Nuprl Lemma : mutual-corec-ext
∀[k:ℕ]. ∀[F:(ℕk ⟶ Type) ⟶ ℕk ⟶ Type].
(mutual-corec(T.F[T]) ≡ F[mutual-corec(T.F[T])]) supposing (k-Monotone(T.F[T]) and k-Continuous(T.F[T]))
Proof
Definitions occuring in Statement :
mutual-corec: mutual-corec(T.F[T]),
k-continuous: k-Continuous(T.F[T]),
k-monotone: k-Monotone(T.F[T]),
k-ext: A ≡ B,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
k-subtype: A ⊆ B,
subtype_rel: A ⊆r B,
top: Top,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
k-monotone: k-Monotone(T.F[T]),
nequal: a ≠ b ∈ T ,
squash: ↓T,
k-ext: A ≡ B,
mutual-corec: mutual-corec(T.F[T]),
so_lambda: λ2x.t[x],
sq_stable: SqStable(P),
k-continuous: k-Continuous(T.F[T]),
int_seg: {i..j-},
lelt: i ≤ j < k,
k-intersection: ⋂n. X[n]
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
int_seg_wf,
primrec0_lemma,
primrec1_lemma,
top_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
primrec-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_antisymmetry_iff,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-subtract-cancel,
primrec_wf,
not-le-2,
not-equal-2,
le_wf,
k-subtype_wf,
squash_wf,
true_wf,
le_weakening2,
le_weakening,
nat_wf,
k-monotone_wf,
k-continuous_wf,
k-intersection_wf,
sq_stable__le,
subtype_rel-equal,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
voidEquality,
applyEquality,
functionExtensionality,
functionEquality,
cumulativity,
because_Cache,
universeEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
intEquality,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
dependent_set_memberEquality,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
isectEquality
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
(k-Monotone(T.F[T]) and
k-Continuous(T.F[T]))
Date html generated:
2018_05_21-PM-00_10_42
Last ObjectModification:
2017_10_18-PM-02_44_21
Theory : co-recursion
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