. . . . . . . . Formalizing a sophisticated defjnition Patrick Massot (Orsay) joint work with Kevin Buzzard (IC London) and Johan Commelin (Freiburg) Formal Methods in Mathematics – Lean Together January 7th 2020
ℝ 𝑟 gives a limit 𝑧 . Set . ̄ Example: (+) ∶ ℚ × ℚ → ℚ ⊂ ℝ extends to (+) ∶ ℝ × ℝ → ℝ . 𝑔 . ̄ 𝑔(𝑦) = 𝑧 . Then prove continuity of ̄ Uniform continuity of 𝑔 ensures 𝑔 ∘ 𝑏 is Cauchy, completeness of For every 𝑦 ∈ ℝ 𝑞 , choose a sequence 𝑏 ∶ ℕ → 𝐵 converging to 𝑦 . But multiplication or inversion are not uniformly continuous. . Theorem Extending functions . . . . . . Let 𝐵 ⊂ ℝ 𝑞 be a dense subset. Every uniformly continuous function 𝑔 ∶ 𝐵 → ℝ 𝑟 extends to a (uniformly) continuous function 𝑔 ∶ ℝ 𝑞 → ℝ 𝑟 .
. . Example: (+) ∶ ℚ × ℚ → ℚ ⊂ ℝ extends to (+) ∶ ℝ × ℝ → ℝ . 𝑔 . ̄ 𝑔(𝑦) = 𝑧 . Then prove continuity of ̄ Uniform continuity of 𝑔 ensures 𝑔 ∘ 𝑏 is Cauchy, completeness of For every 𝑦 ∈ ℝ 𝑞 , choose a sequence 𝑏 ∶ ℕ → 𝐵 converging to 𝑦 . ̄ But multiplication or inversion are not uniformly continuous. Theorem Extending functions . . . . . . Let 𝐵 ⊂ ℝ 𝑞 be a dense subset. Every uniformly continuous function 𝑔 ∶ 𝐵 → ℝ 𝑟 extends to a (uniformly) continuous function 𝑔 ∶ ℝ 𝑞 → ℝ 𝑟 . ℝ 𝑟 gives a limit 𝑧 . Set
. . Example: (+) ∶ ℚ × ℚ → ℚ ⊂ ℝ extends to (+) ∶ ℝ × ℝ → ℝ . 𝑔 . ̄ 𝑔(𝑦) = 𝑧 . Then prove continuity of ̄ Uniform continuity of 𝑔 ensures 𝑔 ∘ 𝑏 is Cauchy, completeness of For every 𝑦 ∈ ℝ 𝑞 , choose a sequence 𝑏 ∶ ℕ → 𝐵 converging to 𝑦 . ̄ But multiplication or inversion are not uniformly continuous. Theorem Extending functions . . . . . . Let 𝐵 ⊂ ℝ 𝑞 be a dense subset. Every uniformly continuous function 𝑔 ∶ 𝐵 → ℝ 𝑟 extends to a (uniformly) continuous function 𝑔 ∶ ℝ 𝑞 → ℝ 𝑟 . ℝ 𝑟 gives a limit 𝑧 . Set
. . . . . . . . Theorem then 𝑔 extends to a continuous function ̄ This applies to multiplication ℚ × ℚ → ℝ . 𝐵 ⊂ ℝ 𝑞 dense subset. If 𝑔 ∶ 𝐵 → ℝ 𝑟 is continuous and ∀𝑦 ∈ ℝ 𝑞 , ∃𝑧 ∈ ℝ 𝑟 , ∀𝑣 ∶ ℕ → 𝐵, 𝑣 𝑜 ⟶ 𝑦 ⇒ 𝑔(𝑣 𝑜 ) ⟶ 𝑧 𝑔 ∶ ℝ 𝑞 → ℝ 𝑟 .
