{"name": "CallElimCorrect.substOldPostSubset", "source": "strata", "statement": "open OldExpressions in\ntheorem substOldPostSubset:\n  (Imperative.HasVarsPure.getVars (P:=Expression)\n    (substOld h2 (Lambda.LExpr.fvar m h1 ty) post)).Subset\n    (Imperative.HasVarsPure.getVars (P:=Expression) post ++ [h1])", "src": "open OldExpressions in\ntheorem substOldPostSubset:\n  (Imperative.HasVarsPure.getVars (P:=Expression)\n    (substOld h2 (Lambda.LExpr.fvar m h1 ty) post)).Subset\n    (Imperative.HasVarsPure.getVars (P:=Expression) post ++ [h1]) := by\n  induction post <;> simp [substOld]\n  case fvar =>\n    intros x Hin\n    rename_i m name ty2\n    simp at m\n    simp at name\n    simp at ty2\n\n    simp_all\n  case op =>\n    intros x Hin\n    rename_i m name ty2\n    simp at m\n    simp at name\n    simp at ty2\n    simp_all\n  case const =>\n    intros x Hin\n    rename_i m name\n    simp at m\n    simp_all\n  case bvar =>\n    intros x Hin\n    rename_i m d\n    simp at m\n    simp_all\n  case abs ih =>\n    exact ih\n  case ite cih tih eih =>\n    simp [Imperative.HasVarsPure.getVars, Lambda.LExpr.LExpr.getVars] at *\n    apply List.Subset.app\n    . apply List.Subset.trans\n      apply cih <;> assumption\n      intros x Hin\n      simp_all\n      cases Hin <;> simp_all\n    apply List.Subset.app\n    . apply List.Subset.trans\n      apply tih <;> assumption\n      intros x Hin\n      simp_all\n      cases Hin <;> simp_all\n    . apply List.Subset.trans\n      apply eih <;> assumption\n      intros x Hin\n      simp_all\n  case app ih1 ih2 =>\n    split\n    . split\n      . simp [Imperative.HasVarsPure.getVars, Lambda.LExpr.LExpr.getVars] at *\n        intros x Hin\n        simp_all\n      . simp [Imperative.HasVarsPure.getVars, Lambda.LExpr.LExpr.getVars] at *\n        intros x Hin\n        simp_all\n    . simp [Imperative.HasVarsPure.getVars, Lambda.LExpr.LExpr.getVars] at *\n      apply List.Subset.app\n      . apply List.Subset.trans\n        apply ih1 <;> assumption\n        intros x Hin\n        simp_all\n        cases Hin <;> simp_all\n      . apply List.Subset.trans\n        apply ih2 <;> assumption\n        intros x Hin\n        simp_all\n  case quant trih eih =>\n    simp [Imperative.HasVarsPure.getVars, Lambda.LExpr.LExpr.getVars] at *\n    apply List.Subset.app\n    . apply List.Subset.trans\n      apply trih <;> assumption\n      intros x Hin\n      rename_i m1 k1 ty1 trigger1 e1\n      have assoc := List.append_assoc (Lambda.LExpr.LExpr.getVars trigger1) (Lambda.LExpr.LExpr.getVars e1) [h1]\n      simp_all\n      cases Hin <;> simp_all\n    . apply List.Subset.trans\n      apply eih <;> assumption\n      intros x Hin\n      simp_all\n  case eq ih1 ih2 =>\n    simp [Imperative.HasVarsPure.getVars, Lambda.LExpr.LExpr.getVars] at *\n    apply List.Subset.app\n    . apply List.Subset.trans\n      apply ih1 <;> assumption\n      intros x Hin\n      simp_all\n      cases Hin <;> simp_all\n    . apply List.Subset.trans\n      apply ih2 <;> assumption\n      intros x Hin\n      simp_all", "proof_length": 313, "num_lines": 97, "header": "", "file_path": "Strata/Transform/CallElimCorrect.lean", "url": "https://github.com/strata-org/Strata", "start_line": 2964, "end_line": 3060, "version_info": [{"v4.26.0": "451e5f047bafa010d178856db76c00029bfa4d7f"}]}
{"name": "CallElimCorrect.extractedOldExprInVars", "source": "strata", "statement": "open OldExpressions in\ntheorem extractedOldExprInVars :\n  NormalizedOldExpr post →\n  (extractOldExprVars post).Subset\n  (Imperative.HasVarsPure.getVars post)", "src": "open OldExpressions in\ntheorem extractedOldExprInVars :\n  NormalizedOldExpr post →\n  (extractOldExprVars post).Subset\n  (Imperative.HasVarsPure.getVars post) := by\n  intros Hnorm\n  induction post <;>\n    simp [Imperative.HasVarsPure.getVars, extractOldExprVars,\n          Lambda.LExpr.LExpr.getVars] at * <;>\n    try simp_all\n  case app fn e fn_ih e_ih =>\n    unfold extractOldExprVars\n    split\n    . simp [Lambda.LExpr.LExpr.getVars]\n      intros x Hin\n      exact Hin\n    . next Hfalse =>\n      cases Hnorm with\n      | app H1 H2 Hn =>\n      exfalso\n      specialize Hn ?_\n      constructor\n      cases Hn\n      apply Hfalse\n      rfl\n    . cases Hnorm with\n      | app H1 H2 Hn =>\n      apply List.Subset.app\n      . apply List.Subset.trans\n        apply fn_ih\n        exact H1\n        intros x Hin\n        simp_all\n      . apply List.Subset.trans\n        apply e_ih\n        exact H2\n        intros x Hin\n        simp_all\n  case abs ih =>\n    cases Hnorm\n    apply ih <;> assumption\n  case quant trih eih =>\n    cases Hnorm\n    rename_i e_normalized\n    rename_i tr_normalized\n    rename_i tr e ty k\n    apply List.Subset.app\n    . apply List.Subset.trans\n      apply trih <;> assumption\n      intros x Hin\n      simp_all\n    . apply List.Subset.trans\n      apply eih <;> assumption\n      intros x Hin\n      simp_all\n  case ite cih tih eih =>\n    cases Hnorm\n    apply List.Subset.app\n    . apply List.Subset.trans\n      apply cih <;> assumption\n      intros x Hin\n      simp_all\n    apply List.Subset.app\n    . apply List.Subset.trans\n      apply tih <;> assumption\n      intros x Hin\n      simp_all\n    . apply List.Subset.trans\n      apply eih <;> assumption\n      intros x Hin\n      simp_all\n  case eq ih1 ih2 =>\n    cases Hnorm\n    apply List.Subset.app\n    . apply List.Subset.trans\n      apply ih1 <;> assumption\n      intros x Hin\n      simp_all\n    . apply List.Subset.trans\n      apply ih2 <;> assumption\n      intros x Hin\n      simp_all", "proof_length": 222, "num_lines": 82, "header": "", "file_path": "Strata/Transform/CallElimCorrect.lean", "url": "https://github.com/strata-org/Strata", "start_line": 2733, "end_line": 2814, "version_info": [{"v4.26.0": "451e5f047bafa010d178856db76c00029bfa4d7f"}]}
{"name": "Core.InitsUpdatesComm", "source": "strata", "statement": "theorem InitsUpdatesComm\n  {P : PureExpr} {σ σ' σ'' : SemanticStore P}\n  {ks ks' : List P.Ident} {vs vs' : List P.Expr} :\n  UpdateStates σ ks vs σ' →\n  InitStates σ' ks' vs' σ'' →\n  ∃ σ₁,\n    σ₁ = (updatedStates σ ks' vs') ∧\n    InitStates σ ks' vs' σ₁ ∧\n    UpdateStates σ₁ ks vs σ''", "src": "theorem InitsUpdatesComm\n  {P : PureExpr} {σ σ' σ'' : SemanticStore P}\n  {ks ks' : List P.Ident} {vs vs' : List P.Expr} :\n  UpdateStates σ ks vs σ' →\n  InitStates σ' ks' vs' σ'' →\n  ∃ σ₁,\n    σ₁ = (updatedStates σ ks' vs') ∧\n    InitStates σ ks' vs' σ₁ ∧\n    UpdateStates σ₁ ks vs σ'' := by\n  intros Hup Hinit\n  exists (updatedStates σ ks' vs')\n  have Hk : (isDefined σ' ks) := UpdateStatesDefined Hup\n  have Hlen1 := InitStatesLength Hinit\n  have Hlen2 := UpdateStatesLength Hup\n  induction Hup generalizing σ''\n  case update_none =>\n    simp_all\n    apply And.intro\n    refine updatedStatesInit Hlen1 ?_ ?_\n    exact InitStatesNotDefined Hinit\n    exact InitStatesNodup Hinit\n    simp [InitStatesUpdated Hinit]\n    constructor\n  case update_some σ x v σ₀ xs vs σ₁ Hup Hups ih =>\n    refine ⟨rfl, ?_, ?_⟩\n    . apply updatedStatesInit Hlen1\n      apply UpdateStateNotDefMonotone' ?_ Hup\n      apply UpdateStatesNotDefMonotone' ?_ Hups\n      exact InitStatesNotDefined Hinit\n      exact InitStatesNodup Hinit\n    . apply UpdateStates.update_some (σ':=updatedStates σ₀ ks' vs')\n      . simp [UpdateStateUpdated Hup, updatedStates]\n        rw [← updatedStateComm']\n        . have Hdef := UpdateStateDefined' Hup\n          simp [isDefined, Option.isSome] at Hdef\n          split at Hdef <;> simp_all\n          next val heq =>\n          apply updatedStateUpdate (v':=val)\n          apply InitStatesSomeMonotone heq\n          apply updatedStatesInit\n          . simp_all\n          . apply UpdateStateNotDefMonotone' ?_ Hup\n            apply UpdateStatesNotDefMonotone' ?_ Hups\n            apply InitStatesNotDefined Hinit\n          . exact InitStatesNodup Hinit\n        . rw [List.unzip_zip] <;> simp_all\n          have Hnd := InitStatesNotDefined Hinit\n          simp [isNotDefined, isDefined] at *\n          apply Not.intro\n          intros Hin\n          specialize Hnd _ Hin\n          simp_all\n      . apply (ih Hinit ?_ ?_).2.2\n        . simp [isDefined] at * <;> simp_all\n        . simp_all", "proof_length": 224, "num_lines": 55, "header": "", "file_path": "Strata/Languages/Core/StatementSemanticsProps.lean", "url": "https://github.com/strata-org/Strata", "start_line": 950, "end_line": 1004, "version_info": [{"v4.29.1": "384e22374e0872095ad9c6f74c5998c09cfd93a9"}, {"v4.27.0": "4348c65a93a3b57b1a4541991ed2d2fcdd9fa35d"}, {"v4.26.0": "451e5f047bafa010d178856db76c00029bfa4d7f"}]}
{"name": "fundamental_theorem_of_variational_calculus'", "source": "physlib", "statement": "lemma fundamental_theorem_of_variational_calculus' {f : Y → V}\n    (μ : Measure Y) [IsFiniteMeasureOnCompacts μ] [μ.IsOpenPosMeasure]\n    [OpensMeasurableSpace Y]\n    (hf : Continuous f) (hg : ∀ g, IsTestFunction g → ∫ x, ⟪f x, g x⟫_ℝ ∂μ = 0) :\n    f = 0", "src": "lemma fundamental_theorem_of_variational_calculus' {f : Y → V}\n    (μ : Measure Y) [IsFiniteMeasureOnCompacts μ] [μ.IsOpenPosMeasure]\n    [OpensMeasurableSpace Y]\n    (hf : Continuous f) (hg : ∀ g, IsTestFunction g → ∫ x, ⟪f x, g x⟫_ℝ ∂μ = 0) :\n    f = 0 := by\n  -- assume ¬(f = 0)\n    rw [funext_iff]; by_contra h₀\n    obtain ⟨x₀, hx0⟩ := not_forall.1 h₀\n    simp at hx0 -- hx0 : f x₀ ≠ 0\n\n  -- [1] Proof that `f` is continuous at `x₀`.\n  -- Embed into the true IP-space `WithLp 2 V`.\n    let f₂ : Y → WithLp 2 V := toL2 ℝ ∘ f\n    let x₂ := f₂ x₀\n  -- x₂ ≠ 0 because `fromL2 (toL2 (f x₀)) = f x₀`\n    have hx2 : x₂ ≠ 0 := by\n      intro h; apply hx0; exact congrArg (fromL2 ℝ) h\n  -- continuity of f₂ at x₀\n    have hcont₂ : ContinuousAt f₂ x₀ := ((toL2 ℝ).continuous.comp hf).continuousAt\n\n  -- [2] find open neighborhood guaranteeing positive inner product with the center, based on\n  -- which the test function `g` will be constructed.\n  -- pick δ₂ so that on B(x₀, δ₂), ‖f₂ x - x₂‖ < ‖x₂‖/2\n    obtain ⟨δ₂, hδ₂_pos, hδ₂⟩ :=\n    Metric.continuousAt_iff.mp hcont₂ (‖x₂‖ / 2)\n      (by simpa [half_pos] using (norm_pos_iff.mpr hx2))\n  -- now the usual “add & subtract” proof inside WithLp 2 V\n    have inner_pos₂ : ∀ x (hx : x ∈ Metric.ball x₀ δ₂), 0 < (⟪f₂ x, x₂⟫_ℝ : ℝ) := by\n      intros x hx\n    -- hx : x ∈ ball x₀ δ₂, so dist x x₀ < δ₂, hence\n    -- this is |⟪u,v⟫| ≤ ‖u‖ * ‖v‖, in the genuine InnerProductSpace on WithLp 2 V\n      have hclose : ‖f₂ x - x₂‖ < ‖x₂‖ / 2 := by\n        convert hδ₂ hx using 1\n        exact mem_sphere_iff_norm.mp rfl\n      have hself : ⟪x₂, x₂⟫_ℝ = ‖x₂‖^2 := real_inner_self_eq_norm_sq (x₂ : WithLp 2 V)\n      -- Cauchy–Schwarz in `WithLp 2 V` bounds the cross term from below\n      have hcs : -(‖f₂ x - x₂‖ * ‖x₂‖) ≤ ⟪f₂ x - x₂, x₂⟫_ℝ :=\n        neg_le_of_abs_le (abs_real_inner_le_norm _ _)\n      have hxp : 0 < ‖x₂‖ := norm_pos_iff.mpr hx2\n      have inner_eq_self_add_diff : ⟪f₂ x, x₂⟫_ℝ = ⟪x₂, x₂⟫_ℝ + ⟪f₂ x - x₂, x₂⟫_ℝ := by\n        rw [← inner_add_left]; simp\n      nlinarith [inner_eq_self_add_diff, hcs, hself, hclose, hxp,\n        mul_lt_mul_of_pos_right hclose hxp]\n  -- pull `inner_pos₂` back to V via `fromL2`:\n    have inner_pos_V : ∀ x ∈ Metric.ball x₀ δ₂, 0 < ⟪f x, f x₀⟫_ℝ := fun x hx => inner_pos₂ x hx\n    -- now we have a genuine positive integrand on a set of positive measure.\n\n  -- [3] `g` construction using bump function.\n    have bump_exists : ∃ φ : Y → ℝ, IsTestFunction φ ∧ φ x₀ > 0 ∧\n        (∀ x ∈ Function.support φ, 0 ≤ φ x) ∧\n        Function.support φ ⊆ Metric.ball x₀ (δ₂/2) ∧\n        (∀ x ∈ Metric.closedBall x₀ (δ₂/4), 0 < φ x) := by\n        -- use `hasContDiffBump_of_innerProductSpace`, leveraging `[innerProductSpace Y]`\n          haveI : HasContDiffBump Y := hasContDiffBump_of_innerProductSpace Y\n          let φ1 : ContDiffBump x₀ :=\n            ⟨δ₂ / 4, δ₂ / 2, by positivity, by linarith⟩\n          refine ⟨φ1.toFun, ⟨φ1.contDiff, φ1.hasCompactSupport⟩,\n            φ1.pos_of_mem_ball (Metric.mem_ball_self φ1.rOut_pos), fun x _ => φ1.nonneg,\n            by rw [ContDiffBump.support_eq], fun x hx => φ1.pos_of_mem_ball ?_⟩\n        -- the closed ball of radius `δ₂/4` lies in the open ball of radius `rOut = δ₂/2`\n          rw [Metric.mem_closedBall] at hx\n          exact Metric.mem_ball.mpr (lt_of_le_of_lt hx (by show δ₂ / 4 < δ₂ / 2; linarith))\n    obtain ⟨φ, hφ_testfun, hφ_pos_x₀, hφ_non_neg, hφ_support_subset, hφ_pos_inner⟩ :=\n      bump_exists\n  -- Define test function g(x) = φ(x) * f(x₀)\n    let g : Y → V := fun x => φ x • f x₀\n  -- Show that g is a test function: `φ` is a test function and `f x₀` is smooth (constant)\n    have hg_test : IsTestFunction g := IsTestFunction.smul_right hφ_testfun contDiff_const\n\n  -- [4] Derive contradiction. First compute the integral ∫ ⟪f x, g x⟫\n  -- [4.1] ∫ φ x * ⟪f x, f x₀⟫ = 0\n    have key_integral := hg g hg_test\n  -- linearity of inner product in the second argument turns the integrand into `φ x * ⟪f x, f x₀⟫`\n    simp only [g, inner_smul_right'] at key_integral\n\n  -- [4.2] 0 < ∫ x, φ x * ⟪f x, f x₀⟫_ℝ ∂μ. Sketch: on the support of φ (which is contained in\n  -- B(x₀, δ/2) ⊆ B(x₀, δ)), we have ⟪f x, f x₀⟫ > ‖f x₀‖²/2 > 0 by our choice of δ.\n  -- Since φ is nonnegative on its support and positive somewhere, this gives the contradiction.