logic.py

#
# Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
#
# SPDX-License-Identifier: BSD-2-Clause
#

import syntax
from syntax import word32T, word8T, boolT, builtinTs, Expr, Node
from syntax import true_term, false_term, mk_num
from syntax import foldr1

(mk_var, mk_plus, mk_uminus, mk_minus, mk_times, mk_modulus, mk_bwand, mk_eq,
mk_less_eq, mk_less, mk_implies, mk_and, mk_or, mk_not, mk_word32, mk_word8,
mk_word32_maybe, mk_cast, mk_memacc, mk_memupd, mk_arr_index, mk_arroffs,
mk_if, mk_meta_typ, mk_pvalid) = syntax.mks

from syntax import structs
from target_objects import trace, printout

def is_int (n):
	return hasattr (n, '__int__')

def mk_eq_with_cast (a, c):
	return mk_eq (a, mk_cast (c, a.typ))

def mk_rodata (m):
	assert m.typ == builtinTs['Mem']
	return Expr ('Op', boolT, name = 'ROData', vals = [m])

def cast_pair (((a, a_addr), (c, c_addr))):
	if a.typ != c.typ and c.typ == boolT:
		c = mk_if (c, mk_word32 (1), mk_word32 (0))
	return ((a, a_addr), (mk_cast (c, a.typ), c_addr))

ghost_assertion_type = syntax.Type ('WordArray', 50, 32)

def split_scalar_globals (vs):
	for i in range (len (vs)):
		if vs[i].typ.kind != 'Word' and vs[i].typ != boolT:
			break
	else:
		i = len (vs)
	scalars = vs[:i]
	global_vars = vs[i:]
	for v in global_vars:
		if v.typ not in [builtinTs['Mem'], builtinTs['Dom'],
				builtinTs['HTD'], builtinTs['PMS'],
				ghost_assertion_type]:
			assert not "scalar_global split expected", vs
	memT = builtinTs['Mem']
	mems = [v for v in global_vars if v.typ == memT]
	others = [v for v in global_vars if v.typ != memT]
	return (scalars, mems, others)

def mk_vars (tups):
	return [mk_var (nm, typ) for (nm, typ) in tups]

def split_scalar_pairs (var_pairs):
	return split_scalar_globals (mk_vars (var_pairs))

def azip (xs, ys):
	assert len (xs) == len (ys)
	return zip (xs, ys)

def mk_mem_eqs (a_imem, c_imem, a_omem, c_omem, tags):
	[a_imem] = a_imem
	a_tag, c_tag = tags
	(c_in, c_out) = (c_tag + '_IN', c_tag + '_OUT')
	(a_in, a_out) = (a_tag + '_IN', a_tag + '_OUT')
	if c_imem:
		[c_imem] = c_imem
		ieqs = [((a_imem, a_in), (c_imem, c_in)),
			((mk_rodata (c_imem), c_in), (true_term, c_in))]
	else:
		ieqs = [((mk_rodata (a_imem), a_in), (true_term, c_in))]
	if c_omem:
		[a_m] = a_omem
		[c_omem] = c_omem
		oeqs = [((a_m, a_out), (c_omem, c_out)),
			((mk_rodata (c_omem), c_out), (true_term, c_out))]
	else:
		oeqs = [((a_m, a_out), (a_imem, a_in)) for a_m in a_omem]

	return (ieqs, oeqs)

def mk_fun_eqs (as_f, c_f, prunes = None):
	(var_a_args, a_imem, glob_a_args) = split_scalar_pairs (as_f.inputs)
	(var_c_args, c_imem, glob_c_args) = split_scalar_pairs (c_f.inputs)
	(var_a_rets, a_omem, glob_a_rets) = split_scalar_pairs (as_f.outputs)
	(var_c_rets, c_omem, glob_c_rets) = split_scalar_pairs (c_f.outputs)

	(mem_ieqs, mem_oeqs) = mk_mem_eqs (a_imem, c_imem, a_omem, c_omem,
		['ASM', 'C'])

	if not prunes:
		prunes = (var_a_args, var_a_args)
	assert len (prunes[0]) == len (var_c_args), (params, var_a_args,
		var_c_args, prunes)
	a_map = dict (azip (prunes[1], var_a_args))
	ivar_pairs = [((a_map[p], 'ASM_IN'), (c, 'C_IN')) for (p, c)
		in azip (prunes[0], var_c_args) if p in a_map]

	ovar_pairs = [((a_ret, 'ASM_OUT'), (c_ret, 'C_OUT')) for (a_ret, c_ret)
		in azip (var_a_rets, var_c_rets)]
	return (map (cast_pair, mem_ieqs + ivar_pairs),
		map (cast_pair, mem_oeqs + ovar_pairs))

def mk_var_list (vs, typ):
	return [syntax.mk_var (v, typ) for v in vs]

def mk_offs_sequence (init, offs, n, do_reverse = False):
	r = range (n)
	if do_reverse:
		r.reverse ()
	def mk_offs (n):
		return Expr ('Num', init.typ, val = offs * n)
	return [mk_plus (init, mk_offs (m)) for m in r]

def mk_stack_sequence (sp, offs, stack, typ, n, do_reverse = False):
	return [(mk_memacc (stack, addr, typ), addr)
		for addr in mk_offs_sequence (sp, offs, n, do_reverse)]

def mk_aligned (w, n):
	assert w.typ.kind == 'Word'
	mask = Expr ('Num', w.typ, val = ((1 << n) - 1))
	return mk_eq (mk_bwand (w, mask), mk_num (0, w.typ))

def mk_eqs_arm_none_eabi_gnu (var_c_args, var_c_rets, c_imem, c_omem,
		min_stack_size):
	arg_regs = mk_var_list (['r0', 'r1', 'r2', 'r3'], word32T)
	r0 = arg_regs[0]
	sp = mk_var ('r13', word32T)
	st = mk_var ('stack', builtinTs['Mem'])
	r0_input = mk_var ('ret_addr_input', word32T)
	sregs = mk_stack_sequence (sp, 4, st, word32T, len (var_c_args) + 1)

	ret = mk_var ('ret', word32T)
	preconds = [mk_aligned (sp, 2), mk_eq (ret, mk_var ('r14', word32T)),
		mk_aligned (ret, 2), mk_eq (r0_input, r0),
		mk_less_eq (min_stack_size, sp)]
	post_eqs = [(x, x) for x in mk_var_list (['r4', 'r5', 'r6', 'r7', 'r8',
			'r9', 'r10', 'r11', 'r13'], word32T)]

