imo.imo1988_q6 ⟷ Archive.Imo.Imo1988Q6

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -137,6 +137,7 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
     -- It contains the image of p.
     use p.2
     apply Set.mem_image_of_mem
+    -- After all, we assumed that the exceptional locus is empty.
     rwa [exceptional_empty, Set.diff_empty]
   -- We are now set for an infinite descent argument.
   -- Let m be the smallest element of the nonempty set S.
@@ -220,7 +221,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
     clear hk a b
   · -- We will now show that the fibers of the solution set are described by a quadratic equation.
     intro x y; dsimp only
-    rw [← Int.coe_nat_inj', ← sub_eq_zero]
+    rw [← Int.natCast_inj, ← sub_eq_zero]
     apply eq_iff_eq_cancel_right.2
     norm_cast
     simp; ring
@@ -284,7 +285,7 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
     clear hk a b
   · -- We will now show that the fibers of the solution set are described by a quadratic equation.
     intro x y; dsimp only
-    rw [← Int.coe_nat_inj', ← sub_eq_zero]
+    rw [← Int.natCast_inj, ← sub_eq_zero]
     apply eq_iff_eq_cancel_right.2
     simp; ring
   ·-- Show that the solution set is symmetric in a and b.

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -82,7 +82,7 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
   -- First of all, we may assume that x ≤ y.
   -- We justify this using H_symm.
   wlog hxy : x ≤ y
-  · rw [H_symm] at h₀ ; apply this y x h₀ B C base _ _ _ _ _ _ (le_of_not_le hxy); assumption'
+  · rw [H_symm] at h₀; apply this y x h₀ B C base _ _ _ _ _ _ (le_of_not_le hxy); assumption'
   -- In fact, we can easily deal with the case x = y.
   by_cases x_eq_y : x = y;
   · subst x_eq_y; exact H_diag h₀
@@ -106,25 +106,25 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
   -- Our assumptions ensure that we can then prove the claim.
   suffices exc : exceptional.nonempty
   · -- Suppose that there exists an element in the exceptional locus.
-    simp [exceptional, -add_comm, Set.Nonempty] at exc 
+    simp [exceptional, -add_comm, Set.Nonempty] at exc
     -- Let (a,b) be such an element, and consider all the possible cases.
     rcases exc with ⟨a, b, hH, hb⟩;
     rcases hb with (_ | rfl | rfl | hB | hB)
     -- The first three cases are rather easy to solve.
     · solve_by_elim
-    · rw [H_symm] at hH ; solve_by_elim
+    · rw [H_symm] at hH; solve_by_elim
     · solve_by_elim
     -- The final two cases are very similar.
     all_goals
       -- Consider the quadratic equation that (a,b) satisfies.
-      rw [H_quad] at hH 
+      rw [H_quad] at hH
       -- We find the other root of the equation, and Vieta's formulas.
       rcases vieta_formula_quadratic hH with ⟨c, h_root, hV₁, hV₂⟩
       -- By substitutions we find that b = 0 or b = a.
-      simp [hB] at hV₁ ;
+      simp [hB] at hV₁;
       subst hV₁
       rw [← Int.ofNat_zero] at *
-      rw [← H_quad] at h_root 
+      rw [← H_quad] at h_root
       -- And hence we are done by H_zero and H_diag.
       solve_by_elim
   -- To finish the main proof, we need to show that the exceptional locus is nonempty.
@@ -137,7 +137,6 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
     -- It contains the image of p.
     use p.2
     apply Set.mem_image_of_mem
-    -- After all, we assumed that the exceptional locus is empty.
     rwa [exceptional_empty, Set.diff_empty]
   -- We are now set for an infinite descent argument.
   -- Let m be the smallest element of the nonempty set S.
@@ -152,20 +151,20 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
   rcases m_mem with ⟨⟨mx, my⟩, ⟨⟨hHm, mx_lt_my⟩, h_base⟩, m_eq⟩
   -- This means that m_y = m,
   -- and the conditions H(m_x, m_y) and m_x < m_y are satisfied.
-  simp [exceptional, hHm] at mx_lt_my h_base m_eq 
-  push_neg at h_base 
+  simp [exceptional, hHm] at mx_lt_my h_base m_eq
+  push_neg at h_base
   -- Finally, it also means that (m_x, m_y) does not lie in the base locus,
   -- that m_x ≠ 0, m_x ≠ m_y, B(m_x) ≠ m_y, and B(m_x) ≠ m_x + m_y.
   rcases h_base with ⟨h_base, hmx, hm_diag, hm_B₁, hm_B₂⟩
   replace hmx : 0 < mx := pos_iff_ne_zero.mpr hmx
   -- Consider the quadratic equation that (m_x, m_y) satisfies.
   have h_quad := hHm;
-  rw [H_quad] at h_quad 
+  rw [H_quad] at h_quad