\n\n  -- [4.2.1] Integrability of the integrand: `integrable_prod` .\n    have support_subset : Function.support φ ⊆ Metric.ball x₀ δ₂ :=\n      hφ_support_subset.trans (Metric.ball_subset_ball (by linarith))\n    have supp_subset2 : Function.support (fun x => φ x * ⟪f x, f x₀⟫_ℝ) ⊆ Function.support φ := by\n      intro x hprod hφ0\n    -- if φ x = 0 then φ x * inner = 0, contradiction\n      simp [hφ0] at hprod\n    have hinner_cont : Continuous (fun x => ⟪f x, f x₀⟫_ℝ) :=\n      Continuous.inner' (f : Y → V) (fun _ => f x₀) hf continuous_const\n    have integrable_prod :\n      Integrable (fun x => φ x * ⟪f x, f x₀⟫_ℝ) μ :=\n    -- (i) build a `HasCompactSupport` witness for the product\n      (Continuous.mul hφ_testfun.smooth.continuous hinner_cont).integrable_of_hasCompactSupport\n        (hφ_testfun.supp.mono supp_subset2)\n\n  -- [4.2.2] Nonnegativity everywhere (`h_nonneg`)\n    have h_nonneg : ∀ x, 0 ≤ φ x * ⟪f x, f x₀⟫_ℝ := by\n      intro x\n      by_cases hx : x ∈ Function.support φ\n      · -- on the support, φ ≥ 0 and ⟪f x, f x₀⟫ > 0\n        exact mul_nonneg (hφ_non_neg x hx) (inner_pos_V x (support_subset hx)).le\n      · -- off the support, φ x = 0 so the product is 0\n        simp [Function.notMem_support.mp hx]\n\n  -- [4.2.3] That closed ball has positive measure, and is contained in the support\n    -- every nonempty open set has positive measure\n    have hμ_ball : 0 < μ (Metric.ball x₀ (δ₂/4)) :=\n      Metric.isOpen_ball.measure_pos μ (Metric.nonempty_ball.mpr (by linarith))\n    have hμ : 0 < μ (Metric.closedBall x₀ (δ₂/4)) :=\n      lt_of_lt_of_le hμ_ball (measure_mono Metric.ball_subset_closedBall)\n    have closedBall_subset_support :\n        Metric.closedBall x₀ (δ₂/4)\n          ⊆ Function.support (fun x => φ x * ⟪f x, f x₀⟫_ℝ) := by\n        intro x hx\n        have hφx := hφ_pos_inner x hx\n        have hin : 0 < ⟪f x, f x₀⟫_ℝ :=\n          inner_pos_V x (Metric.closedBall_subset_ball (by linarith) hx)\n        simp only [Function.support_mul, Set.mem_inter_iff, Function.mem_support, ne_eq]\n        constructor\n        · linarith\n        · linarith\n\n  -- [4.2.4] putting everything together: positivity contradicts `key_integral : ∫ ... = 0`\n    have integral_pos : 0 < ∫ x, φ x * ⟪f x, f x₀⟫_ℝ ∂μ :=\n      (integral_pos_iff_support_of_nonneg h_nonneg integrable_prod).mpr\n        (lt_of_lt_of_le hμ (measure_mono closedBall_subset_support))\n    linarith", "proof_length": 851, "num_lines": 126, "header": "", "file_path": "Physlib/Mathematics/VariationalCalculus/Basic.lean", "url": "https://github.com/leanprover-community/physlib", "start_line": 99, "end_line": 224, "version_info": [{"v4.32.0": "c48433678e8fb6306ebcd48453300c8e16058a62"}]}
{"name": "Electromagnetism.ElectromagneticPotential.time_deriv_time_deriv_electricField_of_isExtrema", "source": "physlib", "statement": "lemma time_deriv_time_deriv_electricField_of_isExtrema {A : ElectromagneticPotential d}\n    {𝓕 : FreeSpace}\n    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d)\n    (hJ : ContDiff ℝ ∞ J) (h : IsExtrema 𝓕 A J)\n    (t : Time) (x : Space d) (i : Fin d) :\n    ∂ₜ (∂ₜ (A.electricField 𝓕.c · x i)) t =\n      𝓕.c ^ 2 * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -\n      𝓕.c ^ 2 / 𝓕.ε₀ * ∂[i] (J.chargeDensity 𝓕.c t ·) x -\n      𝓕.c ^ 2 * 𝓕.μ₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t", "src": "lemma time_deriv_time_deriv_electricField_of_isExtrema {A : ElectromagneticPotential d}\n    {𝓕 : FreeSpace}\n    (hA : ContDiff ℝ ∞ A) (J : LorentzCurrentDensity d)\n    (hJ : ContDiff ℝ ∞ J) (h : IsExtrema 𝓕 A J)\n    (t : Time) (x : Space d) (i : Fin d) :\n    ∂ₜ (∂ₜ (A.electricField 𝓕.c · x i)) t =\n      𝓕.c ^ 2 * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -\n      𝓕.c ^ 2 / 𝓕.ε₀ * ∂[i] (J.chargeDensity 𝓕.c t ·) x -\n      𝓕.c ^ 2 * 𝓕.μ₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n  have hEs : ∀ j, ContDiff ℝ 2 (fun y => A.electricField 𝓕.c t y j) :=\n    fun j => electricField_apply_contDiff_space (i := j) (hA.of_le (right_eq_inf.mp rfl)) t\n  have hEd : ∀ j k, Differentiable ℝ (∂[k] (fun y => A.electricField 𝓕.c t y j)) :=\n    fun j k => Space.deriv_differentiable (hEs j) k\n  have hBt : ∀ j, Differentiable ℝ\n      (fun s => ∂[j] (fun y => A.magneticFieldMatrix 𝓕.c s y (j, i)) x) :=\n    fun j => Space.space_deriv_differentiable_time (i := j)\n      (magneticFieldMatrix_contDiff _ (hA.of_le (right_eq_inf.mp rfl)) (j, i)) x\n  have hJt : Differentiable ℝ (fun s => J.currentDensity 𝓕.c s x i) :=\n    LorentzCurrentDensity.currentDensity_apply_differentiable_time (hJ.differentiable (by simp)) x i\n  calc _\n    _= ∂ₜ (fun t =>\n      1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, Space.deriv j (fun x => magneticFieldMatrix 𝓕.c A t x (j, i)) x -\n      1 / 𝓕.ε₀ * LorentzCurrentDensity.currentDensity 𝓕.c J t x i) t := by\n        conv_lhs =>\n          enter [1]\n          change fun t => ∂ₜ (A.electricField 𝓕.c · x i) t\n          enter [t]\n          rw [Time.deriv_euclid (electricField_differentiable_time\n            (hA.of_le (right_eq_inf.mp rfl)) _),\n            time_deriv_electricField_of_isExtrema hA J hJ h]\n    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∂ₜ (fun t => ∑ j, ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x) t -\n      1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n      rw [Time.deriv_eq]\n      rw [fderiv_fun_sub]\n      simp only [one_div, mul_inv_rev, FunLike.coe_sub, Pi.sub_apply]\n      rw [fderiv_const_mul (Differentiable.fun_sum fun j _ => hBt j).differentiableAt]\n      rw [fderiv_const_mul hJt.differentiableAt]\n      simp [Time.deriv_eq]\n      · exact ((Differentiable.fun_sum fun j _ => hBt j).const_mul _).differentiableAt\n      · exact hJt.differentiableAt.const_mul _\n    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ((∑ j, ∂ₜ (fun t => ∂[j] (A.magneticFieldMatrix 𝓕.c t · (j, i)) x)) t) -\n      1 / 𝓕.ε₀ * (∂ₜ (J.currentDensity 𝓕.c · x i) t) := by\n      congr\n      rw [Time.deriv_eq]\n      rw [fderiv_fun_sum fun i _ => (hBt i).differentiableAt]\n      simp only [FunLike.coe_sum, Finset.sum_apply]\n      rfl\n    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * (∑ j, ∂[j] (fun x => ∂ₜ (A.magneticFieldMatrix 𝓕.c · x (j, i)) t)) x -\n        1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n      congr\n      simp only [Finset.sum_apply]\n      congr\n      funext k\n      rw [Space.time_deriv_comm_space_deriv]\n      apply magneticFieldMatrix_contDiff\n      apply hA.of_le (right_eq_inf.mp rfl)\n    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) *(∑ j, ∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x -\n        ∂[i] (A.electricField 𝓕.c t · j) x)) x -\n        1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n        congr\n        simp only [Finset.sum_apply]\n        congr\n        funext k\n        congr\n        funext x\n        rw [time_deriv_magneticFieldMatrix _ (hA.of_le (ENat.LEInfty.out))]\n    _ = (1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x -\n          ∂[j] (fun x => ∂[i] (A.electricField 𝓕.c t · j) x) x)) -\n          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n        congr\n        simp only [Finset.sum_apply]\n        congr\n        funext j\n        rw [Space.deriv_eq_fderiv_basis]\n        rw [fderiv_fun_sub (hEd _ _).differentiableAt (hEd _ _).differentiableAt]\n        simp [← Space.deriv_eq_fderiv_basis]\n    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -\n          1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[i] (A.electricField 𝓕.c t · j) x) x) -\n          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by simp [mul_sub]\n    _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -\n        1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[i] (fun x => ∂[j] (A.electricField 𝓕.c t · j) x) x) -\n        1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n        congr\n        funext j\n        rw [Space.deriv_commute _ (hEs _), Space.deriv_eq_fderiv_basis]\n      _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -\n          1 / (𝓕.μ₀ * 𝓕.ε₀) * (∂[i] (fun x => ∑ j, ∂[j] (A.electricField 𝓕.c t · j) x) x) -\n          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n        congr\n        rw [Space.deriv_eq_fderiv_basis]\n        rw [fderiv_fun_sum]\n        simp [← Space.deriv_eq_fderiv_basis]\n        intro j _\n        exact (hEd j j).differentiableAt\n      _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (fun x => ∂[j] (A.electricField 𝓕.c t · i) x) x) -\n          1 / (𝓕.μ₀ * 𝓕.ε₀) * (∂[i] (fun x => (∇ ⬝ (A.electricField 𝓕.c t)) x) x) -\n          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n        rfl\n      _ = 1 / (𝓕.μ₀ * 𝓕.ε₀) * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -\n          1 / (𝓕.μ₀ * 𝓕.ε₀ ^ 2) * ∂[i] (J.chargeDensity 𝓕.c t ·) x -\n          1 / 𝓕.ε₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n        congr 2\n        rw [isExtrema_iff_gauss_ampere_magneticFieldMatrix] at h\n\n        conv_lhs =>\n          enter [2, 2, x]\n          rw [(h t x).1]\n        trans 1 / (𝓕.μ₀ * 𝓕.ε₀) * Space.deriv i\n            (fun x => (1/ 𝓕.ε₀) * LorentzCurrentDensity.chargeDensity 𝓕.c J t x) x\n        · congr\n          funext x\n          ring\n        · rw [Space.deriv_eq_fderiv_basis]\n          rw [fderiv_const_mul]\n          simp [← Space.deriv_eq_fderiv_basis]\n          field_simp\n          apply Differentiable.differentiableAt\n          apply LorentzCurrentDensity.chargeDensity_differentiable_space\n          exact hJ.differentiable (by simp)\n        · exact hA\n        · exact hJ\n      _ = 𝓕.c ^ 2 * ∑ j, (∂[j] (∂[j] (A.electricField 𝓕.c t · i)) x) -\n            𝓕.c ^ 2 / 𝓕.ε₀ * ∂[i] (J.chargeDensity 𝓕.c t ·) x -\n            𝓕.c ^ 2 * 𝓕.μ₀ * ∂ₜ (J.currentDensity 𝓕.c · x i) t := by\n          simp [FreeSpace.c_sq]\n          field_simp", "proof_length": 1372, "num_lines": 126, "header": "", "file_path": "Physlib/Electromagnetism/Dynamics/IsExtrema.lean", "url": "https://github.com/leanprover-community/physlib", "start_line": 376, "end_line": 501, "version_info": [{"v4.32.0": "c48433678e8fb6306ebcd48453300c8e16058a62"}]}
{"name": "FieldSpecification.WickAlgebra.ι_timeOrderF_superCommuteF_eq_time", "source": "physlib", "statement": "lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnFieldOp}\n    (hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.FieldOpFreeAlgebra) :\n    ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * b) =\n    ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * 𝓣ᶠ(a * b))", "src": "lemma ι_timeOrderF_superCommuteF_eq_time {φ ψ : 𝓕.CrAnFieldOp}\n    (hφψ : crAnTimeOrderRel φ ψ) (hψφ : crAnTimeOrderRel ψ φ) (a b : 𝓕.FieldOpFreeAlgebra) :\n    ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * b) =\n    ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * 𝓣ᶠ(a * b)) := by\n  let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :\n    Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * b) =\n    ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * 𝓣ᶠ(a * b))\n  change pb b (Basis.mem_span _ b)\n  apply Submodule.span_induction\n  · rintro x ⟨φs, rfl⟩\n    simp only [ofListBasis_eq_ofList, map_mul, pb]\n    let pa (a : 𝓕.FieldOpFreeAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :\n      Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * ofCrAnListF φs) =\n      ι ([ofCrAnOpF φ, ofCrAnOpF ψ]ₛF * 𝓣ᶠ(a* ofCrAnListF φs))\n    change pa a (Basis.mem_span _ a)\n    apply Submodule.span_induction\n    · rintro x ⟨φs', rfl⟩\n      simp only [ofListBasis_eq_ofList, map_mul, pa]\n      calc _\n        /- Split the commutator. -/\n        _ = crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) •\n          ι (ofCrAnListF (crAnTimeOrderList (φs' ++ φ :: ψ :: φs))) -\n            crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) •\n          𝓢(𝓕.crAnStatistics φ, 𝓕.crAnStatistics ψ) •\n            ι (ofCrAnListF (crAnTimeOrderList (φs' ++ ψ :: φ :: φs))) := by\n          conv_lhs =>\n            rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton,\n              superCommuteF_ofCrAnListF_ofCrAnListF]\n            simp [mul_sub, sub_mul, ← ofCrAnListF_append]\n            rw [timeOrderF_ofCrAnListF, timeOrderF_ofCrAnListF]\n          simp only [map_smul, sub_right_inj]\n          module\n        /- Simplify the signs. -/\n        _ = crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) •\n          (ι (ofCrAnListF (crAnTimeOrderList (φs' ++ φ :: ψ :: φs))) -\n          𝓢(𝓕.crAnStatistics φ, 𝓕.crAnStatistics ψ) •\n            ι (ofCrAnListF (crAnTimeOrderList (φs' ++ ψ :: φ :: φs)))) := by\n          rw [crAnTimeOrderSign_swap_eq_time hφψ hψφ]\n          module\n        /- Splitting the time-ordered lists. -/\n        _ = crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs) •\n          (ι (ofCrAnListF (List.takeWhile (fun c => ¬crAnTimeOrderRel φ c)\n                (List.insertionSort crAnTimeOrderRel (φs' ++ φs)))) *\n              ι (ofCrAnListF (List.filter (fun c =>\n                (crAnTimeOrderRel φ c ∧ crAnTimeOrderRel c φ)) φs')) *\n              ι (ofCrAnListF [φ, ψ]) *\n              ι (ofCrAnListF (List.filter (fun c =>\n                (crAnTimeOrderRel φ c ∧ crAnTimeOrderRel c φ)) φs)) *\n              ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧ ¬crAnTimeOrderRel