Nuprl Lemma : flattice-order-meet
∀X:Type. ∀a1,b1,as,bs:(X + X) List List.
(flattice-order(X;a1;b1) ⇒ flattice-order(X;as;bs) ⇒ flattice-order(X;free-dl-meet(a1;as);free-dl-meet(b1;bs)))
Proof
Definitions occuring in Statement :
flattice-order: flattice-order(X;as;bs),
free-dl-meet: free-dl-meet(as;bs),
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
union: left + right,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
flattice-order: flattice-order(X;as;bs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
cand: A c∧ B,
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
l_subset: l_subset(T;as;bs)
Lemmas referenced :
l_all_iff,
list_wf,
l_member_wf,
or_wf,
l_exists_wf,
equal_wf,
flip-union_wf,
l_contains_wf,
free-dl-meet_wf_list,
l_exists_iff,
exists_wf,
append_wf,
member-free-dl-meet,
all_wf,
flattice-order_wf,
length_wf_nat,
nat_wf,
member_append,
l_exists_append,
squash_wf,
true_wf,
iff_weakening_equal,
l_subset-l_contains,
l_subset_wf,
l_subset_append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
unionEquality,
cumulativity,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
setElimination,
rename,
because_Cache,
setEquality,
productElimination,
independent_functionElimination,
allFunctionality,
addLevel,
orFunctionality,
productEquality,
promote_hyp,
impliesFunctionality,
existsFunctionality,
independent_pairFormation,
andLevelFunctionality,
existsLevelFunctionality,
functionEquality,
universeEquality,
unionElimination,
inlFormation,
dependent_set_memberEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
inrFormation,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}X:Type. \mforall{}a1,b1,as,bs:(X + X) List List.
(flattice-order(X;a1;b1)
{}\mRightarrow{} flattice-order(X;as;bs)
{}\mRightarrow{} flattice-order(X;free-dl-meet(a1;as);free-dl-meet(b1;bs)))
Date html generated:
2020_05_20-AM-08_59_33
Last ObjectModification:
2017_07_28-AM-09_18_16
Theory : lattices
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