Step
*
of Lemma
memory-class3-fun-eq
∀[Info,B,A1,A2,A3:Type]. ∀[init:Id ─→ B]. ∀[tr1:Id ─→ A1 ─→ B ─→ B]. ∀[X1:EClass(A1)]. ∀[tr2:Id ─→ A2 ─→ B ─→ B].
∀[X2:EClass(A2)]. ∀[tr3:Id ─→ A3 ─→ B ─→ B]. ∀[X3:EClass(A3)]. ∀[es:EO+(Info)]. ∀[e:E].
(memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(e)
= if first(e) then init loc(e)
if pred(e) ∈b X1 then tr1 loc(e) X1@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
if pred(e) ∈b X2 then tr2 loc(e) X2@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
if pred(e) ∈b X3 then tr3 loc(e) X3@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
else memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi
∈ B) supposing
(disjoint-classrel(es;A1;X1;A2;X2) and
disjoint-classrel(es;A1;X1;A3;X3) and
disjoint-classrel(es;A2;X2;A3;X3) and
single-valued-classrel(es;X1;A1) and
single-valued-classrel(es;X2;A2) and
single-valued-classrel(es;X3;A3))
BY
{ ((UnivCD THENA Auto)
THEN RW (AddrC [2;2] (UnfoldTopAbC)) 0
THEN (RWO "loop-class-memory-fun-eq" 0 THENA (Reduce 0 THEN Auto THEN ProveFunctional THEN Auto))
THEN (Reduce 0 THEN AutoSplit)
THEN Repeat ((RWO "member-parallel-class-bool" 0 THENA Auto))
THEN Repeat ((RWO "member-eclass-eclass1" 0 THENA Auto))
THEN Repeat (AutoSplit)) }
1
1. Info : Type
2. B : Type
3. A1 : Type
4. A2 : Type
5. A3 : Type
6. init : Id ─→ B
7. tr1 : Id ─→ A1 ─→ B ─→ B
8. X1 : EClass(A1)
9. tr2 : Id ─→ A2 ─→ B ─→ B
10. X2 : EClass(A2)
11. tr3 : Id ─→ A3 ─→ B ─→ B
12. X3 : EClass(A3)
13. es : EO+(Info)
14. e : E
15. ¬↑first(e)
16. single-valued-classrel(es;X3;A3)
17. single-valued-classrel(es;X2;A2)
18. single-valued-classrel(es;X1;A1)
19. disjoint-classrel(es;A2;X2;A3;X3)
20. disjoint-classrel(es;A1;X1;A3;X3)
21. disjoint-classrel(es;A1;X1;A2;X2)
22. ↑pred(e) ∈b X1
⊢ ((tr1 o X1) || (tr2 o X2) || (tr3 o X3)(pred(e))
loop-class-memory((tr1 o X1) || (tr2 o X2) || (tr3 o X3);λloc.{init loc})(pred(e)))
= (tr1 loc(e) X1@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e)))
∈ B
2
1. Info : Type
2. B : Type
3. A1 : Type
4. A2 : Type
5. A3 : Type
6. init : Id ─→ B
7. tr1 : Id ─→ A1 ─→ B ─→ B
8. X1 : EClass(A1)
9. tr2 : Id ─→ A2 ─→ B ─→ B
10. X2 : EClass(A2)
11. tr3 : Id ─→ A3 ─→ B ─→ B
12. X3 : EClass(A3)
13. es : EO+(Info)
14. e : E
15. ¬↑pred(e) ∈b X1
16. ¬↑first(e)
17. single-valued-classrel(es;X3;A3)
18. single-valued-classrel(es;X2;A2)
19. single-valued-classrel(es;X1;A1)
20. disjoint-classrel(es;A2;X2;A3;X3)
21. disjoint-classrel(es;A1;X1;A3;X3)
22. disjoint-classrel(es;A1;X1;A2;X2)
23. ↑pred(e) ∈b X2
⊢ ((tr1 o X1) || (tr2 o X2) || (tr3 o X3)(pred(e))
loop-class-memory((tr1 o X1) || (tr2 o X2) || (tr3 o X3);λloc.{init loc})(pred(e)))
= (tr2 loc(e) X2@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e)))
∈ B
3
1. Info : Type
2. B : Type
3. A1 : Type
4. A2 : Type
5. A3 : Type
6. init : Id ─→ B
7. tr1 : Id ─→ A1 ─→ B ─→ B
8. X1 : EClass(A1)
9. tr2 : Id ─→ A2 ─→ B ─→ B
