# Differences

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equations [2017/10/01 21:50]
jipsen
equations [2017/10/02 10:57] (current)
jipsen
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|45  |Moufang3:  | $(x*y)*(z*x) = (x*(y*z))*x$  | Moufang4:  | $(x*y)*(z*x) = x*((y*z)*x)$  | |45  |Moufang3:  | $(x*y)*(z*x) = (x*(y*z))*x$  | Moufang4:  | $(x*y)*(z*x) = x*((y*z)*x)$  |
+Here are the identities in the syntax of the Lean Theorem Prover
+
+<code>
+section identities
+
+variables {α: Type u} {β: Type v}
+variables f g: α → α → α
+variables h k: α → α
+variable  c: α
+local notation a⬝b  := f a b
+local notation a+b := g a b
+local notation a⁻¹ := h a
+local notation 1   := c
+local notation 0   := c
+
+def involutive              := ∀x,     h(h x) = x
+def inverse_operations      := ∀x,     h(k x) = x
+def left_absorption         := ∀x,     h(k x) = k x
+def right_absorption        := ∀x,     h(k x) = h x
+def unary_idempotent        := ∀x,     h(h x) = h x
+def idempotent              := ∀x,     x⬝x = x
+def left_identity           := ∀x,     1⬝x = x
+def right_identity          := ∀x,     x⬝1 = x
+def left_zero               := ∀x,     0⬝x = 0
+def right_zero              := ∀x,     x⬝0 = 0
+def left_inverse            := ∀x,     x⁻¹⬝x = 1
+def right_inverse           := ∀x,     x⬝x⁻¹ = 1
+def left_const_mult         := ∀x,     c⬝x = h x
+def right_const_mult        := ∀x,     x⬝c = h x
+def square_constant         := ∀x,     x⬝x = c
+def square_unary            := ∀x,     x⬝x = h x
+def left_unary_identity     := ∀x,     (h x)⬝x = x
+def right_unary_identity    := ∀x,     x⬝(h x) = x
+def left_unary_const_mult   := ∀x,     h(c⬝x) = c⬝(h x)
+def right_unary_const_mult  := ∀x,     h(x⬝c) = (h x)⬝c
+def commutative             := ∀x y,   x⬝y = y⬝x
+def left_unary_projection   := ∀x y,   x⬝y = h x
+def right_unary_projection  := ∀x y,   x⬝y = h y
+def left_idempotent         := ∀x y,   x⬝(x⬝y) = x⬝y
+def right_idempotent        := ∀x y,   (x⬝y)⬝y = x⬝y
+def left_rectangular        := ∀x y,   (x⬝y)⬝x = x
+def right_rectangular       := ∀x y,   x⬝(y⬝x) = x
+def left_absorption1        := ∀x y,   (x⬝y)+y = y
+def right_absorption1       := ∀x y,   y+(x⬝y) = y
+def left_absorption2        := ∀x y,   (x⬝y)+x = x
+def right_absorption2       := ∀x y,   x+(y⬝x) = x
+def left_subtraction        := ∀x y,   x⬝(x+y) = y
+def right_subtraction       := ∀x y,   (y+x)⬝x = y
+def unary_commutative       := ∀x y,   (h x)⬝(h y) = (h y)⬝(h x)
+def unary_involutive        := ∀x y,   h(x⬝y) = (h y)⬝(h x)
+def interdistributive       := ∀x y,   h(x⬝y) = (h x)+(h y)
+def unary_distributive      := ∀x y,   h(x⬝y) = (h x)⬝(h y)
+def left_twisted            := ∀x y,   (h(x⬝y))⬝x = x⬝(h y)
+def right_twisted           := ∀x y,   x⬝(h(y⬝x)) = (h y)⬝x
+def left_locality           := ∀x y,   h((h x)⬝y) = h(x⬝y)
+def right_locality          := ∀x y,   h(x⬝(h y)) = h(x⬝y)
+def left_unary_distributive := ∀x y,   h((h x)⬝y) = (h x)⬝(h y)
+def right_unary_distributive:= ∀x y,   h(x⬝(h y)) = (h x)⬝(h y)
+def left_absorbtive         := ∀x y,   (h x)⬝(h(x⬝y)) = h(x⬝y)
+def right_absorbtive        := ∀x y,   (h(x⬝y))⬝(h y) = h(x⬝y)
+def flexible                := ∀x y,   (x⬝y)⬝x = x⬝(y⬝x)
+def associative             := ∀x y z, x⬝(y⬝z) = (x⬝y)⬝z
+def left_commutative        := ∀x y z, x⬝(y⬝z) = y⬝(x⬝z)
+def right_commutative       := ∀x y z, (x⬝y)⬝z = (x⬝z)⬝y
+def interassociative1       := ∀x y z, x⬝(y+z) = (x⬝y)+z
+def interassociative2       := ∀x y z, x⬝(y+z) = (x+y)⬝z
+def left_distributive       := ∀x y z, x⬝(y+z) = (x⬝y)+(x⬝z)
+def right_distributive      := ∀x y z, (x+y)⬝z = (x⬝z)+(y⬝z)
+def left_self_distributive  := ∀x y z, x⬝(y⬝z) = (x⬝y)⬝(x⬝z)
+def right_self_distributive := ∀x y z, (x⬝y)⬝z = (x⬝z)⬝(y⬝z)
+def Moufang1                := ∀x y z, ((x⬝y)⬝x)⬝z = x⬝(y⬝(x⬝z))
+def Moufang2                := ∀x y z, ((x⬝y)⬝z)⬝y = x⬝(y⬝(z⬝y))
+def Moufang3                := ∀x y z, (x⬝y)⬝(z⬝x) = (x⬝(y⬝z))⬝x
+def Moufang4                := ∀x y z, (x⬝y)⬝(z⬝x) = x⬝((y⬝z)⬝x)
+def entropic                := ∀x y z w, (x⬝y)⬝(z⬝w) = (x⬝z)⬝(y⬝w)
+def paramedial              := ∀x y z w, (x⬝y)⬝(z⬝w) = (w⬝y)⬝(z⬝x)
+
+
+end identities
+</code>