import macros import subtype using subtype namespace num theorem inhabited_ind : inhabited ind -- We use as the witness for non-emptiness, the value w in ind that is not convered by f. := obtain f His, from infinity, obtain w Hw, from and_elimr His, inhabited_intro w definition S := ε (inhabited_ex_intro infinity) (λ f, injective f ∧ non_surjective f) definition Z := ε inhabited_ind (λ y, ∀ x, ¬ S x = y) theorem injective_S : injective S := and_eliml (exists_to_eps infinity) theorem non_surjective_S : non_surjective S := and_elimr (exists_to_eps infinity) theorem S_ne_Z (i : ind) : S i ≠ Z := obtain w Hw, from non_surjective_S, eps_ax inhabited_ind w Hw i definition N (i : ind) : Bool := ∀ P, P Z → (∀ x, P x → P (S x)) → P i theorem N_Z : N Z := λ P Hz Hi, Hz theorem N_S {i : ind} (H : N i) : N (S i) := λ P Hz Hi, Hi i (H P Hz Hi) theorem N_smallest : ∀ P : ind → Bool, P Z → (∀ x, P x → P (S x)) → (∀ i, N i → P i) := λ P Hz Hi i Hni, Hni P Hz Hi definition num := subtype ind N theorem inhab : inhabited num := subtype_inhabited (exists_intro Z N_Z) definition zero : num := abst Z inhab theorem zero_pred : N Z := N_Z definition succ (n : num) : num := abst (S (rep n)) inhab theorem succ_pred (n : num) : N (S (rep n)) := have N_n : N (rep n), from P_rep n, show N (S (rep n)), from N_S N_n theorem succ_inj (a b : num) : succ a = succ b → a = b := assume Heq1 : succ a = succ b, have Heq2 : S (rep a) = S (rep b), from abst_inj inhab (succ_pred a) (succ_pred b) Heq1, have rep_eq : (rep a) = (rep b), from injective_S (rep a) (rep b) Heq2, show a = b, from rep_inj rep_eq theorem succ_nz (a : num) : succ a ≠ zero := assume R : succ a = zero, have Heq1 : S (rep a) = Z, from abst_inj inhab (succ_pred a) zero_pred R, show false, from absurd Heq1 (S_ne_Z (rep a)) theorem induction {P : num → Bool} (H1 : P zero) (H2 : ∀ n, P n → P (succ n)) : ∀ a, P a := take a, let Q := λ x, N x ∧ P (abst x inhab) in have QZ : Q Z, from and_intro zero_pred H1, have QS : ∀ x, Q x → Q (S x), from take x, assume Qx, have Hp1 : P (succ (abst x inhab)), from H2 (abst x inhab) (and_elimr Qx), have Hp2 : P (abst (S (rep (abst x inhab))) inhab), from Hp1, have Nx : N x, from and_eliml Qx, have rep_eq : rep (abst x inhab) = x, from rep_abst inhab x Nx, show Q (S x), from and_intro (N_S Nx) (subst Hp2 rep_eq), have Qa : P (abst (rep a) inhab), from and_elimr (N_smallest Q QZ QS (rep a) (P_rep a)), have abst_eq : abst (rep a) inhab = a, from abst_rep inhab a, show P a, from subst Qa abst_eq set_opaque num true set_opaque Z true set_opaque S true set_opaque zero true set_opaque succ true end definition num := num::num