10. X2 : EClass(A2)
11. tr3 : Id ─→ A3 ─→ B ─→ B
12. X3 : EClass(A3)
13. es : EO+(Info)
14. e : E
15. ¬↑pred(e) ∈b X2
16. ¬↑pred(e) ∈b X1
17. ¬↑first(e)
18. single-valued-classrel(es;X3;A3)
19. single-valued-classrel(es;X2;A2)
20. single-valued-classrel(es;X1;A1)
21. disjoint-classrel(es;A2;X2;A3;X3)
22. disjoint-classrel(es;A1;X1;A3;X3)
23. disjoint-classrel(es;A1;X1;A2;X2)
24. ↑pred(e) ∈b X3
⊢ ((tr1 o X1) || (tr2 o X2) || (tr3 o X3)(pred(e))
loop-class-memory((tr1 o X1) || (tr2 o X2) || (tr3 o X3);λloc.{init loc})(pred(e)))
= (tr3 loc(e) X3@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e)))
∈ B
4
1. Info : Type
2. B : Type
3. A1 : Type
4. A2 : Type
5. A3 : Type
6. init : Id ─→ B
7. tr1 : Id ─→ A1 ─→ B ─→ B
8. X1 : EClass(A1)
9. tr2 : Id ─→ A2 ─→ B ─→ B
10. X2 : EClass(A2)
11. tr3 : Id ─→ A3 ─→ B ─→ B
12. X3 : EClass(A3)
13. es : EO+(Info)
14. e : E
15. ¬↑pred(e) ∈b X3
16. ¬↑pred(e) ∈b X2
17. ¬↑pred(e) ∈b X1
18. ¬↑first(e)
19. single-valued-classrel(es;X3;A3)
20. single-valued-classrel(es;X2;A2)
21. single-valued-classrel(es;X1;A1)
22. disjoint-classrel(es;A2;X2;A3;X3)
23. disjoint-classrel(es;A1;X1;A3;X3)
24. disjoint-classrel(es;A1;X1;A2;X2)
⊢ loop-class-memory((tr1 o X1) || (tr2 o X2) || (tr3 o X3);λloc.{init loc})(pred(e))
= memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
∈ B
Latex:
Latex:
\mforall{}[Info,B,A1,A2,A3:Type]. \mforall{}[init:Id {}\mrightarrow{} B]. \mforall{}[tr1:Id {}\mrightarrow{} A1 {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[X1:EClass(A1)]. \mforall{}[tr2:Id
{}\mrightarrow{} A2
{}\mrightarrow{} B
{}\mrightarrow{} B].
\mforall{}[X2:EClass(A2)]. \mforall{}[tr3:Id {}\mrightarrow{} A3 {}\mrightarrow{} B {}\mrightarrow{} B]. \mforall{}[X3:EClass(A3)]. \mforall{}[es:EO+(Info)]. \mforall{}[e:E].
(memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(e)
= if first(e) then init loc(e)
if pred(e) \mmember{}\msubb{} X1 then tr1 loc(e) X1@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
if pred(e) \mmember{}\msubb{} X2 then tr2 loc(e) X2@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
if pred(e) \mmember{}\msubb{} X3 then tr3 loc(e) X3@pred(e) memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
else memory-class3(init;tr1;X1;tr2;X2;tr3;X3)(pred(e))
fi ) supposing
(disjoint-classrel(es;A1;X1;A2;X2) and
disjoint-classrel(es;A1;X1;A3;X3) and
disjoint-classrel(es;A2;X2;A3;X3) and
single-valued-classrel(es;X1;A1) and
single-valued-classrel(es;X2;A2) and
single-valued-classrel(es;X3;A3))
By
Latex:
((UnivCD THENA Auto)
THEN RW (AddrC [2;2] (UnfoldTopAbC)) 0
THEN (RWO "loop-class-memory-fun-eq" 0 THENA (Reduce 0 THEN Auto THEN ProveFunctional THEN Auto))
THEN (Reduce 0 THEN AutoSplit)
THEN Repeat ((RWO "member-parallel-class-bool" 0 THENA Auto))
THEN Repeat ((RWO "member-eclass-eclass1" 0 THENA Auto))
THEN Repeat (AutoSplit))
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