   -- We find the other root of the equation, and Vieta's formulas.
   rcases vieta_formula_quadratic h_quad with ⟨c, h_root, hV₁, hV₂⟩
   -- No we rewrite Vietas formulas a bit, and apply the descent step.
   replace hV₁ : c = B mx - my := eq_sub_of_add_eq' hV₁
-  rw [mul_comm] at hV₂ 
+  rw [mul_comm] at hV₂
   have Hc := H_desc hmx mx_lt_my h_base hHm c h_root hV₁ hV₂
   -- This means that we may assume that c ≥ 0 and c ≤ m_x.
   cases' Hc with c_nonneg c_lt
@@ -178,7 +177,7 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
   -- The second condition is rather easy to check, so we do that first.
   constructor;
   swap
-  · rwa [m_eq] at mx_lt_my 
+  · rwa [m_eq] at mx_lt_my
   -- Now we need to show that p' projects onto S. In other words, that c ∈ S.
   -- We do that, by showing that it lies in the upper branch
   -- (which is sufficient, because we assumed that the exceptional locus is empty).
@@ -214,7 +213,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
   by
   rcases h with ⟨k, hk⟩
   rw [hk, Nat.mul_div_cancel_left _ (Nat.succ_pos (a * b))]
-  simp only [sq] at hk 
+  simp only [sq] at hk
   apply
       constant_descent_vieta_jumping a b hk (fun x => k * x) (fun x => x * x - k) fun x y =>
         False <;>
@@ -234,7 +233,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
   · -- Show that the claim is true if a = b.
     intro x hx
     suffices k ≤ 1 by
-      rw [Nat.le_add_one_iff, le_zero_iff] at this 
+      rw [Nat.le_add_one_iff, le_zero_iff] at this
       rcases this with (rfl | rfl)
       · use 0; simp
       · use 1; simp
@@ -257,7 +256,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
         by
         rw [← sub_eq_zero, ← h_root]
         ring
-      rw [hzx] at hpos 
+      rw [hzx] at hpos
       replace hpos : z * x + 1 > 0 := pos_of_mul_pos_left hpos (Int.ofNat_zero_le k)
       replace hpos : z * x ≥ 0 := Int.le_of_lt_add_one hpos
       apply nonneg_of_mul_nonneg_left hpos (by exact_mod_cast hx)
@@ -278,7 +277,7 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
   by
   rcases h with ⟨k, hk⟩
   suffices k = 3 by simp_all; ring
-  simp only [sq] at hk 
+  simp only [sq] at hk
   apply
       constant_descent_vieta_jumping a b hk (fun x => k * x) (fun x => x * x + 1) fun x y =>
         x ≤ 1 <;>
@@ -295,7 +294,7 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
   · -- Show that the claim is true if a = b.
     intro x hx
     have x_sq_dvd : x * x ∣ x * x * k := dvd_mul_right (x * x) k
-    rw [← hx] at x_sq_dvd 
+    rw [← hx] at x_sq_dvd
     obtain ⟨y, hy⟩ : x * x ∣ 1 := by simpa only [Nat.dvd_add_self_left, add_assoc] using x_sq_dvd
     obtain ⟨rfl, rfl⟩ : x = 1 ∧ y = 1 := by simpa [mul_eq_one] using hy.symm
     simpa using hx.symm
@@ -307,7 +306,7 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
       linarith
     · contrapose! hV₀ with x_lt_z
       apply ne_of_gt
-      push_neg at h_base 
+      push_neg at h_base
       calc
         z * y > x * y := by apply mul_lt_mul_of_pos_right <;> linarith
         _ ≥ x * (x + 1) := by apply mul_le_mul <;> linarith
@@ -317,11 +316,11 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
           assumption_mod_cast
   · -- Show the base case.
     intro x y h h_base
-    obtain rfl | rfl : x = 0 ∨ x = 1 := by rwa [Nat.le_add_one_iff, le_zero_iff] at h_base 
+    obtain rfl | rfl : x = 0 ∨ x = 1 := by rwa [Nat.le_add_one_iff, le_zero_iff] at h_base
     · simpa using h
-    · simp only [mul_one, one_mul, add_comm, zero_add] at h 
+    · simp only [mul_one, one_mul, add_comm, zero_add] at h
       have y_dvd : y ∣ y * k := dvd_mul_right y k
-      rw [← h, ← add_assoc, Nat.dvd_add_left (dvd_mul_left y y)] at y_dvd 
+      rw [← h, ← add_assoc, Nat.dvd_add_left (dvd_mul_left y y)] at y_dvd
       obtain rfl | rfl := (Nat.dvd_prime Nat.prime_two).mp y_dvd <;> apply mul_left_cancel₀
       exacts [one_ne_zero, h.symm, two_ne_zero, h.symm]
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -251,7 +251,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
       have hpos : z * z + x * x > 0 :=
         by
         apply add_pos_of_nonneg_of_pos
-        · apply mul_self_nonneg
+        · apply hMul_self_nonneg
         · apply mul_pos <;> exact_mod_cast hx
       have hzx : z * z + x * x = (z * x + 1) * k :=
         by

