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https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeSource_inj | [17, 1] | [21, 31] | case idx.idx => simpa [encodeSource] using λ hhd htl ↦ ⟨encodeSource_inj hhd, encodeSource_inj htl⟩ | s₀ s₁ hd✝¹ tl✝¹ : Source
h₀ : s₀ = Source.idx hd✝¹ tl✝¹
hd✝ tl✝ : Source
h₁ : s₁ = Source.idx hd✝ tl✝
⊢ encodeSource (Source.idx hd✝¹ tl✝¹) = encodeSource (Source.idx hd✝ tl✝) → Source.idx hd✝¹ tl✝¹ = Source.idx hd✝ tl✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeSource_inj | [17, 1] | [21, 31] | all_goals simp[encodeSource] | case nil.nil
s₀ s₁ : Source
h₀ : s₀ = Source.nil
h₁ : s₁ = Source.nil
⊢ encodeSource Source.nil = encodeSource Source.nil → Source.nil = Source.nil
case nil.imm
s₀ s₁ : Source
h₀ : s₀ = Source.nil
hd✝ : Bool
tl✝ : Source
h₁ : s₁ = Source.imm hd✝ tl✝
⊢ encodeSource Source.nil = encodeSource (Source.imm hd✝ tl✝) → Source.nil = Source.imm hd✝ tl✝
case nil.idx
s₀ s₁ : Source
h₀ : s₀ = Source.nil
hd✝ tl✝ : Source
h₁ : s₁ = Source.idx hd✝ tl✝
⊢ encodeSource Source.nil = encodeSource (Source.idx hd✝ tl✝) → Source.nil = Source.idx hd✝ tl✝
case imm.nil
s₀ s₁ : Source
hd✝ : Bool
tl✝ : Source
h₀ : s₀ = Source.imm hd✝ tl✝
h₁ : s₁ = Source.nil
⊢ encodeSource (Source.imm hd✝ tl✝) = encodeSource Source.nil → Source.imm hd✝ tl✝ = Source.nil
case imm.idx
s₀ s₁ : Source
hd✝¹ : Bool
tl✝¹ : Source
h₀ : s₀ = Source.imm hd✝¹ tl✝¹
hd✝ tl✝ : Source
h₁ : s₁ = Source.idx hd✝ tl✝
⊢ encodeSource (Source.imm hd✝¹ tl✝¹) = encodeSource (Source.idx hd✝ tl✝) → Source.imm hd✝¹ tl✝¹ = Source.idx hd✝ tl✝
case idx.nil
s₀ s₁ hd✝ tl✝ : Source
h₀ : s₀ = Source.idx hd✝ tl✝
h₁ : s₁ = Source.nil
⊢ encodeSource (Source.idx hd✝ tl✝) = encodeSource Source.nil → Source.idx hd✝ tl✝ = Source.nil
case idx.imm
s₀ s₁ hd✝¹ tl✝¹ : Source
h₀ : s₀ = Source.idx hd✝¹ tl✝¹
hd✝ : Bool
tl✝ : Source
h₁ : s₁ = Source.imm hd✝ tl✝
⊢ encodeSource (Source.idx hd✝¹ tl✝¹) = encodeSource (Source.imm hd✝ tl✝) → Source.idx hd✝¹ tl✝¹ = Source.imm hd✝ tl✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeSource_inj | [17, 1] | [21, 31] | simpa [encodeSource] using λ hhd htl ↦ ⟨hhd, encodeSource_inj htl⟩ | s₀ s₁ : Source
hd✝¹ : Bool
tl✝¹ : Source
h₀ : s₀ = Source.imm hd✝¹ tl✝¹
hd✝ : Bool
tl✝ : Source
h₁ : s₁ = Source.imm hd✝ tl✝
⊢ encodeSource (Source.imm hd✝¹ tl✝¹) = encodeSource (Source.imm hd✝ tl✝) → Source.imm hd✝¹ tl✝¹ = Source.imm hd✝ tl✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeSource_inj | [17, 1] | [21, 31] | simpa [encodeSource] using λ hhd htl ↦ ⟨encodeSource_inj hhd, encodeSource_inj htl⟩ | s₀ s₁ hd✝¹ tl✝¹ : Source
h₀ : s₀ = Source.idx hd✝¹ tl✝¹
hd✝ tl✝ : Source
h₁ : s₁ = Source.idx hd✝ tl✝
⊢ encodeSource (Source.idx hd✝¹ tl✝¹) = encodeSource (Source.idx hd✝ tl✝) → Source.idx hd✝¹ tl✝¹ = Source.idx hd✝ tl✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeSource_inj | [17, 1] | [21, 31] | simp[encodeSource] | case idx.imm
s₀ s₁ hd✝¹ tl✝¹ : Source
h₀ : s₀ = Source.idx hd✝¹ tl✝¹
hd✝ : Bool
tl✝ : Source
h₁ : s₁ = Source.imm hd✝ tl✝
⊢ encodeSource (Source.idx hd✝¹ tl✝¹) = encodeSource (Source.imm hd✝ tl✝) → Source.idx hd✝¹ tl✝¹ = Source.imm hd✝ tl✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | induction p₀ generalizing p₁ <;> cases p₁ | p₀ p₁ : Program
⊢ encodeProgram p₀ = encodeProgram p₁ → p₀ = p₁ | case exit.exit
⊢ encodeProgram Program.exit = encodeProgram Program.exit → Program.exit = Program.exit
case exit.op
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram Program.exit = encodeProgram (Program.op inst✝ next✝) → Program.exit = Program.op inst✝ next✝
case exit.branch
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram Program.exit = encodeProgram (Program.branch inst✝ next✝) → Program.exit = Program.branch inst✝ next✝
case exit.subroutine
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram Program.exit = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.exit = Program.subroutine dst✝ src✝ func✝ next✝
case exit.recurse
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram Program.exit = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.exit = Program.recurse dst✝ src✝ next✝
case op.exit
inst✝ : OpInstruction
next✝ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝ = encodeProgram p₁ → next✝ = p₁
⊢ encodeProgram (Program.op inst✝ next✝) = encodeProgram Program.exit → Program.op inst✝ next✝ = Program.exit
case op.op
inst✝¹ : OpInstruction
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.op inst✝¹ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.op inst✝¹ next✝¹ = Program.op inst✝ next✝
case op.branch
inst✝¹ : OpInstruction
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.op inst✝¹ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.op inst✝¹ next✝¹ = Program.branch inst✝ next✝
case op.subroutine
inst✝ : OpInstruction
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.op inst✝ next✝¹) = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.op inst✝ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝
case op.recurse
inst✝ : OpInstruction
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.op inst✝ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.op inst✝ next✝¹ = Program.recurse dst✝ src✝ next✝
case branch.exit
inst✝ : BranchInstruction
next✝ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝ a) = encodeProgram p₁ → next✝ a = p₁
⊢ encodeProgram (Program.branch inst✝ next✝) = encodeProgram Program.exit → Program.branch inst✝ next✝ = Program.exit
case branch.op
inst✝¹ : BranchInstruction
next✝¹ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.branch inst✝¹ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.branch inst✝¹ next✝¹ = Program.op inst✝ next✝
case branch.branch
inst✝¹ : BranchInstruction
next✝¹ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.branch inst✝¹ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.branch inst✝¹ next✝¹ = Program.branch inst✝ next✝
case branch.subroutine
inst✝ : BranchInstruction
next✝¹ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.branch inst✝ next✝¹) = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.branch inst✝ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝
case branch.recurse
inst✝ : BranchInstruction
next✝¹ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.branch inst✝ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.branch inst✝ next✝¹ = Program.recurse dst✝ src✝ next✝
case subroutine.exit
dst✝ src✝ : Source
func✝ next✝ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝ = encodeProgram p₁ → func✝ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝ = encodeProgram p₁ → next✝ = p₁
⊢ encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) = encodeProgram Program.exit →
Program.subroutine dst✝ src✝ func✝ next✝ = Program.exit
case subroutine.op
dst✝ src✝ : Source
func✝ next✝¹ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝ = encodeProgram p₁ → func✝ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.subroutine dst✝ src✝ func✝ next✝¹ = Program.op inst✝ next✝
case subroutine.branch
dst✝ src✝ : Source
func✝ next✝¹ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝ = encodeProgram p₁ → func✝ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.subroutine dst✝ src✝ func✝ next✝¹ = Program.branch inst✝ next✝
case subroutine.subroutine
dst✝¹ src✝¹ : Source
func✝¹ next✝¹ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝¹ = encodeProgram p₁ → func✝¹ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.subroutine dst✝¹ src✝¹ func✝¹ next✝¹) =
encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.subroutine dst✝¹ src✝¹ func✝¹ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝
case subroutine.recurse
dst✝¹ src✝¹ : Source
func✝ next✝¹ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝ = encodeProgram p₁ → func✝ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.subroutine dst✝¹ src✝¹ func✝ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.subroutine dst✝¹ src✝¹ func✝ next✝¹ = Program.recurse dst✝ src✝ next✝
case recurse.exit
dst✝ src✝ : Source
next✝ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝ = encodeProgram p₁ → next✝ = p₁
⊢ encodeProgram (Program.recurse dst✝ src✝ next✝) = encodeProgram Program.exit →
Program.recurse dst✝ src✝ next✝ = Program.exit
case recurse.op
dst✝ src✝ : Source
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.recurse dst✝ src✝ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.recurse dst✝ src✝ next✝¹ = Program.op inst✝ next✝
case recurse.branch
dst✝ src✝ : Source
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.recurse dst✝ src✝ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.recurse dst✝ src✝ next✝¹ = Program.branch inst✝ next✝
case recurse.subroutine
dst✝¹ src✝¹ : Source
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.recurse dst✝¹ src✝¹ next✝¹) = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.recurse dst✝¹ src✝¹ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝
case recurse.recurse
dst✝¹ src✝¹ : Source
