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feat(library/standard): add sigma types and subtypes, make inhabited constructive
2014-08-15 03:12:54 +00:00
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
feat(library): changes in eq.lean sigma.lean and category.lean in eq.lean, make rec_on depend on the proof, and add congruence theorems for n-ary functions with n between 2 and 5 in sigma.lean, finish proving equality of triples in category.lean, define the functor category, and make the proofs of the arrow and slice categories easier for the elaborator
2014-09-26 23:36:04 +00:00
-- Author: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
refactor(library/data/sigma): break file into smaller pieces to reduce dependencies
2014-11-17 02:22:03 +00:00
import data.sigma.decl
refactor(data/sigma): move notation from sigma.thms to sigma.decl
2014-11-21 04:22:19 +00:00
open inhabited eq.ops sigma.ops
feat(library/standard): port int, and reorganize a lot
2014-08-20 02:32:44 +00:00
feat(library/standard): add sigma types and subtypes, make inhabited constructive
2014-08-15 03:12:54 +00:00
namespace sigma
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
universe variables u v
variables {A A' : Type.{u}} {B : A → Type.{v}} {B' : A' → Type.{v}}
feat(library/standard): add sigma types and subtypes, make inhabited constructive
2014-08-15 03:12:54 +00:00
refactor(data/sigma): move notation from sigma.thms to sigma.decl
2014-11-21 04:22:19 +00:00
definition unpack {C : (Σa, B a) → Type} {u : Σa, B a} (H : C ⟨u.1 , u.2⟩) : C u :=
feat(hott/types/sigma): port large part of the sigma library from the hott library most importantly, prove the characterization of paths in sigma types
2014-11-09 22:22:42 +00:00
destruct u (λx y H, H) H
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
theorem dpair_eq {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} (H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) :
refactor(library/data/sigma): cleanup module
2014-09-05 14:48:36 +00:00
dpair a₁ b₁ = dpair a₂ b₂ :=
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
dcongr_arg2 dpair H₁ H₂
refactor(library/data/sigma): cleanup module
2014-09-05 14:48:36 +00:00
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
theorem dpair_heq {a : A} {a' : A'} {b : B a} {b' : B' a'}
(HB : B == B') (Ha : a == a') (Hb : b == b') : dpair a b == dpair a' b' :=
hcongr_arg4 @dpair (heq.type_eq Ha) HB Ha Hb
protected theorem equal {p₁ p₂ : Σa : A, B a} :
∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : eq.rec_on H₁ (dpr2 p₁) = dpr2 p₂), p₁ = p₂ :=
refactor(library/data/sigma): cleanup module
2014-09-05 14:48:36 +00:00
destruct p₁ (take a₁ b₁, destruct p₂ (take a₂ b₂ H₁ H₂, dpair_eq H₁ H₂))
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
protected theorem hequal {p : Σa : A, B a} {p' : Σa' : A', B' a'} (HB : B == B') :
∀(H₁ : dpr1 p == dpr1 p') (H₂ : dpr2 p == dpr2 p'), p == p' :=
destruct p (take a₁ b₁, destruct p' (take a₂ b₂ H₁ H₂, dpair_heq HB H₁ H₂))
refactor(library): is_inhabited "theorems" should be "definitions", they are "data"
2014-10-02 16:00:34 +00:00
protected definition is_inhabited [instance] (H₁ : inhabited A) (H₂ : inhabited (B (default A))) :
feat(library/standard): add sigma types and subtypes, make inhabited constructive
2014-08-15 03:12:54 +00:00
inhabited (sigma B) :=
refactor(library/data/sigma): cleanup module
2014-09-05 14:48:36 +00:00
inhabited.destruct H₁ (λa, inhabited.destruct H₂ (λb, inhabited.mk (dpair (default A) b)))
feat(library): add constructions of categories, some changes in eq, sigma and path in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
2014-09-22 18:48:09 +00:00
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
theorem eq_rec_dpair_commute {C : Πa, B a → Type} {a a' : A} (H : a = a') (b : B a) (c : C a b) :
eq.rec_on H (dpair b c) = dpair (eq.rec_on H b) (eq.rec_on (dcongr_arg2 C H rfl) c) :=
eq.drec_on H (dpair_eq rfl (!eq.rec_on_id⁻¹))
refactor(library): remove some unnecessary sections
2014-10-10 23:33:58 +00:00
variables {C : Πa, B a → Type} {D : Πa b, C a b → Type}
feat(library): add constructions of categories, some changes in eq, sigma and path in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
2014-09-22 18:48:09 +00:00
refactor(data/sigma): move notation from sigma.thms to sigma.decl
2014-11-21 04:22:19 +00:00
definition dtrip (a : A) (b : B a) (c : C a b) := ⟨a, b, c⟩
definition dquad (a : A) (b : B a) (c : C a b) (d : D a b c) := ⟨a, b, c, d⟩
feat(library): add constructions of categories, some changes in eq, sigma and path in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
2014-09-22 18:48:09 +00:00
feat(hott/types/sigma): port large part of the sigma library from the hott library most importantly, prove the characterization of paths in sigma types