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,11 +3,11 @@ Copyright (c) 2019 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathbin.Data.Nat.Prime
-import Mathbin.Data.Rat.Defs
-import Mathbin.Order.WellFounded
+import Data.Nat.Prime
+import Data.Rat.Defs
+import Order.WellFounded
 import Mathbin.Tactic.Linarith.Default
-import Mathbin.Tactic.Wlog
+import Tactic.Wlog
 
 #align_import imo.imo1988_q6 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2019 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module imo.imo1988_q6
-! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Nat.Prime
 import Mathbin.Data.Rat.Defs
@@ -14,6 +9,8 @@ import Mathbin.Order.WellFounded
 import Mathbin.Tactic.Linarith.Default
 import Mathbin.Tactic.Wlog
 
+#align_import imo.imo1988_q6 from "leanprover-community/mathlib"@"08b081ea92d80e3a41f899eea36ef6d56e0f1db0"
+
 /-!
 # IMO 1988 Q6 and constant descent Vieta jumping
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bf2428c9486c407ca38b5b3fb10b87dad0bc99fa

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 
 ! This file was ported from Lean 3 source module imo.imo1988_q6
-! leanprover-community/mathlib commit 308826471968962c6b59c7ff82a22757386603e3
+! leanprover-community/mathlib commit 08b081ea92d80e3a41f899eea36ef6d56e0f1db0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.Tactic.Wlog
 /-!
 # IMO 1988 Q6 and constant descent Vieta jumping
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Question 6 of IMO1988 is somewhat (in)famous. Several expert problem solvers
 could not tackle the question within the given time limit.
 The problem lead to the introduction of a new proof technique,

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -244,7 +244,6 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
       x * x + x * x = x * x * 2 := by rw [mul_two]
       _ ≤ x * x * k := (Nat.mul_le_mul_left (x * x) k_lt_one)
       _ < (x * x + 1) * k := by linarith
-      
   · -- Show the descent step.
     intro x y hx x_lt_y hxky h z h_root hV₁ hV₀
     constructor
@@ -267,7 +266,6 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
       calc
         z * y > x * x := by apply mul_lt_mul' <;> linarith
         _ ≥ x * x - k := sub_le_self _ (Int.ofNat_zero_le k)
-        
   ·-- There is no base case in this application of Vieta jumping.
     simp
 #align imo1988_q6 imo1988_q6
@@ -317,7 +315,6 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
           rw [mul_add, mul_one]
           apply add_lt_add_left
           assumption_mod_cast
-        
   · -- Show the base case.
     intro x y h h_base
     obtain rfl | rfl : x = 0 ∨ x = 1 := by rwa [Nat.le_add_one_iff, le_zero_iff] at h_base 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3209ddf94136d36e5e5c624b10b2a347cc9d090

