hcju/isabelleps huggingface dataset:
bidirected_digraph_del_arc: fixes a :: "'b" shows "bidirected_digraph (pre_digraph.del_arc (G.del_arc (edge_rev M a)) a) (perm_restrict (edge_rev M) (arcs G - {a, edge_rev M a}))"
in_rel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a multiset"   and b :: "'b multiset" shows "rel_mset R a b = (\<exists>z. z \<in> {x. set_mset x \<subseteq> {(x, y). R x y}} \<and> image_mset fst z = a \<and> image_mset snd z = b)"
mono_mprefix_DT_set: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> 'a process" assumes "A \<subseteq> B" shows "Mprefix B P \<sqsubseteq>\<^sub>D\<^sub>T Mprefix A P"
SC_WbisT_imp_SC_BisT: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.SC_WbisT tval aval indis c" shows "PL_Indis.SC_BisT tval aval indis c"
meas_3: shows "(\<lambda>x. (- fst x, - snd x)) \<in> borel_measurable borel"
Gen_Shleg_def: fixes n :: "nat"   and k :: "nat"   and x :: "real" shows "Gen_Shleg n k x = poly (of_int_poly (gen_shleg_poly n k)) x"
interleave_recover1: fixes w :: "'b list"   and w1a :: "'b list"   and w1b :: "'b list"   and w2 :: "'b list" assumes "w \<in> w1a @ w1b \<otimes> w2" shows "\<exists>wa wb w2a w2b. w = wa @ wb \<and> w2 = w2a @ w2b \<and> wa \<in> w1a \<otimes> w2a \<and> wb \<in> w1b \<otimes> w2b"
setmonotone_limit: fixes f :: "'a set \<Rightarrow> 'a set" assumes "setmonotone f" shows "setmonotone (limit f)"
assms_3: shows "boruvka_inner_invariant j f h g d"
hyps: fixes p :: "'a list" assumes "poly p = poly cs" shows "pnormalize p = pnormalize cs"
is_sig_GB_uptD3: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_GB_upt d G u"   and "r \<in> dgrad_sig_set' (length fs) d"   and "lt r \<prec>\<^sub>t u" shows "sig_red_zero (\<preceq>\<^sub>t) G r"
neg_Neg: fixes \<phi> :: "fm" shows "{} \<turnstile> neg \<phi> IFF Neg \<phi>"
set_integrable_subset: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "set_integrable M A f"   and "B \<in> sets M"   and "B \<subseteq> A" shows "set_integrable M B f"
cbeta2_gammabm: shows "(b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>) O (e \<union> b \<union> m \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d \<union> d\<inverse>) \<subseteq> e \<union> b \<union> m \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d \<union> d\<inverse>"
hm_iteratei_def: fixes hm :: "('b, 'c) hashmap" shows "hm_iteratei hm \<equiv> HashMap_Impl.iteratei (impl_of_RBT_HM hm)"
infdist_attains_inf: fixes X :: "'b set"   and y :: "'b" assumes "closed X"   and "X \<noteq> {}" obtains x :: "'b" where "x \<in> X" and "infdist y X = dist y x"
side1: shows "{y. 0 < (y - flow0 x t2) \<bullet> rot (flow0 x t2 - flow0 x t1)} = {y. (y - rp) \<bullet> rot (a - b) < 0}"
ccompare_prod_def: shows "ccompare \<equiv> case ID ccompare of None \<Rightarrow> None | Some comp_'a \<Rightarrow> case ID ccompare of None \<Rightarrow> None | Some comp_'b \<Rightarrow> Some (comparator_prod comp_'a comp_'b)"
h4: assumes "Av * Bv \<le> 0 \<and> Av\<^sup>2 = Bv\<^sup>2 * Cv" shows "Av + Bv * sqrt Cv = 0"
list_update_id: fixes xs :: "'c list"   and i :: "nat" shows "xs[i := xs ! i] = xs"
data_dependency_inv_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "data_dependency_inv ts \<equiv> valid_data_dependency ts \<and> load_tmps_fresh ts \<and> valid_sops ts"
