Nuprl Lemma : term-accum-induction-ext
∀[opr,P:Type]. ∀[R:P ⟶ term(opr) ⟶ ℙ].
∀Q:P ⟶ opr ⟶ (varname() List) ⟶ ((t:term(opr) × p:P × R[p;t]) List) ⟶ P
((∀p:P. ∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . R[p;varterm(v)])
⇒ (∀p:P. ∀f:opr. ∀bts:bound-term(opr) List. ∀L:{L:(t:term(opr) × p:P × R[p;t]) List|
(||L|| = ||bts|| ∈ ℤ)
∧ (∀i:ℕ||L||. ((fst(L[i])) = (snd(bts[i])) ∈ term(opr)))
∧ (∀i:ℕ||L||
((fst(snd(L[i]))) = Q[p;f;fst(bts[i]);firstn(i;L)] ∈ P))} .
R[p;mkterm(f;bts)])
⇒ {∀t:term(opr). ∀p:P. R[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(),
firstn: firstn(n;as),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
product: x:A × B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
pi1: fst(t),
so_apply: x[s1;s2;s3;s4],
pi2: snd(t),
nil: [],
it: ⋅,
cons: [a / b],
let: let,
genrec-ap: genrec-ap,
term-accum1: term-accum1,
term-accum-induction,
uniform-comp-nat-induction,
sq_stable__le,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda4,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
term-accum-induction,
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,P:Type]. \mforall{}[R:P {}\mrightarrow{} term(opr) {}\mrightarrow{} \mBbbP{}].
\mforall{}Q:P {}\mrightarrow{} opr {}\mrightarrow{} (varname() List) {}\mrightarrow{} ((t:term(opr) \mtimes{} p:P \mtimes{} R[p;t]) List) {}\mrightarrow{} P
((\mforall{}p:P. \mforall{}v:\{v:varname()| \mneg{}(v = nullvar())\} . R[p;varterm(v)])
{}\mRightarrow{} (\mforall{}p:P. \mforall{}f:opr. \mforall{}bts:bound-term(opr) List. \mforall{}L:\{L:(t:term(opr) \mtimes{} p:P \mtimes{} R[p;t]) List|
(||L|| = ||bts||)
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((fst(L[i])) = (snd(bts[i]))))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
((fst(snd(L[i])))
= Q[p;f;fst(bts[i]);firstn(i;L)]))\} .
R[p;mkterm(f;bts)])
{}\mRightarrow{} \{\mforall{}t:term(opr). \mforall{}p:P. R[p;t]\})
Date html generated:
2020_05_19-PM-09_55_07
Last ObjectModification:
2020_03_11-PM-09_18_42
Theory : terms
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