Nuprl Lemma : stream-lex-iff
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. ∀s1,s2:stream(T).
(s1 stream-lex(T;R) s2 ⇐⇒ (s-hd(s1) R s-hd(s2)) ∧ ((s-hd(s1) = s-hd(s2) ∈ T) ⇒ (s-tl(s1) stream-lex(T;R) s-tl(s2))))
Proof
Definitions occuring in Statement :
stream-lex: stream-lex(T;R),
s-tl: s-tl(s),
s-hd: s-hd(s),
stream: stream(A),
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
infix_ap: x f y,
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
stream-lex: stream-lex(T;R),
rel_implies: R1 => R2,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
guard: {T},
rel-monotone: rel-monotone{i:l}(T;R.F[R]),
rel-continuous: rel-continuous{i:l}(T;R.F[R]),
isect-rel: isect-rel(T;i.R[i]),
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A
Lemmas referenced :
bigrel-iff,
and_wf,
s-hd_wf,
equal_wf,
s-tl_wf,
stream_wf,
stream-lex_wf,
all_wf,
nat_wf,
false_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
dependent_set_memberEquality,
natural_numberEquality
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}s1,s2:stream(T).
(s1 stream-lex(T;R) s2
\mLeftarrow{}{}\mRightarrow{} (s-hd(s1) R s-hd(s2)) \mwedge{} ((s-hd(s1) = s-hd(s2)) {}\mRightarrow{} (s-tl(s1) stream-lex(T;R) s-tl(s2))))
Date html generated:
2016_05_14-AM-06_24_02
Last ObjectModification:
2015_12_26-AM-11_58_36
Theory : co-recursion
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