acquired_reads_all_acquired': fixes sb :: "'b memref list"   and \<O> :: "nat set" shows "acquired_reads True sb \<O> \<subseteq> acquired True sb \<O> \<union> all_shared sb"
tendsto_mult_right: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter"   and c :: "'b" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. f x * c) \<longlongrightarrow> l * c) F"
catalan_closed_form_aux: fixes n :: "nat" shows "catalan n * Suc n = 2 * n choose n"
add_commute: fixes a :: "'a"   and b :: "'a" shows "a + b = b + a"
Discr_undiscr: fixes y :: "'a discr" shows "Discr (undiscr y) = y"
ext: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "g \<bullet> f \<noteq> null" shows "seq g f"
VA: shows "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT (Differential \<theta>)}. FVT (\<sigma> i))"
nfa_Reverse_nfa: shows "nfa (Reverse_nfa M)"
element_ptr_kinds_M_reads: fixes h :: "(_) heap"   and h' :: "(_) heap" shows "reads (\<Union>element_ptr. {preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t element_ptr RObject.nothing)}) element_ptr_kinds_M h h'"
dim_eq: assumes "V.fin_dim"   and "inj_on T (carrier V)"   and "T ` carrier V = carrier W" shows "V.dim = W.dim"
is_stmt_rews_3: fixes x :: "expr list" shows "\<not> is_stmt (In3 x)"
CastThrow: fixes C :: "char list"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>Cast C (throw e),s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
triangle_right: shows "(F g \<star>\<^sub>D D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) \<cdot>\<^sub>D \<a>\<^sub>D[F g, F f, F g] \<cdot>\<^sub>D (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f) \<star>\<^sub>D F g) = \<r>\<^sub>D\<^sup>-\<^sup>1[F g] \<cdot>\<^sub>D \<l>\<^sub>D[F g]"
cases: fixes x :: "recf" obtains "x = Z"   | "x = S"   | m :: "nat" and n :: "nat" where "x = recf.Id m n__"   | n :: "nat" and f :: "recf" and gs :: "recf list" where "x = Cn na__ fa__ gs"   | n :: "nat" and f :: "recf" and g :: "recf" where "x = Pr nb__ fb__ g"   | n :: "nat" and f :: "recf" where "x = Mn n f"
restrict_to_halfspaces_impl: fixes sctns :: "'b sctn set"   and X :: "'b set" shows "ASSUME (finite sctns) \<bind> (\<lambda>_. FOREACH\<^bsup>\<lambda>sctns' Y. Y = X \<inter> below_halfspaces (sctns - sctns')\<^esup> sctns restrict_to_halfspace X) \<le> restrict_to_halfspaces sctns X"
SigmaE: fixes c :: "'e \<times> 'f"   and A :: "'e set"   and B :: "'e \<Rightarrow> 'f set" assumes "c \<in> Sigma A B" obtains x :: "'e" and y :: "'f" where "x \<in> A" and "y \<in> B x" and "c = (x, y)"
rel_4: shows "int_of_uint64 x' \<le> p"
total_quasi_ordered_set_axioms: shows "total_quasi_ordered_set A (\<lambda>x y. y \<sqsubseteq> x)"
pmf_bind_pmf_of_multiset: fixes A :: "'a multiset"   and f :: "'a \<Rightarrow> 'b pmf"   and y :: "'b" assumes "A \<noteq> {#}" shows "pmf (pmf_of_multiset A \<bind> f) y = (\<Sum>x\<in>set_mset A. real (count A x) * pmf (f x) y) / real (size A)"
vcomp_vempty_left: fixes r :: "V" shows "0 \<circ>\<^sub>\<circ> r = 0"
set_less_eq_aux_Compl2_conv_set_less_eq_aux_Compl: fixes xs :: "'a list"   and ys :: "'a list" assumes "finite UNIV"   and "sorted xs"   and "distinct xs"   and "sorted ys"   and "distinct ys" shows "(set xs \<sqsubseteq>' - set ys) = set_less_eq_aux_Compl None xs ys"
states_of_se_assign: fixes c :: "('c, 'd) conf"   and v :: "'c"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and c' :: "('c, 'd) conf" assumes "se c (Assign v e) c'" shows "states c' = {\<sigma>(v := e \<sigma>) |\<sigma>. \<sigma> \<in> states c}"
