From 6f1e5ff85d736b80a6d3490a21a30c8d37ea18de Mon Sep 17 00:00:00 2001 From: Gaëtan Gilbert Date: Wed, 21 Jun 2017 16:48:38 +0200 Subject: Add .v extension to dev/doc/notes-on-conversion This gives syntax highlighting in Coq-aware editors. --- dev/doc/notes-on-conversion | 73 ------------------------------------------- dev/doc/notes-on-conversion.v | 73 +++++++++++++++++++++++++++++++++++++++++++ 2 files changed, 73 insertions(+), 73 deletions(-) delete mode 100644 dev/doc/notes-on-conversion create mode 100644 dev/doc/notes-on-conversion.v (limited to 'dev') diff --git a/dev/doc/notes-on-conversion b/dev/doc/notes-on-conversion deleted file mode 100644 index a81f170c63..0000000000 --- a/dev/doc/notes-on-conversion +++ /dev/null @@ -1,73 +0,0 @@ -(**********************************************************************) -(* A few examples showing the current limits of the conversion algorithm *) -(**********************************************************************) - -(*** We define (pseudo-)divergence from Ackermann function ***) - -Definition ack (n : nat) := - (fix F (n0 : nat) : nat -> nat := - match n0 with - | O => S - | S n1 => - fun m : nat => - (fix F0 (n2 : nat) : nat := - match n2 with - | O => F n1 1 - | S n3 => F n1 (F0 n3) - end) m - end) n. - -Notation OMEGA := (ack 4 4). - -Definition f (x:nat) := x. - -(* Evaluation in tactics can somehow be controlled *) -Lemma l1 : OMEGA = OMEGA. -reflexivity. (* succeed: identity *) -Qed. (* succeed: identity *) - -Lemma l2 : OMEGA = f OMEGA. -reflexivity. (* fail: conversion wants to convert OMEGA with f OMEGA *) -Abort. (* but it reduces the right side first! *) - -Lemma l3 : f OMEGA = OMEGA. -reflexivity. (* succeed: reduce left side first *) -Qed. (* succeed: expected concl (the one with f) is on the left *) - -Lemma l4 : OMEGA = OMEGA. -assert (f OMEGA = OMEGA) by reflexivity. (* succeed *) -unfold f in H. (* succeed: no type-checking *) -exact H. (* succeed: identity *) -Qed. (* fail: "f" is on the left *) - -(* This example would fail whatever the preferred side is *) -Lemma l5 : OMEGA = f OMEGA. -unfold f. -assert (f OMEGA = OMEGA) by reflexivity. -unfold f in H. -exact H. -Qed. (* needs to convert (f OMEGA = OMEGA) and (OMEGA = f OMEGA) *) - -(**********************************************************************) -(* Analysis of the inefficiency in Nijmegen/LinAlg/LinAlg/subspace_dim.v *) -(* (proof of span_ind_uninject_prop *) - -In the proof, a problem of the form (Equal S t1 t2) -is "simpl"ified, then "red"uced to (Equal S' t1 t1) -where the new t1's are surrounded by invisible coercions. -A reflexivity steps conclude the proof. - -The trick is that Equal projects the equality in the setoid S, and -that (Equal S) itself reduces to some (fun x y => Equal S' (f x) (g y)). - -At the Qed time, the problem to solve is (Equal S t1 t2) = (Equal S' t1 t1) -and the algorithm is to first compare S and S', and t1 and t2. -Unfortunately it does not work, and since t1 and t2 involve concrete -instances of algebraic structures, it takes a lot of time to realize that -it is not convertible. - -The only hope to improve this problem is to observe that S' hides -(behind two indirections) a Setoid constructor. This could be the -argument to solve the problem. - - diff --git a/dev/doc/notes-on-conversion.v b/dev/doc/notes-on-conversion.v new file mode 100644 index 0000000000..a81f170c63 --- /dev/null +++ b/dev/doc/notes-on-conversion.v @@ -0,0 +1,73 @@ +(**********************************************************************) +(* A few examples showing the current limits of the conversion algorithm *) +(**********************************************************************) + +(*** We define (pseudo-)divergence from Ackermann function ***) + +Definition ack (n : nat) := + (fix F (n0 : nat) : nat -> nat := + match n0 with + | O => S + | S n1 => + fun m : nat => + (fix F0 (n2 : nat) : nat := + match n2 with + | O => F n1 1 + | S n3 => F n1 (F0 n3) + end) m + end) n. + +Notation OMEGA := (ack 4 4). + +Definition f (x:nat) := x. + +(* Evaluation in tactics can somehow be controlled *) +Lemma l1 : OMEGA = OMEGA. +reflexivity. (* succeed: identity *) +Qed. (* succeed: identity *) + +Lemma l2 : OMEGA = f OMEGA. +reflexivity. (* fail: conversion wants to convert OMEGA with f OMEGA *) +Abort. (* but it reduces the right side first! *) + +Lemma l3 : f OMEGA = OMEGA. +reflexivity. (* succeed: reduce left side first *) +Qed. (* succeed: expected concl (the one with f) is on the left *) + +Lemma l4 : OMEGA = OMEGA. +assert (f OMEGA = OMEGA) by reflexivity. (* succeed *) +unfold f in H. (* succeed: no type-checking *) +exact H. (* succeed: identity *) +Qed. (* fail: "f" is on the left *) + +(* This example would fail whatever the preferred side is *) +Lemma l5 : OMEGA = f OMEGA. +unfold f. +assert (f OMEGA = OMEGA) by reflexivity. +unfold f in H. +exact H. +Qed. (* needs to convert (f OMEGA = OMEGA) and (OMEGA = f OMEGA) *) + +(**********************************************************************) +(* Analysis of the inefficiency in Nijmegen/LinAlg/LinAlg/subspace_dim.v *) +(* (proof of span_ind_uninject_prop *) + +In the proof, a problem of the form (Equal S t1 t2) +is "simpl"ified, then "red"uced to (Equal S' t1 t1) +where the new t1's are surrounded by invisible coercions. +A reflexivity steps conclude the proof. + +The trick is that Equal projects the equality in the setoid S, and +that (Equal S) itself reduces to some (fun x y => Equal S' (f x) (g y)). + +At the Qed time, the problem to solve is (Equal S t1 t2) = (Equal S' t1 t1) +and the algorithm is to first compare S and S', and t1 and t2. +Unfortunately it does not work, and since t1 and t2 involve concrete +instances of algebraic structures, it takes a lot of time to realize that +it is not convertible. + +The only hope to improve this problem is to observe that S' hides +(behind two indirections) a Setoid constructor. This could be the +argument to solve the problem. + + -- cgit v1.2.3