Termination proof of /tmp/tmpsWXQft/PastaA9.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 ITRS

↳5 ITRStoIDPProof (⇔)

↳6 IDP

↳7 UsableRulesProof (⇔)

↳8 IDP

↳9 IDPNonInfProof (⇐)

↳10 IDP

↳11 IDependencyGraphProof (⇔)

↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA9

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaA9 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();
        int z = Random.random();

        if (y > 0) {
            while (x >= z) {
                z += y;
            }
        }
    }
}


public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 253 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Obligation:

ITRS problem:

The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The TRS R consists of the following rules:
Load1620(i14, i94, i87) → Cond_Load1620(i14 >= i87 && i94 > 0, i14, i94, i87)
Cond_Load1620(TRUE, i14, i94, i87) → Load1620(i14, i94, i87 + i94)
The set Q consists of the following terms:
Load1620(x0, x1, x2)
Cond_Load1620(TRUE, x0, x1, x2)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
Load1620(i14, i94, i87) → Cond_Load1620(i14 >= i87 && i94 > 0, i14, i94, i87)
Cond_Load1620(TRUE, i14, i94, i87) → Load1620(i14, i94, i87 + i94)

The integer pair graph contains the following rules and edges:
(0): LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(i14[0] >= i87[0] && i94[0] > 0, i14[0], i94[0], i87[0])
(1): COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], i87[1] + i94[1])

(0) -> (1), if ((i14[0] →^* i14[1])∧(i14[0] >= i87[0] && i94[0] > 0 →^* TRUE)∧(i87[0] →^* i87[1])∧(i94[0] →^* i94[1]))

(1) -> (0), if ((i87[1] + i94[1] →^* i87[0])∧(i94[1] →^* i94[0])∧(i14[1] →^* i14[0]))

The set Q consists of the following terms:
Load1620(x0, x1, x2)
Cond_Load1620(TRUE, x0, x1, x2)

(7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(8) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(i14[0] >= i87[0] && i94[0] > 0, i14[0], i94[0], i87[0])
(1): COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], i87[1] + i94[1])

(0) -> (1), if ((i14[0] →^* i14[1])∧(i14[0] >= i87[0] && i94[0] > 0 →^* TRUE)∧(i87[0] →^* i87[1])∧(i94[0] →^* i94[1]))

(1) -> (0), if ((i87[1] + i94[1] →^* i87[0])∧(i94[1] →^* i94[0])∧(i14[1] →^* i14[0]))

The set Q consists of the following terms:
Load1620(x0, x1, x2)
Cond_Load1620(TRUE, x0, x1, x2)

(9) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair LOAD1620(i14, i94, i87) → COND_LOAD1620(&&(>=(i14, i87), >(i94, 0)), i14, i94, i87) the following chains were created:

We consider the chain LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0]), COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], +(i87[1], i94[1])) which results in the following constraint:
(1)    (i14[0]=i14[1]∧&&(>=(i14[0], i87[0]), >(i94[0], 0))=TRUE∧i87[0]=i87[1]∧i94[0]=i94[1] ⇒ LOAD1620(i14[0], i94[0], i87[0])≥NonInfC∧LOAD1620(i14[0], i94[0], i87[0])≥COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])∧(U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(i14[0], i87[0])=TRUE∧>(i94[0], 0)=TRUE ⇒ LOAD1620(i14[0], i94[0], i87[0])≥NonInfC∧LOAD1620(i14[0], i94[0], i87[0])≥COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])∧(U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i87[0] + [bni_15]i14[0] ≥ 0∧[(-1)bso_16] + i94[0] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i87[0] + [bni_15]i14[0] ≥ 0∧[(-1)bso_16] + i94[0] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i87[0] + [bni_15]i14[0] ≥ 0∧[(-1)bso_16] + i94[0] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (i14[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[(-1)bso_16] + i94[0] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (i14[0] ≥ 0∧i94[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(8)    (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)

(9)    (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)

For Pair COND_LOAD1620(TRUE, i14, i94, i87) → LOAD1620(i14, i94, +(i87, i94)) the following chains were created:

We consider the chain LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0]), COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], +(i87[1], i94[1])), LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0]) which results in the following constraint:
(10)    (i14[0]=i14[1]∧&&(>=(i14[0], i87[0]), >(i94[0], 0))=TRUE∧i87[0]=i87[1]∧i94[0]=i94[1]∧+(i87[1], i94[1])=i87[0]1∧i94[1]=i94[0]1∧i14[1]=i14[0]1 ⇒ COND_LOAD1620(TRUE, i14[1], i94[1], i87[1])≥NonInfC∧COND_LOAD1620(TRUE, i14[1], i94[1], i87[1])≥LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))∧(U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥))

We simplified constraint (10) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(11)    (>=(i14[0], i87[0])=TRUE∧>(i94[0], 0)=TRUE ⇒ COND_LOAD1620(TRUE, i14[0], i94[0], i87[0])≥NonInfC∧COND_LOAD1620(TRUE, i14[0], i94[0], i87[0])≥LOAD1620(i14[0], i94[0], +(i87[0], i94[0]))∧(U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥))

We simplified constraint (11) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(12)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i87[0] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (12) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(13)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i87[0] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (13) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(14)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i87[0] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15)    (i14[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(16)    (i14[0] ≥ 0∧i94[0] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(17)    (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

(18)    (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

LOAD1620(i14, i94, i87) → COND_LOAD1620(&&(>=(i14, i87), >(i94, 0)), i14, i94, i87)
- (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)
- (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)
COND_LOAD1620(TRUE, i14, i94, i87) → LOAD1620(i14, i94, +(i87, i94))
- (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)
- (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (U^Increasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [1]
POL(FALSE) = [3]
POL(LOAD1620(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₁
POL(COND_LOAD1620(x₁, x₂, x₃, x₄)) = [-1]x₄ + [-1]x₃ + x₂ + [-1]x₁
POL(&&(x₁, x₂)) = [1]
POL(>=(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂

The following pairs are in P_>:

LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])

The following pairs are in P_bound:

LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])

The following pairs are in P_≥:

COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE¹ → &&(TRUE, TRUE)¹
FALSE¹ → &&(TRUE, FALSE)¹
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(FALSE, FALSE)¹

(10) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], i87[1] + i94[1])

The set Q consists of the following terms:
Load1620(x0, x1, x2)
Cond_Load1620(TRUE, x0, x1, x2)

(11) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) ITRStoIDPProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) IDPNonInfProof (SOUND transformation)

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE