### Nuprl Lemma : comp_nat_ind_tp

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

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

Proof

Definitions occuring in Statement :  nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` guard: `{T}` all: `∀x:A. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` prop: `ℙ` so_apply: `x[s]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` false: `False` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` not: `¬A` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` sq_stable: `SqStable(P)` squash: `↓T`
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

Latex:
\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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