Merge branch 'main' of github.com:braxtonhall/fib

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+Require Import Coq.Arith.Wf_nat. (* import strong induction on naturals *)
+Require Import Lia. (* Linear Integer Arithmetic solver *)
+
+(* First, simple specification *)
+
+Fixpoint fib_simpl (n : nat) {struct n} : nat :=
+  match n with
+  | 0 => 1
+  | S n => match n with
+           | 0 => 1
+           | S m => fib_simpl n + fib_simpl m
+           end
+  end.
+
+Example fib_3 : (fib_simpl 2 = 2). reflexivity. Qed.
+Example fib_4 : (fib_simpl 3 = 3). reflexivity. Qed.
+Example fib_5 : (fib_simpl 4 = 5). reflexivity. Qed.
+
+Lemma fib_simpl_spec : forall n, (* useful lemma for rewriting *)
+    fib_simpl (S (S n)) = fib_simpl (S n) + fib_simpl n.
+Proof.
+  destruct n.
+  - (* case n := 0 *)
+    reflexivity.
+  - (* case n := (S n) *)
+    destruct n; reflexivity.
+Qed.
+      
+(* For a (slightly) faster version *)
+
+Fixpoint fib_acc_aux (n : nat) (acc : nat) (prev : nat) {struct n} : nat :=
+  match n with
+  | 0 => acc + prev
+  | S n => fib_acc_aux n (acc + prev) acc
+  end.
+
+Definition fib_faster (n : nat) : nat :=
+  match n with
+  | 0 => 1
+  | S n => match n with
+           | 0 => 1
+           | S m => (fib_acc_aux m 1 1)
+           end
+  end.
+
+Example fib_faster_3 : (fib_faster 2 = 2). reflexivity. Qed.
+Example fib_faster_4 : (fib_faster 3 = 3). reflexivity. Qed.
+Example fib_faster_5 : (fib_faster 4 = 5). reflexivity. Qed.
+
+Lemma fib_acc_aux_rewrite : forall n a b c d,
+    fib_acc_aux n a c + fib_acc_aux n b d = fib_acc_aux n (a + b) (c + d).
+Proof.
+  induction n; intros.
+  - (* case n := 0 *)
+    simpl; lia.
+  - (* case n := (S n) *)
+    simpl.
+    rewrite IHn.
+    f_equal.
+    lia.
+Qed.    
+  
+Theorem fib_faster_spec: forall n, fib_simpl n = fib_faster n.
+Proof.
+  induction n using lt_wf_ind. (* use strong induction *)
+  rename H into strong_induction_hypothesis.
+  destruct n. (* lt_wf_ind does not deal immediately with cases *)
+  - (* case n := 0 *)
+    reflexivity.
+  - (* case n := (S n) *)
+    destruct n.
+    + (* case n := 1 *)
+      reflexivity.
+    + (* case n := (S (S n)) *)
+      rewrite ?fib_simpl_spec.
+      do 2 rewrite strong_induction_hypothesis by lia.
+      enough (forall n, fib_faster (S n) + fib_faster n = fib_faster (S (S n))) as lemma by apply lemma.
+      clear.
+      destruct n.
+      * (* for lemma, case n := 0 *)
+        reflexivity.
+      * (* for lemma, case n := S n *)
+        destruct n.
+        -- (* for lemma, case n := 1 *)
+          reflexivity.
+        -- (* for lemma, case n := S (S n) *)
+           simpl. 
+           apply fib_acc_aux_rewrite.
+Qed.
+