feat(Relativity): reverse Cauchy-Schwarz and triangle inequalities, twin paradox age gap nonnegative

Physlib.lean

@@ -468,6 +468,7 @@ public import Physlib.Relativity.Tensors.RealTensor.Units.Basic
 public import Physlib.Relativity.Tensors.RealTensor.Units.Pre
 public import Physlib.Relativity.Tensors.RealTensor.Vector.Basic
 public import Physlib.Relativity.Tensors.RealTensor.Vector.Causality.Basic
+public import Physlib.Relativity.Tensors.RealTensor.Vector.Causality.CausallyFollows
 public import Physlib.Relativity.Tensors.RealTensor.Vector.Causality.LightLike
 public import Physlib.Relativity.Tensors.RealTensor.Vector.Causality.TimeLike
 public import Physlib.Relativity.Tensors.RealTensor.Vector.MinkowskiProduct

Physlib/Relativity/Special/TwinParadox/Basic.lean

@@ -6,6 +6,7 @@ Authors: Joseph Tooby-Smith
 module
 
 public import Physlib.Relativity.Special.ProperTime
+public import Physlib.Relativity.Tensors.RealTensor.Vector.Causality.CausallyFollows
 /-!
 # Twin Paradox
 
@@ -67,11 +68,18 @@ def ageGap : ℝ := T.properTimeTwinA - T.properTimeTwinB
 
 TODO "Find the conditions for which the age gap for the twin paradox is zero."
 
-/-- In the twin paradox with instantaneous acceleration, Twin A is always older
-  then Twin B. -/
-informal_lemma ageGap_nonneg where
-  deps := [``ageGap]
-  tag := "7ROVE"
+/-- In the twin paradox with instantaneous acceleration, Twin A is always at least as old as
+  Twin B: the age gap is nonnegative. This is the reverse triangle inequality of Minkowski space
+  (`sqrt_add_sqrt_le_sqrt_add_of_causallyFollows`) applied to the two legs of Twin B. -/
+lemma ageGap_nonneg : 0 ≤ T.ageGap := by
+  have hu := causallyFollows_zero_sub T.twinBMid_causallyFollows_startPoint
+  have hv := causallyFollows_zero_sub T.endPoint_causallyFollows_twinBMid
+  have h := sqrt_add_sqrt_le_sqrt_add_of_causallyFollows hu hv
+  have hsum : T.endPoint - T.startPoint
+      = (T.twinBMid - T.startPoint) + (T.endPoint - T.twinBMid) := by abel
+  unfold ageGap properTimeTwinA properTimeTwinB properTime
+  rw [hsum]
+  linarith
 
 /-!
 

