Nuprl Lemma : stream-coinduction
∀[A:Type]. ∀[R:stream(A) ⟶ stream(A) ⟶ ℙ].
∀[x,y:stream(A)]. x = y ∈ stream(A) supposing x R y
supposing ∀x,y:stream(A). ((x R y) ⇒ ((s-hd(x) = s-hd(y) ∈ A) ∧ (s-tl(x) R s-tl(y))))
Proof
Definitions occuring in Statement :
s-tl: s-tl(s),
s-hd: s-hd(s),
stream: stream(A),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
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],
stream: stream(A),
so_apply: x[s1;s2],
infix_ap: x f y,
prop: ℙ,
implies: P ⇒ Q,
and: P ∧ Q,
subtype_rel: A ⊆r B,
F-bisimulation: x,y.R[x; y] is an T.F[T]-bisimulation,
all: ∀x:A. B[x],
s-tl: s-tl(s),
s-hd: s-hd(s),
guard: {T},
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
coinduction-principle,
continuous-monotone-product,
continuous-monotone-constant,
continuous-monotone-id,
stream_wf,
all_wf,
equal_wf,
s-hd_wf,
s-tl_wf,
corec_wf,
subtype_rel_wf,
stream-ext,
subtype_rel_product,
subtype_rel_transitivity,
subtype_rel_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
productEquality,
cumulativity,
hypothesisEquality,
universeEquality,
independent_isectElimination,
hypothesis,
applyEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lambdaFormation,
productElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairEquality
Latex:
\mforall{}[A:Type]. \mforall{}[R:stream(A) {}\mrightarrow{} stream(A) {}\mrightarrow{} \mBbbP{}].
\mforall{}[x,y:stream(A)]. x = y supposing x R y
supposing \mforall{}x,y:stream(A). ((x R y) {}\mRightarrow{} ((s-hd(x) = s-hd(y)) \mwedge{} (s-tl(x) R s-tl(y))))
Date html generated:
2017_04_14-AM-07_47_20
Last ObjectModification:
2017_02_27-PM-03_17_41
Theory : co-recursion
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