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -227,7 +227,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
     apply ne_of_lt
     calc
       x * x + x * x = x * x * 2 := by rw [mul_two]
-      _ ≤ x * x * k := (Nat.mul_le_mul_left (x * x) k_lt_one)
+      _ ≤ x * x * k := Nat.mul_le_mul_left (x * x) k_lt_one
       _ < (x * x + 1) * k := by linarith
   · -- Show the descent step.
     intro x y hx x_lt_y _ _ z h_root _ hV₀

Reduce the diff of #11499

Renames

All in the Int namespace:

• ofNat_eq_cast → ofNat_eq_natCast
• cast_eq_cast_iff_Nat → natCast_inj
• natCast_eq_ofNat → ofNat_eq_natCast
• coe_nat_sub → natCast_sub
• coe_nat_nonneg → natCast_nonneg
• sign_coe_add_one → sign_natCast_add_one
• nat_succ_eq_int_succ → natCast_succ
• succ_neg_nat_succ → succ_neg_natCast_succ
• coe_pred_of_pos → natCast_pred_of_pos
• coe_nat_div → natCast_div
• coe_nat_ediv → natCast_ediv
• sign_coe_nat_of_nonzero → sign_natCast_of_ne_zero
• toNat_coe_nat → toNat_natCast
• toNat_coe_nat_add_one → toNat_natCast_add_one
• coe_nat_dvd → natCast_dvd_natCast
• coe_nat_dvd_left → natCast_dvd
• coe_nat_dvd_right → dvd_natCast
• le_coe_nat_sub → le_natCast_sub
• succ_coe_nat_pos → succ_natCast_pos
• coe_nat_modEq_iff → natCast_modEq_iff
• coe_natAbs → natCast_natAbs
• coe_nat_eq_zero → natCast_eq_zero
• coe_nat_ne_zero → natCast_ne_zero
• coe_nat_ne_zero_iff_pos → natCast_ne_zero_iff_pos
• abs_coe_nat → abs_natCast
• coe_nat_nonpos_iff → natCast_nonpos_iff

Also rename Nat.coe_nat_dvd to Nat.cast_dvd_cast

@@ -207,7 +207,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
     clear hk a b
   · -- We will now show that the fibers of the solution set are described by a quadratic equation.
     intro x y
-    rw [← Int.coe_nat_inj', ← sub_eq_zero]
+    rw [← Int.natCast_inj, ← sub_eq_zero]
     apply eq_iff_eq_cancel_right.2
     simp; ring
   · -- Show that the solution set is symmetric in a and b.
@@ -265,7 +265,7 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
     clear hk a b
   · -- We will now show that the fibers of the solution set are described by a quadratic equation.
     intro x y
-    rw [← Int.coe_nat_inj', ← sub_eq_zero]
+    rw [← Int.natCast_inj, ← sub_eq_zero]
     apply eq_iff_eq_cancel_right.2
     simp; ring
   · -- Show that the solution set is symmetric in a and b.

Reconstitute the file Algebra.Order.Monoid.WithZero from three files:

• Algebra.Order.Monoid.WithZero.Defs
• Algebra.Order.Monoid.WithZero.Basic
• Algebra.Order.WithZero

Avoid importing it in many files. Most uses were just to get le_zero_iff to work on Nat.

Before

After

@@ -3,7 +3,6 @@ Copyright (c) 2019 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
-import Mathlib.Algebra.Order.WithZero
 import Mathlib.Data.Nat.Prime
 import Mathlib.Data.Rat.Defs
 import Mathlib.Order.WellFounded
@@ -220,7 +219,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
   · -- Show that the claim is true if a = b.
     intro x hx
     suffices k ≤ 1 by
-      rw [Nat.le_add_one_iff, le_zero_iff] at this
+      rw [Nat.le_add_one_iff, Nat.le_zero] at this
       rcases this with (rfl | rfl)
       · use 0; simp
       · use 1; simp
@@ -298,7 +297,7 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
           assumption_mod_cast
   · -- Show the base case.
     intro x y h h_base
-    obtain rfl | rfl : x = 0 ∨ x = 1 := by rwa [Nat.le_add_one_iff, le_zero_iff] at h_base
+    obtain rfl | rfl : x = 0 ∨ x = 1 := by rwa [Nat.le_add_one_iff, Nat.le_zero] at h_base
     · simp at h
     · rw [mul_one, one_mul, add_right_comm] at h
       have y_dvd : y ∣ y * k := dvd_mul_right y k

