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, ret = Consts('nil ret', fgsort)
nxt = Function('nxt', fgsort, fgsort)
lst = RecFunction('lst', fgsort, boolsort)
rlst = RecFunction('rlst', fgsort, boolsort)
red = Function('red', fgsort, boolsort)
black = Function('black', fgsort, boolsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(
rlst, x,
If(
x == nil, True,
And(
Or(
And(red(x),
And(Not(black(x)), And(black(nxt(x)), Not(red(nxt(x)))))),
And(black(x),
And(Not(red(x)), And(red(nxt(x)), Not(black(nxt(x))))))),
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, SetIntersect, SetComplement, SetAdd, EmptySet, IsMember
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
from naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
# declarations
x, y = Vars('x y', fgsort)
nil = Const('nil', fgsort)
k = Const('k', intsort)
nxt = Function('nxt', fgsort, fgsort)
key = Function('key', fgsort, intsort)
lst = RecFunction('lst', fgsort, boolsort)
hlst = RecFunction('hlst', fgsort, fgsetsort)
lseg = RecFunction('lseg', fgsort, fgsort, boolsort)
AddRecDefinition(lst, x, If(x == nil, True, lst(nxt(x))))
AddRecDefinition(hlst, x, If(x == nil, fgsetsort.lattice_bottom, SetAdd(hlst(nxt(x)), x)))
AddRecDefinition(lseg, (x, y), If(x == y, True, lseg(nxt(x), y)))
AddAxiom((), nxt(nil) == nil)
# vc
goal = Implies(lst(x), Implies(key(x) != k, Implies(IsMember(y, hlst(x)), lseg(x,y))))
# check validity with natural proof solver and no hardcoded lemmas
np_solver = NPSolver()
solution = np_solver.solve(make_pfp_formula(goal))
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)
k = Const('k', intsort)
key = Function('key', fgsort, intsort)
lft = Function('lft', fgsort, fgsort)
rght = Function('rght', fgsort, fgsort)
parent = Function('parent', fgsort, fgsort)
tree = RecFunction('tree', fgsort, boolsort)
tree_p = RecFunction('tree_p', 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)),
import z3
from z3 import And, Or, Not, Implies, If
from z3 import IsSubset, Union, 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 naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
# declarations
x, y, v, z = Vars('x y v z', fgsort)
nil = Const('nil', fgsort)
nxt = Function('nxt', fgsort, fgsort)
lseg = RecFunction('lseg', fgsort, fgsort, boolsort)
hlseg = RecFunction('hlseg', fgsort, fgsort, fgsort, boolsort)
AddRecDefinition(lseg, (x, y),
If(x == nil, False, If(nxt(x) == y, True, lseg(nxt(x), y))))
# v \in hlseg(x, y)
AddRecDefinition(hlseg, (x, y, v),
If(x == nil, False, If(v == x, True, hlseg(nxt(x), y, v))))
AddAxiom((), nxt(nil) == nil)
# Uncomment this line for fixed_depth=1 mode
# config_params['goal_instantiation_mode'] = proveroptions.fixed_depth # Default depth is 1
# Uncomment these two lines for manual instantiation mode
# config_params['goal_instantiation_mode'] = proveroptions.manual_instantiation
# Helper function for normalising conjunctive expressions
def binary_anded(exprs):
if len(exprs) == 0:
return z3.BoolVal(True)
elif len(exprs) == 1:
return exprs[0]
else:
return And(exprs[0], binary_anded(exprs[1:]))
# Signature
x = Var('x', fgsort)
nil = Const('nil', fgsort)
lft = Function('lft', fgsort, fgsort)
rght = Function('rght', fgsort, fgsort)
parent = Function('parent', fgsort, fgsort)
key = Function('key', fgsort, intsort)
tree = RecFunction('tree', fgsort, boolsort)
signature = dict()
signature['x'] = x
signature['nil'] = nil
signature['lft'] = lft
signature['rght'] = rght
signature['parent'] = parent
signature['key'] = key
signature['tree'] = tree
# Additional recursive definitions
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 lemsynth.lemsynth_engine import solveProblem
# Declarations
x = Var('x', fgsort)
nil = Const('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(dlst(x), Implies(x == 0, lst(x)))
# parameters representing the grammar for synth-fun and
# terms on which finite model is extracted
# TODO: extract this automatically from grammar_string
v = Var('v', fgsort)
lemma_grammar_args = [v, nil]
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) == 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,
If(x == s, True, And(reach_pgm(p(x)), And(cond, assign))))
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
x, y, nx, ny = Vars('x y nx ny', fgsort)
# ADT definition of nats
zero = Const('zero', fgsort)
succ = Function('succ', fgsort, fgsort)
# projection function - analogous to tail of list
pred = Function('pred', fgsort, fgsort)
# rec defs
plus = RecFunction('plus', fgsort, fgsort, fgsort)
AddRecDefinition(plus, (x, y), If(x == zero, y, succ(plus(pred(x), y))))
# axioms
AddAxiom(x, pred(succ(x)) == x)
AddAxiom(x, succ(x) != zero)
goal = plus(x, y) == plus(y, x)
# adt pfp of goal
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 = Var('x', fgsort)
nil = Const('nil', fgsort)
k = Const('k', intsort)
key = Function('key', fgsort, intsort)
keys = Function('keys', fgsort, intsetsort)
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,
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 = Var('x', fgsort)
nil, ret = Consts('nil ret', fgsort)
lft = Function('lft', fgsort, fgsort)
rght = Function('rght', fgsort, fgsort)
parent = Function('parent', fgsort, fgsort)
tree = RecFunction('tree', fgsort, boolsort)
tree_p = RecFunction('tree_p', fgsort, boolsort)
htree = RecFunction('htree', fgsort, fgsetsort)
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 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)
nil = Const('nil', fgsort)
k = Const('k', intsort)
key = Function('key', fgsort, intsort)
keys = Function('keys', fgsort, intsetsort)
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)
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(
