### Nuprl Lemma : fun_exp_compose

`∀[T:Type]. ∀[n:ℕ]. ∀[h,f:T ⟶ T].  ((f^n o h) = primrec(n;h;λi,g. (f o g)) ∈ (T ⟶ T))`

Proof

Definitions occuring in Statement :  fun_exp: `f^n` compose: `f o g` primrec: `primrec(n;b;c)` nat: `ℕ` uall: `∀[x:A]. B[x]` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` guard: `{T}` uimplies: `b supposing a` prop: `ℙ` fun_exp: `f^n` all: `∀x:A. B[x]` top: `Top` 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` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` compose: `f o g` sq_type: `SQType(T)` squash: `↓T`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination lambdaFormation natural_numberEquality independent_isectElimination independent_functionElimination voidElimination sqequalRule lambdaEquality dependent_functionElimination isect_memberEquality axiomEquality functionEquality cumulativity voidEquality unionElimination independent_pairFormation productElimination addEquality applyEquality intEquality minusEquality because_Cache universeEquality functionExtensionality instantiate equalityTransitivity equalitySymmetry dependent_set_memberEquality hyp_replacement applyLambdaEquality imageElimination imageMemberEquality baseClosed

Latex:
\mforall{}[T:Type].  \mforall{}[n:\mBbbN{}].  \mforall{}[h,f:T  {}\mrightarrow{}  T].    ((f\^{}n  o  h)  =  primrec(n;h;\mlambda{}i,g.  (f  o  g)))

Date html generated: 2017_04_14-AM-07_34_37
Last ObjectModification: 2017_02_27-PM-03_07_44

Theory : fun_1

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