### Nuprl Lemma : uniform-comp-nat-induction

`∀[P:ℕ ⟶ ℙ]. ((∀[n:ℕ]. ((∀[m:ℕn]. P[m]) `` P[n])) `` (∀[n:ℕ]. P[n]))`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` implies: `P `` Q` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` nat: `ℕ` so_apply: `x[s]` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` uimplies: `b supposing a` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` all: `∀x:A. B[x]` ge: `i ≥ j ` cand: `A c∧ B` less_than: `a < b` squash: `↓T` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` top: `Top` true: `True` sq_stable: `SqStable(P)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  sqequalRule Error :isectIsType,  cut introduction extract_by_obid hypothesis Error :functionIsType,  Error :universeIsType,  sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality applyEquality productElimination independent_isectElimination independent_pairFormation instantiate universeEquality because_Cache intWeakElimination imageElimination independent_functionElimination voidElimination Error :lambdaEquality_alt,  dependent_functionElimination axiomEquality Error :functionIsTypeImplies,  Error :inhabitedIsType,  Error :isect_memberEquality_alt,  Error :dependent_set_memberEquality_alt,  unionElimination addEquality minusEquality Error :productIsType,  equalityTransitivity equalitySymmetry Error :equalityIstype,  imageMemberEquality baseClosed multiplyEquality

Latex:
\mforall{}[P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}].  ((\mforall{}[n:\mBbbN{}].  ((\mforall{}[m:\mBbbN{}n].  P[m])  {}\mRightarrow{}  P[n]))  {}\mRightarrow{}  (\mforall{}[n:\mBbbN{}].  P[n]))

Date html generated: 2019_06_20-AM-11_33_40
Last ObjectModification: 2019_03_12-PM-05_31_41

Theory : int_1

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