import importlib_resources
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsMember, IsSubset, SetUnion, SetIntersect, SetComplement, EmptySet, SetAdd
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort, min_intsort, max_intsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from lemsynth.lemsynth_engine import solveProblem
# Declarations
x = Var('x', fgsort)
y, nil = Consts('y nil', fgsort)
k = Const('k', intsort)
key = Function('key', fgsort, intsort)
lft = Function('lft', fgsort, fgsort)
rght = Function('rght', fgsort, fgsort)
minr = Function('minr', fgsort, intsort)
maxr = Function('maxr', fgsort, intsort)
bst = RecFunction('bst', fgsort, boolsort)
hbst = RecFunction('hbst', fgsort, fgsetsort)
AddRecDefinition(minr, x, If(x == nil, 100, min_intsort(key(x), minr(lft(x)), minr(rght(x)))))
AddRecDefinition(maxr, x, If(x == nil, -1, max_intsort(key(x), maxr(lft(x)), maxr(rght(x)))))
AddRecDefinition(bst, x, If(x == nil, True,
And(0 < key(x),
And(key(x) < 100,
And(bst(lft(x)),
And(bst(rght(x)),
And(maxr(lft(x)) <= key(x),
key(x) <= minr(rght(x)))))))))
import importlib_resources
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
from lemsynth.lemsynth_engine import solveProblem
# Declarations
x, y = Vars('x y', fgsort)
nil, c = Consts('nil c', fgsort)
nxt = Function('nxt', fgsort, fgsort)
prv = Function('prv', fgsort, fgsort)
lst = RecFunction('lst', fgsort, boolsort)
dlst = RecFunction('dlst', fgsort, boolsort)
lseg = RecFunction('lseg', fgsort, fgsort, boolsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(dlst, x, If(x == nil, True,
If(nxt(x) == nil, True,
And(prv(nxt(x)) == x, dlst(nxt(x))))))
AddRecDefinition(lseg, (x, y) , If(x == y, True, lseg(nxt(x), y)))
AddAxiom((), nxt(nil) == nil)
# Problem parameters
goal = Implies(lseg(x, y), Implies(x != c, Implies(dlst(y), lst(x))))
from naturalproofs.prover import NPSolver
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort, min_intsort, max_intsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
def notInChildren(x):
return And(Not(IsMember(x, htree(lft(x)))),
Not(IsMember(x, htree(rght(x)))))
# declarations
x, y = Vars('x y', fgsort)
nil, ret = Consts('nil ret', fgsort)
lft = Function('lft', fgsort, fgsort)
rght = Function('rght', fgsort, fgsort)
tree = RecFunction('tree', fgsort, boolsort)
htree = RecFunction('htree', fgsort, fgsetsort)
reach_lr = RecFunction('reach_lr', fgsort, fgsort, boolsort)
AddRecDefinition(
tree, x,
If(
x == nil, True,
And(
notInChildren(x),
And(
SetIntersect(htree(lft(x)),
htree(rght(x))) == fgsetsort.lattice_bottom,
And(tree(lft(x)), tree(rght(x)))))))
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
from naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
# declarations
x, y = Vars('x y', fgsort)
nil, z = Consts('nil z', fgsort)
k = Const('k', intsort)
nxt = Function('nxt', fgsort, fgsort)
lst = RecFunction('lst', fgsort, boolsort)
lseg = RecFunction('lseg', fgsort, fgsort, boolsort)
key = Function('key', fgsort, intsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(lseg, (x, y), If(x == y, True, lseg(nxt(x), y)))
AddAxiom((), nxt(nil) == nil)
# vc
goal = Implies(lseg(x, y), Implies(And(key(x) != k, nxt(y) == z), lseg(x, z)))
# check validity with natural proof solver and no hardcoded lemmas
np_solver = NPSolver()
solution = np_solver.solve(make_pfp_formula(goal))
import importlib_resources
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.prover import NPSolver