Let κ : kernel α (γ × β) and ν : kernel α γ be two finite kernels with kernel.fst κ ≤ ν, where γ has a countably generated σ-algebra (true in particular for standard Borel spaces). We build a function f : α → γ → Set β → ℝ jointly measurable in the first two arguments such that for all a : α and all measurable sets s : Set β and A : Set γ, ∫ x in A, f a x s ∂(ν a) = (κ a (A ×ˢ s)).toReal.

Co-authored-by: Rémy Degenne <remydegenne@gmail.com>

@@ -3,6 +3,7 @@ Copyright (c) 2019 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
 -/
+import Mathlib.Algebra.Order.WithZero
 import Mathlib.Data.Nat.Prime
 import Mathlib.Data.Rat.Defs
 import Mathlib.Order.WellFounded

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -98,7 +98,7 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
   -- Our assumptions ensure that we can then prove the claim.
   suffices exc : exceptional.Nonempty by
     -- Suppose that there exists an element in the exceptional locus.
-    simp only [Set.Nonempty, Prod.exists, Set.mem_setOf_eq] at exc
+    simp only [Set.Nonempty, Prod.exists, Set.mem_setOf_eq, exceptional] at exc
     -- Let (a,b) be such an element, and consider all the possible cases.
     rcases exc with ⟨a, b, hH, hb⟩
     rcases hb with (_ | rfl | rfl | hB | hB)
@@ -144,7 +144,7 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
   -- This means that m_y = m,
   -- and the conditions H(m_x, m_y) and m_x < m_y are satisfied.
   simp only at mx_lt_my hHm m_eq
-  simp only [hHm, Set.mem_setOf_eq, true_and] at h_base
+  simp only [exceptional, hHm, Set.mem_setOf_eq, true_and] at h_base
   push_neg at h_base
   -- Finally, it also means that (m_x, m_y) does not lie in the base locus,
   -- that m_x ≠ 0, m_x ≠ m_y, B(m_x) ≠ m_y, and B(m_x) ≠ m_x + m_y.

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -96,8 +96,8 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
   -- The strategy is to show that the exceptional locus in nonempty
   -- by running a descent argument that starts with the given point p = (x,y).
   -- Our assumptions ensure that we can then prove the claim.
-  suffices exc : exceptional.Nonempty
-  · -- Suppose that there exists an element in the exceptional locus.
+  suffices exc : exceptional.Nonempty by
+    -- Suppose that there exists an element in the exceptional locus.
     simp only [Set.Nonempty, Prod.exists, Set.mem_setOf_eq] at exc
     -- Let (a,b) be such an element, and consider all the possible cases.
     rcases exc with ⟨a, b, hH, hb⟩
@@ -182,9 +182,7 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
     rw [H_symm, H_quad]
     simpa using h_root
   · -- For the second condition, we note that it suffices to check that c ≠ m_x.
-    suffices hc : c ≠ mx
-    · refine' lt_of_le_of_ne _ hc
-      exact mod_cast c_lt
+    suffices hc : c ≠ mx from lt_of_le_of_ne (mod_cast c_lt) hc
     -- However, recall that B(m_x) ≠ m_x + m_y.
     -- If c = m_x, we can prove B(m_x) = m_x + m_y.
     contrapose! hm_B₂

