43 lines
1.8 KiB
Plaintext
43 lines
1.8 KiB
Plaintext
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/--
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@[ext] structure Cut where
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S : Set ℚ
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not_all : S ≠ Set.univ
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nonempty : S.Nonempty
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lt_elem x y : x ∈ S → y < x → y ∈ S
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no_max x : x ∈ S → ∃ y ∈ S, x < y
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--/
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-- structure == inductive, but structure has one constructor. it is all these things it's an and, it's like a named tuple or record in coq, structs in C
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structure Cut where
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carrier: Set ℚ -- carrier == α in Rudin, the actual cut
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not_empty: carrier ≠ {} -- α ≠ ∅ (I)
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not_all: carrier ≠ Set.univ -- alpha ≠ Set ℚ, carrier ≠ { _a | True} (I)
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p_less_q_in_carrier: ∀ p ∈ carrier, ∀ q : ℚ, q < p -> q ∈ carrier -- (II)
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p_less_q_in_carrier': (p : ℚ) -> p ∈ carrier -> (q : ℚ) -> q < p -> q ∈ carrier -- (III)
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always_exist_a_larger_element: ∀ p ∈ carrier, ∃ r ∈ carrier, p < r -- (III')
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if_q_not_in_carrier_then_grater_than_elem_in_carrier: ∀ p ∈ carrier, ∀ q : ℚ, (q ∉ carrier) → p < q
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-- Now let's debug/test the above structure we defined
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-- 1. let's proof some subset of Q is a (Dedikin) Cut
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-- 2. thm, rfl proof of that p_less_q_in_carrier' == p_less_q_in_carrier'
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-- 3. continue in Rudin!
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-- 4. Show (II) implies --> (IV) & (V)
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def A : Set ℚ := { x | x < 3}
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theorem A_is_a_cut : Cut d
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{
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carrier := A,
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not_empty := sorry, -- Proof that A is not empty
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not_all := sorry, -- Proof that A is not the set of all rationals
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p_less_q_in_carrier := sorry, -- Proof of ∀ p ∈ A, ∀ q : ℚ, q < p -> q ∈ A
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p_less_q_in_carrier' := sorry, -- Alternative proof of the same property
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always_exist_a_larger_element := sorry, -- Proof of ∀ p ∈ A, ∃ r ∈ A, p < r
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if_q_not_in_carrier_then_grater_than_elem_in_carrier := sorry -- Proof of ∀ p ∈ A, ∀ q : ℚ, (q ∉ A) → p < q
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}
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/--
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include private proofs discussed with scott
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--/
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