openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Geometry /ActualRoundedGroupInputs.lean
| import OAI.Combinatorics.EditApproximation.Geometry.RoundedGroupStates | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI | |
| /-! Every actual unskipped rounded group input belongs to a common | |
| small endpoint list determined by the cell, child and envelope label. -/ | |
| namespace EditApproximation | |
| theorem groupRoundSpacing_pos (exponent : ℕ) (h : ℚ) : 0 < groupRoundSpacing exponent h := by | |
| apply Nat.ceil_pos.mpr | |
| exact lt_of_lt_of_le zero_lt_one (le_max_left _ _) | |
| theorem groupRoundSpacing_envelope_bound (exponent : ℕ) (h : ℚ) : | |
| h ≤ (2 : ℚ) ^ exponent * groupRoundSpacing exponent h := by | |
| have hpow : (0 : ℚ) < (2 : ℚ) ^ exponent := by positivity | |
| have hceil : (2 : ℚ) ^ (-(exponent : ℤ)) * h ≤ (groupRoundSpacing exponent h : ℚ) := | |
| (le_max_right _ _).trans (Nat.le_ceil _) | |
| have hmul := mul_le_mul_of_nonneg_left hceil hpow.le | |
| rw [zpow_neg, zpow_natCast] at hmul | |
| simpa only [← mul_assoc, mul_inv_cancel₀ hpow.ne', one_mul] using hmul | |
| theorem bandRoundSpacing_group_spacing {ny M : ℕ} (exponent b : ℕ) | |
| (band : List (Fin M → TargetInterval ny)) (initial : Fin M → TargetInterval ny → ℚ) | |
| (i : Fin M) (h : ℚ) (henvelope : bandChildEnvelope b band initial i = h) : | |
| bandRoundSpacing exponent b band initial i = groupRoundSpacing exponent h := by | |
| have heq := bandRoundSpacing_exact exponent b band initial i | |
| rw [henvelope] at heq | |
| have hceil := congrArg (fun r : ℚ => ⌈r⌉₊) heq | |
| simpa only [Nat.ceil_natCast, groupRoundSpacing] using hceil | |
| theorem gridStateBox_sound {n g radius lo hi : ℕ} (hg : 0 < g) (r : TargetInterval n) | |
| (hr : r ∈ gridStateBox n g radius lo hi) : | |
| Nat.dist lo r.lo ≤ radius ∧ Nat.dist hi r.hi ≤ radius ∧ g ∣ r.lo ∧ g ∣ r.hi := by | |
| obtain ⟨s, hs, hr⟩ := List.mem_flatMap.mp hr | |
| obtain ⟨t, ht, hmake⟩ := List.mem_filterMap.mp hr | |
| have heq : r.lo = s ∧ r.hi = t := by | |
| unfold makeTargetState at hmake | |
| split_ifs at hmake with hvalid | |
| cases hmake | |
| exact ⟨rfl, rfl⟩ | |
| rw [← heq.1] at hs | |
| rw [← heq.2] at ht | |
| have hlo := gridPointsBetween_sound hg hs | |
| have hhi := gridPointsBetween_sound hg ht | |
| refine ⟨?_, ?_, hlo.2.2, hhi.2.2⟩ <;> unfold Nat.dist <;> omega | |
| theorem actual_rounded_group_input_mem {α : Type*} (source target : List α) {M b P exponent : ℕ} | |
| (parent : TargetInterval source.length) (tau a : ℚ) (F : ℕ) | |
| (parentInitial : TargetInterval target.length → ℚ) | |
| (child initial : Fin M → TargetInterval target.length → ℚ) (cell : ℕ × ℕ) | |
| (member : UnskippedBand source target parent b P exponent tau F child initial) | |
| (hb : 1 ≤ b) | |
| (hcenter : member.center ∈ cellRepresentativeStates target.length P F b a parentInitial | |
| ((2 : ℚ) ^ (-(exponent : ℤ))) cell) | |
| (i : Fin M) (h : ℚ) (henvelope : member.envelope i = h) : | |
| member.rounded i ∈ groupRoundedStateList source target parent b exponent cell i h := by | |
| have hspacing := bandRoundSpacing_group_spacing exponent b member.actions initial i h henvelope | |
| have hgrid : groupRoundSpacing exponent h ∣ (member.rounded i).lo ∧ | |
| groupRoundSpacing exponent h ∣ (member.rounded i).hi := by | |
| simpa only [UnskippedBand.rounded, canonicalRoundedBandAction, roundedBandAction, hspacing] using | |
| roundInterval_on_grid (bandRoundSpacing exponent b member.actions initial i) | |
| (member.actions.head member.nonempty i) | |
| have hwide : member.rounded i ∈ wideChildStateList parent member.center b P i := | |
| finiteAssignments_sound M (wideChildStateList parent member.center b P) member.rounded member.wide_guard i | |
| have hwideFirst := (gridStateBox_sound (by unfold seedGridSpacing; omega) (member.rounded i) hwide).1 | |
| have htheta : 0 ≤ (2 : ℚ) ^ (-(exponent : ℤ)) ∧ (2 : ℚ) ^ (-(exponent : ℤ)) ≤ 1 := by | |
| refine ⟨(zpow_pos (by norm_num : (0 : ℚ) < 2) _).le, ?_⟩ | |
| rw [zpow_neg, zpow_natCast] | |
| exact inv_le_one_of_one_le₀ (one_le_pow₀ (by norm_num : (1 : ℚ) ≤ 2)) | |
| have hfirst := wide_firstEndpoint_near_cell cell (by unfold seedGridSpacing; omega) htheta hb | |
| member.center (member.rounded i) ((sourceChild parent i).lo - parent.lo) | |
| (List.mem_filter.mp hcenter).1 hwideFirst | |
| have hgap := member.imbalance_guard i | |
| change |rationalStateGap _ _ (member.rounded i)| ≤ 2 * member.envelope i at hgap | |
| rw [henvelope] at hgap | |
| unfold rationalStateGap at hgap | |
| have hgapDistance := lengthGap_guard_distance (member.rounded i) | |
| (substring source (sourceChild parent i).lo (sourceChild parent i).hi).length h hgap | |
| unfold groupRoundedStateList | |
| apply gridGapStates_complete (groupRoundSpacing_pos exponent h) _ _ hgapDistance hgrid | |
| simpa only [mul_assoc] using hfirst | |
| theorem groupRoundedStateList_length {α : Type*} (source target : List α) {M : ℕ} | |
| (parent : TargetInterval source.length) (b exponent : ℕ) (cell : ℕ × ℕ) | |
| (i : Fin M) (h : ℚ) (hh : 0 < h) : | |
| ((groupRoundedStateList source target parent b exponent cell i h).length : ℚ) ≤ | |
| 208 * ((2 : ℚ) ^ exponent) ^ 2 * (1 + b / h) := by | |
| exact gridGapStates_rounded_count _ _ _ _ b h ((2 : ℚ) ^ exponent) | |
| (groupRoundSpacing_pos exponent h) hh (one_le_pow₀ (by norm_num : (1 : ℚ) ≤ 2)) | |
| (groupRoundSpacing_envelope_bound exponent h) | |
| end EditApproximation | |
| end OAI | |