Nuprl Lemma : term-induction1-ext
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ].
((∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . P[varterm(v)])
⇒ (∀f:opr. ∀bts:bound-term(opr) List. ((∀i:ℕ||bts||. P[snd(bts[i])]) ⇒ P[mkterm(f;bts)]))
⇒ {∀t:term(opr). P[t]})
Proof
Definitions occuring in Statement :
bound-term: bound-term(opr),
mkterm: mkterm(opr;bts),
varterm: varterm(v),
term: term(opr),
nullvar: nullvar(),
varname: varname(),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
pi2: snd(t),
genrec-ap: genrec-ap,
so_apply: x[s1;s2;s3],
so_apply: x[s],
term-ind: term-ind,
term-induction1,
uniform-comp-nat-induction,
sq_stable__le,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
term-induction1,
lifting-strict-decide,
strict4-apply,
lifting-strict-spread,
uniform-comp-nat-induction,
sq_stable__le
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
Error :memTop,
independent_isectElimination
Latex:
\mforall{}[opr:Type]. \mforall{}[P:term(opr) {}\mrightarrow{} \mBbbP{}].
((\mforall{}v:\{v:varname()| \mneg{}(v = nullvar())\} . P[varterm(v)])
{}\mRightarrow{} (\mforall{}f:opr. \mforall{}bts:bound-term(opr) List. ((\mforall{}i:\mBbbN{}||bts||. P[snd(bts[i])]) {}\mRightarrow{} P[mkterm(f;bts)]))
{}\mRightarrow{} \{\mforall{}t:term(opr). P[t]\})
Date html generated:
2020_05_19-PM-09_54_26
Last ObjectModification:
2020_03_11-PM-09_20_28
Theory : terms
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