Cubical Syntax for Reflection-Free Extensional Equality Jonathan - PowerPoint PPT Presentation

FSCD 2019 Cubical Syntax for Reflection-Free Extensional Equality Jonathan Sterling 1 Carlo Angiuli 1 Daniel Gratzer 2 1 Carnegie Mellon University 2 Aarhus University 1 / 32

𝑦 ∶ 𝐵 ⊢ 𝐶(𝑦) type need to consider equality of types: if 𝐵 = 𝐶 type , then elements of 𝐵 should be elements of 𝐶 . types depend on elements, so equality of elements necessary too. not all equations can be made automatic, so a language of proofs must account for coercions . 𝑦 ∶ 𝐵 ⊢ 𝑁(𝑦) ∶ 𝐶(𝑦) dependent type theory ... is about families ! 2 / 32

need to consider equality of types: if 𝐵 = 𝐶 type , then elements of 𝐵 should be elements of 𝐶 . types depend on elements, so equality of elements necessary too. not all equations can be made automatic, so a language of proofs must account for coercions . 𝑦 ∶ 𝐵 ⊢ 𝑁(𝑦) ∶ 𝐶(𝑦) dependent type theory ... is about families ! 𝑦 ∶ 𝐵 ⊢ 𝐶(𝑦) type 2 / 32

need to consider equality of types: if 𝐵 = 𝐶 type , then elements of 𝐵 should be elements of 𝐶 . types depend on elements, so equality of elements necessary too. not all equations can be made automatic, so a language of proofs must account for coercions . 𝑦 ∶ 𝐵 ⊢ 𝑁(𝑦) ∶ 𝐶(𝑦) dependent type theory ... is about families ! 𝑦 ∶ 𝐵 ⊢ 𝐶(𝑦) type 2 / 32

types depend on elements, so equality of elements necessary too. not all equations can be made automatic, so a language of proofs must account for coercions . 𝑦 ∶ 𝐵 ⊢ 𝑁(𝑦) ∶ 𝐶(𝑦) dependent type theory ... is about families ! 𝑦 ∶ 𝐵 ⊢ 𝐶(𝑦) type need to consider equality of types: if 𝐵 = 𝐶 type , then elements of 𝐵 should be elements of 𝐶 . 2 / 32

𝑦 ∶ 𝐵 ⊢ 𝑁(𝑦) ∶ 𝐶(𝑦) dependent type theory ... is about families ! 𝑦 ∶ 𝐵 ⊢ 𝐶(𝑦) type need to consider equality of types: if 𝐵 = 𝐶 type , then elements of 𝐵 should be elements of 𝐶 . types depend on elements, so equality of elements necessary too. not all equations can be made automatic, so a language of proofs must account for coercions . 2 / 32

𝑦 ∶ 𝐵 × 𝐶 ⊢ 𝑁 ∶ 𝐺(𝑦) ifg 𝑦 ∶ 𝐵 × 𝐶 ⊢ 𝑁 ∶ 𝐺(⟨𝑦.1, 𝑦.2⟩) equality in type theory what equations can be made automatic? surely 𝛽/𝜀/𝛾 , and type theorists also know how to automate 𝜃/𝜊/𝜉/ … 3 / 32

𝑦 ∶ 𝐵 × 𝐶 ⊢ 𝑁 ∶ 𝐺(𝑦) ifg 𝑦 ∶ 𝐵 × 𝐶 ⊢ 𝑁 ∶ 𝐺(⟨𝑦.1, 𝑦.2⟩) equality in type theory what equations can be made automatic? surely 𝛽/𝜀/𝛾 , and type theorists also know how to automate 𝜃/𝜊/𝜉/ … 3 / 32

1. is 𝑁 equal to coe 𝐺(−) (𝑄, 𝑁) ? yes, up to a coercion equality in type theory other equations may require explicit coercion. 1 consider a family 𝑜 ∶ nat ⊢ 𝐺(𝑜) type … 1 “equality reflection” just pushes the problem elsewhere and makes it worse! unlike many, I speak from experience. 4 / 32

1. is 𝑁 equal to coe 𝐺(−) (𝑄, 𝑁) ? yes, up to a coercion ifg??? equality in type theory other equations may require explicit coercion. 1 consider a family 𝑜 ∶ nat ⊢ 𝐺(𝑜) type … 𝑜 ∶ nat ⊢ 𝑁 ∶ 𝐺(𝑜 + 1) 𝑜 ∶ nat ⊢ 𝑁 ∶ 𝐺(1 + 𝑜) 1 “equality reflection” just pushes the problem elsewhere and makes it worse! unlike many, I speak from experience. 4 / 32

