| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Entropy.TerminalRates |
| import OAI.LinearAlgebra.MatrixFields.Parameters.ParametersHalfComplement |
| import OAI.LinearAlgebra.MatrixFields.Tensors.CWZeroGeometry |
|
|
| namespace OAI |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| noncomputable section |
| namespace MatrixMultiplication.AllFieldZeroWords |
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory |
| open AllFieldTerminalStatisticLaws AllFieldTerminalRates CWWindowedLeaves |
| open CWCompleteStatistics InheritedMasks |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} |
|
|
| def length (h : TerminalZero K) : ℕ := currentLength (history h).1 |
| def shape (h : TerminalZero K) : Shape := currentShape (history h).1 |
|
|
| def statistic : (h : TerminalZero K) → Raw (length h) → Slot h |
| | .inl _, w => fourStatistic w |
| | .inr _, w => twoStatistic w |
|
|
| def complement : (h : TerminalZero K) → Slot h ≃ Slot h |
| | .inl _ => orderedComplementStatistic |
| | .inr _ => complementStatistic |
|
|
| theorem statistic_complement (h : TerminalZero K) (w : Raw (length h)) : |
| statistic h (complementWord w) = complement h (statistic h w) := by |
| cases h with |
| | inl h => exact four_complement w |
| | inr h => exact two_complement w |
|
|
| def words (allocation : Allocation) (m : ℕ) (h : TerminalZero K) |
| (w : ZeroWords allocation m h) : |
| Fin (population allocation (terminalDilation K m) (history h)) → |
| Alphabet (length h) (shapeMax (shape h)) := by |
| cases h with |
| | inl h => |
| exact fun i => w.val ((finCongr (statisticCounts_repeat_sum allocation m (.inl h))).symm i) |
| | inr h => |
| exact fun i => w.val ((finCongr (statisticCounts_repeat_sum allocation m (.inr h))).symm i) |
|
|
| theorem words_injective (allocation : Allocation) (m : ℕ) (h : TerminalZero K) : |
| Function.Injective (words allocation m h) := by |
| cases h with |
| | inl h => |
| intro x y he |
| apply Subtype.ext |
| exact (finCongr (statisticCounts_repeat_sum allocation m (.inl h))).symm.surjective.injective_comp_right he |
| | inr h => |
| intro x y he |
| apply Subtype.ext |
| exact (finCongr (statisticCounts_repeat_sum allocation m (.inr h))).symm.surjective.injective_comp_right he |
|
|
| theorem repeat_ratio (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| (h : TerminalZero K) (s : Slot h) : |
| ((m * statisticCounts allocation h s : ℕ) : ℝ) / |
| (∑ s, m * statisticCounts allocation h s : ℕ) = (law h s : ℝ) := by |
| have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hm |
| simp only [← Finset.mul_sum, Nat.cast_mul, mul_div_mul_left _ _ hm'] |
| exact statisticCounts_ratio allocation h s |
|
|
| private theorem exact_window_reindex {Slots Letters : Type*} |
| [Fintype Slots] [DecidableEq Slots] (stat : Letters → Slots) (counts : Slots → ℕ) |
| (word : ExactStatisticWords.Words stat counts) (ν : Slots → ℝ) (η : ℝ) |
| (hη : 0 ≤ η) (hratio : ∀ slot, (counts slot : ℝ) / (∑ slot, counts slot : ℕ) = ν slot) |
| {positions : ℕ} (reindex : Fin positions ≃ Fin (∑ slot, counts slot)) : |
| typeWindow ν η ((stat ∘ word.val) ∘ reindex) := by |
| intro slot |
| simp only [empiricalLaw, wordPopulation_reindex, word.property, |
| Fintype.card_congr reindex, Fintype.card_fin, hratio slot, sub_self, abs_zero] |