We can still say that 𝑔(𝑦) converges to 𝑧 when 𝑦 tends to 𝑦 0 while for each 𝑦 0 ∈ 𝑌 , 𝑔(𝑦) tends to a limit in 𝑍 when 𝑦 tends to 𝑦 0 . ̄ while remaining in 𝐵 then 𝑔 extends to a continuous map 𝑔 ∶ 𝐵 → 𝑍 a continuous mapping of 𝐵 into a regular space 𝑍 . If, Let 𝑌 be a topological space, 𝐵 a dense subset of 𝑌 , and Theorem ∀𝑋 ∈ 𝒪 𝑧 , ∃𝑊 ∈ 𝒪 𝑦 , ∀𝑏 ∈ 𝐵 ∩ 𝑊 , 𝑔(𝑏) ∈ 𝑋. remaining in 𝐵 : topological spaces 𝑌 and 𝑍 . . A better framework? . . . . . . 𝑔 ∶ 𝑌 → 𝑍 In order to handle inversion ℚ ∗ → ℝ ∗ and more general spaces, we want a version where ℝ 𝑞 and ℝ 𝑟 are replaced by general
for each 𝑦 0 ∈ 𝑌 , 𝑔(𝑦) tends to a limit in 𝑍 when 𝑦 tends to 𝑦 0 . . ̄ while remaining in 𝐵 then 𝑔 extends to a continuous map 𝑔 ∶ 𝐵 → 𝑍 a continuous mapping of 𝐵 into a regular space 𝑍 . If, Let 𝑌 be a topological space, 𝐵 a dense subset of 𝑌 , and Theorem ∀𝑋 ∈ 𝒪 𝑧 , ∃𝑊 ∈ 𝒪 𝑦 , ∀𝑏 ∈ 𝐵 ∩ 𝑊 , 𝑔(𝑏) ∈ 𝑋. remaining in 𝐵 : topological spaces 𝑌 and 𝑍 . A better framework? . . . . . . 𝑔 ∶ 𝑌 → 𝑍 In order to handle inversion ℚ ∗ → ℝ ∗ and more general spaces, we want a version where ℝ 𝑞 and ℝ 𝑟 are replaced by general We can still say that 𝑔(𝑦) converges to 𝑧 when 𝑦 tends to 𝑦 0 while
. . ̄ while remaining in 𝐵 then 𝑔 extends to a continuous map 𝑔 ∶ 𝐵 → 𝑍 a continuous mapping of 𝐵 into a regular space 𝑍 . If, Let 𝑌 be a topological space, 𝐵 a dense subset of 𝑌 , and Theorem ∀𝑋 ∈ 𝒪 𝑧 , ∃𝑊 ∈ 𝒪 𝑦 , ∀𝑏 ∈ 𝐵 ∩ 𝑊 , 𝑔(𝑏) ∈ 𝑋. remaining in 𝐵 : topological spaces 𝑌 and 𝑍 . A better framework? . . . . . . 𝑔 ∶ 𝑌 → 𝑍 In order to handle inversion ℚ ∗ → ℝ ∗ and more general spaces, we want a version where ℝ 𝑞 and ℝ 𝑟 are replaced by general We can still say that 𝑔(𝑦) converges to 𝑧 when 𝑦 tends to 𝑦 0 while for each 𝑦 0 ∈ 𝑌 , 𝑔(𝑦) tends to a limit in 𝑍 when 𝑦 tends to 𝑦 0
∃? ̄ . 𝑗 𝑗𝑜𝑤 ℝ ℝ ∗ ℝ ∗ ℝ ℚ ℚ ∗ ℚ ∗ ℚ Issue: will we need discussions of 𝑔 𝑔 . 𝑌 𝑍 𝐵 Better framework: Hint: ℚ ⊄ ℝ . Does this theorem really applies to ℚ × ℚ ⊂ ℝ × ℝ ? . . . . . . 𝑗𝑜𝑤
∃? ̄ . 𝑗 𝑗𝑜𝑤 ℝ ℝ ∗ ℝ ∗ ℝ ℚ ℚ ∗ ℚ ∗ ℚ Issue: will we need discussions of 𝑔 𝑔 . 𝑌 𝑍 𝐵 Better framework: Hint: ℚ ⊄ ℝ . Does this theorem really applies to ℚ × ℚ ⊂ ℝ × ℝ ? . . . . . . 𝑗𝑜𝑤
. 𝑗 𝑗𝑜𝑤 ℝ ℝ ∗ ℝ ∗ ℝ ℚ ℚ ∗ ℚ ∗ ℚ Issue: will we need discussions of 𝑔 𝑔 . 𝑌 𝑍 𝐵 Better framework: Hint: ℚ ⊄ ℝ . Does this theorem really applies to ℚ × ℚ ⊂ ℝ × ℝ ? . . . . . . 𝑗𝑜𝑤 ∃? ̄
. 𝑗 𝑗𝑜𝑤 ℝ ℝ ∗ ℝ ∗ ℝ ℚ ℚ ∗ ℚ ∗ ℚ Issue: will we need discussions of 𝑔 𝑔 . 𝑌 𝑍 𝐵 Better framework: Hint: ℚ ⊄ ℝ . Does this theorem really applies to ℚ × ℚ ⊂ ℝ × ℝ ? . . . . . . 𝑗𝑜𝑤 ∃? ̄
This looks clunky. Note that i and f can be inferred from the types of de and h . Should we use extend de h ? A better solution is to defjne an extension operator 𝐹 𝑗 by: Then use de.extend f extend f i de h where de is a proof that 𝑗 is a dense Density of image of 𝑗 is used only to ensure 𝑍 is non-empty! some junk value if no such 𝑧 exists 𝐹 𝑗 (𝑔)(𝑦) = { some 𝑧 such that 𝑔(𝑏) tends to 𝑧 when 𝑏 tends to 𝑦 topological embedding, and h is a proof that 𝑔 admits a limit...? . . ̄ Side issue: how to formally refer to . . . . . . 𝑔 ?