	arg_seq = [(r, None) for r in arg_regs] + sregs
	if len (var_c_rets) > 1:
		# the 'return-too-much' issue.
		# instead r0 is a save-returns-here pointer
		arg_seq.pop (0)
		preconds += [mk_aligned (r0, 2), mk_less_eq (sp, r0)]
		save_seq = mk_stack_sequence (r0_input, 4, st, word32T,
			len (var_c_rets))
		save_addrs = [addr for (_, addr) in save_seq]
		post_eqs += [(r0_input, r0_input)]
		out_eqs = zip (var_c_rets, [x for (x, _) in save_seq])
		out_eqs = [(c, mk_cast (a, c.typ)) for (c, a) in out_eqs]
		init_save_seq = mk_stack_sequence (r0, 4, st, word32T,
			len (var_c_rets))
		(_, last_arg_addr) = arg_seq[len (var_c_args) - 1]
		preconds += [mk_less_eq (r0, addr)
			for (_, addr) in init_save_seq[-1:]]
		if last_arg_addr:
			preconds += [mk_less (last_arg_addr, addr)
				for (_, addr) in init_save_seq[:1]]
	else:
		out_eqs = zip (var_c_rets, [r0])
		save_addrs = []
	arg_seq_addrs = [addr for ((_, addr), _) in zip (arg_seq, var_c_args)
		if addr != None]
	swrap = mk_stack_wrapper (sp, st, arg_seq_addrs)
	swrap2 = mk_stack_wrapper (sp, st, save_addrs)
	post_eqs += [(swrap, swrap2)]

	mem = mk_var ('mem', builtinTs['Mem'])
	(mem_ieqs, mem_oeqs) = mk_mem_eqs ([mem], c_imem, [mem], c_omem,
		['ASM', 'C'])

	addr = None
	arg_eqs = [cast_pair (((a_x, 'ASM_IN'), (c_x, 'C_IN')))
		for (c_x, (a_x, addr)) in zip (var_c_args, arg_seq)]
	if addr:
		preconds += [mk_less_eq (sp, addr)]
	ret_eqs = [cast_pair (((a_x, 'ASM_OUT'), (c_x, 'C_OUT')))
		for (c_x, a_x) in out_eqs]
	preconds = [((a_x, 'ASM_IN'), (true_term, 'ASM_IN')) for a_x in preconds]
	asm_invs = [((vin, 'ASM_IN'), (vout, 'ASM_OUT')) for (vin, vout) in post_eqs]

	return (arg_eqs + mem_ieqs + preconds,
		ret_eqs + mem_oeqs + asm_invs)

known_CPUs = {
	'arm-none-eabi-gnu': mk_eqs_arm_none_eabi_gnu
}

def mk_fun_eqs_CPU (cpu_f, c_f, cpu_name, funcall_depth = 1):
	cpu = known_CPUs[cpu_name]
	(var_c_args, c_imem, glob_c_args) = split_scalar_pairs (c_f.inputs)
	(var_c_rets, c_omem, glob_c_rets) = split_scalar_pairs (c_f.outputs)

	return cpu (var_c_args, var_c_rets, c_imem, c_omem,
		(funcall_depth * 256) + 256)

class Pairing:
	def __init__ (self, tags, funs, eqs, notes = None):
		[l_tag, r_tag] = tags
		self.tags = tags
		assert set (funs) == set (tags)
		self.funs = funs
		self.eqs = eqs

		self.l_f = funs[l_tag]
		self.r_f = funs[r_tag]
		self.name = 'Pairing (%s (%s) <= %s (%s))' % (self.l_f,
			l_tag, self.r_f, r_tag)

		self.notes = {}
		if notes != None:
			self.notes.update (notes)

	def __str__ (self):
		return self.name

	def __hash__ (self):
		return hash (self.name)

	def __eq__ (self, other):
		return self.name == other.name and self.eqs == other.eqs

	def __ne__ (self, other):
		return not other or not self == other

def mk_pairing (functions, c_f, as_f, prunes = None, cpu = None):
	fs = (functions[as_f], functions[c_f])
	if cpu:
		eqs = mk_fun_eqs_CPU (fs[0], fs[1], cpu,
			funcall_depth = funcall_depth (functions, c_f))
	else:
		eqs = mk_fun_eqs (fs[0], fs[1], prunes = prunes)
	return Pairing (['ASM', 'C'], {'C': c_f, 'ASM': as_f}, eqs)

def inst_eqs_pattern (pattern, params):
	(pat_params, inp_eqs, out_eqs) = pattern
	substs = [((x.name, x.typ), y)
		for (pat_vs, vs) in azip (pat_params, params)
		for (x, y) in azip (pat_vs, vs)]
	substs = dict (substs)
	subst = lambda x: var_subst (x, substs)
	return ([(subst (x), subst (y)) for (x, y) in inp_eqs],
		[(subst (x), subst (y)) for (x, y) in out_eqs])

def inst_eqs_pattern_tuples (pattern, params):
	return inst_eqs_pattern (pattern, map (mk_vars, params))

def inst_eqs_pattern_exprs (pattern, params):
	(inp_eqs, out_eqs) = inst_eqs_pattern (pattern, params)
	return (foldr1 (mk_and, [mk_eq (a, c) for (a, c) in inp_eqs]),
		foldr1 (mk_and, [mk_eq (a, c) for (a, c) in out_eqs]))

def var_match (var_exp, conc_exp, assigns):
	if var_exp.typ != conc_exp.typ:
		return False
	if var_exp.kind == 'Var':
		key = (var_exp.name, var_exp.typ)
		if key in assigns:
			return conc_exp == assigns[key]
		else:
			assigns[key] = conc_exp
			return True
	elif var_exp.kind == 'Op':
		if conc_exp.kind != 'Op' or var_exp.name != conc_exp.name:
			return False
		return all ([var_match (a, b, assigns)
			for (a, b) in azip (var_exp.vals, conc_exp.vals)])
	else:
		return False

def var_subst (var_exp, assigns, must_subst = True):
	def substor (var_exp):
		if var_exp.kind == 'Var':
			k = (var_exp.name, var_exp.typ)
			if must_subst or k in assigns:
				return assigns[k]
			else:
				return None
		else:
			return None
	return syntax.do_subst (var_exp, substor)

def recursive_term_subst (eqs, expr):
	if expr in eqs:
		return eqs[expr]
	if expr.kind == 'Op':
		vals = [recursive_term_subst (eqs, x) for x in expr.vals]
		return syntax.adjust_op_vals (expr, vals)
	return expr

def mk_accum_rewrites (typ):
	x = mk_var ('x', typ)
	y = mk_var ('y', typ)
	z = mk_var ('z', typ)
	i = mk_var ('i', typ)
	return [(x, i, mk_plus (x, y), mk_plus (x, mk_times (i, y)),
			y),
		(x, i, mk_plus (y, x), mk_plus (x, mk_times (i, y)),
			y),
		(x, i, mk_minus (x, y), mk_minus (x, mk_times (i, y)),
			mk_uminus (y)),
		(x, i, mk_plus (mk_plus (x, y), z),
			mk_plus (x, mk_times (i, mk_plus (y, z))),
			mk_plus (y, z)),
		(x, i, mk_plus (mk_plus (y, x), z),
			mk_plus (x, mk_times (i, mk_plus (y, z))),
			mk_plus (y, z)),
		(x, i, mk_plus (y, mk_plus (x, z)),
			mk_plus (x, mk_times (i, mk_plus (y, z))),
			mk_plus (y, z)),
		(x, i, mk_plus (y, mk_plus (z, x)),
			mk_plus (x, mk_times (i, mk_plus (y, z))),
			mk_plus (y, z)),
		(x, i, mk_minus (mk_minus (x, y), z),
			mk_minus (x, mk_times (i, mk_plus (y, z))),
			mk_plus (y, z)),
	]

def mk_all_accum_rewrites ():
	return [rew for typ in [word8T, word32T, syntax.word16T,
			syntax.word64T]
		for rew in mk_accum_rewrites (typ)]

