Step
*
1
2
1
2
2
2
of Lemma
satisfies-gcd-reduce-eq-constraints
1. n : ℕ+
2. v : ℤ List
3. ||[1 / v]|| = n ∈ ℤ
4. u : ℤ
5. v2 : ℤ List
6. ||[u / v2]|| = n ∈ ℤ
7. v1 : {L:ℤ List| ||L|| = n ∈ ℤ} List
8. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List
((∀as∈sat.[1 / v] ⋅ as =0)
⇒ (∀as∈v1.[1 / v] ⋅ as =0)
⇒ ((↑isl(gcd-reduce-eq-constraints(sat;v1))) ∧ (∀as∈outl(gcd-reduce-eq-constraints(sat;v1)).[1 / v] ⋅ as =0)))
9. sat : {L:ℤ List| ||L|| = n ∈ ℤ} List
10. (∀as∈sat.[1 / v] ⋅ as =0)
11. (∀as∈[[u / v2] / v1].[1 / v] ⋅ as =0)
12. ¬↑null(v2)
⊢ (↑isl(let s' ⟵ eval g = |gcd-list(v2)| in
if (1) < (g)
then if u rem g=0
then eval L' = eval x = u ÷ g in
eval r = eager-map(λx.(x ÷ g);v2) in
[x / r] in
inl [L' / sat]
else (inr ⋅ )
else (inl [[u / v2] / sat])
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v1)))
∧ (∀as∈outl(let s' ⟵ eval g = |gcd-list(v2)| in
if (1) < (g)
then if u rem g=0
then eval L' = eval x = u ÷ g in
eval r = eager-map(λx.(x ÷ g);v2) in
[x / r] in
inl [L' / sat]
else (inr ⋅ )
else (inl [[u / v2] / sat])
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v1)).[1 / v] ⋅ as =0)
BY
{ ((GenConcl ⌜|gcd-list(v2)| = gg ∈ ℤ⌝⋅ THENA Auto)
THEN (CallByValueReduceOn ⌜gg⌝ 0⋅ THENA Auto)
THEN (Decide 1 < gg THENA Auto)
THEN (Reduce 0 THENA Auto)) }
1
1. n : ℕ+
2. v : ℤ List
3. ||[1 / v]|| = n ∈ ℤ
4. u : ℤ
5. v2 : ℤ List
6. ||[u / v2]|| = n ∈ ℤ
7. v1 : {L:ℤ List| ||L|| = n ∈ ℤ} List
8. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List
((∀as∈sat.[1 / v] ⋅ as =0)
⇒ (∀as∈v1.[1 / v] ⋅ as =0)
⇒ ((↑isl(gcd-reduce-eq-constraints(sat;v1))) ∧ (∀as∈outl(gcd-reduce-eq-constraints(sat;v1)).[1 / v] ⋅ as =0)))
9. sat : {L:ℤ List| ||L|| = n ∈ ℤ} List
10. (∀as∈sat.[1 / v] ⋅ as =0)
11. (∀as∈[[u / v2] / v1].[1 / v] ⋅ as =0)
12. ¬↑null(v2)
13. gg : ℤ
14. |gcd-list(v2)| = gg ∈ ℤ
15. 1 < gg
⊢ (↑isl(let s' ⟵ if u rem gg=0
then eval L' = eval x = u ÷ gg in
eval r = eager-map(λx.(x ÷ gg);v2) in
[x / r] in
inl [L' / sat]
else (inr ⋅ )
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v1)))
∧ (∀as∈outl(let s' ⟵ if u rem gg=0
then eval L' = eval x = u ÷ gg in
eval r = eager-map(λx.(x ÷ gg);v2) in
[x / r] in
inl [L' / sat]
else (inr ⋅ )
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v1)).[1 / v] ⋅ as =0)
2
1. n : ℕ+
2. v : ℤ List
3. ||[1 / v]|| = n ∈ ℤ
4. u : ℤ
5. v2 : ℤ List
6. ||[u / v2]|| = n ∈ ℤ
7. v1 : {L:ℤ List| ||L|| = n ∈ ℤ} List
8. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List
((∀as∈sat.[1 / v] ⋅ as =0)
⇒ (∀as∈v1.[1 / v] ⋅ as =0)
