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