Nuprl Lemma : mutual-corec-ext2
∀[k:ℕ]. ∀[F:(ℕk ⟶ Type) ⟶ ℕk ⟶ Type].
(mutual-corec(T.F[T]) ≡ F[mutual-corec(T.F[T])]) supposing
((∀i,j:ℕk. ∀Z:ℕk ⟶ Type. Continuous(X.F[λi.if (i =z j) then X else Z i fi ] i)) and
k-Monotone(T.F[T]))
Proof
Definitions occuring in Statement :
mutual-corec: mutual-corec(T.F[T]),
k-monotone: k-Monotone(T.F[T]),
k-ext: A ≡ B,
type-continuous: Continuous(T.F[T]),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
k-ext: A ≡ B,
and: P ∧ Q,
k-subtype: A ⊆ B,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
mutual-corec-ext,
implies-k-continuous,
int_seg_wf,
all_wf,
type-continuous_wf,
eq_int_wf,
bool_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
k-monotone_wf,
nat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
because_Cache,
sqequalRule,
independent_functionElimination,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
natural_numberEquality,
setElimination,
rename,
instantiate,
applyEquality,
cumulativity,
universeEquality,
functionEquality,
functionExtensionality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
voidElimination,
isect_memberEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[F:(\mBbbN{}k {}\mrightarrow{} Type) {}\mrightarrow{} \mBbbN{}k {}\mrightarrow{} Type].
(mutual-corec(T.F[T]) \mequiv{} F[mutual-corec(T.F[T])]) supposing
((\mforall{}i,j:\mBbbN{}k. \mforall{}Z:\mBbbN{}k {}\mrightarrow{} Type. Continuous(X.F[\mlambda{}i.if (i =\msubz{} j) then X else Z i fi ] i)) and
k-Monotone(T.F[T]))
Date html generated:
2018_05_21-PM-00_10_50
Last ObjectModification:
2017_10_18-PM-02_45_04
Theory : co-recursion
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