| exact hη |
|
|
| theorem selected_window (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| (h : TerminalZero K) (η : ℝ) (hη : 0 ≤ η) (w : ZeroWords allocation m h) : |
| typeWindow (fun s => (law h s : ℝ)) η |
| (fun i => statistic h (words allocation m h w i).val) := by |
| cases h with |
| | inl h => |
| change AllFieldZeroLeafRates.fourWords (aShape h.1.val) |
| (statisticCounts allocation (.inl h)) m at w |
| intro s |
| change PairSlot at s |
| have hsum := statisticCounts_repeat_sum allocation m (.inl h) |
| let reindex : Fin (population allocation (terminalDilation K m) (history (.inl h))) ≃ |
| Fin (∑ slot : PairSlot, m * statisticCounts allocation (.inl h) slot) := |
| (finCongr hsum).symm |
| have hwindow := exact_window_reindex _ _ w |
| (fun slot : PairSlot => (law (.inl h) slot : ℝ)) η hη |
| (fun slot => repeat_ratio allocation hm (.inl h) slot) reindex s |
| let : DecidableEq PairSlot := fun _ _ => Classical.propDecidable _ |
| change |empiricalLaw ((fun index => fourStatistic (w.val index).val) ∘ reindex) s - |
| (law (.inl h) s : ℝ)| ≤ η |
| simpa only [empiricalLaw, wordPopulation, Fintype.card_eq_nat_card, |
| Function.comp_def] using hwindow |
| | inr h => |
| change AllFieldZeroLeafRates.twoWords (bWeight h) |
| (statisticCounts allocation (.inr h)) m at w |
| intro s |
| change Fin 6 at s |
| have hsum := statisticCounts_repeat_sum allocation m (.inr h) |
| let reindex : Fin (population allocation (terminalDilation K m) (history (.inr h))) ≃ |
| Fin (∑ slot : Fin 6, m * statisticCounts allocation (.inr h) slot) := |
| (finCongr hsum).symm |
| have hwindow := exact_window_reindex _ _ w |
| (fun slot : Fin 6 => (law (.inr h) slot : ℝ)) η hη |
| (fun slot => repeat_ratio allocation hm (.inr h) slot) reindex s |
| let : DecidableEq (Fin 6) := fun _ _ => Classical.propDecidable _ |
| change |empiricalLaw ((fun index => twoStatistic (w.val index).val) ∘ reindex) s - |
| (law (.inr h) s : ℝ)| ≤ η |
| simpa only [empiricalLaw, wordPopulation, Fintype.card_eq_nat_card, |
| Function.comp_def] using hwindow |
|
|
| theorem complementary_window (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| (h : TerminalZero K) (η : ℝ) (hη : 0 ≤ η) (w : ZeroWords allocation m h) : |
| typeWindow ((fun s => (law h s : ℝ)) ∘ (complement h).symm) η |
| (fun i => statistic h (complementWord (words allocation m h w i).val)) := by |
| have hs := selected_window allocation hm h η hη w |
| have he : (fun i => statistic h (complementWord (words allocation m h w i).val)) = |
| complement h ∘ (fun i => statistic h (words allocation m h w i).val) := by |
| funext i |
| exact statistic_complement h _ |
| rw [he] |
| exact (typeWindow_equiv _ η _ (complement h)).mpr hs |
|
|
| end MatrixMultiplication.AllFieldZeroWords |
|
|
| namespace MatrixMultiplication.AllFieldZeroFinish |
| open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory |
| open AllFieldTerminalStatisticLaws AllFieldTerminalRates AllFieldZeroWords |
| open CWWindowedLeaves CWCompleteStatistics CWZeroGeometry CWOrientedZero |
| open AllFieldFiniteFamily InheritedMasks |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {K : ℕ} |
|
|
| def canonicalZero (h : TerminalZero K) : Fin 3 := zeroAxis (shape h) |
