Nuprl Lemma : dM-lift_wf
∀[I,J:fset(ℕ)]. ∀[f:I ⟶ J]. (dM-lift(I;J;f) ∈ {g:dma-hom(dM(J);dM(I))| ∀j:names(J). ((g <j>) = (f j) ∈ Point(dM(I)))} \000C)
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),
dma-hom: dma-hom(dma1;dma2),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
names-hom: I ⟶ J,
dM-lift: dM-lift(I;J;f),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
deq: EqDecider(T),
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
bool: 𝔹,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
assert: ↑b,
rev_implies: P ⇐ Q,
dM_inc: <x>,
so_lambda: λ2x.t[x],
DeMorgan-algebra: DeMorganAlgebra,
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
names-hom_wf,
fset_wf,
nat_wf,
free-dma-lift_wf,
names_wf,
names-deq_wf,
dM_wf,
free-dml-deq_wf,
subtype_rel_self,
dma-hom_wf,
all_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
applyEquality,
dependent_functionElimination,
functionExtensionality,
setEquality,
lambdaEquality,
instantiate,
productEquality,
cumulativity,
independent_isectElimination,
setElimination,
rename
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[f:I {}\mrightarrow{} J].
(dM-lift(I;J;f) \mmember{} \{g:dma-hom(dM(J);dM(I))| \mforall{}j:names(J). ((g <j>) = (f j))\} )
Date html generated:
2018_05_23-AM-08_27_35
Last ObjectModification:
2018_05_20-PM-05_35_37
Theory : cubical!type!theory
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