Nuprl Lemma : State-comb-invariant-or
∀[Info,A,S:Type]. ∀[P:S ⟶ ℙ].
∀init:Id ⟶ bag(S). ∀f:A ⟶ S ⟶ S. ∀X:EClass(A). ∀es:EO+(Info). ∀e:E. ∀v:S.
(((single-valued-bag(init loc(e);S) ∧ single-valued-classrel(es;X;A)) ∨ (∀s:S. SqStable(P[s])))
⇒ (∀s:S. (s ↓∈ init loc(e) ⇒ P[s]))
⇒ (∀a:A. ∀e':E. (e' ≤loc e ⇒ a ∈ X(e') ⇒ (∀s:S. (s ∈ Memory-class(f;init;X)(e') ⇒ P[s] ⇒ P[f a s]))))
⇒ v ∈ State-comb(init;f;X)(e)
⇒ P[v])
Proof
Definitions occuring in Statement :
State-comb: State-comb(init;f;X),
Memory-class: Memory-class(f;init;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-loc: loc(e),
es-E: E,
Id: Id,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
single-valued-bag: single-valued-bag(b;T),
bag-member: x ↓∈ bs,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}[Info,A,S:Type]. \mforall{}[P:S {}\mrightarrow{} \mBbbP{}].
\mforall{}init:Id {}\mrightarrow{} bag(S). \mforall{}f:A {}\mrightarrow{} S {}\mrightarrow{} S. \mforall{}X:EClass(A). \mforall{}es:EO+(Info). \mforall{}e:E. \mforall{}v:S.
(((single-valued-bag(init loc(e);S) \mwedge{} single-valued-classrel(es;X;A)) \mvee{} (\mforall{}s:S. SqStable(P[s])))
{}\mRightarrow{} (\mforall{}s:S. (s \mdownarrow{}\mmember{} init loc(e) {}\mRightarrow{} P[s]))
{}\mRightarrow{} (\mforall{}a:A. \mforall{}e':E.
(e' \mleq{}loc e {}\mRightarrow{} a \mmember{} X(e') {}\mRightarrow{} (\mforall{}s:S. (s \mmember{} Memory-class(f;init;X)(e') {}\mRightarrow{} P[s] {}\mRightarrow{} P[f a s]))))
{}\mRightarrow{} v \mmember{} State-comb(init;f;X)(e)
{}\mRightarrow{} P[v])
Date html generated:
2016_05_17-AM-09_59_50
Last ObjectModification:
2015_12_29-PM-03_55_50
Theory : classrel!lemmas
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