shows "f x = g y"
adds_pp_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_pp pair_of_term t v = (t adds term_powerprod.pp_of_term pair_of_term v)"
vec_of_list_index: fixes xs :: "'b list"   and j :: "nat" shows "vec_of_list xs $v j = xs ! j"
frequently_eq_eventually: fixes F :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "ultrafilter F" shows "frequently P F = eventually P F"
less_eq_tree_def: fixes s :: "'a tree"   and t :: "'a tree" shows "(s \<le> t) = tree_emb UNIV (\<le>) s t"
assms: shows "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"
simps_2: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option" shows "rm_MT_rules C [] = []"
A_RCOSETS_def': fixes G :: "('e, 'f) ring_scheme"   and H :: "'e set" shows "a_rcosets\<^bsub>G\<^esub> H \<equiv> \<Union>a\<in>carrier G. {H +>\<^bsub>G\<^esub> a}"
relation_of_oPrefix: fixes c :: "'d \<Rightarrow> 'c"   and P :: "('c, 'd) action" shows "action.relation_of (oPrefix c P) = (R (true \<turnstile> do c \<and> (\<lambda>(A, A'). alpha_rp.more A' = alpha_rp.more A)) ;; action.relation_of P)"
simps_3: shows "\<bar>\<infinity>\<bar> = \<infinity>"
find_mu_def: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b" shows "fx0.find_mu f x0 \<equiv> (\<lambda>s. s\<lparr>m := 0, tortoise := x0\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) (\<lambda>s. s\<lparr>tortoise := f (tortoise s), hare := f (hare s), m := m s + 1\<rparr>)"
a_host_shadow_root_rel_def: fixes h :: "(_) heap" shows "host_shadow_root_rel h = (\<lambda>(x, y). (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>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^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 y)) ` {(host, shadow_root). host |\<in>| element_ptr_kinds h \<and> |h \<turnstile> get_shadow_root host|\<^sub>r = Some shadow_root}"
Pre_def: fixes sc :: "'c \<times> 'c \<Rightarrow> bool" shows "Pre sc \<equiv> \<lambda>A. \<exists>A'. sc (A, A')"
trans: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "suffix b a"   and "suffix c b" shows "suffix c a"
rep_eq: fixes x :: "real_alg_3"   and xa :: "real_alg_3" shows "compare_3 x xa = compare_2 (rep_real_alg_3 x) (rep_real_alg_3 xa)"
J: fixes ns :: "nat list" shows "offs_pred ns ub xs index key mi ma \<longrightarrow> i < length ns \<longrightarrow> 0 < offs_num (length ns) xs index key mi ma i \<longrightarrow> ns ! i \<le> j \<and> j < offs_next ns ub xs index key mi ma i \<longrightarrow> fill xs ns index key ub mi ma ! j = Some x \<longrightarrow> index key x (length ns) mi ma = i"
add_def: fixes x :: "'a"   and y :: "'a" shows "x + y = Abs ((Rep x + Rep y) mod n)"
lq: shows "component_of_term (lt q) = component_of_term u"