Nuprl Lemma : alle-at-iff
∀es:EO. ∀i:Id.
∀[P:{e:E| loc(e) = i ∈ Id} ⟶ ℙ]
(∀e@i.P[e] ⇐⇒ ∀e@i.P[e] supposing ↑first(e) ∧ ∀e@i.P[pred(e)] ⇒ P[e] supposing ¬↑first(e))
Proof
Definitions occuring in Statement :
alle-at: ∀e@i.P[e],
es-first: first(e),
es-pred: pred(e),
es-loc: loc(e),
es-E: E,
event_ordering: EO,
Id: Id,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
es-E: E,
es-base-E: es-base-E(es),
guard: {T},
subtype_rel: A ⊆r B,
alle-at: ∀e@i.P[e],
not: ¬A,
false: False,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
or: P ∨ Q
Latex:
\mforall{}es:EO. \mforall{}i:Id.
\mforall{}[P:\{e:E| loc(e) = i\} {}\mrightarrow{} \mBbbP{}]
(\mforall{}e@i.P[e] \mLeftarrow{}{}\mRightarrow{} \mforall{}e@i.P[e] supposing \muparrow{}first(e) \mwedge{} \mforall{}e@i.P[pred(e)] {}\mRightarrow{} P[e] supposing \mneg{}\muparrow{}first(e))
Date html generated:
2016_05_16-AM-09_40_22
Last ObjectModification:
2015_12_28-PM-09_43_50
Theory : new!event-ordering
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