number_theory.liouville.basic ⟷ Mathlib.NumberTheory.Liouville.Basic

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

@@ -182,7 +182,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
     rw [show (a + 1 : ℝ) = ((a + 1 : ℕ) : ℤ) by norm_cast] at hq ⊢
     -- key observation: the right-hand side of the inequality is an *integer*.  Therefore,
     -- if its absolute value is not at least one, then it vanishes.  Proceed by contradiction
-    refine' one_le_pow_mul_abs_eval_div (Int.coe_nat_succ_pos a) fun hy => _
+    refine' one_le_pow_mul_abs_eval_div (Int.natCast_succ_pos a) fun hy => _
     -- As the evaluation of the polynomial vanishes, we found a root of `fR` that is rational.
     -- We know that `α` is the only root of `fR` in our interval, and `α` is irrational:
     -- follow your nose.

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

@@ -60,14 +60,14 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
       div_lt_div_iff (abs_pos.mpr (ne_of_gt bq0)) (pow_pos qR0 _), abs_of_pos bq0, one_mul,
       ←-- ... and revert to integers
       Int.cast_pow,
-      ← Int.cast_mul, ← Int.cast_ofNat, ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, ←
+      ← Int.cast_mul, ← Int.cast_natCast, ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, ←
       Int.cast_abs, ← Int.cast_mul, Int.cast_lt] at a1
   -- At a0, clear denominators...
   replace a0 : ¬a * q - ↑b * p = 0;
   ·
     rwa [Ne.def, div_eq_div_iff b0 (ne_of_gt qR0), mul_comm ↑p, ← sub_eq_zero,
       ←-- ... and revert to integers
-      Int.cast_ofNat,
+      Int.cast_natCast,
       ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, Int.cast_eq_zero] at a0
   -- Actually, `q` is a natural number
   lift q to ℕ using (zero_lt_one.trans q1).le

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Jujian Zhang
 -/
 import Analysis.Calculus.MeanValue
-import Data.Polynomial.DenomsClearable
+import Algebra.Polynomial.DenomsClearable
 import Data.Real.Irrational
 
 #align_import number_theory.liouville.basic from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"

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

@@ -57,14 +57,18 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   replace a1 : |a * q - b * p| * q ^ (b + 1) < b * q;
   ·
     rwa [div_sub_div _ _ b0 (ne_of_gt qR0), abs_div,
-      div_lt_div_iff (abs_pos.mpr (ne_of_gt bq0)) (pow_pos qR0 _), abs_of_pos bq0, one_mul, ←
-      Int.cast_pow, ← Int.cast_mul, ← Int.cast_ofNat, ← Int.cast_mul, ← Int.cast_mul, ←
-      Int.cast_sub, ← Int.cast_abs, ← Int.cast_mul, Int.cast_lt] at a1
+      div_lt_div_iff (abs_pos.mpr (ne_of_gt bq0)) (pow_pos qR0 _), abs_of_pos bq0, one_mul,
+      ←-- ... and revert to integers
+      Int.cast_pow,
+      ← Int.cast_mul, ← Int.cast_ofNat, ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, ←
+      Int.cast_abs, ← Int.cast_mul, Int.cast_lt] at a1
   -- At a0, clear denominators...
   replace a0 : ¬a * q - ↑b * p = 0;
   ·
-    rwa [Ne.def, div_eq_div_iff b0 (ne_of_gt qR0), mul_comm ↑p, ← sub_eq_zero, ← Int.cast_ofNat, ←
-      Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, Int.cast_eq_zero] at a0
+    rwa [Ne.def, div_eq_div_iff b0 (ne_of_gt qR0), mul_comm ↑p, ← sub_eq_zero,
+      ←-- ... and revert to integers
+      Int.cast_ofNat,
+      ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, Int.cast_eq_zero] at a0
   -- Actually, `q` is a natural number
   lift q to ℕ using (zero_lt_one.trans q1).le
   -- Looks innocuous, but we now have an integer with non-zero absolute value: this is at

