Nuprl Lemma : dM-lift-is-id2
∀[I:fset(ℕ)]. ∀[J,K:{K:fset(ℕ)| I ⊆ K} ]. ∀[h:K ⟶ J].
∀[x:Point(dM(I))]. ((dM-lift(K;J;h) x) = x ∈ Point(dM(K))) supposing ∀i:names(I). ((h i) = <i> ∈ Point(dM(K)))
Proof
Definitions occuring in Statement :
dM-lift: dM-lift(I;J;f),
names-hom: I ⟶ J,
dM_inc: <x>,
dM: dM(I),
names: names(I),
lattice-point: Point(l),
f-subset: xs ⊆ ys,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
DeMorgan-algebra: DeMorganAlgebra,
prop: ℙ,
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),
names-hom: I ⟶ J,
nat: ℕ,
compose: f o g,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
compose-dma-hom,
dM_wf,
dM-subobject,
sq_stable_from_decidable,
f-subset_wf,
nat_wf,
int-deq_wf,
decidable__f-subset,
dM-lift_wf,
dma-hom_wf,
all_wf,
names_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,
dM_inc_wf,
dM-dM-homs-equal,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
dM-lift_wf2,
names-subtype,
iff_weakening_equal,
dM-lift-inc,
names-hom_wf,
set_wf,
fset_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
because_Cache,
independent_isectElimination,
applyEquality,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
setEquality,
instantiate,
productEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
intEquality,
natural_numberEquality,
lambdaFormation,
productElimination,
isect_memberFormation,
isect_memberEquality,
axiomEquality,
applyLambdaEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[J,K:\{K:fset(\mBbbN{})| I \msubseteq{} K\} ]. \mforall{}[h:K {}\mrightarrow{} J].
\mforall{}[x:Point(dM(I))]. ((dM-lift(K;J;h) x) = x) supposing \mforall{}i:names(I). ((h i) = <i>)
Date html generated:
2018_05_23-AM-08_28_38
Last ObjectModification:
2018_05_20-PM-05_37_16
Theory : cubical!type!theory
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