Physlib/Relativity/Tensors/RealTensor/Vector/Causality/CausallyFollows.lean

@@ -0,0 +1,162 @@
+/-
+Copyright (c) 2026 Philippe Kevorkian. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Philippe Kevorkian
+-/
+module
+
+public import Physlib.Relativity.Tensors.RealTensor.Vector.Causality.TimeLike
+public import Physlib.Relativity.Tensors.RealTensor.Vector.Causality.LightLike
+/-!
+
+# Vectors in the causal future of the origin
+
+## i. Overview
+
+A vector `u` with `causallyFollows 0 u` is future-directed causal: time-like or light-like with a
+nonnegative time component (`causallyFollows_zero_iff`). For two such vectors of Minkowski space
+(signature `+---`), the Minkowski product dominates the product of the Minkowski norms,
+`⟪u, v⟫ₘ ≥ √⟪u, u⟫ₘ √⟪v, v⟫ₘ` (reverse Cauchy-Schwarz inequality), and the Minkowski norm of the
+sum dominates the sum of the norms, `√⟪u + v, u + v⟫ₘ ≥ √⟪u, u⟫ₘ + √⟪v, v⟫ₘ` (reverse triangle
+inequality). The proofs reduce to the elementary inequality `√(a² - c²) √(b² - d²) ≤ a b - c d` for
+`0 ≤ c ≤ a`, `0 ≤ d ≤ b`, applied to the time components and the Euclidean norms of the spatial
+parts, together with the Euclidean Cauchy-Schwarz inequality.
+
+## ii. Key results
+
+- `causallyFollows_zero_iff`: `causallyFollows 0 u ↔ 0 ≤ ⟪u, u⟫ₘ ∧ 0 ≤ u⁰`;
+  `causallyFollows_zero_sub`: `q - p` is in the causal future of `0` when `q` causally follows `p`.
+- `norm_spatialPart_le_timeComponent_of_causallyFollows`: `‖u_spatial‖ ≤ u⁰`.
+- `sqrt_mul_sqrt_le_minkowskiProduct_of_causallyFollows`: the reverse Cauchy-Schwarz inequality.
+- `sqrt_add_sqrt_le_sqrt_add_of_causallyFollows`: the reverse triangle inequality;
+  `causallyFollows_zero_add_of_causallyFollows`.
+
+## iii. Table of contents
+
+- A. The causal future of the origin
+- B. The reverse Cauchy-Schwarz inequality
+- C. The reverse triangle inequality
+
+-/
+
+@[expose] public section
+
+namespace Lorentz
+
+namespace Vector
+
+open InnerProductSpace
+
+/-!
+
+## A. The causal future of the origin
+
+-/
+
+/-- A vector is in the causal future of the origin if and only if it is time-like or light-like
+  (`0 ≤ ⟪u, u⟫ₘ`) with a nonnegative time component. -/
+lemma causallyFollows_zero_iff {d : ℕ} {u : Vector d} :
+    causallyFollows 0 u ↔ 0 ≤ ⟪u, u⟫ₘ ∧ 0 ≤ u.timeComponent := by
+  simp only [causallyFollows, interiorFutureLightCone, futureLightConeBoundary, Set.mem_ofPred_eq,
+    sub_zero, timeLike_iff_norm_sq_pos, lightLike_iff_norm_sq_zero]
+  constructor
+  · rintro (⟨h1, h2⟩ | ⟨h1, h2⟩)
+    · exact ⟨h1.le, h2.le⟩
+    · exact ⟨h1.ge, h2⟩
+  · rintro ⟨h1, h2⟩
+    rcases h1.lt_or_eq with h1 | h1
+    · left
+      refine ⟨h1, lt_of_le_of_ne h2 fun h0 => ?_⟩
+      have h := minkowskiProduct_self_le_timeComponent_sq u
+      have h0' : u.timeComponent = 0 := h0.symm
+      rw [h0'] at h
+      norm_num at h
+      linarith
+    · right
+      exact ⟨h1.symm, h2⟩
+
+/-- The vector from `p` to `q` is in the causal future of the origin when `q` causally
+  follows `p`. -/
+lemma causallyFollows_zero_sub {d : ℕ} {p q : Vector d} (h : causallyFollows p q) :
+    causallyFollows 0 (q - p) := by
+  simpa only [causallyFollows, interiorFutureLightCone, futureLightConeBoundary, Set.mem_ofPred_eq,
+    sub_zero] using h
+
+/-!
+
+## B. The reverse Cauchy-Schwarz inequality
+
+-/
+
+/-- For a vector in the causal future of the origin, the Euclidean norm of the spatial part is at
+  most the time component. -/
+lemma norm_spatialPart_le_timeComponent_of_causallyFollows {d : ℕ} {u : Vector d}
+    (hu : causallyFollows 0 u) : ‖u.spatialPart‖ ≤ u.timeComponent := by
+  obtain ⟨h1, _⟩ := causallyFollows_zero_iff.mp hu
+  rw [minkowskiProduct_self_eq_sq_sub] at h1
+  nlinarith [norm_nonneg u.spatialPart]
+
+/-- The reverse Cauchy-Schwarz inequality: for `u`, `v` in the causal future of the origin,
+  `√⟪u, u⟫ₘ √⟪v, v⟫ₘ ≤ ⟪u, v⟫ₘ`. -/
+lemma sqrt_mul_sqrt_le_minkowskiProduct_of_causallyFollows {d : ℕ} {u v : Vector d}
+    (hu : causallyFollows 0 u) (hv : causallyFollows 0 v) : √⟪u, u⟫ₘ * √⟪v, v⟫ₘ ≤ ⟪u, v⟫ₘ := by
+  have hu' := norm_spatialPart_le_timeComponent_of_causallyFollows hu
+  have hv' := norm_spatialPart_le_timeComponent_of_causallyFollows hv
+  rw [minkowskiProduct_self_eq_sq_sub u, minkowskiProduct_self_eq_sq_sub v,
+    minkowskiProduct_eq_timeComponent_spatialPart u v]
+  have hcs := real_inner_le_norm u.spatialPart v.spatialPart
+  set a := u.timeComponent
+  set c := ‖u.spatialPart‖
+  set b := v.timeComponent
+  set e := ‖v.spatialPart‖
+  have hc0 : 0 ≤ c := norm_nonneg _
+  have he0 : 0 ≤ e := norm_nonneg _
+  have helem : √(a ^ 2 - c ^ 2) * √(b ^ 2 - e ^ 2) ≤ a * b - c * e := by
+    -- the elementary inequality `√(a² - c²) √(b² - e²) ≤ a b - c e` for `0 ≤ c ≤ a`, `0 ≤ e ≤ b`:
+    -- indeed `(a b - c e)² - (a² - c²)(b² - e²) = (a e - b c)²`
+    have h1 : 0 ≤ a ^ 2 - c ^ 2 := by nlinarith
+    have h2 : 0 ≤ a * b - c * e := by nlinarith
+    calc √(a ^ 2 - c ^ 2) * √(b ^ 2 - e ^ 2) = √((a ^ 2 - c ^ 2) * (b ^ 2 - e ^ 2)) :=
+          (Real.sqrt_mul h1 _).symm
+      _ ≤ √((a * b - c * e) ^ 2) := Real.sqrt_le_sqrt (by nlinarith [sq_nonneg (a * e - b * c)])
+      _ = a * b - c * e := Real.sqrt_sq h2
+  linarith
+
+/-!
+
+## C. The reverse triangle inequality
+
+-/
+
+/-- The reverse triangle inequality: for `u`, `v` in the causal future of the origin,
+  `√⟪u, u⟫ₘ + √⟪v, v⟫ₘ ≤ √⟪u + v, u + v⟫ₘ`. -/
+lemma sqrt_add_sqrt_le_sqrt_add_of_causallyFollows {d : ℕ} {u v : Vector d}
+    (hu : causallyFollows 0 u) (hv : causallyFollows 0 v) :
+    √⟪u, u⟫ₘ + √⟪v, v⟫ₘ ≤ √⟪u + v, u + v⟫ₘ := by
+  have hcs := sqrt_mul_sqrt_le_minkowskiProduct_of_causallyFollows hu hv
+  have hu1 := (causallyFollows_zero_iff.mp hu).1
+  have hv1 := (causallyFollows_zero_iff.mp hv).1
+  have hsum : (√⟪u, u⟫ₘ + √⟪v, v⟫ₘ) ^ 2 ≤ ⟪u + v, u + v⟫ₘ := by
+    rw [minkowskiProduct_add_self, add_sq, Real.sq_sqrt hu1, Real.sq_sqrt hv1]
+    linarith
+  exact Real.le_sqrt_of_sq_le hsum
+
+/-- The sum of two vectors in the causal future of the origin is in the causal future of the
+  origin. -/
+lemma causallyFollows_zero_add_of_causallyFollows {d : ℕ} {u v : Vector d}
+    (hu : causallyFollows 0 u) (hv : causallyFollows 0 v) : causallyFollows 0 (u + v) := by
+  obtain ⟨hu1, hu2⟩ := causallyFollows_zero_iff.mp hu
+  obtain ⟨hv1, hv2⟩ := causallyFollows_zero_iff.mp hv
+  refine causallyFollows_zero_iff.mpr ⟨?_, ?_⟩
+  · have hcs := sqrt_mul_sqrt_le_minkowskiProduct_of_causallyFollows hu hv
+    have hsq : 0 ≤ (√⟪u, u⟫ₘ + √⟪v, v⟫ₘ) ^ 2 := sq_nonneg _
+    rw [add_sq, Real.sq_sqrt hu1, Real.sq_sqrt hv1] at hsq
+    rw [minkowskiProduct_add_self]
+    linarith
+  · show 0 ≤ (u + v) (Sum.inl 0)
+    rw [apply_add]
+    exact add_nonneg hu2 hv2
+
+end Vector
+
+end Lorentz

