Documentation

Counterexamples.CanonicallyOrderedCommSemiringTwoMul

A canonically ordered commutative semiring with two different elements a and b such that a ≠ b and 2 * a = 2 * b. Thus, multiplication by a fixed non-zero element of a canonically ordered semiring need not be injective. In particular, multiplying by a strictly positive element need not be strictly monotone.

Recall that a canonically ordered commutative semiring is a commutative semiring with a partial ordering that is "canonical" in the sense that the inequality a ≤ b holds if and only if there is a c such that a + c = b. There are several compatibility conditions among addition/multiplication and the order relation. The point of the counterexample is to show that monotonicity of multiplication cannot be strengthened to strict monotonicity.

Reference: https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/canonically_ordered.20pathology

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theorem Counterexample.mem_zmod_2 (a : ZMod 2) :
a = 0 ∨ a = 1
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theorem Counterexample.add_self_zmod_2 (a : ZMod 2) :
a + a = 0
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instance Counterexample.Nxzmod2.preN2 :
PartialOrder (ℕ × ZMod 2)

The preorder relation on ℕ × ℤ/2ℤ where we only compare the first coordinate, except that we leave incomparable each pair of elements with the same first component. For instance, ∀ α, β ∈ ℤ/2ℤ, the inequality (1,α) ≤ (2,β) holds, whereas, ∀ n ∈ ℤ, the elements (n,0) and (n,1) are incomparable.

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instance Counterexample.Nxzmod2.csrN2 :
CommSemiring (ℕ × ZMod 2)
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instance Counterexample.Nxzmod2.csrN21 :
AddCancelCommMonoid (ℕ × ZMod 2)
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@[simp]
theorem Counterexample.Nxzmod2.lt_def {a b : ℕ × ZMod 2} :
a < b ↔ a.1 < b.1

A strict inequality forces the first components to be different.

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theorem Counterexample.Nxzmod2.add_left_cancel (a b c : ℕ × ZMod 2) :
a + b = a + c → b = c
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theorem Counterexample.Nxzmod2.add_le_add_left (a b : ℕ × ZMod 2) :
a ≤ b → ∀ (c : ℕ × ZMod 2), c + a ≤ c + b
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theorem Counterexample.Nxzmod2.le_of_add_le_add_left (a b c : ℕ × ZMod 2) :
a + b ≤ a + c → b ≤ c
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instance Counterexample.Nxzmod2.instZeroLEOneClassProdNatZModOfNat_counterexamples :
ZeroLEOneClass (ℕ × ZMod 2)
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theorem Counterexample.Nxzmod2.mul_lt_mul_of_pos_left (a b c : ℕ × ZMod 2) :
a < b → 0 < c → c * a < c * b
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theorem Counterexample.Nxzmod2.mul_lt_mul_of_pos_right (a b c : ℕ × ZMod 2) :
a < b → 0 < c → a * c < b * c
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instance Counterexample.Nxzmod2.socsN2 :
StrictOrderedCommSemiring (ℕ × ZMod 2)
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def Counterexample.ExL.L :
Type

Initially, L was defined as the subsemiring closure of (1,0).

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Instances For
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    theorem Counterexample.ExL.add_L {a b : ℕ × ZMod 2} (ha : a ≠ (0, 1)) (hb : b ≠ (0, 1)) :
    a + b ≠ (0, 1)
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    theorem Counterexample.ExL.mul_L {a b : ℕ × ZMod 2} (ha : a ≠ (0, 1)) (hb : b ≠ (0, 1)) :
    a * b ≠ (0, 1)
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    def Counterexample.ExL.lSubsemiring :
    Subsemiring (ℕ × ZMod 2)

    The subsemiring corresponding to the elements of L, used to transfer instances.

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    • One or more equations did not get rendered due to their size.
    Instances For
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      instance Counterexample.ExL.instOrderedCommSemiringL :
      OrderedCommSemiring L
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      instance Counterexample.ExL.inhabited :
      Inhabited L
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      theorem Counterexample.ExL.bot_le (a : L) :
      0 ≤ a
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      instance Counterexample.ExL.orderBot :
      OrderBot L
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      theorem Counterexample.ExL.exists_add_of_le (a b : L) :
      a ≤ b → ∃ (c : L), b = a + c
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      theorem Counterexample.ExL.le_self_add (a b : L) :
      a ≤ a + b
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      theorem Counterexample.ExL.eq_zero_or_eq_zero_of_mul_eq_zero (a b : L) :
      a * b = 0 → a = 0 ∨ b = 0
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      instance Counterexample.ExL.instOrderedCommSemiringL_1 :
      OrderedCommSemiring L
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      instance Counterexample.ExL.instCanonicallyOrderedAddL :
      CanonicallyOrderedAdd L
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      instance Counterexample.ExL.instNoZeroDivisorsL :
      NoZeroDivisors L