test_inner = ->
  run_test [

    # since we don't know anything about
    # a and b, we have to leave this unevaluated
    # turning this into a normal multiplication (which is
    # commutative) would not be OK
    "inner(a,b)",
    "inner(a,b)",

    # this shouldn't ever go wrong
    # but one never knows
    "inner(b,a)",
    "inner(b,a)",

    "inner(2,a)",
    "2*a",

    "inner(a,2)",
    "2*a",

    "inner(a,[b1,b2])",
    "inner(a,[b1,b2])",

    "inner([a1,a2],b)",
    "inner([a1,a2],b)",

    "inner(2,[b1,b2])",
    "[2*b1,2*b2]",

    "inner([b1,b2],2)",
    "[2*b1,2*b2]",

    "inner([[a11,a12],[a21,a22]],[x1,x2])",
    "[a11*x1+a12*x2,a21*x1+a22*x2]",

    "inner([1,2],[3,4])",
    "11",

    # non-invertible matrix
    "inner([[2,2],[2,2]],[[0],[1]])",
    "[[2],[2]]",

    "inner(inner([1,2],[[3,4],[5,6]]),[7,8])",
    "219",

    "inner([1,2],inner([[3,4],[5,6]],[7,8]))",
    "219",

    "inner([1,2],[[3,4],[5,6]],[7,8])",
    "219",

    "inner(c,a1+b1)",
    "inner(c,a1)+inner(c,b1)",

    "inner(b1+c1,a)",
    "inner(b1,a)+inner(c1,a)",

    "inner(inner(a,b),c)",
    "inner(a,inner(b,c))",

    "inner(inner(a,b),c) - inner(a,inner(b,c))",
    "0",

    "inner(inner(a,b),c,d) - inner(a,b,inner(c,d))",
    "0",

    # trying some associativity and distributivity together
    "inner(inner(a,b),c,d+f) - ( (inner(a,inner(b,c),d)) + inner(inner(a,b),c,f) )",
    "0",

    # bring it to a canonical form
    # using associativity
    "inner(a,b,c)",
    "inner(a,inner(b,c))",

    "inner(a,b+c,d)",
    "inner(a,inner(b,d))+inner(a,inner(c,d))",

    "inner(inner(a,b),inner(c,d))",
    "inner(a,inner(b,inner(c,d)))",

    # scalar product of vectors:
    "inner([a, b, c], [x, y, z])",
    "a*x+b*y+c*z",

    # products of matrices and vectors:
    "inner([[a, b], [c,d]], [x, y])",
    "[a*x+b*y,c*x+d*y]",

    "inner([x, y], [[a, b], [c,d]])",
    "[a*x+c*y,b*x+d*y]",

    "inner([x, y], [[a, b], [c,d]], [r, s])",
    "a*r*x+b*s*x+c*r*y+d*s*y",

    # matrix product:

    "inner([[a,b],[c,d]],[[r,s],[t,u]])",
    "[[a*r+b*t,a*s+b*u],[c*r+d*t,c*s+d*u]]",

    # ---------------------------------------
    # using the dot
    # ---------------------------------------

    "a·b",
    "inner(a,b)",

    "a·b·c",
    "inner(a,inner(b,c))",

    "a·b*c",
    "c*inner(a,b)",

    # note how we use associativity to bring it all
    # to a canonical form
    "((a·b)·c)·d",
    "inner(a,inner(b,inner(c,d)))",

    "a*b·c",
    "a*inner(b,c)",

    "2*a·b",
    "2*inner(a,b)",

    "inv(a)·a",
    "I",

    "a·inv(a)",
    "I",

    "b·a·inv(a)·c",
    "inner(b,c)",

    "b·aᵀ·inv(aᵀ)·c",
    "inner(b,c)",

    "b·inv(aᵀ)·aᵀ·c",
    "inner(b,c)",

    "b·inv((a+b)ᵀ)·(b+a)ᵀ·c",
    "inner(b,c)",

    "(-a)·(-b)",
    "inner(a,b)",

    "I·I",
    "I",

    "I·Iᵀ",
    "I",

    "(-I)·(-I)",
    "I",

    "(-Iᵀ)·(-I)",
    "I",

    "c·(b+a)ᵀ·inv((a+b)ᵀ)·d",
    "inner(c,d)",
      
    "c·inv((b+a)ᵀ)·(a+b)ᵀ·d",
    "inner(c,d)",
      
    "c·(b+a)ᵀ·inv((a+b)ᵀ)·inv(c)",
    "I",

    "c·inv((b+a)ᵀ)·(a+b)ᵀ·inv(c)",
    "I",
      
    "inv(c)·(b+a)ᵀ·inv((a+b)ᵀ)·c",
    "I",

    "inv(c)·inv((b+a)ᵀ)·(a+b)ᵀ·c",
    "I",
      
      
    "c·d·(b+a)ᵀ·inv((a+b)ᵀ)",
    "inner(c,d)",

    "d·(b+a)ᵀ·inv((a+b)ᵀ)",
    "d",
      
    "(b+a)ᵀ·inv((a+b)ᵀ)",
    "I",
      
    "c·d·(b+a)ᵀ·inv((a+b)ᵀ)·inv(d)",
    "c",
          
    "c·d·(b+a)ᵀ·inv((a+b)ᵀ)·inv(d)·inv(c)",
    "I",
      
    "c·d·(b+a)ᵀ·inv((a+b)ᵀ)·inv(c·d)",
    "I",
      
    "c·d·(a+b)ᵀ·inv(c·d·(b+a)ᵀ)",
    "I",
      
    "inv(c)·c",
    "I",
      
    "inv(c·a)·c·a",
    "I",
      
    "inv(c·b·a)·c·b·a",
    "I",
      
    "inv(c)·d·(a+b)ᵀ·inv(inv(c)·d·(b+a)ᵀ)",
    "I",
      
    "inv(c+f)·d·(a+b)ᵀ·inv(inv(f+c)·d·(b+a)ᵀ)",
    "I",
      
    "c·d·inv(a)·inv(c·d·inv(a))",
    "I",

  ]