Nuprl Lemma : global-order-compat-combine
∀[Info:Type]
∀L1,L2:(Id × Info) List.
(L1 || L2
⇒ (∃L:(Id × Info) List
(L1 ≤ L
∧ (∃f:ℕ||L2|| ─→ ℕ||L||
((∀i,j:ℕ||L2||. (i < j ⇒ (f i < f j ∨ f j < ||L1||)))
∧ (∀j:ℕ||L2||
((L2[j] = L[f j] ∈ (Id × Info))
∧ (filter(λx.fst(x) = fst(L2[j]);firstn(j + 1;L2))
= filter(λx.fst(x) = fst(L2[j]);firstn((f j) + 1;L))
∈ ((Id × Info) List))))
∧ (∀x:ℕ||L||. (x < ||L1|| ∨ (∃i:ℕ||L2||. (x = (f i) ∈ ℤ))))))
∧ L2 ≤ L, locally
∧ (∀i:Id
((filter(λx.fst(x) = i;L) = filter(λx.fst(x) = i;L1) ∈ ((Id × Info) List))
∨ (filter(λx.fst(x) = i;L) = filter(λx.fst(x) = i;L2) ∈ ((Id × Info) List)))))))
Proof
Definitions occuring in Statement :
global-order-compat: L1 || L2,
global-order-iseg: L1 ≤ L2, locally,
eq_id: a = b,
Id: Id,
iseg: l1 ≤ l2,
filter: filter(P;l),
select: L[n],
firstn: firstn(n;as),
length: ||as||,
list: T List,
int_seg: {i..j-},
less_than: a < b,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
product: x:A × B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Lemmas :
last_induction,
global-order-compat_wf,
exists_wf,
iseg_wf,
int_seg_wf,
length_wf,
all_wf,
less_than_wf,
or_wf,
select_wf,
sq_stable__le,
zero-le-nat,
int_seg_subtype-nat,
false_wf,
non_neg_length,
length_wf_nat,
filter_wf5,
firstn_wf,
l_member_wf,
eq_id_wf,
equal_wf,
global-order-iseg_wf,
nil_wf,
append_wf,
cons_wf,
list_wf,
Id_wf,
iseg_weakening,
length_of_nil_lemma,
less_than_transitivity1,
less_than_irreflexivity,
lelt_wf,
stuck-spread,
base_wf,
list_ind_nil_lemma,
filter_nil_lemma,
decidable__lt,
equal-wf-base-T,
int_subtype_base,
less-iff-le,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
pi1_wf_top,
nil_iseg,
length_nil,
length_wf_nil,
subtype_rel_product,
top_wf,
filter_append_sq,
iseg_append_iff,
filter_cons_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
iseg_append0,
assert_wf,
bnot_wf,
not_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert-eq-id,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
iseg_transitivity,
decidable__le,
subtype_rel_list,
filter_iseg,
iseg_length,
iseg_ge_length,
nat_wf,
ge_wf,
le_weakening,
length_cons,
length_append,
lt_int_wf,
assert_of_lt_int,
subtype_rel-int_seg,
bool_cases_sqequal,
assert-bnot,
add-associates,
select-append,
iff_imp_equal_bool,
btrue_wf,
iff_wf,
firstn_append,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
add-zero,
le-add-cancel2,
le_wf,
subtract_wf,
select-cons-hd,
minus-zero,
add-mul-special,
zero-mul,
firstn_all,
append-nil,
iseg_append_single,
atom2_subtype_base,
iseg_weakening2,
iseg_filter_last,
last_singleton_append,
length-append,
last_wf,
list-cases,
null_nil_lemma,
length_of_cons_lemma,
product_subtype_list,
null_cons_lemma,
minus-minus,
le-add-cancel-alt,
iseg_select,
subtract-is-less,
set_wf,
iseg-iff-firstn,
and_wf,
append_back_nil
Latex:
\mforall{}[Info:Type]
\mforall{}L1,L2:(Id \mtimes{} Info) List.
(L1 || L2
{}\mRightarrow{} (\mexists{}L:(Id \mtimes{} Info) List
(L1 \mleq{} L
\mwedge{} (\mexists{}f:\mBbbN{}||L2|| {}\mrightarrow{} \mBbbN{}||L||
((\mforall{}i,j:\mBbbN{}||L2||. (i < j {}\mRightarrow{} (f i < f j \mvee{} f j < ||L1||)))
\mwedge{} (\mforall{}j:\mBbbN{}||L2||
((L2[j] = L[f j])
\mwedge{} (filter(\mlambda{}x.fst(x) = fst(L2[j]);firstn(j + 1;L2))
= filter(\mlambda{}x.fst(x) = fst(L2[j]);firstn((f j) + 1;L)))))
\mwedge{} (\mforall{}x:\mBbbN{}||L||. (x < ||L1|| \mvee{} (\mexists{}i:\mBbbN{}||L2||. (x = (f i)))))))
\mwedge{} L2 \mleq{} L, locally
\mwedge{} (\mforall{}i:Id
((filter(\mlambda{}x.fst(x) = i;L) = filter(\mlambda{}x.fst(x) = i;L1))
\mvee{} (filter(\mlambda{}x.fst(x) = i;L) = filter(\mlambda{}x.fst(x) = i;L2)))))))
Date html generated:
2015_07_21-PM-04_39_47
Last ObjectModification:
2015_02_04-PM-06_02_35
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