Step
*
1
1
2
1
1
of Lemma
unsat-omega_start
1. n : ℕ
2. u : {L:ℤ List| ||L|| = (n + 1) ∈ ℤ}
3. v : {L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List
4. ineqs : {L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List
5. xs : ℤ List
6. (∀as∈[u / v].xs ⋅ as =0)
7. (∀bs∈ineqs.xs ⋅ bs ≥0)
8. ||xs|| = ||u|| ∈ ℤ
9. 0 < ||xs||
10. hd(xs) = 1 ∈ ℤ
11. u ⋅ xs = 0 ∈ ℤ
12. x : ℤ List+ List
13. gcd-reduce-eq-constraints([];[u / v]) = (inl x) ∈ (ℤ List+ List?)
14. True
15. (∀as∈x.xs ⋅ as =0)
⊢ xs |= case gcd-reduce-ineq-constraints([];ineqs) of inl(ineqs') => inl <x, ineqs'> | inr(x) => inr x
BY
{ ((InstLemma `satisfies-gcd-reduce-ineq-constraints` [⌜n + 1⌝;⌜ineqs⌝;⌜[]⌝;⌜xs⌝]⋅ THENA Auto)
THEN D -1
THEN Thin (-1)
THEN (D -1 THENA Auto)
THEN MoveToConcl (-1)
THEN (GenConclTerm ⌜gcd-reduce-ineq-constraints([];ineqs)⌝⋅ THENA Auto)
THEN (D -2 THEN Reduce 0)
THEN Auto) }
1
1. n : ℕ
2. u : {L:ℤ List| ||L|| = (n + 1) ∈ ℤ}
3. v : {L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List
4. ineqs : {L:ℤ List| ||L|| = (n + 1) ∈ ℤ} List
5. xs : ℤ List
6. (∀as∈[u / v].xs ⋅ as =0)
7. (∀bs∈ineqs.xs ⋅ bs ≥0)
8. ||xs|| = ||u|| ∈ ℤ
9. 0 < ||xs||
10. hd(xs) = 1 ∈ ℤ
11. u ⋅ xs = 0 ∈ ℤ
12. x : ℤ List+ List
13. gcd-reduce-eq-constraints([];[u / v]) = (inl x) ∈ (ℤ List+ List?)
14. True
15. (∀as∈x.xs ⋅ as =0)
16. x1 : ℤ List+ List
17. gcd-reduce-ineq-constraints([];ineqs) = (inl x1) ∈ (ℤ List+ List?)
18. True
19. (∀as∈x1.xs ⋅ as ≥0)
⊢ xs |= inl <x, x1>
Latex:
Latex:
1. n : \mBbbN{}
2. u : \{L:\mBbbZ{} List| ||L|| = (n + 1)\}
3. v : \{L:\mBbbZ{} List| ||L|| = (n + 1)\} List
4. ineqs : \{L:\mBbbZ{} List| ||L|| = (n + 1)\} List
5. xs : \mBbbZ{} List
6. (\mforall{}as\mmember{}[u / v].xs \mcdot{} as =0)
7. (\mforall{}bs\mmember{}ineqs.xs \mcdot{} bs \mgeq{}0)
8. ||xs|| = ||u||
9. 0 < ||xs||
10. hd(xs) = 1
11. u \mcdot{} xs = 0
12. x : \mBbbZ{} List\msupplus{} List
13. gcd-reduce-eq-constraints([];[u / v]) = (inl x)
14. True
15. (\mforall{}as\mmember{}x.xs \mcdot{} as =0)
\mvdash{} xs |= case gcd-reduce-ineq-constraints([];ineqs) of inl(ineqs') => inl <x, ineqs'> | inr(x) => inr\000C x
By
Latex:
((InstLemma `satisfies-gcd-reduce-ineq-constraints` [\mkleeneopen{}n + 1\mkleeneclose{};\mkleeneopen{}ineqs\mkleeneclose{};\mkleeneopen{}[]\mkleeneclose{};\mkleeneopen{}xs\mkleeneclose{}]\mcdot{} THENA Auto)
THEN D -1
THEN Thin (-1)
THEN (D -1 THENA Auto)
THEN MoveToConcl (-1)
THEN (GenConclTerm \mkleeneopen{}gcd-reduce-ineq-constraints([];ineqs)\mkleeneclose{}\mcdot{} THENA Auto)
THEN (D -2 THEN Reduce 0)
THEN Auto)
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