c φ))\n                (List.insertionSort crAnTimeOrderRel (φs' ++ φs)))) -\n          𝓢(𝓕.crAnStatistics φ, 𝓕.crAnStatistics ψ) •\n          ι (ofCrAnListF (List.takeWhile (fun c => ¬crAnTimeOrderRel φ c)\n                      (List.insertionSort crAnTimeOrderRel (φs' ++ φs)))) *\n              ι (ofCrAnListF (List.filter (fun c =>\n                (crAnTimeOrderRel φ c ∧ crAnTimeOrderRel c φ)) φs')) *\n              ι (ofCrAnListF [ψ, φ]) *\n              ι (ofCrAnListF (List.filter (fun c =>\n                (crAnTimeOrderRel φ c ∧ crAnTimeOrderRel c φ)) φs)) *\n              ι (ofCrAnListF\n              (List.filter (fun c => decide (crAnTimeOrderRel φ c ∧ ¬crAnTimeOrderRel c φ))\n                (List.insertionSort crAnTimeOrderRel (φs' ++ φs))))) := by\n          have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [φ, ψ] φs\n            (by simp_all)\n          rw [crAnTimeOrderList, show φs' ++ φ :: ψ :: φs = φs' ++ [φ, ψ] ++ φs by simp, h1]\n          have h2 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [ψ, φ] φs\n            (by simp_all)\n          rw [crAnTimeOrderList, show φs' ++ ψ :: φ :: φs = φs' ++ [ψ, φ] ++ φs by simp, h2]\n          repeat rw [ofCrAnListF_append]\n          simp\n        _ = crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs) •\n            (ι (ofCrAnListF (List.takeWhile (fun c => ¬crAnTimeOrderRel φ c)\n                (List.insertionSort crAnTimeOrderRel (φs' ++ φs)))) *\n              ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧\n                crAnTimeOrderRel c φ)) φs')) *\n              (ι (ofCrAnListF [φ, ψ]) - 𝓢(𝓕.crAnStatistics φ, 𝓕.crAnStatistics ψ) •\n                ι (ofCrAnListF [ψ, φ])) *\n              ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧\n                crAnTimeOrderRel c φ)) φs)) *\n              ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧ ¬crAnTimeOrderRel c φ))\n                (List.insertionSort crAnTimeOrderRel (φs' ++ φs))))) := by\n            simp [mul_sub, sub_mul]\n        _ = crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs) •\n            (ι (ofCrAnListF (List.takeWhile (fun c => ¬crAnTimeOrderRel φ c)\n                (List.insertionSort crAnTimeOrderRel (φs' ++ φs)))) *\n              ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧\n                crAnTimeOrderRel c φ)) φs')) *\n              ι [ofCrAnOpF φ, ofCrAnOpF ψ]ₛF *\n              ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧\n                crAnTimeOrderRel c φ)) φs)) *\n              ι (ofCrAnListF (List.filter (fun c => (crAnTimeOrderRel φ c ∧ ¬crAnTimeOrderRel c φ))\n                (List.insertionSort crAnTimeOrderRel (φs' ++ φs))))) := by\n            congr\n            rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, ← ofCrAnListF_singleton,\n              ← ofCrAnListF_singleton]\n            simp [← ofCrAnListF_append]\n      rw [Subalgebra.mem_center_iff.mp (ι_superCommuteF_ofCrAnOpF_ofCrAnOpF_mem_center φ ψ)]\n      repeat rw [mul_assoc]\n      rw [← map_mul, ← map_mul, ← map_mul, ← ofCrAnListF_append, ← ofCrAnListF_append,\n        ← ofCrAnListF_append]\n      have h1 := insertionSort_of_takeWhile_filter 𝓕.crAnTimeOrderRel φ φs' φs\n      simp [decide_not, Bool.decide_and, List.append_assoc, List.cons_append] at h1 ⊢\n      rw [← h1, ← crAnTimeOrderList]\n      by_cases hq : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)\n      · simp [ι_superCommuteF_of_diff_statistic hq]\n      · rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _ hφψ hψφ (by simp_all),\n          ← crAnTimeOrderSign, ← ofCrAnListF_append, timeOrderF_ofCrAnListF]\n        simp only [map_smul, Algebra.mul_smul_comm]\n    · simp only [map_mul, zero_mul, map_zero, mul_zero, pa]\n    · intro x y hx hy hpx hpy\n      simp_all [pa,mul_add, add_mul]\n    · intro x hx hpx\n      simp_all [pa]\n  · simp only [map_mul, mul_zero, map_zero, pb]\n  · intro x y hx hy hpx hpy\n    simp_all [pb,mul_add]\n  · intro x hx hpx\n    simp_all [pb]", "proof_length": 1217, "num_lines": 117, "header": "", "file_path": "Physlib/QFT/PerturbationTheory/WickAlgebra/TimeOrder.lean", "url": "https://github.com/leanprover-community/physlib", "start_line": 162, "end_line": 278, "version_info": [{"v4.32.0": "c48433678e8fb6306ebcd48453300c8e16058a62"}]}
{"name": "Cslib.LambdaCalculus.LocallyNameless.Fsub.Typing.progress", "source": "cslib", "statement": "lemma Typing.progress (der : Typing [] t τ) : t.Value ∨ ∃ t', t ⭢βᵛ t'", "src": "lemma Typing.progress (der : Typing [] t τ) : t.Value ∨ ∃ t', t ⭢βᵛ t' := by\n  generalize eq : [] = Γ at der\n  have der' : Typing Γ t τ := der\n  induction der <;> subst eq\n  case var mem => grind\n  case app t₁ _ _ t₂ l r ih_l ih_r =>\n    right\n    cases ih_l rfl l with\n    | inl val_l =>\n        cases ih_r rfl r with\n        | inl val_r =>\n            have ⟨σ, t₁, eq⟩ := l.canonical_form_abs val_l\n            exists t₁ ^ᵗᵗ t₂\n            grind\n        | inr red_r =>\n            obtain ⟨t₂', _⟩ := red_r\n            exists t₁.app t₂'\n            grind\n    | inr red_l =>\n        obtain ⟨t₁', _⟩ := red_l\n        exists t₁'.app t₂\n        grind\n  case tapp σ' der _ ih =>\n    right\n    specialize ih rfl der\n    cases ih with\n    | inl val =>\n        obtain ⟨_, t, _⟩ := der.canonical_form_tabs val\n        exists t ^ᵗᵞ σ'\n        grind\n    | inr red =>\n        obtain ⟨t', _⟩ := red\n        exists .tapp t' σ'\n        grind\n  case let' t₁ σ t₂ τ L der _ ih _ =>\n    right\n    cases ih rfl der with\n    | inl _ =>\n        exists t₂ ^ᵗᵗ t₁\n        grind\n    | inr red =>\n        obtain ⟨t₁', _⟩ := red\n        exists t₁'.let' t₂\n        grind\n  case inl der _ ih =>\n    cases (ih rfl der) with\n    | inl val => grind\n    | inr red =>\n        right\n        obtain ⟨t', _⟩ := red\n        exists .inl t'\n        grind\n  case inr der _ ih =>\n    cases (ih rfl der) with\n    | inl val => grind\n    | inr red =>\n        right\n        obtain ⟨t', _⟩ := red\n        exists .inr t'\n        grind\n  case case t₁ _ _ t₂ _ t₃ _ der _ _ ih _ _ =>\n    right\n    cases ih rfl der with\n    | inl val =>\n        have ⟨t₁, lr⟩ := der.canonical_form_sum val\n        cases lr <;> [exists t₂ ^ᵗᵗ t₁; exists t₃ ^ᵗᵗ t₁] <;> grind\n    | inr red =>\n        obtain ⟨t₁', _⟩ := red\n        exists t₁'.case t₂ t₃\n        grind\n  case sub => grind\n  case abs σ _ τ L _ _=>\n    left\n    constructor\n    apply LC.abs L\n    · grind only [→ wf, cases Term.LC]\n    · grind only [→ wf]\n  case tabs L _ _=>\n    left\n    constructor\n    apply LC.tabs L\n    · grind only [→ wf, cases Term.LC]\n    · grind only [→ wf]", "proof_length": 408, "num_lines": 83, "header": "", "file_path": "Cslib/Languages/LambdaCalculus/LocallyNameless/Fsub/Safety.lean", "url": "https://github.com/leanprover/cslib", "start_line": 76, "end_line": 158, "version_info": [{"v4.33.0-rc2": "3aa9d4416c185e0b9faeb72bbd65abe85b95dbcc"}, {"v4.32.0": "197a7be621263b84c67ca4f803f69205b36d06df"}, {"v4.31.0": "8ea71c8df91acd63285515a04da952b23b78d266"}]}
{"name": "Cslib.SKI.parallelReduction_diamond", "source": "cslib", "statement": "theorem parallelReduction_diamond : Diamond ParallelReduction", "src": "theorem parallelReduction_diamond : Diamond ParallelReduction := by\n  intro a a₁ a₂ h₁ h₂\n  cases h₁\n  case refl => exact ⟨a₂, h₂, .refl a₂⟩\n  case par a a' b b' ha' hb' =>\n    cases h₂\n    case refl =>\n      use a' ⬝ b'\n      exact ⟨.refl (a' ⬝ b'), .par ha' hb'⟩\n    case par a'' b'' ha'' hb'' =>\n      let ⟨a₃, ha⟩ := parallelReduction_diamond ha' ha''\n      let ⟨b₃, hb⟩ := parallelReduction_diamond hb' hb''\n      use a₃ ⬝ b₃\n      constructor\n      · exact .par ha.1 hb.1\n      · exact .par ha.2 hb.2\n    case red_I =>\n      rw [I_irreducible a' ha']\n      use b', .red_I b'\n    case red_K =>\n      let ⟨a₂', ha₂'⟩ := Ka_irreducible a₂ a' ha'\n      rw [ha₂'.2]\n      use a₂'\n      exact ⟨.red_K a₂' b', ha₂'.1⟩\n    case red_S a c =>\n      let ⟨a'', c', h⟩ := Sab_irreducible a c a' ha'\n      rw [h.2.2]\n      use a'' ⬝ b' ⬝ (c' ⬝ b'), .red_S a'' c' b'\n      apply ParallelReduction.par <;>\n        apply ParallelReduction.par <;>\n        grind\n  case red_I =>\n    cases h₂\n    case refl => use a₁; exact ⟨.refl a₁, .red_I a₁⟩\n    case par c a₁' hc ha =>\n      rw [I_irreducible c hc]\n      use a₁'\n      exact ⟨ha, .red_I a₁'⟩\n    case red_I => use a₁; exact ⟨.refl a₁, .refl a₁⟩\n  case red_K c =>\n    cases h₂\n    case refl => use a₁; exact ⟨.refl a₁, .red_K a₁ c⟩\n    case par a' c' ha hc =>\n      let ⟨a₁', h'⟩ := Ka_irreducible a₁ a' ha\n      rw [h'.2]\n      use a₁'\n      exact ⟨h'.1, .red_K a₁' c'⟩\n    case red_K =>\n      use a₁; exact ⟨.refl a₁, .refl a₁⟩\n  case red_S a b c =>\n    cases h₂\n    case refl =>\n      use a ⬝ c ⬝ (b ⬝ c)\n      exact ⟨.refl _, .red_S ..⟩\n    case par d c' hd hc =>\n      let ⟨a', b', h⟩ := Sab_irreducible a b d hd\n      rw [h.2.2]\n      use a' ⬝ c' ⬝ (b' ⬝ c')\n      constructor\n      · apply ParallelReduction.par\n        · exact .par h.left hc\n        · exact .par h.2.1 hc\n      · exact .red_S ..\n    case red_S => exact ⟨a ⬝ c ⬝ (b ⬝ c), .refl _, .refl _,⟩", "proof_length": 407, "num_lines": 64, "header": "", "file_path": "Cslib/Languages/CombinatoryLogic/Confluence.lean", "url": "https://github.com/leanprover/cslib", "start_line": 142, "end_line": 205, "version_info": [{"v4.33.0-rc2": "3aa9d4416c185e0b9faeb72bbd65abe85b95dbcc"}, {"v4.32.0": "197a7be621263b84c67ca4f803f69205b36d06df"}, {"v4.31.0": "8ea71c8df91acd63285515a04da952b23b78d266"}, {"v4.30.0": "c7944a9fb44c3298f1a960a5e574ab23a6ab8ed5"}]}
{"name": "Cslib.CCS.bisimilarity_congr_choice", "source": "cslib", "statement": "theorem bisimilarity_congr_choice :\n    (p ~[lts (defs := defs)] q) → (choice p r) ~[lts (defs := defs)] (choice q r)", "src": "theorem bisimilarity_congr_choice :\n    (p ~[lts (defs := defs)] q) → (choice p r) ~[lts (defs := defs)] (choice q r) := by\n  intro h\n  exists @ChoiceBisim _ _ defs\n  constructor\n  · constructor; assumption\n  intro s1 s2 r μ\n  constructor\n  case left =>\n    intro s1' htr\n    cases r\n    case choice p q r hbisim =>\n      obtain ⟨rel, hr, hb⟩ := hbisim\n      cases htr\n      case choiceL a b c htr =>\n        obtain ⟨s2', htr2, hr2⟩ := hb.follow_fst hr htr\n        exists s2'\n        constructor\n        · apply Tr.choiceL htr2\n        · constructor\n          apply hb.le_bisimilarity _ _ hr2\n      case choiceR a b c htr =>\n        exists s1'\n        constructor\n        · apply Tr.choiceR htr\n        · constructor\n          apply HomBisimilarity.refl\n    case bisim hbisim =>\n      obtain ⟨rel, hr, hb⟩ := hbisim\n      obtain ⟨s2', htr2, hr2⟩ := hb.follow_fst hr htr\n      exists s2'\n      constructor\n      · assumption\n      constructor\n      apply hb.le_bisimilarity _ _ hr2\n  case right =>\n    intro s2' htr\n    cases r\n    case choice p q r hbisim =>\n      obtain ⟨rel, hr, hb⟩ := hbisim\n      cases htr\n      case choiceL a b c htr =>\n        obtain ⟨s1', htr1, hr1⟩ := hb.follow_snd hr htr\n        exists s1'\n        constructor\n        · apply Tr.choiceL htr1\n        · constructor\n          apply hb.le_bisimilarity _ _ hr1\n      case choiceR a b c htr =>\n        exists s2'\n        constructor\n        · apply Tr.choiceR htr\n        · constructor\n          apply HomBisimilarity.refl\n    case bisim hbisim =>\n      obtain ⟨rel, hr, hb⟩ := hbisim\n      obtain ⟨s1', htr1, hr1⟩ := hb.follow_snd hr htr\n      exists s1'\n      constructor\n      · assumption\n      · constructor\n        apply hb.le_bisimilarity _ _ hr1", "proof_length": 251, "num_lines": 62, "header": "", "file_path": "Cslib/Languages/CCS/BehaviouralTheory.lean", "url": "https://github.com/leanprover/cslib", "start_line": 311, "end_line": 372, "version_info": [{"v4.33.0-rc2": "3aa9d4416c185e0b9faeb72bbd65abe85b95dbcc"}, {"v4.32.0": "197a7be621263b84c67ca4f803f69205b36d06df"}, {"v4.31.0": "8ea71c8df91acd63285515a04da952b23b78d266"}]}
