[Imprint][Data Privacy Statement][E-Mail]



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 1 from proof Uniform-list (conclusion) (back to tree)

INDHYP-list([]), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x) ⊦ ∀ x. INDHYP-list(x)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof Uniform-list (previous step) (back to tree)

INDHYP-list([]), ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x) ⊦

User applied rule induction

And obtained the following new goals

Rule argument

indvar: # x wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: # x

Proof step 3 from proof Uniform-list (previous step) (back to tree)

INDHYP-list([]), ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list([]) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: array-listops, name:nil, seq: ⊦ [] = mkarray(0), subst:, <all paths>

Additional information

Used terms:

Proof step 4 from proof Uniform-list (previous step) (back to tree)

⊦ [] = mkarray(0)

A spec-lemma

Proof step 5 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [x0, INDHYP-list] → [x.rest, INDHYP-list]

Additional information

Used terms: x.rest, INDHYP-list

Proof step 6 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), ¬ # x.rest < # x, ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 7 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), # x ≤ # x.rest, ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: array-listops, name:rest, seq: ⊦ ar.rest = subarray(ar, 1, # ar - 1), subst:[ar] → [x], <all paths>

Additional information

Used terms: x

Proof step 8 from proof Uniform-list (previous step) (back to tree)

⊦ ar.rest = subarray(ar, 1, # ar - 1)

A spec-lemma

Proof step 9 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), # x ≤ # subarray(x, 1, # x - 1), ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ 0 ≤ m; (s) leprf: ⊦ m ≤ m - 1 → m = 0; (forward) from specification nat ⊦ # subarray(ar, m, n) = n; (s)

Proof step 10 from proof Uniform-list (previous step) (back to tree)

# x = 0, INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x), ∀ x0, INDHYP-list. # x0 < 0 → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: x

Additional information

Used simplifier rules: array generated by mkarray, [ ];; ⊦ # ar[n, a] = # ar; (s) ⊦ ar[n, a] = ar ↔ ¬ n < # ar ∨ ar[n] = a; (s) ⊦ m < n → n ≠ 0; (s) ⊦ # mkarray(n) = n; (s)

Proof step 11 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦ INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x.rest) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), INDHYP-list(x.rest), ¬ INDHYP-list(x), ∀ a, x. INDHYP-list(x) → INDHYP-list(a + x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: x.first, x.rest abort: F discard: T computed: F

Additional information

Used terms: x.first, x.rest

Proof step 13 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(x.rest), INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), INDHYP-list(x.rest) → INDHYP-list(x.first + x.rest), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 14 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), INDHYP-list(x.rest), INDHYP-list(x.first + x.rest), ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: x.first + x.rest = x

Additional information

Used formulas: x.first + x.rest = x

Proof step 15 from proof Uniform-list (previous step) (back to tree)

x.first + x.rest ≠ x, INDHYP-list(x.first + x.rest), INDHYP-list(x.rest), INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: array-listops, name:rest, seq: ⊦ ar.rest = subarray(ar, 1, # ar - 1), subst:[ar] → [x], <all paths>

Additional information

Used terms: x

Proof step 16 from proof Uniform-list (previous step) (back to tree)

⊦ ar.rest = subarray(ar, 1, # ar - 1)

A spec-lemma

Proof step 17 from proof Uniform-list (previous step) (back to tree)

x.first + subarray(x, 1, # x - 1) ≠ x, INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(x.first + subarray(x, 1, # x - 1)), ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: array-listops, name:first, seq: ⊦ ar.first = ar[0], subst:[ar] → [x], <all paths>

Additional information

Used terms: x

Proof step 18 from proof Uniform-list (previous step) (back to tree)

⊦ ar.first = ar[0]

A spec-lemma

Proof step 19 from proof Uniform-list (previous step) (back to tree)

x[0] + subarray(x, 1, # x - 1) ≠ x, INDHYP-list(x[0] + subarray(x, 1, # x - 1)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: array, name:Extension, seq:⊦ ar1 = ar2 ↔ # ar1 = # ar2 ∧ (∀ n. n < # ar1 → ar1[n] = ar2[n]), subst:[ar1, ar2] → [x[0] + subarray(x, 1, # x - 1), x], <all paths>

Additional information

Used terms: x[0] + subarray(x, 1, # x - 1), x

Proof step 20 from proof Uniform-list (previous step) (back to tree)

⊦ ar1 = ar2 ↔ # ar1 = # ar2 ∧ (∀ n. n < # ar1 → ar1[n] = ar2[n])

