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
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