### Nuprl Lemma : primrec-wf-int_seg

`∀[a,b:ℤ].`
`  ∀[P:{a..b-} ⟶ ℙ]. ∀[init:P[a]]. ∀[s:∀n:ℕb - 1 - a. (P[a + n] `` P[a + n + 1])]. ∀[n:{a..b-}].`
`    (primrec(n - a;init;s) ∈ P[n]) `
`  supposing a < b`

Proof

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

Latex:
\mforall{}[a,b:\mBbbZ{}].
\mforall{}[P:\{a..b\msupminus{}\}  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[init:P[a]].  \mforall{}[s:\mforall{}n:\mBbbN{}b  -  1  -  a.  (P[a  +  n]  {}\mRightarrow{}  P[a  +  n  +  1])].  \mforall{}[n:\{a..b\msupminus{}\}].
(primrec(n  -  a;init;s)  \mmember{}  P[n])
supposing  a  <  b

Date html generated: 2019_06_20-AM-11_27_36
Last ObjectModification: 2019_01_28-PM-05_23_48

Theory : call!by!value_2

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