### Nuprl Lemma : corec-ext

`∀[F:Type ⟶ Type]. corec(T.F[T]) ≡ F[corec(T.F[T])] supposing ContinuousMonotone(T.F[T])`

Proof

Definitions occuring in Statement :  corec: `corec(T.F[T])` continuous-monotone: `ContinuousMonotone(T.F[T])` ext-eq: `A ≡ B` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` guard: `{T}` prop: `ℙ` subtype_rel: `A ⊆r B` top: `Top` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` not: `¬A` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` continuous-monotone: `ContinuousMonotone(T.F[T])` type-monotone: `Monotone(T.F[T])` nequal: `a ≠ b ∈ T ` squash: `↓T` ext-eq: `A ≡ B` so_lambda: `λ2x.t[x]` corec: `corec(T.F[T])` type-continuous: `Continuous(T.F[T])` sq_stable: `SqStable(P)`
Lemmas referenced :  nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf less_than_wf primrec0_lemma primrec1_lemma top_wf decidable__le subtract_wf false_wf not-ge-2 less-iff-le condition-implies-le minus-one-mul zero-add minus-one-mul-top minus-add minus-minus add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel primrec-unroll eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int le_antisymmetry_iff eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int add-subtract-cancel primrec_wf le_weakening2 le_wf int_seg_wf not-le-2 not-equal-2 subtract-add-cancel subtype_rel_self subtype_rel_transitivity subtype_rel_wf squash_wf true_wf le_weakening nat_wf continuous-monotone_wf sq_stable__le subtype_rel-equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination axiomEquality isect_memberEquality voidEquality applyEquality functionExtensionality universeEquality unionElimination independent_pairFormation productElimination addEquality intEquality minusEquality because_Cache equalityElimination equalityTransitivity equalitySymmetry dependent_pairFormation promote_hyp instantiate cumulativity dependent_set_memberEquality hyp_replacement imageElimination imageMemberEquality baseClosed independent_pairEquality functionEquality isectEquality

Latex:
\mforall{}[F:Type  {}\mrightarrow{}  Type].  corec(T.F[T])  \mequiv{}  F[corec(T.F[T])]  supposing  ContinuousMonotone(T.F[T])

Date html generated: 2017_04_14-AM-07_41_52
Last ObjectModification: 2017_02_27-PM-03_14_15

Theory : co-recursion

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