Idris2/samples/proofs/induction.idr

nat_induction :
    (prop : Nat -> Type) ->                -- Property to show
    (prop Z) ->                            -- Base case
    ((k : Nat) -> prop k -> prop (S k)) -> -- Inductive step
    (x : Nat) ->                           -- Show for all x
    prop x
nat_induction prop p_Z p_S Z = p_Z
nat_induction prop p_Z p_S (S k) = p_S k (nat_induction prop p_Z p_S k)

plus_ind : Nat -> Nat -> Nat
plus_ind n m
   = nat_induction (\x => Nat)
                   m                      -- Base case, plus_ind Z m
                   (\k, k_rec => S k_rec) -- Inductive step plus_ind (S k) m
                                          -- where k_rec = plus_ind k m
                   n
Remove trailing whitespaces and add trailing newlines 2021-01-16 10:03:45 +03:00			`nat_induction :`
Copy samples directory from Idris2-boot This is referred to in the documentation, so should be there 2020-06-30 12:51:09 +03:00			`(prop : Nat -> Type) -> -- Property to show`
			`(prop Z) -> -- Base case`
			`((k : Nat) -> prop k -> prop (S k)) -> -- Inductive step`
			`(x : Nat) -> -- Show for all x`
			`prop x`
			`nat_induction prop p_Z p_S Z = p_Z`
			`nat_induction prop p_Z p_S (S k) = p_S k (nat_induction prop p_Z p_S k)`

			`plus_ind : Nat -> Nat -> Nat`
			`plus_ind n m`
			`= nat_induction (\x => Nat)`
			`m -- Base case, plus_ind Z m`
			`(\k, k_rec => S k_rec) -- Inductive step plus_ind (S k) m`
			`-- where k_rec = plus_ind k m`
			`n`