Enhance README with usage examples and syntax

README.md

@@ -2,3 +2,153 @@
 ![Logo](assets/logo.png)
 
 tryCoq provides interactive equational proof with ADT type
+
+tryCoq is an **interactive proof assistant** that takes **OCaml code** as input and allows users to **perform proofs interactively**.  
+It is designed for reasoning about functional programs using techniques from **structural induction**, **equational reasoning**.
+
+---
+
+## Example
+
+Input (OCaml function):
+```ocaml
+type nat = Z | S of nat
+
+let rec add x y =
+  match x with
+  | Z -> y
+  | S n -> S (add n y)
+```
+
+Goal (to prove interactively):
+```
+forall (x:nat), add x Z = x
+```
+
+Interactive Proof Sequences:
+```
+>>> induction x
+>>> simpl
+>>> reflexivity
+>>> simpl
+>>> rewrite IH1 in goal at *
+>>> reflexivity
+```
+
+---
+
+## Getting Started
+### Requirements
+tryCoq use OCaml package manager(OPAM) and dune
+
+### Dependencies
+First, run:
+```bash
+dune build
+```
+
+To install the dependency, run:
+```bash
+opam install . --deps-only
+```
+
+### Run
+```bash
+$ dune exec tryCoq
+Enter the definition file path (1/2):
+>
+```
+
+---
+
+## Interactive Proving
+First, insert ocaml code location having ADT type and function definitions
+tryCoq take 2 file paths
+```
+Enter the definition file path (1/2):
+> exam.ml
+```
+
+Second, choose the proof mode
+```
+Choose the proof type :
+1) Interactive Mode      2) Auto Mode
+1
+```
+
+Second, insert conjecture that you want to prove
+```
+No conjecture
+>>> assert forall (x:nat), add x Z = x
+```
+
+Now, you can proof interactively with tryCoq
+```
+1st goal of : forall (nat1:nat), add (nat1) (0) = nat1
+
+---------------------------------------
+forall (nat1:nat), add (nat1) (0) = nat1
+
+0 goal(s) more...
+
+0 conjecture(s) more...
+
+>>> induction nat
+...
+```
+
+## Proposition Language
+```
+<prop>  ::= "forall" <prop>
+          | <expr> "=" <expr>
+          | <prop>* "->" <prop>
+
+<expr>  ::= <fun_name> <expr>*
+          | <constructor> <expr>*
+          | <var>
+```
+
+## Proof Language(tactic)
+```
+tactic           ::=
+                  | "assert" <prop>
+                  | "simpl"
+                  | "simpl in" <target_label> 
+                  | "rewrite" <rewrite_label> "in" <target_label> "at" <loc>
+                  | "induction" <var>
+                  | "case" <expr>
+                  | "intro" <var>
+                  | "intro" <fact_label>
+                  | "reflexivity"
+                  | "discriminate"
+
+<target_label>  ::= <fact_label> | "goal"
+
+<rewrite_label> ::= <fact_label> | <lemma_label>
+
+<loc>           ::= "*" | "1" | "2" | ...
+
+<fact_label>    ::= "IH1" | "Case1" | "Cond1" | ...
+
+<lemma_label>   ::= "lemma1" | ...
+
+<var>           ::= "x" | "y" | ...
+
+```
+
+
+```markdown
+| No. | tactic      | Description                 |
+|------|------------------|----------------------------------|
+| 7    | `assert`        | Establish lemma or theroem |
+| 3    | `simpl`         | Simplify the target.           |
+| 4    | `rewrite IH1`   | Rewrite goal using assumption or lemma.   |
+| 2    | `induction x`   | Perform structural induction on `x`.          |
+| 6    | `case x`        | Split goal into two subgoal x is true and x is false |
+| 1    | `intro`        | Introduced variable or implication condtion into the context .         |
+| 5    | `reflexivity`   | Solve equalities of the form `a=a`               |
+| 8    | `discriminate`   | Solve propostion if assumption leads contradiction |
+```
+
+
+