accum_rewrites = mk_all_accum_rewrites ()

def default_val (typ):
	if typ.kind == 'Word':
		return Expr ('Num', typ, val = 0)
	elif typ == boolT:
		return false_term
	else:
		assert not 'default value for type %s created', typ

trace_accumulators = []

def accumulator_closed_form (expr, (nm, typ), add_mask = None):
	expr = toplevel_split_out_cast (expr)
	n = get_bwand_mask (expr)
	if n and not add_mask:
		return accumulator_closed_form (expr.vals[0], (nm, typ),
			add_mask = n)

	for (x, i, pattern, rewrite, offset) in accum_rewrites:
		var = mk_var (nm, typ)
		ass = {(x.name, x.typ): var}
		m = var_match (pattern, expr, ass)
		if m:
			x2_def = default_val (typ)
			i2_def = default_val (word32T)
			def do_rewrite (x2 = x2_def, i2 = i2_def):
				ass[(x.name, x.typ)] = x2
				ass[(i.name, i.typ)] = i2
				vs = var_subst (rewrite, ass)
				if add_mask:
					vs = mk_bwand_mask (vs, add_mask)
				return vs
			offs = var_subst (offset, ass)
			return (do_rewrite, offs)
	if trace_accumulators:
		trace ('no accumulator %s' % ((expr, nm, typ), ))
	return (None, None)

def split_out_cast (expr, target_typ, bits):
	"""given a word-type expression expr (of any word length),
	compute a simplified expression expr' of the target type, which will
	have the property that expr' && mask bits = cast expr,
	where && is bitwise-and (BWAnd), mask n is the bitpattern set at the
	bottom n bits, e.g. (1 << n) - 1, and cast is WordCast."""
	if expr.is_op (['WordCast', 'WordCastSigned']):
		[x] = expr.vals
		if x.typ.num >= bits and expr.typ.num >= bits:
			return split_out_cast (x, target_typ, bits)
		else:
			return mk_cast (expr, target_typ)
	elif expr.is_op ('BWAnd'):
		[x, y] = expr.vals
		if y.kind == 'Num':
			val = y.val
		else:
			val = 0
		full_mask = (1 << bits) - 1
		if val & full_mask == full_mask:
			return split_out_cast (x, target_typ, bits)
		else:
			return mk_cast (expr, target_typ)
	elif expr.is_op (['Plus', 'Minus']):
		# rounding issues will appear if this arithmetic is done
		# at a smaller number of bits than we'll eventually report
		if expr.typ.num >= bits:
			vals = [split_out_cast (x, target_typ, bits)
				for x in expr.vals]
			if expr.is_op ('Plus'):
				return mk_plus (vals[0], vals[1])
			else:
				return mk_minus (vals[0], vals[1])
		else:
			return mk_cast (expr, target_typ)
	else:
		return mk_cast (expr, target_typ)

def toplevel_split_out_cast (expr):
	bits = None
	if expr.is_op (['WordCast', 'WordCastSigned']):
		bits = min ([expr.typ.num, expr.vals[0].typ.num])
	elif expr.is_op ('BWAnd'):
		bits = get_bwand_mask (expr)

	if bits:
		expr = split_out_cast (expr, expr.typ, bits)
		return mk_bwand_mask (expr, bits)
	else:
		return expr

two_powers = {}

def get_bwand_mask (expr):
	"""recognise (x && mask) opers, where mask = ((1 << n) - 1)
	for some n"""
	if not expr.is_op ('BWAnd'):
		return
	[x, y] = expr.vals
	if not y.kind == 'Num':
		return
	val = y.val & ((1 << (y.typ.num)) - 1)
	if not two_powers:
		for i in range (129):
			two_powers[1 << i] = i
	return two_powers.get (val + 1)

def mk_bwand_mask (expr, n):
	return mk_bwand (expr, mk_num (((1 << n) - 1), expr.typ))

def end_addr (p, typ):
	if typ[0] == 'Array':
		(_, typ, n) = typ
		sz = mk_times (mk_word32 (typ.size ()), n)
	else:
		assert typ[0] == 'Type', typ
		(_, typ) = typ
		sz = mk_word32 (typ.size ())
	return mk_plus (p, mk_minus (sz, mk_word32 (1)))

def pvalid_assertion1 ((typ, k, p, pv), (typ2, k2, p2, pv2)):
	"""first pointer validity assertion: incompatibility.
	pvalid1 & pvalid2 --> non-overlapping OR somehow-contained.
	typ/typ2 is ('Type', syntax.Type) or ('Array', Type, Expr) for
	dynamically sized arrays.
	"""
	offs1 = mk_minus (p, p2)
	cond1 = get_styp_condition (offs1, typ, typ2)
	offs2 = mk_minus (p2, p)
	cond2 = get_styp_condition (offs2, typ2, typ)

	out1 = mk_less (end_addr (p, typ), p2)
	out2 = mk_less (end_addr (p2, typ2), p)
	return mk_implies (mk_and (pv, pv2), foldr1 (mk_or,
		[cond1, cond2, out1, out2]))

def pvalid_assertion2 ((typ, k, p, pv), (typ2, k2, p2, pv2)):
	"""second pointer validity assertion: implication.
	pvalid1 & strictly-contained --> pvalid2
	"""
	if typ[0] == 'Array' and typ2[0] == 'Array':
		# this is such a vague notion it's not worth it
		return true_term
	offs1 = mk_minus (p, p2)
	cond1 = get_styp_condition (offs1, typ, typ2)
	imp1 = mk_implies (mk_and (cond1, pv2), pv)
	offs2 = mk_minus (p2, p)
	cond2 = get_styp_condition (offs2, typ2, typ)
	imp2 = mk_implies (mk_and (cond2, pv), pv2)
	return mk_and (imp1, imp2)

def sym_distinct_assertion ((typ, p, pv), (start, end)):
	out1 = mk_less (mk_plus (p, mk_word32 (typ.size () - 1)), mk_word32 (start))
	out2 = mk_less (mk_word32 (end), p)
	return mk_implies (pv, mk_or (out1, out2))

def norm_array_type (t):
	if t[0] == 'Type' and t[1].kind == 'Array':
		(_, atyp) = t
		return ('Array', atyp.el_typ_symb, mk_word32 (atyp.num), 'Strong')
	elif t[0] == 'Array' and len (t) == 3:
		(_, typ, l) = t
		# these derive from PArrayValid assertions. we know the array is
		# at least this long, but it might be longer.
		return ('Array', typ, l, 'Weak')
	else:
		return t

stored_styp_conditions = {}

def get_styp_condition (offs, inner_typ, outer_typ):
	r = get_styp_condition_inner1 (inner_typ, outer_typ)
	if not r:
		return false_term
	else:
		return r (offs)

def get_styp_condition_inner1 (inner_typ, outer_typ):
	inner_typ = norm_array_type (inner_typ)
	outer_typ = norm_array_type (outer_typ)
	k = (inner_typ, outer_typ)
	if k in stored_styp_conditions:
		return stored_styp_conditions[k]
	r = get_styp_condition_inner2 (inner_typ, outer_typ)
	stored_styp_conditions[k] = r
	return r

def array_typ_size ((kind, el_typ, num, _)):
	el_size = mk_word32 (el_typ.size ())
	return mk_times (num, el_size)