A better solution is to defjne an extension operator 𝐹 𝑗 by: Then use de.extend f . Density of image of 𝑗 is used only to ensure 𝑍 is non-empty! some junk value if no such 𝑧 exists 𝐹 𝑗 (𝑔)(𝑦) = { some 𝑧 such that 𝑔(𝑏) tends to 𝑧 when 𝑏 tends to 𝑦 topological embedding, and h is a proof that 𝑔 admits a limit...? extend f i de h where de is a proof that 𝑗 is a dense . ̄ Side issue: how to formally refer to . . . . . . 𝑔 ? This looks clunky. Note that i and f can be inferred from the types of de and h . Should we use extend de h ?
. ̄ Density of image of 𝑗 is used only to ensure 𝑍 is non-empty! some junk value if no such 𝑧 exists 𝐹 𝑗 (𝑔)(𝑦) = { some 𝑧 such that 𝑔(𝑏) tends to 𝑧 when 𝑏 tends to 𝑦 topological embedding, and h is a proof that 𝑔 admits a limit...? extend f i de h where de is a proof that 𝑗 is a dense . 𝑔 ? Side issue: how to formally refer to . . . . . . This looks clunky. Note that i and f can be inferred from the types of de and h . Should we use extend de h ? A better solution is to defjne an extension operator 𝐹 𝑗 by: Then use de.extend f
build some ̂ natural map 𝑗 ∶ 𝑆 → ̂ . general topological ring 𝑆 (not necessarily metric, or even multiplication. 𝑆 . We want to extend addition and 𝑆 which is a (minimal) complete separated space and a There is a notion of completeness of a topological ring. One can separated). We want to generalize the story going from ℚ to ℝ , starting with a . Separation issues . . . . . . The 𝑗 map is not injective if {0} is not closed in 𝑆 .
. . multiplication. 𝑆 . We want to extend addition and 𝑆 which is a (minimal) complete separated space and a There is a notion of completeness of a topological ring. One can separated). general topological ring 𝑆 (not necessarily metric, or even We want to generalize the story going from ℚ to ℝ , starting with a Separation issues . . . . . . The 𝑗 map is not injective if {0} is not closed in 𝑆 . build some ̂ natural map 𝑗 ∶ 𝑆 → ̂
. . multiplication. 𝑆 . We want to extend addition and 𝑆 which is a (minimal) complete separated space and a There is a notion of completeness of a topological ring. One can separated). general topological ring 𝑆 (not necessarily metric, or even We want to generalize the story going from ℚ to ℝ , starting with a Separation issues . . . . . . The 𝑗 map is not injective if {0} is not closed in 𝑆 . build some ̂ natural map 𝑗 ∶ 𝑆 → ̂
Then 𝑗 𝑆 ′ ∶ 𝑆 ′ → ̂ 𝑆 ′ is injective and ̂ 𝑆 ′ is isomorphic to Then redefjne ̂ 𝑆 = ̂ Note: Even in ZFC, if 𝑆 is already separated, 𝑆 ′ ≠ 𝑆 . but ok. 𝑆 ′ . 𝑆 . ̂ . . Standard solution . . . . . . inherits addition and multiplication. Continuity is slightly tricky, Defjne 𝑆 ′ = 𝑆/{0} which is separated. Since {0} is an ideal, 𝑆 ′
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