Nuprl Lemma : fl-lift-unique2
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[L:BoundedDistributiveLattice]. ∀[eqL:EqDecider(Point(L))]. ∀[f0,f1:T ⟶ Point(L)].
  ∀g:Hom(face-lattice(T;eq);L)
    ∀x:Point(face-lattice(T;eq)). ((fl-lift(T;eq;L;eqL;f0;f1) x) = (g x) ∈ Point(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]
, 
squash: ↓T
, 
prop: ℙ
, 
subtype_rel: A ⊆r B
, 
bdd-distributive-lattice: BoundedDistributiveLattice
, 
so_lambda: λ2x.t[x]
, 
and: P ∧ Q
, 
so_apply: x[s]
, 
bounded-lattice-hom: Hom(l1;l2)
, 
lattice-hom: Hom(l1;l2)
, 
true: True
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
Lemmas referenced : 
fl-lift-unique, 
equal_wf, 
squash_wf, 
true_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, 
uall_wf, 
lattice-meet_wf, 
lattice-join_wf, 
fl-lift_wf, 
bounded-lattice-hom_wf, 
face-lattice_wf, 
all_wf, 
face-lattice0_wf, 
face-lattice1_wf, 
iff_weakening_equal, 
bdd-distributive-lattice_wf, 
lattice-0_wf, 
deq_wf
Rules used in proof : 
cut, 
introduction, 
extract_by_obid, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
hypothesis, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
independent_isectElimination, 
lambdaFormation, 
applyEquality, 
lambdaEquality, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
universeEquality, 
sqequalRule, 
instantiate, 
productEquality, 
because_Cache, 
cumulativity, 
functionExtensionality, 
setElimination, 
rename, 
setEquality, 
dependent_functionElimination, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
productElimination, 
independent_functionElimination, 
axiomEquality, 
isect_memberEquality, 
functionEquality
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)
        \mforall{}x:Point(face-lattice(T;eq)).  ((fl-lift(T;eq;L;eqL;f0;f1)  x)  =  (g  x)) 
        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:
2017_10_05-AM-00_41_26
Last ObjectModification:
2017_07_28-AM-09_16_34
Theory : lattices
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