Nuprl Lemma : face-lattice-property
∀T:Type. ∀eq:EqDecider(T). ∀L:BoundedDistributiveLattice. ∀eqL:EqDecider(Point(L)). ∀f0,f1:T ⟶ Point(L).
∃g:Hom(face-lattice(T;eq);L) [(∀x:T. (((g (x=0)) = (f0 x) ∈ Point(L)) ∧ ((g (x=1)) = (f1 x) ∈ Point(L))))]
supposing ∀x:T. (f0 x ∧ f1 x = 0 ∈ Point(L))
Proof
Definitions occuring in Statement :
face-lattice1: (x=1),
face-lattice0: (x=0),
face-lattice: face-lattice(T;eq),
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-0: 0,
lattice-meet: a ∧ b,
lattice-point: Point(l),
deq: EqDecider(T),
uimplies: b supposing a,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
face-lattice: face-lattice(T;eq),
exists: ∃x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
bdd-distributive-lattice: BoundedDistributiveLattice,
face-lattice-constraints: face-lattice-constraints(x),
uiff: uiff(P;Q),
fset-pair: {a,b},
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
empty-fset: {},
true: True,
lattice-fset-meet: /\(s),
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bdd-lattice: BoundedLattice,
compose: f o g
Lemmas referenced :
free-dist-lattice-with-constraints-property,
union-deq_wf,
face-lattice-constraints_wf,
fset-member_wf,
fset_wf,
deq-fset_wf,
lattice-point_wf,
face-lattice0_wf,
face-lattice1_wf,
lattice-meet_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
equal_wf,
lattice-join_wf,
lattice-0_wf,
deq_wf,
bdd-distributive-lattice_wf,
istype-universe,
member-fset-singleton,
fset-pair_wf,
list_accum_cons_lemma,
list_accum_nil_lemma,
lattice-fset-meet_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
decidable-equal-deq,
fset-image_wf,
fset-union_wf,
empty-fset_wf,
fset-singleton_wf,
reduce_nil_lemma,
squash_wf,
true_wf,
lattice-fset-meet-union,
lattice-fset-meet-singleton,
subtype_rel_self,
iff_weakening_equal,
lattice-1-meet,
face-lattice0-is-inc,
face-lattice1-is-inc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
rename,
extract_by_obid,
isectElimination,
unionEquality,
unionIsType,
universeIsType,
equalityTransitivity,
equalitySymmetry,
unionElimination,
applyEquality,
equalityIstype,
independent_functionElimination,
independent_isectElimination,
because_Cache,
productElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
functionIsType,
productIsType,
setElimination,
instantiate,
productEquality,
cumulativity,
isectEquality,
universeEquality,
inlEquality_alt,
inrEquality_alt,
Error :memTop,
hyp_replacement,
applyLambdaEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}L:BoundedDistributiveLattice. \mforall{}eqL:EqDecider(Point(L)).
\mforall{}f0,f1:T {}\mrightarrow{} Point(L).
\mexists{}g:Hom(face-lattice(T;eq);L) [(\mforall{}x:T. (((g (x=0)) = (f0 x)) \mwedge{} ((g (x=1)) = (f1 x))))]
supposing \mforall{}x:T. (f0 x \mwedge{} f1 x = 0)
Date html generated:
2020_05_20-AM-08_51_37
Last ObjectModification:
2020_02_03-PM-03_08_10
Theory : lattices
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