### 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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