### Nuprl Lemma : prec-size-induction-ext

`∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[Q:i:P ⟶ prec(lbl,p.a[lbl;p];i) ⟶ TYPE].`
`  ((∀i:P. ∀x:prec(lbl,p.a[lbl;p];i).  ((∀j:P. ∀z:{z:prec(lbl,p.a[lbl;p];j)| ||j;z|| < ||i;x||} .  Q[j;z]) `` Q[i;x]))`
`  `` (∀i:P. ∀x:prec(lbl,p.a[lbl;p];i).  Q[i;x]))`

Proof

Definitions occuring in Statement :  prec-size: `||i;x||` prec: `prec(lbl,p.a[lbl; p];i)` list: `T List` less_than: `a < b` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` union: `left + right` atom: `Atom` universe: `Type`
Definitions unfolded in proof :  member: `t ∈ T` genrec-ap: genrec-ap prec-size-induction
Lemmas referenced :  prec-size-induction
Rules used in proof :  introduction sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut instantiate extract_by_obid hypothesis sqequalRule thin sqequalHypSubstitution equalityTransitivity equalitySymmetry

Latex:
\mforall{}[P:Type].  \mforall{}[a:Atom  {}\mrightarrow{}  P  {}\mrightarrow{}  ((P  +  P  +  Type)  List)].  \mforall{}[Q:i:P  {}\mrightarrow{}  prec(lbl,p.a[lbl;p];i)  {}\mrightarrow{}  TYPE].
((\mforall{}i:P.  \mforall{}x:prec(lbl,p.a[lbl;p];i).
((\mforall{}j:P.  \mforall{}z:\{z:prec(lbl,p.a[lbl;p];j)|  ||j;z||  <  ||i;x||\}  .    Q[j;z])  {}\mRightarrow{}  Q[i;x]))
{}\mRightarrow{}  (\mforall{}i:P.  \mforall{}x:prec(lbl,p.a[lbl;p];i).    Q[i;x]))

Date html generated: 2019_06_20-PM-02_05_03
Last ObjectModification: 2019_03_25-PM-05_19_00

Theory : tuples

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