### Nuprl Lemma : rec-nat-induction

`∀[P:ℕ ⟶ ℙ]. (∀[n:ℕ]. (P[n] `` P[n + 1])) `` (∀[n:ℕ]. P[n]) supposing Top ⊆r P[0]`

Proof

Definitions occuring in Statement :  nat: `ℕ` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` top: `Top` prop: `ℙ` so_apply: `x[s]` implies: `P `` Q` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` member: `t ∈ T` subtype_rel: `A ⊆r B` implies: `P `` Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` not: `¬A` rev_implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` sq_stable: `SqStable(P)` squash: `↓T` subtract: `n - m` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` ge: `i ≥ j ` guard: `{T}`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut introduction sqequalRule axiomEquality hypothesis thin rename lambdaFormation lemma_by_obid sqequalHypSubstitution isectElimination lambdaEquality functionEquality applyEquality because_Cache hypothesisEquality dependent_set_memberEquality addEquality setElimination natural_numberEquality dependent_functionElimination unionElimination independent_pairFormation voidElimination productElimination independent_functionElimination independent_isectElimination imageMemberEquality baseClosed imageElimination isect_memberEquality voidEquality intEquality minusEquality cumulativity universeEquality intWeakElimination equalityTransitivity equalitySymmetry

Latex:
\mforall{}[P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  (\mforall{}[n:\mBbbN{}].  (P[n]  {}\mRightarrow{}  P[n  +  1]))  {}\mRightarrow{}  (\mforall{}[n:\mBbbN{}].  P[n])  supposing  Top  \msubseteq{}r  P[0]

Date html generated: 2016_05_13-PM-04_02_53
Last ObjectModification: 2016_01_14-PM-07_24_38

Theory : int_1

Home Index