Nuprl Lemma : dM-lift-is-id
∀[I:fset(ℕ)]. ∀[J:{J:fset(ℕ)| I ⊆ J} ]. ∀[h:I ⟶ J].
∀[x:Point(dM(I))]. ((dM-lift(I;J;h) x) = x ∈ Point(dM(I))) supposing ∀i:names(I). ((h i) = <i> ∈ Point(dM(I)))
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],
f-subset: xs ⊆ ys,
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
names-hom: I ⟶ J,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
DeMorgan-algebra: DeMorganAlgebra,
nat: ℕ
Lemmas referenced :
sq_stable__all,
nat_wf,
isect_wf,
fset-member_wf,
int-deq_wf,
sq_stable__uall,
sq_stable__fset-member,
fset-member_witness,
squash_wf,
sq_stable_from_decidable,
f-subset_wf,
decidable__f-subset,
dM-point-subtype,
dM-dma-hom-invariant,
dM-lift_wf,
dma-hom_wf,
dM_wf,
all_wf,
names_wf,
equal_wf,
dM_inc_wf,
true_wf,
names-subtype,
iff_weakening_equal,
dM-lift-inc,
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,
names-hom_wf,
set_wf,
fset_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
hypothesisEquality,
setElimination,
rename,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
setEquality,
universeEquality,
natural_numberEquality,
productElimination,
instantiate,
productEquality,
cumulativity,
axiomEquality,
intEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[J:\{J:fset(\mBbbN{})| I \msubseteq{} J\} ]. \mforall{}[h:I {}\mrightarrow{} J].
\mforall{}[x:Point(dM(I))]. ((dM-lift(I;J;h) x) = x) supposing \mforall{}i:names(I). ((h i) = <i>)
Date html generated:
2017_10_05-AM-01_00_42
Last ObjectModification:
2017_07_28-AM-09_25_51
Theory : cubical!type!theory
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