leD: fixes x :: "'a list"   and y :: "'a list" assumes "x \<le>lex y" shows "\<not> y <lex x"
rep_eq: fixes x :: "uint8" shows "int_of_uint8 x = uint (Rep_uint8 x)"
homotopic_neighbourhood_extension: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and g :: "'a \<Rightarrow> 'b"   and T :: "'a set" assumes "continuous_on S f"   and "f ` S \<subseteq> U"   and "continuous_on S g"   and "g ` S \<subseteq> U"   and "closedin (top_of_set S) T"   and "ANR U"   and "homotopic_with_canon (\<lambda>x. True) T U f g" obtains V :: "'a set" where "T \<subseteq> V" and "openin (top_of_set S) V" and "homotopic_with_canon (\<lambda>x. True) V U f g"
simps_3: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and tst :: "'c"   and c1 :: "('c, 'b) com"   and c2 :: "('c, 'b) com" assumes "PL_Indis indis" shows "PL_Indis.SC_discr aval indis (if tst then c1 else c2) = (PL_Indis.SC_discr aval indis c1 \<and> PL_Indis.SC_discr aval indis c2)"
app_vconverseD: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ>" shows "\<langle>b, a\<rangle> \<in>\<^sub>\<circ> r"
assms_1: shows "expectation_gpv2 \<equiv> expectation_gpv 0 \<I>'"
indicator_le_1: fixes S :: "'c set"   and x :: "'c" shows "indicator S x \<le> (1::'b)"
pareto_losers_lift: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and agents' :: "'a set"   and alts' :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "scf_lowering agents alts scf agents' alts'"   and "pref_profile_wf agents' alts' R" shows "pareto_losers (lift_pref_profile agents' alts' agents alts R) = pareto_losers R \<union> (alts - alts')"
Ptl_LCons: shows "ltl P = LCons comp.w0 (ltl (ltl P))"
right_minus: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "a \<cdot> (- b) = - a \<cdot> b"
a_get_root_node_si_locs_def: fixes get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_si_locs get_parent_locs get_host_locs = l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_si_locs get_parent_locs get_host_locs"
\<phi>'\<psi>': shows "meta_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G' \<phi>' \<psi>'"
F_disjnt: fixes p :: "'a set"   and q :: "'a set" assumes "p \<in> P"   and "q \<in> P"   and "disjnt p q" shows "disjnt (F p) (F q)"
triples_valid2_Suc: fixes ts :: "'b triple set"   and G :: "prog"   and n :: "nat" assumes "Ball ts (triple_valid2 G (Suc n))" shows "Ball ts (triple_valid2 G n)"
row_echelon_form_berlekamp_resulting_mat: shows "row_echelon_form (berlekamp_resulting_mat u)"
comp_arr_dom: fixes f :: "'d"   and a :: "'d" assumes "Dop.arr f"   and "Dop.dom f = a" shows "Dop.comp f a = f"
hnorm_ge_zero: fixes x :: "'a star" shows "0 \<le> hnorm x"
distinct_26: fixes x51 :: "'b ptrm"   and x52 :: "'b ptrm"   and x31 :: "'b ptrm"   and x32 :: "'b ptrm" shows "PPair x51 x52 \<noteq> PApp x31 x32"
perp_distinct: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D" shows "A \<noteq> B \<and> C \<noteq> D"
filter_map: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "filter P (map f xs) = map f (filter (P \<circ> f) xs)"
gpv_stop_parametric': fixes A :: "'f \<Rightarrow> 'i \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'k \<Rightarrow> bool" shows "(rel_gpv'' A C R ===> rel_gpv'' (rel_option A) C (rel_option R)) gpv_stop gpv_stop"
