feat: port Data.MvPolynomial.Funext (#3225)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
Co-authored-by: Johan Commelin <johan@commelin.net>

Author: 3 people

Mathlib.lean

@@ -921,8 +921,10 @@ import Mathlib.Data.MvPolynomial.Counit
 import Mathlib.Data.MvPolynomial.Division
 import Mathlib.Data.MvPolynomial.Equiv
 import Mathlib.Data.MvPolynomial.Expand
+import Mathlib.Data.MvPolynomial.Funext
 import Mathlib.Data.MvPolynomial.Invertible
 import Mathlib.Data.MvPolynomial.Monad
+import Mathlib.Data.MvPolynomial.Polynomial
 import Mathlib.Data.MvPolynomial.Rename
 import Mathlib.Data.MvPolynomial.Supported
 import Mathlib.Data.MvPolynomial.Variables

Mathlib/Data/MvPolynomial/Basic.lean

@@ -1148,6 +1148,17 @@ theorem eval_assoc {τ} (f : σ → MvPolynomial τ R) (g : τ → R) (p : MvPol
   congr with a; simp
 #align mv_polynomial.eval_assoc MvPolynomial.eval_assoc
 
+-- Porting note: new theorem
+theorem eval_eval₂ [CommSemiring R] [CommSemiring S]
+    (f : R →+* MvPolynomial τ S) (g : σ → MvPolynomial τ S) (p : MvPolynomial σ R) :
+    eval x (eval₂ f g p) = eval₂ ((eval x).comp f) (fun s => eval x (g s)) p := by
+  apply induction_on p
+  · simp
+  · intro p q hp hq
+    simp [hp, hq]
+  · intro p n hp
+    simp [hp]
+
 end Eval
 
 section Map
@@ -1204,6 +1215,12 @@ theorem eval₂_eq_eval_map (g : σ → S₁) (p : MvPolynomial σ R) : p.eval
     simp only [comp_apply, eval₂_X]
 #align mv_polynomial.eval₂_eq_eval_map MvPolynomial.eval₂_eq_eval_map
 
+-- Porting note: new theorem
+-- This probably belongs earlier, but it breaks the fragile proof of `eval₂_eq_eval_map`
+@[simp]
+theorem eval₂_id (p : MvPolynomial σ R) : eval₂ (RingHom.id _) g p = eval g p :=
+  rfl
+
 theorem eval₂_comp_right {S₂} [CommSemiring S₂] (k : S₁ →+* S₂) (f : R →+* S₁) (g : σ → S₁) (p) :
     k (eval₂ f g p) = eval₂ k (k ∘ g) (map f p) := by
   apply MvPolynomial.induction_on p