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.recurse dst✝¹ src✝¹ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.recurse dst✝¹ src✝¹ next✝¹ = Program.recurse dst✝ src✝ next✝ |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | case op.op ih _ _ => simpa [encodeProgram] using λ x y ↦ ⟨x, ih y⟩ | inst✝¹ : OpInstruction
next✝¹ : Program
ih : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.op inst✝¹ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.op inst✝¹ next✝¹ = Program.op inst✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | case branch.branch nih _ _ =>
simpa [encodeProgram] using λ hop hf ht ↦
⟨ hop, funext λ | false => nih _ hf | true => nih _ ht ⟩ | inst✝¹ : BranchInstruction
next✝¹ : Bool → Program
nih : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.branch inst✝¹ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.branch inst✝¹ next✝¹ = Program.branch inst✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | case subroutine.subroutine fih nih _ _ _ _ =>
simpa [encodeProgram] using λ hdst hsrc hfunc hnext ↦ ⟨ hdst, hsrc, fih hfunc, nih hnext ⟩ | dst✝¹ src✝¹ : Source
func✝¹ next✝¹ : Program
fih : ∀ {p₁ : Program}, encodeProgram func✝¹ = encodeProgram p₁ → func✝¹ = p₁
nih : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.subroutine dst✝¹ src✝¹ func✝¹ next✝¹) =
encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.subroutine dst✝¹ src✝¹ func✝¹ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | case recurse.recurse ih _ _ _ =>
simpa [encodeProgram] using λ hdst hsrc hnext ↦ ⟨hdst, hsrc, ih hnext⟩ | dst✝¹ src✝¹ : Source
next✝¹ : Program
ih : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.recurse dst✝¹ src✝¹ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.recurse dst✝¹ src✝¹ next✝¹ = Program.recurse dst✝ src✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | all_goals simp [encodeProgram] | case exit.exit
⊢ encodeProgram Program.exit = encodeProgram Program.exit → Program.exit = Program.exit
case exit.op
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram Program.exit = encodeProgram (Program.op inst✝ next✝) → Program.exit = Program.op inst✝ next✝
case exit.branch
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram Program.exit = encodeProgram (Program.branch inst✝ next✝) → Program.exit = Program.branch inst✝ next✝
case exit.subroutine
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram Program.exit = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.exit = Program.subroutine dst✝ src✝ func✝ next✝
case exit.recurse
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram Program.exit = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.exit = Program.recurse dst✝ src✝ next✝
case op.exit
inst✝ : OpInstruction
next✝ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝ = encodeProgram p₁ → next✝ = p₁
⊢ encodeProgram (Program.op inst✝ next✝) = encodeProgram Program.exit → Program.op inst✝ next✝ = Program.exit
case op.branch
inst✝¹ : OpInstruction
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.op inst✝¹ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.op inst✝¹ next✝¹ = Program.branch inst✝ next✝
case op.subroutine
inst✝ : OpInstruction
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.op inst✝ next✝¹) = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.op inst✝ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝
case op.recurse
inst✝ : OpInstruction
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.op inst✝ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.op inst✝ next✝¹ = Program.recurse dst✝ src✝ next✝
case branch.exit
inst✝ : BranchInstruction
next✝ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝ a) = encodeProgram p₁ → next✝ a = p₁
⊢ encodeProgram (Program.branch inst✝ next✝) = encodeProgram Program.exit → Program.branch inst✝ next✝ = Program.exit
case branch.op
inst✝¹ : BranchInstruction
next✝¹ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.branch inst✝¹ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.branch inst✝¹ next✝¹ = Program.op inst✝ next✝
case branch.subroutine
inst✝ : BranchInstruction
next✝¹ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.branch inst✝ next✝¹) = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.branch inst✝ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝
case branch.recurse
inst✝ : BranchInstruction
next✝¹ : Bool → Program
next_ih✝ : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.branch inst✝ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.branch inst✝ next✝¹ = Program.recurse dst✝ src✝ next✝
case subroutine.exit
dst✝ src✝ : Source
func✝ next✝ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝ = encodeProgram p₁ → func✝ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝ = encodeProgram p₁ → next✝ = p₁
⊢ encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) = encodeProgram Program.exit →
Program.subroutine dst✝ src✝ func✝ next✝ = Program.exit
case subroutine.op
dst✝ src✝ : Source
func✝ next✝¹ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝ = encodeProgram p₁ → func✝ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.subroutine dst✝ src✝ func✝ next✝¹ = Program.op inst✝ next✝
case subroutine.branch
dst✝ src✝ : Source
func✝ next✝¹ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝ = encodeProgram p₁ → func✝ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.subroutine dst✝ src✝ func✝ next✝¹ = Program.branch inst✝ next✝
case subroutine.recurse
dst✝¹ src✝¹ : Source
func✝ next✝¹ : Program
func_ih✝ : ∀ {p₁ : Program}, encodeProgram func✝ = encodeProgram p₁ → func✝ = p₁
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.subroutine dst✝¹ src✝¹ func✝ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.subroutine dst✝¹ src✝¹ func✝ next✝¹ = Program.recurse dst✝ src✝ next✝
case recurse.exit
dst✝ src✝ : Source
next✝ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝ = encodeProgram p₁ → next✝ = p₁
⊢ encodeProgram (Program.recurse dst✝ src✝ next✝) = encodeProgram Program.exit →
Program.recurse dst✝ src✝ next✝ = Program.exit
case recurse.op
dst✝ src✝ : Source
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.recurse dst✝ src✝ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.recurse dst✝ src✝ next✝¹ = Program.op inst✝ next✝
case recurse.branch
dst✝ src✝ : Source
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.recurse dst✝ src✝ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.recurse dst✝ src✝ next✝¹ = Program.branch inst✝ next✝
case recurse.subroutine
dst✝¹ src✝¹ : Source
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.recurse dst✝¹ src✝¹ next✝¹) = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.recurse dst✝¹ src✝¹ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | simpa [encodeProgram] using λ x y ↦ ⟨x, ih y⟩ | inst✝¹ : OpInstruction
next✝¹ : Program
ih : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
inst✝ : OpInstruction
next✝ : Program
⊢ encodeProgram (Program.op inst✝¹ next✝¹) = encodeProgram (Program.op inst✝ next✝) →
Program.op inst✝¹ next✝¹ = Program.op inst✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | simpa [encodeProgram] using λ hop hf ht ↦
⟨ hop, funext λ | false => nih _ hf | true => nih _ ht ⟩ | inst✝¹ : BranchInstruction
next✝¹ : Bool → Program
nih : ∀ (a : Bool) {p₁ : Program}, encodeProgram (next✝¹ a) = encodeProgram p₁ → next✝¹ a = p₁
inst✝ : BranchInstruction
next✝ : Bool → Program
⊢ encodeProgram (Program.branch inst✝¹ next✝¹) = encodeProgram (Program.branch inst✝ next✝) →
Program.branch inst✝¹ next✝¹ = Program.branch inst✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | simpa [encodeProgram] using λ hdst hsrc hfunc hnext ↦ ⟨ hdst, hsrc, fih hfunc, nih hnext ⟩ | dst✝¹ src✝¹ : Source
func✝¹ next✝¹ : Program
fih : ∀ {p₁ : Program}, encodeProgram func✝¹ = encodeProgram p₁ → func✝¹ = p₁
nih : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.subroutine dst✝¹ src✝¹ func✝¹ next✝¹) =
encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.subroutine dst✝¹ src✝¹ func✝¹ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | simpa [encodeProgram] using λ hdst hsrc hnext ↦ ⟨hdst, hsrc, ih hnext⟩ | dst✝¹ src✝¹ : Source
next✝¹ : Program
ih : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
next✝ : Program
⊢ encodeProgram (Program.recurse dst✝¹ src✝¹ next✝¹) = encodeProgram (Program.recurse dst✝ src✝ next✝) →
Program.recurse dst✝¹ src✝¹ next✝¹ = Program.recurse dst✝ src✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Encoding/Emulator.lean | HMem.Encoding.encodeProgram_inj | [64, 1] | [74, 33] | simp [encodeProgram] | case recurse.subroutine
dst✝¹ src✝¹ : Source
next✝¹ : Program
next_ih✝ : ∀ {p₁ : Program}, encodeProgram next✝¹ = encodeProgram p₁ → next✝¹ = p₁
dst✝ src✝ : Source
func✝ next✝ : Program
⊢ encodeProgram (Program.recurse dst✝¹ src✝¹ next✝¹) = encodeProgram (Program.subroutine dst✝ src✝ func✝ next✝) →
Program.recurse dst✝¹ src✝¹ next✝¹ = Program.subroutine dst✝ src✝ func✝ next✝ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.toNatHelper_acc | [253, 1] | [258, 59] | simpa [toNatHelper, bound_zero] using Nat.add_comm _ _ | val✝ : Fin 2