clock_simAction_jAction: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agents :: "'b odlist"   and t :: "'e Trace"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and aec :: "'e odlist"   and a :: "'b"   and cec :: "'e odlist" assumes "FiniteLinorderEnvironment jkbp agents"   and "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal (Environment.clock_jview envObs)"   and "toSet aec = FiniteLinorderEnvironment.agent_abs jkbp envInit envAction envTrans envVal envObs a t"   and "toSet cec = FiniteLinorderEnvironment.common_abs jkbp envInit envAction envTrans envVal envObs t" shows "set (ClockView.clock_simAction jkbp envVal envObs a (cec, aec)) = set (JKBP.jAction jkbp (mkKripke (toSet cec) (FiniteLinorderEnvironment.clock_repRels envObs) envVal) (tLast t) a)"
RedStaticCastNull: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char 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" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>null,s\<rangle> \<rightarrow> \<langle>null,s\<rangle>"
states_of_exec_reachable: fixes ex :: "(nat \<Rightarrow> 'e option) \<times> (nat \<Rightarrow> 'f)"   and A :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set" assumes "ex \<in> executions A" shows "\<forall>n. reachable A (snd ex n)"
pre: shows "check_precond_r2 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta>"
bQ: fixes x :: "'a" shows "bounded_by 1 (Q x)"
measure_induct: fixes f :: "'g \<Rightarrow> 'h"   and P :: "'g \<Rightarrow> bool"   and a :: "'g" assumes "\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
c_sum_as_b_least: shows "c_sum = (\<lambda>u. b_least2 (\<lambda>u z. sgn1 (sf (z + 1) - u)) u (Suc u))"
CLAMP_5: fixes f :: "nat \<Rightarrow> real" shows "\<Theta>(f) \<equiv> \<Theta>(CLAMP f)"
dominated_convergence_at_top_2: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'c"   and s :: "'b set"   and h :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>k. f k integrable_on s"   and "h integrable_on s"   and "\<And>k x. x \<in> s \<Longrightarrow> norm (f k x) \<le> h x"   and "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) at_top" shows "((\<lambda>k. integral s (f k)) \<longlongrightarrow> integral s g) at_top"
normalX: shows "distributed M lborel X (\<lambda>x. ennreal (normal_density \<mu> \<sigma> x))"
eq_linorder_on_axioms: shows "eq_linorder_on D1 cmp1"
foldr_append: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list"   and a :: "'b" shows "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
antisym: fixes b :: "'g multiset"   and a :: "'g multiset" assumes "b \<subseteq># a"   and "a \<subseteq># b" shows "a = b"
simps_2: fixes t :: "term"   and ts :: "term list" shows "linears (t # ts) = (linear t \<and> fdisjnt (frees t) (freess ts) \<and> linears ts)"
bex_cong: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> P x = Q x" shows "(\<exists>x\<in>A. P x) = (\<exists>x\<in>B. Q x)"
exec: shows "steps0 (s, l, r) A stp = (s', l', r')"
add_divide_eq_if_simps_1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and z :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "plus a (divide b z) = (if z = zero then a else divide (plus (times a z) b) z)"
Accessible_language: shows "dfa.language ST.Accessible_dfa = ST.language"
iteratei_def: fixes t :: "('d, 'e) trie" shows "iteratei t = set_iterator_image trie_reverse_key (Trie_Impl.iteratei (impl_of t))"
