Nuprl Lemma : cond_rel_equivalent
∀[T:Type]. ∀[R,Q:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ].
(Trans(T;x,y.Q x y)
⇒ R => Q
⇒ (∀x,y:T. (((P x) ∧ (P y)) ⇒ (((R x y) ∨ (x = y ∈ T)) ∨ (R y x))))
⇒ (∀x,y:T. (((P x) ∧ (P y)) ⇒ (R x y ⇐⇒ Q x y)))
supposing ∀x,y:T. ((Q x y) ⇒ (¬(Q y x))))
Proof
Definitions occuring in Statement :
rel_implies: R1 => R2,
trans: Trans(T;x,y.E[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rel_implies: R1 => R2,
infix_ap: x f y,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
or: P ∨ Q
Latex:
\mforall{}[T:Type]. \mforall{}[R,Q:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
(Trans(T;x,y.Q x y)
{}\mRightarrow{} R => Q
{}\mRightarrow{} (\mforall{}x,y:T. (((P x) \mwedge{} (P y)) {}\mRightarrow{} (((R x y) \mvee{} (x = y)) \mvee{} (R y x))))
{}\mRightarrow{} (\mforall{}x,y:T. (((P x) \mwedge{} (P y)) {}\mRightarrow{} (R x y \mLeftarrow{}{}\mRightarrow{} Q x y)))
supposing \mforall{}x,y:T. ((Q x y) {}\mRightarrow{} (\mneg{}(Q y x))))
Date html generated:
2016_05_16-AM-10_34_16
Last ObjectModification:
2015_12_28-PM-09_17_40
Theory : new!event-ordering
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