### Nuprl Lemma : apply_larger_list

`∀[B:Type]. ∀[n:ℕ]. ∀[m:ℕn + 1]. ∀[q:ℕm + 1]. ∀[A:ℕn ⟶ Type]. ∀[lst:k:{q..n-} ⟶ (A k)]. ∀[r:ℕm]. ∀[a:A r].`
`∀[f:funtype(n - m;λx.(A (x + m));B)].`
`  ((apply_gen(n;λx.if (x =z r) then a else lst x fi ) m f) = (apply_gen(n;lst) m f) ∈ B)`

Proof

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

Latex:
\mforall{}[B:Type].  \mforall{}[n:\mBbbN{}].  \mforall{}[m:\mBbbN{}n  +  1].  \mforall{}[q:\mBbbN{}m  +  1].  \mforall{}[A:\mBbbN{}n  {}\mrightarrow{}  Type].  \mforall{}[lst:k:\{q..n\msupminus{}\}  {}\mrightarrow{}  (A  k)].  \mforall{}[r:\mBbbN{}m].
\mforall{}[a:A  r].  \mforall{}[f:funtype(n  -  m;\mlambda{}x.(A  (x  +  m));B)].
((apply\_gen(n;\mlambda{}x.if  (x  =\msubz{}  r)  then  a  else  lst  x  fi  )  m  f)  =  (apply\_gen(n;lst)  m  f))

Date html generated: 2017_10_01-AM-09_03_48
Last ObjectModification: 2017_07_26-PM-04_44_35

Theory : bags

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