Nuprl Lemma : fix_wf-pcorec-partial-nat
∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[f:⋂X:P ⟶ Type
((i:P ⟶ (X i) ⟶ partial(ℕ))
⟶ i:P
⟶ (ptuple(lbl,p.a[lbl;p];X) i)
⟶ partial(ℕ))].
(fix(f) ∈ i:P ⟶ (pcorec(lbl,p.a[lbl;p]) i) ⟶ partial(ℕ))
Proof
Definitions occuring in Statement :
pcorec: pcorec(lbl,p.a[lbl; p]),
ptuple: ptuple(lbl,p.a[lbl; p];X),
list: T List,
partial: partial(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
member: t ∈ T,
apply: f a,
fix: fix(F),
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
union: left + right,
atom: Atom,
universe: Type
Definitions unfolded in proof :
pcorec: pcorec(lbl,p.a[lbl; p]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
so_apply: x[s]
Lemmas referenced :
fix_wf_corec-family-partial-nat,
ptuple_wf,
istype-atom,
ptuple-monotone,
ptuple-continuous,
partial_wf,
nat_wf,
list_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaEquality_alt,
because_Cache,
applyEquality,
Error :universeIsType,
hypothesis,
Error :functionIsType,
Error :inhabitedIsType,
independent_isectElimination,
independent_pairFormation,
Error :isect_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
Error :isectIsType,
axiomEquality,
Error :isectIsTypeImplies,
instantiate,
unionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[P:Type]. \mforall{}[a:Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)]. \mforall{}[f:\mcap{}X:P {}\mrightarrow{} Type
((i:P {}\mrightarrow{} (X i) {}\mrightarrow{} partial(\mBbbN{}))
{}\mrightarrow{} i:P
{}\mrightarrow{} (ptuple(lbl,p.a[lbl;p];X) i)
{}\mrightarrow{} partial(\mBbbN{}))].
(fix(f) \mmember{} i:P {}\mrightarrow{} (pcorec(lbl,p.a[lbl;p]) i) {}\mrightarrow{} partial(\mBbbN{}))
Date html generated:
2019_06_20-PM-02_04_09
Last ObjectModification:
2019_02_28-PM-02_06_36
Theory : tuples
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