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Mathlib.Algebra.Ring.MinimalAxioms

Minimal Axioms for a Ring #

This file defines constructors to define a Ring or CommRing structure on a Type, while proving a minimum number of equalities.

Main Definitions #

source
@[reducible, inline]
abbrev Ring.ofMinimalAxioms {R : Type u} [Add R] [Mul R] [Neg R] [Zero R] [One R] (add_assoc : ∀ (a b c : R), a + b + c = a + (b + c)) (zero_add : ∀ (a : R), 0 + a = a) (neg_add_cancel : ∀ (a : R), -a + a = 0) (mul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)) (one_mul : ∀ (a : R), 1 * a = a) (mul_one : ∀ (a : R), a * 1 = a) (left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c) (right_distrib : ∀ (a b c : R), (a + b) * c = a * c + b * c) :
Ring R

Define a Ring structure on a Type by proving a minimized set of axioms. Note that this uses the default definitions for npow, nsmul, zsmul and sub See note [reducible non-instances].

Equations
  • Ring.ofMinimalAxioms add_assoc zero_add neg_add_cancel mul_assoc one_mul mul_one left_distrib right_distrib = Ring.mk ⋯ (fun (x1 : ℤ) (x2 : R) => x1 • x2) ⋯ ⋯ ⋯ neg_add_cancel ⋯ ⋯
Instances For
    source
    @[reducible, inline]
    abbrev CommRing.ofMinimalAxioms {R : Type u} [Add R] [Mul R] [Neg R] [Zero R] [One R] (add_assoc : ∀ (a b c : R), a + b + c = a + (b + c)) (zero_add : ∀ (a : R), 0 + a = a) (neg_add_cancel : ∀ (a : R), -a + a = 0) (mul_assoc : ∀ (a b c : R), a * b * c = a * (b * c)) (mul_comm : ∀ (a b : R), a * b = b * a) (one_mul : ∀ (a : R), 1 * a = a) (left_distrib : ∀ (a b c : R), a * (b + c) = a * b + a * c) :
    CommRing R

    Define a CommRing structure on a Type by proving a minimized set of axioms. Note that this uses the default definitions for npow, nsmul, zsmul and sub See note [reducible non-instances].

    Equations
    Instances For