bat/tests/syntax-tests/highlighted/Lean/test.lean

import MIL.Common
import Mathlib.Topology.Instances.Real.Defs

open Set Filter Topology

variable {α : Type*}
variable (s t : Set ℕ)
variable (ssubt : s ⊆ t)
variable {α : Type*} (s : Set (Set α))
-- Apostrophes are allowed in variable names
variable (f'_x x' : ℕ)
variable (bangwI' jablu'DI' QaQqu' nay' Ghay'cha' he' : ℕ)

-- In the next example we could use `tauto` in each proof instead of knowing the lemmas
example {α : Type*} (s : Set α) : Filter α :=
  { sets := { t | s ⊆ t }
    univ_sets := subset_univ s
    sets_of_superset := fun hU hUV ↦ Subset.trans hU hUV
    inter_sets := fun hU hV ↦ subset_inter hU hV }


namespace chess.utils

section repr

@[class] structure One₂ (α : Type) where
  /-- The element one -/
  one : α

structure StandardTwoSimplex where
  x : ℝ
  y : ℝ
  z : ℝ
  x_nonneg : 0 ≤ x
  y_nonneg : 0 ≤ y
  z_nonneg : 0 ≤ z
  sum_eq : x + y + z = 1

#check Pi.ringHom
#check ker_Pi_Quotient_mk
#eval 1 + 1

/-- The homomorphism from ``R ⧸ ⨅ i, I i`` to ``Π i, R ⧸ I i`` featured in the Chinese
  Remainder Theorem. -/
def chineseMap (I : ι → Ideal R) : (R ⧸ ⨅ i, I i) →+* Π i, R ⧸ I i :=
  Ideal.Quotient.lift (⨅ i, I i) (Pi.ringHom fun i : ι ↦ Ideal.Quotient.mk (I i))
    (by simp [← RingHom.mem_ker, ker_Pi_Quotient_mk])

lemma chineseMap_mk (I : ι → Ideal R) (x : R) :
    chineseMap I (Quotient.mk _ x) = fun i : ι ↦ Ideal.Quotient.mk (I i) x :=
  rfl

theorem isCoprime_Inf {I : Ideal R} {J : ι → Ideal R} {s : Finset ι}
    (hf : ∀ j ∈ s, IsCoprime I (J j)) : IsCoprime I (⨅ j ∈ s, J j) := by
  classical
  simp_rw [isCoprime_iff_add] at *
  induction s using Finset.induction with
  | empty =>
      simp
  | @insert i s _ hs =>
      rw [Finset.iInf_insert, inf_comm, one_eq_top, eq_top_iff, ← one_eq_top]
      set K := ⨅ j ∈ s, J j
      calc
        1 = I + K                  := (hs fun j hj ↦ hf j (Finset.mem_insert_of_mem hj)).symm
        _ = I + K * (I + J i)      := by rw [hf i (Finset.mem_insert_self i s), mul_one]
        _ = (1 + K) * I + K * J i  := by ring
        _ ≤ I + K ⊓ J i            := by gcongr ; apply mul_le_left ; apply mul_le_inf


class Ring₃ (R : Type) extends AddGroup₃ R, Monoid₃ R, MulZeroClass R where
  /-- Multiplication is left distributive over addition -/
  left_distrib : ∀ a b c : R, a * (b + c) = a * b + a * c
  /-- Multiplication is right distributive over addition -/
  right_distrib : ∀ a b c : R, (a + b) * c = a * c + b * c

instance {R : Type} [Ring₃ R] : AddCommGroup₃ R :=
{ Ring₃.toAddGroup₃ with
  add_comm := by
    sorry }

end repr

end chess.utils