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

@@ -45,7 +45,7 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   -- By contradiction, `x = a / b`, with `a ∈ ℤ`, `0 < b ∈ ℕ` is a Liouville number,
   rintro ⟨⟨a, b, bN0, cop⟩, rfl⟩
   -- clear up the mess of constructions of rationals
-  rw [Rat.cast_mk', ← div_eq_mul_inv] at h 
+  rw [Rat.cast_mk', ← div_eq_mul_inv] at h
   -- Since `a / b` is a Liouville number, there are `p, q ∈ ℤ`, with `q1 : 1 < q`,
   -- `a0 : a / b ≠ p / q` and `a1 : |a / b - p / q| < 1 / q ^ (b + 1)`
   rcases h (b + 1) with ⟨p, q, q1, a0, a1⟩
@@ -57,18 +57,14 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   replace a1 : |a * q - b * p| * q ^ (b + 1) < b * q;
   ·
     rwa [div_sub_div _ _ b0 (ne_of_gt qR0), abs_div,
-      div_lt_div_iff (abs_pos.mpr (ne_of_gt bq0)) (pow_pos qR0 _), abs_of_pos bq0, one_mul,
-      ←-- ... and revert to integers
-      Int.cast_pow,
-      ← Int.cast_mul, ← Int.cast_ofNat, ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, ←
-      Int.cast_abs, ← Int.cast_mul, Int.cast_lt] at a1 
+      div_lt_div_iff (abs_pos.mpr (ne_of_gt bq0)) (pow_pos qR0 _), abs_of_pos bq0, one_mul, ←
+      Int.cast_pow, ← Int.cast_mul, ← Int.cast_ofNat, ← Int.cast_mul, ← Int.cast_mul, ←
+      Int.cast_sub, ← Int.cast_abs, ← Int.cast_mul, Int.cast_lt] at a1
   -- At a0, clear denominators...
   replace a0 : ¬a * q - ↑b * p = 0;
   ·
-    rwa [Ne.def, div_eq_div_iff b0 (ne_of_gt qR0), mul_comm ↑p, ← sub_eq_zero,
-      ←-- ... and revert to integers
-      Int.cast_ofNat,
-      ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, Int.cast_eq_zero] at a0 
+    rwa [Ne.def, div_eq_div_iff b0 (ne_of_gt qR0), mul_comm ↑p, ← sub_eq_zero, ← Int.cast_ofNat, ←
+      Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, Int.cast_eq_zero] at a0
   -- Actually, `q` is a natural number
   lift q to ℕ using (zero_lt_one.trans q1).le
   -- Looks innocuous, but we now have an integer with non-zero absolute value: this is at
@@ -77,7 +73,7 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   -- Actually, the absolute value of an integer is a natural number
   lift |a * ↑q - ↑b * p| to ℕ using abs_nonneg (a * ↑q - ↑b * p)
   -- At a1, revert to natural numbers
-  rw [← Int.ofNat_mul, ← Int.coe_nat_pow, ← Int.ofNat_mul, Int.ofNat_lt] at a1 
+  rw [← Int.ofNat_mul, ← Int.coe_nat_pow, ← Int.ofNat_mul, Int.ofNat_lt] at a1
   -- Recall this is by contradiction: we obtained the inequality `b * q ≤ x * q ^ (b + 1)`, so
   -- we are done.
   exact not_le.mpr a1 (Nat.mul_lt_mul_pow_succ (int.coe_nat_pos.mp ap) (int.coe_nat_lt.mp q1)).le
@@ -170,7 +166,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
   · exact fun a => one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg) _
   -- 2: the polynomial `fR` is Lipschitz at `α` -- as its derivative continuous;
   · rw [mul_comm]
-    rw [Real.closedBall_eq_Icc] at hy 
+    rw [Real.closedBall_eq_Icc] at hy
     -- apply the Mean Value Theorem: the bound on the derivative comes from differentiability.
     refine'
       Convex.norm_image_sub_le_of_norm_deriv_le (fun _ _ => fR.differentiable_at)
@@ -202,7 +198,7 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
   rintro ⟨f : ℤ[X], f0, ef0⟩
   -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.
   replace ef0 : (f.map (algebraMap ℤ ℝ)).eval x = 0;
-  · rwa [aeval_def, ← eval_map] at ef0 
+  · rwa [aeval_def, ← eval_map] at ef0
   -- There is a "large" real number `A` such that `(b + 1) ^ (deg f) * |f (x - a / (b + 1))| * A`
   -- is at least one.  This is obtained from lemma `exists_pos_real_of_irrational_root`.
   obtain ⟨A, hA, h⟩ :
@@ -219,7 +215,7 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
   -- recall, this is a proof by contradiction!
   refine' lt_irrefl ((b : ℝ) ^ f.nat_degree * |x - ↑a / ↑b|) _
   -- clear denominators at `a1`
-  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1 
+  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1
   -- split the inequality via `1 / A`.
   refine' (_ : (b : ℝ) ^ f.nat_degree * |x - a / b| < 1 / A).trans_le _
   -- This branch of the proof uses the Liouville condition and the Archimedean property
@@ -234,7 +230,7 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
   · lift b to ℕ using zero_le_one.trans b1.le
     specialize h a b.pred
     rwa [← Nat.cast_succ, Nat.succ_pred_eq_of_pos (zero_lt_one.trans _), ← mul_assoc, ←
-      div_le_iff hA] at h 
+      div_le_iff hA] at h
     exact int.coe_nat_lt.mp b1
 #align liouville.transcendental Liouville.transcendental
 -/

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

@@ -227,7 +227,7 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
     refine' lt_of_le_of_lt _ a1
     refine' mul_le_mul_of_nonneg_right _ (mul_nonneg (pow_nonneg b0.le _) (abs_nonneg _))
     refine' hn.le.trans _
-    refine' pow_le_pow_of_le_left zero_le_two _ _
+    refine' pow_le_pow_left zero_le_two _ _
     exact int.cast_two.symm.le.trans (int.cast_le.mpr (int.add_one_le_iff.mpr b1))
   -- this branch of the proof exploits the "integrality" of evaluations of polynomials
   -- at ratios of integers.

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

@@ -3,9 +3,9 @@ Copyright (c) 2020 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Jujian Zhang
 -/
-import Mathbin.Analysis.Calculus.MeanValue
-import Mathbin.Data.Polynomial.DenomsClearable
-import Mathbin.Data.Real.Irrational
+import Analysis.Calculus.MeanValue
+import Data.Polynomial.DenomsClearable
+import Data.Real.Irrational
 
 #align_import number_theory.liouville.basic from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Jujian Zhang
-
-! This file was ported from Lean 3 source module number_theory.liouville.basic
-! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Calculus.MeanValue
 import Mathbin.Data.Polynomial.DenomsClearable
 import Mathbin.Data.Real.Irrational
 
+#align_import number_theory.liouville.basic from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
+
 /-!
 
