Step
*
3
1
2
2
2
1
1
1
of Lemma
b-almost-full-intersection
1. R : ℕ ⟶ ℕ ⟶ ℙ
2. T : ℕ ⟶ ℕ ⟶ ℙ
3. b-almost-full(n,m.R[n;m])
4. b-almost-full(n,m.T[n;m])
5. n : ℕ
6. a : {s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)}
7. ∀m:{m:ℕ| strictly-increasing-seq(n + 1;a.m@n)} . ∀s@0:StrictInc.
∃n@0:ℕ. (baf-bar(n,m.R[n;m];n,m.T[n;m];n + 2;a.m@n.s@0 n@0@n + 1) ∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
8. gamma : StrictInc
9. N : ℕ+
10. ∀g:ℕN ⟶ ℕN ⟶ ℕn. ∃i,j,k:ℕN. (i < j ∧ j < k ∧ ((g i j) = (g i k) ∈ ℤ) ∧ ((g i k) = (g j k) ∈ ℤ))
11. s : ℕN ⟶ ℕ
12. ∀i:ℕN. ∀j:{i + 1..N-}.
(baf-bar(n,m.R[n;m];n,m.T[n;m];n + 2;a.gamma (s i)@n.gamma (s j)@n + 1)
∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
⊢ ∃n@0:ℕ. (baf-bar(n,m.R[n;m];n,m.T[n;m];n + 1;a.gamma n@0@n) ∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
BY
{ TACTIC:(RenameVar `h' (-1) THEN (Decide ∃i:ℕN. ∃j:{i + 1..N-}. (↑isr(h i j)) THENA Auto)) }
1
1. R : ℕ ⟶ ℕ ⟶ ℙ
2. T : ℕ ⟶ ℕ ⟶ ℙ
3. b-almost-full(n,m.R[n;m])
4. b-almost-full(n,m.T[n;m])
5. n : ℕ
6. a : {s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)}
7. ∀m:{m:ℕ| strictly-increasing-seq(n + 1;a.m@n)} . ∀s@0:StrictInc.
∃n@0:ℕ. (baf-bar(n,m.R[n;m];n,m.T[n;m];n + 2;a.m@n.s@0 n@0@n + 1) ∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
8. gamma : StrictInc
9. N : ℕ+
10. ∀g:ℕN ⟶ ℕN ⟶ ℕn. ∃i,j,k:ℕN. (i < j ∧ j < k ∧ ((g i j) = (g i k) ∈ ℤ) ∧ ((g i k) = (g j k) ∈ ℤ))
11. s : ℕN ⟶ ℕ
12. h : ∀i:ℕN. ∀j:{i + 1..N-}.
(baf-bar(n,m.R[n;m];n,m.T[n;m];n + 2;a.gamma (s i)@n.gamma (s j)@n + 1)
∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
13. ∃i:ℕN. ∃j:{i + 1..N-}. (↑isr(h i j))
⊢ ∃n@0:ℕ. (baf-bar(n,m.R[n;m];n,m.T[n;m];n + 1;a.gamma n@0@n) ∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
2
1. R : ℕ ⟶ ℕ ⟶ ℙ
2. T : ℕ ⟶ ℕ ⟶ ℙ
3. b-almost-full(n,m.R[n;m])
4. b-almost-full(n,m.T[n;m])
5. n : ℕ
6. a : {s:ℕn ⟶ ℕ| strictly-increasing-seq(n;s)}
7. ∀m:{m:ℕ| strictly-increasing-seq(n + 1;a.m@n)} . ∀s@0:StrictInc.
∃n@0:ℕ. (baf-bar(n,m.R[n;m];n,m.T[n;m];n + 2;a.m@n.s@0 n@0@n + 1) ∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
8. gamma : StrictInc
9. N : ℕ+
10. ∀g:ℕN ⟶ ℕN ⟶ ℕn. ∃i,j,k:ℕN. (i < j ∧ j < k ∧ ((g i j) = (g i k) ∈ ℤ) ∧ ((g i k) = (g j k) ∈ ℤ))
11. s : ℕN ⟶ ℕ
12. h : ∀i:ℕN. ∀j:{i + 1..N-}.
(baf-bar(n,m.R[n;m];n,m.T[n;m];n + 2;a.gamma (s i)@n.gamma (s j)@n + 1)
∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
13. ¬(∃i:ℕN. ∃j:{i + 1..N-}. (↑isr(h i j)))
⊢ ∃n@0:ℕ. (baf-bar(n,m.R[n;m];n,m.T[n;m];n + 1;a.gamma n@0@n) ∨ (∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
Latex:
Latex:
1. R : \mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}
2. T : \mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}
3. b-almost-full(n,m.R[n;m])
4. b-almost-full(n,m.T[n;m])
5. n : \mBbbN{}
6. a : \{s:\mBbbN{}n {}\mrightarrow{} \mBbbN{}| strictly-increasing-seq(n;s)\}
7. \mforall{}m:\{m:\mBbbN{}| strictly-increasing-seq(n + 1;a.m@n)\} . \mforall{}s@0:StrictInc.
\mexists{}n@0:\mBbbN{}
(baf-bar(n,m.R[n;m];n,m.T[n;m];n + 2;a.m@n.s@0 n@0@n + 1)
\mvee{} (\mexists{}n:\mBbbN{}. \mexists{}m:\{n + 1...\}. (R[n;m] \mwedge{} T[n;m])))
8. gamma : StrictInc
9. N : \mBbbN{}\msupplus{}
10. \mforall{}g:\mBbbN{}N {}\mrightarrow{} \mBbbN{}N {}\mrightarrow{} \mBbbN{}n. \mexists{}i,j,k:\mBbbN{}N. (i < j \mwedge{} j < k \mwedge{} ((g i j) = (g i k)) \mwedge{} ((g i k) = (g j k)))
11. s : \mBbbN{}N {}\mrightarrow{} \mBbbN{}
12. \mforall{}i:\mBbbN{}N. \mforall{}j:\{i + 1..N\msupminus{}\}.
(baf-bar(n,m.R[n;m];n,m.T[n;m];n + 2;a.gamma (s i)@n.gamma (s j)@n + 1)
\mvee{} (\mexists{}n:\mBbbN{}. \mexists{}m:\{n + 1...\}. (R[n;m] \mwedge{} T[n;m])))
\mvdash{} \mexists{}n@0:\mBbbN{}
(baf-bar(n,m.R[n;m];n,m.T[n;m];n + 1;a.gamma n@0@n) \mvee{} (\mexists{}n:\mBbbN{}. \mexists{}m:\{n + 1...\}. (R[n;m] \mwedge{} T[n;m])))
By
Latex:
TACTIC:(RenameVar `h' (-1) THEN (Decide \mexists{}i:\mBbbN{}N. \mexists{}j:\{i + 1..N\msupminus{}\}. (\muparrow{}isr(h i j)) THENA Auto))
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