Nuprl Lemma : Accum-class-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])
⇒ (∀s1,s2:B. SqStable(R[s1;s2]))
⇒ (∀a:A. ∀e:E. ∀s:B.
((e1 <loc e)
⇒ e ≤loc e2
⇒ a ∈ X(e)
⇒ s ∈ Prior(Accum-class(f;init;X))?init(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 ∈ Accum-class(f;init;X)(e1)
⇒ v2 ∈ Accum-class(f;init;X)(e2)
⇒ (e1 <loc e2)
⇒ (((∃e:E. ∃a:A. ∃s:B. ((e1 <loc e) ∧ e ≤loc e2 ∧ s ∈ Prior(Accum-class(f;init;X))?init(e) ∧ a ∈ X(e) ∧ P[a;s]))
⇒ R[v1;v2])
∧ ((∀e:E. ∀s:B.
((e1 <loc e)
⇒ e ≤loc e2
⇒ s ∈ Prior(Accum-class(f;init;X))?init(e)
⇒ (∀a:A. ((¬a ∈ X(e)) ∨ (¬P[a;s])))))
⇒ (v1 = v2 ∈ B))))
Proof
Definitions occuring in Statement :
Accum-class: Accum-class(f;init;X),
primed-class-opt: Prior(X)?b,
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-loc: loc(e),
es-E: E,
Id: Id,
trans: Trans(T;x,y.E[x; y]),
sq_stable: SqStable(P),
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,
or: 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)
Lemmas :
es-causl-swellfnd,
less_than_wf,
decidable__lt,
subtract_wf,
false_wf,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
add-commutes,
less-iff-le,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
es-causl_wf,
rec-combined-class-opt-1-classrel,
simple-comb-2-classrel,
primed-class-opt_wf,
rec-combined-class-opt-1_wf,
lifting-2_wf,
sq_stable__and,
exists_wf,
es-locl_wf,
es-le_wf,
classrel_wf,
all_wf,
or_wf,
not_wf,
sq_stable__all,
equal_wf,
sq_stable__equal,
squash_wf,
primed-class-opt-classrel,
es-locl-trichotomy,
Accum-class_wf,
single-valued-bag_wf,
es-loc_wf,
single-valued-classrel_wf,
sq_stable_wf,
trans_wf,
decidable_wf,
nat_wf,
set_wf,
primrec-wf2,
zero-le-nat,
le_wf,
add-mul-special,
zero-mul,
add-zero,
es-E_wf,
event-ordering+_subtype,
eclass_wf,
event-ordering+_wf,
Id_wf,
bag_wf,
es-causl_weakening,
es-locl_transitivity1,
es-le_weakening,
es-le-self,
and_wf,
Prior-Accum-class-single-val0,
decidable__existse-between3,
assert_wf,
es-first_wf2,
Memory-class_wf,
bag-member_wf,
iterated_classrel_wf,
es-pred_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_imp_equal_bool,
bfalse_wf,
decidable__exists_iterated_classrel,
es-loc-pred,
decidable_functionality,
Memory-classrel,
and_false_l,
or_false_l,
decidable__exists-single-valued-classrel,
iterated_classrel-single-val,
Accum-class-single-val0,
es-le_weakening_eq
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{}s1,s2:B. SqStable(R[s1;s2]))
{}\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{} Prior(Accum-class(f;init;X))?init(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{} Accum-class(f;init;X)(e1)
{}\mRightarrow{} v2 \mmember{} Accum-class(f;init;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{} Prior(Accum-class(f;init;X))?init(e)
\mwedge{} a \mmember{} X(e)
\mwedge{} P[a;s]))
{}\mRightarrow{} R[v1;v2])
\mwedge{} ((\mforall{}e:E. \mforall{}s:B.
((e1 <loc e)
{}\mRightarrow{} e \mleq{}loc e2
{}\mRightarrow{} s \mmember{} Prior(Accum-class(f;init;X))?init(e)
{}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s])))))
{}\mRightarrow{} (v1 = v2))))
Date html generated:
2015_07_22-PM-00_19_07
Last ObjectModification:
2015_07_16-AM-09_39_15
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