def get_styp_condition_inner2 (inner_typ, outer_typ):
	if inner_typ[0] == 'Array' and outer_typ[0] == 'Array':
		(_, ityp, inum, _) = inner_typ
		(_, otyp, onum, outer_bound) = outer_typ
		# array fits in another array if the starting element is
		# a sub-element, and if the size of the left array plus
		# the offset fits in the right array
		cond = get_styp_condition_inner1 (('Type', ityp), outer_typ)
		isize = array_typ_size (inner_typ)
		osize = array_typ_size (outer_typ)
		if outer_bound == 'Strong' and cond:
			return lambda offs: mk_and (cond (offs),
				mk_less_eq (mk_plus (isize, offs), osize))
		else:
			return cond
	elif inner_typ == outer_typ:
		return lambda offs: mk_eq (offs, mk_word32 (0))
	elif outer_typ[0] == 'Type' and outer_typ[1].kind == 'Struct':
		conds = [(get_styp_condition_inner1 (inner_typ,
				('Type', sf_typ)), mk_word32 (offs2))
			for (_, offs2, sf_typ)
			in structs[outer_typ[1].name].fields.itervalues()]
		conds = [cond for cond in conds if cond[0]]
		if conds:
			return lambda offs: foldr1 (mk_or,
				[c (mk_minus (offs, offs2))
					for (c, offs2) in conds])
		else:
			return None
	elif outer_typ[0] == 'Array':
		(_, el_typ, n, bound) = outer_typ
		cond = get_styp_condition_inner1 (inner_typ, ('Type', el_typ))
		el_size = mk_word32 (el_typ.size ())
		size = mk_times (n, el_size)
		if bound == 'Strong' and cond:
			return lambda offs: mk_and (mk_less (offs, size),
				cond (mk_modulus (offs, el_size)))
		elif cond:
			return lambda offs: cond (mk_modulus (offs, el_size))
		else:
			return None
	else:
		return None

def all_vars_have_prop (expr, prop):
	class Failed (Exception):
		pass
	def visit (expr):
		if expr.kind != 'Var':
			return
		v2 = (expr.name, expr.typ)
		if not prop (v2):
			raise Failed ()
	try:
		expr.visit (visit)
		return True
	except Failed:
		return False

def all_vars_in_set (expr, var_set):
	return all_vars_have_prop (expr, lambda v: v in var_set)

def var_not_in_expr (var, expr):
	v2 = (var.name, var.typ)
	return all_vars_have_prop (expr, lambda v: v != v2)

def mk_array_size_ineq (typ, num, p):
	align = typ.align ()
	size = mk_times (mk_word32 (typ.size ()), num)
	size_lim = ((2 ** 32) - 4) / typ.size ()
	return mk_less_eq (num, mk_word32 (size_lim))

def mk_align_valid_ineq (typ, p):
	if typ[0] == 'Type':
		(_, typ) = typ
		align = typ.align ()
		size = mk_word32 (typ.size ())
		size_req = []
	else:
		assert typ[0] == 'Array', typ
		(kind, typ, num) = typ
		align = typ.align ()
		size = mk_times (mk_word32 (typ.size ()), num)
		size_req = [mk_array_size_ineq (typ, num, p)]
	assert align in [1, 4, 8]
	w0 = mk_word32 (0)
	if align > 1:
		align_req = [mk_eq (mk_bwand (p, mk_word32 (align - 1)), w0)]
	else:
		align_req = []
	return foldr1 (mk_and, align_req + size_req + [mk_not (mk_eq (p, w0)),
		mk_implies (mk_less (w0, size),
			mk_less_eq (p, mk_uminus (size)))])

# generic operations on function/problem graphs
def dict_list (xys, keys = None):
	"""dict_list ([(1, 2), (1, 3), (2, 4)]) = {1: [2, 3], 2: [4]}"""
	d = {}
	for (x, y) in xys:
		d.setdefault (x, [])
		d[x].append (y)
	if keys:
		for x in keys:
			d.setdefault (x, [])
	return d

def compute_preds (nodes):
	preds = dict_list ([(c, n) for n in nodes
			for c in nodes[n].get_conts ()],
		keys = nodes)
	for n in ['Ret', 'Err']:
		preds.setdefault (n, [])
	preds = dict ([(n, sorted (set (ps)))
		for (n, ps) in preds.iteritems ()])
	return preds

def simplify_node_elementary(node):
	if node.kind == 'Cond' and node.cond == true_term:
		return Node ('Basic', node.left, [])
	elif node.kind == 'Cond' and node.cond == false_term:
		return Node ('Basic', node.right, [])
	elif node.kind == 'Cond' and node.left == node.right:
		return Node ('Basic', node.left, [])
	else:
		return node

def compute_var_flows (nodes, outputs, preds, override_lvals_rvals = {}):
	# compute a graph of reverse var flows to pass to tarjan's algorithm
	graph = {}
	entries = ['Ret']
	for (n, node) in nodes.iteritems ():
		if node.kind == 'Basic':
			for (lv, rv) in node.upds:
				graph[(n, 'Post', lv)] = [(n, 'Pre', v)
					for v in syntax.get_expr_var_set (rv)]
		elif node.is_noop ():
			pass
		else:
			if n in override_lvals_rvals:
				(lvals, rvals) = override_lvals_rvals[n]
			else:
				rvals = syntax.get_node_rvals (node)
				rvals = set (rvals.iteritems ())
				lvals = set (node.get_lvals ())
			if node.kind != 'Basic':
				lvals = list (lvals) + ['PC']
				entries.append ((n, 'Post', 'PC'))
			for lv in lvals:
				graph[(n, 'Post', lv)] = [(n, 'Pre', rv)
					for rv in rvals]
	graph['Ret'] = [(n, 'Post', v)
		for n in preds['Ret'] for v in outputs (n)]
	vs = set ([v for k in graph for (_, _, v) in graph[k]])
	for v in vs:
		for n in nodes:
			graph.setdefault ((n, 'Post', v), [(n, 'Pre', v)])
			graph[(n, 'Pre', v)] = [(n2, 'Post', v)
				for n2 in preds[n]]

	comps = tarjan (graph, entries)
	return (graph, comps)

def mk_not_red (v):
	if v.is_op ('Not'):
		[v] = v.vals
		return v
	else:
		return syntax.mk_not (v)

def cont_with_conds (nodes, n, conds):
	while True:
		if n not in nodes or nodes[n].kind != 'Cond':
			return n
		cond = nodes[n].cond
		if cond in conds:
			n = nodes[n].left
		elif mk_not_red (cond) in conds:
			n = nodes[n].right
		else:
			return n

def contextual_conds (nodes, preds):
	"""computes a collection of conditions that can be assumed true
	at any point in the node graph."""
	pre_conds = {}
	arc_conds = {}
	visit = [n for n in nodes if not (preds[n])]
	while visit:
		n = visit.pop ()
		if n not in nodes:
			continue
		in_arc_conds = [arc_conds.get ((pre, n), set ())
			for pre in preds[n]]
		if not in_arc_conds:
			conds = set ()
		else:
			conds = set.intersection (* in_arc_conds)
		if pre_conds.get (n) == conds:
			continue
		pre_conds[n] = conds
		if n not in nodes:
			continue
		if nodes[n].kind == 'Cond' and nodes[n].left == nodes[n].right:
			c_conds = [conds, conds]
		elif nodes[n].kind == 'Cond':
			c_conds = [nodes[n].cond, mk_not_red (nodes[n].cond)]
			c_conds = [set.union (set ([c]), conds)
				for c in c_conds]
		else:
			upds = set (nodes[n].get_lvals ())
			c_conds = [set ([c for c in conds if
				not set.intersection (upds,
					syntax.get_expr_var_set (c))])]
		for (cont, conds) in zip (nodes[n].get_conts (), c_conds):
			arc_conds[(n, cont)] = conds
			visit.append (cont)
	return (arc_conds, pre_conds)

