| import OAI.Optimization.Tropical.TruffetModel |
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| namespace OAI |
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| /-! A forced pair of tropical substitutions with a strictly better feasible point. -/ |
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| noncomputable section |
| namespace TruffetV4Counterexample |
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| private lemma fin3_two_ne_one : (2 : Fin 3) ≠ 1 := by decide |
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| private lemma fin3_two_ne_zero : (2 : Fin 3) ≠ 0 := by decide |
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| private lemma coeff_one_add_neg_one : (1 : Coeff) + ((-1 : ℝ) : Coeff) = 0 := by |
| change (((1 : ℝ) + (-1) : ℝ) : Coeff) = (0 : Coeff) |
| norm_num |
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| theorem Row.ext {a b : Row} (hl : a.lhs = b.lhs) (hr : a.rhs = b.rhs) : a = b := by |
| cases a |
| cases b |
| cases hl |
| cases hr |
| rfl |
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| theorem State.ext {a b : State} (hr : a.rows = b.rows) (hc : a.cost = b.cost) |
| (hv : a.remaining = b.remaining) : a = b := by |
| cases a |
| cases b |
| cases hr |
| cases hc |
| cases hv |
| rfl |
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| lemma initial_crossRowRestriction : crossRowRestriction initial := by |
| intro i r hir α |
| fin_cases i <;> fin_cases r |
| · exact (hir rfl).elim |
| · refine ⟨![0, 0, 3-α], ?_⟩ |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, eval, initial] |
| have hreal : (0 : ℝ) < α + max 1 (-2 + (3-α)) := by |
| linarith [le_max_right (1 : ℝ) (-2 + (3-α))] |
| exact WithBot.coe_lt_coe.mpr hreal |
| · refine ⟨![0, 0, 1-α], ?_⟩ |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, eval, initial] |
| · exact (hir rfl).elim |
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| lemma advance_first : advance initial 0 1 = middle := by |
| classical |
| apply State.ext |
| · funext i |
| apply Row.ext <;> funext k <;> fin_cases i <;> fin_cases k <;> |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, advance, initial, middle, cleanup, substitute, lowerForm, column, |
| nullForm, Fin.forall_fin_succ] |
| · funext k |
| fin_cases k <;> norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, advance, initial, middle, substitute, lowerForm, column] |
| · funext j |
| apply propext |
| fin_cases j <;> simp [advance, initial, middle] |
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| lemma advance_second : advance middle 1 0 = finalState := by |
| classical |
| apply State.ext |
| · funext i |
| apply Row.ext <;> funext k <;> fin_cases i <;> fin_cases k <;> |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, advance, middle, finalState, cleanup, substitute, lowerForm, column, |
| nullForm, Fin.forall_fin_succ] |
| · funext k |
| fin_cases k <;> norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, advance, middle, finalState, substitute, lowerForm, column] |
| · funext j |
| apply propext |
| fin_cases j <;> simp [advance, middle, finalState] |
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| lemma first_forced : ForcedSingletonStep initial middle 0 1 := by |
| classical |
| unfold ForcedSingletonStep |
| refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, advance_first.symm⟩ |
| all_goals |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, wellFormed, bounded, stop, switch, failure, eligible, dominating, lower, upper, |
| initial, middle, finalState, nullForm, substitute, lowerForm, column, Fin.forall_fin_succ, Fin.exists_fin_succ] |
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| lemma second_forced : ForcedSingletonStep middle finalState 1 0 := by |
| classical |
| unfold ForcedSingletonStep |
| refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, advance_second.symm⟩ |
| all_goals |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, wellFormed, bounded, stop, switch, failure, eligible, dominating, lower, upper, |
| middle, finalState, nullForm, substitute, lowerForm, column, Fin.forall_fin_succ, |
| Fin.exists_fin_succ] |
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| lemma final_terminal : (¬ ∃ j, finalState.remaining j) ∧ stop finalState := by |
| simp [finalState, stop, nullForm] |
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| lemma finite_search_domain : |
| (∀ j : Var, (⊥ : Coeff) ≤ (output (column j) : Coeff)) ∧ |
| (∀ j : Var, (⊥ : Coeff) ≤ (better (column j) : Coeff)) ∧ |
| (0 : ℝ) ≤ output 2 ∧ (0 : ℝ) ≤ better 2 ∧ |
| (⊥ : Coeff) < (output 2 : Coeff) ∧ (⊥ : Coeff) < (better 2 : Coeff) := by |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, output, better] |
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| lemma output_backsubstitution : |
| (output 1 : Coeff) = eval (lowerForm initial 0 1) output ∧ |
| (output 0 : Coeff) = eval (lowerForm middle 1 0) output ∧ output 2 = 0 := by |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, output, eval, lowerForm, initial, middle, column] |
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| lemma output_cost_and_feasible : feasible initial output ∧ |
| eval initial.cost output = (1 : Coeff) ∧ |
| eval finalState.cost output = (1 : Coeff) := by |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, feasible, initial, finalState, output, eval, Fin.forall_fin_succ] |
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| lemma better_feasible_and_cost : feasible initial better ∧ |
| eval initial.cost better = (0 : Coeff) ∧ better 2 = 0 := by |
| norm_num [fin3_two_ne_one, fin3_two_ne_zero, coeff_one_add_neg_one, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail, |
| ← WithBot.coe_one, ← WithBot.coe_add, feasible, initial, better, eval, Fin.forall_fin_succ] |
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| theorem finite_execution_not_optimal : |
| crossRowRestriction initial ∧ |
| ForcedSingletonStep initial middle 0 1 ∧ |
| ForcedSingletonStep middle finalState 1 0 ∧ |
| (¬ ∃ j, finalState.remaining j) ∧ stop finalState ∧ |
| feasible initial output ∧ feasible initial better ∧ |
| output 2 = 0 ∧ better 2 = 0 ∧ |
| (output 1 : Coeff) = eval (lowerForm initial 0 1) output ∧ |
| (output 0 : Coeff) = eval (lowerForm middle 1 0) output ∧ |
| eval initial.cost output = eval finalState.cost output ∧ |
| eval initial.cost better < eval finalState.cost output := by |
| refine ⟨initial_crossRowRestriction, first_forced, second_forced, |
| final_terminal.1, final_terminal.2, output_cost_and_feasible.1, |
| better_feasible_and_cost.1, rfl, rfl, output_backsubstitution.1, |
| output_backsubstitution.2.1, ?_, ?_⟩ |
| · rw [output_cost_and_feasible.2.1, output_cost_and_feasible.2.2] |
| · rw [better_feasible_and_cost.2.1, output_cost_and_feasible.2.2] |
| norm_num |
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| end TruffetV4Counterexample |
| end |
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| end OAI |
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