{"name": "Binius.BinaryBasefold.fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius", "source": "arklib", "statement": "theorem fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius (i : Fin ℓ) (steps : ℕ)\n    [NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)\n    (f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)\n  (h_fw_dist_lt : fiberwiseClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)\n    (i := i) (steps := steps) (h_i_add_steps := h_i_add_steps) (f := f)) :\n  hammingClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩ f", "src": "theorem fiberwise_dist_lt_imp_dist_lt_unique_decoding_radius (i : Fin ℓ) (steps : ℕ)\n    [NeZero steps] (h_i_add_steps : i.val + steps ≤ ℓ)\n    (f : OracleFunction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)\n  (h_fw_dist_lt : fiberwiseClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)\n    (i := i) (steps := steps) (h_i_add_steps := h_i_add_steps) (f := f)) :\n  hammingClose 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩ f := by\n  unfold fiberwiseClose at h_fw_dist_lt\n  unfold hammingClose\n  -- 2 * Δ₀(f, ↑(BBF_Code 𝔽q β ⟨↑i, ⋯⟩)) < ↑(BBF_CodeDistance ℓ 𝓡 ⟨↑i, ⋯⟩)\n  let d_fw := fiberwiseDistance 𝔽q β (i := i) steps h_i_add_steps f\n  let C_i := (BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i, by omega⟩)\n  let d_H := Code.distFromCode f C_i\n  let d_i := BBF_CodeDistance ℓ 𝓡 (⟨i, by omega⟩)\n  let d_i_plus_steps := BBF_CodeDistance ℓ 𝓡 ⟨i.val + steps, by omega⟩\n\n  have h_d_i_gt_0 : d_i > 0 := by\n    dsimp [d_i, BBF_CodeDistance] -- ⊢ 2 ^ (ℓ + 𝓡 - ↑i) - 2 ^ (ℓ - ↑i) + 1 > 0\n    have h_exp_lt : ℓ - i.val < ℓ + 𝓡 - i.val := by\n      exact Nat.sub_lt_sub_right (a := ℓ) (b := ℓ + 𝓡) (c := i.val) (by omega) (by\n        apply Nat.lt_add_of_pos_right; exact pos_of_neZero 𝓡)\n    have h_pow_lt : 2 ^ (ℓ - i.val) < 2 ^ (ℓ + 𝓡 - i.val) := by\n      exact Nat.pow_lt_pow_right (by norm_num) h_exp_lt\n    omega\n\n  have h_C_i_nonempty : Nonempty C_i := by\n    simp only [nonempty_subtype, C_i]\n    exact Submodule.nonempty (BBF_Code 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) ⟨i.val, by omega⟩)\n\n  -- 1. Relate Hamming distance `d_H` to fiber-wise distance `d_fw`.\n  obtain ⟨g', h_g'_mem, h_g'_min_card⟩ : ∃ g' ∈ C_i, d_fw\n    = (fiberwiseDisagreementSet 𝔽q β i steps h_i_add_steps f g').ncard := by\n    -- Let `S` be the set of all possible fiber-wise disagreement sizes.\n    let S := (fun (g : C_i) => (fiberwiseDisagreementSet 𝔽q β i steps h_i_add_steps\n      f g).ncard) '' Set.univ\n    -- The code `C_i` (a submodule) is non-empty, so `S` is also non-empty.\n    have hS_nonempty : S.Nonempty := by\n      refine Set.image_nonempty.mpr ?_\n\n      exact Set.univ_nonempty\n    -- For a non-empty set of natural numbers, `sInf` is an element of the set.\n    have h_sInf_mem : sInf S ∈ S := Nat.sInf_mem hS_nonempty\n    -- By definition, `d_fw = sInf S`.\n    unfold d_fw at h_sInf_mem\n    -- Since `sInf S` is in the image set `S`, there must be an element `g_subtype` in the domain\n    -- (`C_i`) that maps to it. This `g_subtype` is the codeword we're looking for.\n    rw [Set.mem_image] at h_sInf_mem\n    rcases h_sInf_mem with ⟨g_subtype, _, h_eq⟩\n    -- Extract the codeword and its membership proof.\n    exact ⟨g_subtype.val, g_subtype.property, by exact id (Eq.symm h_eq)⟩\n\n  -- The Hamming distance to any codeword `g'` is bounded by `d_fw * 2 ^ steps`.\n  have h_dist_le_fw_dist_times_fiber_size : (hammingDist f g' : ℕ∞) ≤ d_fw * 2 ^ steps := by\n    -- This proves `dist f g' ≤ (fiberwiseDisagreementSet ... f g').ncard * 2 ^ steps`\n    -- and lifts to ℕ∞. We prove the `Nat` version `hammingDist f g' ≤ ...`,\n    -- which is equivalent.\n    change (Δ₀(f, g') : ℕ∞) ≤ ↑d_fw * ((2 ^ steps : ℕ) : ℕ∞)\n    rw [←ENat.coe_mul, ENat.coe_le_coe, h_g'_min_card]\n    -- Let ΔH be the finset of actually bad x points where f and g' disagree.\n    set ΔH := Finset.filter (fun x => f x ≠ g' x) Finset.univ\n    have h_dist_eq_card : hammingDist f g' = ΔH.card := by\n      simp only [hammingDist, ne_eq, ΔH]\n    rw [h_dist_eq_card]\n    -- Y_bad is the set of quotient points y that THERE EXISTS a bad fiber point x\n    set Y_bad := fiberwiseDisagreementSet 𝔽q β i steps h_i_add_steps f g'\n    simp only at * -- simplify domain indices everywhere\n\n    -- ⊢ #ΔH ≤ Y_bad.ncard * 2 ^ steps\n\n    have hFinType_Y_bad : Fintype Y_bad := by exact Fintype.ofFinite ↑Y_bad\n    -- Every point of disagreement `x` must belong to a fiber over some `y` in `Y_bad`,\n    -- BY DEFINITION of `Y_bad`. Therefore, `ΔH` is a subset of the union of the fibers\n    -- of `Y_bad`\n    have h_ΔH_subset_bad_fiber_points : ΔH ⊆ Finset.biUnion Y_bad.toFinset\n        (t := fun y => ((qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n          (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y)) ''\n          (Finset.univ : Finset (Fin ((2:ℕ)^steps)))).toFinset) := by\n      -- ⊢ If any x ∈ ΔH, then x ∈ Union(qMap_total_fiber(y), ∀ y ∈ Y_bad)\n      intro x hx_in_ΔH; -- ⊢ x ∈ Union(qMap_total_fiber(y), ∀ y ∈ Y_bad)\n      simp only [ΔH, Finset.mem_filter] at hx_in_ΔH\n      -- Now we actually apply iterated qMap into x to get y_of_x,\n      -- then x ∈ qMap_total_fiber(y_of_x) by definition\n      let y_of_x := iteratedQuotientMap 𝔽q β h_ℓ_add_R_rate i steps h_i_add_steps x\n      apply Finset.mem_biUnion.mpr; use y_of_x\n      -- ⊢ y_of_x ∈ Y_bad.toFinset ∧ x ∈ qMap_total_fiber(y_of_x)\n      have h_elemenet_Y_bad : y_of_x ∈ Y_bad.toFinset := by\n        -- ⊢ y ∈ Y_bad.toFinset\n        simp only [fiberwiseDisagreementSet, iteratedQuotientMap, ne_eq, Subtype.exists,\n          Set.toFinset_setOf, mem_filter, mem_univ, true_and, Y_bad]\n        -- one bad fiber point of y_of_x is x itself\n        let X := x.val\n        have h_X_in_source : X ∈ sDomain 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) := by\n          exact Submodule.coe_mem x\n        use X\n        use h_X_in_source\n        -- ⊢ Ŵ_steps⁽ⁱ⁾(X) = y (iterated quotient map) ∧ ¬f ⟨X, ⋯⟩ = g' ⟨X, ⋯⟩\n        have h_forward_iterated_qmap : Polynomial.eval X\n            (intermediateNormVpoly 𝔽q β h_ℓ_add_R_rate ⟨↑i, by omega⟩\n              ⟨steps, by simp only; omega⟩) = y_of_x := by\n          simp only [iteratedQuotientMap, X, y_of_x];\n        have h_eval_diff : f ⟨X, by omega⟩ ≠ g' ⟨X, by omega⟩ := by\n          unfold X\n          simp only [Subtype.coe_eta, ne_eq, hx_in_ΔH, not_false_eq_true]\n        simp only [h_forward_iterated_qmap, Subtype.coe_eta, h_eval_diff,\n          not_false_eq_true, and_self]\n      simp only [h_elemenet_Y_bad, true_and]\n\n      set qMapFiber := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y_of_x)\n      simp only [coe_univ, Set.image_univ, Set.toFinset_range, mem_image, mem_univ, true_and]\n      use (pointToIterateQuotientIndex (i := ⟨i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by omega) (x := x))\n      have h_res := is_fiber_iff_generates_quotient_point 𝔽q β i steps (by omega)\n        (x := x) (y := y_of_x).mp (by rfl)\n      exact h_res\n    -- ⊢ #ΔH ≤ Y_bad.ncard * 2 ^ steps\n    -- The cardinality of a subset is at most the cardinality of the superset.\n    apply (Finset.card_le_card h_ΔH_subset_bad_fiber_points).trans\n    -- The cardinality of a disjoint union is the sum of cardinalities.\n    rw [Finset.card_biUnion]\n    · -- The size of the sum is the number of bad fibers (`Y_bad.ncard`) times\n      -- the size of each fiber (`2 ^ steps`).\n      simp only [Set.toFinset_card]\n      have h_card_fiber_per_quotient_point := card_qMap_total_fiber 𝔽q β\n        (h_ℓ_add_R_rate := h_ℓ_add_R_rate) i steps h_i_add_steps\n      simp only [Set.image_univ, Fintype.card_ofFinset,\n        Subtype.forall] at h_card_fiber_per_quotient_point\n      have h_card_fiber_of_each_y : ∀ y ∈ Y_bad.toFinset,\n          Fintype.card ((qMap_total_fiber 𝔽q β (i := ⟨↑i, by omega⟩) (steps := steps)\n            (h_i_add_steps := by apply Nat.lt_add_of_pos_right_of_le; omega) (y := y)) ''\n            ↑(Finset.univ : Finset (Fin ((2:ℕ)^steps)))) = 2 ^ steps := by\n        intro y hy_in_Y_bad\n        have hy_card_fiber_of_y := h_card_fiber_per_quotient_point (a := y) (b := by\n          exact Submodule.coe_mem y)\n        simp only [coe_univ, Set.image_univ, Fintype.card_ofFinset, hy_card_fiber_of_y]\n      rw [Finset.sum_congr rfl h_card_fiber_of_each_y]\n      -- ⊢ ∑ x ∈ Y_bad.toFinset, 2 ^ steps ≤ Y_bad.encard.toNat * 2 ^ steps\n      simp only [sum_const, Set.toFinset_card, smul_eq_mul, ofNat_pos, pow_pos,\n        _root_.mul_le_mul_right, ge_iff_le]\n      conv_rhs => rw [←_root_.Nat.card_coe_set_eq] -- convert .ncard back to .card\n      -- ⊢ Fintype.card ↑Y_bad ≤ Nat.card ↑Y_bad\n      simp only [card_eq_fintype_card, le_refl]\n    · -- Prove that the fibers for distinct quotient points y₁, y₂ are disjoint.\n      intro y₁ hy₁ y₂ hy₂ hy_ne\n      have h_disjoint := qMap_total_fiber_disjoint (i := ⟨↑i, by omega⟩) (steps := steps)\n        (h_i_add_steps := by omega) (y₁ := y₁) (y₂ := y₂) (hy_ne := hy_ne)\n      simp only [Function.onFun, coe_univ]\n      exact h_disjoint\n\n  -- The minimum distance `d_H` is bounded by the distance to this specific `g'`.\n  have h_dist_bridge : d_H ≤ d_fw * 2 ^ steps := by\n    -- exact h_dist_le_fw_dist_times_fiber_size\n    apply le_trans (a := d_H) (c := d_fw * 2 ^ steps) (b := hammingDist f g')\n    · -- ⊢ d_H ≤ ↑Δ₀(f, g')\n      simp only [distFromCode, SetLike.mem_coe, hammingDist, ne_eq, d_H];\n      -- ⊢ Δ₀(f, C_i) ≤ ↑Δ₀(f, g')\n      -- ⊢ sInf {d | ∃ v ∈ C_i, ↑(#{i | f i ≠ v i}) ≤ d} ≤ ↑(#{i | f i ≠ g' i})\n      apply sInf_le\n      use g'\n    · exact h_dist_le_fw_dist_times_fiber_size\n\n  -- 2. Use the premise : `2 * d_fw < d_{i+steps}`.\n  -- As a `Nat` inequality, this is equivalent to `2 * d_fw ≤ d_{i+steps} - 1`.\n  have h_fw_bound : 2 * d_fw ≤ d_i_plus_steps - 1 := by\n    -- Convert the ENat inequality to a Nat inequality using `a < b ↔ a + 1 ≤ b`.\n    exact Nat.le_of_lt_succ (WithTop.coe_lt_coe.1 h_fw_dist_lt)\n\n  -- 3. The Algebraic Identity.\n  -- The core of the proof is the identity : `(d_{i+steps} - 1) * 2 ^ steps = d_i - 1`.\n  have h_algebraic_identity : (d_i_plus_steps - 1) * 2 ^ steps = d_i - 1 := by\n    dsimp [d_i, d_i_plus_steps, BBF_CodeDistance]\n    rw [Nat.sub_mul, ←Nat.pow_add, ←Nat.pow_add];\n    have h1 : ℓ + 𝓡 - (↑i + steps) + steps = ℓ + 𝓡 - i := by\n      rw [Nat.sub_add_eq_sub_sub_rev (h1 := by omega) (h2 := by omega),\n        Nat.add_sub_cancel (n := i) (m := steps)]\n    have h2 : (ℓ - (↑i + steps) + steps) = ℓ - i := by\n      rw [Nat.sub_add_eq_sub_sub_rev (h1 := by omega) (h2 := by omega),\n        Nat.add_sub_cancel (n := i) (m := steps)]\n    rw [h1, h2]\n\n  -- 4. Conclusion : Chain the inequalities to prove `2 * d_H < d_i`.\n  -- We know `d_H` is finite, since `C_i` is nonempty.\n  have h_dH_ne_top : d_H ≠ ⊤ := by\n    simp only [ne_eq, d_H]\n    rw [Code.distFromCode_eq_top_iff_empty f C_i]\n    exact Set.nonempty_iff_ne_empty'.mp h_C_i_nonempty\n\n  -- We can now work with the `Nat` value of `d_H`.\n  let d_H_nat := ENat.toNat d_H\n  have h_dH_eq : d_H = d_H_nat := (ENat.coe_toNat h_dH_ne_top).symm\n\n  -- The calculation is now done entirely in `Nat`.\n  have h_final_inequality : 2 * d_H_nat ≤ d_i - 1 := by\n    have h_bridge_nat : d_H_nat ≤ d_fw * 2 ^ steps := by\n        rw [←ENat.coe_le_coe]\n        exact le_of_eq_of_le (id (Eq.symm h_dH_eq)) h_dist_bridge\n    calc 2 * d_H_nat\n      _ ≤ 2 * (d_fw * 2 ^ steps) := by gcongr\n      _ = (2 * d_fw) * 2 ^ steps := by rw [mul_assoc]\n      _ ≤ (d_i_plus_steps - 1) * 2 ^ steps := by gcongr;\n      _ = d_i - 1 := h_algebraic_identity\n\n  simp only [d_H, d_H_nat] at h_dH_eq\n  -- This final line is equivalent to the goal statement.\n  rw [h_dH_eq]\n  -- ⊢ 2 * ↑Δ₀(f, C_i).toNat < ↑(BBF_CodeDistance ℓ 𝓡 ⟨↑i, ⋯⟩)\n  change ((2 : ℕ) : ℕ∞) * ↑Δ₀(f, C_i).toNat < ↑(BBF_CodeDistance ℓ 𝓡 ⟨↑i, by omega⟩)\n  rw [←ENat.coe_mul, ENat.coe_lt_coe]\n  apply Nat.lt_of_le_pred (n := 2 * Δ₀(f, C_i).toNat) (m := d_i) (h := h_d_i_gt_0)\n    (h_final_inequality)", "proof_length": 1463, "num_lines": 209, "header": "", "file_path": "ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean", "url": "https://github.com/Verified-zkEVM/ArkLib", "start_line": 1265, "end_line": 1473, "version_info": [{"v4.31.0": "c1712f6c496d0aef04f09bae42aec13811b3273f"}, {"v4.30.0": "e6d77b18ca1334a91faf4d1ccf9f96d854d58ba4"}, {"v4.29.0": "de71a9dfd2f5a343e8895517b7efe857d9f1ce5c"}, {"v4.28.0": "673fa440714eb1949b8b50a28f2a4f52b1837e9b"}]}
{"name": "Binius.BinaryBasefold.fold_advances_evaluation_poly", "source": "arklib", "statement": "theorem fold_advances_evaluation_poly\n  (i : Fin (ℓ)) (h_i_succ_lt : i + 1 < ℓ + 𝓡)\n  (coeffs : Fin (2 ^ (ℓ - ↑i)) → L) (r_chal : L) :\n  let P_i : L[X] := intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by\n    exact Nat.lt_trans (n := i) (k := ℓ+1) (m := ℓ) (h₁ := i.isLt) (by exact Nat.lt_add_one ℓ)\n  ⟩) coeffs\n  let f_i := fun (x : (sDomain 𝔽q β h_ℓ_add_R_rate)\n      ⟨i, by exact Nat.lt_trans (n := i) (k := r) (m := ℓ) (h₁ := by omega) (by omega)⟩) =>\n    P_i.eval (x.val : L)\n  let f_i_plus_1 := fold (i := ⟨i, by omega⟩) (h_i := by omega) (f := f_i) (r_chal := r_chal)\n  let new_coeffs := fun j : Fin (2^(ℓ - (i + 1))) =>\n    (1 - r_chal) * (coeffs ⟨j.val * 2, by\n      rw [←Nat.add_zero (j.val * 2)]\n      apply mul_two_add_bit_lt_two_pow (c := ℓ - i) (a := j) (b := ℓ - (↑i + 1))\n        (i := 0) (by omega) (by omega)\n    ⟩) +\n    r_chal * (coeffs ⟨j.val * 2 + 1, by\n      apply mul_two_add_bit_lt_two_pow (c := ℓ - i) (a := j) (b := ℓ - (↑i + 1))\n        (i := 1) (by omega) (by omega)\n    ⟩)\n  let P_i_plus_1 :=\n    intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i+1, by omega⟩) new_coeffs\n  ∀ (y : (sDomain 𝔽q β h_ℓ_add_R_rate)\n    ⟨i+1, by omega⟩), f_i_plus_1 y = P_i_plus_1.eval y.val", "src": "theorem fold_advances_evaluation_poly\n  (i : Fin (ℓ)) (h_i_succ_lt : i + 1 < ℓ + 𝓡)\n  (coeffs : Fin (2 ^ (ℓ - ↑i)) → L) (r_chal : L) :\n  let P_i : L[X] := intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by\n    exact Nat.lt_trans (n := i) (k := ℓ+1) (m := ℓ) (h₁ := i.isLt) (by exact Nat.lt_add_one ℓ)\n  ⟩) coeffs\n  let f_i := fun (x : (sDomain 𝔽q β h_ℓ_add_R_rate)\n      ⟨i, by exact Nat.lt_trans (n := i) (k := r) (m := ℓ) (h₁ := by omega) (by omega)⟩) =>\n    P_i.eval (x.val : L)\n  let f_i_plus_1 := fold (i := ⟨i, by omega⟩) (h_i := by omega) (f := f_i) (r_chal := r_chal)\n  let new_coeffs := fun j : Fin (2^(ℓ - (i + 1))) =>\n    (1 - r_chal) * (coeffs ⟨j.val * 2, by\n      rw [←Nat.add_zero (j.val * 2)]\n      apply mul_two_add_bit_lt_two_pow (c := ℓ - i) (a := j) (b := ℓ - (↑i + 1))\n        (i := 0) (by