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
# declarations
x = Var('x', fgsort)
c, s, nil = Consts('c s nil', fgsort)
v1 = Function('v1', fgsort, fgsort)
v2 = Function('v2', fgsort, fgsort)
p = Function('p', fgsort, fgsort)
n = Function('n', fgsort, fgsort)
reach_pgm = RecFunction('reach_pgm', fgsort, boolsort)
# precondition
AddAxiom((), v1(s) == n(v2(s)))
cond = v1(p(x)) != nil
assign1 = v1(x) == n(v1(p(x)))
assign2 = v2(x) == n(v2(p(x)))
assign = And(assign1, assign2)
AddRecDefinition(reach_pgm, x,
from z3 import And, Or, Not, Implies, If
from z3 import IsMember, IsSubset, SetUnion, SetIntersect, SetComplement, EmptySet, SetAdd
from naturalproofs.prover import NPSolver
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort, min_intsort, max_intsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
def notInChildren(x):
return And(Not(IsMember(x, hbst(lft(x)))), Not(IsMember(x, hbst(rght(x)))))
# declarations
x, y = Vars('x y', fgsort)
z, nil = Consts('z nil', fgsort)
key = Function('key', fgsort, intsort)
lft = Function('lft', fgsort, fgsort)
rght = Function('rght', fgsort, fgsort)
minr = RecFunction('minr', fgsort, intsort)
maxr = RecFunction('maxr', fgsort, intsort)
bst = RecFunction('bst', fgsort, boolsort)
hbst = RecFunction('hbst', fgsort, fgsetsort)
AddRecDefinition(minr, x, If(x == nil, 100, min_intsort(key(x), minr(lft(x)), minr(rght(x)))))
AddRecDefinition(maxr, x, If(x == nil, -1, max_intsort(key(x), maxr(lft(x)), maxr(rght(x)))))
AddRecDefinition(bst, x, If(x == nil, True,
And(0 < key(x),
And(key(x) < 100,
And(bst(lft(x)),
And(bst(rght(x)),
And(maxr(lft(x)) <= key(x),
import importlib_resources
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
from lemsynth.lemsynth_engine import solveProblem
# Declarations
x, y = Vars('x y', fgsort)
c, nil, z = Consts('c nil z', fgsort)
nxt = Function('nxt', fgsort, fgsort)
lst = RecFunction('lst', fgsort, boolsort)
lseg = RecFunction('lseg', fgsort, fgsort, boolsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(lseg, (x, y) , If(x == y, True, lseg(nxt(x), y)))
AddAxiom((), nxt(nil) == nil)
# Problem parameters
pgm = And(lst(y), And(x != c, nxt(y) == z))
goal = Implies(lseg(x, y), Implies(pgm, And(lseg(x, z), lst(z))))
# parameters representing the grammar for synth-fun and
# terms on which finite model is extracted
# TODO: extract this automatically from grammar_string
v1, v2 = Vars('v1 v2', fgsort)
# Only importing this for writing this file as a test
import unittest
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.pfp import make_pfp_formula
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
# Declarations
a, b, c, zero = Consts('a b c zero', fgsort)
suc = Function('suc', fgsort, fgsort)
pred = Function('pred', fgsort, fgsort)
AddAxiom(a, pred(suc(a)) == a)
nat = RecFunction('nat', fgsort, boolsort)
add = RecFunction('add', fgsort, fgsort, fgsort, boolsort)
AddRecDefinition(nat, a, If(a == zero, True, nat(pred(a))))
AddRecDefinition(add, (a, b, c), If(a == zero, b == c, add(pred(a), suc(b), c)))
# Problem parameters
goal = Implies(add(a, b, c), Implies(And(nat(a), nat(b)), nat(c)))
pfp_of_goal = make_pfp_formula(goal)
# Call a natural proofs solver
npsolver = NPSolver()
# Configure the solver
npsolver.options.instantiation_mode = proveroptions.manual_instantiation
npsolver.options.terms_to_instantiate = {a, b, c, zero, pred(b), suc(b)}
# Ask for proof
npsolution = npsolver.solve(pfp_of_goal)
"""
# Only importing this for writing this file as a test
import unittest
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.pfp import make_pfp_formula
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
# Declarations
a, zero = Consts('a zero', fgsort)
suc = Function('suc', fgsort, fgsort)
pred = Function('pred', fgsort, fgsort)
AddAxiom(a, pred(suc(a)) == a)
nat = RecFunction('nat', fgsort, boolsort)
even = RecFunction('even', fgsort, boolsort)
odd = RecFunction('odd', fgsort, boolsort)
AddRecDefinition(nat, a, If(a == zero, True, nat(pred(a))))
AddRecDefinition(even, a, If(a == zero, True, odd(pred(a))))
AddRecDefinition(odd, a, If(a == zero, False, If(a == suc(zero), True, even(pred(a)))))
# Problem parameters
goal = Implies(nat(a), Or(even(a), odd(a)))
pfp_of_goal = make_pfp_formula(goal)
# Call a natural proofs solver
npsolver = NPSolver()
# Ask for proof
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
from naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
# declarations
x = Var('x', fgsort)
nil, yc, zc = Consts('nil yc zc', fgsort)
k = Const('k', intsort)
nxt = Function('nxt', fgsort, fgsort)
lsegy = RecFunction('lsegy', fgsort, boolsort)
lsegz = RecFunction('lsegz', fgsort, boolsort)
lsegy_p = RecFunction('lsegy_p', fgsort, boolsort)
lsegz_p = RecFunction('lsegz_p', fgsort, boolsort)
key = Function('key', fgsort, intsort)
AddRecDefinition(lsegy, x, If(x == yc, True, lsegy(nxt(x))))
AddRecDefinition(lsegz, x, If(x == zc, True, lsegz(nxt(x))))
# lsegy_p, lsegz_p defs reflect change of nxt(y) == z
AddRecDefinition(lsegy_p, x, If(x == yc, True, lsegy_p(If(x == yc, zc,
nxt(x)))))
AddRecDefinition(lsegz_p, x, If(x == zc, True, lsegz_p(If(x == yc, zc,
nxt(x)))))
AddAxiom((), nxt(nil) == nil)
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
from naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
# declarations
x, y = Vars('x y', fgsort)
nil, c, z = Consts('nil c z', fgsort)
nxt = Function('nxt', fgsort, fgsort)
lst = RecFunction('lst', fgsort, boolsort)
lseg = RecFunction('lseg', fgsort, fgsort, boolsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(lseg, (x, y), If(x == y, True, lseg(nxt(x), y)))
AddAxiom((), nxt(nil) == nil)
# vc
rhs = Implies(And(lst(y), And(x != c, nxt(y) == z)), And(lseg(x, z), lst(z)))
goal = Implies(lseg(x, y), rhs)
# check validity with natural proof solver and no hardcoded lemmas
np_solver = NPSolver()
solution = np_solver.solve(make_pfp_formula(goal))
if not solution.if_sat:
import importlib_resources
import z3
from z3 import And, Or, Not, Implies, If, Exists
from z3 import IsSubset, IsMember, SetUnion, SetIntersect, SetComplement, EmptySet, SetAdd
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
from lemsynth.lemsynth_engine import solveProblem
l, tl = Vars('l tl', fgsort)
k, h = Consts('k h', intsort)
x = Var('x', intsort)
# ADT definition of lists
nil = Const('nil', fgsort)
cons = Function('cons', intsort, fgsort, fgsort)
# projections for cons
head = Function('head', fgsort, intsort)
tail = Function('tail', fgsort, fgsort)
# rec defs
slst = RecFunction('slst', fgsort, boolsort)
elems = RecFunction('elems', fgsort, fgsetsort)
member = RecFunction('member', fgsort, boolsort)
AddRecDefinition(
slst, l,
from naturalproofs.prover import NPSolver
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort, min_intsort, max_intsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
