Nuprl Lemma : State-comb-trans-refl
∀[Info,B,A:Type]. ∀[R:B ⟶ B ⟶ ℙ].
∀f:A ⟶ B ⟶ B. ∀init:Id ⟶ bag(B). ∀X:EClass(A). ∀es:EO+(Info). ∀e1,e2:E. ∀v1,v2:B.
(Refl(B;x,y.R[x;y])
⇒ Trans(B;x,y.R[x;y])
⇒ (∀a:A. ∀e:E.
((e1 <loc e) ⇒ e ≤loc e2 ⇒ a ∈ X(e) ⇒ (∀s:B. (s ∈ State-comb(init;f;X)(pred(e)) ⇒ R[s;f a s]))))
⇒ 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
⇒ 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]),
refl: Refl(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
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,
es-le: e ≤loc e' ,
or: P ∨ Q,
member: t ∈ T,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
sq_stable: SqStable(P),
es-locl: (e <loc e'),
decidable: Dec(P),
exists: ∃x:A. B[x],
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
guard: {T},
refl: Refl(T;x,y.E[x; y])
Latex:
\mforall{}[Info,B,A:Type]. \mforall{}[R:B {}\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.
(Refl(B;x,y.R[x;y])
{}\mRightarrow{} Trans(B;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} a \mmember{} X(e)
{}\mRightarrow{} (\mforall{}s:B. (s \mmember{} State-comb(init;f;X)(pred(e)) {}\mRightarrow{} R[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 \mleq{}loc e2
{}\mRightarrow{} R[v1;v2])
Date html generated:
2016_05_17-AM-10_00_12
Last ObjectModification:
2016_01_17-PM-11_06_09
Theory : classrel!lemmas
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