lean2/library/init/prod.lean

/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Jeremy Avigad
-/
prelude
import init.num init.relation

definition pair [constructor] := @prod.mk
notation A × B := prod A B
-- notation for n-ary tuples
notation `(` h `, ` t:(foldl `, ` (e r, prod.mk r e) h) `)` := t

namespace prod
  notation `pr₁` := pr1
  notation `pr₂` := pr2

  namespace ops
  postfix `.1`:(max+1) := pr1
  postfix `.2`:(max+1) := pr2
  end ops

  definition destruct [reducible] := @prod.cases_on

  section
  variables {A B : Type}
  lemma pr1.mk (a : A) (b : B) : pr1 (mk a b) = a := rfl
  lemma pr2.mk (a : A) (b : B) : pr2 (mk a b) = b := rfl
  lemma eta : ∀ (p : A × B), mk (pr1 p) (pr2 p) = p
  | (a, b) := rfl
  end

end prod
-												refactor(library): add 'init' folder

											
										
										
											2014-12-01 04:34:12 +00:00
+								/-
 								Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 								Released under Apache 2.0 license as described in the file LICENSE.
 								Author: Leonardo de Moura, Jeremy Avigad
 								-/
 								prelude
-												feat(init/wf): port from standard library to HoTT library

After this commit we need some more advanced theorems in init/wf, notably function extenstionality.
For this reason I had to refactor the init folder a little bit.
To keep the init folders in both libraries similar, I did the same refactorization in the standard library, even though that was not required for the standard library

											
										
										
											2016-02-09 01:07:44 +00:00
+								import init.num init.relation
-												refactor(library): add 'init' folder

											
										
										
											2014-12-01 04:34:12 +00:00
-												feat(hott): port more from chapters 4 and 6 of the book

											
										
										
											2015-09-22 16:01:55 +00:00
+								definition pair [constructor] := @prod.mk
-												fix(library/init/{prod,sigma},library/data/sum): move notation in/out of namespaces

											
										
										
											2015-02-01 19:14:01 +00:00
+								notation A × B := prod A B
 								-- notation for n-ary tuples
-												feat(library): add some spacing hints

											
										
										
											2015-09-30 15:06:31 +00:00
+								notation `(` h `, ` t:(foldl `, ` (e r, prod.mk r e) h) `)` := t
-												feat(library/data/prod/wf): lex of well-founded relations is well-founded

											
										
										
											2014-11-11 08:29:21 +00:00
 								namespace prod
-												refactor(library): add 'init' folder

											
										
										
											2014-12-01 04:34:12 +00:00
+								  notation `pr₁` := pr1
 								  notation `pr₂` := pr2
-												feat(library/data,init/prod,sigma,sum): make more notation available at top level

											
										
										
											2015-02-01 16:39:47 +00:00
+								  namespace ops
 								  postfix `.1`:(max+1) := pr1
 								  postfix `.2`:(max+1) := pr2
 								  end ops
-												refactor(library/init): define prod as an inductive datatype

Motivation: prod is used internally in the definitional package.
If we define prod as a structure, then Lean will tag pr1 and pr2 as
projections. This creates problems when we add special support for
projections in the elaborator. The heuristics avoid some case-splits
that are currently performed, and without them some files break.

											
										
										
											2015-06-26 00:15:16 +00:00
+								  definition destruct [reducible] := @prod.cases_on
 								  section
 								  variables {A B : Type}
 								  lemma pr1.mk (a : A) (b : B) : pr1 (mk a b) = a := rfl
 								  lemma pr2.mk (a : A) (b : B) : pr2 (mk a b) = b := rfl
 								  lemma eta : ∀ (p : A × B), mk (pr1 p) (pr2 p) = p
 								  | (a, b) := rfl
 								  end
-												feat(library/data/prod/wf): lex of well-founded relations is well-founded

											
										
										
											2014-11-11 08:29:21 +00:00
+								end prod