⇒ ((↑isl(gcd-reduce-eq-constraints(sat;v1))) ∧ (∀as∈outl(gcd-reduce-eq-constraints(sat;v1)).[1 / v] ⋅ as =0)))
9. sat : {L:ℤ List| ||L|| = n ∈ ℤ} List
10. (∀as∈sat.[1 / v] ⋅ as =0)
11. (∀as∈[[u / v2] / v1].[1 / v] ⋅ as =0)
12. ¬↑null(v2)
13. gg : ℤ
14. |gcd-list(v2)| = gg ∈ ℤ
15. ¬1 < gg
⊢ (↑isl(let s' ⟵ inl [[u / v2] / sat]
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v1)))
∧ (∀as∈outl(let s' ⟵ inl [[u / v2] / sat]
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v1)).[1 / v] ⋅ as =0)
Latex:
Latex:
1. n : \mBbbN{}\msupplus{}
2. v : \mBbbZ{} List
3. ||[1 / v]|| = n
4. u : \mBbbZ{}
5. v2 : \mBbbZ{} List
6. ||[u / v2]|| = n
7. v1 : \{L:\mBbbZ{} List| ||L|| = n\} List
8. \mforall{}sat:\{L:\mBbbZ{} List| ||L|| = n\} List
((\mforall{}as\mmember{}sat.[1 / v] \mcdot{} as =0)
{}\mRightarrow{} (\mforall{}as\mmember{}v1.[1 / v] \mcdot{} as =0)
{}\mRightarrow{} ((\muparrow{}isl(gcd-reduce-eq-constraints(sat;v1)))
\mwedge{} (\mforall{}as\mmember{}outl(gcd-reduce-eq-constraints(sat;v1)).[1 / v] \mcdot{} as =0)))
9. sat : \{L:\mBbbZ{} List| ||L|| = n\} List
10. (\mforall{}as\mmember{}sat.[1 / v] \mcdot{} as =0)
11. (\mforall{}as\mmember{}[[u / v2] / v1].[1 / v] \mcdot{} as =0)
12. \mneg{}\muparrow{}null(v2)
\mvdash{} (\muparrow{}isl(let s' \mleftarrow{}{} eval g = |gcd-list(v2)| in
if (1) < (g)
then if u rem g=0
then eval L' = eval x = u \mdiv{} g in
eval r = eager-map(\mlambda{}x.(x \mdiv{} g);v2) in
[x / r] in
inl [L' / sat]
else (inr \mcdot{} )
else (inl [[u / v2] / sat])
in accumulate\_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr \mcdot{} )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(\mlambda{}x.(x \mdiv{} g);L) in
inl [L' / Ls]
else (inr \mcdot{} )
else (inl [L / Ls]);s';v1)))
\mwedge{} (\mforall{}as\mmember{}outl(let s' \mleftarrow{}{} eval g = |gcd-list(v2)| in
if (1) < (g)
then if u rem g=0
then eval L' = eval x = u \mdiv{} g in
eval r = eager-map(\mlambda{}x.(x \mdiv{} g);v2) in
[x / r] in
inl [L' / sat]
else (inr \mcdot{} )
else (inl [[u / v2] / sat])
in accumulate\_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr \mcdot{} )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(\mlambda{}x.(x \mdiv{} g);L) in
inl [L' / Ls]
else (inr \mcdot{} )
else (inl [L / Ls]);s';v1)).[1 / v] \mcdot{} as =0)
By
Latex:
((GenConcl \mkleeneopen{}|gcd-list(v2)| = gg\mkleeneclose{}\mcdot{} THENA Auto)
THEN (CallByValueReduceOn \mkleeneopen{}gg\mkleeneclose{} 0\mcdot{} THENA Auto)
THEN (Decide 1 < gg THENA Auto)
THEN (Reduce 0 THENA Auto))
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