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: uall: [x:A]. B[x] prop: so_apply: x[s] implies:  Q function: x:A ⟶ B[x] natural_number: $n
Definitions unfolded in proof :  uall: [x:A]. B[x] implies:  Q member: t ∈ T nat: so_apply: x[s] subtype_rel: A ⊆B int_seg: {i..j-} lelt: i ≤ j < k and: P ∧ Q le: A ≤ B uimplies: supposing a less_than': less_than'(a;b) false: False not: ¬A prop: all: x:A. B[x] ge: i ≥  cand: c∧ B less_than: a < b squash: T guard: {T} decidable: Dec(P) or: P ∨ Q iff: ⇐⇒ Q rev_implies:  Q uiff: uiff(P;Q) subtract: m top: Top true: True sq_stable: SqStable(P)
Lemmas referenced :  istype-nat int_seg_wf int_seg_subtype_nat istype-false subtype_rel_self nat_properties less_than_transitivity1 less_than_irreflexivity ge_wf istype-less_than subtract-1-ge-0 decidable__lt subtract_wf not-lt-2 condition-implies-le minus-add istype-void minus-minus minus-one-mul add-swap minus-one-mul-top add-commutes less-iff-le add_functionality_wrt_le add-associates le-add-cancel istype-le decidable__le not-le-2 sq_stable__le zero-add add-zero add-mul-special zero-mul
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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