# LTS Termination Proof

by T2Cert

## Input

Integer Transition System
• Initial Location: 2
• Transitions: (pre-variables and post-variables)  0 0 1: 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 0 ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ − a_1 + x2_post ≤ 0 ∧ a_1 − x2_post ≤ 0 ∧ ret_incr3_post − x2_post ≤ 0 ∧ − ret_incr3_post + x2_post ≤ 0 ∧ a_post − ret_incr3_post ≤ 0 ∧ − a_post + ret_incr3_post ≤ 0 ∧ a_0 − a_post ≤ 0 ∧ − a_0 + a_post ≤ 0 ∧ ret_incr3_0 − ret_incr3_post ≤ 0 ∧ − ret_incr3_0 + ret_incr3_post ≤ 0 ∧ x2_0 − x2_post ≤ 0 ∧ − x2_0 + x2_post ≤ 0 2 1 0: − x2_post + x2_post ≤ 0 ∧ x2_post − x2_post ≤ 0 ∧ − x2_0 + x2_0 ≤ 0 ∧ x2_0 − x2_0 ≤ 0 ∧ − ret_incr3_post + ret_incr3_post ≤ 0 ∧ ret_incr3_post − ret_incr3_post ≤ 0 ∧ − ret_incr3_0 + ret_incr3_0 ≤ 0 ∧ ret_incr3_0 − ret_incr3_0 ≤ 0 ∧ − a_post + a_post ≤ 0 ∧ a_post − a_post ≤ 0 ∧ − a_1 + a_1 ≤ 0 ∧ a_1 − a_1 ≤ 0 ∧ − a_0 + a_0 ≤ 0 ∧ a_0 − a_0 ≤ 0

## Proof

### 1 Invariant Updates

The following invariants are asserted.

 0: TRUE 1: 1 + a_post ≤ 0 ∧ 1 + ret_incr3_post ≤ 0 ∧ 1 + x2_post ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ 1 + a_0 ≤ 0 ∧ 1 + ret_incr3_0 ≤ 0 ∧ 1 + x2_0 ≤ 0 2: TRUE

The invariants are proved as follows.

### IMPACT Invariant Proof

• nodes (location) invariant:  0 (0) TRUE 1 (1) 1 + a_post ≤ 0 ∧ 1 + ret_incr3_post ≤ 0 ∧ 1 + x2_post ≤ 0 ∧ 1 + a_1 ≤ 0 ∧ −1 − a_1 ≤ 0 ∧ 1 + a_0 ≤ 0 ∧ 1 + ret_incr3_0 ≤ 0 ∧ 1 + x2_0 ≤ 0 2 (2) TRUE
• initial node: 2
• cover edges:
• transition edges:  0 0 1 2 1 0

### 2 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
and for every transition t, a duplicate t is considered.

### 3 Transition Removal

We remove transitions 0, 1 using the following ranking functions, which are bounded by −8.

 2: 0 0: 0 1: 0 2: −4 0: −5 1: −6
Hints:
 0 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ] 1 lexStrict[ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] , [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ]

### 4 SCC Decomposition

There exist no SCC in the program graph.

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• version: 1.0