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -184,7 +184,7 @@ theorem constant_descent_vieta_jumping (x y : ℕ) {claim : Prop} {H : ℕ → 
   · -- For the second condition, we note that it suffices to check that c ≠ m_x.
     suffices hc : c ≠ mx
     · refine' lt_of_le_of_ne _ hc
-      exact_mod_cast c_lt
+      exact mod_cast c_lt
     -- However, recall that B(m_x) ≠ m_x + m_y.
     -- If c = m_x, we can prove B(m_x) = m_x + m_y.
     contrapose! hm_B₂
@@ -237,14 +237,14 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
     · have hpos : z * z + x * x > 0 := by
         apply add_pos_of_nonneg_of_pos
         · apply mul_self_nonneg
-        · apply mul_pos <;> exact_mod_cast hx
+        · apply mul_pos <;> exact mod_cast hx
       have hzx : z * z + x * x = (z * x + 1) * k := by
         rw [← sub_eq_zero, ← h_root]
         ring
       rw [hzx] at hpos
       replace hpos : z * x + 1 > 0 := pos_of_mul_pos_left hpos (Int.ofNat_zero_le k)
       replace hpos : z * x ≥ 0 := Int.le_of_lt_add_one hpos
-      apply nonneg_of_mul_nonneg_left hpos (by exact_mod_cast hx)
+      apply nonneg_of_mul_nonneg_left hpos (mod_cast hx)
     · contrapose! hV₀ with x_lt_z
       apply ne_of_gt
       calc
@@ -284,7 +284,7 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
   · -- Show the descent step.
     intro x y _ hx h_base _ z _ _ hV₀
     constructor
-    · have zy_pos : z * y ≥ 0 := by rw [hV₀]; exact_mod_cast Nat.zero_le _
+    · have zy_pos : z * y ≥ 0 := by rw [hV₀]; exact mod_cast Nat.zero_le _
       apply nonneg_of_mul_nonneg_left zy_pos
       linarith
     · contrapose! hV₀ with x_lt_z

I appreciate that this is "going backwards" in the sense of requiring more explicit imports of tactics (and making it more likely that users writing new files will need to import tactics rather than finding them already available).

However it's useful for me while I'm intensively working on refactoring and upstreaming tactics if I can minimise the dependencies between tactics. I'm very much aware that in the long run we don't want users to have to manage imports of tactics, and I am definitely going to tidy this all up again once I'm finished with this project!

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -7,6 +7,7 @@ import Mathlib.Data.Nat.Prime
 import Mathlib.Data.Rat.Defs
 import Mathlib.Order.WellFounded
 import Mathlib.Tactic.Linarith
+import Mathlib.Tactic.Ring
 import Mathlib.Tactic.WLOG
 
 #align_import imo.imo1988_q6 from "leanprover-community/mathlib"@"308826471968962c6b59c7ff82a22757386603e3"

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -207,7 +207,7 @@ theorem imo1988_q6 {a b : ℕ} (h : a * b + 1 ∣ a ^ 2 + b ^ 2) :
       hk (fun x => k * x) (fun x => x * x - k) fun _ _ => False <;>
     clear hk a b
   · -- We will now show that the fibers of the solution set are described by a quadratic equation.
-    intro x y; dsimp only
+    intro x y
     rw [← Int.coe_nat_inj', ← sub_eq_zero]
     apply eq_iff_eq_cancel_right.2
     simp; ring
@@ -265,7 +265,7 @@ example {a b : ℕ} (h : a * b ∣ a ^ 2 + b ^ 2 + 1) : 3 * a * b = a ^ 2 + b ^
       hk (fun x => k * x) (fun x => x * x + 1) fun x _ => x ≤ 1 <;>
     clear hk a b
   · -- We will now show that the fibers of the solution set are described by a quadratic equation.
-    intro x y; dsimp only
+    intro x y
     rw [← Int.coe_nat_inj', ← sub_eq_zero]
     apply eq_iff_eq_cancel_right.2
     simp; ring

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2019 Johan Commelin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Johan Commelin
-
-! This file was ported from Lean 3 source module imo.imo1988_q6
-! leanprover-community/mathlib commit 308826471968962c6b59c7ff82a22757386603e3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Nat.Prime
 import Mathlib.Data.Rat.Defs
@@ -14,6 +9,8 @@ import Mathlib.Order.WellFounded
 import Mathlib.Tactic.Linarith
 import Mathlib.Tactic.WLOG
 
+#align_import imo.imo1988_q6 from "leanprover-community/mathlib"@"308826471968962c6b59c7ff82a22757386603e3"
+
 /-!
 # IMO 1988 Q6 and constant descent Vieta jumping