Step
*
2
1
1
2
2
of Lemma
satisfies-gcd-reduce-eq-constraints
1. n : ℕ+
2. xs : {L:ℤ List| ||L|| = n ∈ ℤ}
3. 0 < ||xs|| ∧ (hd(xs) = 1 ∈ ℤ)
4. u : ℤ
5. v1 : ℤ List
6. [%4] : ||[u / v1]|| = n ∈ ℤ
7. v : {L:ℤ List| ||L|| = n ∈ ℤ} List
8. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List
((↑isl(gcd-reduce-eq-constraints(sat;v)))
⇒ (∀as∈outl(gcd-reduce-eq-constraints(sat;v)).xs ⋅ as =0)
⇒ (∀as∈v.xs ⋅ as =0))
9. sat : {L:ℤ List| ||L|| = n ∈ ℤ} List
⊢ (↑isl(gcd-reduce-eq-constraints(sat;[[u / v1] / v])))
⇒ (∀as∈outl(gcd-reduce-eq-constraints(sat;[[u / v1] / v])).xs ⋅ as =0)
⇒ (∀as∈[[u / v1] / v].xs ⋅ as =0)
BY
{ (Unfold `gcd-reduce-eq-constraints` 0
THEN (RWW "accumulate_abort_cons_lemma" 0 THENA Auto)
THEN Reduce 0
THEN (Decide ⌜↑null(v1)⌝⋅ THENA Auto)
THEN RW (SweepDnC IsAxiomC) 0) }
1
1. n : ℕ+
2. xs : {L:ℤ List| ||L|| = n ∈ ℤ}
3. 0 < ||xs|| ∧ (hd(xs) = 1 ∈ ℤ)
4. u : ℤ
5. v1 : ℤ List
6. [%4] : ||[u / v1]|| = n ∈ ℤ
7. v : {L:ℤ List| ||L|| = n ∈ ℤ} List
8. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List
((↑isl(gcd-reduce-eq-constraints(sat;v)))
⇒ (∀as∈outl(gcd-reduce-eq-constraints(sat;v)).xs ⋅ as =0)
⇒ (∀as∈v.xs ⋅ as =0))
9. sat : {L:ℤ List| ||L|| = n ∈ ℤ} List
10. ↑null(v1)
⊢ (↑isl(let s' ⟵ if u=0 then inl [[u / v1] / 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';v)))
⇒ (∀as∈outl(let s' ⟵ if u=0 then inl [[u / v1] / 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';v)).xs ⋅ as =0)
⇒ (∀as∈[[u / v1] / v].xs ⋅ as =0)
2
1. n : ℕ+
2. xs : {L:ℤ List| ||L|| = n ∈ ℤ}
3. 0 < ||xs|| ∧ (hd(xs) = 1 ∈ ℤ)
4. u : ℤ
5. v1 : ℤ List
6. [%4] : ||[u / v1]|| = n ∈ ℤ
7. v : {L:ℤ List| ||L|| = n ∈ ℤ} List
8. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List
((↑isl(gcd-reduce-eq-constraints(sat;v)))
⇒ (∀as∈outl(gcd-reduce-eq-constraints(sat;v)).xs ⋅ as =0)
⇒ (∀as∈v.xs ⋅ as =0))
9. sat : {L:ℤ List| ||L|| = n ∈ ℤ} List
10. ¬↑null(v1)
⊢ (↑isl(let s' ⟵ eval g = |gcd-list(v1)| 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);v1) in
[x / r] in
inl [L' / sat]
else (inr ⋅ )
else (inl [[u / v1] / 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';v)))
⇒ (∀as∈outl(let s' ⟵ eval g = |gcd-list(v1)| 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);v1) in
[x / r] in
inl [L' / sat]
else (inr ⋅ )
else (inl [[u / v1] / 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';v)).xs ⋅ as =0)
⇒ (∀as∈[[u / v1] / v].xs ⋅ as =0)
Latex:
Latex:
1. n : \mBbbN{}\msupplus{}
2. xs : \{L:\mBbbZ{} List| ||L|| = n\}
3. 0 < ||xs|| \mwedge{} (hd(xs) = 1)
4. u : \mBbbZ{}
5. v1 : \mBbbZ{} List
6. [\%4] : ||[u / v1]|| = n
7. v : \{L:\mBbbZ{} List| ||L|| = n\} List
8. \mforall{}sat:\{L:\mBbbZ{} List| ||L|| = n\} List
((\muparrow{}isl(gcd-reduce-eq-constraints(sat;v)))
{}\mRightarrow{} (\mforall{}as\mmember{}outl(gcd-reduce-eq-constraints(sat;v)).xs \mcdot{} as =0)
{}\mRightarrow{} (\mforall{}as\mmember{}v.xs \mcdot{} as =0))
9. sat : \{L:\mBbbZ{} List| ||L|| = n\} List
\mvdash{} (\muparrow{}isl(gcd-reduce-eq-constraints(sat;[[u / v1] / v])))
{}\mRightarrow{} (\mforall{}as\mmember{}outl(gcd-reduce-eq-constraints(sat;[[u / v1] / v])).xs \mcdot{} as =0)
{}\mRightarrow{} (\mforall{}as\mmember{}[[u / v1] / v].xs \mcdot{} as =0)
By
Latex:
(Unfold `gcd-reduce-eq-constraints` 0
THEN (RWW "accumulate\_abort\_cons\_lemma" 0 THENA Auto)
THEN Reduce 0
THEN (Decide \mkleeneopen{}\muparrow{}null(v1)\mkleeneclose{}\mcdot{} THENA Auto)
THEN RW (SweepDnC IsAxiomC) 0)
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