def contextual_cond_simps (nodes, preds):
	"""a common pattern in architectures with conditional operations is
	a sequence of instructions with the same condition.
	we can usually then reduce to a single contional block.
	  b   e    =>    b-e
	 / \ / \   =>   /   \
	a-c-d-f-g  =>  a-c-f-g
	this is sometimes important if b calculates a register that e uses
	since variable dependency analysis will see this register escape via
	the impossible path a-c-d-e
	"""
	(arc_conds, pre_conds) = contextual_conds (nodes, preds)
	nodes = dict (nodes)
	for n in nodes:
		if nodes[n].kind == 'Cond':
			continue
		cont = nodes[n].cont
		conds = arc_conds[(n, cont)]
		cont2 = cont_with_conds (nodes, cont, conds)
		if cont2 != cont:
			nodes[n] = syntax.copy_rename (nodes[n],
				({}, {cont: cont2}))
	return nodes

def minimal_loop_node_set (p):
	"""discover a minimal set of loop addresses, excluding some operations
	using conditional instructions which are syntactically within the
	loop but semantically must always be followed by an immediate loop
	exit.

	amounts to rerunning loop detection after contextual_cond_simps."""

	loop_ns = set (p.loop_data)
	really_in_loop = {}
	nodes = contextual_cond_simps (p.nodes, p.preds)
	def is_really_in_loop (n):
		if n in really_in_loop:
			return really_in_loop[n]
		ns = []
		r = None
		while r == None:
			ns.append (n)
			if n not in loop_ns:
				r = False
			elif n in p.splittable_points (n):
				r = True
			else:
				conts = [n2 for n2 in nodes[n].get_conts ()
					if n2 != 'Err']
				if len (conts) > 1:
					r = True
				else:
					[n] = conts
		for n in ns:
			really_in_loop[n] = r
		return r
	return set ([n for n in loop_ns if is_really_in_loop (n)])

def possible_graph_divs (p, min_cost = 20, max_cost = 20, ratio = 0.85,
		trace = None):
	es = [e[0] for e in p.entries]
	divs = []
	direct_costs = {}
	future_costs = {'Ret': set (), 'Err': set ()}
	prev_costs = {}
	int_costs = {}
	fracs = {}
	for n in p.nodes:
		node = p.nodes[n]
		if node.kind == 'Call':
			cost = set ([(n, 20)])
		elif p.loop_id (n):
			cost = set ([(p.loop_id (n), 50)])
		else:
			cost = set ([(n, len (node.get_mem_accesses ()))])
			cost.discard ((n, 0))
		direct_costs[n] = cost
	for n in p.tarjan_order:
		prev_costs[n] = set.union (* ([direct_costs[n]]
			+ [prev_costs.get (c, set ()) for c in p.preds[n]]))
	for n in reversed (p.tarjan_order):
		cont_costs = [future_costs.get (c, set ())
			for c in p.nodes[n].get_conts ()]
		cost = set.union (* ([direct_costs[n]] + cont_costs))
		p_ct = sum ([c for (_, c) in prev_costs[n]])
		future_costs[n] = cost
		if p.nodes[n].kind != 'Cond' or p_ct > max_cost:
			continue
		ct = sum ([c for (_, c) in set.union (cost, prev_costs[n])])
		if ct < min_cost:
			continue
		[c1, c2] = [sum ([c for (_, c)
				in set.union (cs, prev_costs[n])])
			for cs in cont_costs]
		fracs[n] = ((c1 * c1) + (c2 * c2)) / (ct * ct * 1.0)
		if fracs[n] < ratio:
			divs.append (n)
	divs.reverse ()
	if trace != None:
		trace[0] = (direct_costs, future_costs, prev_costs,
			int_costs, fracs)
	return divs

def compute_var_deps (nodes, outputs, preds, override_lvals_rvals = {},
		trace = None):
	# outs = list of (outname, retvars)
	var_deps = {}
	visit = set ()
	visit.update (preds['Ret'])
	visit.update (preds['Err'])

	nodes = contextual_cond_simps (nodes, preds)

	while visit:
		n = visit.pop ()

		node = simplify_node_elementary (nodes[n])
		if n in override_lvals_rvals:
			(lvals, rvals) = override_lvals_rvals[n]
			lvals = set (lvals)
			rvals = set (rvals)
		elif node.is_noop ():
			lvals = set ([])
			rvals = set ([])
		else:
			rvals = syntax.get_node_rvals (node)
			rvals = set (rvals.iteritems ())
			lvals = set (node.get_lvals ())
		cont_vs = set ()

		for c in node.get_conts ():
			if c == 'Ret':
				cont_vs.update (outputs (n))
			elif c == 'Err':
				pass
			else:
				cont_vs.update (var_deps.get (c, []))
		vs = set.union (rvals, cont_vs - lvals)

		if n in var_deps and vs <= var_deps[n]:
			continue
		if trace and n in trace:
			diff = vs - var_deps.get (n, set())
			printout ('add %s at %d' % (diff, n))
			printout ('  %s, %s, %s, %s' % (len (vs), len (cont_vs), len (lvals), len (rvals)))
		var_deps[n] = vs
		visit.update (preds[n])

	return var_deps

def compute_loop_var_analysis (p, var_deps, n, override_nodes = None):
	if override_nodes == None:
		nodes = p.nodes
	else:
		nodes = override_nodes

	upd_vs = set ([v for n2 in p.loop_body (n)
		if not nodes[n2].is_noop ()
		for v in nodes[n2].get_lvals ()])
	const_vs = set ([v for n2 in p.loop_body (n)
		for v in var_deps[n2] if v not in upd_vs])

	vca = compute_var_cycle_analysis (p, nodes, n,
		const_vs, set (var_deps[n]))
	vca = [(syntax.mk_var (nm, typ), data)
		for ((nm, typ), data) in vca.items ()]
	return vca

cvca_trace = []
cvca_diag = [False]
no_accum_expressions = set ()

def compute_var_cycle_analysis (p, nodes, n, const_vars, vs, diag = None):

	if diag == None:
		diag = cvca_diag[0]

	cache = {}
	del cvca_trace[:]
	impossible_nodes = {}
	loop = p.loop_body (n)

	def warm_cache_before (n2, v):
		cvca_trace.append ((n2, v))
		cvca_trace.append ('(')
		arc = []
		for i in range (100000):
			opts = [n3 for n3 in p.preds[n2] if n3 in loop
				if v not in nodes[n3].get_lvals ()
				if n3 != n
				if (n3, v) not in cache]
			if not opts:
				break
			n2 = opts[0]
			arc.append (n2)
		if not (len (arc) < 100000):
			trace ('warmup overrun in compute_var_cycle_analysis')
			trace ('chasing %s in %s' % (v, set (arc)))
			assert False, (v, arc[-500:])
		for n2 in reversed (arc):
			var_eval_before (n2, v)
		cvca_trace.append (')')