| def canonicalSelected : TerminalZero K → Fin 3 |
| | .inl h => halfMaxAxis (aShape h.1.val) |
| | .inr h => maxAxis (bShape h.1.val) |
| def physicalZero (h : TerminalZero K) : Fin 3 := (history h).2 (canonicalZero h) |
| def physicalSelected (h : TerminalZero K) : Fin 3 := (history h).2 (canonicalSelected h) |
|
|
| theorem shape_has_zero (h : TerminalZero K) : ∃ s, shape h s = 0 := by |
| cases h with |
| | inl h => exact not_positive_has_zero _ (aShape_not_positive h) |
| | inr h => exact not_positive_has_zero _ h.1.property |
|
|
| theorem canonicalZero_spec (h : TerminalZero K) : shape h (canonicalZero h) = 0 := |
| zeroAxis_spec (shape h) (shape_has_zero h) |
|
|
| theorem canonicalSelected_spec (h : TerminalZero K) : |
| shape h (canonicalSelected h) = shapeMax (shape h) := by |
| cases h with |
| | inl h => exact halfMaxAxis_spec _ (aShape_mem_shapes h) |
| | inr h => exact maxAxis_spec _ |
|
|
| theorem shape_total (h : TerminalZero K) : shapeTotal (shape h) = 2 * length h := |
| (currentShape_spec (history h).1).2 |
|
|
| theorem shape_total_pos (h : TerminalZero K) : 0 < shapeTotal (shape h) := by |
| rw [shape_total] |
| cases h <;> norm_num [length, history, currentLength] |
|
|
| theorem axes_distinct (h : TerminalZero K) : canonicalZero h ≠ canonicalSelected h := |
| zero_ne_selected (shape h) _ _ (shape_total_pos h) |
| (canonicalZero_spec h) (canonicalSelected_spec h) |
|
|
| theorem physical_axes_distinct (h : TerminalZero K) : physicalZero h ≠ physicalSelected h := |
| (history h).2.injective.ne (axes_distinct h) |
|
|
| theorem physical_shape (h : TerminalZero K) : |
| currentPhysicalShape (history h) = placedShape (physicalZero h) (physicalSelected h) |
| (shapeMax (shape h)) (shapeTotal (shape h) - shapeMax (shape h)) := |
| physicalShape_eq_placedShape (shape h) (history h).2 _ _ |
| (canonicalZero_spec h) (canonicalSelected_spec h) (axes_distinct h) |
|
|
| def nativeLaw : (h : TerminalZero K) → Fin 3 → Slot h → ℚ |
| | .inl h, side => halfLaw (aShape h.1.val) side |
| | .inr h, side => littleLaw (bParent h) (bShape h.1.val) side |
|
|
| theorem selected_law (h : TerminalZero K) (s : Slot h) : |
| nativeLaw h (canonicalSelected h) s = law h s := by |
| cases h with |
| | inl h => exact halfLaw_at_maximum _ (aShape_mem_zeroSecond h) s |
| | inr h => |
| change Fin 6 at s |
| change littleLaw (bParent h) (bShape h.1.val) (maxAxis (bShape h.1.val)) s = _ |
| rw [littleLaw_eq_statisticLaw _ _ _ (child_statistic_weight_bounded _ |
| (bParent_mem_positiveSecond h) _ (bShape_mem_below h) _)] |
| congr 1 |
| apply Fin.ext |
| exact maxAxis_spec _ |
|
|
| theorem zero_law (h : TerminalZero K) (s : Slot h) : |
| nativeLaw h (canonicalZero h) s = |
| if statistic h (zeroWord (length h)) = s then 1 else 0 := by |
| have hz := canonicalZero_spec h |
| cases h with |
| | inl h => |
| change PairSlot at s |
| let : DecidableEq PairSlot := fun _ _ => Classical.propDecidable _ |
| change halfLaw (aShape h.1.val) (canonicalZero (.inl h)) s = |
| if ((0, 0) : PairSlot) = s then 1 else 0 |
| rw [halfLaw_weight_zero _ (aShape_mem_shapes h) _ hz s] |
| by_cases hs : s = (0,0) |
| · rw [ite_eq_left hs, ite_eq_left hs.symm] |
| · rw [ite_eq_right hs, ite_eq_right (Ne.symm hs)] |
| | inr h => |
