53 lines
1.4 KiB
Text
53 lines
1.4 KiB
Text
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import tactic
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universe U >= 2
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variable f (A : (Type 1)) : (Type 1)
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axiom Ax1 (a : Type) : f a = a
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rewrite_set S
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add_rewrite Ax1 eq_id : S
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theorem T1a (A : Type) : f A = A
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:= by simp S
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-- The following theorem should not be provable.
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-- The axiom Ax1 is only for arguments convertible to Type (i.e., Type 0)
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-- The argument A in the following theorem lives in (Type 1)
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theorem T1b (A : (Type 1)) : f A = A
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:= by simp S
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variable g (A : Type → (Type 1)) : (Type 1)
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axiom Ax2 (a : Type → Type) : g a = a Bool
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add_rewrite Ax2 : S
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theorem T2a (A : Type → Type) : g A = A Bool
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:= by simp S
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-- The following theorem should not be provable.
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-- See T1b
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theorem T2b (A : Type → (Type 1)) : g A = A Bool
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:= by simp S
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variable h (A : Type) (B : (Type 1)) : (Type 1)
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axiom Ax3 (a : Type) : h a a = a
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add_rewrite Ax3 : S
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theorem T3a (A : Type) : h A A = A
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:= by simp S
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axiom Ax4 (a b : Type) : h a b = b
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add_rewrite Ax4 : S
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theorem T4a (A : Type) (B : Type) : h A B = B
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:= by simp S
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-- The following theorem should not be provable.
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-- See T1b
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theorem T4b (A : Type) (B : (Type 1)) : h A B = B
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:= by simp S
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variable h2 (A : Type) (B : (Type 1)) : Type
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axiom Ax5 (a b : Type) : f a = a -> h2 a b = a
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add_rewrite Ax5 : S
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theorem T5a (A B : Type) : h2 A B = A
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:= by simp S
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-- The following theorem should not be provable.
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-- See T1b
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theorem T5b (A : Type) (B : (Type 1)) : h2 A B = A
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:= by simp S
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print environment 1
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