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 :
btrue: tt,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
free-dist-lattice: free-dist-lattice(T; eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
bfalse: ff,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
record-update: r[x := v],
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
record-select: r.x,
lattice-point: Point(l),
lattice-hom: Hom(l1;l2),
bounded-lattice-hom: Hom(l1;l2),
dma-hom: dma-hom(dma1;dma2),
so_apply: x[s],
guard: {T},
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
DeMorgan-algebra: DeMorganAlgebra,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
istype-universe,
DeMorgan-algebra_wf,
deq_wf,
free-DeMorgan-algebra_wf,
dma-hom_wf,
dminc_wf,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
lattice-point_wf,
free-DeMorgan-algebra-hom-unique,
free-dma-lift_wf,
squash_wf,
true_wf,
free-dma-lift-inc,
trivial-equal,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
universeEquality,
dependent_functionElimination,
inhabitedIsType,
isectIsTypeImplies,
axiomEquality,
isect_memberEquality_alt,
rename,
setElimination,
because_Cache,
isectEquality,
independent_isectElimination,
cumulativity,
productEquality,
lambdaEquality_alt,
instantiate,
applyEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
equalityIstype,
hypothesisEquality,
universeIsType,
functionIsType,
sqequalRule,
hypothesis,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
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:
2020_05_20-AM-08_57_13
Last ObjectModification:
2020_02_04-PM-02_03_33
Theory : lattices
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