| change Fin 6 at s |
| let : DecidableEq (Fin 6) := fun _ _ => Classical.propDecidable _ |
| change bShape h.1.val (canonicalZero (.inr h)) = 0 at hz |
| change littleLaw (bParent h) (bShape h.1.val) (canonicalZero (.inr h)) s = |
| if (0 : Fin 6) = s then 1 else 0 |
| simp only [littleLaw, hz, show (0 : ℕ) ≠ 2 from by decide, ite_false, singletonSlot] |
| by_cases hs : s = 0 |
| · simp [hs] |
| · simp [hs, Ne.symm hs] |
|
|
| theorem remaining_law (h : TerminalZero K) (side : Fin 3) |
| (hz : side ≠ canonicalZero h) (hs : side ≠ canonicalSelected h) (s : Slot h) : |
| nativeLaw h side s = law h ((AllFieldZeroWords.complement h).symm s) := by |
| have hsum := selected_add_remaining (shape h) (canonicalZero h) (canonicalSelected h) side |
| (canonicalZero_spec h) (canonicalSelected_spec h) (axes_distinct h) hz hs |
| have he : nativeLaw h (canonicalSelected h) ((AllFieldZeroWords.complement h).symm s) = |
| nativeLaw h side s := by |
| cases h with |
| | inl h => |
| exact halfLaw_complementary_sides _ (aShape_mem_shapes h) _ _ (Ne.symm hs) |
| (hsum.trans (shape_total (.inl h))) s |
| | inr h => |
| exact littleLaw_complement _ _ _ _ |
| (child_statistic_weight_bounded _ (bParent_mem_positiveSecond h) _ (bShape_mem_below h) _) |
| (child_statistic_weight_bounded _ (bParent_mem_positiveSecond h) _ (bShape_mem_below h) _) |
| (hsum.trans (shape_total (.inr h))) s |
| rw [selected_law] at he |
| exact he.symm |
|
|
| def embed (allocation : Allocation) (m : ℕ) (h : TerminalZero K) |
| (side : Fin 3) (w : ZeroWords allocation m h) : |
| HistoryWord allocation (terminalDilation K m) (history h) := |
| fun i => sideWord (physicalZero h) (physicalSelected h) side (words allocation m h w i) |
|
|
| theorem embed_window (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| (h : TerminalZero K) (side : Fin 3) (η : ℝ) (hη : 0 ≤ η) |
| (w : ZeroWords allocation m h) : |
| typeWindow (fun s => (nativeLaw h ((history h).2.symm side) s : ℝ)) η |
| (fun i => statistic h (embed allocation m h side w i)) := by |
| by_cases hz : side = physicalZero h |
| · have hc : (history h).2.symm side = canonicalZero h := by |
| simp only [hz, physicalZero, Equiv.symm_apply_apply] |
| simp only [embed, sideWord, ite_eq_left hz, hc] |
| have hpop : 0 < population allocation (terminalDilation K m) (history h) := by |
| rw [← statisticCounts_repeat_sum allocation m h, ← Finset.mul_sum] |
| exact Nat.mul_pos hm (statisticCounts_pos allocation h) |
| apply zero_window (length h) _ hpop (statistic h) _ η hη |
| intro s |
| rw [zero_law] |
| split_ifs <;> norm_num |
| · by_cases hs : side = physicalSelected h |
| · have hc : (history h).2.symm side = canonicalSelected h := by |
| simp only [hs, physicalSelected, Equiv.symm_apply_apply] |
| simpa only [embed, sideWord, ite_eq_right hz, ite_eq_left hs, hc, selected_law] using |
| selected_window allocation hm h η hη w |
| · have hcz : (history h).2.symm side ≠ canonicalZero h := by |
| intro hh |
| apply hz |
| exact ((history h).2.apply_symm_apply side).symm.trans (congrArg (history h).2 hh) |
| have hcs : (history h).2.symm side ≠ canonicalSelected h := by |
| intro hh |
| apply hs |
| exact ((history h).2.apply_symm_apply side).symm.trans (congrArg (history h).2 hh) |
