Theorem: BitTree.karatsuba_eq_mul
Proof state before:
case h₂.h.h₂.h₂.h
H : List Unit
xhi xlo yhi ylo : BitTree
lhs✝¹ : Perfect H xhi
rhs✝¹ : Perfect H xlo
lhs✝ : Perfect H yhi
rhs✝ : Perfect H ylo
h :
  ∀ {H : List Unit} {x y : BitTree} [inst : Perfect H x] [inst : Perfect H y],
    toNat H x * toNat H y % bound (() :: H) = toNat H x * toNat H y
karatsuba_toNat :
  ∀ {x y : BitTree} [inst : Perfect H x] [inst : Perfect H y], toNat (() :: H) (karatsuba H x y) = toNat H x * toNat H y
h₃ :
  karatsubaHelper₃ H (karatsuba H (addWithCarry xhi xlo 0).2 (addWithCarry yhi ylo 0).2) (addWithCarry xhi xlo 0)
      (addWithCarry yhi ylo 0) =
    ofNat (() :: () :: H) ((toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo))
h₁ :
  karatsubaHelper₁ (() :: H) (karatsuba H xlo ylo) (karatsuba H xhi yhi)
      (ofNat (() :: () :: H) ((toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo))) =
    ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)
⊢ bound H * bound H * toNat H xhi * toNat H yhi + bound H * toNat H xlo * toNat H yhi +
      bound H * toNat H xhi * toNat H ylo <
    bound H * bound H * bound H * bound H
Tactic:
apply lt_of_le_of_lt (b := (node xhi xlo).toNat (()::H) * (node yhi ylo).toNat (()::H))
Proof state after:
case h₂.h.h₂.h₂.h.a
H : List Unit
xhi xlo yhi ylo : BitTree
lhs✝¹ : Perfect H xhi
rhs✝¹ : Perfect H xlo
lhs✝ : Perfect H yhi
rhs✝ : Perfect H ylo
h :
  ∀ {H : List Unit} {x y : BitTree} [inst : Perfect H x] [inst : Perfect H y],
    toNat H x * toNat H y % bound (() :: H) = toNat H x * toNat H y
karatsuba_toNat :
  ∀ {x y : BitTree} [inst : Perfect H x] [inst : Perfect H y], toNat (() :: H) (karatsuba H x y) = toNat H x * toNat H y
h₃ :
  karatsubaHelper₃ H (karatsuba H (addWithCarry xhi xlo 0).2 (addWithCarry yhi ylo 0).2) (addWithCarry xhi xlo 0)
      (addWithCarry yhi ylo 0) =
    ofNat (() :: () :: H) ((toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo))
h₁ :
  karatsubaHelper₁ (() :: H) (karatsuba H xlo ylo) (karatsuba H xhi yhi)
      (ofNat (() :: () :: H) ((toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo))) =
    ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)
⊢ bound H * bound H * toNat H xhi * toNat H yhi + bound H * toNat H xlo * toNat H yhi +
      bound H * toNat H xhi * toNat H ylo ≤
    toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)

case h₂.h.h₂.h₂.h.a
H : List Unit
xhi xlo yhi ylo : BitTree
lhs✝¹ : Perfect H xhi
rhs✝¹ : Perfect H xlo
lhs✝ : Perfect H yhi
rhs✝ : Perfect H ylo
h :
  ∀ {H : List Unit} {x y : BitTree} [inst : Perfect H x] [inst : Perfect H y],
    toNat H x * toNat H y % bound (() :: H) = toNat H x * toNat H y
karatsuba_toNat :
  ∀ {x y : BitTree} [inst : Perfect H x] [inst : Perfect H y], toNat (() :: H) (karatsuba H x y) = toNat H x * toNat H y
h₃ :
  karatsubaHelper₃ H (karatsuba H (addWithCarry xhi xlo 0).2 (addWithCarry yhi ylo 0).2) (addWithCarry xhi xlo 0)
      (addWithCarry yhi ylo 0) =
    ofNat (() :: () :: H) ((toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo))
h₁ :
  karatsubaHelper₁ (() :: H) (karatsuba H xlo ylo) (karatsuba H xhi yhi)
      (ofNat (() :: () :: H) ((toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo))) =
    ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)
⊢ toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) < bound H * bound H * bound H * bound H