add(closure): reflexive transitive closure of reduction and its algebra (M2 support)

theory/Closure.v

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+From Stdlib Require Import List Bool Arith Lia PeanoNat.
+Import ListNotations.
+From LambdaSub Require Import ExecReducer Binding Reduction Metatheory.
+
+Inductive Star (R : trm -> trm -> Prop) : trm -> trm -> Prop :=
+| star_refl : forall t, Star R t t
+| star_step : forall t u v, R t u -> Star R u v -> Star R t v.
+
+Lemma Star_trans : forall R a b c, Star R a b -> Star R b c -> Star R a c.
+Proof.
+  intros R a b c H. induction H; intros Hbc.
+  - exact Hbc.
+  - apply star_step with (u := u). exact H. apply IHStar. exact Hbc.
+Qed.
+
+Lemma red1_star : forall t t', red1 t t' -> Star red1 t t'.
+Proof.
+  intros t t' H. apply star_step with (u := t'). exact H. apply star_refl.
+Qed.
+
+Lemma Plus_to_Star : forall R t u, Plus R t u -> Star R t u.
+Proof.
+  intros R t u H. induction H.
+  - apply star_step with (u := u). exact H. apply star_refl.
+  - apply star_step with (u := u). exact H. exact IHPlus.
+Qed.
+
+Lemma Star_red1_context : forall C t t', Star red1 t t' -> Star red1 (plug C t) (plug C t').
+Proof.
+  intros C t t' H. induction H.
+  - apply star_refl.
+  - apply star_step with (u := plug C u).
+    + destruct H as [r Hr]. exists r. apply rctx. exact Hr.
+    + exact IHStar.
+Qed.
+
+Lemma star_one_trans : forall R a b c, R a b -> Star R b c -> Star R a c.
+Proof. intros. apply star_step with (u := b). exact H. exact H0. Qed.
+
+Print Assumptions Star_trans.
+Print Assumptions red1_star.
+Print Assumptions Plus_to_Star.
+Print Assumptions Star_red1_context.
+Print Assumptions star_one_trans.