Nuprl Lemma : fl-lift-unique
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))]. ∀[f0,f1:T ⟶ Point(L)].
∀g:Hom(face-lattice(T;eq);L)
fl-lift(T;eq;L;eqL;f0;f1) = g ∈ Hom(face-lattice(T;eq);L)
supposing ∀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 :
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
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,
uall: ∀[x:A]. B[x],
all: ∀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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
face-lattice-hom-unique,
fl-lift_wf,
lattice-point_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-meet_wf,
lattice-join_wf,
face-lattice0_wf,
face-lattice1_wf,
bounded-lattice-hom_wf,
face-lattice_wf,
lattice-0_wf,
deq_wf,
bdd-distributive-lattice_wf,
istype-universe,
iff_weakening_equal,
squash_wf,
true_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
independent_isectElimination,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
equalityIstype,
independent_functionElimination,
because_Cache,
independent_pairFormation,
functionIsType,
universeIsType,
productIsType,
instantiate,
productEquality,
cumulativity,
isectEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsTypeImplies,
universeEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[L:BoundedDistributiveLattice]. \mforall{}[eqL:EqDecider(Point(L))].
\mforall{}[f0,f1:T {}\mrightarrow{} Point(L)].
\mforall{}g:Hom(face-lattice(T;eq);L)
fl-lift(T;eq;L;eqL;f0;f1) = g supposing \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_53_31
Last ObjectModification:
2020_02_03-PM-03_52_32
Theory : lattices
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