### Nuprl Lemma : weakly-safe-extension

`∀[R:ℕ ⟶ ℕ ⟶ ℙ]. ∀[n:ℕ]. ∀[s:ℕn ⟶ ℕ].  (weakly-safe-seq(R;n;s) `` (¬¬(∃p:ℕ. weakly-safe-seq(R;n + 1;s.p@n))))`

Proof

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

Latex:
\mforall{}[R:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[n:\mBbbN{}].  \mforall{}[s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}].
(weakly-safe-seq(R;n;s)  {}\mRightarrow{}  (\mneg{}\mneg{}(\mexists{}p:\mBbbN{}.  weakly-safe-seq(R;n  +  1;s.p@n))))

Date html generated: 2019_06_20-AM-11_29_19
Last ObjectModification: 2018_10_18-PM-03_54_59

Theory : bar-induction

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