2014-11-09 22:22:42 +00:00
definition dpr1' (x : Σ a, B a) := x.1
definition dpr2' (x : Σ a b, C a b) := x.2.1
definition dpr3 (x : Σ a b, C a b) := x.2.2
definition dpr3' (x : Σ a b c, D a b c) := x.2.2.1
definition dpr4 (x : Σ a b c, D a b c) := x.2.2.2
feat(library): add constructions of categories, some changes in eq, sigma and path in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
2014-09-22 18:48:09 +00:00
feat(library): changes in eq.lean sigma.lean and category.lean in eq.lean, make rec_on depend on the proof, and add congruence theorems for n-ary functions with n between 2 and 5 in sigma.lean, finish proving equality of triples in category.lean, define the functor category, and make the proofs of the arrow and slice categories easier for the elaborator
2014-09-26 23:36:04 +00:00
theorem dtrip_eq {a₁ a₂ : A} {b₁ : B a₁} {b₂ : B a₂} {c₁ : C a₁ b₁} {c₂ : C a₂ b₂}
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
(H₁ : a₁ = a₂) (H₂ : eq.rec_on H₁ b₁ = b₂) (H₃ : cast (dcongr_arg2 C H₁ H₂) c₁ = c₂) :
refactor(data/sigma): move notation from sigma.thms to sigma.decl
2014-11-21 04:22:19 +00:00
⟨a₁, b₁, c₁⟩ = ⟨a₂, b₂, c₂⟩ :=
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
dcongr_arg3 dtrip H₁ H₂ H₃
feat(library): add constructions of categories, some changes in eq, sigma and path in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
2014-09-22 18:48:09 +00:00
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
theorem ndtrip_eq {A B : Type} {C : A → B → Type} {a₁ a₂ : A} {b₁ b₂ : B}
{c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (H₁ : a₁ = a₂) (H₂ : b₁ = b₂)
refactor(data/sigma): move notation from sigma.thms to sigma.decl
2014-11-21 04:22:19 +00:00
(H₃ : cast (congr_arg2 C H₁ H₂) c₁ = c₂) : ⟨a₁, b₁, c₁⟩ = ⟨a₂, b₂, c₂⟩ :=
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
hdcongr_arg3 dtrip H₁ (heq.from_eq H₂) H₃
feat(library): changes in eq.lean sigma.lean and category.lean in eq.lean, make rec_on depend on the proof, and add congruence theorems for n-ary functions with n between 2 and 5 in sigma.lean, finish proving equality of triples in category.lean, define the functor category, and make the proofs of the arrow and slice categories easier for the elaborator
2014-09-26 23:36:04 +00:00
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
theorem ndtrip_equal {A B : Type} {C : A → B → Type} {p₁ p₂ : Σa b, C a b} :
feat(library): add constructions of categories, some changes in eq, sigma and path in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
2014-09-22 18:48:09 +00:00
∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : dpr2' p₁ = dpr2' p₂)
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
(H₃ : eq.rec_on (congr_arg2 C H₁ H₂) (dpr3 p₁) = dpr3 p₂), p₁ = p₂ :=
feat(library): add constructions of categories, some changes in eq, sigma and path in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
2014-09-22 18:48:09 +00:00
destruct p₁ (take a₁ q₁, destruct q₁ (take b₁ c₁, destruct p₂ (take a₂ q₂, destruct q₂
refactor(library): major changes in the library I made some major changes in the library. I wanted to wait with pushing until I had finished the formalization of the slice functor, but for some reason that is very hard to formalize, requiring a lot of casts and manipulation of casts. So I've not finished that yet. Changes: - in multiple files make more use of variables - move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts). - prove some more theorems involving heq, e.g. hcongr_arg3 (which do not require piext) - in theorems where casts are used in the statement use eq.rec_on instead of eq.drec_on - in category split basic into basic, functor and natural_transformation - change the definition of functor to use fully bundled categories. @avigad: this means that the file semisimplicial.lean will also need changes (but I'm quite sure nothing major). You want to define the fully bundled category Delta, and use only fully bundled categories (type and ᵒᵖ are notations for the fully bundled Type_category and Opposite if you open namespace category.ops). If you want I can make the changes. - lots of minor changes
2014-11-04 00:22:30 +00:00
(take b₂ c₂ H₁ H₂ H₃, ndtrip_eq H₁ H₂ H₃))))
feat(library): add constructions of categories, some changes in eq, sigma and path in eq, add theorem for proof irrelevance and congruence for binary functions in sigma, add some support for triplets in path, comment out two unneccesary definitions in category, add Cat, slice, coslice, product and arrow categories, also add fully bundled approach
2014-09-22 18:48:09 +00:00
feat(library/standard): add sigma types and subtypes, make inhabited constructive
2014-08-15 03:12:54 +00:00
end sigma
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