 # Liouville's theorem

mathlib commit https://github.com/leanprover-community/mathlib/commit/9240e8be927a0955b9a82c6c85ef499ee3a626b8

@@ -167,7 +167,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
       (continuous_abs.comp fR.derivative.continuous_aeval).ContinuousOn
   -- Use the key lemma `exists_one_le_pow_mul_dist`: we are left to show that ...
   refine'
-    @exists_one_le_pow_mul_dist ℤ ℕ ℝ _ _ _ (fun y => fR.eval y) α ζ (|fR.derivative.eval xm|) _ z0
+    @exists_one_le_pow_mul_dist ℤ ℕ ℝ _ _ _ (fun y => fR.eval y) α ζ |fR.derivative.eval xm| _ z0
       (fun y hy => _) fun z a hq => _
   -- 1: the denominators are positive -- essentially by definition;
   · exact fun a => one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg) _

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -91,6 +91,7 @@ open Polynomial Metric Set Real RingHom
 
 open scoped Polynomial
 
+#print Liouville.exists_one_le_pow_mul_dist /-
 /-- Let `Z, N` be types, let `R` be a metric space, let `α : R` be a point and let
 `j : Z → N → R` be a function.  We aim to estimate how close we can get to `α`, while staying
 in the image of `j`.  The points `j z a` of `R` in the image of `j` come with a "cost" equal to
@@ -137,7 +138,9 @@ theorem exists_one_le_pow_mul_dist {Z N R : Type _} [PseudoMetricSpace R] {d : N
     refine' mul_le_mul_of_nonneg_left ((B this).trans _) (zero_le_one.trans (d0 a))
     exact mul_le_mul_of_nonneg_left (le_max_right _ M) dist_nonneg
 #align liouville.exists_one_le_pow_mul_dist Liouville.exists_one_le_pow_mul_dist
+-/
 
+#print Liouville.exists_pos_real_of_irrational_root /-
 theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f : ℤ[X]} (f0 : f ≠ 0)
     (fa : eval α (map (algebraMap ℤ ℝ) f) = 0) :
     ∃ A : ℝ, 0 < A ∧ ∀ a : ℤ, ∀ b : ℕ, (1 : ℝ) ≤ (b + 1) ^ f.natDegree * (|α - a / (b + 1)| * A) :=
@@ -191,7 +194,9 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
     refine' ⟨hq, finset.mem_coe.mp (multiset.mem_to_finset.mpr _)⟩
     exact (mem_roots fR0).mpr (is_root.def.mpr hy)
 #align liouville.exists_pos_real_of_irrational_root Liouville.exists_pos_real_of_irrational_root
+-/
 
+#print Liouville.transcendental /-
 /-- **Liouville's Theorem** -/
 protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x :=
   by
@@ -235,6 +240,7 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
       div_le_iff hA] at h 
     exact int.coe_nat_lt.mp b1
 #align liouville.transcendental Liouville.transcendental
+-/
 
 end Liouville
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Jujian Zhang
 
 ! This file was ported from Lean 3 source module number_theory.liouville.basic
-! leanprover-community/mathlib commit 04e80bb7e8510958cd9aacd32fe2dc147af0b9f1
+! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -16,6 +16,9 @@ import Mathbin.Data.Real.Irrational
 
 # Liouville's theorem
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file contains a proof of Liouville's theorem stating that all Liouville numbers are
 transcendental.
 
@@ -27,6 +30,7 @@ Liouville numbers.
 -/
 
 
+#print Liouville /-
 /-- A Liouville number is a real number `x` such that for every natural number `n`, there exist
 `a, b ∈ ℤ` with `1 < b` such that `0 < |x - a/b| < 1/bⁿ`.
 In the implementation, the condition `x ≠ a/b` replaces the traditional equivalent `0 < |x - a/b|`.
@@ -34,9 +38,11 @@ In the implementation, the condition `x ≠ a/b` replaces the traditional equiva
 def Liouville (x : ℝ) :=
   ∀ n : ℕ, ∃ a b : ℤ, 1 < b ∧ x ≠ a / b ∧ |x - a / b| < 1 / b ^ n
 #align liouville Liouville
+-/
 
 namespace Liouville
 
+#print Liouville.irrational /-
 protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   by
   -- By contradiction, `x = a / b`, with `a ∈ ℤ`, `0 < b ∈ ℕ` is a Liouville number,
@@ -79,6 +85,7 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   -- we are done.
   exact not_le.mpr a1 (Nat.mul_lt_mul_pow_succ (int.coe_nat_pos.mp ap) (int.coe_nat_lt.mp q1)).le
 #align liouville.irrational Liouville.irrational
+-/
 
 open Polynomial Metric Set Real RingHom
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/34ebaffc1d1e8e783fc05438ec2e70af87275ac9

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Jujian Zhang
 
 ! This file was ported from Lean 3 source module number_theory.liouville.basic
-! leanprover-community/mathlib commit 62c0a4ef1441edb463095ea02a06e87f3dfe135c
+! leanprover-community/mathlib commit 04e80bb7e8510958cd9aacd32fe2dc147af0b9f1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -37,8 +37,7 @@ def Liouville (x : ℝ) :=
 
 namespace Liouville
 
-@[protected]
-theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
+protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   by
   -- By contradiction, `x = a / b`, with `a ∈ ℤ`, `0 < b ∈ ℕ` is a Liouville number,
   rintro ⟨⟨a, b, bN0, cop⟩, rfl⟩
@@ -187,7 +186,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
 #align liouville.exists_pos_real_of_irrational_root Liouville.exists_pos_real_of_irrational_root
 
 /-- **Liouville's Theorem** -/
-theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x :=
+protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x :=
   by
   -- Proceed by contradiction: if `x` is algebraic, then `x` is the root (`ef0`) of a
   -- non-zero (`f0`) polynomial `f`

