Nuprl Lemma : def-cont-induction-lemma-ext

`∀[P:ℕ ⟶ ℙ]`
`  ((∀n:ℕ. (P[n] `` P[n + 1])) `` (∀x:ℤ List. ∀[n,m:ℕ].  P[n] `` P[m] supposing (x = [n, m) ∈ (ℤ List)) ∧ (n ≤ m)))`

Proof

Definitions occuring in Statement :  from-upto: `[n, m)` list: `T List` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` def-cont-induction-lemma list_induction sq_stable__and sq_stable__equal
Lemmas referenced :  def-cont-induction-lemma list_induction sq_stable__and sq_stable__equal
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution equalityTransitivity equalitySymmetry

Latex:
\mforall{}[P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}]
((\mforall{}n:\mBbbN{}.  (P[n]  {}\mRightarrow{}  P[n  +  1]))
{}\mRightarrow{}  (\mforall{}x:\mBbbZ{}  List.  \mforall{}[n,m:\mBbbN{}].    P[n]  {}\mRightarrow{}  P[m]  supposing  (x  =  [n,  m))  \mwedge{}  (n  \mleq{}  m)))

Date html generated: 2018_05_21-PM-07_00_03
Last ObjectModification: 2018_05_19-PM-04_42_26

Theory : general

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