added test file

src/Test.idr

@@ -0,0 +1,128 @@
+module Test
+
+import Signature
+import Term
+import Data.Vect
+import Formula
+import Context
+import Proof
+
+namespace Sig
+  public export
+  zero : FuncSym
+  zero = MkFuncSym "0" 0
+
+  public export
+  suc : FuncSym
+  suc = MkFuncSym "S" 1
+
+  public export
+  plus : FuncSym
+  plus = MkFuncSym "+" 2
+
+  public export
+  times : FuncSym
+  times = MkFuncSym "*" 2
+
+  public export
+  leq : RelSym
+  leq = MkRelSym "<=" 2
+
+  public export
+  NT : Signature 4 1
+  NT = MkSignature
+    [ zero, suc, plus, times ]
+    [ leq ]
+
+s0ps0 : Term NT 0
+s0ps0 = Func 2
+  [ Func 1 [ Func 0 [] ]
+  , Func 1 [ Func 0 [] ]
+  ]
+
+O : Term NT d
+O = Func 0 []
+
+S : Term NT d -> Term NT d
+S t = Func 1 [ t ]
+
+export infixl 8 +
+(+) : Term NT d -> Term NT d -> Term NT d
+s + t = Func 2 [ s, t ]
+
+export infixl 9 *
+(*) : Term NT d -> Term NT d -> Term NT d
+s * t = Func 3 [ s, t ]
+
+Y : Term NT (S d)
+Y = Var 0
+
+X : Term NT (S (S d))
+X = Var 1
+
+private infix 3 .=
+(.=) : Term sig n -> Term sig n -> Formula sig n
+(.=) = Equal
+
+(<=) : Term NT n -> Term NT n -> Formula NT n
+t1 <= t2 = Rel 0 [ t1, t2 ]
+
+addComm : Formula NT 0
+addComm = Forall $ Forall $ Y + X .= X + Y
+
+private infixr 2 ==>
+(==>) : Formula sig n -> Formula sig n -> Formula sig n
+(==>) = Imp
+
+Not : Formula sig n -> Formula sig n
+Not phi = phi ==> Bot
+
+(&&) : Formula sig n -> Formula sig n -> Formula sig n
+phi && psi = Not (phi ==> Not psi)
+
+(||) : Formula sig n -> Formula sig n -> Formula sig n
+phi || psi = Not (Not phi && Not psi)
+
+zeroNotSuc : Formula NT d
+zeroNotSuc = Forall $ Not $ O .= S (Var 0)
+
+sucInj : Formula NT d
+sucInj = Forall $ Forall $ S (Var 0) .= S (Var 1) ==> Var 0 .= Var 1
+
+addZeroR : Formula NT d
+addZeroR = Forall $ Var 0 + O .= Var 0
+
+addSR : Formula NT d
+addSR = Forall $ Forall $ X + S Y .= S (X + Y)
+
+mulZeroR : Formula NT d
+mulZeroR = Forall $ Y * O .= O
+
+mulSucR : Formula NT d
+mulSucR = Forall $ Forall $ X * S Y .= X * Y + X
+
+induction : {d : _} -> Formula NT (S d) -> Formula NT d
+induction phi =
+  elimBinder O phi ==>
+  (Forall $ phi ==> (sub 0 (S Y) phi)) ==>
+  Forall phi
+
+doubleNeg :
+  {d : _} -> {c : _} ->
+  {phi : Formula sig d} ->
+  Proof c (Not (Not phi)) -> Proof c phi
+doubleNeg pf =
+  Contra $
+  ImpE (Weaken pf (Missing Same)) $
+  Hyp 0
+  
+AndI : {d : _} -> {c : _} ->
+  {phi, psi : Formula sig d} ->
+  Proof c phi -> Proof c psi ->
+  Proof c (phi && psi)
+AndI pfPhi pfPsi =
+  ImpI $
+  ImpE
+    (ImpE (Hyp 0) $
+     Weaken pfPhi $ Missing Same)
+    (Weaken pfPsi $ Missing Same)