37 lines
1.3 KiB
Text
37 lines
1.3 KiB
Text
|
import macros -- loads the take, assume, obtain macros
|
||
|
|
|
||
|
|
using Nat -- using the Nat namespace (it allows us to suppress the Nat:: prefix)
|
||
|
|
|
||
|
|
axiom Induction : ∀ P : Nat → Bool, P 0 ⇒ (∀ n, P n ⇒ P (n + 1)) ⇒ ∀ n, P n.
|
||
|
|
|
||
|
|
-- induction on n
|
||
|
|
|
||
|
|
theorem Comm1 : ∀ n m, n + m = m + n
|
||
|
|
:= Induction
|
||
|
|
◂ _ -- I use a placeholder because I do not want to write the P
|
||
|
|
◂ (take m, -- Base case
|
||
|
|
calc 0 + m = m : add::zerol m
|
||
|
|
... = m + 0 : symm (add::zeror m))
|
||
|
|
◂ (take n, -- Inductive case
|
||
|
|
assume (iH : ∀ m, n + m = m + n),
|
||
|
|
take m,
|
||
|
|
calc n + 1 + m = (n + m) + 1 : add::succl n m
|
||
|
|
... = (m + n) + 1 : { iH ◂ m }
|
||
|
|
... = m + (n + 1) : symm (add::succr m n))
|
||
|
|
|
||
|
|
-- indunction on m
|
||
|
|
|
||
|
|
theorem Comm2 : ∀ n m, n + m = m + n
|
||
|
|
:= take n,
|
||
|
|
Induction
|
||
|
|
◂ _
|
||
|
|
◂ (calc n + 0 = n : add::zeror n
|
||
|
|
... = 0 + n : symm (add::zerol n))
|
||
|
|
◂ (take m,
|
||
|
|
assume (iH : n + m = m + n),
|
||
|
|
calc n + (m + 1) = (n + m) + 1 : add::succr n m
|
||
|
|
... = (m + n) + 1 : { iH }
|
||
|
|
... = (m + 1) + n : symm (add::succl m n))
|
||
|
|
|
||
|
|
|
||
|
|
print environment 1
|