| simp only [embed, sideWord, ite_eq_right hz, ite_eq_right hs, remaining_law h _ hcz hcs] |
| exact complementary_window allocation hm h η hη w |
|
|
| theorem embed_mask (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| {ε : ℝ} (hε : 0 ≤ ε) (h : TerminalZero K) (side : Fin 3) |
| (w : ZeroWords allocation m h) : |
| residentMask allocation (terminalDilation K m) ε (history h) side |
| (embed allocation m h side w) := by |
| cases h with |
| | inl h => |
| intro _ |
| exact embed_window allocation hm (.inl h) side (ε/8) (by positivity) w |
| | inr h => |
| intro _ |
| exact embed_window allocation hm (.inr h) side (ε/128) (by positivity) w |
|
|
| variable (F : Type*) [Field F] |
|
|
| theorem coefficient (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| {ε : ℝ} (hε : 0 ≤ ε) (h : TerminalZero K) (x y z : ZeroWords allocation m h) : |
| historyTensor F allocation (terminalDilation K m) ε (history h) |
| (embed allocation m h 0 x) (embed allocation m h 1 y) (embed allocation m h 2 z) = |
| if matched (physicalZero h) x y z then 1 else 0 := by |
| simp only [historyTensor, ExactRecovery.delete, embed_mask allocation hm hε, |
| and_self, ite_true] |
| rw [physical_shape] |
| exact power_coefficient F (length h) _ _ _ |
| ((max_add_complement (shape h)).trans (shape_total h)) |
| (physicalZero h) (physicalSelected h) (physical_axes_distinct h) |
| (words allocation m h) (words_injective allocation m h) x y z |
|
|
| abbrev Row (allocation : Allocation) (m : ℕ) (h : TerminalZero K) := |
| CWOrientedZero.Row (physicalZero h) (ZeroWords allocation m h) |
| abbrev Middle (allocation : Allocation) (m : ℕ) (h : TerminalZero K) := |
| CWOrientedZero.Middle (physicalZero h) (ZeroWords allocation m h) |
| abbrev Column (allocation : Allocation) (m : ℕ) (h : TerminalZero K) := |
| CWOrientedZero.Column (physicalZero h) (ZeroWords allocation m h) |
|
|
| def localMap (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| {ε : ℝ} (hε : 0 ≤ ε) (h : TerminalZero K) : |
| LocalMap (historyTensor F allocation (terminalDilation K m) ε (history h)) |
| (Tensor.matrixCoefficients (Row allocation m h) (Middle allocation m h) (Column allocation m h)) := by |
| letI : Nonempty (ZeroWords allocation m h) := |
| Fintype.card_pos_iff.mp (zeroWords_card_pos allocation m h) |
| exact ofMatching F _ (physicalZero h) (embed allocation m h) |
| (coefficient F allocation hm hε h) |
|
|
| theorem volume (allocation : Allocation) (m : ℕ) (h : TerminalZero K) : |
| Fintype.card (Row allocation m h) * Fintype.card (Middle allocation m h) * |
| Fintype.card (Column allocation m h) = Fintype.card (ZeroWords allocation m h) := |
| CWOrientedZero.volume (physicalZero h) (ZeroWords allocation m h) |
|
|
| theorem dimensions_pos (allocation : Allocation) (m : ℕ) (h : TerminalZero K) : |
| 0 < Fintype.card (Row allocation m h) ∧ |
| 0 < Fintype.card (Middle allocation m h) ∧ 0 < Fintype.card (Column allocation m h) := by |
| have hp := zeroWords_card_pos allocation m h |
| rw [← volume allocation m h] at hp |
| have hab := Nat.pos_of_mul_pos_right hp |
| exact ⟨Nat.pos_of_mul_pos_right hab, Nat.pos_of_mul_pos_left hab, |
| Nat.pos_of_mul_pos_left hp⟩ |
|
|
| end MatrixMultiplication.AllFieldZeroFinish |
|
|
| end |
|
|
| end MatrixAllFields |
|
|
| end OAI |
|
|
