Nuprl Lemma : stream-lex-monotonic
∀T:Type
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀R:T ⟶ T ⟶ ℙ
(Trans(T;x,y.x R y)
⇒ AntiSym(T;x,y.x R y)
⇒ (∀f:T ⟶ T ⟶ T
((∀x,y,u,v:T. ((x R y) ⇒ (u R v) ⇒ ((f x u) R (f y v))))
⇒ (∀x,y,u,v:T. ((x R y) ⇒ (u R v) ⇒ (¬((x = y ∈ T) ∧ (u = v ∈ T))) ⇒ (¬((f x u) = (f y v) ∈ T))))
⇒ (∀a,b,c,d:stream(T).
((a stream-lex(T;R) b)
⇒ (c stream-lex(T;R) d)
⇒ (stream-zip(f;a;c) stream-lex(T;R) stream-zip(f;b;d)))))))))
Proof
Definitions occuring in Statement :
stream-lex: stream-lex(T;R),
stream-zip: stream-zip(f;as;bs),
stream: stream(A),
anti_sym: AntiSym(T;x,y.R[x; y]),
trans: Trans(T;x,y.E[x; y]),
decidable: Dec(P),
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
rel-monotone: rel-monotone{i:l}(T;R.F[R]),
rel_implies: R1 => R2,
infix_ap: x f y,
cand: A c∧ B,
stream-lex: stream-lex(T;R),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
top: Top,
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False
Lemmas referenced :
implies-bigrel,
infix_ap_wf,
s-hd_wf,
equal_wf,
stream_wf,
s-tl_wf,
all_wf,
stream-lex_wf,
not_wf,
anti_sym_wf,
trans_wf,
decidable_wf,
exists_wf,
stream-zip_wf,
stream-lex-iff,
stream-subtype,
top_wf,
hd-stream-zip,
tl-stream-zip,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
productEquality,
instantiate,
cumulativity,
hypothesisEquality,
universeEquality,
functionExtensionality,
applyEquality,
hypothesis,
functionEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
unionElimination,
dependent_pairFormation
Latex:
\mforall{}T:Type
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}
(Trans(T;x,y.x R y)
{}\mRightarrow{} AntiSym(T;x,y.x R y)
{}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} T
((\mforall{}x,y,u,v:T. ((x R y) {}\mRightarrow{} (u R v) {}\mRightarrow{} ((f x u) R (f y v))))
{}\mRightarrow{} (\mforall{}x,y,u,v:T.
((x R y) {}\mRightarrow{} (u R v) {}\mRightarrow{} (\mneg{}((x = y) \mwedge{} (u = v))) {}\mRightarrow{} (\mneg{}((f x u) = (f y v)))))
{}\mRightarrow{} (\mforall{}a,b,c,d:stream(T).
((a stream-lex(T;R) b)
{}\mRightarrow{} (c stream-lex(T;R) d)
{}\mRightarrow{} (stream-zip(f;a;c) stream-lex(T;R) stream-zip(f;b;d)))))))))
Date html generated:
2017_04_14-AM-07_48_23
Last ObjectModification:
2017_02_27-PM-03_21_13
Theory : co-recursion
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