Nuprl Lemma : free-dma-lift-unique
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[dm:DeMorganAlgebra]. ∀[eq2:EqDecider(Point(dm))]. ∀[f:T ⟶ Point(dm)].
∀[g:dma-hom(free-DeMorgan-algebra(T;eq);dm)].
free-dma-lift(T;eq;dm;eq2;f) = g ∈ dma-hom(free-DeMorgan-algebra(T;eq);dm)
supposing ∀i:T. ((g <i>) = (f i) ∈ Point(dm))
Proof
Definitions occuring in Statement :
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
dma-hom: dma-hom(dma1;dma2),
DeMorgan-algebra: DeMorganAlgebra,
dminc: <i>,
lattice-point: Point(l),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
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,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
guard: {T},
so_apply: x[s],
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
all: ∀x:A. B[x],
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
all_wf,
equal_wf,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dminc_wf,
free-dma-point-subtype,
dma-hom_wf,
free-DeMorgan-algebra_wf,
deq_wf,
DeMorgan-algebra_wf,
free-DeMorgan-algebra-hom-unique,
free-dma-lift_wf,
squash_wf,
true_wf,
free-dma-lift-inc,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
instantiate,
productEquality,
independent_isectElimination,
universeEquality,
because_Cache,
setElimination,
rename,
functionExtensionality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
functionEquality,
setEquality,
lambdaFormation,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[dm:DeMorganAlgebra]. \mforall{}[eq2:EqDecider(Point(dm))].
\mforall{}[f:T {}\mrightarrow{} Point(dm)]. \mforall{}[g:dma-hom(free-DeMorgan-algebra(T;eq);dm)].
free-dma-lift(T;eq;dm;eq2;f) = g supposing \mforall{}i:T. ((g <i>) = (f i))
Date html generated:
2017_10_05-AM-00_43_03
Last ObjectModification:
2017_07_28-AM-09_17_44
Theory : lattices
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