x✝ : ℕ
⊢ toNatHelper [] (bit val✝) x✝ = bound [] * x✝ + toNatHelper [] (bit val✝) 0 | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.toNatHelper_acc | [253, 1] | [258, 59] | simp [bound_cons, toNatHelper, toNatHelper_acc H _ x, toNatHelper_acc H _ (_ + _),
toNatHelper_acc H _ (toNatHelper _ _ _),
left_distrib, right_distrib, ← mul_assoc, ← add_assoc] | H : List Unit
left✝ right✝ : BitTree
lhs✝ : Perfect H left✝
rhs✝ : Perfect H right✝
x : ℕ
⊢ toNatHelper (() :: H) (node left✝ right✝) x = bound (() :: H) * x + toNatHelper (() :: H) (node left✝ right✝) 0 | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.addWithCarry_eq_ofNat_add | [394, 1] | [412, 72] | simp [addWithCarry, toNat, toNatHelper, bound_zero] | val✝¹ val✝ x✝ : Fin 2
⊢ addWithCarry (bit val✝¹) (bit val✝) x✝ =
({ val := (toNat [] (bit val✝¹) + toNat [] (bit val✝) + ↑x✝) / bound [],
isLt := (_ : (toNat [] (bit val✝¹) + toNat [] (bit val✝) + ↑x✝) / bound [] < 2) },
ofNat [] (toNat [] (bit val✝¹) + toNat [] (bit val✝) + ↑x✝)) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.addWithCarry_eq_ofNat_add | [394, 1] | [412, 72] | intro _ _ _ _ _ _ | H : List Unit
l✝¹ r✝¹ : BitTree
lhs✝¹ : Perfect H l✝¹
rhs✝¹ : Perfect H r✝¹
l✝ r✝ : BitTree
lhs✝ : Perfect H l✝
rhs✝ : Perfect H r✝
x✝ : Fin 2
⊢ ∀ {m xh xl yh yl c : ℕ}, m * xh + xl + (m * yh + yl) + c = m * (xh + yh) + (xl + yl + c) | H : List Unit
l✝¹ r✝¹ : BitTree
lhs✝¹ : Perfect H l✝¹
rhs✝¹ : Perfect H r✝¹
l✝ r✝ : BitTree
lhs✝ : Perfect H l✝
rhs✝ : Perfect H r✝
x✝ : Fin 2
m✝ xh✝ xl✝ yh✝ yl✝ c✝ : ℕ
⊢ m✝ * xh✝ + xl✝ + (m✝ * yh✝ + yl✝) + c✝ = m✝ * (xh✝ + yh✝) + (xl✝ + yl✝ + c✝) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.addWithCarry_eq_ofNat_add | [394, 1] | [412, 72] | ring | H : List Unit
l✝¹ r✝¹ : BitTree
lhs✝¹ : Perfect H l✝¹
rhs✝¹ : Perfect H r✝¹
l✝ r✝ : BitTree
lhs✝ : Perfect H l✝
rhs✝ : Perfect H r✝
x✝ : Fin 2
m✝ xh✝ xl✝ yh✝ yl✝ c✝ : ℕ
⊢ m✝ * xh✝ + xl✝ + (m✝ * yh✝ + yl✝) + c✝ = m✝ * (xh✝ + yh✝) + (xl✝ + yl✝ + c✝) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.addWithCarry_eq_ofNat_add | [394, 1] | [412, 72] | simp [addWithCarry, @addWithCarry_eq_ofNat_add H, rearrange, toNat_node, bound_cons,
← Nat.div_div_eq_div_mul, Nat.mul_add_div (bound_pos _), divBound] | H : List Unit
l✝¹ r✝¹ : BitTree
lhs✝¹ : Perfect H l✝¹
rhs✝¹ : Perfect H r✝¹
l✝ r✝ : BitTree
lhs✝ : Perfect H l✝
rhs✝ : Perfect H r✝
x✝ : Fin 2
rearrange : ∀ {m xh xl yh yl c : ℕ}, m * xh + xl + (m * yh + yl) + c = m * (xh + yh) + (xl + yl + c)
⊢ addWithCarry (node l✝¹ r✝¹) (node l✝ r✝) x✝ =
({ val := (toNat (() :: H) (node l✝¹ r✝¹) + toNat (() :: H) (node l✝ r✝) + ↑x✝) / bound (() :: H),
isLt := (_ : (toNat (() :: H) (node l✝¹ r✝¹) + toNat (() :: H) (node l✝ r✝) + ↑x✝) / bound (() :: H) < 2) },
ofNat (() :: H) (toNat (() :: H) (node l✝¹ r✝¹) + toNat (() :: H) (node l✝ r✝) + ↑x✝)) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.addWithCarry_negate | [436, 1] | [439, 78] | simp [addWithCarry, addWithCarry_negate (H := H)] | H : List Unit
l✝ r✝ : BitTree
lhs✝ : Perfect H l✝
rhs✝ : Perfect H r✝
⊢ addWithCarry (node l✝ r✝) (negate (node l✝ r✝)) 1 = (1, ofNat (() :: H) 0) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.add_toNat_negate | [441, 1] | [448, 111] | intro _ _ _ _ _ _ | H✝ : List Unit
l✝ r✝ : BitTree
lhs✝ : Perfect H✝ l✝
rhs✝ : Perfect H✝ r✝
⊢ ∀ {m a b c d e : ℕ}, m * a + b + (m * c + d) + e = m * (a + c) + (b + d + e) | H✝ : List Unit
l✝ r✝ : BitTree
lhs✝ : Perfect H✝ l✝
rhs✝ : Perfect H✝ r✝
m✝ a✝ b✝ c✝ d✝ e✝ : ℕ
⊢ m✝ * a✝ + b✝ + (m✝ * c✝ + d✝) + e✝ = m✝ * (a✝ + c✝) + (b✝ + d✝ + e✝) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.add_toNat_negate | [441, 1] | [448, 111] | ring | H✝ : List Unit
l✝ r✝ : BitTree
lhs✝ : Perfect H✝ l✝
rhs✝ : Perfect H✝ r✝
m✝ a✝ b✝ c✝ d✝ e✝ : ℕ
⊢ m✝ * a✝ + b✝ + (m✝ * c✝ + d✝) + e✝ = m✝ * (a✝ + c✝) + (b✝ + d✝ + e✝) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.add_toNat_negate | [441, 1] | [448, 111] | simp [toNat_node, rearrange, negate, add_toNat_negate _, ← Nat.mul_succ, Nat.succ_eq_add_one, bound_cons] | H✝ : List Unit
l✝ r✝ : BitTree
lhs✝ : Perfect H✝ l✝
rhs✝ : Perfect H✝ r✝
rearrange : ∀ {m a b c d e : ℕ}, m * a + b + (m * c + d) + e = m * (a + c) + (b + d + e)
⊢ toNat (() :: H✝) (node l✝ r✝) + toNat (() :: H✝) (negate (node l✝ r✝)) + 1 = bound (() :: H✝) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.sub_toNat | [492, 1] | [496, 68] | intro _ _ _ _ | H : List Unit
x y : BitTree
inst✝¹ : Perfect H x
inst✝ : Perfect H y
⊢ ∀ {a b c d : ℕ}, a + b + c + d = a + (d + b + c) | H : List Unit
x y : BitTree
inst✝¹ : Perfect H x
inst✝ : Perfect H y
a✝ b✝ c✝ d✝ : ℕ
⊢ a✝ + b✝ + c✝ + d✝ = a✝ + (d✝ + b✝ + c✝) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.sub_toNat | [492, 1] | [496, 68] | ring | H : List Unit
x y : BitTree
inst✝¹ : Perfect H x
inst✝ : Perfect H y
a✝ b✝ c✝ d✝ : ℕ
⊢ a✝ + b✝ + c✝ + d✝ = a✝ + (d✝ + b✝ + c✝) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.lshift2_toNat | [506, 1] | [513, 27] | simp only [
toNat_node, lo, hi, lshift2, bound_cons,
left_distrib, right_distrib, ← mul_assoc,
mul_right_comm _ _ (bound H), mul_comm (toNat _ _) (bound H),
Nat.add_assoc, Nat.mul_add_mod,
toNat_zero, add_zero] | H : List Unit
left✝¹ right✝¹ left✝ right✝ : BitTree
lhs✝¹ : Perfect H left✝¹
rhs✝¹ : Perfect H right✝¹
lhs✝ : Perfect H left✝
rhs✝ : Perfect H right✝
⊢ toNat (() :: () :: H) (lshift2 H (node (node left✝¹ right✝¹) (node left✝ right✝))) % bound (() :: () :: H) =
toNat (() :: () :: H) (node (node left✝¹ right✝¹) (node left✝ right✝)) * bound H % bound (() :: () :: H) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsubaHelper₃_toNat | [524, 1] | [557, 63] | have h₀: ∀ {x y: Fin 2} {H: List Unit}, x.val * y.val % (bound H * bound H) = x.val * y.val :=
Nat.mod_eq_of_lt (Nat.mul_lt_mul_for_real
(lt_of_lt_of_le (Fin.isLt _) (two_le_bound _))
(lt_of_lt_of_le (Fin.isLt _) (two_le_bound _))) | H : List Unit
xy : BitTree
xc : Fin 2
x : BitTree
yc : Fin 2
y : BitTree
x✝² : Perfect (() :: H) xy
x✝¹ : Perfect H (xc, x).2
x✝ : Perfect H (yc, y).2
h : toNat (() :: H) xy = toNat H (xc, x).2 * toNat H (yc, y).2
⊢ toNat (() :: () :: H) (karatsubaHelper₃ H xy (xc, x) (yc, y)) =
(bound H * ↑(xc, x).1 + toNat H (xc, x).2) * (bound H * ↑(yc, y).1 + toNat H (yc, y).2) | H : List Unit
xy : BitTree
xc : Fin 2
x : BitTree
yc : Fin 2
y : BitTree
x✝² : Perfect (() :: H) xy
x✝¹ : Perfect H (xc, x).2
x✝ : Perfect H (yc, y).2
h : toNat (() :: H) xy = toNat H (xc, x).2 * toNat H (yc, y).2
h₀ : ∀ {x y : Fin 2} {H : List Unit}, ↑x * ↑y % (bound H * bound H) = ↑x * ↑y
⊢ toNat (() :: () :: H) (karatsubaHelper₃ H xy (xc, x) (yc, y)) =
(bound H * ↑(xc, x).1 + toNat H (xc, x).2) * (bound H * ↑(yc, y).1 + toNat H (yc, y).2) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsubaHelper₃_toNat | [524, 1] | [557, 63] | have h₁: ∀ {x: Fin 2} {H: List Unit} {y: BitTree} [Perfect H y], x.val * y.toNat H % bound H = x.val * y.toNat H
| 0, _, _, _ => Nat.mod_eq_of_lt (lt_of_eq_of_lt (zero_mul _) (bound_pos _))
| 1, _, _, _ => Nat.mod_eq_of_lt (lt_of_eq_of_lt (one_mul _) (toNat_lt _)) | H : List Unit
xy : BitTree
xc : Fin 2
x : BitTree
yc : Fin 2
y : BitTree
x✝² : Perfect (() :: H) xy
x✝¹ : Perfect H (xc, x).2
x✝ : Perfect H (yc, y).2
h : toNat (() :: H) xy = toNat H (xc, x).2 * toNat H (yc, y).2
h₀ : ∀ {x y : Fin 2} {H : List Unit}, ↑x * ↑y % (bound H * bound H) = ↑x * ↑y
⊢ toNat (() :: () :: H) (karatsubaHelper₃ H xy (xc, x) (yc, y)) =
(bound H * ↑(xc, x).1 + toNat H (xc, x).2) * (bound H * ↑(yc, y).1 + toNat H (yc, y).2) | H : List Unit
xy : BitTree
xc : Fin 2
x : BitTree
yc : Fin 2
y : BitTree
x✝² : Perfect (() :: H) xy
x✝¹ : Perfect H (xc, x).2
x✝ : Perfect H (yc, y).2
h : toNat (() :: H) xy = toNat H (xc, x).2 * toNat H (yc, y).2
h₀ : ∀ {x y : Fin 2} {H : List Unit}, ↑x * ↑y % (bound H * bound H) = ↑x * ↑y
h₁ : ∀ {x : Fin 2} {H : List Unit} {y : BitTree} [inst : Perfect H y], ↑x * toNat H y % bound H = ↑x * toNat H y
⊢ toNat (() :: () :: H) (karatsubaHelper₃ H xy (xc, x) (yc, y)) =
(bound H * ↑(xc, x).1 + toNat H (xc, x).2) * (bound H * ↑(yc, y).1 + toNat H (yc, y).2) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsubaHelper₃_toNat | [524, 1] | [557, 63] | simpa only [karatsubaHelper₃,
add_toNat (()::()::H), mulBit_ofNat, mulBit'_ofNat,
bound_cons, ← mul_assoc, mul_zero,
mul_right_comm _ _ (bound H),
mul_right_comm _ (toNat _ _) (Fin.val _),
mul_comm (toNat _ _) (bound H),
Nat.add_right_comm _ (_ * toNat H y) (_ * toNat H x),
Nat.add_right_comm _ (toNat _ _ * toNat _ _),
left_distrib, right_distrib,
toNat_ofNat, toNat_node, h, toNat_zero,
add_zero, zero_add, Nat.mod_add_mod, ← Nat.add_assoc,
h₀, h₁] using Nat.mod_eq_of_lt (lt_of_eq_of_lt
(Nat.add_assoc _ _ _)
(add_lt_mul_bound
(add_le_mul_bound
(le_trans
(mul_le_of_le_one_right' (Nat.le_of_lt_succ (Fin.isLt _)))
(mul_le_of_le_one_right' (Nat.le_of_lt_succ (Fin.isLt _))))
(Nat.mul_le_mul
(mul_le_of_le_one_right' (Nat.le_of_lt_succ (Fin.isLt _)))
(le_of_lt (toNat_lt _))))
(add_lt_mul_bound
(Nat.mul_le_mul (mul_le_of_le_one_right' (Nat.le_of_lt_succ (Fin.isLt _))) (le_of_lt (toNat_lt _)))
