49 lines
1.6 KiB
Text
49 lines
1.6 KiB
Text
prelude constant A : Type.{1}
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definition bool : Type.{1} := Type.{0}
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constant 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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constant 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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attribute [subst] subst
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attribute [refl] eq_refl
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attribute [refl] le_refl
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attribute [trans] eq_trans
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attribute [trans] le_trans
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attribute [trans] eq_le_trans
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attribute [trans] le_eq_trans
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constants 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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constant 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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attribute [trans] le_lt_trans
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check calc b ≤ c : H2
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... < d : H5
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constant le2 : A → A → bool
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infixl ` ≤ `:50 := le2
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constant le2_trans (a b c : A) (H1 : le2 a b) (H2 : le2 b c) : le2 a c
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attribute [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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