Physlib/Relativity/Tensors/RealTensor/Vector/MinkowskiProduct.lean

@@ -172,6 +172,18 @@ lemma minkowskiProduct_self_le_timeComponent_sq {d : ℕ} (p : Vector d) :
   rw [minkowskiProduct_self_eq_timeComponent_spatialPart]
   simp
 
+/-- `⟪u, u⟫ₘ = (u⁰)² - ‖u_spatial‖²`. -/
+lemma minkowskiProduct_self_eq_sq_sub {d : ℕ} (u : Vector d) :
+    ⟪u, u⟫ₘ = u.timeComponent ^ 2 - ‖u.spatialPart‖ ^ 2 := by
+  rw [minkowskiProduct_self_eq_timeComponent_spatialPart, Real.norm_eq_abs, sq_abs]
+
+/-- `⟪u + v, u + v⟫ₘ = ⟪u, u⟫ₘ + 2 ⟪u, v⟫ₘ + ⟪v, v⟫ₘ`. -/
+lemma minkowskiProduct_add_self {d : ℕ} (u v : Vector d) :
+    ⟪u + v, u + v⟫ₘ = ⟪u, u⟫ₘ + 2 * ⟪u, v⟫ₘ + ⟪v, v⟫ₘ := by
+  simp only [map_add, _root_.add_apply]
+  rw [minkowskiProduct_symm v u]
+  ring
+
 @[simp]
 lemma minkowskiProduct_basis_left {d : ℕ} (μ : Fin 1 ⊕ Fin d) (p : Vector d) :
     ⟪basis μ, p⟫ₘ = minkowskiMatrix μ μ * p μ := by