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(*                                                             *)
(* This file contains the formal verification of BST insert    *)
(* and the alternative proofs using traditional definitions of *)
(* partial tree predicates. Those proofs and tactics outside   *)
(* Modules are shared.                                         *)
(*                                                             *)
(* Magic-wand-as-frame proof is in                             *)
(*  Module insert_by_WandQFrame_Func_Hole.                     *)
(*                                                             *)
(* Quantifier free version of Magic-wand-as-frame proof is in  *)
(*  Module insert_by_WandFrame.                                *)
(*                                                             *)
(* Another version of Magic-wand-as-frame proof is in          *)
(*  Module insert_by_WandQFrame_Ind_Hole.                      *)
(*                                                             *)
(* The proof using traditionally defined partial_treebox_rep   *)
(* is in                                                       *)
(*  Module insert_by_Ind_Pred_Ind_Hole.                        *)
(*                                                             *)
(*                                                             *)
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Lemma if_trueb: forall {A: Type} b (a1 a2: A), b = true -> (if b then a1 else a2) = a1.