Nuprl Lemma : stream-pointwise_transitivity
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. (Trans(T;x,y.x R y) ⇒ Trans(stream(T);s1,s2.s1 stream-pointwise(R) s2))
Proof
Definitions occuring in Statement :
stream-pointwise: stream-pointwise(R),
stream: stream(A),
trans: Trans(T;x,y.E[x; y]),
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
stream-pointwise: stream-pointwise(R),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
prop: ℙ,
so_apply: x[s],
trans: Trans(T;x,y.E[x; y]),
isect-rel: isect-rel(T;i.R[i]),
guard: {T},
and: P ∧ Q,
cand: A c∧ B,
true: True
Lemmas referenced :
bigrel-induction,
stream_wf,
trans_wf,
and_wf,
s-hd_wf,
s-tl_wf,
all_wf,
nat_wf,
isect-rel_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
productElimination,
independent_pairFormation,
natural_numberEquality
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. (Trans(T;x,y.x R y) {}\mRightarrow{} Trans(stream(T);s1,s2.s1 stream-pointwise(R) s2))
Date html generated:
2016_05_14-AM-06_24_40
Last ObjectModification:
2015_12_26-AM-11_58_19
Theory : co-recursion
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