### Nuprl Lemma : natrec_wf_intseg

`∀[k:ℤ]. ∀[T:{k...} ⟶ Type]. ∀[g:n:{k...} ⟶ (m:{k..n-} ⟶ T[m]) ⟶ T[n]].  (letrec f(n)=g[n;f] in f ∈ n:{k...} ⟶ T[n])`

Proof

Definitions occuring in Statement :  natrec: natrec int_upper: `{i...}` int_seg: `{i..j-}` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]` int: `ℤ` universe: `Type`
Definitions unfolded in proof :  natrec: natrec uall: `∀[x:A]. B[x]` member: `t ∈ T` int_upper: `{i...}` so_apply: `x[s]` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` lelt: `i ≤ j < k` and: `P ∧ Q` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` nat: `ℕ` false: `False` ge: `i ≥ j ` guard: `{T}` genrec: genrec so_apply: `x[s1;s2]` sq_stable: `SqStable(P)` uiff: `uiff(P;Q)` subtract: `n - m` squash: `↓T` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` top: `Top` sq_type: `SQType(T)` nat_plus: `ℕ+` less_than: `a < b`
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut sqequalHypSubstitution hypothesis axiomEquality equalityTransitivity equalitySymmetry functionEquality lemma_by_obid isectElimination thin hypothesisEquality because_Cache setElimination rename applyEquality intEquality lambdaEquality independent_isectElimination setEquality lambdaFormation productElimination isect_memberEquality cumulativity universeEquality intWeakElimination natural_numberEquality independent_functionElimination voidElimination dependent_functionElimination addEquality minusEquality imageMemberEquality baseClosed imageElimination unionElimination independent_pairFormation voidEquality dependent_set_memberEquality instantiate multiplyEquality

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

Date html generated: 2016_05_13-PM-04_03_14
Last ObjectModification: 2016_01_14-PM-07_24_45

Theory : int_1

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