Nuprl Lemma : es-first-at-implies-first-at
∀es:EO. ∀i:Id.
∀[P:{e:E| loc(e) = i ∈ Id} ⟶ ℙ]
∀e:E
(e is first@ i s.t. e.P[e]
⇒ {∀[Q:{e:E| loc(e) = i ∈ Id} ⟶ ℙ]
(e is first@ i s.t. e.Q[e] ⇐⇒ Q[e] ∧ ∀e'<e.e' is first@ i s.t. e.Q[e] ⇒ P[e'])})
Proof
Definitions occuring in Statement :
es-first-at: e is first@ i s.t. e.P[e],
alle-lt: ∀e<e'.P[e],
es-loc: loc(e),
es-E: E,
event_ordering: EO,
Id: Id,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
alle-lt: ∀e<e'.P[e],
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
es-locl: (e <loc e'),
es-first-at: e is first@ i s.t. e.P[e],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
cand: A c∧ B,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T
Latex:
\mforall{}es:EO. \mforall{}i:Id.
\mforall{}[P:\{e:E| loc(e) = i\} {}\mrightarrow{} \mBbbP{}]
\mforall{}e:E
(e is first@ i s.t. e.P[e]
{}\mRightarrow{} \{\mforall{}[Q:\{e:E| loc(e) = i\} {}\mrightarrow{} \mBbbP{}]
(e is first@ i s.t. e.Q[e] \mLeftarrow{}{}\mRightarrow{} Q[e] \mwedge{} \mforall{}e'<e.e' is first@ i s.t. e.Q[e] {}\mRightarrow{} P[e'])\})
Date html generated:
2016_05_16-AM-09_49_32
Last ObjectModification:
2016_01_17-PM-01_26_33
Theory : new!event-ordering
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