Step
*
4
1
1
of Lemma
State-comb-fun-eq
1. Info : Type
2. B : Type
3. A : Type
4. f : A ⟶ B ⟶ B
5. init : Id ⟶ bag(B)
6. X : EClass(A)
7. es : EO+(Info)
8. e : E
9. ¬↑first(e)
10. ¬↑e ∈b X
11. ∀l:Id. (1 ≤ #(init l))
12. ∀l:Id. single-valued-bag(init l;B)
13. single-valued-classrel(es;X;A)
14. (X es e) = {} ∈ bag(A)
15. v : (∃e':{E| ((e' <loc e)
∧ (↑((λe'.0 <z #(State-comb(init;f;X) es e')) e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑((λe'.0 <z #(State-comb(init;f;X) es e')) e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑((λe'.0 <z #(State-comb(init;f;X) es e')) e')))}))@i
16. (last(λe'.0 <z #(State-comb(init;f;X) es e')) e)
= v
∈ ((∃e':{E| ((e' <loc e)
∧ (↑((λe'.0 <z #(State-comb(init;f;X) es e')) e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑((λe'.0 <z #(State-comb(init;f;X) es e')) e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑((λe'.0 <z #(State-comb(init;f;X) es e')) e')))})))@i
⊢ sv-bag-only(case v of inl(e') => State-comb(init;f;X) es e' | inr(x) => init loc(e))
= sv-bag-only(State-comb(init;f;X) es pred(e))
∈ B
BY
{ (Reduce (-2)
THEN DVar `v'
THEN ExRepD
THEN Reduce 0
THEN RecUnfold `es-local-pred` (-1)
THEN Reduce (-1)
THEN SplitOnHypITE (-1)
THEN Auto
THEN (SplitOnHypITE (-2) THENA (Auto THEN Try (Fold `eclass` 0) THEN Auto))) }
1
.....truecase.....
1. Info : Type
2. B : Type
3. A : Type
4. f : A ⟶ B ⟶ B
5. init : Id ⟶ bag(B)
6. X : EClass(A)
7. es : EO+(Info)
8. e : E
9. ¬↑first(e)
10. ¬↑e ∈b X
11. ∀l:Id. (1 ≤ #(init l))
12. ∀l:Id. single-valued-bag(init l;B)
13. single-valued-classrel(es;X;A)
14. (X es e) = {} ∈ bag(A)
15. x : E@i
16. (x <loc e)@i
17. ↑0 <z #(State-comb(init;f;X) es x)@i
18. ∀e'':E. ((x <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(State-comb(init;f;X) es e'')))@i
19. (inl pred(e))
= (inl x)
∈ ((∃e':{E| ((e' <loc e)
∧ (↑0 <z #(State-comb(init;f;X) es e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(State-comb(init;f;X) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(State-comb(init;f;X) es e')))})))@i
20. ¬↑first(e)
21. 0 < #(State-comb(init;f;X) es pred(e))
⊢ sv-bag-only(State-comb(init;f;X) es x) = sv-bag-only(State-comb(init;f;X) es pred(e)) ∈ B
2
.....falsecase.....
1. Info : Type
2. B : Type
3. A : Type
4. f : A ⟶ B ⟶ B
5. init : Id ⟶ bag(B)
6. X : EClass(A)
7. es : EO+(Info)
8. e : E
9. ¬↑first(e)
10. ¬↑e ∈b X
11. ∀l:Id. (1 ≤ #(init l))
12. ∀l:Id. single-valued-bag(init l;B)
13. single-valued-classrel(es;X;A)
14. (X es e) = {} ∈ bag(A)
15. x : E@i
16. (x <loc e)@i
17. ↑0 <z #(State-comb(init;f;X) es x)@i
18. ∀e'':E. ((x <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(State-comb(init;f;X) es e'')))@i
19. (last(λe'.0 <z #(State-comb(init;f;X) es e')) pred(e))
= (inl x)
∈ ((∃e':{E| ((e' <loc e)
∧ (↑0 <z #(State-comb(init;f;X) es e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(State-comb(init;f;X) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(State-comb(init;f;X) es e')))})))@i
20. ¬↑first(e)
21. ¬0 < #(State-comb(init;f;X) es pred(e))
⊢ sv-bag-only(State-comb(init;f;X) es x) = sv-bag-only(State-comb(init;f;X) es pred(e)) ∈ B
3
.....truecase.....
