### Nuprl Lemma : implies-strictly-increasing-seq

`∀[n:ℕ]. ∀[s:ℕn ⟶ ℤ].  ((∀i:ℕn - 1. s i < s (i + 1)) `` strictly-increasing-seq(n;s))`

Proof

Definitions occuring in Statement :  strictly-increasing-seq: `strictly-increasing-seq(n;s)` int_seg: `{i..j-}` nat: `ℕ` less_than: `a < b` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` 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` implies: `P `` Q` strictly-increasing-seq: `strictly-increasing-seq(n;s)` prop: `ℙ` nat: `ℕ` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` 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)` uimplies: `b supposing a` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` less_than': `less_than'(a;b)` true: `True` so_apply: `x[s]` guard: `{T}` ge: `i ≥ j ` less_than: `a < b` exists: `∃x:A. B[x]` nat_plus: `ℕ+` squash: `↓T` sq_type: `SQType(T)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename because_Cache hypothesis sqequalRule lambdaEquality applyEquality functionExtensionality hypothesisEquality dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination voidElimination independent_functionElimination independent_isectElimination addEquality minusEquality isect_memberEquality voidEquality intEquality functionEquality intWeakElimination dependent_pairFormation sqequalIntensionalEquality equalityTransitivity equalitySymmetry promote_hyp baseApply closedConclusion baseClosed addLevel multiplyEquality levelHypothesis imageMemberEquality setEquality instantiate cumulativity

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[s:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    ((\mforall{}i:\mBbbN{}n  -  1.  s  i  <  s  (i  +  1))  {}\mRightarrow{}  strictly-increasing-seq(n;s))

Date html generated: 2017_04_14-AM-07_26_19
Last ObjectModification: 2017_02_27-PM-02_56_14

Theory : bar-induction

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