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -43,7 +43,7 @@ theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   -- By contradiction, `x = a / b`, with `a ∈ ℤ`, `0 < b ∈ ℕ` is a Liouville number,
   rintro ⟨⟨a, b, bN0, cop⟩, rfl⟩
   -- clear up the mess of constructions of rationals
-  rw [Rat.cast_mk', ← div_eq_mul_inv] at h
+  rw [Rat.cast_mk', ← div_eq_mul_inv] at h 
   -- Since `a / b` is a Liouville number, there are `p, q ∈ ℤ`, with `q1 : 1 < q`,
   -- `a0 : a / b ≠ p / q` and `a1 : |a / b - p / q| < 1 / q ^ (b + 1)`
   rcases h (b + 1) with ⟨p, q, q1, a0, a1⟩
@@ -59,14 +59,14 @@ theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
       ←-- ... and revert to integers
       Int.cast_pow,
       ← Int.cast_mul, ← Int.cast_ofNat, ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, ←
-      Int.cast_abs, ← Int.cast_mul, Int.cast_lt] at a1
+      Int.cast_abs, ← Int.cast_mul, Int.cast_lt] at a1 
   -- At a0, clear denominators...
   replace a0 : ¬a * q - ↑b * p = 0;
   ·
     rwa [Ne.def, div_eq_div_iff b0 (ne_of_gt qR0), mul_comm ↑p, ← sub_eq_zero,
       ←-- ... and revert to integers
       Int.cast_ofNat,
-      ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, Int.cast_eq_zero] at a0
+      ← Int.cast_mul, ← Int.cast_mul, ← Int.cast_sub, Int.cast_eq_zero] at a0 
   -- Actually, `q` is a natural number
   lift q to ℕ using (zero_lt_one.trans q1).le
   -- Looks innocuous, but we now have an integer with non-zero absolute value: this is at
@@ -75,7 +75,7 @@ theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
   -- Actually, the absolute value of an integer is a natural number
   lift |a * ↑q - ↑b * p| to ℕ using abs_nonneg (a * ↑q - ↑b * p)
   -- At a1, revert to natural numbers
-  rw [← Int.ofNat_mul, ← Int.coe_nat_pow, ← Int.ofNat_mul, Int.ofNat_lt] at a1
+  rw [← Int.ofNat_mul, ← Int.coe_nat_pow, ← Int.ofNat_mul, Int.ofNat_lt] at a1 
   -- Recall this is by contradiction: we obtained the inequality `b * q ≤ x * q ^ (b + 1)`, so
   -- we are done.
   exact not_le.mpr a1 (Nat.mul_lt_mul_pow_succ (int.coe_nat_pos.mp ap) (int.coe_nat_lt.mp q1)).le
@@ -151,7 +151,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
     @exists_closed_ball_inter_eq_singleton_of_discrete _ _ _ discrete_of_t1_of_finite _ ar
   -- Since `fR` is continuous, it is bounded on the interval above.
   obtain ⟨xm, -, hM⟩ :
-    ∃ (xm : ℝ)(H : xm ∈ Icc (α - ζ) (α + ζ)),
+    ∃ (xm : ℝ) (H : xm ∈ Icc (α - ζ) (α + ζ)),
       ∀ y : ℝ, y ∈ Icc (α - ζ) (α + ζ) → |fR.derivative.eval y| ≤ |fR.derivative.eval xm| :=
     IsCompact.exists_forall_ge is_compact_Icc
       ⟨α, (sub_lt_self α z0).le, (lt_add_of_pos_right α z0).le⟩
@@ -164,7 +164,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
   · exact fun a => one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg) _
   -- 2: the polynomial `fR` is Lipschitz at `α` -- as its derivative continuous;
   · rw [mul_comm]
-    rw [Real.closedBall_eq_Icc] at hy
+    rw [Real.closedBall_eq_Icc] at hy 
     -- apply the Mean Value Theorem: the bound on the derivative comes from differentiability.
     refine'
       Convex.norm_image_sub_le_of_norm_deriv_le (fun _ _ => fR.differentiable_at)
@@ -173,7 +173,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
   -- 3: the weird inequality of Liouville type with powers of the denominators.
   · show 1 ≤ (a + 1 : ℝ) ^ f.nat_degree * |eval α fR - eval (z / (a + 1)) fR|
     rw [fa, zero_sub, abs_neg]
-    rw [show (a + 1 : ℝ) = ((a + 1 : ℕ) : ℤ) by norm_cast] at hq⊢
+    rw [show (a + 1 : ℝ) = ((a + 1 : ℕ) : ℤ) by norm_cast] at hq ⊢
     -- key observation: the right-hand side of the inequality is an *integer*.  Therefore,
     -- if its absolute value is not at least one, then it vanishes.  Proceed by contradiction
     refine' one_le_pow_mul_abs_eval_div (Int.coe_nat_succ_pos a) fun hy => _
@@ -194,7 +194,7 @@ theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x :=
   rintro ⟨f : ℤ[X], f0, ef0⟩
   -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.
   replace ef0 : (f.map (algebraMap ℤ ℝ)).eval x = 0;
-  · rwa [aeval_def, ← eval_map] at ef0
+  · rwa [aeval_def, ← eval_map] at ef0 
   -- There is a "large" real number `A` such that `(b + 1) ^ (deg f) * |f (x - a / (b + 1))| * A`
   -- is at least one.  This is obtained from lemma `exists_pos_real_of_irrational_root`.
   obtain ⟨A, hA, h⟩ :
@@ -211,7 +211,7 @@ theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x :=
   -- recall, this is a proof by contradiction!
   refine' lt_irrefl ((b : ℝ) ^ f.nat_degree * |x - ↑a / ↑b|) _
   -- clear denominators at `a1`
-  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1
+  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1 
   -- split the inequality via `1 / A`.
   refine' (_ : (b : ℝ) ^ f.nat_degree * |x - a / b| < 1 / A).trans_le _
   -- This branch of the proof uses the Liouville condition and the Archimedean property
@@ -226,7 +226,7 @@ theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x :=
   · lift b to ℕ using zero_le_one.trans b1.le
     specialize h a b.pred
     rwa [← Nat.cast_succ, Nat.succ_pred_eq_of_pos (zero_lt_one.trans _), ← mul_assoc, ←
-      div_le_iff hA] at h
+      div_le_iff hA] at h 
     exact int.coe_nat_lt.mp b1
 #align liouville.transcendental Liouville.transcendental
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -83,7 +83,7 @@ theorem irrational {x : ℝ} (h : Liouville x) : Irrational x :=
 
