Proof step 1 from proof unequal (conclusion) (back to tree)

# x = # y, x ≠ y ⊦ ∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y = z + a0 ' + z1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 2:
# x = # y, x ≠ y ⊦ ∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y = z + a0 ' + z1

Rule argument

Additional information

Proof step 2 from proof unequal (previous step) (back to tree)

# x = # y, x ≠ y ⊦ ∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y = z + a0 ' + z1

User applied rule structural induction

And obtained the following new goals

Proof step 3:
# x = #(a + y0), x ≠ a + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ x = z + a0 ' + z0 ∧ a + y0 = z + a1 ' + z1

Rule argument

indvar: y

Additional information

Used simplifier rules: list('a) freely generated by [], +;; ⊦ # [] = 0; (s) ⊦ # x = 0 ↔ x = []; (s)

Proof step 3 from proof unequal (previous step) (back to tree)

# x = #(a + y0), x ≠ a + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ x = z + a0 ' + z0 ∧ a + y0 = z + a1 ' + z1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
# x = # y0 + 1, x ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ x = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Rule argument

Additional information

Used simplifier rules: len-05: ⊦ #(a ' + x) = (# x)+1; a: ⊦ (x + y) + z = x + y + z; c: ⊦ a + x = a ' + x; ⊦ m +1 = m + 1; (s)

Proof step 4 from proof unequal (previous step) (back to tree)

# x = # y0 + 1, x ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ x = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

User applied rule constructor cut

And obtained the following new goals

Proof step 5:
# x = # y0 + 1, x = a2 + x0, x ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ x = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Rule argument

var: x

Additional information

Used simplifier rules: list('a) freely generated by [], +;; ⊦ # [] = 0; (s) ⊦ m + 1 ≠ 0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 5 from proof unequal (previous step) (back to tree)

# x = # y0 + 1, x = a2 + x0, x ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ x = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
# x0 = # y0, a2 ' + x0 ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Rule argument

Additional information

Used simplifier rules: len-05: ⊦ #(a ' + x) = (# x)+1; a: ⊦ (x + y) + z = x + y + z; c: ⊦ a + x = a ' + x; c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ m +1 = m + 1; (s)

Proof step 6 from proof unequal (previous step) (back to tree)

# x0 = # y0, a2 ' + x0 ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 7:
# x0 = # y0, a ' ≠ [], a2 ' + x0 ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Rule argument

fma: a ' = []

Additional information

Used simplifier rules: e: ⊦ a ' ≠ []; Used formulas: a ' = []

Proof step 7 from proof unequal (previous step) (back to tree)

# x0 = # y0, a ' ≠ [], a2 ' + x0 ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
# x0 = # y0, a2 ' + x0 ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Rule argument

Additional information

Used simplifier rules: e: ⊦ a ' ≠ [];

Proof step 8 from proof unequal (previous step) (back to tree)

User applied rule cut formula

And obtained the following new goals

Proof step 9:
# x0 = # y0, a2 = a, a2 ' + x0 ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1
Proof step 13:
# x0 = # y0, a2 ' + x0 ≠ a ' + y0, a2 ≠ a, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Rule argument

fma: a2 = a

Additional information

Used formulas: a2 = a

Proof step 9 from proof unequal (previous step) (back to tree)

# x0 = # y0, a2 = a, a2 ' + x0 ≠ a ' + y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
# x0 = # y0, x0 ≠ y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Rule argument

Additional information

Used simplifier rules: app-08: ⊦ a ' + x = a ' + y ↔ x = y; a: ⊦ (x + y) + z = x + y + z;

Proof step 10 from proof unequal (previous step) (back to tree)

# x0 = # y0, x0 ≠ y0, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

User applied rule all left

And obtained the following new goals

Proof step 11:
# x0 = # y0, x0 ≠ y0, # x0 = # y0 ∧ x0 ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x0 = z + a ' + z0 ∧ y0 = z + a0 ' + z1), ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Rule argument

pos: left 3 terms: x0 abort: F discard: F computed: T

Additional information

Used terms: x0

Proof step 11 from proof unequal (previous step) (back to tree)

# x0 = # y0, x0 ≠ y0, # x0 = # y0 ∧ x0 ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x0 = z + a ' + z0 ∧ y0 = z + a0 ' + z1), ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
# a0 ' + # z0 = # a1 ' + # z1, a0 ' + z0 ≠ a1 ' + z1, a0 ≠ a1, ∀ x. # x = # z + # a1 ' + # z1 ∧ x ≠ z + a1 ' + z1 → (∃ z0, a, a0, z2, z3. a ≠ a0 ∧ x = z0 + a ' + z2 ∧ z + a1 ' + z1 = z0 + a0 ' + z3) ⊦ ∃ z2, a2, a3, z3, z4. a2 ≠ a3 ∧ a ' + z + a0 ' + z0 = z2 + a2 ' + z3 ∧ a ' + z + a1 ' + z1 = z2 + a3 ' + z4

Rule argument

Additional information

Used simplifier rules: app-01: ⊦ x + y = x + z ↔ y = z; len: ⊦ #(x + y) = # x + # y; a: ⊦ (x + y) + z = x + y + z; c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s)

Proof step 12 from proof unequal (previous step) (back to tree)

# a0 ' + # z0 = # a1 ' + # z1, a0 ' + z0 ≠ a1 ' + z1, a0 ≠ a1, ∀ x. # x = # z + # a1 ' + # z1 ∧ x ≠ z + a1 ' + z1 → (∃ z0, a, a0, z2, z3. a ≠ a0 ∧ x = z0 + a ' + z2 ∧ z + a1 ' + z1 = z0 + a0 ' + z3) ⊦ ∃ z2, a2, a3, z3, z4. a2 ≠ a3 ∧ a ' + z + a0 ' + z0 = z2 + a2 ' + z3 ∧ a ' + z + a1 ' + z1 = z2 + a3 ' + z4

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: a ' + z, a0, a1, z0, z1

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; Used terms: a ' + z, a0, a1, z0, z1

Proof step 13 from proof unequal (previous step) (back to tree)

# x0 = # y0, a2 ' + x0 ≠ a ' + y0, a2 ≠ a, ∀ x. # x = # y0 ∧ x ≠ y0 → (∃ z, a, a0, z0, z1. a ≠ a0 ∧ x = z + a ' + z0 ∧ y0 = z + a0 ' + z1) ⊦ ∃ z, a0, a1, z0, z1. a0 ≠ a1 ∧ a2 ' + x0 = z + a0 ' + z0 ∧ a ' + y0 = z + a1 ' + z1

User applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: [], a2, a, x0, y0 abort: F discard: T computed: F

Additional information

Used simplifier rules: Nil: ⊦ [] + x = x; c-06: ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; a: ⊦ (x + y) + z = x + y + z; Used terms: [], a2, a, x0, y0

Proof Tree for unequal (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.