### Nuprl Lemma : fix_wf_corec-partial1

`∀[A:Type]`
`  (∀[F:Type ⟶ Type]`
`     ∀[f:⋂T:Type. ((T ⟶ partial(A)) ⟶ F[T] ⟶ partial(A))]. (fix(f) ∈ corec(T.F[T]) ⟶ partial(A)) `
`     supposing ContinuousMonotone(T.F[T])) supposing `
`     (mono(A) and `
`     value-type(A))`

Proof

Definitions occuring in Statement :  corec: `corec(T.F[T])` partial: `partial(T)` mono: `mono(T)` continuous-monotone: `ContinuousMonotone(T.F[T])` value-type: `value-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` member: `t ∈ T` fix: `fix(F)` isect: `⋂x:A. B[x]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` implies: `P `` Q` continuous-monotone: `ContinuousMonotone(T.F[T])` and: `P ∧ Q` prop: `ℙ`
Lemmas referenced :  fix-corec-partial1 corec_wf partial_wf subtype_rel_dep_function corec_subtype subtype_rel_self equal_wf continuous-monotone_wf mono_wf value-type_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis sqequalRule lambdaEquality applyEquality functionExtensionality universeEquality cumulativity equalityTransitivity equalitySymmetry functionEquality lambdaFormation productElimination because_Cache dependent_functionElimination independent_functionElimination isectEquality axiomEquality isect_memberEquality

Latex:
\mforall{}[A:Type]
(\mforall{}[F:Type  {}\mrightarrow{}  Type]
\mforall{}[f:\mcap{}T:Type.  ((T  {}\mrightarrow{}  partial(A))  {}\mrightarrow{}  F[T]  {}\mrightarrow{}  partial(A))]
(fix(f)  \mmember{}  corec(T.F[T])  {}\mrightarrow{}  partial(A))
supposing  ContinuousMonotone(T.F[T]))  supposing
(mono(A)  and
value-type(A))

Date html generated: 2017_04_14-AM-07_48_35
Last ObjectModification: 2017_02_27-PM-03_18_06

Theory : co-recursion

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