Step
*
2
1
1
of Lemma
priority-select-tt
1. T : Type
2. as : T List
3. f : T ⟶ 𝔹
4. g : T ⟶ 𝔹
5. T ⊆r ℤ
6. ∀i:ℕ||as||. ∀j:ℕi. (as[j] ≤ as[i])
7. no_repeats(T;as)
8. (priority-select(f;g;as) = (inl tt) ∈ (𝔹?))
⇒ (∃i:ℕ||as||. ((↑(f as[i])) ∧ (∀j:ℕi. (¬↑(g as[j])))))
9. (priority-select(f;g;as) = (inl tt) ∈ (𝔹?))
⇐ ∃i:ℕ||as||. ((↑(f as[i])) ∧ (∀j:ℕi. (¬↑(g as[j]))))
10. i : ℕ||as||
11. ↑(f as[i])
12. ∀b:T. ((b ∈ as)
⇒ ¬↑(g b) supposing b < as[i])
13. ↑(f as[i])
14. j : ℕi
⊢ as[j] < as[i]
BY
{ (Decide ⌜i = j ∈ ℤ⌝⋅ THENA Auto) }
1
1. T : Type
2. as : T List
3. f : T ⟶ 𝔹
4. g : T ⟶ 𝔹
5. T ⊆r ℤ
6. ∀i:ℕ||as||. ∀j:ℕi. (as[j] ≤ as[i])
7. no_repeats(T;as)
8. (priority-select(f;g;as) = (inl tt) ∈ (𝔹?))
⇒ (∃i:ℕ||as||. ((↑(f as[i])) ∧ (∀j:ℕi. (¬↑(g as[j])))))
9. (priority-select(f;g;as) = (inl tt) ∈ (𝔹?))
⇐ ∃i:ℕ||as||. ((↑(f as[i])) ∧ (∀j:ℕi. (¬↑(g as[j]))))
10. i : ℕ||as||
11. ↑(f as[i])
12. ∀b:T. ((b ∈ as)
⇒ ¬↑(g b) supposing b < as[i])
13. ↑(f as[i])
14. j : ℕi
15. i = j ∈ ℤ
⊢ as[j] < as[i]
2
1. T : Type
2. as : T List
3. f : T ⟶ 𝔹
4. g : T ⟶ 𝔹
5. T ⊆r ℤ
6. ∀i:ℕ||as||. ∀j:ℕi. (as[j] ≤ as[i])
7. no_repeats(T;as)
8. (priority-select(f;g;as) = (inl tt) ∈ (𝔹?))
⇒ (∃i:ℕ||as||. ((↑(f as[i])) ∧ (∀j:ℕi. (¬↑(g as[j])))))
9. (priority-select(f;g;as) = (inl tt) ∈ (𝔹?))
⇐ ∃i:ℕ||as||. ((↑(f as[i])) ∧ (∀j:ℕi. (¬↑(g as[j]))))
10. i : ℕ||as||
11. ↑(f as[i])
12. ∀b:T. ((b ∈ as)
⇒ ¬↑(g b) supposing b < as[i])
13. ↑(f as[i])
14. j : ℕi
15. ¬(i = j ∈ ℤ)
⊢ as[j] < as[i]
Latex:
Latex:
1. T : Type
2. as : T List
3. f : T {}\mrightarrow{} \mBbbB{}
4. g : T {}\mrightarrow{} \mBbbB{}
5. T \msubseteq{}r \mBbbZ{}
6. \mforall{}i:\mBbbN{}||as||. \mforall{}j:\mBbbN{}i. (as[j] \mleq{} as[i])
7. no\_repeats(T;as)
8. (priority-select(f;g;as) = (inl tt)) {}\mRightarrow{} (\mexists{}i:\mBbbN{}||as||. ((\muparrow{}(f as[i])) \mwedge{} (\mforall{}j:\mBbbN{}i. (\mneg{}\muparrow{}(g as[j])))))
9. (priority-select(f;g;as) = (inl tt)) \mLeftarrow{}{} \mexists{}i:\mBbbN{}||as||. ((\muparrow{}(f as[i])) \mwedge{} (\mforall{}j:\mBbbN{}i. (\mneg{}\muparrow{}(g as[j]))))
10. i : \mBbbN{}||as||
11. \muparrow{}(f as[i])
12. \mforall{}b:T. ((b \mmember{} as) {}\mRightarrow{} \mneg{}\muparrow{}(g b) supposing b < as[i])
13. \muparrow{}(f as[i])
14. j : \mBbbN{}i
\mvdash{} as[j] < as[i]
By
Latex:
(Decide \mkleeneopen{}i = j\mkleeneclose{}\mcdot{} THENA Auto)
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