Mathlib/Data/MvPolynomial/Funext.lean

@@ -0,0 +1,70 @@
+/-
+Copyright (c) 2020 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 data.mv_polynomial.funext
+! leanprover-community/mathlib commit da01792ca4894d4f3a98d06b6c50455e5ed25da3
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Data.Polynomial.RingDivision
+import Mathlib.Data.MvPolynomial.Rename
+import Mathlib.Data.MvPolynomial.Polynomial
+import Mathlib.RingTheory.Polynomial.Basic
+
+/-!
+## Function extensionality for multivariate polynomials
+
+In this file we show that two multivariate polynomials over an infinite integral domain are equal
+if they are equal upon evaluating them on an arbitrary assignment of the variables.
+
+# Main declaration
+
+* `MvPolynomial.funext`: two polynomials `φ ψ : MvPolynomial σ R`
+  over an infinite integral domain `R` are equal if `eval x φ = eval x ψ` for all `x : σ → R`.
+
+-/
+
+namespace MvPolynomial
+
+variable {R : Type _} [CommRing R] [IsDomain R] [Infinite R]
+
+private theorem funext_fin {n : ℕ} {p : MvPolynomial (Fin n) R}
+    (h : ∀ x : Fin n → R, eval x p = 0) : p = 0 := by
+  induction' n with n ih
+  · apply (MvPolynomial.isEmptyRingEquiv R (Fin 0)).injective
+    rw [RingEquiv.map_zero]
+    convert h finZeroElim
+  · apply (finSuccEquiv R n).injective
+    simp only [AlgEquiv.map_zero]
+    refine Polynomial.funext fun q => ?_
+    rw [Polynomial.eval_zero]
+    apply ih fun x => ?_
+    calc _ = _ := eval_polynomial_eval_finSuccEquiv p _
+         _ = 0 := h _
+
+/-- Two multivariate polynomials over an infinite integral domain are equal
+if they are equal upon evaluating them on an arbitrary assignment of the variables. -/
+theorem funext {σ : Type _} {p q : MvPolynomial σ R} (h : ∀ x : σ → R, eval x p = eval x q) :
+    p = q := by
+  suffices ∀ p, (∀ x : σ → R, eval x p = 0) → p = 0 by
+    rw [← sub_eq_zero, this (p - q)]
+    simp only [h, RingHom.map_sub, forall_const, sub_self]
+  clear h p q
+  intro p h
+  obtain ⟨n, f, hf, p, rfl⟩ := exists_fin_rename p
+  suffices p = 0 by rw [this, AlgHom.map_zero]
+  apply funext_fin
+  intro x
+  classical
+    convert h (Function.extend f x 0)
+    simp only [eval, eval₂Hom_rename, Function.extend_comp hf]
+#align mv_polynomial.funext MvPolynomial.funext
+
+theorem funext_iff {σ : Type _} {p q : MvPolynomial σ R} :
+    p = q ↔ ∀ x : σ → R, eval x p = eval x q :=
+  ⟨by rintro rfl; simp only [forall_const, eq_self_iff_true], funext⟩
+#align mv_polynomial.funext_iff MvPolynomial.funext_iff
+
+end MvPolynomial

Mathlib/Data/MvPolynomial/Polynomial.lean

@@ -0,0 +1,36 @@
+/-
+Copyright (c) 2023 Scott Morrison. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Scott Morrison
+-/
+import Mathlib.Data.MvPolynomial.Equiv
+import Mathlib.Data.Polynomial.Eval
+
+/-!
+# Some lemmas relating polynomials and multivariable polynomials.
+-/
+
+namespace MvPolynomial
+
+theorem polynomial_eval_eval₂ [CommSemiring R] [CommSemiring S]
+    (f : R →+* Polynomial S) (g : σ → Polynomial S) (p : MvPolynomial σ R) :
+    Polynomial.eval x (eval₂ f g p) =
+      eval₂ ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) p := by
+  apply induction_on p
+  · simp
+  · intro p q hp hq
+    simp [hp, hq]
+  · intro p n hp
+    simp [hp]
+
+theorem eval_polynomial_eval_finSuccEquiv
+    [CommSemiring R] (f : MvPolynomial (Fin (n + 1)) R) (q : MvPolynomial (Fin n) R) :
+    (eval x) (Polynomial.eval q (finSuccEquiv R n f)) = eval (Fin.cases (eval x q) x) f := by
+  simp only [finSuccEquiv_apply, coe_eval₂Hom, polynomial_eval_eval₂, eval_eval₂]
+  conv in RingHom.comp _ _ =>
+  { refine @RingHom.ext _ _ _ _ _ (RingHom.id _) fun r => ?_
+    simp }
+  simp only [eval₂_id]
+  congr
+  funext i
+  refine Fin.cases (by simp) (by simp) i

Mathlib/Tactic/Conv.lean

@@ -50,6 +50,9 @@ syntax (name := dischargeConv) "discharge" (" => " tacticSeq)? : conv
       setGoals (m.mvarId! :: gs)
   | _ => Elab.throwUnsupportedSyntax
 
+/-- Use `refine` in `conv` mode. -/
+macro "refine " e:term : conv => `(conv| tactic => refine $e)
+
 open Elab Tactic
 /--
 The command `#conv tac => e` will run a conv tactic `tac` on `e`, and display the resulting