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diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_cir.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_cir.ma
index 5cee2e207..99846d67a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_cir.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_cir.ma
@@ -17,9 +17,9 @@ include "basic_2/reduction/cpr.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
 
-(* Advanced forward lemmas on context-sensitive irreducible terms ***********)
+(* Advanced forward lemmas on irreducibility ********************************)
 
-lemma cpr_fwd_cir: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ 𝐈⦃T1⦄ → T2 = T1.
+lemma cpr_fwd_cir: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ ➡ 𝐈⦃T1⦄ → T2 = T1.
 #G #L #T1 #T2 #H elim H -G -L -T1 -T2
 [ //
 | #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
@@ -28,15 +28,15 @@ lemma cpr_fwd_cir: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ 
   [ elim (cir_inv_bind … H) -H #HV1 #HT1 * #H destruct
     lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
     lapply (IHT1 … HT1) -IHT1 #H destruct //
-  | elim (cir_inv_ib2 … H) -H /2 width=1/ /3 width=2/
+  | elim (cir_inv_ib2 … H) -H /3 width=2 by or_introl, eq_f2/
   ]
 | * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
   [ elim (cir_inv_appl … H) -H #HV1 #HT1 #_
     >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
-  | elim (cir_inv_ri2 … H) /2 width=1/
+  | elim (cir_inv_ri2 … H) /2 width=1 by/
   ]
 | #G #L #V1 #T1 #T #T2 #_ #_ #_ #H
-  elim (cir_inv_ri2 … H) /2 width=1/
+  elim (cir_inv_ri2 … H) /2 width=1 by or_introl/
 | #G #L #V1 #T1 #T2 #_ #_ #H
   elim (cir_inv_ri2 … H) /2 width=1/
 | #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H