Nuprl Lemma : hereditarily_functionality_wrt_subterm
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ].  ∀t,s:term(opr).  (s << t 
⇒ hereditarily(opr;s.P[s];t) 
⇒ hereditarily(opr;s.P[s];s))
Proof
Definitions occuring in Statement : 
hereditarily: hereditarily(opr;s.P[s];t)
, 
subterm: s << t
, 
term: term(opr)
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
function: x:A ⟶ B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
hereditarily: hereditarily(opr;s.P[s];t)
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
member: t ∈ T
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
guard: {T}
Lemmas referenced : 
subterm_transitivity, 
subterm_wf, 
hereditarily_wf, 
term_wf, 
istype-universe
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
sqequalHypSubstitution, 
productElimination, 
thin, 
cut, 
independent_pairFormation, 
hypothesis, 
dependent_functionElimination, 
hypothesisEquality, 
independent_functionElimination, 
introduction, 
extract_by_obid, 
isectElimination, 
universeIsType, 
because_Cache, 
sqequalRule, 
lambdaEquality_alt, 
applyEquality, 
inhabitedIsType, 
functionIsType, 
universeEquality, 
instantiate
Latex:
\mforall{}[opr:Type].  \mforall{}[P:term(opr)  {}\mrightarrow{}  \mBbbP{}].
    \mforall{}t,s:term(opr).    (s  <<  t  {}\mRightarrow{}  hereditarily(opr;s.P[s];t)  {}\mRightarrow{}  hereditarily(opr;s.P[s];s))
Date html generated:
2020_05_19-PM-09_54_34
Last ObjectModification:
2020_03_10-PM-01_24_28
Theory : terms
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