 open Polynomial Metric Set Real RingHom
 
-open Polynomial
+open scoped Polynomial
 
 /-- Let `Z, N` be types, let `R` be a metric space, let `α : R` be a point and let
 `j : Z → N → R` be a function.  We aim to estimate how close we can get to `α`, while staying

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -168,10 +168,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
     -- apply the Mean Value Theorem: the bound on the derivative comes from differentiability.
     refine'
       Convex.norm_image_sub_le_of_norm_deriv_le (fun _ _ => fR.differentiable_at)
-        (fun y h => by
-          rw [fR.deriv]
-          exact hM _ h)
-        (convex_Icc _ _) hy (mem_Icc_iff_abs_le.mp _)
+        (fun y h => by rw [fR.deriv]; exact hM _ h) (convex_Icc _ _) hy (mem_Icc_iff_abs_le.mp _)
     exact @mem_closed_ball_self ℝ _ α ζ (le_of_lt z0)
   -- 3: the weird inequality of Liouville type with powers of the denominators.
   · show 1 ≤ (a + 1 : ℝ) ^ f.nat_degree * |eval α fR - eval (z / (a + 1)) fR|
@@ -209,11 +206,7 @@ theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental ℤ x :=
   -- Use the Liouville property, with exponent `r +  deg f`.
   obtain ⟨a, b, b1, -, a1⟩ :
     ∃ a b : ℤ, 1 < b ∧ x ≠ a / b ∧ |x - a / b| < 1 / b ^ (r + f.nat_degree) := lx (r + f.nat_degree)
-  have b0 : (0 : ℝ) < b :=
-    zero_lt_one.trans
-      (by
-        rw [← Int.cast_one]
-        exact int.cast_lt.mpr b1)
+  have b0 : (0 : ℝ) < b := zero_lt_one.trans (by rw [← Int.cast_one]; exact int.cast_lt.mpr b1)
   -- Prove that `b ^ f.nat_degree * abs (x - a / b)` is strictly smaller than itself
   -- recall, this is a proof by contradiction!
   refine' lt_irrefl ((b : ℝ) ^ f.nat_degree * |x - ↑a / ↑b|) _

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

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

@@ -132,7 +132,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
       (fR0.trans (Polynomial.map_zero _).symm)
   -- reformulating assumption `fa`: `α` is a root of `fR`.
   have ar : α ∈ (fR.roots.toFinset : Set ℝ) :=
-    Finset.mem_coe.mpr (Multiset.mem_toFinset.mpr ((mem_roots fR0).mpr (IsRoot.definition.mpr fa)))
+    Finset.mem_coe.mpr (Multiset.mem_toFinset.mpr ((mem_roots fR0).mpr (IsRoot.def.mpr fa)))
   -- Since the polynomial `fR` has finitely many roots, there is a closed interval centered at `α`
   -- such that `α` is the only root of `fR` in the interval.
   obtain ⟨ζ, z0, U⟩ : ∃ ζ > 0, closedBall α ζ ∩ fR.roots.toFinset = {α} :=
@@ -170,7 +170,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
     refine' (irrational_iff_ne_rational α).mp ha z (a + 1) (mem_singleton_iff.mp _).symm
     refine' U.subset _
     refine' ⟨hq, Finset.mem_coe.mp (Multiset.mem_toFinset.mpr _)⟩
-    exact (mem_roots fR0).mpr (IsRoot.definition.mpr hy)
+    exact (mem_roots fR0).mpr (IsRoot.def.mpr hy)
 #align liouville.exists_pos_real_of_irrational_root Liouville.exists_pos_real_of_irrational_root
 
 /-- **Liouville's Theorem** -/

This is less exhaustive than its sibling #11486 because edge cases are harder to classify. No fundamental difficulty, just me being a bit fast and lazy.

Reduce the diff of #11203

@@ -68,7 +68,7 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x := by
   rw [← Int.ofNat_mul, ← Int.coe_nat_pow, ← Int.ofNat_mul, Int.ofNat_lt] at a1
   -- Recall this is by contradiction: we obtained the inequality `b * q ≤ x * q ^ (b + 1)`, so
   -- we are done.
-  exact not_le.mpr a1 (Nat.mul_lt_mul_pow_succ (Int.coe_nat_pos.mp ap) (Int.ofNat_lt.mp q1)).le
+  exact not_le.mpr a1 (Nat.mul_lt_mul_pow_succ (Int.natCast_pos.mp ap) (Int.ofNat_lt.mp q1)).le
 #align liouville.irrational Liouville.irrational
 
 open Polynomial Metric Set Real RingHom
@@ -163,7 +163,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
     rw [show (a + 1 : ℝ) = ((a + 1 : ℕ) : ℤ) by norm_cast] at hq ⊢
     -- key observation: the right-hand side of the inequality is an *integer*.  Therefore,
     -- if its absolute value is not at least one, then it vanishes.  Proceed by contradiction
-    refine' one_le_pow_mul_abs_eval_div (Int.coe_nat_succ_pos a) fun hy => _
+    refine' one_le_pow_mul_abs_eval_div (Int.natCast_succ_pos a) fun hy => _
     -- As the evaluation of the polynomial vanishes, we found a root of `fR` that is rational.
     -- We know that `α` is the only root of `fR` in our interval, and `α` is irrational:
     -- follow your nose.