	def var_eval_before (n2, v, do_cmp = True):
		if (n2, v) in cache and do_cmp:
			return cache[(n2, v)]
		if n2 == n and do_cmp:
			var_exp = mk_var (v[0], v[1])
			vs = set ([v for v in [v] if v not in const_vars])
			return (vs, var_exp)
		warm_cache_before (n2, v)
		ps = [n3 for n3 in p.preds[n2] if n3 in loop
			if not node_impossible (n3)]
		if not ps:
			return None
		vs = [var_eval_after (n3, v) for n3 in ps]
		if not all ([v3 == vs[0] for v3 in vs]):
			if diag:
				trace ('vs disagree for %s @ %d: %s' % (v, n2, vs))
			r = None
		else:
			r = vs[0]
		if do_cmp:
			cache[(n2, v)] = r
		return r
	def var_eval_after (n2, v):
		node = nodes[n2]
		if node.kind == 'Call' and v in node.rets:
			if diag:
				trace ('fetched %s from call at %d' % (v, n2))
			return None
		elif node.kind == 'Basic':
			for (lv, val) in node.upds:
				if lv == v:
					return expr_eval_before (n2, val)
			return var_eval_before (n2, v)
		else:
			return var_eval_before (n2, v)
	def expr_eval_before (n2, expr):
		if expr.kind == 'Op':
			if expr.vals == []:
				return (set(), expr)
			vals = [expr_eval_before (n2, v)
				for v in expr.vals]
			if None in vals:
				return None
			s = set.union (* [s for (s, v) in vals])
			if len(s) > 1:
				if diag:
					trace ('too many vars for %s @ %d: %s' % (expr, n2, s))
				return None
			return (s, Expr ('Op', expr.typ,
				name = expr.name,
				vals = [v for (s, v) in vals]))
		elif expr.kind == 'Num':
			return (set(), expr)
		elif expr.kind == 'Var':
			return var_eval_before (n2,
				(expr.name, expr.typ))
		else:
			if diag:
				trace ('Unwalkable expr %s' % expr)
			return None
	def node_impossible (n2):
		if n2 in impossible_nodes:
			return impossible_nodes[n2]
		if n2 == n or n2 in p.get_loop_splittables (n):
			imposs = False
		else:
			pres = [n3 for n3 in p.preds[n2]
				if n3 in loop if not node_impossible (n3)]
			if n2 in impossible_nodes:
				imposs = impossible_nodes[n2]
			else:
				imposs = not bool (pres)
		impossible_nodes[n2] = imposs
		node = nodes[n2]
		if imposs or node.kind != 'Cond':
			return imposs
		if 1 >= len ([n3 for n3 in node.get_conts ()
				if n3 in loop]):
			return imposs
		c = expr_eval_before (n2, node.cond)
		if c != None:
			c = try_eval_expr (c[1])
		if c != None:
			trace ('determined loop inner cond at %d equals %s'
				% (n2, c == syntax.true_term))
		if c == syntax.true_term:
			impossible_nodes[node.right] = True
		elif c == syntax.false_term:
			impossible_nodes[node.left] = True
		return imposs

	vca = {}
	for v in vs:
		rv = var_eval_before (n, v, do_cmp = False)
		if rv == None:
			vca[v] = 'LoopVariable'
			continue
		(s, expr) = rv
		if expr == mk_var (v[0], v[1]):
			vca[v] = 'LoopConst'
			continue
		if all_vars_in_set (expr, const_vars):
			# a repeatedly evaluated const expression, is const
			vca[v] = 'LoopConst'
			continue
		if var_not_in_expr (mk_var (v[0], v[1]), expr):
			# leaf calculations do not have data flow to
			# themselves. the search algorithm doesn't
			# have to worry about these.
			vca[v] = 'LoopLeaf'
			continue
		(form, offs) = accumulator_closed_form (expr, v)
		if form != None and all_vars_in_set (form (), const_vars):
			vca[v] = ('LoopLinearSeries', form, offs)
		else:
			if diag:
				trace ('No accumulator %s => %s'
					% (v, expr))
			no_accum_expressions.add ((v, expr))
			vca[v] = 'LoopVariable'
	return vca

eval_expr_solver = [None]

def try_eval_expr (expr):
	"""attempt to reduce an expression to a single result, vaguely like
	what constant propagation would do. it might work!"""
	import search
	if not eval_expr_solver[0]:
		import solver
		eval_expr_solver[0] = solver.Solver ()
	try:
		return search.eval_model_expr ({}, eval_expr_solver[0], expr)
	except KeyboardInterrupt, e:
		raise e
	except Exception, e:
		return None

expr_linear_sum = set (['Plus', 'Minus'])
expr_linear_cast = set (['WordCast', 'WordCastSigned'])

expr_linear_all = set.union (expr_linear_sum, expr_linear_cast,
	['Times', 'ShiftLeft'])

def possibly_linear (expr):
	if expr.kind in set (['Var', 'Num', 'Symbol', 'Type', 'Token']):
		return True
	elif expr.is_op (expr_linear_all):
		return all ([possibly_linear (x) for x in expr.vals])
	else:
		return False

def lv_expr (expr, env):
	if expr in env:
		return env[expr]
	elif expr.kind in set (['Num', 'Symbol', 'Type', 'Token']):
		return (expr, 'LoopConst', None, set ())
	elif expr.kind == 'Var':
		return (None, None, None, None)
	elif expr.kind != 'Op':
		assert expr in env, expr

	lvs = [lv_expr (v, env) for v in expr.vals]
	rs = [lv[1] for lv in lvs]
	mk_offs = lambda vals: syntax.adjust_op_vals (expr, vals)
	if None in rs:
		return (None, None, None, None)
	if set (rs) == set (['LoopConst']):
		return (expr, 'LoopConst', None, set ())
	offs_set = set.union (* ([lv[3] for lv in lvs] + [set ()]))
	arg_offs = []
	for (expr2, k, offs, _) in lvs:
		if k == 'LoopConst' and expr2.typ.kind == 'Word':
			arg_offs.append (syntax.mk_num (0, expr2.typ))
		else:
			arg_offs.append (offs)
	if expr.is_op (expr_linear_sum):
		if set (rs) == set (['LoopConst', 'LoopLinearSeries']):
			return (expr, 'LoopLinearSeries', mk_offs (arg_offs),
				offs_set)
	elif expr.is_op ('Times'):
		if set (rs) == set (['LoopLinearSeries', 'LoopConst']):
			# the new offset is the product of the linear offset
			# and the constant value
			[linear_offs] = [offs for (_, k, offs, _) in lvs
				if k == 'LoopLinearSeries']
			[const_value] = [v for (v, k, _, _) in lvs
				if k == 'LoopConst']
			return (expr, 'LoopLinearSeries',
				mk_offs ([linear_offs, const_value]), offs_set)
	if expr.is_op ('ShiftLeft'):
		if rs == ['LoopLinearSeries', 'LoopConst']:
			return (expr, 'LoopLinearSeries',
				mk_offs ([arg_offs[0], lvs[1][0]]), offs_set)
	if expr.is_op (expr_linear_cast):
		if rs == ['LoopLinearSeries']:
			return (expr, 'LoopLinearSeries', mk_offs (arg_offs),
				offs_set)
	return (None, None, None, None)