1. Info : Type
2. B : Type
3. A : Type
4. f : A ⟶ B ⟶ B
5. init : Id ⟶ bag(B)
6. X : EClass(A)
7. es : EO+(Info)
8. e : E
9. ¬↑first(e)
10. ¬↑e ∈b X
11. ∀l:Id. (1 ≤ #(init l))
12. ∀l:Id. single-valued-bag(init l;B)
13. single-valued-classrel(es;X;A)
14. (X es e) = {} ∈ bag(A)
15. y : ¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(State-comb(init;f;X) es e')))})@i
16. (inl pred(e))
= (inr y )
∈ ((∃e':{E| ((e' <loc e)
∧ (↑0 <z #(State-comb(init;f;X) es e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(State-comb(init;f;X) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(State-comb(init;f;X) es e')))})))@i
17. ¬↑first(e)
18. 0 < #(State-comb(init;f;X) es pred(e))
⊢ sv-bag-only(init loc(e)) = sv-bag-only(State-comb(init;f;X) es pred(e)) ∈ B
4
.....falsecase.....
1. Info : Type
2. B : Type
3. A : Type
4. f : A ⟶ B ⟶ B
5. init : Id ⟶ bag(B)
6. X : EClass(A)
7. es : EO+(Info)
8. e : E
9. ¬↑first(e)
10. ¬↑e ∈b X
11. ∀l:Id. (1 ≤ #(init l))
12. ∀l:Id. single-valued-bag(init l;B)
13. single-valued-classrel(es;X;A)
14. (X es e) = {} ∈ bag(A)
15. y : ¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(State-comb(init;f;X) es e')))})@i
16. (last(λe'.0 <z #(State-comb(init;f;X) es e')) pred(e))
= (inr y )
∈ ((∃e':{E| ((e' <loc e)
∧ (↑0 <z #(State-comb(init;f;X) es e'))
∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ (¬↑0 <z #(State-comb(init;f;X) es e'')))))})
∨ (¬(∃e':{E| ((e' <loc e) ∧ (↑0 <z #(State-comb(init;f;X) es e')))})))@i
17. ¬↑first(e)
18. ¬0 < #(State-comb(init;f;X) es pred(e))
⊢ sv-bag-only(init loc(e)) = sv-bag-only(State-comb(init;f;X) es pred(e)) ∈ B
Latex:
Latex:
1. Info : Type
2. B : Type
3. A : Type
4. f : A {}\mrightarrow{} B {}\mrightarrow{} B
5. init : Id {}\mrightarrow{} bag(B)
6. X : EClass(A)
7. es : EO+(Info)
8. e : E
9. \mneg{}\muparrow{}first(e)
10. \mneg{}\muparrow{}e \mmember{}\msubb{} X
11. \mforall{}l:Id. (1 \mleq{} \#(init l))
12. \mforall{}l:Id. single-valued-bag(init l;B)
13. single-valued-classrel(es;X;A)
14. (X es e) = \{\}
15. v : (\mexists{}e':\{E| ((e' <loc e)
\mwedge{} (\muparrow{}((\mlambda{}e'.0 <z \#(State-comb(init;f;X) es e')) e'))
\mwedge{} (\mforall{}e'':E
((e' <loc e'')
{}\mRightarrow{} (e'' <loc e)
{}\mRightarrow{} (\mneg{}\muparrow{}((\mlambda{}e'.0 <z \#(State-comb(init;f;X) es e')) e'')))))\})
\mvee{} (\mneg{}(\mexists{}e':\{E| ((e' <loc e) \mwedge{} (\muparrow{}((\mlambda{}e'.0 <z \#(State-comb(init;f;X) es e')) e')))\}))@i
16. (last(\mlambda{}e'.0 <z \#(State-comb(init;f;X) es e')) e) = v@i
\mvdash{} sv-bag-only(case v of inl(e') => State-comb(init;f;X) es e' | inr(x) => init loc(e))
= sv-bag-only(State-comb(init;f;X) es pred(e))
By
Latex:
(Reduce (-2)
THEN DVar `v'
THEN ExRepD
THEN Reduce 0
THEN RecUnfold `es-local-pred` (-1)
THEN Reduce (-1)
THEN SplitOnHypITE (-1)
THEN Auto
THEN (SplitOnHypITE (-2) THENA (Auto THEN Try (Fold `eclass` 0) THEN Auto)))
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