(Nat.mul_lt_mul_for_real (toNat_lt _) (toNat_lt _))))) | H : List Unit
xy : BitTree
xc : Fin 2
x : BitTree
yc : Fin 2
y : BitTree
x✝² : Perfect (() :: H) xy
x✝¹ : Perfect H (xc, x).2
x✝ : Perfect H (yc, y).2
h : toNat (() :: H) xy = toNat H (xc, x).2 * toNat H (yc, y).2
h₀ : ∀ {x y : Fin 2} {H : List Unit}, ↑x * ↑y % (bound H * bound H) = ↑x * ↑y
h₁ : ∀ {x : Fin 2} {H : List Unit} {y : BitTree} [inst : Perfect H y], ↑x * toNat H y % bound H = ↑x * toNat H y
⊢ toNat (() :: () :: H) (karatsubaHelper₃ H xy (xc, x) (yc, y)) =
(bound H * ↑(xc, x).1 + toNat H (xc, x).2) * (bound H * ↑(yc, y).1 + toNat H (yc, y).2) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsubaHelper₁_toNat | [563, 1] | [609, 39] | intro h₀ h₂ h₃ | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
⊢ toNat (() :: H) z₀ = toNat H xlo * toNat H ylo →
toNat (() :: H) z₂ = toNat H xhi * toNat H yhi →
toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo) →
toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) =
toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
h₀ : toNat (() :: H) z₀ = toNat H xlo * toNat H ylo
h₂ : toNat (() :: H) z₂ = toNat H xhi * toNat H yhi
h₃ : toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo)
⊢ toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) = toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsubaHelper₁_toNat | [563, 1] | [609, 39] | have h: ∀ {H: List Unit} {a b c d e f: BitTree} [Perfect H a] [Perfect H a] [Perfect H b] [Perfect H c] [Perfect H d] [Perfect H e] [Perfect H f],
(a.toNat H * b.toNat H + c.toNat H * d.toNat H + e.toNat H * f.toNat H) % (bound H * bound H * bound H * bound H) =
a.toNat H * b.toNat H + c.toNat H * d.toNat H + e.toNat H * f.toNat H :=
Nat.mod_eq_of_lt (add_lt_mul_bound
(add_le_mul_bound
(Nat.mul_le_mul
(le_of_lt (toNat_lt _))
(le_of_lt (toNat_lt _)))
(Nat.mul_le_mul
(le_of_lt (toNat_lt _))
(le_of_lt (toNat_lt _))))
(add_lt_mul_bound
(Nat.mul_le_mul
(le_of_lt (toNat_lt _))
(le_of_lt (toNat_lt _)))
(Nat.mul_lt_mul_for_real
(bound_pos _)
(bound_pos _)))) | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
h₀ : toNat (() :: H) z₀ = toNat H xlo * toNat H ylo
h₂ : toNat (() :: H) z₂ = toNat H xhi * toNat H yhi
h₃ : toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo)
⊢ toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) = toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
h₀ : toNat (() :: H) z₀ = toNat H xlo * toNat H ylo
h₂ : toNat (() :: H) z₂ = toNat H xhi * toNat H yhi
h₃ : toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo)
h :
∀ {H : List Unit} {a b c d e f : BitTree} [inst : Perfect H a] [inst : Perfect H a] [inst : Perfect H b]
[inst : Perfect H c] [inst : Perfect H d] [inst : Perfect H e] [inst : Perfect H f],
(toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f
⊢ toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) = toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsubaHelper₁_toNat | [563, 1] | [609, 39] | have h': ∀ {H: List Unit} {a b c d: BitTree} [Perfect H a] [Perfect H b] [Perfect H c] [Perfect H d],
(a.toNat H * b.toNat H + c.toNat H * d.toNat H) % (bound H * bound H * bound H * bound H) =
a.toNat H * b.toNat H + c.toNat H * d.toNat H :=
(congrArg₂ _ (congrArg _ (congrArg₂ _
toNat_zero.symm toNat_zero.symm)) rfl).trans
(h.trans (congrArg _ (congrArg₂ _
toNat_zero toNat_zero))) | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
h₀ : toNat (() :: H) z₀ = toNat H xlo * toNat H ylo
h₂ : toNat (() :: H) z₂ = toNat H xhi * toNat H yhi
h₃ : toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo)
h :
∀ {H : List Unit} {a b c d e f : BitTree} [inst : Perfect H a] [inst : Perfect H a] [inst : Perfect H b]
[inst : Perfect H c] [inst : Perfect H d] [inst : Perfect H e] [inst : Perfect H f],
(toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f
⊢ toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) = toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
h₀ : toNat (() :: H) z₀ = toNat H xlo * toNat H ylo
h₂ : toNat (() :: H) z₂ = toNat H xhi * toNat H yhi
h₃ : toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo)
h :
∀ {H : List Unit} {a b c d e f : BitTree} [inst : Perfect H a] [inst : Perfect H a] [inst : Perfect H b]
[inst : Perfect H c] [inst : Perfect H d] [inst : Perfect H e] [inst : Perfect H f],
(toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f
h' :
∀ {H : List Unit} {a b c d : BitTree} [inst : Perfect H a] [inst : Perfect H b] [inst : Perfect H c]
[inst : Perfect H d],
(toNat H a * toNat H b + toNat H c * toNat H d) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d
⊢ toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) = toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsubaHelper₁_toNat | [563, 1] | [609, 39] | have hs: ∀ {a b c: ℕ}, a + b + c - b = a + c :=
(congrArg₂ _ (Nat.add_right_comm _ _ _) rfl).trans (Nat.add_sub_cancel _ _) | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
h₀ : toNat (() :: H) z₀ = toNat H xlo * toNat H ylo
h₂ : toNat (() :: H) z₂ = toNat H xhi * toNat H yhi
h₃ : toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo)
h :
∀ {H : List Unit} {a b c d e f : BitTree} [inst : Perfect H a] [inst : Perfect H a] [inst : Perfect H b]
[inst : Perfect H c] [inst : Perfect H d] [inst : Perfect H e] [inst : Perfect H f],
(toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f
h' :
∀ {H : List Unit} {a b c d : BitTree} [inst : Perfect H a] [inst : Perfect H b] [inst : Perfect H c]
[inst : Perfect H d],
(toNat H a * toNat H b + toNat H c * toNat H d) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d
⊢ toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) = toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
h₀ : toNat (() :: H) z₀ = toNat H xlo * toNat H ylo
h₂ : toNat (() :: H) z₂ = toNat H xhi * toNat H yhi
h₃ : toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo)
h :
∀ {H : List Unit} {a b c d e f : BitTree} [inst : Perfect H a] [inst : Perfect H a] [inst : Perfect H b]
[inst : Perfect H c] [inst : Perfect H d] [inst : Perfect H e] [inst : Perfect H f],
(toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f
h' :
∀ {H : List Unit} {a b c d : BitTree} [inst : Perfect H a] [inst : Perfect H b] [inst : Perfect H c]
[inst : Perfect H d],
(toNat H a * toNat H b + toNat H c * toNat H d) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d
hs : ∀ {a b c : ℕ}, a + b + c - b = a + c
⊢ toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) = toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsubaHelper₁_toNat | [563, 1] | [609, 39] | simpa only [karatsubaHelper₁,
sub_toNat (()::()::H), h₃, left_distrib, right_distrib, add_comm (xhi.toNat _ * yhi.toNat _), ←
add_assoc, add_right_comm _ (xhi.toNat _ * yhi.toNat _), bound_cons, ← mul_assoc, toNat_node,
toNat_zero, mul_zero, h₂, zero_add, add_tsub_cancel_right, toNat_ofNat, Nat.add_mod_right, h,
h₀, hs, h'] using Nat.add_comm _ _ | H : List Unit
xhi xlo yhi ylo : BitTree
inst✝⁶ : Perfect H xhi
inst✝⁵ : Perfect H xlo
inst✝⁴ : Perfect H yhi
inst✝³ : Perfect H ylo
z₀ z₂ z₃ : BitTree
inst✝² : Perfect (() :: H) z₀
inst✝¹ : Perfect (() :: H) z₂
inst✝ : Perfect (() :: () :: H) z₃
h₀ : toNat (() :: H) z₀ = toNat H xlo * toNat H ylo
h₂ : toNat (() :: H) z₂ = toNat H xhi * toNat H yhi
h₃ : toNat (() :: () :: H) z₃ = (toNat H xhi + toNat H xlo) * (toNat H yhi + toNat H ylo)
h :
∀ {H : List Unit} {a b c d e f : BitTree} [inst : Perfect H a] [inst : Perfect H a] [inst : Perfect H b]
[inst : Perfect H c] [inst : Perfect H d] [inst : Perfect H e] [inst : Perfect H f],
(toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d + toNat H e * toNat H f
h' :
∀ {H : List Unit} {a b c d : BitTree} [inst : Perfect H a] [inst : Perfect H b] [inst : Perfect H c]
[inst : Perfect H d],
(toNat H a * toNat H b + toNat H c * toNat H d) % (bound H * bound H * bound H * bound H) =
toNat H a * toNat H b + toNat H c * toNat H d
hs : ∀ {a b c : ℕ}, a + b + c - b = a + c
⊢ toNat (() :: () :: H) (karatsubaHelper₁ (() :: H) z₀ z₂ z₃) = toNat H xhi * toNat H ylo + toNat H xlo * toNat H yhi | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | have h: ∀ {H: List Unit} {x y: BitTree} [Perfect H x] [Perfect H y], x.toNat H * y.toNat H % (bound (()::H)) = x.toNat H * y.toNat H :=
Nat.mod_eq_of_lt (lt_of_lt_of_eq (Nat.mul_lt_mul_for_real (toNat_lt _) (toNat_lt _)) (bound_cons _).symm) | 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
⊢ karatsuba (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) | 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 (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | have karatsuba_toNat: ∀ {x y} [Perfect H x] [Perfect H y], toNat (()::H) (karatsuba H x y) = x.toNat H * y.toNat H :=
(congrArg _ (karatsuba_eq_mul _ _ _)).trans ((toNat_ofNat _ _).trans 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 (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) | 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
