Nuprl Lemma : wf-term-hereditarily-correct-sort-arity
∀[opr:Type]
  ∀sort:term(opr) ⟶ ℕ. ∀arity:opr ⟶ ((ℕ × ℕ) List). ∀t:term(opr).
    (↑wf-term(arity;sort;t) 
⇐⇒ hereditarily(opr;s.correct-sort-arity(sort;arity;s);t))
Proof
Definitions occuring in Statement : 
wf-term: wf-term(arity;sort;t)
, 
correct-sort-arity: correct-sort-arity(sort;arity;t)
, 
hereditarily: hereditarily(opr;s.P[s];t)
, 
term: term(opr)
, 
list: T List
, 
nat: ℕ
, 
assert: ↑b
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
function: x:A ⟶ B[x]
, 
product: x:A × B[x]
, 
universe: Type
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
implies: P 
⇒ Q
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
correct-sort-arity: correct-sort-arity(sort;arity;t)
, 
varterm: varterm(v)
, 
isvarterm: isvarterm(t)
, 
isl: isl(x)
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
not: ¬A
, 
true: True
, 
false: False
, 
uimplies: b supposing a
, 
wf-term: wf-term(arity;sort;t)
, 
prop: ℙ
, 
cand: A c∧ B
, 
bound-term: bound-term(opr)
, 
uiff: uiff(P;Q)
, 
l_member: (x ∈ l)
, 
exists: ∃x:A. B[x]
, 
pi2: snd(t)
, 
squash: ↓T
, 
label: ...$L... t
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
rev_uimplies: rev_uimplies(P;Q)
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
lelt: i ≤ j < k
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
mkterm: mkterm(opr;bts)
, 
term-opr: term-opr(t)
, 
let: let, 
term-bts: term-bts(t)
, 
outr: outr(x)
, 
bfalse: ff
, 
pi1: fst(t)
, 
hereditarily: hereditarily(opr;s.P[s];t)
, 
le: A ≤ B
Lemmas referenced : 
list_wf, 
nat_wf, 
term_wf, 
istype-nat, 
istype-universe, 
term-induction, 
iff_wf, 
assert_wf, 
wf-term_wf, 
hereditarily_wf, 
correct-sort-arity_wf, 
hereditarily-varterm, 
istype-true, 
istype-void, 
istype-assert, 
varterm_wf, 
nullvar_wf, 
assert_witness, 
varname_wf, 
hereditarily-mkterm, 
assert-wf-mkterm, 
assert_functionality_wrt_uiff, 
squash_wf, 
true_wf, 
equal_wf, 
subtype_rel_self, 
iff_weakening_equal, 
l_member_wf, 
bound-term_wf, 
mkterm_wf, 
nat_properties, 
decidable__le, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
istype-int, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
istype-le, 
istype-less_than, 
int_seg_wf, 
length_wf, 
select_wf, 
int_seg_properties, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
select_member, 
trivial-subterm, 
hereditarily_functionality_wrt_subterm
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaFormation_alt, 
functionIsType, 
universeIsType, 
hypothesisEquality, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
productEquality, 
hypothesis, 
instantiate, 
universeEquality, 
sqequalRule, 
lambdaEquality_alt, 
inhabitedIsType, 
independent_functionElimination, 
independent_pairFormation, 
because_Cache, 
productElimination, 
natural_numberEquality, 
voidElimination, 
setElimination, 
rename, 
independent_isectElimination, 
equalityIstype, 
setIsType, 
dependent_functionElimination, 
equalityTransitivity, 
equalitySymmetry, 
applyEquality, 
imageElimination, 
dependent_set_memberEquality_alt, 
productIsType, 
applyLambdaEquality, 
imageMemberEquality, 
baseClosed, 
unionElimination, 
approximateComputation, 
dependent_pairFormation_alt, 
int_eqEquality, 
Error :memTop, 
independent_pairEquality, 
axiomEquality, 
functionIsTypeImplies
Latex:
\mforall{}[opr:Type]
    \mforall{}sort:term(opr)  {}\mrightarrow{}  \mBbbN{}.  \mforall{}arity:opr  {}\mrightarrow{}  ((\mBbbN{}  \mtimes{}  \mBbbN{})  List).  \mforall{}t:term(opr).
        (\muparrow{}wf-term(arity;sort;t)  \mLeftarrow{}{}\mRightarrow{}  hereditarily(opr;s.correct-sort-arity(sort;arity;s);t))
Date html generated:
2020_05_19-PM-09_58_25
Last ObjectModification:
2020_03_11-PM-04_29_04
Theory : terms
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