Nuprl Lemma : bfs-equiv-implies
∀[S:Type]. ∀[K:RngSig]. ∀[E:basic-formal-sum(K;S) ⟶ basic-formal-sum(K;S) ⟶ ℙ].
((∀x,y:basic-formal-sum(K;S). (bfs-reduce(K;S;x;y) ⇒ (E x y)))
⇒ EquivRel(basic-formal-sum(K;S);x,y.E x y)
⇒ (∀x,y:basic-formal-sum(K;S). (bfs-equiv(K;S;x;y) ⇒ (E x y))))
Proof
Definitions occuring in Statement :
bfs-equiv: bfs-equiv(K;S;fs1;fs2),
bfs-reduce: bfs-reduce(K;S;as;bs),
basic-formal-sum: basic-formal-sum(K;S),
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
rng_sig: RngSig
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rel_implies: R1 => R2,
infix_ap: x f y,
bfs-equiv: bfs-equiv(K;S;fs1;fs2)
Lemmas referenced :
least-equiv-implies,
basic-formal-sum_wf,
bfs-reduce_wf,
rng_sig_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
sqequalRule,
universeEquality
Latex:
\mforall{}[S:Type]. \mforall{}[K:RngSig]. \mforall{}[E:basic-formal-sum(K;S) {}\mrightarrow{} basic-formal-sum(K;S) {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:basic-formal-sum(K;S). (bfs-reduce(K;S;x;y) {}\mRightarrow{} (E x y)))
{}\mRightarrow{} EquivRel(basic-formal-sum(K;S);x,y.E x y)
{}\mRightarrow{} (\mforall{}x,y:basic-formal-sum(K;S). (bfs-equiv(K;S;x;y) {}\mRightarrow{} (E x y))))
Date html generated:
2018_05_22-PM-09_45_07
Last ObjectModification:
2018_05_20-PM-10_42_29
Theory : linear!algebra
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