143 lines
4.3 KiB
Text
143 lines
4.3 KiB
Text
import Lean
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open Lean
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open Lean.Meta
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open Lean.Elab.Tactic
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universes u
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axiom elimEx (motive : Nat → Nat → Sort u) (x y : Nat)
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(diag : (a : Nat) → motive a a)
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(upper : (delta a : Nat) → motive a (a + delta.succ))
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(lower : (delta a : Nat) → motive (a + delta.succ) a)
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: motive y x
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theorem ex1 (p q : Nat) : p ≤ q ∨ p > q := by
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cases p, q using elimEx with
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| diag => apply Or.inl; apply Nat.leRefl
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| lower d => apply Or.inl; show p ≤ p + d.succ; admit
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| upper d => apply Or.inr; show q + d.succ > q; admit
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theorem ex2 (p q : Nat) : p ≤ q ∨ p > q := by
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cases p, q using elimEx
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case lower => admit
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case upper => admit
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case diag => apply Or.inl; apply Nat.leRefl
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axiom Nat.parityElim (motive : Nat → Sort u)
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(even : (n : Nat) → motive (2*n))
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(odd : (n : Nat) → motive (2*n+1))
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(n : Nat)
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: motive n
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theorem time2Eq (n : Nat) : 2*n = n + n := by
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rw Nat.mul_comm
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show (0 + n) + n = n+n
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simp
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theorem ex3 (n : Nat) : Exists (fun m => n = m + m ∨ n = m + m + 1) := by
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cases n using Nat.parityElim with
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| even i =>
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apply Exists.intro i
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apply Or.inl
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rw time2Eq
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| odd i =>
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apply Exists.intro i
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apply Or.inr
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rw time2Eq
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open Nat in
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theorem ex3b (n : Nat) : Exists (fun m => n = m + m ∨ n = m + m + 1) := by
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cases n using parityElim with
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| even i =>
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apply Exists.intro i
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apply Or.inl
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rw time2Eq
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| odd i =>
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apply Exists.intro i
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apply Or.inr
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rw time2Eq
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def ex4 {α} (xs : List α) (h : xs = [] → False) : α := by
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cases he:xs with
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| nil => contradiction
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| cons x _ => exact x
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def ex5 {α} (xs : List α) (h : xs = [] → False) : α := by
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cases he:xs using List.casesOn with
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| nil => contradiction
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| cons x _ => exact x
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theorem ex6 {α} (f : List α → Bool) (h₁ : {xs : List α} → f xs = true → xs = []) (xs : List α) (h₂ : xs ≠ []) : f xs = false :=
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match he:f xs with
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| true => False.elim (h₂ (h₁ he))
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| false => rfl
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theorem ex7 {α} (f : List α → Bool) (h₁ : {xs : List α} → f xs = true → xs = []) (xs : List α) (h₂ : xs ≠ []) : f xs = false := by
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cases he:f xs with
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| true => exact False.elim (h₂ (h₁ he))
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| false => rfl
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theorem ex8 {α} (f : List α → Bool) (h₁ : {xs : List α} → f xs = true → xs = []) (xs : List α) (h₂ : xs ≠ []) : f xs = false := by
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cases he:f xs using Bool.casesOn with
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| true => exact False.elim (h₂ (h₁ he))
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| false => rfl
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theorem ex9 (xs : List α) (h : xs = [] → False) : Nonempty α := by
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cases xs using List.rec with
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| nil => contradiction
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| cons x _ => apply Nonempty.intro; assumption
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theorem modLt (x : Nat) {y : Nat} (h : y > 0) : x % y < y := by
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induction x, y using Nat.mod.inductionOn with
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| ind x y h₁ ih =>
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rw [Nat.mod_eq_sub_mod h₁.2]
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exact ih h
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| base x y h₁ =>
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match Iff.mp (Decidable.notAndIffOrNot ..) h₁ with
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| Or.inl h₁ => contradiction
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| Or.inr h₁ =>
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have hgt := Nat.gtOfNotLe h₁
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have heq := Nat.mod_eq_of_lt hgt
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rw [← heq] at hgt
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assumption
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theorem ex11 {p q : Prop } (h : p ∨ q) : q ∨ p := by
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induction h using Or.casesOn with
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| inr h => ?myright
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| inl h => ?myleft
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case myleft => exact Or.inr h
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case myright => exact Or.inl h
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theorem ex12 {p q : Prop } (h : p ∨ q) : q ∨ p := by
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cases h using Or.casesOn with
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| inr h => ?myright
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| inl h => ?myleft
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case myleft => exact Or.inr h
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case myright => exact Or.inl h
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theorem ex13 (p q : Nat) : p ≤ q ∨ p > q := by
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cases p, q using elimEx with
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| diag => ?hdiag
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| lower d => ?hlower
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| upper d => ?hupper
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case hdiag => apply Or.inl; apply Nat.leRefl
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case hlower => apply Or.inl; show p ≤ p + d.succ; admit
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case hupper => apply Or.inr; show q + d.succ > q; admit
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theorem ex14 (p q : Nat) : p ≤ q ∨ p > q := by
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cases p, q using elimEx with
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| diag => ?hdiag
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| lower d => _
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| upper d => ?hupper
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case hdiag => apply Or.inl; apply Nat.leRefl
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case lower => apply Or.inl; show p ≤ p + d.succ; admit
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case hupper => apply Or.inr; show q + d.succ > q; admit
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theorem ex15 (p q : Nat) : p ≤ q ∨ p > q := by
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cases p, q using elimEx with
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| diag => ?hdiag
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| lower d => _
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| upper d => ?hupper
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{ apply Or.inl; apply Nat.leRefl }
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{ apply Or.inr; show q + d.succ > q; admit }
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{ apply Or.inl; show p ≤ p + d.succ; admit }
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