-- p2.org

— Activity is two pages!

-- Problem 1

g(α,a) = g(b,β),
h(γ,γ) = h(f(α),γ)

decompose g

h(γ,γ) = h(f(α),γ)
α = b
a = β

decompose h

α = b
a = β
γ = f(α)
γ = γ

eliminate α

a = β
γ = f(b)
γ = γ
α ↦ b

orient, then eliminate β

γ = f(b)
γ = γ
α ↦ b
β ↦ a

eliminate γ

f(b) = f(b)
α ↦ b
β ↦ a
γ ↦ f(b)

delete f(b) = f(b)

α ↦ b
β ↦ a
γ ↦ f(b)

-- Problem 2

f(α,α) = f(γ,x)
h(β,g(γ)) = h(y,δ)

decompose f

h(β,g(γ)) = h(y,δ)
α = γ
α = x

decompose h

β = y
g(γ) = δ
α = γ
α = x

eliminate β

g(γ) = δ
α = γ
α = x
β ↦ y

orient δ, then eliminate it

α = γ
α = x
β ↦ y
δ ↦ g(γ)

eliminate α

γ = x
β ↦ y
δ ↦ g(γ)
α ↦ γ

eliminate γ

β ↦ y
δ ↦ g(x)
α ↦ x
γ  ↦ x

-- Problem 3

f(α) = f(x)
g(α) = g(β)
h(γ,x) = h(β,α)

decompose f

α = x
g(α) = g(β)
h(γ,x) = h(β,α)

eliminate α

g(x) = g(β)
h(γ,x) = h(β,x)
α ↦ x

decompose g

x = β
h(γ,x) = h(β,x)
α ↦ x

orient and eliminate β

h(γ,x) = h(x,x)
α ↦ x
β ↦ x

decompose h

γ = x
x = x
α ↦ x
β ↦ x

eliminate γ

x = x
α ↦ x
β ↦ x
γ ↦ x

delete x = x

α ↦ x
β ↦ x
γ ↦ x