Nuprl Lemma : mk-eo_wf
∀[E:Type]. ∀[dom:E ─→ 𝔹]. ∀[l:E ─→ Id]. ∀[R:E ─→ E ─→ ℙ]. ∀[locless:E ─→ E ─→ 𝔹]. ∀[pred:E ─→ E]. ∀[rank:E ─→ ℕ].
mk-eo(E;dom;l;R;locless;pred;rank) ∈ EO
supposing (∀x,y:E. ((↓x R y) ⇒ rank x < rank y))
∧ (∀e:E. ((l (pred e)) = (l e) ∈ Id))
∧ (∀e:E. (¬↓e R (pred e)))
∧ (∀e,x:E. ((↓x R e) ⇒ ((l x) = (l e) ∈ Id) ⇒ ((↓(pred e) R e) ∧ (¬↓(pred e) R x))))
∧ (∀x,y,z:E. ((↓x R y) ⇒ (↓y R z) ⇒ (↓x R z)))
∧ (∀e1,e2:E.
(↓e1 R e2 ⇐⇒ ↑(e1 locless e2)) ∧ ((¬↓e1 R e2) ⇒ (¬↓e2 R e1) ⇒ (e1 = e2 ∈ E)) supposing (l e1) = (l e2) ∈ Id)
Proof
Definitions occuring in Statement :
mk-eo: mk-eo(E;dom;l;R;locless;pred;rank),
event_ordering: EO,
Id: Id,
nat: ℕ,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
mk-eo-record_wf,
rec_select_update_lemma,
eo_axioms_wf,
all_wf,
squash_wf,
infix_ap_wf,
less_than_wf,
nat_wf,
equal_wf,
Id_wf,
not_wf,
isect_wf,
iff_wf,
assert_wf,
bool_wf
\mforall{}[E:Type]. \mforall{}[dom:E {}\mrightarrow{} \mBbbB{}]. \mforall{}[l:E {}\mrightarrow{} Id]. \mforall{}[R:E {}\mrightarrow{} E {}\mrightarrow{} \mBbbP{}]. \mforall{}[locless:E {}\mrightarrow{} E {}\mrightarrow{} \mBbbB{}]. \mforall{}[pred:E {}\mrightarrow{} E].
\mforall{}[rank:E {}\mrightarrow{} \mBbbN{}].
mk-eo(E;dom;l;R;locless;pred;rank) \mmember{} EO
supposing (\mforall{}x,y:E. ((\mdownarrow{}x R y) {}\mRightarrow{} rank x < rank y))
\mwedge{} (\mforall{}e:E. ((l (pred e)) = (l e)))
\mwedge{} (\mforall{}e:E. (\mneg{}\mdownarrow{}e R (pred e)))
\mwedge{} (\mforall{}e,x:E. ((\mdownarrow{}x R e) {}\mRightarrow{} ((l x) = (l e)) {}\mRightarrow{} ((\mdownarrow{}(pred e) R e) \mwedge{} (\mneg{}\mdownarrow{}(pred e) R x))))
\mwedge{} (\mforall{}x,y,z:E. ((\mdownarrow{}x R y) {}\mRightarrow{} (\mdownarrow{}y R z) {}\mRightarrow{} (\mdownarrow{}x R z)))
\mwedge{} (\mforall{}e1,e2:E.
(\mdownarrow{}e1 R e2 \mLeftarrow{}{}\mRightarrow{} \muparrow{}(e1 locless e2)) \mwedge{} ((\mneg{}\mdownarrow{}e1 R e2) {}\mRightarrow{} (\mneg{}\mdownarrow{}e2 R e1) {}\mRightarrow{} (e1 = e2))
supposing (l e1) = (l e2))
Date html generated:
2015_07_17-AM-08_33_57
Last ObjectModification:
2015_01_27-PM-03_00_03
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