49 lines
1.5 KiB
Text
49 lines
1.5 KiB
Text
variable A : Type.{1}
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definition bool [inline] : Type.{1} := Type.{0}
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variable eq : A → A → bool
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infixl `=` 50 := eq
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axiom subst (P : A → bool) (a b : A) (H1 : a = b) (H2 : P a) : P b
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axiom eq_trans (a b c : A) (H1 : a = b) (H2 : b = c) : a = c
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axiom eq_refl (a : A) : a = a
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variable le : A → A → bool
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infixl `≤` 50 := le
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axiom le_trans (a b c : A) (H1 : a ≤ b) (H2 : b ≤ c) : a ≤ c
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axiom le_refl (a : A) : a ≤ a
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axiom eq_le_trans (a b c : A) (H1 : a = b) (H2 : b ≤ c) : a ≤ c
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axiom le_eq_trans (a b c : A) (H1 : a ≤ b) (H2 : b = c) : a ≤ c
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calc_subst subst
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calc_refl eq_refl
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calc_refl le_refl
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calc_trans eq_trans
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calc_trans le_trans
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calc_trans eq_le_trans
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calc_trans le_eq_trans
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variables a b c d e f : A
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axiom H1 : a = b
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axiom H2 : b ≤ c
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axiom H3 : c ≤ d
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axiom H4 : d = e
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check calc a = b : H1
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... ≤ c : H2
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... ≤ d : H3
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... = e : H4
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variable lt : A → A → bool
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infixl `<` 50 := lt
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axiom lt_trans (a b c : A) (H1 : a < b) (H2 : b < c) : a < c
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axiom le_lt_trans (a b c : A) (H1 : a ≤ b) (H2 : b < c) : a < c
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axiom lt_le_trans (a b c : A) (H1 : a < b) (H2 : b ≤ c) : a < c
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axiom H5 : c < d
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check calc b ≤ c : H2
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... < d : H5 -- Error le_lt_trans was not registered yet
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calc_trans le_lt_trans
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check calc b ≤ c : H2
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... < d : H5
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variable le2 : A → A → bool
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infixl `≤` 50 := le2
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variable le2_trans (a b c : A) (H1 : le2 a b) (H2 : le2 b c) : le2 a c
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calc_trans le2_trans
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print raw calc b ≤ c : H2
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... ≤ d : H3
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... ≤ e : H4
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