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,
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
x, y, tx = Vars('x y tx', fgsort)
hx = Var('hx', 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
append = RecFunction('append', fgsort, fgsort, fgsort)
length = RecFunction('length', fgsort, intsort)
AddRecDefinition(append, (x, y),
If(x == nil, y, cons(head(x), append(tail(x), y))))
AddRecDefinition(length, x, If(x == nil, 0, length(tail(x)) + 1))
import importlib_resources
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
# hardcoded lemmas
lemma_params = (x,y)
lemma_body = Implies(bst(x), Implies(IsMember(y, hbst(x)), bst(y)))
lemmas = {(lemma_params, lemma_body)}
# check validity of lemmas
solution = np_solver.solve(make_pfp_formula(lemma_body))
if not solution.if_sat:
print('lemma is valid')
else:
print('lemma is invalid')
# check validity with natural proof solver and hardcoded lemmas
# TODO: lemmas not sufficient
solution = np_solver.solve(goal, lemmas)
if not solution.if_sat:
print('goal (with lemmas) is valid')
else:
print('goal (with lemmas) is invalid')
# lemma synthesis
v1, v2 = Vars('v1 v2', fgsort)
lemma_grammar_args = [v1, v2, k, nil]
lemma_grammar_terms = {v1, v2, k}
name = 'bst-maximal'
grammar_string = importlib_resources.read_text('grammars', 'grammar_{}.sy'.format(name))
solveProblem(lemma_grammar_args, lemma_grammar_terms, goal, name, grammar_string)
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 = Consts('nil ret', 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)
dag = RecFunction('dag', fgsort, boolsort)
maxheap = RecFunction('maxheap', fgsort, boolsort)
htree = RecFunction('htree', fgsort, fgsetsort)
AddRecDefinition(
dag, x,
If(x == nil, True, And(notInChildren(x), And(dag(lft(x)), dag(rght(x))))))
AddRecDefinition(
maxheap, x,
If(
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, y, z = Vars('x y z', fgsort)
c = Const('c', fgsort)
nxt = Function('nxt', fgsort, fgsort)
lseg = RecFunction('lseg', fgsort, fgsort, boolsort)
AddRecDefinition(lseg, (x, y), If(x == y, True, lseg(nxt(x), y)))
# Problem parameters
goal = Implies(lseg(x, y), Implies(x != c, Implies(nxt(y) == z, lseg(x, 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, v3 = Vars('v1 v2 v3', fgsort)
lemma_grammar_args = [v1, v2, v3]
lemma_grammar_terms = {
nxt(v1),
nxt(nxt(v1)),
nxt(v2),
nxt(nxt(v2)),
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
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)
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 naturalproofs.pfp import make_pfp_formula
from lemsynth.lemsynth_engine import solveProblem
# declarations
x, y, z = Vars('x y z', fgsort)
nil = Const('nil', 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, lseg(y, z)), lseg(x, z)))
# check validity with natural proof solver and no hardcoded lemmas
np_solver = NPSolver()