{"action":"created","comment":{"url":"https://api.github.com/repos/UniMath/UniMath/pulls/comments/45415359","id":45415359,"diff_hunk":"@@ -1752,10 +1755,45 @@ assert ( egf : ∀ xy : _ , ( g ( f xy ) ) = xy ) . intro . induction xy . apply\n assert ( efg : ∀ yx : _ , ( f ( g yx ) ) = yx ) . intro . induction yx . apply idpath .\n split with f . apply ( gradth _ _ egf  efg ) . Defined .\n \n+Definition weqtotal2dirprodcomm {X Y:UU} (P: X × Y -> UU) : (Σ xy : X×Y, P xy) ≃ (Σ xy : Y×X, P (weqdirprodcomm _ _ xy)).\n+Proof.\n+  intros.\n+  refine (weqgradth _ _ _ _).\n+  - intros xyp. induction xyp as [xy p]. induction xy as [x y]. exact ((y,,x),,p).\n+  - intros yxp. induction yxp as [yx p]. induction yx as [y x]. exact ((x,,y),,p).\n+  - intros xyp. induction xyp as [xy p]. induction xy as [x y]. reflexivity.\n+  - intros yxp. induction yxp as [yx p]. induction yx as [y x]. reflexivity.\n+Defined.\n \n+Definition weqtotal2dirprodassoc  {X Y:UU} (P: X × Y -> UU) : (Σ xy : X×Y, P xy) ≃ (Σ (x:X) (y:Y), P (x,,y)).\n+  intros.\n+  refine (weqgradth _ _ _ _).\n+  - intros xyp. induction xyp as [xy p]. induction xy as [x y]. exact (x,,y,,p).\n+  - intros xyp. induction xyp as [x yp]. induction yp as [y p]. exact ((x,,y),,p).\n+  - intros xyp. induction xyp as [xy p]. induction xy as [x y]. reflexivity.\n+  - intros xyp. induction xyp as [x yp]. induction yp as [y p]. reflexivity.\n+Defined.\n \n+Definition weqtotal2dirprodassoc' {X Y:UU} (P: X × Y -> UU) : (Σ xy : X×Y, P xy) ≃ (Σ (y:Y) (x:X), P (x,,y)).\n+Proof.\n+  intros.\n+  refine (weqgradth _ _ _ _).\n+  - intros xyp. induction xyp as [xy p]. induction xy as [x y]. exact (y,,x,,p).\n+  - intros yxp. induction yxp as [x yp]. induction yp as [y p]. exact ((y,,x),,p).\n+  - intros xyp. induction xyp as [xy p]. induction xy as [x y]. reflexivity.\n+  - intros yxp. induction yxp as [x yp]. induction yp as [y p]. reflexivity.\n+Defined.\n \n-\n+Definition weqtotal2comm12 {X} (P Q : X -> UU) :\n+  (Σ (w : Σ x, P x), Q (pr1 w)) ≃ (Σ (w : Σ x, Q x), P (pr1 w)).\n+Proof.\n+  intros.\n+  refine (weqgradth _ _ _ _).\n+  - intros [[x p] q]. exact ((x,,q),,p).","path":"UniMath/Foundations/Basics/PartA.v","position":189,"original_position":189,"commit_id":"b805dbf8bd2f11632270905f4a59559928aef7c0","original_commit_id":"b805dbf8bd2f11632270905f4a59559928aef7c0","user":{"login":"benediktahrens","id":1106102,"avatar_url":"https://avatars.githubusercontent.com/u/1106102?v=3","gravatar_id":"","url":"https://api.github.com/users/benediktahrens","html_url":"https://github.com/benediktahrens","followers_url":"https://api.github.com/users/benediktahrens/followers","following_url":"https://api.github.com/users/benediktahrens/following{/other_user}","gists_url":"https://api.github.com/users/benediktahrens/gists{/gist_id}","starred_url":"https://api.github.com/users/benediktahrens/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/benediktahrens/subscriptions","organizations_url":"https://api.github.com/users/benediktahrens/orgs","repos_url":"https://api.github.com/users/benediktahrens/repos","events_url":"https://api.github.com/users/benediktahrens/events{/privacy}","received_events_url":"https://api.github.com/users/benediktahrens/received_events","type":"User","site_admin":false},"body":"I seem to remember that nested brackets in \"intros ...\" were forbidden? ","created_at":"2015-11-19T23:13:07Z","updated_at":"2015-11-19T23:13:07Z","html_url":"https://github.com/UniMath/UniMath/pull/240#discussion_r45415359","pull_request_url":"https://api.github.com/repos/UniMath/UniMath/pulls/240","_links":{"self":{"href":"https://api.github.com/repos/UniMath/UniMath/pulls/comments/45415359"},"html":{"href":"https://github.com/UniMath/UniMath/pull/240#discussion_r45415359"},"pull_request":{"href":"https://api.github.com/repos/UniMath/UniMath/pulls/240"}}},"pull_request":{"url":"https://api.github.com/repos/UniMath/UniMath/pulls/240","id":51009847,"html_url":"https://github.com/UniMath/UniMath/pull/240","diff_url":"https://github.com/UniMath/UniMath/pull/240.diff","patch_url":"https://github.com/UniMath/UniMath/pull/240.patch","issue_url":"https://api.github.com/repos/UniMath/UniMath/issues/240","number":240,"state":"open","locked":false,"title":"Postponing use of funextempty","user":{"login":"DanGrayson","id":700228,"avatar_url":"https://avatars.githubusercontent.com/u/700228?v=3","gravatar_id":"","url":"https://api.github.com/users/DanGrayson","html_url":"https://github.com/DanGrayson","followers_url":"https://api.github.com/users/DanGrayson/followers","following_url":"https://api.github.com/users/DanGrayson/following{/other_user}","gists_url":"https://api.github.com/users/DanGrayson/gists{/gist_id}","starred_url":"https://api.github.com/users/DanGrayson/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/DanGrayson/subscriptions","organizations_url":"https://api.github.com/users/DanGrayson/orgs","repos_url":"https://api.github.com/users/DanGrayson/repos","events_url":"https://api.github.com/users/DanGrayson/events{/privacy}","received_events_url":"https://api.github.com/users/DanGrayson/received_events","type":"User","site_admin":false},"body":"Here are some changes that remove early uses of the axiom funextempty, in order\r\nto improve computation.  The axiom is used, for example, to show that a negation is\r\na proposition, so I've rooted out various propositions that were phrased as negations and\r\nreplaced them by axiom-free propositions.  This includes m ≤ n on nat.  We downplay the\r\nuse of x != y, and introduce on nat the notation m ≠ n for an equivalent proposition.\r\nWe should do the same for hz and hq, which I left 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