univ_poly_zero_closed: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" shows "[] \<in> carrier (K [X]\<^bsub>R\<^esub>)"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a OR a OR b = a OR b"
connected_closedD: fixes s :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "connected s"   and "A \<inter> B \<inter> s = {}"   and "s \<subseteq> A \<union> B"   and "closed A"   and "closed B" shows "A \<inter> s = {} \<or> B \<inter> s = {}"
\<I>_deduct: fixes M :: "('fun, 'var) Term.term set"   and x :: "'var" shows "M \<turnstile>\<^sub>c \<I> x"
tuple_pr_arr: fixes a :: "'a"   and b :: "'a"   and h :: "'a" assumes "ide a"   and "ide b"   and "seq \<pp>\<^sub>0[a, b] h" shows "\<langle>\<pp>\<^sub>1[a, b] \<cdot> h, \<pp>\<^sub>0[a, b] \<cdot> h\<rangle> = h"
P: shows "P = (\<lambda>((s, l, r), ss, x). s = start_of ly as + 2 * n + 16)"
pr_simps_5: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "local.dom \<pp>\<^sub>1[a, b] = a \<otimes> b"
case_prod_app: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'c"   and x :: "'d \<times> 'e"   and y :: "'f" shows "(case x of (x, xa) \<Rightarrow> f x xa) y = (case x of (l, r) \<Rightarrow> f l r y)"
simps_2: shows "PX5d D2 = linord_of_list [X5.Xd2, X5.Xd2', Xd2'']"
integrableI_simple_bochner_integrable: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f" shows "integrable M f"
DgrDataRefinement2_def: fixes P :: "'a \<Rightarrow> 'b"   and D :: "'a \<times> 'a \<Rightarrow> 'b \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and D' :: "'a \<times> 'a \<Rightarrow> 'c \<Rightarrow> 'c" shows "DgrDataRefinement2 P D R D' = (\<forall>i j. DataRefinement ({. P i .} \<circ> D (i, j)) (R i) (R j) (D' (i, j)))"
ifexd_valid_def: fixes bdd :: "bdd"   and e :: "('a, nat) IFEXD" shows "ifexd_valid bdd e \<equiv> case e of IFD x t e \<Rightarrow> bdd_node_valid bdd t \<and> bdd_node_valid bdd e | _ \<Rightarrow> True"
simps_2: shows "norm_instr Inca.instr.IPop = Std.instr.IPop"
bind_bind: fixes m :: "'g\<cdot>'e"   and f :: "'g \<rightarrow> 'f\<cdot>'e"   and g :: "'f \<rightarrow> 'd\<cdot>'e" shows "m \<bind> f \<bind> g = m \<bind> (\<Lambda> x. f\<cdot>x \<bind> g)"
simps: fixes xs :: "'b list"   and ys :: "'c list"   and z :: "'c"   and zs :: "'c list" shows "proj_list_3 xs (ys, z, zs) = (take (length ys) xs, xs ! length ys, take (length zs) (drop (length ys + 1) xs))"
inner_zero_left: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner zero x = 0"
assms: shows "x \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> A"
disjunctiveI: fixes a :: "'b \<Rightarrow> 'b" assumes "\<And>b c. a \<circ> b \<squnion> c = (a \<circ> b) \<squnion> (a \<circ> c)" shows "a \<in> Comp.disjunctive"
fair_imp_Liminf_minus_Rf_subset_ground_Liminf_state: fixes Gs :: "'a literal multiset set llist" assumes "chain (\<leadsto>) Sts"   and "fair_state_seq Sts"   and "Gs = lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts" shows "Liminf_llist Gs - sr.Rf (Liminf_llist Gs) \<subseteq> grounding_of_clss (Q_of_state (Liminf_state Sts))"
empty_def: shows "RBT.empty \<equiv> RBT rbt.Empty"
strict_mono_Suc_iff: fixes f :: "nat \<Rightarrow> 'a" shows "strict_mono f = (\<forall>n. f n < f (Suc n))"
update_tableau_valuated: fixes s :: "('i, 'a) state"   and x :: "nat"   and v :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<nabla> (update x v s)"