1. is 𝑁 equal to coe 𝐺(−) (𝑄, 𝑁) ? yes, up to a coercion ifg equality in type theory other equations may require explicit coercion. 1 consider a family 𝑜 ∶ nat ⊢ 𝐺(𝑜) type … 𝑜 ∶ nat ⊢ 𝑁 ∶ 𝐺(𝑜 + 1) 𝑜 ∶ nat ⊢ coe 𝐺(−) (𝑄, 𝑁) ∶ 𝐺(1 + 𝑜) 1 “equality reflection” just pushes the problem elsewhere and makes it worse! unlike many, I speak from experience. 4 / 32

ifg equality in type theory other equations may require explicit coercion. 1 consider a family 𝑜 ∶ nat ⊢ 𝐺(𝑜) type … 𝑜 ∶ nat ⊢ 𝑁 ∶ 𝐺(𝑜 + 1) 𝑜 ∶ nat ⊢ coe 𝐺(−) (𝑄, 𝑁) ∶ 𝐺(1 + 𝑜) 1. is 𝑁 equal to coe 𝐺(−) (𝑄, 𝑁) ? yes, up to a coercion 1 “equality reflection” just pushes the problem elsewhere and makes it worse! unlike many, I speak from experience. 4 / 32

ifg equality in type theory other equations may require explicit coercion. 1 consider a family 𝑜 ∶ nat ⊢ 𝐺(𝑜) type … 𝑜 ∶ nat ⊢ 𝑁 ∶ 𝐺(𝑜 + 1) 𝑜 ∶ nat ⊢ coe 𝐺(−) (𝑄, 𝑁) ∶ 𝐺(1 + 𝑜) 1. is 𝑁 equal to coe 𝐺(−) (𝑄, 𝑁) ? yes, up to a coercion 2. is coe 𝐺(−) (𝑄, 𝑁) equal to coe 𝐺(−) (𝑅, 𝑁) ? maybe 1 “equality reflection” just pushes the problem elsewhere and makes it worse! unlike many, I speak from experience. 4 / 32

• hierarchy of closed / inductive universes of sets, props • heterogeneous equality type Eq (𝑁 ∶ 𝐵, 𝑂 ∶ 𝐶) defined as generic program , by recursion on type codes 𝐵, 𝐶 Eq (𝐺 0 ∶ 𝐵 0 → 𝐶 0 , 𝐺 1 ∶ 𝐵 1 → 𝐶 1 ) = 𝑦 ∶ Eq (𝑦 0 ∶ 𝐵 0 , 𝑦 1 ∶ 𝐵 1 )) (funext) (𝑦 0 ∶ 𝐵 0 )(𝑦 1 ∶ 𝐵 1 )(̃ → Eq (𝐺 0 (𝑦 0 ) ∶ 𝐶 0 , 𝐺 1 (𝑦 1 ) ∶ 𝐶 1 ) • judgmental UIP (proof irrelevance): always have 𝑄 = 𝑅 ∶ Eq (𝑁 0 ∶ 𝐵 0 , 𝑁 1 ∶ 𝐵 1 ) • many primitives: reflexivity, respect, coercion, coherence, heterogeneous irrelevance (see Altenkirch, McBride, and Swierstra [AMS07]) Altenkirch, McBride, and Swierstra [AMS07]. “Observational Equality, Now!” Observational Type Theory Altenkirch and McBride [AM06]. Towards Observational Type Theory . 5 / 32

• heterogeneous equality type Eq (𝑁 ∶ 𝐵, 𝑂 ∶ 𝐶) defined as generic program , by recursion on type codes 𝐵, 𝐶 Eq (𝐺 0 ∶ 𝐵 0 → 𝐶 0 , 𝐺 1 ∶ 𝐵 1 → 𝐶 1 ) = 𝑦 ∶ Eq (𝑦 0 ∶ 𝐵 0 , 𝑦 1 ∶ 𝐵 1 )) (funext) (𝑦 0 ∶ 𝐵 0 )(𝑦 1 ∶ 𝐵 1 )(̃ → Eq (𝐺 0 (𝑦 0 ) ∶ 𝐶 0 , 𝐺 1 (𝑦 1 ) ∶ 𝐶 1 ) • judgmental UIP (proof irrelevance): always have 𝑄 = 𝑅 ∶ Eq (𝑁 0 ∶ 𝐵 0 , 𝑁 1 ∶ 𝐵 1 ) • many primitives: reflexivity, respect, coercion, coherence, heterogeneous irrelevance (see Altenkirch, McBride, and Swierstra [AMS07]) Altenkirch, McBride, and Swierstra [AMS07]. “Observational Equality, Now!” Observational Type Theory Altenkirch and McBride [AM06]. Towards Observational Type Theory . • hierarchy of closed / inductive universes of sets, props 5 / 32

