### Nuprl Lemma : primrec-wf-nsub

`∀[b:ℕ+]. ∀[P:ℕb ⟶ ℙ]. ∀[init:P[0]]. ∀[s:∀n:ℕb - 1. (P[n] `` P[n + 1])]. ∀[n:ℕb].  (primrec(n;init;s) ∈ P[n])`

Proof

Definitions occuring in Statement :  primrec: `primrec(n;b;c)` int_seg: `{i..j-}` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat_plus: `ℕ+` uimplies: `b supposing a` int_seg: `{i..j-}` and: `P ∧ Q` less_than: `a < b` squash: `↓T` cand: `A c∧ B` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` implies: `P `` Q` prop: `ℙ` so_apply: `x[s]` le: `A ≤ B` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` subtract: `n - m` top: `Top` less_than': `less_than'(a;b)` true: `True` sq_stable: `SqStable(P)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache setElimination rename hypothesis independent_isectElimination independent_pairFormation imageElimination productElimination hypothesisEquality applyEquality closedConclusion natural_numberEquality sqequalRule Error :lambdaEquality_alt,  functionEquality Error :dependent_set_memberEquality_alt,  dependent_functionElimination unionElimination Error :lambdaFormation_alt,  voidElimination independent_functionElimination addEquality minusEquality Error :isect_memberEquality_alt,  Error :productIsType,  Error :universeIsType,  imageMemberEquality baseClosed axiomEquality equalityTransitivity equalitySymmetry Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsType,  universeEquality

Latex:
\mforall{}[b:\mBbbN{}\msupplus{}].  \mforall{}[P:\mBbbN{}b  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[init:P[0]].  \mforall{}[s:\mforall{}n:\mBbbN{}b  -  1.  (P[n]  {}\mRightarrow{}  P[n  +  1])].  \mforall{}[n:\mBbbN{}b].
(primrec(n;init;s)  \mmember{}  P[n])

Date html generated: 2019_06_20-AM-11_27_37
Last ObjectModification: 2019_01_28-PM-05_24_18

Theory : call!by!value_2

Home Index