A spec-lemma

Proof step 21 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(x[0] + subarray(x, 1, # x - 1)), ¬ INDHYP-list(x), #(x[0] + subarray(x, 1, # x - 1)) ≠ # x ∨ (∃ n. ¬(n < #(x[0] + subarray(x, 1, # x - 1)) → (x[0] + subarray(x, 1, # x - 1))[n] = x[n])), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 22 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(x[0] + subarray(x, 1, # x - 1)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), #(x[0] + subarray(x, 1, # x - 1)) ≠ # x ∨ ¬(n < #(x[0] + subarray(x, 1, # x - 1)) → (x[0] + subarray(x, 1, # x - 1))[n] = x[n]), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:size-add, seq: ⊦ #(a + x) = (# x)+1, subst:[a, x] → [x[0], subarray(x, 1, # x - 1)], <all paths>

Additional information

Used terms: x[0], subarray(x, 1, # x - 1)

Proof step 23 from proof Uniform-list (previous step) (back to tree)

⊦ #(a + x) = (# x)+1

Lemma size-add

Proof step 24 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(x[0] + subarray(x, 1, # x - 1)), ¬ INDHYP-list(x), (# subarray(x, 1, # x - 1))+1 ≠ # x ∨ ¬(n < (# subarray(x, 1, # x - 1))+1 → (x[0] + subarray(x, 1, # x - 1))[n] = x[n]), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ # subarray(ar, m, n) = n; (s)

Proof step 25 from proof Uniform-list (previous step) (back to tree)

INDHYP-list(x[0] + subarray(x, 1, # x - 1)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), (# x - 1) + 1 ≠ # x ∨ ¬(¬ # x - 1 < n → (x[0] + subarray(x, 1, # x - 1))[n] = x[n]), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: nat, name:num-02, seq:n ≠ 0 ⊦ (n - 1) + 1 = n, subst:[n] → [# x], <all paths>

Additional information

Used terms: # x

Proof step 26 from proof Uniform-list (previous step) (back to tree)

n ≠ 0 ⊦ (n - 1) + 1 = n

A spec-lemma

Proof step 27 from proof Uniform-list (previous step) (back to tree)

# x = 0, INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(x[0] + subarray(x, 1, # x - 1)), ¬ INDHYP-list(x), (# x - 1) + 1 ≠ # x ∨ ¬(¬ # x - 1 < n → (x[0] + subarray(x, 1, # x - 1))[n] = x[n]), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≠ 0; (s)

Proof step 28 from proof Uniform-list (previous step) (back to tree)

# x = 0, INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, 0 - 1)), INDHYP-list(x[0] + subarray(x, 1, 0 - 1)), ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < 0 → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: x

Additional information

Used simplifier rules: array generated by mkarray, [ ];; ⊦ # ar[n, a] = # ar; (s) ⊦ ar[n, a] = ar ↔ ¬ n < # ar ∨ ar[n] = a; (s) ⊦ m < n → n ≠ 0; (s) ⊦ # mkarray(n) = n; (s)

Proof step 29 from proof Uniform-list (previous step) (back to tree)

# x ≠ 0, INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(x[0] + subarray(x, 1, # x - 1)), ¬ INDHYP-list(x), # x ≠ # x ∨ ¬(¬ # x - 1 < n → (x[0] + subarray(x, 1, # x - 1))[n] = x[n]), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ 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 < n + 1 ↔ ¬ n < m; (s) n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 30 from proof Uniform-list (previous step) (back to tree)

(x[0] + subarray(x, 1, # x - 1))[n] ≠ x[n], n < # x, INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, # x - 1)), INDHYP-list(x[0] + subarray(x, 1, # x - 1)), ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [# x, n0]

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # x, m

Proof step 31 from proof Uniform-list (previous step) (back to tree)

# x = n0 + 1, # x ≠ 0, (x[0] + subarray(x, 1, n0))[n] ≠ x[n], n < # x, INDHYP-list(x[0] + subarray(x, 1, n0)), INDHYP-list(subarray(x, 1, n0)), INDHYP-list(mkarray(0)), ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < # x → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m + 1 ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s)

Proof step 32 from proof Uniform-list (previous step) (back to tree)

