@@ -101,7 +101,7 @@ Abort.
101101Local Parameter P Q R : Prop.
102102Local Parameter HPQ : P <-> Q.
103103
104- Waterproof Register Unfold Apply "notation" "for " "P" P ; (HPQ).
104+ Waterproof Register Unfold Apply "notation" "for " "P" P ; "Alternative characterization of P"; (HPQ).
105105
106106(* Test 8: Use alternative characterization, with concept in conclusion,
107107but without having the automation able to prove the alternative
@@ -150,7 +150,8 @@ Proof.
150150 (fun () => Unfold the definition of notation for P)
151151"Remove this line in the final version of your proof.")
152152 Info
153- ["Applied alternative characterizations in statements where applicable.";
153+ ["Alternative characterization of P";
154+ "Applied alternative characterizations in statements where applicable.";
154155"Hint, replace with: It suffices to show that Q."].
155156It suffices to show that Q.
156157Abort .
@@ -165,7 +166,8 @@ Proof.
165166 (fun () => Unfold the definition of notation for P)
166167"Remove this line in the final version of your proof.")
167168 Info
168- ["Applied alternative characterizations in statements where applicable.";
169+ ["Alternative characterization of P";
170+ "Applied alternative characterizations in statements where applicable.";
169171"Hint, replace with: It holds that Q."].
170172It holds that Q.
171173Abort .
@@ -178,7 +180,7 @@ Proof.
178180assert_constr_equal (tactic_in_constr constr:(HPR) constr:(P -> Q)) constr:(R -> Q).
179181Abort .
180182
181- Waterproof Register Unfold Rewrite "characterization" "of" "P" P ; HPR.
183+ Waterproof Register Unfold Rewrite "characterization" "of" "P" P ; "Characterization of P"; HPR.
182184
183185Local Hint Extern 1 => rewrite HPR : core.
184186
@@ -193,8 +195,10 @@ intros.
193195 (fun () => Unfold the definition of notation for P)
194196"Remove this line in the final version of your proof.")
195197 Info
196- ["Applied alternative characterizations in statements where applicable.";
198+ ["Characterization of P";
199+ "Applied alternative characterizations in statements where applicable.";
197200"Hint, replace with: It suffices to show that R.";
201+ "Alternative characterization of P";
198202"Applied alternative characterizations in statements where applicable.";
199203"Hint, replace with: It suffices to show that Q."].
200204It suffices to show that Q.
@@ -223,11 +227,13 @@ assert_feedback_with_strings
223227 (fun () => Unfold the definition of infimum)
224228"Remove this line in the final version of your proof.")
225229 Info
226- ["Expanded definition in statements where applicable.";
230+ ["Definition infimum";
231+ "Expanded definition in statements where applicable.";
227232"Hint, insert: We need to show that (3 is a _lower bound_ for A
228233 ∧ (∀ l ∈ ℝ, l is a _lower bound_ for A ⇨ l ≤ 3)).";
229234"Hint, insert: It holds that (4 is a _lower bound_ for A
230235 ∧ (∀ l ∈ ℝ, l is a _lower bound_ for A ⇨ l ≤ 4)).";
236+ "Alternative characterization infimum";
231237"Applied alternative characterizations in statements where applicable.";
232238"Hint, insert: It suffices to show that (3 is a _lower bound_ for A
233239 ∧ (∀ ε > 0, ∃ a ∈ A, a < 3 + ε)).";
@@ -244,11 +250,13 @@ assert_feedback_with_strings
244250 (fun () => Unfold the definition of supremum)
245251"Remove this line in the final version of your proof.")
246252 Info
247- ["Expanded definition in statements where applicable.";
253+ ["Definition supremum";
254+ "Expanded definition in statements where applicable.";
248255"Hint, insert: We need to show that (3 is an _upper bound_ for A
249256 ∧ (∀ L ∈ ℝ, L is an _upper bound_ for A ⇨ 3 ≤ L)).";
250257"Hint, insert: It holds that (4 is an _upper bound_ for A
251258 ∧ (∀ L ∈ ℝ, L is an _upper bound_ for A ⇨ 4 ≤ L)).";
259+ "Alternative characterization supremum";
252260"Applied alternative characterizations in statements where applicable.";
253261"Hint, insert: It suffices to show that (3 is an _upper bound_ for A
254262 ∧ (∀ ε > 0, ∃ a ∈ A, 3 - ε < a)).";
@@ -265,15 +273,20 @@ assert_feedback_with_strings
265273 (fun () => Unfold All)
266274"Remove this line in the final version of your proof.")
267275 Info
268- ["Expanded definition in statements where applicable.";
276+ [
277+ "Definition infimum";
278+ "Expanded definition in statements where applicable.";
269279"Hint, replace with: We need to show that (3 is a _lower bound_ for A
270280 ∧ (∀ l ∈ ℝ, l is a _lower bound_ for A ⇨ l ≤ 3)).";
281+ "Alternative characterization infimum";
271282"Applied alternative characterizations in statements where applicable.";
272283"Hint, replace with: It suffices to show that (3 is a _lower bound_ for A
273284 ∧ (∀ ε > 0, ∃ a ∈ A, a < 3 + ε)).";
285+ "Definition supremum";
274286"Expanded definition in statements where applicable.";
275287"Hint, replace with: It holds that (4 is an _upper bound_ for A
276288 ∧ (∀ L ∈ ℝ, L is an _upper bound_ for A ⇨ 4 ≤ L)).";
289+ "Alternative characterization supremum";
277290"Applied alternative characterizations in statements where applicable.";
278291"Hint, replace with: It holds that (4 is an _upper bound_ for A ∧ (∀ ε > 0, ∃ a ∈ A, 4 - ε < a))."].
279292Abort .
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