Nuprl Lemma : hereditarily-varterm
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ]. ∀[v:{v:varname()| ¬(v = nullvar() ∈ varname())} ].
(hereditarily(opr;s.P[s];varterm(v)) ⇐⇒ P[varterm(v)])
Proof
Definitions occuring in Statement :
hereditarily: hereditarily(opr;s.P[s];t),
varterm: varterm(v),
term: term(opr),
nullvar: nullvar(),
varname: varname(),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
not: ¬A,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
false: False,
prop: ℙ,
rev_implies: P ⇐ Q,
hereditarily: hereditarily(opr;s.P[s];t),
all: ∀x:A. B[x]
Lemmas referenced :
hereditarily_wf,
term_wf,
varterm_wf,
nullvar_wf,
subterm_wf,
varname_wf,
istype-void,
istype-universe,
subterm-varterm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
independent_isectElimination,
independent_functionElimination,
voidElimination,
equalityIstype,
inhabitedIsType,
setIsType,
functionIsType,
universeEquality,
instantiate,
productElimination
Latex:
\mforall{}[opr:Type]. \mforall{}[P:term(opr) {}\mrightarrow{} \mBbbP{}]. \mforall{}[v:\{v:varname()| \mneg{}(v = nullvar())\} ].
(hereditarily(opr;s.P[s];varterm(v)) \mLeftarrow{}{}\mRightarrow{} P[varterm(v)])
Date html generated:
2020_05_19-PM-09_54_35
Last ObjectModification:
2020_03_12-AM-11_08_47
Theory : terms
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