• judgmental UIP (proof irrelevance): always have 𝑄 = 𝑅 ∶ Eq (𝑁 0 ∶ 𝐵 0 , 𝑁 1 ∶ 𝐵 1 ) • many primitives: reflexivity, respect, coercion, coherence, heterogeneous irrelevance (see Altenkirch, McBride, and Swierstra [AMS07]) Altenkirch, McBride, and Swierstra [AMS07]. “Observational Equality, Now!” Observational Type Theory Altenkirch and McBride [AM06]. Towards Observational Type Theory . • hierarchy of closed / inductive universes of sets, props • heterogeneous equality type Eq (𝑁 ∶ 𝐵, 𝑂 ∶ 𝐶) defined as generic program , by recursion on type codes 𝐵, 𝐶 Eq (𝐺 0 ∶ 𝐵 0 → 𝐶 0 , 𝐺 1 ∶ 𝐵 1 → 𝐶 1 ) = 𝑦 ∶ Eq (𝑦 0 ∶ 𝐵 0 , 𝑦 1 ∶ 𝐵 1 )) (funext) (𝑦 0 ∶ 𝐵 0 )(𝑦 1 ∶ 𝐵 1 )(̃ → Eq (𝐺 0 (𝑦 0 ) ∶ 𝐶 0 , 𝐺 1 (𝑦 1 ) ∶ 𝐶 1 ) 5 / 32

• many primitives: reflexivity, respect, coercion, coherence, heterogeneous irrelevance (see Altenkirch, McBride, and Swierstra [AMS07]) Altenkirch, McBride, and Swierstra [AMS07]. “Observational Equality, Now!” Observational Type Theory Altenkirch and McBride [AM06]. Towards Observational Type Theory . • hierarchy of closed / inductive universes of sets, props • heterogeneous equality type Eq (𝑁 ∶ 𝐵, 𝑂 ∶ 𝐶) defined as generic program , by recursion on type codes 𝐵, 𝐶 Eq (𝐺 0 ∶ 𝐵 0 → 𝐶 0 , 𝐺 1 ∶ 𝐵 1 → 𝐶 1 ) = 𝑦 ∶ Eq (𝑦 0 ∶ 𝐵 0 , 𝑦 1 ∶ 𝐵 1 )) (funext) (𝑦 0 ∶ 𝐵 0 )(𝑦 1 ∶ 𝐵 1 )(̃ → Eq (𝐺 0 (𝑦 0 ) ∶ 𝐶 0 , 𝐺 1 (𝑦 1 ) ∶ 𝐶 1 ) • judgmental UIP (proof irrelevance): always have 𝑄 = 𝑅 ∶ Eq (𝑁 0 ∶ 𝐵 0 , 𝑁 1 ∶ 𝐵 1 ) 5 / 32

Altenkirch, McBride, and Swierstra [AMS07]. “Observational Equality, Now!” Observational Type Theory Altenkirch and McBride [AM06]. Towards Observational Type Theory . • hierarchy of closed / inductive universes of sets, props • heterogeneous equality type Eq (𝑁 ∶ 𝐵, 𝑂 ∶ 𝐶) defined as generic program , by recursion on type codes 𝐵, 𝐶 Eq (𝐺 0 ∶ 𝐵 0 → 𝐶 0 , 𝐺 1 ∶ 𝐵 1 → 𝐶 1 ) = 𝑦 ∶ Eq (𝑦 0 ∶ 𝐵 0 , 𝑦 1 ∶ 𝐵 1 )) (funext) (𝑦 0 ∶ 𝐵 0 )(𝑦 1 ∶ 𝐵 1 )(̃ → Eq (𝐺 0 (𝑦 0 ) ∶ 𝐶 0 , 𝐺 1 (𝑦 1 ) ∶ 𝐶 1 ) • judgmental UIP (proof irrelevance): always have 𝑄 = 𝑅 ∶ Eq (𝑁 0 ∶ 𝐵 0 , 𝑁 1 ∶ 𝐵 1 ) • many primitives: reflexivity, respect, coercion, coherence, heterogeneous irrelevance (see Altenkirch, McBride, and Swierstra [AMS07]) 5 / 32

cubical reconstruction: XTT goal: find smaller set of primitives which systematically generate (something in the spirit of) OTT idea: start with Cartesian cubical type theory [ABCFHL], restrict to Bishop sets à la Coquand [Coq17] the XTT paper Sterling, Angiuli, and Gratzer [SAG19]. “Cubical Syntax for Reflection-Free Extensional Equality”. Formal Structures for Computation and Deduction (FSCD 2019). 6 / 32