Nuprl Lemma : State-comb-progress
∀[Info,B,A:Type].
∀R:B ⟶ B ⟶ ℙ. ∀P:A ⟶ B ⟶ ℙ. ∀f:A ⟶ B ⟶ B. ∀init:Id ⟶ bag(B). ∀X:EClass(A). ∀es:EO+(Info). ∀e1,e2:E. ∀v1,v2:B.
((∀a:A. ∀s:B. Dec(P[a;s]))
⇒ Trans(B;x,y.R[x;y])
⇒ (∀a:A. ∀e:E. ∀s:B.
((e1 <loc e)
⇒ e ≤loc e2
⇒ a ∈ X(e)
⇒ s ∈ State-comb(init;f;X)(pred(e))
⇒ ((P[a;s] ⇒ R[s;f a s]) ∧ ((¬P[a;s]) ⇒ (s = (f a s) ∈ B)))))
⇒ single-valued-classrel(es;X;A)
⇒ single-valued-bag(init loc(e1);B)
⇒ v1 ∈ State-comb(init;f;X)(e1)
⇒ v2 ∈ State-comb(init;f;X)(e2)
⇒ (e1 <loc e2)
⇒ (∃e:E. ∃a:A. ∃s:B. ((e1 <loc e) ∧ e ≤loc e2 ∧ s ∈ State-comb(init;f;X)(pred(e)) ∧ a ∈ X(e) ∧ P[a;s]))
⇒ R[v1;v2])
Proof
Definitions occuring in Statement :
State-comb: State-comb(init;f;X),
single-valued-classrel: single-valued-classrel(es;X;T),
classrel: v ∈ X(e),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-locl: (e <loc e'),
es-pred: pred(e),
es-loc: loc(e),
es-E: E,
Id: Id,
trans: Trans(T;x,y.E[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
single-valued-bag: single-valued-bag(b;T),
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
not: ¬A,
subtype_rel: A ⊆r B,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
squash: ↓T,
so_apply: x[s1;s2],
sq_stable: SqStable(P),
es-locl: (e <loc e'),
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y]
Latex:
\mforall{}[Info,B,A:Type].
\mforall{}R:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}. \mforall{}P:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}. \mforall{}f:A {}\mrightarrow{} B {}\mrightarrow{} B. \mforall{}init:Id {}\mrightarrow{} bag(B). \mforall{}X:EClass(A). \mforall{}es:EO+(Info).
\mforall{}e1,e2:E. \mforall{}v1,v2:B.
((\mforall{}a:A. \mforall{}s:B. Dec(P[a;s]))
{}\mRightarrow{} Trans(B;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E. \mforall{}s:B.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} s \mmember{} State-comb(init;f;X)(pred(e))
{}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (s = (f a s))))))
{}\mRightarrow{} single-valued-classrel(es;X;A)
{}\mRightarrow{} single-valued-bag(init loc(e1);B)
{}\mRightarrow{} v1 \mmember{} State-comb(init;f;X)(e1)
{}\mRightarrow{} v2 \mmember{} State-comb(init;f;X)(e2)
{}\mRightarrow{} (e1 <loc e2)
{}\mRightarrow{} (\mexists{}e:E
\mexists{}a:A
\mexists{}s:B. ((e1 <loc e) \mwedge{} e \mleq{}loc e2 \mwedge{} s \mmember{} State-comb(init;f;X)(pred(e)) \mwedge{} a \mmember{} X(e) \mwedge{} P[a;s]))
{}\mRightarrow{} R[v1;v2])
Date html generated:
2016_05_17-AM-10_00_19
Last ObjectModification:
2016_01_17-PM-11_07_41
Theory : classrel!lemmas
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