omega) (by omega)\n    ⟩) +\n    r_chal * (coeffs ⟨j.val * 2 + 1, by\n      apply mul_two_add_bit_lt_two_pow (c := ℓ - i) (a := j) (b := ℓ - (↑i + 1))\n        (i := 1) (by omega) (by omega)\n    ⟩)\n  let P_i_plus_1 :=\n    intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i+1, by omega⟩) new_coeffs\n  ∀ (y : (sDomain 𝔽q β h_ℓ_add_R_rate)\n    ⟨i+1, by omega⟩), f_i_plus_1 y = P_i_plus_1.eval y.val := by\n  simp only\n  intro y\n  set fiberMap := qMap_total_fiber 𝔽q β (i := ⟨i, by omega⟩) (steps := 1)\n    (h_i_add_steps := by simp only; omega) (y := y)\n  set x₀ := fiberMap 0\n  set x₁ := fiberMap 1\n  set P_i := intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i, by omega⟩) coeffs\n  set new_coeffs := fun j : Fin (2^(ℓ - (i + 1))) =>\n    (1 - r_chal) * (coeffs ⟨j.val * 2, by\n      have h : j.val * 2 < 2^(ℓ - (i + 1)) * 2 := by omega\n      have h2 : 2^(ℓ - i) = 2^(ℓ - (i + 1)) * 2 := by\n        conv_rhs => enter[2]; rw [←Nat.pow_one 2]\n        rw [←pow_add]; congr\n        rw [Nat.sub_add_eq_sub_sub_rev (h1 := by omega) (h2 := by omega)]\n        -- ⊢ ℓ - ↑i = ℓ - (↑i + 1 - 1)\n        rw [Nat.add_sub_cancel (n := i) (m := 1)]\n      omega\n    ⟩) +\n    r_chal * (coeffs ⟨j.val * 2 + 1, by\n      apply mul_two_add_bit_lt_two_pow (c := ℓ - i) (a := j) (b := ℓ - (↑i + 1)) (i := 1)\n      · omega\n      · omega\n    ⟩)\n  have h_eval_qMap_x₀ : (AdditiveNTT.qMap 𝔽q β ⟨i, by omega⟩).eval x₀.val = y := by\n    have h := iteratedQuotientMap_k_eq_1_is_qMap 𝔽q β h_ℓ_add_R_rate i (by omega) x₀\n    simp only [Subtype.eq_iff] at h\n    rw [h.symm]\n    have h_res := is_fiber_iff_generates_quotient_point 𝔽q β i (steps := 1) (by omega)\n      (x := x₀) (y := y).mpr (by rw [pointToIterateQuotientIndex_qMap_total_fiber_eq_self])\n    rw [h_res]\n    -- exact qMap_eval_fiber_eq_self ⟦L⟧ ⟨i + 1, by omega⟩ (by simp only; omega) h_i_succ_lt y 0\n  have h_eval_qMap_x₁ : (AdditiveNTT.qMap 𝔽q β ⟨i, by omega⟩).eval x₁.val = y := by\n    have h := iteratedQuotientMap_k_eq_1_is_qMap 𝔽q β h_ℓ_add_R_rate i (by omega) x₁\n    simp only [Subtype.eq_iff] at h\n    rw [h.symm]\n    have h_res := is_fiber_iff_generates_quotient_point 𝔽q β i (steps := 1) (by omega)\n      (x := x₁) (y := y).mpr (by rw [pointToIterateQuotientIndex_qMap_total_fiber_eq_self])\n    rw [h_res]\n  have hx₀ := qMap_total_fiber_basis_sum_repr 𝔽q β i (steps := 1)\n    (h_i_add_steps := by omega) y 0\n  have hx₁ := qMap_total_fiber_basis_sum_repr 𝔽q β i (steps := 1)\n    (h_i_add_steps := by omega) y 1\n  simp only [Fin.isValue] at hx₀ hx₁\n\n  have h_fiber_diff : x₁.val - x₀.val = 1 := by\n    simp only [Fin.isValue, x₁, x₀, fiberMap]\n    rw [hx₁, hx₀]\n    simp only [Fin.isValue, AddSubmonoidClass.coe_finset_sum, SetLike.val_smul]\n    have h_index : ℓ + 𝓡 - i = (ℓ + 𝓡 - (i.val + 1)) + 1 := by omega\n    rw! (castMode := .all) [h_index]\n    rw [Fin.sum_univ_succ, Fin.sum_univ_succ] -- (free_term + y_repr) - (free_term + y_repr) = 1\n    -- First, simplify the free terms\n    simp only [fiber_coeff, eqRec_eq_cast, lt_one_iff, reducePow, Fin.isValue,\n      Fin.coe_ofNat_eq_mod, mod_succ, dite_smul, ite_smul, zero_smul, one_smul, zero_mod]\n    have h_cast_0 :\n        (cast (Eq.symm h_index ▸ rfl : Fin (ℓ + 𝓡 - (↑i + 1) + 1) = Fin (ℓ + 𝓡 - ↑i)) 0).val =\n        0 := by\n      rw [←Fin.cast_eq_cast (h := by omega)]\n      rw [Fin.cast_val_eq_val (h_eq := by omega)]\n      simp only [Fin.coe_ofNat_eq_mod, mod_succ_eq_iff_lt, succ_eq_add_one, lt_add_iff_pos_left]\n      omega\n    have h_cast_1 :\n        (cast (Eq.symm h_index ▸ rfl : Fin (ℓ + 𝓡 - (↑i + 1) + 1) = Fin (ℓ + 𝓡 - ↑i)) 1).val =\n        1 := by\n      rw [←Fin.cast_eq_cast (h := by omega)]\n      rw [Fin.cast_val_eq_val (h_eq := by omega)]\n      simp only [Fin.coe_ofNat_eq_mod, mod_succ_eq_iff_lt, succ_eq_add_one,\n        lt_add_iff_pos_left, tsub_pos_iff_lt]\n      omega\n    simp only [h_cast_0, ↓reduceDIte]\n    have h_getBit_0_of_0 : Nat.getBit (k := 0) (n := 0) = 0 := by\n      simp only [getBit, shiftRight_zero, and_one_is_mod, zero_mod]\n    have h_getBit_0_of_1 : Nat.getBit (k := 0) (n := 1) = 1 := by\n      simp only [getBit, shiftRight_zero, Nat.and_self]\n    simp only [h_getBit_0_of_1, one_ne_zero, ↓reduceIte, h_getBit_0_of_0, zero_add]\n    rw! (castMode := .all) [←h_index]\n    rw [cast_eq]\n    simp only [get_sDomain_basis, Fin.coe_ofNat_eq_mod, zero_mod, add_zero, cast_eq]\n    rw [normalizedWᵢ_eval_βᵢ_eq_1 𝔽q β]\n    ring_nf\n    conv_rhs => rw [←add_zero (a := 1)]\n    rw [add_sub_assoc]\n    congr 1\n    rw [sub_eq_zero]\n    apply Finset.sum_congr (h := by rfl)\n    simp only [mem_univ, congr_eqRec, Fin.val_succ, Nat.add_eq_zero, one_ne_zero, and_false,\n      ↓reduceDIte, add_tsub_cancel_right, Fin.eta, imp_self, implies_true]\n  set P_i_plus_1 :=\n    intermediateEvaluationPoly 𝔽q β h_ℓ_add_R_rate (i := ⟨i+1, by omega⟩) new_coeffs\n  -- Set up the even and odd refinement polynomials\n  set P₀_coeffs := fun j : Fin (2^(ℓ - (i + 1))) => coeffs ⟨j.val * 2, by\n    have h1 : ℓ - (i + 1) + 1 = ℓ - i := by omega\n    have h2 : 2^(ℓ - (i + 1) + 1) = 2^(ℓ - i) := by rw [h1]\n    have h3 : 2^(ℓ - (i + 1)) * 2 = 2^(ℓ - (i + 1) + 1) := by rw [pow_succ]\n    rw [← h2, ← h3]; omega⟩\n  set P₁_coeffs := fun j : Fin (2^(ℓ - (i + 1))) => coeffs ⟨j.val * 2 + 1, by\n    have h1 : ℓ - (i + 1) + 1 = ℓ - i := by omega\n    have h2 : 2^(ℓ - (i + 1) + 1) = 2^(ℓ - i) := by rw [h1]\n    have h3 : 2^(ℓ - (i + 1)) * 2 = 2^(ℓ - (i + 1) + 1) := by rw [pow_succ]\n    rw [← h2, ← h3]; omega⟩\n  set P₀ := evenRefinement 𝔽q β h_ℓ_add_R_rate i coeffs\n  set P₁ := oddRefinement 𝔽q β h_ℓ_add_R_rate i coeffs\n  have h_P_i_eval := evaluation_poly_split_identity 𝔽q β h_ℓ_add_R_rate ⟨i, by omega⟩ coeffs\n  -- Equation 39 : P^(i)(X) = P₀^(i+1)(q^(i)(X)) + X · P₁^(i+1)(q^(i)(X))\n  have h_equation_39_x₀ : P_i.eval x₀.val = P₀.eval y.val + x₀.val * P₁.eval y.val := by\n    simp only [h_P_i_eval, Fin.eta, Polynomial.eval_add, eval_comp,\n      h_eval_qMap_x₀, Polynomial.eval_mul, Polynomial.eval_X, P_i, P₀, P₁]\n  have h_equation_39_x₁ : P_i.eval x₁.val = P₀.eval y.val + x₁.val * P₁.eval y.val := by\n    simp only [h_P_i_eval, Fin.eta, Polynomial.eval_add, eval_comp,\n      h_eval_qMap_x₁, Polynomial.eval_mul, Polynomial.eval_X, P_i, P₀, P₁]\n  set f_i := fun (x : (sDomain 𝔽q β h_ℓ_add_R_rate) ⟨i, by omega⟩) => P_i.eval (x.val : L)\n  set f_i_plus_1 := fold (i := ⟨i, by omega⟩) (h_i := by omega) (f := f_i) (r_chal := r_chal)\n  -- Unfold the definition of f_i_plus_1 using the fold function\n  have h_fold_def : f_i_plus_1 y =\n      f_i x₀ * ((1 - r_chal) * x₁.val - r_chal) +\n      f_i x₁ * (r_chal - (1 - r_chal) * x₀.val) := rfl\n  -- Main calculation following the outline\n  calc f_i_plus_1 y\n    = f_i x₀ * ((1 - r_chal) * x₁.val - r_chal) +\n        f_i x₁ * (r_chal - (1 - r_chal) * x₀.val) := h_fold_def\n    _ = P_i.eval x₀.val * ((1 - r_chal) * x₁.val - r_chal) +\n        P_i.eval x₁.val * (r_chal - (1 - r_chal) * x₀.val) := by simp only [f_i]\n    _ = (P₀.eval y.val + x₀.val * P₁.eval y.val) * ((1 - r_chal) * x₁.val - r_chal) +\n        (P₀.eval y.val + x₁.val * P₁.eval y.val) * (r_chal - (1 - r_chal) * x₀.val) := by\n      rw [h_equation_39_x₀, h_equation_39_x₁]\n    _ = P₀.eval y.val * ((1 - r_chal) * x₁.val - r_chal + r_chal - (1 - r_chal) * x₀.val) +\n        P₁.eval y.val * (x₀.val * ((1 - r_chal) * x₁.val - r_chal) +\n          x₁.val * (r_chal - (1 - r_chal) * x₀.val)) := by ring\n    _ = P₀.eval y.val * ((1 - r_chal) * (x₁.val - x₀.val)) +\n        P₁.eval y.val * ((x₁.val - x₀.val) * r_chal) := by ring\n    _ = P₀.eval y.val * (1 - r_chal) + P₁.eval y.val * r_chal := by rw [h_fiber_diff]; ring\n    _ = P_i_plus_1.eval y.val := by\n      simp only [P_i_plus_1, P₀, P₁, new_coeffs, evenRefinement, oddRefinement,\n        intermediateEvaluationPoly]\n      conv_lhs => enter [1]; rw [mul_comm, ←Polynomial.eval_C_mul]\n      conv_lhs => enter [2]; rw [mul_comm, ←Polynomial.eval_C_mul]\n      -- ⊢ eval y (C (1-r) * ∑...) + eval y (C r * ∑...) = eval y (∑...)\n      rw [←Polynomial.eval_add]\n      -- ⊢ poly_left.eval y = poly_right.eval y\n      congr\n      simp_rw [mul_sum, ←Finset.sum_add_distrib]\n      -- We now prove that the terms inside the sums are equal for each index.\n      apply Finset.sum_congr rfl\n      intro j hj\n      have h_j_lt : j.val < 2 ^ (ℓ - (↑i + 1)) := by\n        rw [Nat.sub_add_eq]\n        omega\n      conv_lhs => enter [1]; rw [mul_comm (a := Polynomial.C (coeffs ⟨j.val * 2, by\n        rw [←Nat.add_zero (j.val * 2)]\n        apply mul_two_add_bit_lt_two_pow (c := ℓ - i) (a := j) (b := ℓ - (↑i + 1))\n          (i := 0) (by omega) (by omega)⟩)), ←mul_assoc,\n        mul_comm (a := Polynomial.C (1 - r_chal))]; rw [mul_assoc]\n      conv_lhs => enter [2]; rw [mul_comm (a := Polynomial.C (coeffs ⟨j.val * 2 + 1, by\n        apply mul_two_add_bit_lt_two_pow (c := ℓ - i) (a := j) (b := ℓ - (↑i + 1))\n          (i := 1) (by omega) (by omega)⟩)), ←mul_assoc,\n        mul_comm (a := Polynomial.C r_chal)]; rw [mul_assoc]\n      conv_rhs => rw [mul_comm]\n      rw [←mul_add]\n      congr\n      simp only [←Polynomial.C_mul, ←Polynomial.C_add]", "proof_length": 1906, "num_lines": 184, "header": "", "file_path": "ArkLib/ProofSystem/Binius/BinaryBasefold/Prelude.lean", "url": "https://github.com/Verified-zkEVM/ArkLib", "start_line": 815, "end_line": 998, "version_info": [{"v4.31.0": "c1712f6c496d0aef04f09bae42aec13811b3273f"}, {"v4.30.0": "e6d77b18ca1334a91faf4d1ccf9f96d854d58ba4"}, {"v4.29.0": "de71a9dfd2f5a343e8895517b7efe857d9f1ce5c"}]}
{"name": "interleaved_affine_gaps_imply_tensor_gaps", "source": "arklib", "statement": "theorem interleaved_affine_gaps_imply_tensor_gaps\n    (hC_proximityGapAffineLines : e_ε_correlatedAgreementAffineLinesNat\n      (F := F) (A := A) (ι := ι) (C := MC) (e := e) (ε := ε))\n    (h_interleaved_gaps : ∀ m : ℕ, m ≥ 1 →\n      e_ε_correlatedAgreementAffineLinesNat (F := F)\n        (A := InterleavedSymbol A (Fin m)) (ι := ι) (C := MC ^⋈ (Fin m)) e ε) :\n    ∀ (ϑ : ℕ), (hϑ_gt_0 : ϑ > 0) → δ_ε_multilinearCorrelatedAgreement_Nat (F := F) (A := A)\n      (ι := ι) (C := MC) (ϑ := ϑ) (e := e) (ε := ε)", "src": "theorem interleaved_affine_gaps_imply_tensor_gaps\n    (hC_proximityGapAffineLines : e_ε_correlatedAgreementAffineLinesNat\n      (F := F) (A := A) (ι := ι) (C := MC) (e := e) (ε := ε))\n    (h_interleaved_gaps : ∀ m : ℕ, m ≥ 1 →\n      e_ε_correlatedAgreementAffineLinesNat (F := F)\n        (A := InterleavedSymbol A (Fin m)) (ι := ι) (C := MC ^⋈ (Fin m)) e ε) :\n    ∀ (ϑ : ℕ), (hϑ_gt_0 : ϑ > 0) → δ_ε_multilinearCorrelatedAgreement_Nat (F := F) (A := A)\n      (ι := ι) (C := MC) (ϑ := ϑ) (e := e) (ε := ε) := by\n    classical\n    intro ϑ\n    induction ϑ with\n    | zero =>\n      intro u\n      contradiction\n    | succ ϑ_sub_1 ih =>\n      cases ϑ_sub_1 with\n      | zero =>\n        -- `ϑ = 1`\n        simp only [zero_add, gt_iff_lt, zero_lt_one, ModuleCode, forall_const]\n        intro u\n        let toAffineLineEval := multilinearCombine₁_eq_affineLineEvaluation (F := F) (u := u)\n        intro hprob_gt\n        simp_rw [toAffineLineEval] at hprob_gt\n        let prob_eq := prob_uniform_singleton_finFun_eq (F := F)\n          (P := fun r => Δ₀(affineLineEvaluation (u 0) (u 1) r, MC) ≤ e)\n        -- Convert sampling (r ← (Fin 1 → F)) into sampling (r ← F)\n        simp_rw [prob_eq, Nat.cast_one, ENNReal.coe_one, one_mul] at hprob_gt\n        have h_correlated_agreement := hC_proximityGapAffineLines (u 0) (u 1) hprob_gt\n        simp only [jointProximityNat₂, Fin.isValue] at h_correlated_agreement\n        have h_u_eq: u = finMapTwoWords (u 0) (u 1) := by\n          funext rowIdx\n          match rowIdx with\n          | 0 => rfl\n          | 1 => rfl\n        rw [h_u_eq.symm] at h_correlated_agreement\n        exact h_correlated_agreement\n      | succ ϑ_sub_2 =>\n        intro hϑ_gt_0 u hP_multilinearCombine_close_gt\n        -- `ϑ ≥ 2`\n        -- set ϑ_sub_1 := ϑ_sub_2 + 1\n        let ϑ := ϑ_sub_2 + 1 + 1\n        have hϑ : ϑ = ϑ_sub_2 + 1 + 1 := by rfl\n        -- have hfinϑ : Fin ϑ = Fin (ϑ_sub_2 + 1 + 1) := by rfl\n        set U₀ : Fin (2 ^ (ϑ_sub_2 + 1)) → Word A ι :=\n          (splitHalfRowWiseInterleavedWords (ϑ := ϑ_sub_2 + 1) u).1\n        set U₁ : Fin (2 ^ (ϑ_sub_2 + 1)) → Word A ι :=\n          (splitHalfRowWiseInterleavedWords (ϑ := ϑ_sub_2 + 1) u).2\n        unfold jointProximityNat\n        simp only\n        -- intro hP_multilinearCombine_close_gt\n        have h_finsnoc_eq_r: ∀ r: Fin (ϑ_sub_2 + 1 + 1) → F,\n          Fin.snoc (fun (i : Fin (ϑ_sub_2 + 1)) ↦ r i.castSucc)\n            (r (Fin.last (ϑ_sub_2 + 1))) = r := fun r => by\n          funext i\n          have hi_le_lt := i.isLt\n          by_cases h : i = Fin.last (ϑ_sub_2 + 1)\n          · simp only [h, Fin.snoc_last]\n          · have h_i_ne_last: i ≠ Fin.last (ϑ_sub_2 + 1) := by omega\n            have h_i_ne_last_val: i.val ≠ Fin.last (ϑ_sub_2 + 1) := by omega\n            rw [Fin.val_last] at h_i_ne_last_val\n            have h_i_lt: i.val < (ϑ_sub_2 + 1) := by omega\n            simp only [Fin.snoc, Fin.castSucc_castLT, cast_eq, dite_eq_ite, ite_eq_left_iff, not_lt]\n            simp only [isEmpty_Prop, not_le, h_i_lt, IsEmpty.forall_iff]\n        let P : F → (Fin (ϑ_sub_2 + 1) → F) → Prop := fun r_last r_init =>\n          Δ₀(multilinearCombine (u:=u) (r:=Fin.snoc r_init r_last), MC) ≤ e\n        let hP_split_r_last := prob_split_last_uniform_sampling_of_finFun\n          (ϑ := ϑ_sub_2 + 1) (F := F) (P := P)\n        unfold P at hP_split_r_last\n        simp_rw [h_finsnoc_eq_r] at hP_split_r_last\n        rw [hP_split_r_last] at hP_multilinearCombine_close_gt\n        -- Now we have two randomness sampling in hP_multilinearCombine_close_gt :\n        -- `((ϑ_sub_2 + 1 + 1) * ε) / |𝔽|\n          -- < Pr_{ r_last; r_init }[  Δ₀((Fin.snoc r_init r_last)|⨂|u, ↑MC) ≤ ↑e)) ]` (0)\n        -- We need to achieve the upperbound for hP_multilinearCombine_close_gt probability by:\n        -- i.e. `Pr_{ r_last; r_init }[  Δ₀((Fin.snoc r_init r_last)|⨂|u, ↑MC) ≤ ↑e)) ]`\n        -- `= Pr_{ r_last; r_init }[ Δ₀((r_init)|⨂|affineCombine(U₀, U₁, r_last), ↑MC) ≤ ↑e)) ]`\n        -- `= PR_{ r_last }[ r_init; Δ₀((r_init)|⨂|affineCombine(U₀, U₁, r_last), ↑MC) ≤ ↑e)) ]`\n        -- Divide into two cases: r_last ∈ R* and r_last ∉ R*\n        -- `= Pr_{ r_last }[ r_last ∈ R* ∧\n          -- Pr_{ r_init }[ Δ₀((r_init)|⨂|affineCombine(U₀, U₁, r_last), ↑MC) ≤ ↑e)) ]` (1)\n        -- `+ Pr_{ r_last }[ r_last ∉ R* ∧\n          -- Pr_{ r_init }[ Δ₀((r_init)|⨂|affineCombine(U₀, U₁, r_last), ↑MC) ≤ ↑e)) ]` (2)\n        -- `(1) = Pr_{ r_last }[ r_last ∈ R* ]` (3)\n        -- `(2): ∀ r ∉ R*, it's trivial that the probability\n          -- ≤ ((ϑ_sub_2 + 1) * ε) / |𝔽|, due to definition of membership of R*` (4)\n\n        -- Combine (0), (3), (4): we have `Pr_{ r_last }[ r_last ∈ R* ] > ε/|𝔽|` (5)\n\n        -- Applying `correlatedAgreement_of_mem_R_star_tensor` to (5), we get\n          -- `Δ₀((r_last)|⨂|affineCombine(U₀, U₁, r_last), ↑MC) ≤ ↑e)) ] > ε/|𝔽|` (6)\n        -- This is premise for affineProxmityGaps of interleaved code (`h_interleaved_gaps`)\n          -- for `m = 2^{ϑ_sub_2 + 1}`, which directly leads to Q.E.D.