Theory Commutative_Ring_Ex

theory Commutative_Ring_Ex
imports Reflective_Field
```(*  Title:      HOL/Decision_Procs/ex/Commutative_Ring_Ex.thy
Author:     Bernhard Haeupler, Stefan Berghofer
*)

section ‹Some examples demonstrating the ring and field methods›

theory Commutative_Ring_Ex
imports "../Reflective_Field"
begin

lemma "4 * (x::int) ^ 5 * y ^ 3 * x ^ 2 * 3 + x * z + 3 ^ 5 = 12 * x ^ 7 * y ^ 3 + z * x + 243"
by ring

lemma (in cring)
assumes "x ∈ carrier R" "y ∈ carrier R" "z ∈ carrier R"
shows "«4» ⊗ x [^] (5::nat) ⊗ y [^] (3::nat) ⊗ x [^] (2::nat) ⊗ «3» ⊕ x ⊗ z ⊕ «3» [^] (5::nat) =
«12» ⊗ x [^] (7::nat) ⊗ y [^] (3::nat) ⊕ z ⊗ x ⊕ «243»"
by ring

lemma "((x::int) + y) ^ 2  = x ^ 2 + y ^ 2 + 2 * x * y"
by ring

lemma (in cring)
assumes "x ∈ carrier R" "y ∈ carrier R"
shows "(x ⊕ y) [^] (2::nat)  = x [^] (2::nat) ⊕ y [^] (2::nat) ⊕ «2» ⊗ x ⊗ y"
by ring

lemma "((x::int) + y) ^ 3 = x ^ 3 + y ^ 3 + 3 * x ^ 2 * y + 3 * y ^ 2 * x"
by ring

lemma (in cring)
assumes "x ∈ carrier R" "y ∈ carrier R"
shows "(x ⊕ y) [^] (3::nat) =
x [^] (3::nat) ⊕ y [^] (3::nat) ⊕ «3» ⊗ x [^] (2::nat) ⊗ y ⊕ «3» ⊗ y [^] (2::nat) ⊗ x"
by ring

lemma "((x::int) - y) ^ 3 = x ^ 3 + 3 * x * y ^ 2 + (- 3) * y * x ^ 2 - y ^ 3"
by ring

lemma (in cring)
assumes "x ∈ carrier R" "y ∈ carrier R"
shows "(x ⊖ y) [^] (3::nat) =
x [^] (3::nat) ⊕ «3» ⊗ x ⊗ y [^] (2::nat) ⊕ (⊖ «3») ⊗ y ⊗ x [^] (2::nat) ⊖ y [^] (3::nat)"
by ring

lemma "((x::int) - y) ^ 2 = x ^ 2 + y ^ 2 - 2 * x * y"
by ring

lemma (in cring)
assumes "x ∈ carrier R" "y ∈ carrier R"
shows "(x ⊖ y) [^] (2::nat) = x [^] (2::nat) ⊕ y [^] (2::nat) ⊖ «2» ⊗ x ⊗ y"
by ring

lemma " ((a::int) + b + c) ^ 2 = a ^ 2 + b ^ 2 + c ^ 2 + 2 * a * b + 2 * b * c + 2 * a * c"
by ring

lemma (in cring)
assumes "a ∈ carrier R" "b ∈ carrier R" "c ∈ carrier R"
shows " (a ⊕ b ⊕ c) [^] (2::nat) =
a [^] (2::nat) ⊕ b [^] (2::nat) ⊕ c [^] (2::nat) ⊕ «2» ⊗ a ⊗ b ⊕ «2» ⊗ b ⊗ c ⊕ «2» ⊗ a ⊗ c"
by ring

lemma "((a::int) - b - c) ^ 2 = a ^ 2 + b ^ 2 + c ^ 2 - 2 * a * b + 2 * b * c - 2 * a * c"