⊢ karatsuba (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | have 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) ((xhi.toNat H + xlo.toNat H) * (yhi.toNat H + ylo.toNat H)) :=
toNat_inj (H := (()::()::H))
(((karatsubaHelper₃_toNat _ _ _ karatsuba_toNat).trans
(congrArg₂ _
(addWithCarry_eq_add₀ _ _)
(addWithCarry_eq_add₀ _ _))).trans
(toNat_ofNat_lt (mul_lt_bound_cons
(add_lt_bound_cons (toNat_lt _) (toNat_lt _))
(add_lt_bound_cons (toNat_lt _) (toNat_lt _)))).symm) | 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
⊢ karatsuba (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) | 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))
⊢ karatsuba (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | have 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) (xlo.toNat H * yhi.toNat H + xhi.toNat H * ylo.toNat H) :=
toNat_inj (H := (()::()::H))
(((karatsubaHelper₁_toNat xhi xlo yhi ylo _ _ _ karatsuba_toNat karatsuba_toNat
(toNat_ofNat_lt (mul_lt_bound_cons
(add_lt_bound_cons (toNat_lt _) (toNat_lt _))
(add_lt_bound_cons (toNat_lt _) (toNat_lt _))))).trans
(add_comm _ _)).trans
(toNat_ofNat_lt (add_lt_bound_cons
(mul_lt_bound_cons (toNat_lt _) (toNat_lt _))
(mul_lt_bound_cons (toNat_lt _) (toNat_lt _)))).symm) | 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))
⊢ karatsuba (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) | 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)
⊢ karatsuba (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | simp only [karatsuba, h₃, h₁, mkPerfect_id, hi, lo] | 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)
⊢ karatsuba (() :: H) (node xhi xlo) (node yhi ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) | 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)
⊢ node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) +
node (zero (() :: H)) (karatsuba H xlo ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Eq.trans | 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)
⊢ node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) +
node (zero (() :: H)) (karatsuba H xlo ylo) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) | case h₁
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)
⊢ node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) +
node (zero (() :: H)) (karatsuba H xlo ylo) =
?b
case h₂
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)
⊢ ?b = ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo))
case b
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)
⊢ BitTree |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply add_toNat (()::()::H) | case h₁
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)
⊢ node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) +
node (zero (() :: H)) (karatsuba H xlo ylo) =
?b
case h₂
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)
⊢ ?b = ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo))
case b
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)
⊢ BitTree | case h₂
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)
⊢ ofNat (() :: () :: H)
(toNat (() :: () :: H)
(node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) +
toNat (() :: () :: H) (node (zero (() :: H)) (karatsuba H xlo ylo))) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply congrArg | case h₂
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)
⊢ ofNat (() :: () :: H)
(toNat (() :: () :: H)
(node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) +
toNat (() :: () :: H) (node (zero (() :: H)) (karatsuba H xlo ylo))) =
ofNat (() :: () :: H) (toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)) | case 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)
⊢ toNat (() :: () :: H)
(node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) +
toNat (() :: () :: H) (node (zero (() :: H)) (karatsuba H xlo ylo)) =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Eq.trans | case 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)
⊢ toNat (() :: () :: H)
(node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) +
toNat (() :: () :: H) (node (zero (() :: H)) (karatsuba H xlo ylo)) =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₁
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 (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) +
toNat (() :: () :: H) (node (zero (() :: H)) (karatsuba H xlo ylo)) =
?h₂.h.b
case h₂.h.h₂
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)
⊢ ?h₂.h.b = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)
case h₂.h.b
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)
⊢ ℕ |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply congrArg₂ | case h₂.h.h₁
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 (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) +
toNat (() :: () :: H) (node (zero (() :: H)) (karatsuba H xlo ylo)) =
?h₂.h.b
case h₂.h.h₂
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)
⊢ ?h₂.h.b = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo)
case h₂.h.b
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)
⊢ ℕ | case h₂.h.h₁.hx
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 (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) =
?h₂.h.h₁.x'
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.x'
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)
⊢ ℕ
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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)
⊢ ?h₂.h.h₁.x' + ?h₂.h.h₁.y' = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply congrArg | case h₂.h.h₁.hx
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 (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) =
?h₂.h.h₁.x'
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.x'
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)
⊢ ℕ
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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)
⊢ ?h₂.h.h₁.x' + ?h₂.h.h₁.y' = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₁.hx.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)
⊢ node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) =
?h₂.h.h₁.hx.a₂
case h₂.h.h₁.hx.a₂
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)
⊢ BitTree
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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) ?h₂.h.h₁.hx.a₂ + ?h₂.h.h₁.y' = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Eq.trans | case h₂.h.h₁.hx.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)
⊢ node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) =
?h₂.h.h₁.hx.a₂
case h₂.h.h₁.hx.a₂
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)
⊢ BitTree
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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) ?h₂.h.h₁.hx.a₂ + ?h₂.h.h₁.y' = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₁.hx.h.h₁
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)
⊢ node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) =
?h₂.h.h₁.hx.h.b
case h₂.h.h₁.hx.h.h₂
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)
⊢ ?h₂.h.h₁.hx.h.b = ?h₂.h.h₁.hx.a₂
case h₂.h.h₁.hx.h.b
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)
⊢ BitTree
case h₂.h.h₁.hx.a₂
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)
⊢ BitTree
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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) ?h₂.h.h₁.hx.a₂ + ?h₂.h.h₁.y' = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply add_toNat (()::()::H) | case h₂.h.h₁.hx.h.h₁
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)
⊢ node (karatsuba H xhi yhi) (zero (() :: H)) +
lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) =
?h₂.h.h₁.hx.h.b
case h₂.h.h₁.hx.h.h₂
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)
⊢ ?h₂.h.h₁.hx.h.b = ?h₂.h.h₁.hx.a₂
case h₂.h.h₁.hx.h.b
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)
⊢ BitTree
case h₂.h.h₁.hx.a₂
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)
⊢ BitTree
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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) ?h₂.h.h₁.hx.a₂ + ?h₂.h.h₁.y' = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₁.hx.h.h₂
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)
⊢ ofNat (() :: () :: H)
(toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) +
toNat (() :: () :: H)
(lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)))) =
?h₂.h.h₁.hx.a₂
case h₂.h.h₁.hx.a₂
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)
⊢ BitTree
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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) ?h₂.h.h₁.hx.a₂ + ?h₂.h.h₁.y' = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply congrArg | case h₂.h.h₁.hx.h.h₂
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)
⊢ ofNat (() :: () :: H)
(toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) +
toNat (() :: () :: H)
(lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)))) =
?h₂.h.h₁.hx.a₂
case h₂.h.h₁.hx.a₂
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)
⊢ BitTree
