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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