Nuprl Lemma : fix_wf_mutual-corec-partial-nat
∀[k:ℕ]. ∀[F:(ℕk ⟶ Type) ⟶ ℕk ⟶ Type].
∀[f:⋂T:ℕk ⟶ Type. ((i:ℕk ⟶ (T i) ⟶ partial(ℕ)) ⟶ i:ℕk ⟶ (F[T] i) ⟶ partial(ℕ))]
(fix(f) ∈ i:ℕk ⟶ m-corec(T.F[T];i) ⟶ partial(ℕ))
supposing k-Monotone(T.F[T]) ∧ (∀i,j:ℕk. ∀Z:ℕk ⟶ Type. Continuous(X.F[λi.if (i =z j) then X else Z i fi ] i))
Proof
Definitions occuring in Statement :
m-corec: m-corec(T.F[T];i),
k-monotone: k-Monotone(T.F[T]),
partial: partial(T),
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],
and: P ∧ Q,
member: t ∈ T,
apply: f a,
fix: fix(F),
lambda: λx.A[x],
isect: ⋂x:A. B[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,
and: P ∧ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
cand: A c∧ B,
prop: ℙ,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
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 :
fix_wf_mutual-corec-partial1,
nat_wf,
set-value-type,
le_wf,
int-value-type,
nat-mono,
int_seg_wf,
partial_wf,
k-monotone_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
functionEquality,
setElimination,
rename,
cumulativity,
universeEquality,
because_Cache,
independent_pairFormation,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isectEquality,
productEquality,
instantiate,
lambdaFormation,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
independent_functionElimination,
voidElimination
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[F:(\mBbbN{}k {}\mrightarrow{} Type) {}\mrightarrow{} \mBbbN{}k {}\mrightarrow{} Type].
\mforall{}[f:\mcap{}T:\mBbbN{}k {}\mrightarrow{} Type. ((i:\mBbbN{}k {}\mrightarrow{} (T i) {}\mrightarrow{} partial(\mBbbN{})) {}\mrightarrow{} i:\mBbbN{}k {}\mrightarrow{} (F[T] i) {}\mrightarrow{} partial(\mBbbN{}))]
(fix(f) \mmember{} i:\mBbbN{}k {}\mrightarrow{} m-corec(T.F[T];i) {}\mrightarrow{} partial(\mBbbN{}))
supposing k-Monotone(T.F[T])
\mwedge{} (\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))
Date html generated:
2018_05_21-PM-00_18_20
Last ObjectModification:
2017_10_18-PM-02_49_32
Theory : co-recursion
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