Nuprl Lemma : corec-subtype-corec2
∀[F,G:Type ⟶ Type]. corec(T.F[T]) ⊆r corec(T.G[T]) supposing ∀A,B:Type. ((A ⊆r B) ⇒ (F[A] ⊆r G[B]))
Proof
Definitions occuring in Statement :
corec: corec(T.F[T]),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
corec: corec(T.F[T]),
subtype_rel: A ⊆r B,
so_apply: x[s],
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
all: ∀x:A. B[x],
top: Top,
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,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_wf,
primrec_wf,
top_wf,
int_seg_wf,
all_wf,
subtype_rel_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
primrec0_lemma,
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,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-le-2,
not-equal-2,
le_wf,
primrec-unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
isect_memberEquality,
extract_by_obid,
hypothesis,
isectEquality,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
universeEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
cumulativity,
natural_numberEquality,
setElimination,
rename,
sqequalRule,
axiomEquality,
functionEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
intWeakElimination,
lambdaFormation,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
voidEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
intEquality,
minusEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
dependent_set_memberEquality
Latex:
\mforall{}[F,G:Type {}\mrightarrow{} Type].
corec(T.F[T]) \msubseteq{}r corec(T.G[T]) supposing \mforall{}A,B:Type. ((A \msubseteq{}r B) {}\mRightarrow{} (F[A] \msubseteq{}r G[B]))
Date html generated:
2017_04_14-AM-07_46_59
Last ObjectModification:
2017_02_27-PM-03_17_32
Theory : co-recursion
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