Fill in some of 3.5, fix numbering #268
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gaearon
commented
Aug 4, 2025
| @@ -67,7 +67,7 @@ abbrev SetTheory.Set.slice (x:Object) (Y:Set) : Set := | |||
| theorem SetTheory.Set.mem_slice (x z:Object) (Y:Set) : | |||
| z ∈ (SetTheory.Set.slice x Y) ↔ ∃ y:Y, z = (⟨x, y⟩:OrderedPair) := replacement_axiom _ _ | |||
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| /-- Definition 3.5.2 (Cartesian product) -/ | |||
| /-- Definition 3.5.4 (Cartesian product) -/ | |||
| apply ext; | ||
| aesop | ||
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| /-- Example 3.5.5 / Exercise 3.6.5. There is a bijection between `X ×ˢ Y` and `Y ×ˢ X`. -/ |
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Calling out the future exercise explicitly so it's clear why we're not filling this in.
| invFun := sorry | ||
| left_inv := sorry | ||
| right_inv := sorry | ||
| /-- Example 3.5.5. A function of two variables can be thought of as a function of a pair. -/ |
gaearon
commented
Aug 4, 2025
| @@ -180,8 +172,27 @@ theorem SetTheory.Set.tuple_mem_iProd {I: Set} {X: I → Set} (a: ∀ i, X i) : | |||
| theorem SetTheory.Set.tuple_inj {I:Set} {X: I → Set} (a b: ∀ i, X i) : | |||
| tuple a = tuple b ↔ a = b := by sorry | |||
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| /-- Example 3.5.8. There is a bijection between `(X ×ˢ Y) ×ˢ Z` and `X ×ˢ (Y ×ˢ Z)`. -/ | |||
| noncomputable abbrev SetTheory.Set.prod_associator (X Y Z:Set) : (X ×ˢ Y) ×ˢ Z ≃ X ×ˢ (Y ×ˢ Z) where | |||
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I've fully filled this in because it wasn't a "why" and it wasn't part of an exercise.
gaearon
commented
Aug 4, 2025
| @@ -151,20 +130,45 @@ noncomputable abbrev SetTheory.Set.prod_associator (X Y Z:Set) : (X ×ˢ Y) ×ˢ | |||
| noncomputable abbrev SetTheory.Set.prod_equiv_prod (X Y:Set) : | |||
| ((X ×ˢ Y):_root_.Set Object) ≃ (X:_root_.Set Object) ×ˢ (Y:_root_.Set Object) where | |||
| toFun := fun z ↦ ⟨(fst z, snd z), by simp⟩ | |||
| invFun := fun z ↦ mk_cartesian ⟨z.val.1, z.prop.1⟩ ⟨z.val.2, z.prop.2⟩ | |||
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Filled these in because it's auxiliary to the book.
gaearon
commented
Aug 4, 2025
| invFun := sorry | ||
| left_inv := sorry | ||
| right_inv := sorry | ||
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| /-- Example 3.5.5. A function of two variables can be thought of as a function of a pair. -/ | ||
| noncomputable abbrev SetTheory.Set.curry_equiv {X Y Z:Set} : (X → Y → Z) ≃ (X ×ˢ Y → Z) where |
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Filled this in because it isn't a "why" or an exercise.
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There's a few changes in here:
mem_insertlemma so we can unwrap >3-item sets without mentioning the instance.curry,uncurry,curry_uncurry, anduncurry_curryinto an equivalence. This matches how other equivalences are shown and is IMO a bit more intuitive. It took me a while to see why they're defined at all.