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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) ?h₂.h.h₁.hx.a₂ + ?h₂.h.h₁.y' = toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₁.hx.h.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)
⊢ toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) +
toNat (() :: () :: H)
(lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) =
?h₂.h.h₁.hx.h.h₂.a₂
case h₂.h.h₁.hx.h.h₂.a₂
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)
⊢ ℕ
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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) (ofNat (() :: () :: H) ?h₂.h.h₁.hx.h.h₂.a₂) + ?h₂.h.h₁.y' =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply congrArg | case h₂.h.h₁.hx.h.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)
⊢ toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) +
toNat (() :: () :: H)
(lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) =
?h₂.h.h₁.hx.h.h₂.a₂
case h₂.h.h₁.hx.h.h₂.a₂
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)
⊢ ℕ
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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) (ofNat (() :: () :: H) ?h₂.h.h₁.hx.h.h₂.a₂) + ?h₂.h.h₁.y' =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₁.hx.h.h₂.h.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)
⊢ toNat (() :: () :: H) (lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) =
?h₂.h.h₁.hx.h.h₂.h.a₂
case h₂.h.h₁.hx.h.h₂.h.a₂
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)
⊢ ℕ
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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)
(ofNat (() :: () :: H)
(toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) + ?h₂.h.h₁.hx.h.h₂.h.a₂)) +
?h₂.h.h₁.y' =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply lshift2_toNat | case h₂.h.h₁.hx.h.h₂.h.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)
⊢ toNat (() :: () :: H) (lshift2 H (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo))) =
?h₂.h.h₁.hx.h.h₂.h.a₂
case h₂.h.h₁.hx.h.h₂.h.a₂
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)
⊢ ℕ
case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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)
(ofNat (() :: () :: H)
(toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) + ?h₂.h.h₁.hx.h.h₂.h.a₂)) +
?h₂.h.h₁.y' =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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)
(ofNat (() :: () :: H)
(toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) +
toNat (() :: () :: H) (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) *
bound H %
bound (() :: () :: H))) +
?h₂.h.h₁.y' =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | rfl | case h₂.h.h₁.hy
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 (zero (() :: H)) (karatsuba H xlo ylo)) = ?h₂.h.h₁.y'
case h₂.h.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂
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)
(ofNat (() :: () :: H)
(toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) +
toNat (() :: () :: H) (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) *
bound H %
bound (() :: () :: H))) +
?h₂.h.h₁.y' =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₂
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)
(ofNat (() :: () :: H)
(toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) +
toNat (() :: () :: H) (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) *
bound H %
bound (() :: () :: H))) +
toNat (() :: () :: H) (node (zero (() :: H)) (karatsuba H xlo ylo)) =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | simp only [toNat_ofNat, toNat_node, toNat_zero, add_zero, karatsuba_toNat,
mul_zero, zero_add, left_distrib, right_distrib,
bound_cons, ← mul_assoc, ← add_assoc,
mul_right_comm _ _ (bound H), mul_comm (toNat _ _) (bound H), mul_comm (_ % _) (bound H),
Nat.add_mod_mod, add_left_inj] | case h₂.h.h₂
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)
(ofNat (() :: () :: H)
(toNat (() :: () :: H) (node (karatsuba H xhi yhi) (zero (() :: H))) +
toNat (() :: () :: H) (ofNat (() :: () :: H) (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo)) *
bound H %
bound (() :: () :: H))) +
toNat (() :: () :: H) (node (zero (() :: H)) (karatsuba H xlo ylo)) =
toNat (() :: H) (node xhi xlo) * toNat (() :: H) (node yhi ylo) | case h₂.h.h₂
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 + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H))) %
(bound H * bound H * bound H * bound H) =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Eq.trans | case h₂.h.h₂
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 + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H))) %
(bound H * bound H * bound H * bound H) =
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 | case h₂.h.h₂.h₁
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 + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H))) %
(bound H * bound H * bound H * bound H) =
?h₂.h.h₂.b
case h₂.h.h₂.h₂
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)
⊢ ?h₂.h.h₂.b =
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
case h₂.h.h₂.b
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)
⊢ ℕ |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply congrArg₂ | case h₂.h.h₂.h₁
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 + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H))) %
(bound H * bound H * bound H * bound H) =
?h₂.h.h₂.b
case h₂.h.h₂.h₂
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)
⊢ ?h₂.h.h₂.b =
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
case h₂.h.h₂.b
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)
⊢ ℕ | case h₂.h.h₂.h₁.hx
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 + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H)) =
?h₂.h.h₂.h₁.x'
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.x'
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)
⊢ ℕ
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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)
⊢ ?h₂.h.h₂.h₁.x' % ?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply congrArg | case h₂.h.h₂.h₁.hx
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 + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H)) =
?h₂.h.h₂.h₁.x'
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.x'
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)
⊢ ℕ
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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)
⊢ ?h₂.h.h₂.h₁.x' % ?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₁.hx.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 * ((toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H)) =
?h₂.h.h₂.h₁.hx.a₂
case h₂.h.h₂.h₁.hx.a₂
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)
⊢ ℕ
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + ?h₂.h.h₂.h₁.hx.a₂) % ?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply congrArg | case h₂.h.h₂.h₁.hx.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 * ((toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H)) =
?h₂.h.h₂.h₁.hx.a₂
case h₂.h.h₂.h₁.hx.a₂
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)
⊢ ℕ
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + ?h₂.h.h₂.h₁.hx.a₂) % ?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₁.hx.h.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)
⊢ (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H) =
?h₂.h.h₂.h₁.hx.h.a₂
case h₂.h.h₂.h₁.hx.h.a₂
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)
⊢ ℕ
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 * ?h₂.h.h₂.h₁.hx.h.a₂) % ?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Nat.mod_eq_of_lt | case h₂.h.h₂.h₁.hx.h.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)
⊢ (toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo) % (bound H * bound H * bound H * bound H) =
?h₂.h.h₂.h₁.hx.h.a₂
case h₂.h.h₂.h₁.hx.h.a₂
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)
⊢ ℕ
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 * ?h₂.h.h₂.h₁.hx.h.a₂) % ?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₁.hx.h.h.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)
⊢ toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo < bound H * bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply add_lt_mul_bound | case h₂.h.h₂.h₁.hx.h.h.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)
⊢ toNat H xlo * toNat H yhi + toNat H xhi * toNat H ylo < bound H * bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₁.hx.h.h.h.ha
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 xlo * toNat H yhi ≤ bound H * bound H * bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply le_trans _ | case h₂.h.h₂.h₁.hx.h.h.h.ha
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 xlo * toNat H yhi ≤ bound H * bound H * bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₁.hx.h.h.h.ha
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)
⊢ ?m.127684 ≤ bound H * bound H * bound 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)
⊢ ℕ
