### Nuprl Lemma : apply-nth_wf

`∀[T:Type]. ∀[n:ℕ]. ∀[A:ℕn + 1 ⟶ Type]. ∀[f:funtype(n + 1;A;T)]. ∀[x:A[n]].  (apply-nth(n; f; x) ∈ funtype(n;A;T))`

Proof

Definitions occuring in Statement :  apply-nth: `apply-nth(n; f; x)` funtype: `funtype(n;A;T)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` apply-nth: `apply-nth(n; f; x)` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` so_apply: `x[s]` nat: `ℕ` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` le: `A ≤ B` ge: `i ≥ j ` funtype: `funtype(n;A;T)` eq_int: `(i =z j)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` subtract: `n - m` sq_type: `SQType(T)` guard: `{T}` true: `True` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than': `less_than'(a;b)` less_than: `a < b` squash: `↓T` subtype_rel: `A ⊆r B` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` so_lambda: `λ2x.t[x]`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin functionEquality extract_by_obid sqequalHypSubstitution isectElimination because_Cache hypothesis lambdaFormation applyEquality functionExtensionality hypothesisEquality equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination sqequalRule axiomEquality natural_numberEquality addEquality setElimination rename dependent_set_memberEquality independent_pairFormation unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll productElimination cumulativity universeEquality intWeakElimination equalityElimination baseClosed instantiate promote_hyp impliesFunctionality imageMemberEquality baseApply closedConclusion multiplyEquality hyp_replacement imageElimination minusEquality

Latex:
\mforall{}[T:Type].  \mforall{}[n:\mBbbN{}].  \mforall{}[A:\mBbbN{}n  +  1  {}\mrightarrow{}  Type].  \mforall{}[f:funtype(n  +  1;A;T)].  \mforall{}[x:A[n]].
(apply-nth(n;  f;  x)  \mmember{}  funtype(n;A;T))

Date html generated: 2018_05_21-PM-08_02_22
Last ObjectModification: 2017_07_26-PM-05_38_55

Theory : general

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