\n        let  ϑ_pred := ϑ_sub_2 + 1\n        have h_ϑ_pred : ϑ_pred = ϑ_sub_2 + 1 := by rfl\n        have h_ϑ : ϑ = ϑ_pred + 1 := by rfl\n        -- Step 1: Apply multilinearCombine recursive form\n        -- We need the lemma `multilinearCombine u r = multilinearCombine`\n          -- `(affineLineEvaluation U₀ U₁ r_last) r_init` where `r = Fin.snoc r_init r_last`\n        have multilinearCombine_snoc_eq_multilinearCombine_affine\n          (r_init : Fin ϑ_pred → F) (r_last : F) : multilinearCombine u (Fin.snoc r_init r_last) =\n            multilinearCombine (affineLineEvaluation U₀ U₁ r_last) r_init := by\n          rw [multilinearCombine_recursive_form]\n          simp only [Fin.snoc_last, Fin.init_snoc]\n          rfl\n        -- Rewrite the probability using this identity\n        simp_rw [multilinearCombine_snoc_eq_multilinearCombine_affine]\n          at hP_multilinearCombine_close_gt\n        -- hP_multilinearCombine_close_gt now looks like:\n        -- Pr_{ r_last ← $ᵖ F; r_init ← $ᵖ (Fin ϑ_pred → F) }[ Δ₀(multilinearCombine\n            -- (affineLineEvaluation U₀ U₁ r_last) r_init, ↑MC) ≤ ↑e ] > ↑(↑ϑ * ↑ε) / q\n        -- Step 2 & 3: Define R* and apply Law of Total Probability\n        let R_star_set := R_star_tensor MC (m:=ϑ_pred) (e:=e) (ε:=ε) U₀ U₁\n        -- Step 5: Show Pr[R*] > ε / q\n        have h_prob_Rstar_gt_eps_div_q : Pr_{ let r ← $ᵖ F }[ r ∈ R_star_set ]\n          > (↑ε : ENNReal) / (Fintype.card F : ENNReal) := by\n          let res := prob_R_star_gt_threshold (MC := MC) (ε := ε) (ϑ := ϑ_sub_2 + 1) (u := u)\n            (e := e) (hP_multilinearCombine_close_gt)\n          exact res\n        -- Convert Pr_{}[] to cardinality\n        have h_R_star_card_gt_eps : R_star_set.card > ε := by\n          rw [prob_uniform_eq_card_filter_div_card] at h_prob_Rstar_gt_eps_div_q -- Needs NNReal\n          simp only [ENNReal.coe_natCast] at h_prob_Rstar_gt_eps_div_q\n          rw [gt_iff_lt] at h_prob_Rstar_gt_eps_div_q\n          have h_cancel_q_denom := ENNReal.div_lt_div_iff_left\n            (a := (ε : ENNReal))\n            (b := (#(filter (fun r => r ∈ R_star_set) Finset.univ)))\n            (c := (Fintype.card F : ENNReal)) (hc₀ := by simp only [ne_eq,\n            Nat.cast_eq_zero, Fintype.card_ne_zero, not_false_eq_true]) (hc := by simp only [ne_eq,\n              ENNReal.natCast_ne_top, not_false_eq_true])\n          rw [h_cancel_q_denom] at h_prob_Rstar_gt_eps_div_q\n          simp only [filter_univ_mem, Nat.cast_lt] at h_prob_Rstar_gt_eps_div_q\n          exact h_prob_Rstar_gt_eps_div_q\n        -- Step 6: Apply Inductive Hypothesis for r_last ∈ R*\n        have h_line_close_to_C_m : ∀ (r : R_star_set),\n          jointProximityNat (u := affineLineEvaluation U₀ U₁ r.val) (e := e) (C := MC) := by\n          intro r_in_Rstar\n          -- Use the lemma proven earlier\n          apply correlatedAgreement_of_mem_R_star_tensor\n            (ih := fun u_prev => ih (by omega) u_prev ) (u := u)\n        -- Step 7: Apply Affine Gap of Interleaved Code C^(m = 2^(ϑ_sub_2 + 1))\n        have h_C_m_gap := h_interleaved_gaps (2^(ϑ_sub_2 + 1)) (Nat.one_le_two_pow)\n        -- Need the hypothesis Pr[...] > ε/q for h_C_m_gap\n        have h_prob_line_gt_eps_div_q :\n          Pr_{ let r ← $ᵖ F }[\n            Δ₀(affineLineEvaluation (u₀ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₀))\n              (u₁ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₁)) (r := r),\n              ((MC ^⋈ (Fin (2 ^ (ϑ_sub_2 + 1)))))) ≤ e\n          ] > (↑ε : ENNReal) / (Fintype.card F : ENNReal) := by\n            -- This is implied by h_prob_Rstar_gt_eps_div_q becuz R* is a subset of the success set\n            apply lt_of_le_of_lt' _ h_prob_Rstar_gt_eps_div_q\n            have h_r_implies: ∀ (r : F), r ∈ R_star_set →\n              Δ₀(affineLineEvaluation\n                (u₀ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₀))\n                (u₁ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₁)) (r := r),\n                ((MC ^⋈ (Fin (2 ^ (ϑ_sub_2 + 1)))))) ≤ e := by\n              intro r hr_in_Rstar\n              specialize h_line_close_to_C_m ⟨r, hr_in_Rstar⟩\n              unfold jointProximityNat at h_line_close_to_C_m\n              exact h_line_close_to_C_m\n            let Pr_le := Pr_le_Pr_of_implies (D := $ᵖ F)\n              (g := fun r => Δ₀(affineLineEvaluation\n                (u₀ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₀))\n                (u₁ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₁)) (r := r),\n                ((MC ^⋈ (Fin (2 ^ (ϑ_sub_2 + 1)))))) ≤ e)\n              (f := fun r => r ∈ R_star_set) (h_imp := h_r_implies)\n            simp only at Pr_le\n            exact Pr_le\n        -- Apply the gap property of C^m\n        have h_final_gap : jointProximityNat₂\n          (u₀ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₀))\n          (u₁ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₁)) (e := e)\n          (C := (MC ^⋈ (Fin (2^(ϑ_sub_2 + 1))))) := by\n          apply h_C_m_gap (u₀ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₀))\n            (u₁ := ⋈|(show WordStack A (Fin (2 ^ (ϑ_sub_2 + 1))) ι from U₁))\n            (h_prob_line_gt_eps_div_q)\n        apply CA_split_rowwise_implies_CA (u := u) (e := e)\n        exact h_final_gap", "proof_length": 1353, "num_lines": 177, "header": "", "file_path": "ArkLib/Data/CodingTheory/ProximityGap/DG25/MainResults.lean", "url": "https://github.com/Verified-zkEVM/ArkLib", "start_line": 1207, "end_line": 1383, "version_info": [{"v4.31.0": "c1712f6c496d0aef04f09bae42aec13811b3273f"}, {"v4.30.0": "e6d77b18ca1334a91faf4d1ccf9f96d854d58ba4"}, {"v4.29.0": "de71a9dfd2f5a343e8895517b7efe857d9f1ce5c"}]}
{"name": "putnam_1964_a4", "source": "putnambench", "statement": "theorem putnam_1964_a4\n(u : ℕ → ℤ)\n(boundedu : ∃ B T : ℤ, ∀ n : ℕ, B ≤ u n ∧ u n ≤ T)\n(hu : ∀ n ≥ 4, u n = ((u (n - 1) + u (n - 2) + u (n - 3) * u (n - 4)) : ℝ) / (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ∧ (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ≠ 0)\n: (∃ N c : ℕ, c > 0 ∧ ∀ n ≥ N, u (n + c) = u n)", "src": "theorem putnam_1964_a4\n(u : ℕ → ℤ)\n(boundedu : ∃ B T : ℤ, ∀ n : ℕ, B ≤ u n ∧ u n ≤ T)\n(hu : ∀ n ≥ 4, u n = ((u (n - 1) + u (n - 2) + u (n - 3) * u (n - 4)) : ℝ) / (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ∧ (u (n - 1) * u (n - 2) + u (n - 3) + u (n - 4)) ≠ 0)\n: (∃ N c : ℕ, c > 0 ∧ ∀ n ≥ N, u (n + c) = u n) := by \n  have h_main : ∃ (i j : ℕ), i < j ∧ (u i = u j ∧ u (i + 1) = u (j + 1) ∧ u (i + 2) = u (j + 2) ∧ u (i + 3) = u (j + 3)) := by\n    obtain ⟨B, T, hB⟩ := boundedu\n    have h₁ : ∃ (i j : ℕ), i < j ∧ (u i = u j ∧ u (i + 1) = u (j + 1) ∧ u (i + 2) = u (j + 2) ∧ u (i + 3) = u (j + 3)) := by\n      \n      classical\n      \n      let f : ℕ → (ℤ × ℤ × ℤ × ℤ) := fun n => (u n, u (n + 1), u (n + 2), u (n + 3))\n      have h₂ : Set.Finite (Set.range f) := by\n        have h₃ : Set.range f ⊆ Set.Icc (B, B, B, B) (T, T, T, T) := by\n          intro x hx\n          rcases hx with ⟨n, rfl⟩\n          have h₄ := hB n\n          have h₅ := hB (n + 1)\n          have h₆ := hB (n + 2)\n          have h₇ := hB (n + 3)\n          simp only [Set.mem_Icc, Prod.le_def] at h₄ h₅ h₆ h₇ ⊢\n          constructor <;>\n          (try constructor) <;>\n          (try constructor) <;>\n          (try constructor) <;>\n          (try simp_all [f]) <;>\n          (try nlinarith)\n        \n        have h₄ : Set.Finite (Set.Icc (B, B, B, B) (T, T, T, T)) := by\n          apply Set.Finite.subset (Set.finite_mem_finset (Finset.Icc (B, B, B, B) (T, T, T, T)))\n          intro x hx\n          simp_all [Set.mem_Icc]\n          <;>\n          (try aesop)\n        exact Set.Finite.subset h₄ h₃\n      \n      have h₃ : ∃ (i j : ℕ), i < j ∧ f i = f j := by\n        by_contra! h₄\n        have h₅ : Function.Injective f := by\n          intro i j h₆\n          by_cases h₇ : i = j\n          · exact h₇\n          · have h₈ : i < j ∨ j < i := by omega\n            cases h₈ with\n            | inl h₈ =>\n              have h₉ := h₄ i j h₈\n              simp_all\n            | inr h₈ =>\n              have h₉ := h₄ j i h₈\n              simp_all\n        have h₆ : Set.Infinite (Set.range f) := Set.infinite_range_of_injective h₅\n        exact Set.not_infinite.mpr h₂ h₆\n      \n      obtain ⟨i, j, h₄, h₅⟩ := h₃\n      refine' ⟨i, j, h₄, _⟩\n      simp [f] at h₅ ⊢\n      <;>\n      (try simp_all [Prod.ext_iff]) <;>\n      (try omega) <;>\n      (try aesop)\n    exact h₁\n  \n  have h_periodic : ∃ N c : ℕ, c > 0 ∧ ∀ n ≥ N, u (n + c) = u n := by\n    obtain ⟨i, j, hij, h_eq⟩ := h_main\n    have h₁ : ∀ m : ℕ, u (i + m) = u (j + m) := by\n      intro m\n      have h₂ : ∀ m : ℕ, u (i + m) = u (j + m) := by\n        intro m\n        induction m using Nat.strong_induction_on with\n        | h m ih =>\n          match m with\n          | 0 =>\n            have h₃ := h_eq.1\n            simp at h₃ ⊢\n            <;> simp_all [add_assoc]\n          | 1 =>\n            have h₃ := h_eq.2.1\n            simp at h₃ ⊢\n            <;> simp_all [add_assoc]\n            <;> ring_nf at *\n            <;> omega\n          | 2 =>\n            have h₃ := h_eq.2.2.1\n            simp at h₃ ⊢\n            <;> simp_all [add_assoc]\n            <;> ring_nf at *\n            <;> omega\n          | 3 =>\n            have h₃ := h_eq.2.2.2\n            simp at h₃ ⊢\n            <;> simp_all [add_assoc]\n            <;> ring_nf at *\n            <;> omega\n          | m + 4 =>\n            have h₃ : i + (m + 4) ≥ 4 := by\n              have h₄ : i ≥ 0 := by linarith\n              omega\n            have h₄ : u (i + (m + 4)) = ((u (i + (m + 4) - 1) + u (i + (m + 4) - 2) + u (i + (m + 4) - 3) * u (i + (m + 4) - 4)) : ℝ) / (u (i + (m + 4) - 1) * u (i + (m + 4) - 2) + u (i + (m + 4) - 3) + u (i + (m + 4) - 4)) := by\n              have h₅ := hu (i + (m + 4)) (by omega)\n              have h₆ := h₅.1\n              norm_cast at h₆ ⊢\n              <;> simp_all [add_assoc]\n              <;> ring_nf at *\n              <;> norm_num at *\n              <;> linarith\n            have h₅ : u (j + (m + 4)) = ((u (j + (m + 4) - 1) + u (j + (m + 4) - 2) + u (j + (m + 4) - 3) * u (j + (m + 4) - 4)) : ℝ) / (u (j + (m + 4) - 1) * u (j + (m + 4) - 2) + u (j + (m + 4) - 3) + u (j + (m + 4) - 4)) := by\n              have h₆ := hu (j + (m + 4)) (by\n                have h₇ : j ≥ i + 1 := by omega\n                omega\n              )\n              have h₇ := h₆.1\n              norm_cast at h₇ ⊢\n              <;> simp_all [add_assoc]\n              <;> ring_nf at *\n              <;> norm_num at *\n              <;> linarith\n            have h₆ : (i + (m + 4) - 1 : ℕ) = i + (m + 3) := by\n              omega\n            have h₇ : (i + (m + 4) - 2 : ℕ) = i + (m + 2) := by\n              omega\n            have h₈ : (i + (m + 4) - 3 : ℕ) = i + (m + 1) := by\n              omega\n            have h₉ : (i + (m + 4) - 4 : ℕ) = i + m := by\n              omega\n            have h₁₀ : (j + (m + 4) - 1 : ℕ) = j + (m + 3) := by\n              omega\n            have h₁₁ : (j + (m + 4) - 2 : ℕ) = j + (m + 2) := by\n              omega\n            have h₁₂ : (j + (m + 4) - 3 : ℕ) = j + (m + 1) := by\n              omega\n            have h₁₃ : (j + (m + 4) - 4 : ℕ) = j + m := by\n              omega\n            have h₁₄ : u (i + (m + 3)) = u (j + (m + 3)) := by\n              have h₁₅ := ih (m + 3) (by omega)\n              simp [add_assoc] at h₁₅ ⊢\n              <;> omega\n            have h₁₅ : u (i + (m + 2)) = u (j + (m + 2)) := by\n              have h₁₆ := ih (m + 2) (by omega)\n              simp [add_assoc] at h₁₆ ⊢\n              <;> omega\n            have h₁₆ : u (i + (m + 1)) = u (j + (m + 1)) := by\n              have h₁₇ := ih (m + 1) (by omega)\n              simp [add_assoc] at h₁₇ ⊢\n              <;> omega\n            have h₁₇ : u (i + m) = u (j + m) := by\n              have h₁₈ := ih m (by omega)\n              simp [add_assoc] at h₁₈ ⊢\n              <;> omega\n            have h₁₈ : (u (i + (m + 4)) : ℝ) = (u (j + (m + 4)) : ℝ) := by\n              rw [h₄, h₅]\n              simp [h₆, h₇, h₈, h₉, h₁₀, h₁₁, h₁₂, h₁₃] at *\n              norm_cast at *\n              <;>\n              (try simp_all [h₁₄, h₁₅, h₁₆, h₁₇]) <;>\n              (try ring_nf at *) <;>\n              (try field_simp at *) <;>\n              (try norm_cast at *) <;>\n              (try linarith)\n            norm_cast at h₁₈ ⊢\n            <;>\n            (try simp_all) <;>\n            (try linarith)\n      exact h₂ m\n    have h₂ : ∃ N c : ℕ, c > 0 ∧ ∀ n ≥ N, u (n + c) = u n := by\n      use i, (j - i)\n      have h₃ : j - i > 0 := by\n        omega\n      constructor\n      · exact h₃\n      · intro n hn\n        have h₄ : u (n + (j - i)) = u n := by\n          have h₅ : u (i + (n - i)) = u (j + (n - i)) := by\n            have h₆ := h₁ (n - i)\n            simp [add_assoc] at h₆ ⊢\n            <;> omega\n          have h₆ : i + (n - i) = n := by\n            omega\n          have h₇ : j + (n - i) = n + (j - i) := by\n            omega\n          rw [h₆] at h₅\n          rw [h₇] at h₅\n          linarith\n        exact h₄\n    exact h₂\n  \n  exact h_periodic", "proof_length": 1769, "num_lines": 186, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Set Function\n\n\n", "file_path": "", "url": "", "version_info": [{"v4.25.0": "1ccd71f89cbbd82ae7d097723ce1722ca7b01c33"}, {"v4.26.0": "2df2f0150c275ad53cb3c90f7c98ec15a56a1a67"}, {"v4.27.0": "a3a10db0e9d66acbebf76c5e6a135066525ac900"}]}
{"name": "putnam_1964_b2", "source": "putnambench", "statement": "theorem putnam_1964_b2\n(S : Type*) [Fintype S] [Nonempty S]\n(P : Finset (Set S))\n(hPP : ∀ T ∈ P, ∀ U ∈ P, T ∩ U ≠ ∅)\n(hPS : ¬∃ T : Set S, T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅))\n: (P.card = 2 ^ (Fintype.card S - 1))", "src": "theorem putnam_1964_b2\n(S : Type*) [Fintype S] [Nonempty S]\n(P : Finset (Set S))\n(hPP : ∀ T ∈ P, ∀ U ∈ P, T ∩ U ≠ ∅)\n(hPS : ¬∃ T : Set S, T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅))\n: (P.card = 2 ^ (Fintype.card S - 1)) := by \n  have h_no_empty : ∀ (T : Set S), T ∈ P → T.Nonempty := by\n    intro T hT\n    have h₁ : T ∩ T ≠ ∅ := hPP T hT T hT\n    have h₂ : T ∩ T = T := by simp [Set.ext_iff]\n    rw [h₂] at h₁\n    exact Set.nonempty_iff_ne_empty.mpr h₁\n  \n  have h_subset_or_compl : ∀ (A : Set S), A ∈ P ∨ Aᶜ ∈ P := by\n    intro A\n    by_contra! h\n    have h₁ : A ∉ P := h.1\n    have h₂ : Aᶜ ∉ P := h.2\n    \n    have h₃ : ∃ (U : Set S), U ∈ P ∧ A ∩ U = ∅ := by\n      have h₄ : ¬(A ∉ P ∧ ∀ (U : Set S), U ∈ P → A ∩ U ≠ ∅) := by\n        intro h₅\n        have h₆ : ∃ (T : Set S), T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅) := by\n          refine' ⟨A, _⟩\n          constructor\n          · exact h₅.1\n          · intro U hU\n            have h₇ := h₅.2 U hU\n            have h₈ : A ∩ U ≠ ∅ := h₇\n            have h₉ : A ∩ U = U ∩ A := by\n              ext x\n              simp [Set.mem_inter_iff]\n              <;> tauto\n            have h₁₀ : U ∩ A ≠ ∅ := by\n              intro h₁₁\n              apply h₈\n              simp_all [Set.ext_iff]\n              <;> aesop\n            simpa [Set.inter_comm] using h₁₀\n        exact hPS h₆\n      by_cases h₅ : A ∉ P\n      · have h₆ : ¬(∀ (U : Set S), U ∈ P → A ∩ U ≠ ∅) := by\n          intro h₇\n          exact h₄ ⟨h₅, h₇⟩\n        push_neg at h₆\n        obtain ⟨U, hU, h₈⟩ := h₆\n        refine' ⟨U, hU, _⟩\n        have h₉ : A ∩ U = ∅ := by\n          by_contra h₁₀\n          have h₁₁ : A ∩ U ≠ ∅ := h₁₀\n          simp_all [Set.ext_iff]\n          <;> aesop\n        exact h₉\n      · exfalso\n        simp_all\n    \n    have h₄ : ∃ (V : Set S), V ∈ P ∧ Aᶜ ∩ V = ∅ := by\n      have h₅ : ¬(Aᶜ ∉ P ∧ ∀ (U : Set S), U ∈ P → Aᶜ ∩ U ≠ ∅) := by\n        intro h₆\n        have h₇ : ∃ (T : Set S), T ∉ P ∧ (∀ U ∈ P, T ∩ U ≠ ∅) := by\n          refine' ⟨Aᶜ, _⟩\n          constructor\n          · exact h₆.1\n          · intro U hU\n            have h₈ := h₆.2 U hU\n            have h₉ : Aᶜ ∩ U ≠ ∅ := h₈\n            have h₁₀ : Aᶜ ∩ U = U ∩ Aᶜ := by\n              ext x\n              simp [Set.mem_inter_iff]\n              <;> tauto\n            have h₁₁ : U ∩ Aᶜ ≠ ∅ := by\n              intro h₁₂\n              apply h₉\n              simp_all [Set.ext_iff]\n              <;> aesop\n            simpa [Set.inter_comm] using h₁₁\n        exact hPS h₇\n      by_cases h₆ : Aᶜ ∉ P\n      · have h₇ : ¬(∀ (U : Set S), U ∈ P → Aᶜ ∩ U ≠ ∅) := by\n          intro h₈\n          exact h₅ ⟨h₆, h₈⟩\n        push_neg at h₇\n        obtain ⟨V, hV, h₈⟩ := h₇\n        refine' ⟨V, hV, _⟩\n        have h₉ : Aᶜ ∩ V = ∅ := by\n          by_contra h₁₀\n          have h₁₁ : Aᶜ ∩ V ≠ ∅ := h₁₀\n          simp_all [Set.ext_iff]\n          <;> aesop\n        exact h₉\n      · exfalso\n        simp_all\n    \n    obtain ⟨U, hU, hU'⟩ := h₃\n    obtain ⟨V, hV, hV'⟩ := h₄\n    \n    have h₅ : U ⊆ Aᶜ := by\n      intro x hx\n      by_contra h₆\n      have h₇ : x ∈ A := by simp_all [Set.mem_compl_iff]\n      have h₈ : x ∈ A ∩ U := Set.mem_inter h₇ hx\n      rw [hU'] at h₈\n      simp at h₈\n      <;> aesop\n    have h₆ : V ⊆ A := by\n      intro x hx\n      by_contra h₇\n      have h₈ : x ∈ Aᶜ := by simp_all [Set.mem_compl_iff]\n      have h₉ : x ∈ Aᶜ ∩ V := Set.mem_inter h₈ hx\n      rw [hV'] at h₉\n      simp at h₉\n      <;> aesop\n    \n    have h₇ : U ∩ V = ∅ := by\n      apply Set.eq_empty_of_forall_not_mem\n      intro x hx\n      have h₈ : x ∈ U := Set.mem_of_mem_inter_left hx\n      have h₉ : x ∈ V := Set.mem_of_mem_inter_right hx\n      have h₁₀ : x ∈ Aᶜ := h₅ h₈\n      have h₁₁ : x ∈ A := h₆ h₉\n      simp_all [Set.mem_compl_iff]\n      <;> aesop\n    \n    have h₈ : U ∩ V ≠ ∅ := hPP U hU V hV\n    simp_all [Set.ext_iff]\n    <;> aesop\n  \n  have h_not_both : ∀ (A : Set S), A ∈ P → Aᶜ ∉ P := by\n    intro A hA\n    by_contra h\n    have h₁ : Aᶜ ∈ P := h\n    have h₂ : A ∩ Aᶜ = ∅ := by\n      ext x\n      simp [Set.mem_inter_iff]\n      <;> tauto\n    have h₃ : A ∩ Aᶜ ≠ ∅ := hPP A hA Aᶜ h₁\n    rw [h₂] at h₃\n    contradiction\n  \n  have h_card : P.card = 2 ^ (Fintype.card S - 1) := by\n    classical\n    \n    let Q : Finset (Set S) := (Finset.univ : Finset (Set S)) \\ P\n    \n    have h₁ : P.card = Q.card := by\n      \n      have h₂ : ∀ (A : Set S), A ∈ P → Aᶜ ∈ Q := by\n        intro A hA\n        have h₃ : Aᶜ ∈ (Finset.univ : Finset (Set S)) := Finset.mem_univ _\n        have h₄ : Aᶜ ∉ P := h_not_both A hA\n        simp only [Q, Finset.mem_sdiff] at *\n        tauto\n      have h₃ : ∀ (A : Set S), A ∈ Q → Aᶜ ∈ P := by\n        intro A hA\n        have h₄ : A ∈ (Finset.univ : Finset (Set S)) := by\n          simp only [Q, Finset.mem_sdiff] at hA\n          tauto\n        have h₅ : A ∉ P := by\n          simp only [Q, Finset.mem_sdiff] at hA\n          tauto\n        have h₆ : A ∈ (Finset.univ : Finset (Set S)) := by tauto\n        have h₇ : A ∈ P ∨ Aᶜ ∈ P := h_subset_or_compl A\n        cases h₇ with\n        | inl h₈ =>\n          exfalso\n          tauto\n        | inr h₈ =>\n          exact h₈\n      \n      have h₄ : P.card = Q.card := by\n        \n        have h₅ : P.card = Q.card := by\n          \n          have h₆ : ∀ (A : Set S), A ∈ P → Aᶜ ∈ Q := h₂\n          have h₇ : ∀ (A : Set S), A ∈ Q → Aᶜ ∈ P := h₃\n          \n          have h₈ : P.card = Q.card := by\n            \n            apply Finset.card_bij' (fun A _ => Aᶜ) (fun A _ => Aᶜ)\n            <;> simp_all [Set.ext_iff]\n            <;>\n            (try\n              {\n                aesop\n              })\n            <;>\n            (try\n              {\n                tauto\n              })\n            <;>\n            (try\n              {\n                intros\n                <;>\n                simp_all [Set.ext_iff]\n                <;>\n                tauto\n              })\n            <;>\n            (try\n              {\n                intros\n                <;>\n                simp_all [Set.ext_iff]\n                <;>\n                tauto\n              })\n          exact h₈\n        exact h₅\n      exact h₄\n    \n    have h₂ : P.card + Q.card = 2 ^ Fintype.card S := by\n      have h₃ : P.card + Q.card = (Finset.univ : Finset (Set S)).card := by\n        have h₄ : Disjoint P Q := by\n          rw [Finset.disjoint_left]\n          intro A hA hA'\n          simp only [Q, Finset.mem_sdiff] at hA'\n          tauto\n        have h₅ : P ∪ Q = (Finset.univ : Finset (Set S)) := by\n          apply Finset.ext\n          intro A\n          simp only [Q, Finset.mem_union, Finset.mem_sdiff, Finset.mem_univ, true_and]\n          <;>\n          by_cases h₆ : A ∈ P <;> simp_all [h_subset_or_compl]\n          <;>\n          (try tauto)\n          <;>\n          (try\n            {\n              have h₇ := h_subset_or_compl A\n              cases h₇ with\n              | inl h₈ => tauto\n              | inr h₈ =>\n                have h₉ : Aᶜ ∈ P := h₈\n                have h₁₀ : A ∈ P ∨ Aᶜ ∈ P := by tauto\n                tauto\n            })\n        have h₆ : P.card + Q.card = (P ∪ Q).card := by\n          rw [← Finset.card_union_add_card_inter P Q]\n          have h₇ : P ∩ Q = ∅ := Finset.disjoint_iff_inter_eq_empty.mp h₄\n          rw [h₇]\n          simp\n        rw [h₅] at h₆\n        exact h₆\n      have h₄ : (Finset.univ : Finset (Set S)).card = 2 ^ Fintype.card S := by\n        simp [Fintype.card_fun]\n        <;>\n        simp_all [Fintype.card_fun]\n        <;>\n        ring_nf\n        <;>\n        simp_all [Fintype.card_fun]\n        <;>\n        norm_num\n        <;>\n        aesop\n      linarith\n    \n    have h₃ : P.card * 2 = 2 ^ Fintype.card S := by\n      have h₄ : P.card + Q.card = 2 ^ Fintype.card S := h₂\n      have h₅ : P.card = Q.card := h₁\n      linarith\n    \n    have h₄ : Fintype.card S ≥ 1 := by\n      have h₅ : Nonempty S := inferInstance\n      have h₆ : 0 < Fintype.card S := by\n        apply Fintype.card_pos_iff.mpr\n        exact ⟨Classical.choice h₅⟩\n      omega\n    have h₅ : P.card = 2 ^ (Fintype.card S - 1) := by\n      have h₆ : P.card * 2 = 2 ^ Fintype.card S := h₃\n      have h₇ : Fintype.card S ≥ 1 := h₄\n      have h₈ : 2 ^ (Fintype.card S - 1) * 2 = 2 ^ Fintype.card S := by\n        have h₉ : Fintype.card S - 1 + 1 = Fintype.card S := by\n          have h₁₀ : Fintype.card S ≥ 1 := h₄\n          omega\n        calc\n          2 ^ (Fintype.card S - 1) * 2 = 2 ^ (Fintype.card S - 1) * 2 ^ 1 := by norm_num\n          _ = 2 ^ ((Fintype.card S - 1) + 1) := by\n            rw [← pow_add]\n          _ = 2 ^ Fintype.card S := by\n            rw [h₉]\n            <;> simp [add_comm]\n      have h₉ : P.card = 2 ^ (Fintype.card S - 1) := by\n        have h₁₀ : P.card * 2 = 2 ^ Fintype.card S := h₃\n        have h₁₁ : 2 ^ (Fintype.card S - 1) * 2 = 2 ^ Fintype.card S := h₈\n        have h₁₂ : P.card = 2 ^ (Fintype.card S - 1) := by\n          nlinarith\n        exact h₁₂\n      exact h₉\n    exact h₅\n  \n  exact h_card", "proof_length": 1754, "num_lines": 294, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Set Function Filter Topology\n\n\n", "file_path": "", "url": "", "version_info": [{"v4.25.0": "1ccd71f89cbbd82ae7d097723ce1722ca7b01c33"}, {"v4.26.0": "2df2f0150c275ad53cb3c90f7c98ec15a56a1a67"}]}
