Nuprl Lemma : dM-lift-dMpair
∀[I,J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x,y:names(I)]. ((dM-lift(J;I;f) dMpair(x;y)) = f x ∧ f y ∈ Point(dM(J)))
Proof
Definitions occuring in Statement :
dM-lift: dM-lift(I;J;f),
names-hom: I ⟶ J,
dMpair: dMpair(i;j),
dM: dM(I),
names: names(I),
lattice-meet: a ∧ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
names: names(I),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
all: ∀x:A. B[x],
names-hom: I ⟶ J,
prop: ℙ,
DeMorgan-algebra: DeMorganAlgebra,
and: P ∧ Q,
guard: {T},
dma-hom: dma-hom(dma1;dma2),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
dMpair-eq-meet,
sq_stable__fset-member,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
names_wf,
names-hom_wf,
fset_wf,
dM-lift_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
lattice-point_wf,
dM_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,
lattice-hom-meet,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
dM_inc_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
independent_isectElimination,
applyEquality,
intEquality,
because_Cache,
sqequalRule,
lambdaEquality_alt,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType,
lambdaFormation_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
instantiate,
universeEquality,
productEquality,
cumulativity,
productElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[x,y:names(I)]. ((dM-lift(J;I;f) dMpair(x;y)) = f x \mwedge{} f y)
Date html generated:
2019_11_04-PM-05_30_46
Last ObjectModification:
2018_12_13-AM-10_02_48
Theory : cubical!type!theory
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