Nuprl Lemma : es-pstar-q_functionality_wrt_implies
∀es:EO. ∀e1:E. ∀e2:{e:E| loc(e) = loc(e1) ∈ Id} .
∀[p,q,p',q':{e:E| loc(e) = loc(e1) ∈ Id} ⟶ {e:E| loc(e) = loc(e1) ∈ Id} ⟶ ℙ].
((∀a,b:{e:E| loc(e) = loc(e1) ∈ Id} . ((a ∈ [e1, e2]) ⇒ (b ∈ [e1, e2]) ⇒ {p[a;b] ⇒ p'[a;b]}))
⇒ (∀a,b:{e:E| loc(e) = loc(e1) ∈ Id} . ((a ∈ [e1, e2]) ⇒ (b ∈ [e1, e2]) ⇒ {q[a;b] ⇒ q'[a;b]}))
⇒ {[e1;e2]~([a,b].p[a;b])*[a,b].q[a;b] ⇒ [e1;e2]~([a,b].p'[a;b])*[a,b].q'[a;b]})
Proof
Definitions occuring in Statement :
es-pstar-q: [e1;e2]~([a,b].p[a; b])*[a,b].q[a; b],
es-interval: [e, e'],
es-loc: loc(e),
es-E: E,
event_ordering: EO,
Id: Id,
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: 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,
member: t ∈ T,
es-pstar-q: [e1;e2]~([a,b].p[a; b])*[a,b].q[a; b],
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
es-le: e ≤loc e' ,
sq_type: SQType(T)
Latex:
\mforall{}es:EO. \mforall{}e1:E. \mforall{}e2:\{e:E| loc(e) = loc(e1)\} .
\mforall{}[p,q,p',q':\{e:E| loc(e) = loc(e1)\} {}\mrightarrow{} \{e:E| loc(e) = loc(e1)\} {}\mrightarrow{} \mBbbP{}].
((\mforall{}a,b:\{e:E| loc(e) = loc(e1)\} . ((a \mmember{} [e1, e2]) {}\mRightarrow{} (b \mmember{} [e1, e2]) {}\mRightarrow{} \{p[a;b] {}\mRightarrow{} p'[a;b]\}))
{}\mRightarrow{} (\mforall{}a,b:\{e:E| loc(e) = loc(e1)\} . ((a \mmember{} [e1, e2]) {}\mRightarrow{} (b \mmember{} [e1, e2]) {}\mRightarrow{} \{q[a;b] {}\mRightarrow{} q'[a;b]\}))
{}\mRightarrow{} \{[e1;e2]\msim{}([a,b].p[a;b])*[a,b].q[a;b] {}\mRightarrow{} [e1;e2]\msim{}([a,b].p'[a;b])*[a,b].q'[a;b]\})
Date html generated:
2016_05_16-AM-09_55_47
Last ObjectModification:
2016_01_17-PM-01_31_17
Theory : new!event-ordering
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