by ring

lemma (in cring)
assumes "a ∈ carrier R" "b ∈ carrier R" "c ∈ carrier R"
shows "(a ⊖ b ⊖ c) [^] (2::nat) =
a [^] (2::nat) ⊕ b [^] (2::nat) ⊕ c [^] (2::nat) ⊖ «2» ⊗ a ⊗ b ⊕ «2» ⊗ b ⊗ c ⊖ «2» ⊗ a ⊗ c"
by ring

lemma "(a::int) * b + a * c = a * (b + c)"
by ring

lemma (in cring)
assumes "a ∈ carrier R" "b ∈ carrier R" "c ∈ carrier R"
shows "a ⊗ b ⊕ a ⊗ c = a ⊗ (b ⊕ c)"
by ring

lemma "(a::int) ^ 2 - b ^ 2 = (a - b) * (a + b)"
by ring

lemma (in cring)
assumes "a ∈ carrier R" "b ∈ carrier R"
shows "a [^] (2::nat) ⊖ b [^] (2::nat) = (a ⊖ b) ⊗ (a ⊕ b)"
by ring

lemma "(a::int) ^ 3 - b ^ 3 = (a - b) * (a ^ 2 + a * b + b ^ 2)"
by ring

lemma (in cring)
assumes "a ∈ carrier R" "b ∈ carrier R"
shows "a [^] (3::nat) ⊖ b [^] (3::nat) = (a ⊖ b) ⊗ (a [^] (2::nat) ⊕ a ⊗ b ⊕ b [^] (2::nat))"
by ring

lemma "(a::int) ^ 3 + b ^ 3 = (a + b) * (a ^ 2 - a * b + b ^ 2)"
by ring

lemma (in cring)
assumes "a ∈ carrier R" "b ∈ carrier R"
shows "a [^] (3::nat) ⊕ b [^] (3::nat) = (a ⊕ b) ⊗ (a [^] (2::nat) ⊖ a ⊗ b ⊕ b [^] (2::nat))"
by ring

lemma "(a::int) ^ 4 - b ^ 4 = (a - b) * (a + b) * (a ^ 2 + b ^ 2)"
by ring

lemma (in cring)
assumes "a ∈ carrier R" "b ∈ carrier R"
shows "a [^] (4::nat) ⊖ b [^] (4::nat) = (a ⊖ b) ⊗ (a ⊕ b) ⊗ (a [^] (2::nat) ⊕ b [^] (2::nat))"
by ring

lemma "(a::int) ^ 10 - b ^ 10 =
(a - b) * (a ^ 9 + a ^ 8 * b + a ^ 7 * b ^ 2 + a ^ 6 * b ^ 3 + a ^ 5 * b ^ 4 +
a ^ 4 * b ^ 5 + a ^ 3 * b ^ 6 + a ^ 2 * b ^ 7 + a * b ^ 8 + b ^ 9)"
by ring

lemma (in cring)
assumes "a ∈ carrier R" "b ∈ carrier R"
shows "a [^] (10::nat) ⊖ b [^] (10::nat) =
(a ⊖ b) ⊗ (a [^] (9::nat) ⊕ a [^] (8::nat) ⊗ b ⊕ a [^] (7::nat) ⊗ b [^] (2::nat) ⊕
a [^] (6::nat) ⊗ b [^] (3::nat) ⊕ a [^] (5::nat) ⊗ b [^] (4::nat) ⊕
a [^] (4::nat) ⊗ b [^] (5::nat) ⊕ a [^] (3::nat) ⊗ b [^] (6::nat) ⊕
a [^] (2::nat) ⊗ b [^] (7::nat) ⊕ a ⊗ b [^] (8::nat) ⊕ b [^] (9::nat))"
by ring

lemma "(x::'a::field) ≠ 0 ⟹ (1 - 1 / x) * x - x + 1 = 0"
by field

lemma (in field)
assumes "x ∈ carrier R"
shows "x ≠ 𝟬 ⟹ (𝟭 ⊖ 𝟭 ⊘ x) ⊗ x ⊖ x ⊕ 𝟭 = 𝟬"
by field

end
```