Nuprl Lemma : very-dep-fun-eta
∀[A,B:Type]. ∀[C:A ⟶ B ⟶ Type]. ∀[f:very-dep-fun(A;B;a,b.C[a;b])].  (f = (λL,b. (f L b)) ∈ very-dep-fun(A;B;a,b.C[a;b]\000C))
Proof
Definitions occuring in Statement : 
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b])
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s1;s2]
, 
apply: f a
, 
lambda: λx.A[x]
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
very-dep-fun: very-dep-fun(A;B;a,b.C[a; b])
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
nat: ℕ
, 
vdf: vdf(A;B;a,b.C[a; b];n)
, 
implies: P 
⇒ Q
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
false: False
, 
not: ¬A
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
prop: ℙ
, 
so_apply: x[s1;s2]
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
so_lambda: λ2x y.t[x; y]
, 
ge: i ≥ j 
, 
lt_int: i <z j
, 
subtract: n - m
Lemmas referenced : 
decidable__le, 
istype-le, 
lt_int_wf, 
eqtt_to_assert, 
assert_of_lt_int, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
assert-bnot, 
iff_weakening_uiff, 
assert_wf, 
less_than_wf, 
istype-less_than, 
full-omega-unsat, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_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_less_lemma, 
int_formula_prop_wf, 
eta_conv, 
list_wf, 
length_wf_nat, 
set_subtype_base, 
le_wf, 
int_subtype_base, 
very-dep-fun_wf, 
istype-universe, 
nat_properties, 
ge_wf, 
subtract-1-ge-0, 
istype-nat, 
equal-wf-base, 
le_int_wf, 
bnot_wf, 
uiff_transitivity, 
assert_functionality_wrt_uiff, 
bnot_of_lt_int, 
assert_of_le_int, 
not_wf, 
istype-assert, 
istype-void, 
bool_cases, 
iff_transitivity, 
assert_of_bnot, 
length_wf, 
vdf-eq_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
sqequalHypSubstitution, 
isect_memberEquality_alt, 
extract_by_obid, 
dependent_functionElimination, 
thin, 
natural_numberEquality, 
hypothesisEquality, 
hypothesis, 
unionElimination, 
dependent_set_memberEquality_alt, 
isectElimination, 
because_Cache, 
sqequalRule, 
inhabitedIsType, 
lambdaFormation_alt, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
independent_isectElimination, 
dependent_pairFormation_alt, 
equalityIstype, 
promote_hyp, 
instantiate, 
cumulativity, 
independent_functionElimination, 
voidElimination, 
approximateComputation, 
lambdaEquality_alt, 
int_eqEquality, 
Error :memTop, 
independent_pairFormation, 
universeIsType, 
functionExtensionality_alt, 
applyEquality, 
setIsType, 
productEquality, 
intEquality, 
baseClosed, 
sqequalBase, 
axiomEquality, 
isectIsTypeImplies, 
functionIsType, 
universeEquality, 
setElimination, 
rename, 
intWeakElimination, 
functionIsTypeImplies, 
baseApply, 
closedConclusion, 
dependentIntersection_memberEquality, 
dependentIntersectionEqElimination, 
functionExtensionality, 
setEquality, 
applyLambdaEquality
Latex:
\mforall{}[A,B:Type].  \mforall{}[C:A  {}\mrightarrow{}  B  {}\mrightarrow{}  Type].  \mforall{}[f:very-dep-fun(A;B;a,b.C[a;b])].    (f  =  (\mlambda{}L,b.  (f  L  b)))
Date html generated:
2020_05_19-PM-09_40_33
Last ObjectModification:
2020_03_10-PM-00_32_13
Theory : co-recursion-2
Home
Index