Nuprl Lemma : interval-type-ap-inc
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. ∀[x:names(J)]. ∀[rho:Top]. ((<x> rho f) = (f x) ∈ Point(dM(I)))
Proof
Definitions occuring in Statement :
interval-type: 𝕀,
cubical-type-ap-morph: (u a f),
names-hom: I ⟶ J,
dM_inc: <x>,
dM: dM(I),
names: names(I),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
names-hom: I ⟶ J,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
interval-type-ap-morph,
equal_wf,
squash_wf,
true_wf,
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,
dM-lift-inc,
iff_weakening_equal,
top_wf,
names_wf,
names-hom_wf,
fset_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
instantiate,
productEquality,
independent_isectElimination,
cumulativity,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[f:I {}\mrightarrow{} J]. \mforall{}[x:names(J)]. \mforall{}[rho:Top]. ((<x> rho f) = (f x))
Date html generated:
2017_10_05-AM-01_32_32
Last ObjectModification:
2017_07_28-AM-09_43_30
Theory : cubical!type!theory
Home
Index