Nuprl Lemma : bfs-equiv-implies2
∀[S:Type]. ∀[K:RngSig].
∀x,y:basic-formal-sum(K;S).
(bfs-equiv(K;S;x;y)
⇒ {∀[P:basic-formal-sum(K;S) ⟶ ℙ]
((∀x,y:basic-formal-sum(K;S). (bfs-reduce(K;S;x;y) ⇒ (P[x] ⇐⇒ P[y]))) ⇒ P[x] ⇒ P[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),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
rng_sig: RngSig
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
infix_ap: x f y,
bfs-equiv: bfs-equiv(K;S;fs1;fs2),
guard: {T},
prop: ℙ
Lemmas referenced :
least-equiv-induction2,
bfs-reduce_wf,
bfs-equiv_wf,
basic-formal-sum_wf,
rng_sig_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
lambdaEquality_alt,
hypothesisEquality,
hypothesis,
inhabitedIsType,
sqequalRule,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
instantiate,
universeEquality
Latex:
\mforall{}[S:Type]. \mforall{}[K:RngSig].
\mforall{}x,y:basic-formal-sum(K;S).
(bfs-equiv(K;S;x;y)
{}\mRightarrow{} \{\mforall{}[P:basic-formal-sum(K;S) {}\mrightarrow{} \mBbbP{}]
((\mforall{}x,y:basic-formal-sum(K;S). (bfs-reduce(K;S;x;y) {}\mRightarrow{} (P[x] \mLeftarrow{}{}\mRightarrow{} P[y])))
{}\mRightarrow{} P[x]
{}\mRightarrow{} P[y])\})
Date html generated:
2019_10_31-AM-06_28_34
Last ObjectModification:
2019_08_14-PM-05_53_45
Theory : linear!algebra
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