57 lines
2.7 KiB
Text
57 lines
2.7 KiB
Text
import macros
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-- Some theorems from Pricipia Mathematica
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theorem p1 {A : TypeU} (p : Bool) (φ : A → Bool) : (∀ x, p ∨ φ x) = (p ∨ ∀ x, φ x)
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:= boolext
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(assume H : (∀ x, p ∨ φ x),
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or_elim (em p)
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(λ Hp : p, or_introl Hp (∀ x, φ x))
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(λ Hnp : ¬ p, or_intror p (take x,
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resolve1 (H x) Hnp)))
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(assume H : (p ∨ ∀ x, φ x),
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take x,
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or_elim H
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(λ H1 : p, or_introl H1 (φ x))
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(λ H2 : (∀ x, φ x), or_intror p (H2 x)))
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theorem p2 {A : TypeU} (p : Bool) (φ : A → Bool) : (∀ x, φ x ∨ p) = ((∀ x, φ x) ∨ p)
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:= calc (∀ x, φ x ∨ p) = (∀ x, p ∨ φ x) : allext (λ x, or_comm (φ x) p)
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... = (p ∨ ∀ x, φ x) : p1 p φ
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... = ((∀ x, φ x) ∨ p) : or_comm p (∀ x, φ x)
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theorem p3 {A : TypeU} (φ ψ : A → Bool) : (∀ x, φ x ∧ ψ x) = ((∀ x, φ x) ∧ (∀ x, ψ x))
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:= boolext
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(assume H : (∀ x, φ x ∧ ψ x),
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and_intro (take x, and_eliml (H x)) (take x, and_elimr (H x)))
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(assume H : (∀ x, φ x) ∧ (∀ x, ψ x),
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take x, and_intro (and_eliml H x) (and_elimr H x))
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theorem p4 {A : TypeU} (p : Bool) (φ : A → Bool) : (∃ x, p ∧ φ x) = (p ∧ ∃ x, φ x)
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:= boolext
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(assume H : (∃ x, p ∧ φ x),
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obtain (w : A) (Hw : p ∧ φ w), from H,
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and_intro (and_eliml Hw) (exists_intro w (and_elimr Hw)))
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(assume H : (p ∧ ∃ x, φ x),
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obtain (w : A) (Hw : φ w), from (and_elimr H),
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exists_intro w (and_intro (and_eliml H) Hw))
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theorem p5 {A : TypeU} (p : Bool) (φ : A → Bool) : (∃ x, φ x ∧ p) = ((∃ x, φ x) ∧ p)
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:= calc (∃ x, φ x ∧ p) = (∃ x, p ∧ φ x) : eq_exists_intro (λ x, and_comm (φ x) p)
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... = (p ∧ (∃ x, φ x)) : p4 p φ
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... = ((∃ x, φ x) ∧ p) : and_comm p (∃ x, φ x)
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theorem p6 {A : TypeU} (φ ψ : A → Bool) : (∃ x, φ x ∨ ψ x) = ((∃ x, φ x) ∨ (∃ x, ψ x))
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:= boolext
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(assume H : (∃ x, φ x ∨ ψ x),
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obtain (w : A) (Hw : φ w ∨ ψ w), from H,
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or_elim Hw
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(λ Hw1 : φ w, or_introl (exists_intro w Hw1) (∃ x, ψ x))
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(λ Hw2 : ψ w, or_intror (∃ x, φ x) (exists_intro w Hw2)))
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(assume H : (∃ x, φ x) ∨ (∃ x, ψ x),
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or_elim H
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(λ H1 : (∃ x, φ x),
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obtain (w : A) (Hw : φ w), from H1,
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exists_intro w (or_introl Hw (ψ w)))
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(λ H2 : (∃ x, ψ x),
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obtain (w : A) (Hw : ψ w), from H2,
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exists_intro w (or_intror (φ w) Hw)))
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