### Nuprl Lemma : int_seg_decide_wf

`∀[i,j:ℤ]. ∀[F:{i..j-} ⟶ ℙ{u}]. ∀[d:∀k:{i..j-}. Dec(F[k])].  (int_seg_decide(d;i;j) ∈ Dec(∃k:{i..j-}. F[k]))`

Proof

Definitions occuring in Statement :  int_seg_decide: `int_seg_decide(d;i;j)` int_seg: `{i..j-}` decidable: `Dec(P)` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` member: `t ∈ T` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` so_apply: `x[s]` prop: `ℙ` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` guard: `{T}` uimplies: `b supposing a` and: `P ∧ Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` exists: `∃x:A. B[x]` le: `A ≤ B` subtract: `n - m` subtype_rel: `A ⊆r B` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` sq_stable: `SqStable(P)` cand: `A c∧ B` int_upper: `{i...}` int_seg_decide: `int_seg_decide(d;i;j)` nat_plus: `ℕ+`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalHypSubstitution hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry Error :functionIsType,  Error :universeIsType,  extract_by_obid isectElimination thin hypothesisEquality instantiate applyEquality Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  Error :inhabitedIsType,  universeEquality Error :lambdaFormation_alt,  setElimination rename intWeakElimination natural_numberEquality independent_isectElimination independent_functionElimination voidElimination Error :lambdaEquality_alt,  dependent_functionElimination Error :functionIsTypeImplies,  productElimination unionElimination equalityElimination because_Cache lessCases axiomSqEquality independent_pairFormation imageMemberEquality baseClosed imageElimination functionExtensionality Error :dependent_set_memberEquality_alt,  Error :productIsType,  minusEquality addEquality Error :inlEquality_alt,  Error :dependent_pairEquality_alt,  Error :equalityIsType1,  Error :dependent_pairFormation_alt,  Error :equalityIsType4,  baseApply closedConclusion promote_hyp cumulativity Error :inrEquality_alt,  multiplyEquality intEquality lambdaEquality lambdaFormation voidEquality isect_memberEquality dependent_set_memberEquality

Latex:
\mforall{}[i,j:\mBbbZ{}].  \mforall{}[F:\{i..j\msupminus{}\}  {}\mrightarrow{}  \mBbbP{}\{u\}].  \mforall{}[d:\mforall{}k:\{i..j\msupminus{}\}.  Dec(F[k])].
(int\_seg\_decide(d;i;j)  \mmember{}  Dec(\mexists{}k:\{i..j\msupminus{}\}.  F[k]))

Date html generated: 2019_06_20-AM-11_28_08
Last ObjectModification: 2018_10_27-PM-05_54_54

Theory : call!by!value_2

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