Nuprl Lemma : comp_nat_ind_tp

[P:ℕ ⟶ ℙ{k}]. ((∀i:ℕ((∀j:ℕP[j] supposing j < i)  P[i]))  {∀i:ℕP[i]})

This theorem is one of freek's list of 100 theorems


Definitions occuring in Statement :  nat: less_than: a < b uimplies: supposing a uall: [x:A]. B[x] prop: guard: {T} so_apply: x[s] all: x:A. B[x] implies:  Q function: x:A ⟶ B[x]
Definitions unfolded in proof :  uall: [x:A]. B[x] implies:  Q guard: {T} all: x:A. B[x] member: t ∈ T uimplies: supposing a nat: prop: so_apply: x[s] subtype_rel: A ⊆B so_lambda: λ2x.t[x] false: False decidable: Dec(P) or: P ∨ Q iff: ⇐⇒ Q and: P ∧ Q not: ¬A rev_implies:  Q uiff: uiff(P;Q) subtract: m top: Top le: A ≤ B less_than': less_than'(a;b) true: True sq_stable: SqStable(P) squash: T
Lemmas referenced :  nat_wf less_than_wf subtype_rel_self subtract_wf istype-int primrec-wf2 all_wf isect_wf member-less_than less_than_transitivity1 less_than_irreflexivity decidable__lt istype-false 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 decidable__le not-le-2 sq_stable__le zero-add add-zero le_wf add-mul-special zero-mul
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  sqequalRule Error :functionIsType,  Error :universeIsType,  cut introduction extract_by_obid hypothesis Error :inhabitedIsType,  hypothesisEquality Error :isectIsType,  sqequalHypSubstitution isectElimination thin setElimination rename applyEquality instantiate universeEquality because_Cache natural_numberEquality Error :setIsType,  Error :lambdaEquality_alt,  cumulativity equalityTransitivity equalitySymmetry independent_isectElimination independent_functionElimination voidElimination dependent_functionElimination unionElimination independent_pairFormation productElimination addEquality minusEquality Error :isect_memberEquality_alt,  Error :dependent_set_memberEquality_alt,  imageMemberEquality baseClosed imageElimination multiplyEquality

\mforall{}[P:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}\{k\}].  ((\mforall{}i:\mBbbN{}.  ((\mforall{}j:\mBbbN{}.  P[j]  supposing  j  <  i)  {}\mRightarrow{}  P[i]))  {}\mRightarrow{}  \{\mforall{}i:\mBbbN{}.  P[i]\})

Date html generated: 2019_06_20-AM-11_27_58
Last ObjectModification: 2018_10_06-AM-10_12_47

Theory : call!by!value_2

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