Calculational HoTT International Conference on Homotopy Type Theory - PowerPoint PPT Presentation

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Calculational HoTT International Conference on Homotopy Type Theory (HoTT 2019) Carnegie Mellon University August 12 to 17, 2019 Bernarda Aldana, Jaime Bohorquez, Ernesto Acosta Escuela Colombiana de Ingenier´ ıa Bogot´ a, Colombia

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Content 1 A few initial words 2 Brief description of CL 3 The problem 4 Deductive chains 5 Calculational HoTT 6 A deduction 7 Conclusions

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Presentation What we do is to rewrite math topics using Calculational Logic (CL),

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Presentation What we do is to rewrite math topics using Calculational Logic (CL), as there is a large community rewriting math in terms of HoTT.

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Presentation What we do is to rewrite math topics using Calculational Logic (CL), as there is a large community rewriting math in terms of HoTT. We ended up trying to interpret HoTT in terms of CL.

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Presentation What we do is to rewrite math topics using Calculational Logic (CL), as there is a large community rewriting math in terms of HoTT. We ended up trying to interpret HoTT in terms of CL. The result: “Calculational HoTT”(arXiv:1901.08883v2), a joint work with Bernarda Aldana and Jaime Bohorquez.

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Equational axioms and Leibniz rules Brief description of CL. Main feature: CL axioms are A ≡ B, C ≡ D, . . . logical equations CL is an equational logical system E [ x/A ] A ≡ B E [ x/B ] CL inference rules are Leibniz’s rules E [ x/B ] A ≡ B E [ x/A ]

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Calculations Derivations in CL are deduction trees of the form: E 1 A ≡ B E 2 C ≡ D E 3 E ≡ F E 4 where A through F are subformulas of the corresponding E i .

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Calculations Derivations in CL are deduction trees of the form: E 1 A ≡ B E 2 C ≡ D E 3 E ≡ F E 4 where A through F are subformulas of the corresponding E i . This deduction tree, written vertically, is what Lifschitz called ‘Calculation’[Lifs]: E 1 ⇔ � A ≡ B � which derives E 1 ≡ E 4 E 2 ⇔ � C ≡ D � Double arrows stand for the bidi- E 3 rectionality of Leibniz rules ⇔ � E ≡ F � E 4

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Calculations Derivations in CL are deduction trees of the form: E 1 A ≡ B E 2 C ≡ D E 3 E ≡ F E 4 where A through F are subformulas of the corresponding E i . This deduction tree, written vertically, is what Lifschitz called ‘Calculation’[Lifs]: E 1 ⇔ � A ≡ B � which derives E 1 ≡ E 4 E 2 ⇔ � C ≡ D � Double arrows stand for the bidi- E 3 rectionality of Leibniz rules ⇔ � E ≡ F � E 4 There are sound and complete calculational versions of both, classical (CCL) and intuitionistic (ICL) first order logic.

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Embeddings The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Embeddings The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory We pose ourself the following question: Is it possible to embed ICL into HoTT?

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Embeddings The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory We pose ourself the following question: Is it possible to embed ICL into HoTT? We concentrated in - establishing a linear calculation format as an instrument to understand proofs in HoTT book, and

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Embeddings The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory We pose ourself the following question: Is it possible to embed ICL into HoTT? We concentrated in - establishing a linear calculation format as an instrument to understand proofs in HoTT book, and - identify and derive equational judgments in HoTT.

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Embeddings The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory We pose ourself the following question: Is it possible to embed ICL into HoTT? We concentrated in - establishing a linear calculation format as an instrument to understand proofs in HoTT book, and - identify and derive equational judgments in HoTT. Note: We expected to be more comfortable with a linear calculation format as an instrument to understand proofs in HoTT book.

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains First : Definition of deductive chains.

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains First : Definition of deductive chains. A → B < : stands temporarily A ❀ B for one of the A ≡ B (read A leads to B ) judgments or A ≃ B < :

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains First : Definition of deductive chains. A → B < : stands temporarily A ❀ B for one of the A ≡ B (read A leads to B ) judgments or A ≃ B < : It is easy to prove the following transitivity rule scheme A 1 ❀ A 2 A 2 ❀ A 3 A 1 ❀ A 3 where the conclusion corresponds to

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains First : Definition of deductive chains. A → B < : stands temporarily A ❀ B for one of the A ≡ B (read A leads to B ) judgments or A ≃ B < : It is easy to prove the following transitivity rule scheme A 1 ❀ A 2 A 2 ❀ A 3 A 1 ❀ A 3 where the conclusion corresponds to if at least one of the premises is a judgment of the A 1 → A 3 < : form A → B < :

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains First : Definition of deductive chains. A → B < : stands temporarily A ❀ B for one of the A ≡ B (read A leads to B ) judgments or A ≃ B < : It is easy to prove the following transitivity rule scheme A 1 ❀ A 2 A 2 ❀ A 3 A 1 ❀ A 3 where the conclusion corresponds to if at least one of the premises is a judgment of the A 1 → A 3 < : form A → B < : if none of the premises is of the form A → B < : A 1 ≃ A 3 < : and at least one is of the form A ≃ B < :

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains First : Definition of deductive chains. A → B < : stands temporarily A ❀ B for one of the A ≡ B (read A leads to B ) judgments or A ≃ B < : It is easy to prove the following transitivity rule scheme A 1 ❀ A 2 A 2 ❀ A 3 A 1 ❀ A 3 where the conclusion corresponds to if at least one of the premises is a judgment of the A 1 → A 3 < : form A → B < : if none of the premises is of the form A → B < : A 1 ≃ A 3 < : and at least one is of the form A ≃ B < : A 1 ≡ A 3 if all the premises are of the form A ≡ B

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains By induction we have the following derivation . . . . . . . . . a : A 1 A 1 ❀ A 2 · · · A n − 1 ❀ A n A n < : .

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains By induction we have the following derivation . . . . . . . . . a : A 1 A 1 ❀ A 2 · · · A n − 1 ❀ A n A n < : . which may be represented vertically by the following format-scheme A n ⇆ � · · · � A n − 1 . . . A 2 � · · · � ⇆ A 1 ∧ � · · · � : a which we called a deductive chain .

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions Deductive chains The links in this format-scheme are B � � ⇆ A