Nuprl Lemma : pes-axioms
∀the_es:EO
(Trans(E;e,e'.(e <loc e'))
∧ SWellFounded((e <loc e'))
∧ (∀e,e':E. (loc(e) = loc(e') ∈ Id ⇐⇒ (e <loc e') ∨ (e = e' ∈ E) ∨ (e' <loc e)))
∧ (∀e:E. (↑first(e) ⇐⇒ ∀e':E. (¬(e' <loc e))))
∧ (∀e:E. (pred(e) <loc e) ∧ (∀e':E. (¬((pred(e) <loc e') ∧ (e' <loc e)))) supposing ¬↑first(e))
∧ Trans(E;e,e'.(e < e'))
∧ SWellFounded((e < e'))
∧ (∀e,e':E. ((e <loc e') ⇒ (e < e'))))
Proof
Definitions occuring in Statement :
es-locl: (e <loc e'),
es-first: first(e),
es-pred: pred(e),
es-causl: (e < e'),
es-loc: loc(e),
es-E: E,
event_ordering: EO,
strongwellfounded: SWellFounded(R[x; y]),
Id: Id,
trans: Trans(T;x,y.E[x; y]),
assert: ↑b,
uimplies: b supposing a,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
trans: Trans(T;x,y.E[x; y]),
es-E: E,
es-base-E: es-base-E(es),
implies: P ⇒ Q,
es-locl: (e <loc e'),
prop: ℙ,
uall: ∀[x:A]. B[x],
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
iff: P ⇐⇒ Q,
uimplies: b supposing a,
rev_implies: P ⇐ Q,
or: P ∨ Q,
guard: {T},
cand: A c∧ B,
not: ¬A,
false: False,
uiff: uiff(P;Q),
es-causl: (e < e'),
squash: ↓T,
true: True
Latex:
\mforall{}the$_{es}$:EO
(Trans(E;e,e'.(e <loc e'))
\mwedge{} SWellFounded((e <loc e'))
\mwedge{} (\mforall{}e,e':E. (loc(e) = loc(e') \mLeftarrow{}{}\mRightarrow{} (e <loc e') \mvee{} (e = e') \mvee{} (e' <loc e)))
\mwedge{} (\mforall{}e:E. (\muparrow{}first(e) \mLeftarrow{}{}\mRightarrow{} \mforall{}e':E. (\mneg{}(e' <loc e))))
\mwedge{} (\mforall{}e:E. (pred(e) <loc e) \mwedge{} (\mforall{}e':E. (\mneg{}((pred(e) <loc e') \mwedge{} (e' <loc e)))) supposing \mneg{}\muparrow{}first(e))
\mwedge{} Trans(E;e,e'.(e < e'))
\mwedge{} SWellFounded((e < e'))
\mwedge{} (\mforall{}e,e':E. ((e <loc e') {}\mRightarrow{} (e < e'))))
Date html generated:
2016_05_16-AM-09_18_17
Last ObjectModification:
2016_01_17-PM-01_31_07
Theory : new!event-ordering
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