def notInChildren(x):
return And(Not(IsMember(x, htree(lft(x)))),
Not(IsMember(x, htree(rght(x)))))
# declarations
x, y = Vars('x y', fgsort)
nil, ret, z = Consts('nil ret z', fgsort)
k = Const('k', intsort)
key = Function('key', fgsort, intsort)
lft = Function('lft', fgsort, fgsort)
rght = Function('rght', fgsort, fgsort)
tree = RecFunction('tree', fgsort, boolsort)
htree = RecFunction('htree', fgsort, fgsetsort)
reach_lr = RecFunction('reach_lr', fgsort, fgsort, boolsort)
AddRecDefinition(
tree, x,
If(
x == nil, True,
And(
notInChildren(x),
And(
SetIntersect(htree(lft(x)),
"""
# Only importing this for writing this file as a test
import unittest
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
from naturalproofs.extensions.finitemodel import extract_finite_model, add_fg_element_offset, get_fg_elements
# Declarations
x, y, nil = Consts('x y nil', fgsort)
nxt = Function('nxt', fgsort, fgsort)
lst = RecFunction('lst', fgsort, boolsort)
ls = RecFunction('ls', fgsort, fgsort, boolsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(ls, (x, y), If(x == nil, True, ls(nxt(x), y)))
# Problem parameters
goal = Implies(ls(x, nil), lst(x))
lemma1_params = (x, y)
lemma1_body = Implies(And(ls(x, y), lst(y)), lst(x))
lemmas = {(lemma1_params, lemma1_body)}
# Call a natural proofs solver
npsolver = NPSolver()
# Configure the solver
npsolver.options.instantiation_mode = proveroptions.manual_instantiation
npsolver.options.terms_to_instantiate = {x, y, nil}
import importlib_resources
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, SetUnion, SetIntersect, SetComplement, EmptySet, SetAdd
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Var, Vars, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
from lemsynth.lemsynth_engine import solveProblem
# Declarations
x, y = Vars('x y', fgsort)
c, nil = Consts('c nil', fgsort)
key = Function('key', fgsort, intsort)
nxt = Function('nxt', fgsort, fgsort)
lst = RecFunction('lst', fgsort, boolsort)
lseg = RecFunction('lseg', fgsort, fgsort, boolsort)
keys = RecFunction('keys', fgsort, intsetsort)
lsegkeys = RecFunction('lsegkeys', fgsort, fgsort, intsetsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(lseg, (x, y), If(x == y, True, lseg(nxt(x), y)))
AddRecDefinition(
keys, x,
If(x == nil, intsetsort.lattice_bottom, SetAdd(keys(nxt(x)), key(x))))
AddRecDefinition(
lsegkeys, (x, y),
# Only importing this for writing this file as a test
import unittest
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, EmptySet
from naturalproofs.uct import fgsort, fgsetsort, intsort, intsetsort, boolsort
from naturalproofs.decl_api import Const, Consts, Function, RecFunction, AddRecDefinition, AddAxiom
from naturalproofs.prover import NPSolver
import naturalproofs.proveroptions as proveroptions
# Declarations
x, nil = Consts('x nil', fgsort)
nxt = Function('nxt', fgsort, fgsort)
prv = Function('prv', fgsort, fgsort)
lst = RecFunction('lst', fgsort, boolsort)
dlst = RecFunction('dlst', fgsort, boolsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(
dlst, x,
If(x == nil, True,
If(nxt(x) == nil, True, And(prv(nxt(x)) == x, dlst(nxt(x))))))
# Problem parameters
goal = Implies(And(dlst(x), x == nil), lst(x))
lemma1_params = (x, )
lemma1_body = Implies(dlst(x), lst(x))
lemmas = {(lemma1_params, lemma1_body)}
# Call a natural proofs solver
npsolver = NPSolver()
# Configure the solver