{"name": "putnam_1995_a3", "source": "putnambench", "statement": "theorem putnam_1995_a3\n(relation : (Fin 9 → ℤ) → (Fin 9 → ℤ) → Prop)\n(digits_to_num : (Fin 9 → ℤ) → ℤ)\n(hdigits_to_num : digits_to_num = fun dig => ∑ i : Fin 9, (dig i) * 10^i.1)\n(hrelation : ∀ d e : (Fin 9 → ℤ), relation d e ↔ (∀ i : Fin 9, d i < 10 ∧ d i ≥ 0 ∧ e i < 10 ∧ e i ≥ 0) ∧ (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j))))\n: ∀ d e f : (Fin 9 → ℤ), ((relation d e) ∧ (relation e f)) → (∀ i : Fin 9, 7 ∣ d i - f i)", "src": "theorem putnam_1995_a3\n(relation : (Fin 9 → ℤ) → (Fin 9 → ℤ) → Prop)\n(digits_to_num : (Fin 9 → ℤ) → ℤ)\n(hdigits_to_num : digits_to_num = fun dig => ∑ i : Fin 9, (dig i) * 10^i.1)\n(hrelation : ∀ d e : (Fin 9 → ℤ), relation d e ↔ (∀ i : Fin 9, d i < 10 ∧ d i ≥ 0 ∧ e i < 10 ∧ e i ≥ 0) ∧ (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j))))\n: ∀ d e f : (Fin 9 → ℤ), ((relation d e) ∧ (relation e f)) → (∀ i : Fin 9, 7 ∣ d i - f i) := by \n  have h_main : ∀ (d e f : (Fin 9 → ℤ)), ((relation d e) ∧ (relation e f)) → (∀ i : Fin 9, 7 ∣ d i - f i) := by\n    intro d e f h\n    have h₁ : relation d e := h.1\n    have h₂ : relation e f := h.2\n    have h₃ : ∀ (i : Fin 9), 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j)) := by\n      have h₄ : (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j))) := (hrelation d e).mp h₁ |>.2\n      exact h₄\n    have h₄ : ∀ (i : Fin 9), 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then f j else e j)) := by\n      have h₅ : (∀ i : Fin 9, 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then f j else e j))) := (hrelation e f).mp h₂ |>.2\n      exact h₅\n    \n    have h₅ : ∀ (i : Fin 9), 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := by\n      intro i\n      have h₆ : 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then e j else d j)) := h₃ i\n      have h₇ : (digits_to_num (fun j : Fin 9 => if j = i then e j else d j)) = digits_to_num d + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n        calc\n          (digits_to_num (fun j : Fin 9 => if j = i then e j else d j)) = ∑ j : Fin 9, (if j = i then e j else d j) * (10 : ℤ) ^ (j : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n          _ = ∑ j : Fin 9, (if j = i then e j else d j) * (10 : ℤ) ^ (j : ℕ) := rfl\n          _ = (∑ j : Fin 9, d j * (10 : ℤ) ^ (j : ℕ)) + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            calc\n              (∑ j : Fin 9, (if j = i then e j else d j) * (10 : ℤ) ^ (j : ℕ)) = ∑ j : Fin 9, (d j * (10 : ℤ) ^ (j : ℕ) + (if j = i then (e j - d j) else 0) * (10 : ℤ) ^ (j : ℕ)) := by\n                apply Finset.sum_congr rfl\n                intro j _\n                by_cases h : j = i\n                · simp [h]\n                  <;> ring\n                · simp [h]\n                  <;> ring\n              _ = (∑ j : Fin 9, d j * (10 : ℤ) ^ (j : ℕ)) + ∑ j : Fin 9, (if j = i then (e j - d j) else 0) * (10 : ℤ) ^ (j : ℕ) := by\n                rw [Finset.sum_add_distrib]\n              _ = (∑ j : Fin 9, d j * (10 : ℤ) ^ (j : ℕ)) + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n                have h₈ : ∑ j : Fin 9, (if j = i then (e j - d j) else 0) * (10 : ℤ) ^ (j : ℕ) = (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n                  calc\n                    _ = ∑ j : Fin 9, (if j = i then (e j - d j) else 0) * (10 : ℤ) ^ (j : ℕ) := rfl\n                    _ = (if i = i then (e i - d i) else 0) * (10 : ℤ) ^ (i : ℕ) := by\n                      \n                      simp [Finset.sum_ite_eq', Finset.mem_univ, i.is_lt]\n                    _ = (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by simp\n                rw [h₈]\n                <;> simp [hdigits_to_num]\n                <;>\n                rfl\n          _ = digits_to_num d + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n      have h₈ : 7 ∣ digits_to_num d + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n        rw [h₇] at h₆\n        exact h₆\n      \n      have h₉ : 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := by\n        \n        have h₁₀ : (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d = digits_to_num d + (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by ring\n        rw [h₁₀]\n        exact h₈\n      exact h₉\n    \n    have h₆ : ∀ (i : Fin 9), 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e := by\n      intro i\n      have h₇ : 7 ∣ (digits_to_num (fun j : Fin 9 => if j = i then f j else e j)) := h₄ i\n      have h₈ : (digits_to_num (fun j : Fin 9 => if j = i then f j else e j)) = digits_to_num e + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n        calc\n          (digits_to_num (fun j : Fin 9 => if j = i then f j else e j)) = ∑ j : Fin 9, (if j = i then f j else e j) * (10 : ℤ) ^ (j : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n          _ = ∑ j : Fin 9, (if j = i then f j else e j) * (10 : ℤ) ^ (j : ℕ) := rfl\n          _ = (∑ j : Fin 9, e j * (10 : ℤ) ^ (j : ℕ)) + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n            calc\n              (∑ j : Fin 9, (if j = i then f j else e j) * (10 : ℤ) ^ (j : ℕ)) = ∑ j : Fin 9, (e j * (10 : ℤ) ^ (j : ℕ) + (if j = i then (f j - e j) else 0) * (10 : ℤ) ^ (j : ℕ)) := by\n                apply Finset.sum_congr rfl\n                intro j _\n                by_cases h : j = i\n                · simp [h]\n                  <;> ring\n                · simp [h]\n                  <;> ring\n              _ = (∑ j : Fin 9, e j * (10 : ℤ) ^ (j : ℕ)) + ∑ j : Fin 9, (if j = i then (f j - e j) else 0) * (10 : ℤ) ^ (j : ℕ) := by\n                rw [Finset.sum_add_distrib]\n              _ = (∑ j : Fin 9, e j * (10 : ℤ) ^ (j : ℕ)) + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n                have h₉ : ∑ j : Fin 9, (if j = i then (f j - e j) else 0) * (10 : ℤ) ^ (j : ℕ) = (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n                  calc\n                    _ = ∑ j : Fin 9, (if j = i then (f j - e j) else 0) * (10 : ℤ) ^ (j : ℕ) := rfl\n                    _ = (if i = i then (f i - e i) else 0) * (10 : ℤ) ^ (i : ℕ) := by\n                      \n                      simp [Finset.sum_ite_eq', Finset.mem_univ, i.is_lt]\n                    _ = (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by simp\n                rw [h₉]\n                <;> simp [hdigits_to_num]\n                <;>\n                rfl\n          _ = digits_to_num e + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n      have h₉ : 7 ∣ digits_to_num e + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by\n        rw [h₈] at h₇\n        exact h₇\n      \n      have h₁₀ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e := by\n        \n        have h₁₁ : (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e = digits_to_num e + (f i - e i) * (10 : ℤ) ^ (i : ℕ) := by ring\n        rw [h₁₁]\n        exact h₉\n      exact h₁₀\n    \n    have h₇ : 7 ∣ digits_to_num e + digits_to_num d := by\n      have h₈ : digits_to_num e = digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n        calc\n          digits_to_num e = ∑ i : Fin 9, e i * (10 : ℤ) ^ (i : ℕ) := by simp [hdigits_to_num]\n          _ = ∑ i : Fin 9, (d i + (e i - d i)) * (10 : ℤ) ^ (i : ℕ) := by\n            apply Finset.sum_congr rfl\n            intro i _\n            ring\n          _ = ∑ i : Fin 9, (d i * (10 : ℤ) ^ (i : ℕ) + (e i - d i) * (10 : ℤ) ^ (i : ℕ)) := by\n            apply Finset.sum_congr rfl\n            intro i _\n            ring\n          _ = (∑ i : Fin 9, d i * (10 : ℤ) ^ (i : ℕ)) + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            rw [Finset.sum_add_distrib]\n          _ = digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            simp [hdigits_to_num]\n            <;>\n            rfl\n      have h₉ : ∀ i : Fin 9, 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := h₅\n      have h₁₀ : 7 ∣ ∑ i : Fin 9, ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) := by\n        apply Finset.dvd_sum\n        intro i _\n        exact h₉ i\n      have h₁₁ : ∑ i : Fin 9, ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) = ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d := by\n        calc\n          ∑ i : Fin 9, ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) = ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + ∑ i : Fin 9, digits_to_num d := by\n            rw [Finset.sum_add_distrib]\n          _ = ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d := by\n            simp [Finset.sum_const, Finset.card_fin]\n            <;> ring\n            <;> norm_num\n      have h₁₂ : 7 ∣ ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d := by\n        rw [h₁₁] at h₁₀\n        exact h₁₀\n      have h₁₃ : 7 ∣ digits_to_num e + digits_to_num d := by\n        have h₁₄ : digits_to_num e + digits_to_num d = 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n          linarith\n        rw [h₁₄]\n        have h₁₅ : 7 ∣ 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n          have h₁₆ : 7 ∣ ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d := h₁₂\n          have h₁₇ : 7 ∣ 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n            \n            have h₁₈ : (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) + 9 * digits_to_num d) = (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 9 * digits_to_num d := by ring\n            rw [h₁₈] at h₁₆\n            \n            \n            \n            \n            \n            \n            have h₁₉ : 7 ∣ (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 2 * digits_to_num d := by\n              \n              \n              \n              have h₂₀ : 7 ∣ (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 9 * digits_to_num d := h₁₆\n              have h₂₁ : 7 ∣ 7 * digits_to_num d := by\n                use digits_to_num d\n                <;> ring\n              have h₂₂ : 7 ∣ (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 2 * digits_to_num d := by\n                \n                have h₂₃ : (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 2 * digits_to_num d = ((∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 9 * digits_to_num d) - 7 * digits_to_num d := by ring\n                rw [h₂₃]\n                exact dvd_sub h₂₀ h₂₁\n              exact h₂₂\n            \n            have h₂₀ : 7 ∣ 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n              \n              have h₂₁ : 2 * digits_to_num d + ∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ) = (∑ i : Fin 9, (e i - d i) * (10 : ℤ) ^ (i : ℕ)) + 2 * digits_to_num d := by ring\n              rw [h₂₁]\n              exact h₁₉\n            exact h₂₀\n          exact h₁₇\n        exact h₁₅\n      exact h₁₃\n    \n    have h₈ : ∀ (i : Fin 9), 7 ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n      intro i\n      have h₉ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e := h₆ i\n      have h₁₀ : 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := h₅ i\n      have h₁₁ : 7 ∣ digits_to_num e + digits_to_num d := h₇\n      \n      have h₁₂ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d := by\n        \n        \n        have h₁₃ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e := h₆ i\n        have h₁₄ : 7 ∣ digits_to_num e + digits_to_num d := h₇\n        \n        have h₁₅ : 7 ∣ ((f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e) - (digits_to_num e + digits_to_num d) := by\n          exact dvd_sub h₁₃ h₁₄\n        \n        have h₁₆ : ((f i - e i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num e) - (digits_to_num e + digits_to_num d) = (f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d := by\n          ring\n        rw [h₁₆] at h₁₅\n        exact h₁₅\n      \n      have h₁₃ : 7 ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n        have h₁₄ : 7 ∣ (f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d := h₁₂\n        have h₁₅ : 7 ∣ (e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d := h₅ i\n        \n        have h₁₆ : 7 ∣ ((f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d) + ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) := by\n          exact dvd_add h₁₄ h₁₅\n        \n        have h₁₇ : ((f i - e i) * (10 : ℤ) ^ (i : ℕ) - digits_to_num d) + ((e i - d i) * (10 : ℤ) ^ (i : ℕ) + digits_to_num d) = (f i - d i) * (10 : ℤ) ^ (i : ℕ) := by\n          ring\n        rw [h₁₇] at h₁₆\n        exact h₁₆\n      exact h₁₃\n    \n    have h₉ : ∀ (i : Fin 9), 7 ∣ d i - f i := by\n      intro i\n      have h₁₀ : 7 ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₈ i\n      have h₁₁ : 7 ∣ (f i - d i) := by\n        \n        have h₁₂ : (7 : ℤ) ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₁₀\n        \n        have h₁₃ : (7 : ℕ).Prime := by decide\n        have h₁₄ : ¬(7 : ℤ) ∣ (10 : ℤ) ^ (i : ℕ) := by\n          \n          intro h\n          have h₁₅ : (7 : ℕ) ∣ (10 : ℕ) ^ (i : ℕ) := by\n            norm_cast at h ⊢\n            <;> simpa [Int.coe_nat_dvd_left] using h\n          have h₁₆ : (7 : ℕ) ∣ 10 := by\n            \n            exact Nat.Prime.dvd_of_dvd_pow h₁₃ h₁₅\n          norm_num at h₁₆\n          <;> contradiction\n        \n        have h₁₅ : (7 : ℤ) ∣ (f i - d i) := by\n          \n          have h₁₆ : (7 : ℤ) ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₁₂\n          have h₁₇ : ¬(7 : ℤ) ∣ (10 : ℤ) ^ (i : ℕ) := h₁₄\n          \n          have h₁₈ : (7 : ℤ) ∣ (f i - d i) := by\n            \n            have h₁₉ : (7 : ℤ) ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₁₂\n            have h₂₀ : (7 : ℕ).Prime := by decide\n            have h₂₁ : (7 : ℤ) ∣ (f i - d i) ∨ (7 : ℤ) ∣ (10 : ℤ) ^ (i : ℕ) := by\n              \n              have h₂₂ : (7 : ℤ) ∣ (f i - d i) * (10 : ℤ) ^ (i : ℕ) := h₁₂\n              have h₂₃ : (7 : ℤ) ∣ (f i - d i) ∨ (7 : ℤ) ∣ (10 : ℤ) ^ (i : ℕ) := by\n                \n                apply (Int.prime_iff_natAbs_prime.mpr (by decide)).dvd_mul.mp\n                exact h₂₂\n              exact h₂₃\n            cases h₂₁ with\n            | inl h₂₁ =>\n              exact h₂₁\n            | inr h₂₁ =>\n              exfalso\n              exact h₁₄ h₂₁\n          exact h₁₈\n        exact h₁₅\n      \n      have h₁₂ : 7 ∣ d i - f i := by\n        have h₁₃ : 7 ∣ (f i - d i) := h₁₁\n        have h₁₄ : 7 ∣ -(f i - d i) := by\n          exact dvd_neg.mpr h₁₃\n        have h₁₅ : -(f i - d i) = d i - f i := by ring\n        rw [h₁₅] at h₁₄\n        exact h₁₄\n      exact h₁₂\n    \n    intro i\n    have h₁₀ : 7 ∣ d i - f i := h₉ i\n    exact h₁₀\n  exact h_main", "proof_length": 4085, "num_lines": 282, "header": "import Mathlib\nimport Aesop\n\nset_option maxHeartbeats 0\n\nopen Filter Topology Real\n\n\n", "file_path": "", "url": "", "version_info": [{"v4.25.0": "1ccd71f89cbbd82ae7d097723ce1722ca7b01c33"}, {"v4.26.0": "2df2f0150c275ad53cb3c90f7c98ec15a56a1a67"}, {"v4.27.0": "a3a10db0e9d66acbebf76c5e6a135066525ac900"}]}