# FIXME: this should probably be unified with compute_var_cycle_analysis,
# but doing so is complicated
def linear_series_exprs (p, loop, va):
	def lv_init (v, data):
		if data[0] == 'LoopLinearSeries':
			return (v, 'LoopLinearSeries', data[2], set ([data[2]]))
		elif data == 'LoopConst':
			return (v, 'LoopConst', None, set ())
		else:
			return (None, None, None, None)
	cache = {loop: dict ([(v, lv_init (v, data)) for (v, data) in va])}
	post_cache = {}
	loop_body = p.loop_body (loop)
	frontier = [n2 for n2 in p.nodes[loop].get_conts ()
		if n2 in loop_body]
	def lv_merge ((v1, lv1, offs1, oset1), (v2, lv2, offs2, oset2)):
		if v1 != v2:
			return (None, None, None, None)
		assert lv1 == lv2 and offs1 == offs2
		return (v1, lv1, offs1, oset1)
	def compute_post (n):
		if n in post_cache:
			return post_cache[n]
		pre_env = cache[n]
		env = dict (cache[n])
		if p.nodes[n].kind == 'Basic':
			for ((v, typ), rexpr) in p.nodes[n].upds:
				env[mk_var (v, typ)] = lv_expr (rexpr, pre_env)
		elif p.nodes[n].kind == 'Call':
			for (v, typ) in p.nodes[n].get_lvals ():
				env[mk_var (v, typ)] = (None, None, None, None)
		post_cache[n] = env
		return env
	while frontier:
		n = frontier.pop ()
		if [n2 for n2 in p.preds[n] if n2 in loop_body
				if n2 not in cache]:
			continue
		if n in cache:
			continue
		envs = [compute_post (n2) for n2 in p.preds[n]
			if n2 in loop_body]
		all_vs = set.union (* [set (env) for env in envs])
		cache[n] = dict ([(v, foldr1 (lv_merge,
				[env.get (v, (None, None, None, None))
					for env in envs]))
			for v in all_vs])
		frontier.extend ([n2 for n2 in p.nodes[n].get_conts ()
			if n2 in loop_body])
	return cache

def get_loop_linear_offs (p, loop_head):
	import search
	va = search.get_loop_var_analysis_at (p, loop_head)
	exprs = linear_series_exprs (p, loop_head, va)
	def offs_fn (n, expr):
		assert p.loop_id (n) == loop_head
		env = exprs[n]
		rv = lv_expr (expr, env)
		if rv[1] == None:
			return None
		elif rv[1] == 'LoopConst':
			return mk_num (0, expr.typ)
		elif rv[1] == 'LoopLinearSeries':
			return rv[2]
		else:
			assert not 'lv_expr kind understood', rv
	return offs_fn

def interesting_node_exprs (p, n, tags = None, use_pairings = True):
	if tags == None:
		tags = p.pairing.tags
	node = p.nodes[n]
	memaccs = node.get_mem_accesses ()
	vs = [(kind, ptr) for (kind, ptr, v, m) in memaccs]
	vs += [('MemUpdateArg', v) for (kind, ptr, v, m) in memaccs
		if kind == 'MemUpdate']

	if node.kind == 'Call' and use_pairings:
		tag = p.node_tags[n][0]
		from target_objects import functions, pairings
		import solver
		fun = functions[node.fname]
		arg_input_map = dict (azip (fun.inputs, node.args))
		pairs = [pair for pair in pairings.get (node.fname, [])
			if pair.tags == tags]
		if not pairs:
			return vs
		[pair] = pairs
		in_eq_vs = [(('Call', pair.name, i),
				var_subst (v, arg_input_map))
			for (i, ((lhs, l_s), (rhs, r_s)))
				in enumerate (pair.eqs[0])
			if l_s.endswith ('_IN') and r_s.endswith ('_IN')
			if l_s != r_s
			if solver.typ_representable (lhs.typ)
			for (v, site) in [(lhs, l_s), (rhs, r_s)]
			if site == '%s_IN' % tag]
		vs.extend (in_eq_vs)
	return vs

def interesting_linear_series_exprs (p, loop, va, tags = None,
		use_pairings = True):
	if tags == None:
		tags = p.pairing.tags
	expr_env = linear_series_exprs (p, loop, va)
	res_env = {}
	for (n, env) in expr_env.iteritems ():
		vs = interesting_node_exprs (p, n)

		vs = [(kind, v, lv_expr (v, env)) for (kind, v) in vs]
		vs = [(kind, v, offs, offs_set)
			for (kind, v, (_, lv, offs, offs_set)) in vs
			if lv == 'LoopLinearSeries']
		if vs:
			res_env[n] = vs
	return res_env

def mk_var_renames (xs, ys):
	renames = {}
	for (x, y) in azip (xs, ys):
		assert x.kind == 'Var' and y.kind == 'Var'
		assert x.name not in renames
		renames[x.name] = y.name
	return renames

def first_aligned_address (nodes, radix):
	ks = [k for k in nodes
		if k % radix == 0]
	if ks:
		return min (ks)
	else:
		return None

def entry_aligned_address (fun, radix):
	n = fun.entry
	while n % radix != 0:
		ns = fun.nodes[n].get_conts ()
		assert len (ns) == 1, (fun.name, n)
		[n] = ns
	return n

def aligned_address_sanity (functions, symbols, radix):
	for (f, func) in functions.iteritems ():
		if f not in symbols:
			# happens for static or invented functions sometimes
			continue
		if func.entry:
			addr = first_aligned_address (func.nodes, radix)
			if addr == None:
				printout ('Warning: %s: no aligned instructions' % f)
				continue
			addr2 = symbols[f][0]
			if addr != addr2:
				printout ('target mismatch on func %s' % f)
				printout ('  (starts at 0x%x not 0x%x)' % (addr, addr2))
				return False
			addr3 = entry_aligned_address (func, radix)
			if addr3 != addr2:
				printout ('entry mismatch on func %s' % f)
				printout ('  (enters at 0x%x not 0x%x)' % (addr3, addr2))
				return False
	return True

# variant of tarjan's strongly connected component algorithm
def tarjan (graph, entries):
	"""tarjan (graph, entries)
	variant of tarjan's strongly connected component algorithm
	e.g. tarjan ({1: [2, 3], 3: [4, 5]}, [1])
	entries should not be reachable"""
	data = {}
	comps = []
	for v in entries:
		assert v not in data
		tarjan1 (graph, v, data, [], set ([]), comps)
	return comps

def tarjan1 (graph, v, data, stack, stack_set, comps):
	vs = []
	while True:
		# skip through nodes with single successors
		data[v] = [len(data), len(data)]
		stack.append(v)
		stack_set.add(v)
		cs = graph[v]
		if len (cs) != 1 or cs[0] in data:
			break
		vs.append ((v, cs[0]))
		[v] = cs

	for c in graph[v]:
		if c not in data:
			tarjan1 (graph, c, data, stack, stack_set, comps)
			data[v][1] = min (data[v][1], data[c][1])
		elif c in stack_set:
			data[v][1] = min (data[v][1], data[c][0])

	vs.reverse ()
	for (v2, c) in vs:
		data[v2][1] = min (data[v2][1], data[c][1])

	for (v2, _) in [(v, 0)] + vs:
		if data[v2][1] == data[v2][0]:
			comp = []
			while True:
				x = stack.pop ()
				stack_set.remove (x)
				if x == v2:
					break
				comp.append (x)
			comps.append ((v2, comp))

def divides_loop (graph, split_set):
	graph2 = dict (graph)
	for n in split_set:
		graph2[n] = []
	assert 'ENTRY_POINT' not in graph2
	graph2['ENTRY_POINT'] = list (graph)
	comps = tarjan (graph2, ['ENTRY_POINT'])
	return not ([(h, t) for (h, t) in comps if t])

def strongly_connected_split_points1 (graph):
	"""find the nodes of a strongly connected
	component which, when removed, disconnect the component.
	complex loops lack such a split point."""

