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)
Lemmas :
iterated-classrel-trans,
State-comb-classrel,
es-pred_wf,
iterated-classrel-iff,
iterated-classrel_wf,
es-locl-first,
assert_elim,
btrue_neq_bfalse,
assert_wf,
es-first_wf2,
classrel_wf,
es-le_wf,
es-locl_wf,
sq_stable__single-valued-iterated-classrel,
and_wf,
equal_wf,
Id_wf,
bag_wf,
single-valued-bag_wf,
decidable__exists-classrel-between3-sv,
exists_wf,
State-comb-single-val0,
es-E_wf,
event-ordering+_subtype,
es-loc_wf,
State-comb_wf,
single-valued-classrel_wf,
all_wf,
trans_wf,
refl_wf,
event-ordering+_wf,
eclass_wf
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:
2015_07_22-PM-00_22_40
Last ObjectModification:
2015_01_28-AM-10_12_32
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