Proof step 1 from proof subset-size (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  bag0 ⊆ bag1, ¬ # bag0 ≤ # bag1 ⊦

Rule argument

Additional information

Proof step 2 from proof subset-size (previous step) (back to tree)

User applied rule structural induction

And obtained the following new goals

• Proof step 3:

  bag ++ a ⊆ bag1, ¬ #(bag ++ a) ≤ # bag1, ∀ bag1. ¬(bag ⊆ bag1 ∧ ¬ # bag ≤ # bag1) ⊦

Rule argument

Additional information

Used simplifier rules: size-empty: ⊦ # ∅ = 0; bag generated by ∅, ++;; ⊦ 0 ≤ m; (s)

Proof step 3 from proof subset-size (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

  # bag2 < # bag, bag ⊆ bag2, ∀ bag0. ¬ bag ⊆ bag0 ∨ # bag ≤ # bag0 ⊦

Rule argument

Additional information

Used simplifier rules: subset-plus: ⊦ bag0 ++ a ⊆ bag1 ↔ (∃ bag2. bag1 = bag2 ++ a ∧ bag0 ⊆ bag2); size-rec: ⊦ #(bag ++ a) = # bag + 1; ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 4 from proof subset-size (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: bag2