# x = n0 + 1, (x[0] + subarray(x, 1, n0))[n] ≠ x[n], INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, n0)), INDHYP-list(x[0] + subarray(x, 1, n0)), n ≤ n0, ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < n0 + 1 → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: array-listops, name:cons, seq: ⊦ a + ar = copy(ar, 0, mkarray(# ar + 1), 1, # ar)[0, a], subst:[a, ar] → [x[0], subarray(x, 1, n0)], <all paths>

Additional information

Used terms: x[0], subarray(x, 1, n0)

Proof step 33 from proof Uniform-list (previous step) (back to tree)

⊦ a + ar = copy(ar, 0, mkarray(# ar + 1), 1, # ar)[0, a]

A spec-lemma

Proof step 34 from proof Uniform-list (previous step) (back to tree)

# x = n0 + 1, copy(subarray(x, 1, n0), 0, mkarray(# subarray(x, 1, n0) + 1), 1, # subarray(x, 1, n0))[0, x[0]][n] ≠ x[n], INDHYP-list(copy(subarray(x, 1, n0), 0, mkarray(# subarray(x, 1, n0) + 1), 1, # subarray(x, 1, n0))[0, x[0]]), INDHYP-list(subarray(x, 1, n0)), INDHYP-list(mkarray(0)), n ≤ n0, ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < n0 + 1 → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic n0 + n1 ≤ # ar0, m + m1 ≤ n1 ⊦ copy(subarray(ar0, n0, n1), m, ar1, m0, m1) = copy(ar0, n0 + m, ar1, m0, m1); (s) ⊦ m ≤ m; (s) ⊦ n + 0 = n; (s) ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ # subarray(ar, m, n) = n; (s)

Proof step 35 from proof Uniform-list (previous step) (back to tree)

# x = n0 + 1, copy(x, 1, mkarray(n0 + 1), 1, n0)[0, x[0]][n] ≠ x[n], INDHYP-list(mkarray(0)), INDHYP-list(subarray(x, 1, n0)), INDHYP-list(copy(x, 1, mkarray(n0 + 1), 1, n0)[0, x[0]]), n ≤ n0, ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < n0 + 1 → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: array, name:Put-other, seq: ⊦ m ≠ n → ar[n, a][m] = ar[m], subst:[ar, n, a, m] → [copy(x, 1, mkarray(n0 + 1), 1, n0), 0, x[0], n], <all paths>

Additional information

Used simplifier rules: ⊦ # copy(ar, m, ar0, m0, n) = # ar0; (s) ⊦ # mkarray(n) = n; (s) ⊦ 0 < m ↔ m ≠ 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 m < # ar ⊦ ar[m, a][m] = a; (s) Used terms: copy(x, 1, mkarray(n0 + 1), 1, n0), 0, x[0], n

Proof step 36 from proof Uniform-list (previous step) (back to tree)

⊦ m ≠ n → ar[n, a][m] = ar[m]

A spec-lemma

Proof step 37 from proof Uniform-list (previous step) (back to tree)

# x = n0 + 1, copy(x, 1, mkarray(n0 + 1), 1, n0)[n] ≠ x[n], n ≠ 0, INDHYP-list(copy(x, 1, mkarray(n0 + 1), 1, n0)[0, x[0]]), INDHYP-list(subarray(x, 1, n0)), INDHYP-list(mkarray(0)), n ≤ n0, ¬ INDHYP-list(x), ∀ x0, INDHYP-list. # x0 < n0 + 1 → INDHYP-list(mkarray(0)) ∧ ¬ INDHYP-list(x0) ∧ (∀ a, x. INDHYP-list(x) → INDHYP-list(a + x)) → false ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: array-copy, name:at-copy, seq:⊦ m0 < # ar1 → copy(ar0, n, ar1, n0, m)[m0] = (m0 < n0 ∨ n0 + m ≤ m0 ⊃ ar1[m0];ar0[m0 + n - n0]), subst:[n0, n, m, ar0, m0, ar1] → [1, 1, n0, x, n, mkarray(n0 + 1)], <all paths>

Additional information

Used simplifier rules: ⊦ m < 1 ↔ m = 0; (s) ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s) ⊦ n + 0 = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ # mkarray(n) = n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) Used terms: 1, 1, n0, x, n, mkarray(n0 + 1)

Proof step 38 from proof Uniform-list (previous step) (back to tree)

⊦ m0 < # ar1 → copy(ar0, n, ar1, n0, m)[m0] = (m0 < n0 ∨ n0 + m ≤ m0 ⊃ ar1[m0];ar0[m0 + n - n0])

A spec-lemma

Proof Tree for Uniform-list (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.

the proof tree