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 xlo * toNat H yhi ≤ ?m.127684
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Nat.le_mul_of_pos_right (bound_pos _) | case h₂.h.h₂.h₁.hx.h.h.h.ha
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)
⊢ ?m.127684 ≤ bound H * bound H * bound 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)
⊢ ℕ
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 xlo * toNat H yhi ≤ ?m.127684
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | 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 xlo * toNat H yhi ≤ bound H * bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply le_of_lt | 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 xlo * toNat H yhi ≤ bound H * bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case 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 xlo * toNat H yhi < bound H * bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Nat.mul_lt_mul_for_real | case 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 xlo * toNat H yhi < bound H * bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case a.hax
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 xlo < bound H
case a.hby
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 yhi < bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply toNat_lt _ | case a.hax
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 xlo < bound H
case a.hby
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 yhi < bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case a.hby
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 yhi < bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply toNat_lt _ | case a.hby
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 yhi < bound H
case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply lt_of_lt_of_le _ | case h₂.h.h₂.h₁.hx.h.h.h.hb
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 xhi * toNat H ylo < bound H * bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₁.hx.h.h.h.hb
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)
⊢ ?m.127933 ≤ bound H * bound H * bound 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)
⊢ ℕ
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 xhi * toNat H ylo < ?m.127933
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Nat.le_mul_of_pos_right (bound_pos _) | case h₂.h.h₂.h₁.hx.h.h.h.hb
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)
⊢ ?m.127933 ≤ bound H * bound H * bound 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)
⊢ ℕ
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 xhi * toNat H ylo < ?m.127933
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | 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 xhi * toNat H ylo < bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Nat.mul_lt_mul_for_real | 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 xhi * toNat H ylo < bound H * bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case hax
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 xhi < bound H
case hby
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 ylo < bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply toNat_lt _ | case hax
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 xhi < bound H
case hby
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 ylo < bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case hby
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 ylo < bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply toNat_lt _ | case hby
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 ylo < bound H
case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | rfl | case h₂.h.h₂.h₁.hy
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 * bound H * bound H = ?h₂.h.h₂.h₁.y'
case h₂.h.h₂.h₁.y'
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)
⊢ ℕ
case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
?h₂.h.h₂.h₁.y' =
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 | case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
(bound H * bound H * bound H * bound H) =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | simp only [left_distrib, ← add_assoc, ← mul_assoc] | case h₂.h.h₂.h₂
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 + toNat H xhi * toNat H ylo)) %
(bound H * bound H * bound H * bound H) =
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 | case h₂.h.h₂.h₂
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) =
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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Nat.mod_eq_of_lt | case h₂.h.h₂.h₂
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) =
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 | 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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply lt_of_le_of_lt (b := (node xhi xlo).toNat (()::H) * (node yhi ylo).toNat (()::H)) | 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 | 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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | { simp [toNat_node, left_distrib, right_distrib, add_assoc, ← mul_assoc, mul_right_comm _ (toNat _ _) (bound _), mul_comm (toNat _ _) (bound _)] } | 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 | 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 |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply lt_of_lt_of_eq _ (mul_assoc _ _ _).symm | 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 | 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) |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply Nat.mul_lt_mul_for_real | 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) | case hax
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) < bound H * bound H
case hby
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 yhi ylo) < bound H * bound H |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply lt_of_lt_of_eq (toNat_lt _) (bound_cons _) | case hax
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) < bound H * bound H
case hby
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 yhi ylo) < bound H * bound H | case hby
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 yhi ylo) < bound H * bound H |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | apply lt_of_lt_of_eq (toNat_lt _) (bound_cons _) | case hby
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 yhi ylo) < bound H * bound H | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | Karatsuba.lean | BitTree.karatsuba_eq_mul | [651, 1] | [730, 51] | simp [toNat_node, left_distrib, right_distrib, add_assoc, ← mul_assoc, mul_right_comm _ (toNat _ _) (bound _), mul_comm (toNat _ _) (bound _)] | 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) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Memory/Basic.lean | HMem.Memory.setm_setm | [133, 9] | [133, 91] | cases b <;> simp | m : Memory
b : Bool
ma₀ ma₁ : Memory
⊢ setm (setm m b ma₀) b ma₁ = setm m b ma₁ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Memory/Basic.lean | HMem.Memory.getv_setv | [135, 1] | [135, 50] | simp | m : Memory
v : Bool
⊢ getv (setv m v) = v | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Memory/Basic.lean | HMem.Memory.getm_setv | [136, 9] | [136, 85] | funext b | m : Memory
v : Bool
⊢ getm (setv m v) = getm m | case h
m : Memory
v b : Bool
⊢ getm (setv m v) b = getm m b |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Memory/Basic.lean | HMem.Memory.getm_setv | [136, 9] | [136, 85] | cases b <;> simp | case h
m : Memory
v b : Bool
⊢ getm (setv m v) b = getm m b | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Memory/Basic.lean | HMem.Memory.getv_setm | [137, 9] | [137, 78] | cases b <;> simp | m : Memory
b : Bool
ma : Memory
⊢ getv (setm m b ma) = getv m | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Memory/Basic.lean | HMem.Memory.getm_setm_eq | [138, 9] | [138, 79] | cases b <;> simp | m : Memory
b : Bool
ma : Memory
⊢ getm (setm m b ma) b = ma | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Memory/Basic.lean | HMem.Memory.setmp_setmp | [156, 9] | [158, 49] | simp [setmp_setmp (bs := bs)] | ma₀ ma₁ m : Memory
b : Bool
bs : List Bool
⊢ setmp (setmp m (b :: bs) ma₀) (b :: bs) ma₁ = setmp m (b :: bs) ma₁ | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Program/Step.lean | HMem.Stack.exit_of_step_result | [88, 1] | [89, 34] | cases is <;> cases cs <;> simp | m : Memory
is p : Program
cs : List (Thunk × List Bool)
r : Memory
⊢ step (execution { state := m, current := is, function := p } cs) = result r → m = r ∧ is = Program.exit ∧ cs = [] | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Program/Step.lean | HMem.Stack.exit_of_step_result' | [91, 1] | [92, 50] | simp [exit_of_step_result h] | cs : List (Thunk × List Bool)
r state✝ : Memory
current✝ function✝ : Program
h : step (execution { state := state✝, current := current✝, function := function✝ } cs) = result r
⊢ execution { state := state✝, current := current✝, function := function✝ } cs =
execution
