Nuprl Lemma : map-interface-accum-equal
∀[Info,A1,B1,A2,B2,C:Type]. ∀[X1:EClass(A1)]. ∀[X2:EClass(A2)]. ∀[b1:B1]. ∀[b2:B2]. ∀[acc1:B1 ─→ A1 ─→ B1].
∀[acc2:B2 ─→ A2 ─→ B2]. ∀[F1:B1 ─→ C]. ∀[F2:B2 ─→ C].
((F1[x] where x from es-interface-accum(acc1;b1;X1))
= (F2[x] where x from es-interface-accum(acc2;b2;X2))
∈ EClass(C)) supposing
((∀a:B1. ∀b:B2.
(((F1 a) = (F2 b) ∈ C)
⇒ (∀es:EO+(Info). ∀e:E. ((↑e ∈b X1) ⇒ (↑e ∈b X1) ⇒ (F1[acc1 a X1(e)] = F2[acc2 b X2(e)] ∈ C))))) and
(F1[b1] = F2[b2] ∈ C) and
(∀es:EO+(Info). ∀e:E. (↑e ∈b X1 ⇐⇒ ↑e ∈b X2)))
Proof
Definitions occuring in Statement :
es-interface-accum: es-interface-accum(f;x;X),
map-class: (f[v] where v from X),
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-E: E,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
all_wf,
es-E_wf,
event-ordering+_subtype,
assert_wf,
in-eclass_wf,
es-interface-subtype_rel2,
top_wf,
eclass-val_wf,
es-E-interface-property,
equal_wf,
iff_wf,
eclass_wf,
event-ordering+_wf,
map-class_functionality,
es-interface-accum_wf,
is-interface-accum,
list_accum_wf,
es-E-interface_wf,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
Id_wf,
es-loc_wf,
es-interface-predecessors_wf,
es-interface-accum-val,
es-interface-predecessors-equal,
list_accum_equality,
es-E-interface_functionality,
subtype_rel_list,
subtype_rel_set,
and_wf
Latex:
\mforall{}[Info,A1,B1,A2,B2,C:Type]. \mforall{}[X1:EClass(A1)]. \mforall{}[X2:EClass(A2)]. \mforall{}[b1:B1]. \mforall{}[b2:B2]. \mforall{}[acc1:B1
{}\mrightarrow{} A1
{}\mrightarrow{} B1].
\mforall{}[acc2:B2 {}\mrightarrow{} A2 {}\mrightarrow{} B2]. \mforall{}[F1:B1 {}\mrightarrow{} C]. \mforall{}[F2:B2 {}\mrightarrow{} C].
((F1[x] where x from es-interface-accum(acc1;b1;X1))
= (F2[x] where x from es-interface-accum(acc2;b2;X2))) supposing
((\mforall{}a:B1. \mforall{}b:B2.
(((F1 a) = (F2 b))
{}\mRightarrow{} (\mforall{}es:EO+(Info). \mforall{}e:E.
((\muparrow{}e \mmember{}\msubb{} X1) {}\mRightarrow{} (\muparrow{}e \mmember{}\msubb{} X1) {}\mRightarrow{} (F1[acc1 a X1(e)] = F2[acc2 b X2(e)]))))) and
(F1[b1] = F2[b2]) and
(\mforall{}es:EO+(Info). \mforall{}e:E. (\muparrow{}e \mmember{}\msubb{} X1 \mLeftarrow{}{}\mRightarrow{} \muparrow{}e \mmember{}\msubb{} X2)))
Date html generated:
2015_07_21-PM-03_33_46
Last ObjectModification:
2015_01_27-PM-06_41_49
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