Soon, there will be NNRat analogs of the Rat fields in the definition of Field. NNRat is less nicely a structure than Rat, hence there is a need to reduce the dependency of Field on the internals of Rat.

This PR achieves this by restating Field.ratCast_mk' in terms of Rat.num, Rat.den. This requires fixing a few downstream instances.

Reduce the diff of #11203.

Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com>

@@ -40,7 +40,7 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x := by
   -- By contradiction, `x = a / b`, with `a ∈ ℤ`, `0 < b ∈ ℕ` is a Liouville number,
   rintro ⟨⟨a, b, bN0, cop⟩, rfl⟩
   -- clear up the mess of constructions of rationals
-  rw [Rat.cast_mk', ← div_eq_mul_inv] at h
+  rw [Rat.cast_mk'] at h
   -- Since `a / b` is a Liouville number, there are `p, q ∈ ℤ`, with `q1 : 1 < q`,
   -- `a0 : a / b ≠ p / q` and `a1 : |a / b - p / q| < 1 / q ^ (b + 1)`
   rcases h (b + 1) with ⟨p, q, q1, a0, a1⟩

Polynomial and MvPolynomial are algebraic objects, hence should be under Algebra (or at least not under Data)

@@ -3,9 +3,9 @@ Copyright (c) 2020 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Jujian Zhang
 -/
+import Mathlib.Algebra.Polynomial.DenomsClearable
 import Mathlib.Analysis.Calculus.MeanValue
 import Mathlib.Analysis.Calculus.Deriv.Polynomial
-import Mathlib.Data.Polynomial.DenomsClearable
 import Mathlib.Data.Real.Irrational
 import Mathlib.Topology.Algebra.Polynomial
 

@@ -55,7 +55,7 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x := by
     exact mod_cast a1
   -- At a0, clear denominators...
   replace a0 : a * q - ↑b * p ≠ 0 := by
-    rw [Ne.def, div_eq_div_iff b0 qR0.ne', mul_comm (p : ℝ), ← sub_eq_zero] at a0
+    rw [Ne, div_eq_div_iff b0 qR0.ne', mul_comm (p : ℝ), ← sub_eq_zero] at a0
     exact mod_cast a0
   -- Actually, `q` is a natural number
   lift q to ℕ using (zero_lt_one.trans q1).le

This will become an error in 2024-03-16 nightly, possibly not permanently.

Co-authored-by: Scott Morrison <scott@tqft.net>

@@ -132,7 +132,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
       (fR0.trans (Polynomial.map_zero _).symm)
   -- reformulating assumption `fa`: `α` is a root of `fR`.
   have ar : α ∈ (fR.roots.toFinset : Set ℝ) :=
-    Finset.mem_coe.mpr (Multiset.mem_toFinset.mpr ((mem_roots fR0).mpr (IsRoot.def.mpr fa)))
+    Finset.mem_coe.mpr (Multiset.mem_toFinset.mpr ((mem_roots fR0).mpr (IsRoot.definition.mpr fa)))
   -- Since the polynomial `fR` has finitely many roots, there is a closed interval centered at `α`
   -- such that `α` is the only root of `fR` in the interval.
   obtain ⟨ζ, z0, U⟩ : ∃ ζ > 0, closedBall α ζ ∩ fR.roots.toFinset = {α} :=
@@ -170,7 +170,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
     refine' (irrational_iff_ne_rational α).mp ha z (a + 1) (mem_singleton_iff.mp _).symm
     refine' U.subset _
     refine' ⟨hq, Finset.mem_coe.mp (Multiset.mem_toFinset.mpr _)⟩
-    exact (mem_roots fR0).mpr (IsRoot.def.mpr hy)
+    exact (mem_roots fR0).mpr (IsRoot.definition.mpr hy)
 #align liouville.exists_pos_real_of_irrational_root Liouville.exists_pos_real_of_irrational_root
 
 /-- **Liouville's Theorem** -/

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>

@@ -49,13 +49,13 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x := by
   have b0 : (b : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr bN0
   have bq0 : (0 : ℝ) < b * q := mul_pos (Nat.cast_pos.mpr bN0.bot_lt) qR0
   -- At a1, clear denominators...
-  replace a1 : |a * q - b * p| * q ^ (b + 1) < b * q
-  · rw [div_sub_div _ _ b0 qR0.ne', abs_div, div_lt_div_iff (abs_pos.mpr bq0.ne') (pow_pos qR0 _),
+  replace a1 : |a * q - b * p| * q ^ (b + 1) < b * q := by
+    rw [div_sub_div _ _ b0 qR0.ne', abs_div, div_lt_div_iff (abs_pos.mpr bq0.ne') (pow_pos qR0 _),
       abs_of_pos bq0, one_mul] at a1
     exact mod_cast a1
   -- At a0, clear denominators...
-  replace a0 : a * q - ↑b * p ≠ 0;
-  · rw [Ne.def, div_eq_div_iff b0 qR0.ne', mul_comm (p : ℝ), ← sub_eq_zero] at a0
+  replace a0 : a * q - ↑b * p ≠ 0 := by
+    rw [Ne.def, div_eq_div_iff b0 qR0.ne', mul_comm (p : ℝ), ← sub_eq_zero] at a0
     exact mod_cast a0
   -- Actually, `q` is a natural number
   lift q to ℕ using (zero_lt_one.trans q1).le
@@ -179,8 +179,8 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
   -- non-zero (`f0`) polynomial `f`
   rintro ⟨f : ℤ[X], f0, ef0⟩
   -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.
-  replace ef0 : (f.map (algebraMap ℤ ℝ)).eval x = 0;
-  · rwa [aeval_def, ← eval_map] at ef0
+  replace ef0 : (f.map (algebraMap ℤ ℝ)).eval x = 0 := by
+    rwa [aeval_def, ← eval_map] at ef0
   -- There is a "large" real number `A` such that `(b + 1) ^ (deg f) * |f (x - a / (b + 1))| * A`
   -- is at least one.  This is obtained from lemma `exists_pos_real_of_irrational_root`.
   obtain ⟨A, hA, h⟩ : ∃ A : ℝ, 0 < A ∧ ∀ (a : ℤ) (b : ℕ),