FIN_subseteq_CP: shows "FIN \<subseteq> CP"
assms_4: fixes x :: "'a" assumes "x \<in> space M"   and "x \<notin> X" shows "f x = g x"
0: shows "round (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j) = 0"
replicate_app_Cons_same: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "x \<up> n @ x # xs = x # x \<up> n @ xs"
proj_preserves: fixes x :: "'b"   and A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "x \<in> A" shows "Equiv_Relations.proj r x \<in> A // r"
unfold_foldrS: fixes s :: "'b"   and f :: "'d \<rightarrow> 'c \<rightarrow> 'c"   and z :: "'c"   and h :: "'b \<rightarrow> ('d, 'b) Step" assumes "s \<noteq> \<bottom>" shows "foldrS\<cdot>f\<cdot>z\<cdot>(Stream\<cdot>h\<cdot>s) = foldrL\<cdot>f\<cdot>z\<cdot>(unfold\<cdot>h\<cdot>s)"
assms_2: fixes m :: "'a match_expr" assumes "normalized_nnf_match m" shows "matches \<gamma> (replace_primitive_matchexpr disc_sel replace_f m) a p = matches \<gamma> m a p"
sets_extend_measure: fixes G :: "'d \<Rightarrow> 'c set"   and I :: "'d set"   and \<Omega> :: "'c set"   and \<mu> :: "'d \<Rightarrow> ennreal" assumes "G ` I \<subseteq> Pow \<Omega>" shows "sets (extend_measure \<Omega> I G \<mu>) = sigma_sets \<Omega> (G ` I)"
round_suc_suc_def: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and ws :: "'c list"   and n :: "nat"   and n' :: "nat"   and u :: "nat" shows "round_suc_suc index key ws n n' u \<equiv> let nmi = mini ws key; nma = maxi ws key; xmi = ws ! nmi; xma = ws ! nma; mi = key xmi; ma = key xma in if mi = ma then (u + n' - n, replicate (Suc (Suc n)) (Suc 0), ws) else let k = case n of 0 \<Rightarrow> n | Suc 0 \<Rightarrow> n | Suc (Suc i) \<Rightarrow> u + n'; zs = nths ws (- {nmi, nma}); ms = enum zs index key k mi ma in (u + n' - k, Suc 0 # ms @ [Suc 0], xmi # map the (fill zs (offs ms 0) index key n mi ma) @ [xma])"
minus_Real: fixes X :: "nat \<Rightarrow> rat" assumes "cauchy X" shows "- Real X = Real (\<lambda>n. - X n)"
ide_is_iso: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "B.VV.iso a"
assms_3: shows "zs \<noteq> ys \<or> z \<noteq> y"
\<epsilon>': shows "\<guillemotleft>unit (src\<^sub>C g) \<cdot>\<^sub>D \<psi> \<cdot>\<^sub>D (D.inv (F f) \<star>\<^sub>D g') \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv \<mu>) \<cdot>\<^sub>D D.inv (\<Phi> (f, g)) : F (f \<star>\<^sub>C g) \<Rightarrow>\<^sub>D F (trg\<^sub>C f)\<guillemotright>"
c_sources_aux_append: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and U :: "'e set"   and xs :: "'f list"   and x :: "'f" shows "c_sources_aux I D U (xs @ [x]) = (if D x \<in> c_sources_aux I D U [x] then c_sources_aux I D (insert (D x) U) xs else c_sources_aux I D U xs)"
prime_elem_multiplicity_eq_zero_iff: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "x \<noteq> zero" shows "(factorial_semiring.multiplicity times one p x = 0) = (\<not> dvd.dvd times p x)"
exec_mbindFStop_failure: fixes ioprog :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and a :: "'f"   and \<sigma> :: "'e"   and S :: "'f list"   and M :: "'d list \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "ioprog a \<sigma> = None" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p (a # S) ioprog) M) = False"
take_is_prefix: fixes n :: "nat"   and xs :: "'b list" shows "prefix (take n xs) xs"