diff options
| -rw-r--r-- | src/Algebra.v | 37 |
1 files changed, 27 insertions, 10 deletions
diff --git a/src/Algebra.v b/src/Algebra.v index 473571824..62b216b92 100644 --- a/src/Algebra.v +++ b/src/Algebra.v @@ -1038,32 +1038,49 @@ Ltac super_nsatz_post_clean_inequalities := try assumption; prensatz_contradict; nsatz_inequality_to_equality; try nsatz. +Ltac nsatz_equality_to_inequality_by_decide_equality := + lazymatch goal with + | [ H : not (?R _ _) |- ?R _ _ ] => idtac + | [ H : (?R _ _ -> False)%type |- ?R _ _ ] => idtac + | [ |- ?R _ _ ] => fail 0 "No hypothesis exists which negates the relation" R + | [ |- ?G ] => fail 0 "The goal is not a binary relation:" G + end; + lazymatch goal with + | [ |- ?R ?x ?y ] + => destruct (@dec (R x y) _); [ assumption | exfalso ] + end. (** Handles inequalities and fractions *) -Ltac super_nsatz := +Ltac super_nsatz_internal nsatz_alternative := (* [nsatz] gives anomalies on duplicate hypotheses, so we strip them *) clear_algebraic_duplicates; prensatz_contradict; (* Each goal left over by [prensatz_contradict] is separate (and there might not be any), so we handle them all separately *) [ try conservative_common_denominator_equality_inequality_all; - [ try nsatz_inequality_to_equality; try nsatz; - (* [nstaz] might leave over side-conditions; we handle them if they are inequalities *) - try super_nsatz_post_clean_inequalities + [ try nsatz_inequality_to_equality; + try first [ nsatz; + (* [nstaz] might leave over side-conditions; we handle them if they are inequalities *) + try super_nsatz_post_clean_inequalities + | nsatz_alternative ] | super_nsatz_post_clean_inequalities.. ].. ]. +Ltac super_nsatz := + super_nsatz_internal + (* if [nsatz] fails, we try turning the goal equality into an inequality and trying again *) + ltac:(nsatz_equality_to_inequality_by_decide_equality; + super_nsatz_internal idtac). + Section ExtraLemmas. Context {F eq zero one opp add sub mul inv div} `{F_field:field F eq zero one opp add sub mul inv div}. Local Infix "+" := add. Local Infix "*" := mul. Local Infix "-" := sub. Local Infix "/" := div. Local Notation "0" := zero. Local Notation "1" := one. Local Infix "=" := eq : type_scope. Local Notation "a <> b" := (not (a = b)) : type_scope. + Example _only_two_square_roots_test x y : x * x = y * y -> x <> opp y -> x = y. + Proof. intros; super_nsatz. Qed. + Lemma only_two_square_roots' x y : x * x = y * y -> x <> y -> x <> opp y -> False. - Proof. - intros. - canonicalize_field_equalities; canonicalize_field_inequalities. - assert (H' : (x + y) * (x - y) <> 0) by (apply mul_nonzero_nonzero; assumption). - apply H'; nsatz. - Qed. + Proof. intros; super_nsatz. Qed. Lemma only_two_square_roots x y z : x * x = z -> y * y = z -> x <> y -> x <> opp y -> False. Proof. |