### Nuprl Lemma : primrec-wf-upper

`∀[k:ℤ]. ∀[P:{k...} ⟶ ℙ]. ∀[b:P[k]]. ∀[s:∀n:{k...}. (P[n] `` P[n + 1])]. ∀[n:{k...}].`
`  (primrec(n - k;b;λi,x. (s (i + k) x)) ∈ P[n])`

Proof

Definitions occuring in Statement :  primrec: `primrec(n;b;c)` int_upper: `{i...}` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  nat: `ℕ` ge: `i ≥ j ` cand: `A c∧ B` less_than: `a < b` guard: `{T}` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` sq_type: `SQType(T)` bfalse: `ff` exists: `∃x:A. B[x]` bnot: `¬bb` assert: `↑b` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` implies: `P `` Q` so_apply: `x[s]` subtype_rel: `A ⊆r B` int_upper: `{i...}` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` not: `¬A` rev_implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` uimplies: `b supposing a` sq_stable: `SqStable(P)` squash: `↓T` subtract: `n - m` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` prop: `ℙ`
Rules used in proof :  intWeakElimination Error :lambdaEquality_alt,  Error :functionIsTypeImplies,  equalityElimination instantiate cumulativity intEquality Error :dependent_pairFormation_alt,  Error :equalityIstype,  promote_hyp functionExtensionality baseApply closedConclusion multiplyEquality sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule sqequalHypSubstitution axiomEquality equalityTransitivity hypothesis equalitySymmetry because_Cache Error :isect_memberEquality_alt,  isectElimination thin hypothesisEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsType,  Error :universeIsType,  applyEquality Error :dependent_set_memberEquality_alt,  addEquality setElimination rename natural_numberEquality extract_by_obid dependent_functionElimination unionElimination independent_pairFormation Error :lambdaFormation_alt,  voidElimination productElimination independent_functionElimination independent_isectElimination imageMemberEquality baseClosed imageElimination minusEquality universeEquality

Latex:
\mforall{}[k:\mBbbZ{}].  \mforall{}[P:\{k...\}  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[b:P[k]].  \mforall{}[s:\mforall{}n:\{k...\}.  (P[n]  {}\mRightarrow{}  P[n  +  1])].  \mforall{}[n:\{k...\}].
(primrec(n  -  k;b;\mlambda{}i,x.  (s  (i  +  k)  x))  \mmember{}  P[n])

Date html generated: 2019_06_20-PM-01_04_41
Last ObjectModification: 2019_06_20-PM-01_01_23

Theory : call!by!value_2

Home Index