### Nuprl Lemma : natrec_wf

`∀[T:ℕ ⟶ Type]. ∀[g:n:ℕ ⟶ (m:ℕn ⟶ T[m]) ⟶ T[n]].  (letrec f(n)=g[n;f] in f ∈ n:ℕ ⟶ T[n])`

Proof

Definitions occuring in Statement :  natrec: natrec int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  natrec: natrec uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` so_apply: `x[s]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` all: `∀x:A. B[x]` ge: `i ≥ j ` guard: `{T}` genrec: genrec so_apply: `x[s1;s2]` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` top: `Top` true: `True` so_lambda: `λ2x.t[x]` sq_stable: `SqStable(P)` squash: `↓T`
Lemmas referenced :  subtype_rel_self sq_stable__le not-le-2 int_seg_subtype subtype_rel_dep_function le-add-cancel add-zero add_functionality_wrt_le add-commutes add-swap add-associates minus-minus minus-add minus-one-mul-top zero-add minus-one-mul condition-implies-le less-iff-le not-ge-2 subtract_wf decidable__le less_than_wf ge_wf less_than_irreflexivity less_than_transitivity1 nat_properties false_wf int_seg_subtype_nat int_seg_wf nat_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut sqequalHypSubstitution hypothesis axiomEquality equalityTransitivity equalitySymmetry functionEquality lemma_by_obid isectElimination thin natural_numberEquality setElimination rename hypothesisEquality applyEquality independent_isectElimination independent_pairFormation lambdaFormation isect_memberEquality because_Cache cumulativity universeEquality intWeakElimination independent_functionElimination voidElimination lambdaEquality dependent_functionElimination productElimination unionElimination addEquality voidEquality intEquality minusEquality imageMemberEquality baseClosed imageElimination

Latex:
\mforall{}[T:\mBbbN{}  {}\mrightarrow{}  Type].  \mforall{}[g:n:\mBbbN{}  {}\mrightarrow{}  (m:\mBbbN{}n  {}\mrightarrow{}  T[m])  {}\mrightarrow{}  T[n]].    (letrec  f(n)=g[n;f]  in  f  \mmember{}  n:\mBbbN{}  {}\mrightarrow{}  T[n])

Date html generated: 2016_05_13-PM-04_03_11
Last ObjectModification: 2016_01_14-PM-07_24_29

Theory : int_1

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