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>

@@ -52,11 +52,11 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x := by
   replace a1 : |a * q - b * p| * q ^ (b + 1) < b * q
   · rw [div_sub_div _ _ b0 qR0.ne', abs_div, div_lt_div_iff (abs_pos.mpr bq0.ne') (pow_pos qR0 _),
       abs_of_pos bq0, one_mul] at a1
-    exact_mod_cast a1
+    exact mod_cast a1
   -- At a0, clear denominators...
   replace a0 : a * q - ↑b * p ≠ 0;
   · rw [Ne.def, div_eq_div_iff b0 qR0.ne', mul_comm (p : ℝ), ← sub_eq_zero] at a0
-    exact_mod_cast a0
+    exact mod_cast a0
   -- Actually, `q` is a natural number
   lift q to ℕ using (zero_lt_one.trans q1).le
   -- Looks innocuous, but we now have an integer with non-zero absolute value: this is at

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -92,7 +92,7 @@ It is stated in more general form than needed: in the intended application, `Z =
 root of `f`, `ε` is small, `M` is a bound on the Lipschitz constant of `f` near `α`, `n` is
 the degree of the polynomial `f`.
 -/
-theorem exists_one_le_pow_mul_dist {Z N R : Type _} [PseudoMetricSpace R] {d : N → ℝ}
+theorem exists_one_le_pow_mul_dist {Z N R : Type*} [PseudoMetricSpace R] {d : N → ℝ}
     {j : Z → N → R} {f : R → R} {α : R} {ε M : ℝ}
     -- denominators are positive
     (d0 : ∀ a : N, 1 ≤ d a)

Also make f/f' arguments implicit in all versions of Rolle's Theorem.

Fixes #4830

API changes

• exists_Ioo_extr_on_Icc:
  • generalize from functions f : ℝ → ℝ to functions from a conditionally complete linear order to a linear order.
  • make f implicit;
• exists_local_extr_Ioo:
  • rename to exists_isLocalExtr_Ioo;
  • generalize as above;
  • make f implicit;
• exists_isExtrOn_Ioo_of_tendsto, exists_isLocalExtr_Ioo_of_tendsto: new lemmas extracted from the proof of exists_hasDerivAt_eq_zero';
• exists_hasDerivAt_eq_zero, exists_hasDerivAt_eq_zero': make f and f' implicit;
• exists_deriv_eq_zero, exists_deriv_eq_zero': make f implicit.

@@ -4,8 +4,10 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Jujian Zhang
 -/
 import Mathlib.Analysis.Calculus.MeanValue
+import Mathlib.Analysis.Calculus.Deriv.Polynomial
 import Mathlib.Data.Polynomial.DenomsClearable
 import Mathlib.Data.Real.Irrational
+import Mathlib.Topology.Algebra.Polynomial
 
 #align_import number_theory.liouville.basic from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
 

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Jujian Zhang. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Damiano Testa, Jujian Zhang
-
-! This file was ported from Lean 3 source module number_theory.liouville.basic
-! leanprover-community/mathlib commit 04e80bb7e8510958cd9aacd32fe2dc147af0b9f1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Calculus.MeanValue
 import Mathlib.Data.Polynomial.DenomsClearable
 import Mathlib.Data.Real.Irrational
 
+#align_import number_theory.liouville.basic from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
+
 /-!
 
 # Liouville's theorem

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

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

@@ -89,7 +89,7 @@ involving the cost function `d`.
 
 This lemma collects the properties used in the proof of `exists_pos_real_of_irrational_root`.
 It is stated in more general form than needed: in the intended application, `Z = ℤ`, `N = ℕ`,
-`R = ℝ`, `d a = (a + 1) ^ f.nat_degree`, `j z a  = z / (a + 1)`, `f ∈ ℤ[x]`, `α` is an irrational
+`R = ℝ`, `d a = (a + 1) ^ f.nat_degree`, `j z a = z / (a + 1)`, `f ∈ ℤ[x]`, `α` is an irrational
 root of `f`, `ε` is small, `M` is a bound on the Lipschitz constant of `f` near `α`, `n` is
 the degree of the polynomial `f`.
 -/
@@ -189,7 +189,7 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
     exists_pos_real_of_irrational_root lx.irrational f0 ef0
   -- Since the real numbers are Archimedean, a power of `2` exceeds `A`: `hn : A < 2 ^ r`.
   rcases pow_unbounded_of_one_lt A (lt_add_one 1) with ⟨r, hn⟩
-  -- Use the Liouville property, with exponent `r +  deg f`.
+  -- Use the Liouville property, with exponent `r + deg f`.
   obtain ⟨a, b, b1, -, a1⟩ : ∃ a b : ℤ, 1 < b ∧ x ≠ a / b ∧
       |x - a / b| < 1 / (b : ℝ) ^ (r + f.natDegree) :=
     lx (r + f.natDegree)

@@ -146,7 +146,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
       (continuous_abs.comp fR.derivative.continuous_aeval).continuousOn
   -- Use the key lemma `exists_one_le_pow_mul_dist`: we are left to show that ...
   refine'
-    @exists_one_le_pow_mul_dist ℤ ℕ ℝ _ _ _ (fun y => fR.eval y) α ζ (|fR.derivative.eval xm|) _ z0
+    @exists_one_le_pow_mul_dist ℤ ℕ ℝ _ _ _ (fun y => fR.eval y) α ζ |fR.derivative.eval xm| _ z0
       (fun y hy => _) fun z a hq => _
   -- 1: the denominators are positive -- essentially by definition;
   · exact fun a => one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg) _

Following on from #4702, another hundred sample uses of the gcongr tactic.