{ state := r, current := Program.exit,
function := { state := state✝, current := current✝, function := function✝ }.function }
[] | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Program/Step.lean | HMem.Stack.step_eq_execution | [109, 1] | [110, 19] | cases s <;> simp | s : Stack
t : Thunk
cs : List (Thunk × List Bool)
⊢ step s = execution t cs → ∃ t' cs', s = execution t' cs' | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Program/Step.lean | HMem.Stack.step_execution_append | [116, 1] | [118, 83] | cases t₀ with | _ m c p => cases c <;> cases cs₀ <;> simp [List.append_left_inj] | t₀ : Thunk
cs₀ : List (Thunk × List Bool)
t₁ : Thunk
cs₁ cs : List (Thunk × List Bool)
⊢ step (execution t₀ cs₀) = execution t₁ cs₁ → step (execution t₀ (cs₀ ++ cs)) = execution t₁ (cs₁ ++ cs) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Program/Step.lean | HMem.Stack.step_execution_append | [116, 1] | [118, 83] | cases c <;> cases cs₀ <;> simp [List.append_left_inj] | case mk
cs₀ : List (Thunk × List Bool)
t₁ : Thunk
cs₁ cs : List (Thunk × List Bool)
m : Memory
c p : Program
⊢ step (execution { state := m, current := c, function := p } cs₀) = execution t₁ cs₁ →
step (execution { state := m, current := c, function := p } (cs₀ ++ cs)) = execution t₁ (cs₁ ++ cs) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Program/Step.lean | HMem.Stack.istep_execution_append | [120, 1] | [127, 69] | simp | cs : List (Thunk × List Bool)
x✝³ x✝² : Thunk
x✝¹ x✝ : List (Thunk × List Bool)
⊢ step^[0] (execution x✝³ x✝¹) = execution x✝² x✝ → step^[0] (execution x✝³ (x✝¹ ++ cs)) = execution x✝² (x✝ ++ cs) | no goals |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Trace/Cost.lean | HMem.Trace.CostedProgram.splitTimeCost | [244, 1] | [349, 16] | apply Stack.hasTimeCost_subroutine_le _ | α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := m, current := toProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next),
function := toProgram fcp }
[])
((subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a) | case hr
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasResult
(Stack.execution
{ state := Memory.getms m src✝, current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } [])
?m.173014
case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.getms m src✝, current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } [])
?m.173011
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.setms m dst✝ ?m.173014, current := TracedProgram.toProgram (toTracedProgram next),
function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Memory
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Trace/Cost.lean | HMem.Trace.CostedProgram.splitTimeCost | [244, 1] | [349, 16] | rw [TracedProgram.sound_subroutine_arg (TracedProgram.sound_of_sound' h _ _ hm)] | case hr
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasResult
(Stack.execution
{ state := Memory.getms m src✝, current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } [])
?m.173014
case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.getms m src✝, current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } [])
?m.173011
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.setms m dst✝ ?m.173014, current := TracedProgram.toProgram (toTracedProgram next),
function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Memory
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a | case hr
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasResult
(Stack.execution
{
state :=
Complexity.encode
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)),
current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ }
[])
?m.173014
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Memory
case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.getms m src✝, current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } [])
?m.173011
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.setms m dst✝ ?m.173014, current := TracedProgram.toProgram (toTracedProgram next),
function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Memory
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Trace/Cost.lean | HMem.Trace.CostedProgram.splitTimeCost | [244, 1] | [349, 16] | apply hcost.toComputable.has_result | case hr
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasResult
(Stack.execution
{
state :=
Complexity.encode
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)),
current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ }
[])
?m.173014
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Memory
case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.getms m src✝, current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } [])
?m.173011
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.setms m dst✝ ?m.173014, current := TracedProgram.toProgram (toTracedProgram next),
function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Memory
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a | case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.getms m src✝, current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } [])
?m.173011
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{
state :=
Memory.setms m dst✝
(Complexity.encode
(fs✝
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))),
current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Trace/Cost.lean | HMem.Trace.CostedProgram.splitTimeCost | [244, 1] | [349, 16] | rw [TracedProgram.sound_subroutine_arg (TracedProgram.sound_of_sound' h _ _ hm)] | case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{ state := Memory.getms m src✝, current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ } [])
?m.173011
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{
state :=
Memory.setms m dst✝
(Complexity.encode
(fs✝
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))),
current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a | case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{
state :=
Complexity.encode
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)),
current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ }
[])
?m.173011
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{
state :=
Memory.setms m dst✝
(Complexity.encode
(fs✝
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))),
current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a |
https://github.com/calcu16/lean_complexity.git | d267eb4f3952c654f130a119311692c1c1a7cd71 | HMem/Trace/Cost.lean | HMem.Trace.CostedProgram.splitTimeCost | [244, 1] | [349, 16] | apply (Stack.timeCost_le_iff _ _).mpr | case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{
state :=
Complexity.encode
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)),
current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ }
[])
?m.173011
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{
state :=
Memory.setms m dst✝
(Complexity.encode
(fs✝
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))),
current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a | case h₀
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.timeCost
(Stack.execution
{
state :=
Complexity.encode
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)),
current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ }
[])
?m.173938 ≤
?m.173011
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.halts
(Stack.execution
{
state :=
Complexity.encode
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)),
current := Encoding.getProgram fs✝, function := Encoding.getProgram fs✝ }
[])
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
case h₁
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ Stack.hasTimeCost
(Stack.execution
{
state :=
Memory.setms m dst✝
(Complexity.encode
(fs✝
(Option.get (Complexity.decode γ✝ (Memory.getms m src✝))
(_ : Option.isSome (Complexity.decode γ✝ (Memory.getms m src✝)) = true)))),
current := TracedProgram.toProgram (toTracedProgram next), function := toProgram fcp }
[])
?m.173010
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ℕ
α : Type u_1
inst✝⁷ : Complexity.Coding α Memory
β : Type u_2
inst✝⁶ : Complexity.Encoding β Memory
f : α → β
tp : TracedProgram
γ δ : Type
inst✝⁵ : Complexity.Coding γ Memory
inst✝⁴ : Complexity.Encoding δ Memory
fs : γ → δ
cs : Complexity.CostFunction γ ℕ
inst✝³ : Complexity.Computable Encoding.Model fs
inst✝² : Complexity Encoding.Model fs cs
size : α → ℕ
fm : α → Option Memory
fcp : CostedProgram
hf : Program.sound (TracedProgram.toProgram (toTracedProgram fcp)) f size
dst✝ src✝ : Source
γ✝ δ✝ : Type
inst✝¹ : Complexity.Coding γ✝ Memory
inst✝ : Complexity.Encoding δ✝ Memory
fs✝ : γ✝ → δ✝
cs✝ : Complexity.CostFunction γ✝ ℕ
hcost : Complexity Encoding.Model fs✝ cs✝
next : CostedProgram
h : TracedProgram.sound' f size (toTracedProgram (subroutine' dst✝ src✝ inst✝¹ inst✝ fs✝ cs✝ hcost next)) fm
a : α
m : Memory
hm : m ∈ fm a
⊢ ?m.173010 + ?m.173011 + 1 ≤ (subroutineTimeCost h + recurseTimeCost hf h + localTimeCost h) a |
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