@@ -41,7 +41,7 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x := by
   -- By contradiction, `x = a / b`, with `a ∈ ℤ`, `0 < b ∈ ℕ` is a Liouville number,
   rintro ⟨⟨a, b, bN0, cop⟩, rfl⟩
   -- clear up the mess of constructions of rationals
-  rw [Rat.cast_mk', ← div_eq_mul_inv] at h 
+  rw [Rat.cast_mk', ← div_eq_mul_inv] at h
   -- Since `a / b` is a Liouville number, there are `p, q ∈ ℤ`, with `q1 : 1 < q`,
   -- `a0 : a / b ≠ p / q` and `a1 : |a / b - p / q| < 1 / q ^ (b + 1)`
   rcases h (b + 1) with ⟨p, q, q1, a0, a1⟩
@@ -66,7 +66,7 @@ protected theorem irrational {x : ℝ} (h : Liouville x) : Irrational x := by
   -- Actually, the absolute value of an integer is a natural number
   lift |a * ↑q - ↑b * p| to ℕ using abs_nonneg (a * ↑q - ↑b * p) with e he
   -- At a1, revert to natural numbers
-  rw [← Int.ofNat_mul, ← Int.coe_nat_pow, ← Int.ofNat_mul, Int.ofNat_lt] at a1 
+  rw [← Int.ofNat_mul, ← Int.coe_nat_pow, ← Int.ofNat_mul, Int.ofNat_lt] at a1
   -- Recall this is by contradiction: we obtained the inequality `b * q ≤ x * q ^ (b + 1)`, so
   -- we are done.
   exact not_le.mpr a1 (Nat.mul_lt_mul_pow_succ (Int.coe_nat_pos.mp ap) (Int.ofNat_lt.mp q1)).le
@@ -152,7 +152,7 @@ theorem exists_pos_real_of_irrational_root {α : ℝ} (ha : Irrational α) {f :
   · exact fun a => one_le_pow_of_one_le ((le_add_iff_nonneg_left 1).mpr a.cast_nonneg) _
   -- 2: the polynomial `fR` is Lipschitz at `α` -- as its derivative continuous;
   · rw [mul_comm]
-    rw [Real.closedBall_eq_Icc] at hy 
+    rw [Real.closedBall_eq_Icc] at hy
     -- apply the Mean Value Theorem: the bound on the derivative comes from differentiability.
     refine'
       Convex.norm_image_sub_le_of_norm_deriv_le (fun _ _ => fR.differentiableAt)
@@ -181,7 +181,7 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
   rintro ⟨f : ℤ[X], f0, ef0⟩
   -- Change `aeval x f = 0` to `eval (map _ f) = 0`, who knew.
   replace ef0 : (f.map (algebraMap ℤ ℝ)).eval x = 0;
-  · rwa [aeval_def, ← eval_map] at ef0 
+  · rwa [aeval_def, ← eval_map] at ef0
   -- There is a "large" real number `A` such that `(b + 1) ^ (deg f) * |f (x - a / (b + 1))| * A`
   -- is at least one.  This is obtained from lemma `exists_pos_real_of_irrational_root`.
   obtain ⟨A, hA, h⟩ : ∃ A : ℝ, 0 < A ∧ ∀ (a : ℤ) (b : ℕ),
@@ -198,23 +198,23 @@ protected theorem transcendental {x : ℝ} (lx : Liouville x) : Transcendental 
   -- recall, this is a proof by contradiction!
   refine' lt_irrefl ((b : ℝ) ^ f.natDegree * |x - ↑a / ↑b|) _
   -- clear denominators at `a1`
-  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1 
+  rw [lt_div_iff' (pow_pos b0 _), pow_add, mul_assoc] at a1
   -- split the inequality via `1 / A`.
   refine' (_ : (b : ℝ) ^ f.natDegree * |x - a / b| < 1 / A).trans_le _
   -- This branch of the proof uses the Liouville condition and the Archimedean property
   · refine' (lt_div_iff' hA).mpr _
     refine' lt_of_le_of_lt _ a1
-    refine' mul_le_mul_of_nonneg_right _ (mul_nonneg (pow_nonneg b0.le _) (abs_nonneg _))
+    gcongr
     refine' hn.le.trans _
     rw [one_add_one_eq_two]
-    refine' pow_le_pow_of_le_left zero_le_two _ _
+    gcongr
     exact Int.cast_two.symm.le.trans (Int.cast_le.mpr (Int.add_one_le_iff.mpr b1))
   -- this branch of the proof exploits the "integrality" of evaluations of polynomials
   -- at ratios of integers.
   · lift b to ℕ using zero_le_one.trans b1.le
     specialize h a b.pred
     rwa [← Nat.cast_succ, Nat.succ_pred_eq_of_pos (zero_lt_one.trans _), ← mul_assoc, ←
-      div_le_iff hA] at h 
+      div_le_iff hA] at h
     exact Int.ofNat_lt.mp b1
 #align liouville.transcendental Liouville.transcendental