	# find one simple cycle in the graph
	walk = []
	walk_set = set ()
	n = min (graph)
	while n not in walk_set:
		walk.append (n)
		walk_set.add (n)
		n = graph[n][0]
	i = walk.index (n)
	cycle = walk[i:]

	def subgraph_test (subgraph):
		graph2 = dict ([(n, [n2 for n2 in graph[n] if n2 in subgraph])
			for n in subgraph])
		graph2['HEAD'] = list (subgraph)
		comps = tarjan (graph2, ['HEAD'])
		return bool ([h for (h, t) in comps if t])

	cycle_set = set (cycle)
	cycle = [('Node', set ([n]), False,
			[n2 for n2 in graph[n] if n2 != graph[n][0]])
		for n in cycle]
	i = 0
	while i < len (cycle):
		print i, cycle
		(kind, ns, test, unvisited) = cycle[i]
		if not unvisited:
			i += 1
			continue
		n = unvisited.pop ()
		arc_set = set ()
		while n not in cycle_set:
			if n in arc_set:
				# found two totally disjoint loops, so there
				# are no splitting points
				return set ()
			arc_set.add (n)
			n = graph[n][0]
		if n in ns:
			if kind == 'Node':
				# only this node can be a splittable now.
				if subgraph_test (set (graph) - set ([n])):
					return set ()
				else:
					return set ([n])
			else:
				cycle[i] = (kind, ns, True, unvisited)
				ns.update (arc_set)
				continue
		j = (i + 1) % len (cycle)
		new_ns = set ()
		new_unvisited = set ()
		new_test = False
		while n not in cycle[j][1]:
			new_ns.update (cycle[j][1])
			new_unvisited.update (cycle[j][3])
			new_test = cycle[j][2] or new_test
			j = (j + 1) % len (cycle)
		new_ns.update (arc_set)
		new_unvisited.update ([n3 for n2 in arc_set for n3 in graph[n2]])
		new_v = ('Group', new_ns, new_test, list (new_unvisited - new_ns))
		print i, j, n
		if j > i:
			cycle[i + 1:j] = [new_v]
		else:
			cycle = [cycle[i], new_v] + cycle[j:i]
			i = 0
		cycle_set.update (new_ns)
	for (kind, ns, test, unvisited) in cycle:
		if test and subgraph_test (ns):
			return set ()
	return set ([n for (kind, ns, _, _) in cycle
		if kind == 'Node' for n in ns])

def strongly_connected_split_points (graph):
	res = strongly_connected_split_points1 (graph)
	res2 = set ()
	for n in graph:
		graph2 = dict (graph)
		graph2[n] = []
		graph2['ENTRY'] = list (graph)
		comps = tarjan (graph2, ['ENTRY'])
		if not [comp for comp in comps if comp[1]]:
			res2.add (n)
	assert res == res2, (graph, res, res2)
	return res

def get_one_loop_splittable (p, loop_set):
	"""discover a component of a strongly connected
	component which, when removed, disconnects the component.
	complex loops lack such a split point."""
	candidates = set (loop_set)
	graph = dict ([(x, [y for y in p.nodes[x].get_conts ()
		if y in loop_set]) for x in loop_set])
	while candidates:
		loop2 = find_loop_avoiding (graph, loop_set, candidates)
		candidates = set.intersection (loop2, candidates)
		if not candidates:
			return None
		n = candidates.pop ()
		graph2 = dict ([(x, [y for y in graph[x] if y != n])
			for x in graph])
		comps = tarjan (graph2, [n])
		comps = [(h, t) for (h, t) in comps if t]
		if not comps:
			return n
		for (h, t) in comps:
			s = set ([h] + t)
			candidates = set.intersection (s, candidates)
	return None

def find_loop_avoiding (graph, loop, avoid):
	n = (list (loop - avoid) + list (loop))[0]
	arc = [n]
	visited = set ([n])
	while True:
		cs = set (graph[n])
		acs = cs - avoid
		vcs = set.intersection (cs, visited)
		if vcs:
			n = vcs.pop ()
			break
		elif acs:
			n = acs.pop ()
		else:
			n = cs.pop ()
		visited.add (n)
		arc.append (n)
	[i] = [i for (i, n2) in enumerate (arc) if n2 == n]
	return set (arc[i:])

# non-equality relations in proof hypotheses are recorded as a pretend
# equality and reverted to their 'real' meaning here.
def mk_stack_wrapper (stack_ptr, stack, excepts):
	return syntax.mk_rel_wrapper ('StackWrapper',
		[stack_ptr, stack] + excepts)

def mk_mem_acc_wrapper (addr, v):
	return syntax.mk_rel_wrapper ('MemAccWrapper', [addr, v])

def mk_mem_wrapper (m):
	return syntax.mk_rel_wrapper ('MemWrapper', [m])

def tm_with_word32_list (xs):
	if xs:
		return foldr1 (mk_plus, map (mk_word32, xs))
	else:
		return mk_uminus (mk_word32 (0))

def word32_list_from_tm (t):
	xs = []
	while t.is_op ('Plus'):
		[x, t] = t.vals
		assert x.kind == 'Num' and x.typ == word32T
		xs.append (x.val)
	if t.kind == 'Num':
		xs.append (t.val)
	return xs

def mk_eq_selective_wrapper (v, (xs, ys)):
	# this is a huge hack, but we need to put these lists somewhere
	xs = tm_with_word32_list (xs)
	ys = tm_with_word32_list (ys)
	return syntax.mk_rel_wrapper ('EqSelectiveWrapper', [v, xs, ys])

def apply_rel_wrapper (lhs, rhs):
	assert lhs.typ == syntax.builtinTs['RelWrapper']
	assert rhs.typ == syntax.builtinTs['RelWrapper']
	assert lhs.kind == 'Op'
	assert rhs.kind == 'Op'
	ops = set ([lhs.name, rhs.name])
	if ops == set (['StackWrapper']):
		[sp1, st1] = lhs.vals[:2]
		[sp2, st2] = rhs.vals[:2]
		excepts = list (set (lhs.vals[2:] + rhs.vals[2:]))
		for p in excepts:
			st1 = syntax.mk_memupd (st1, p, syntax.mk_word32 (0))
			st2 = syntax.mk_memupd (st2, p, syntax.mk_word32 (0))
		return syntax.Expr ('Op', boolT, name = 'StackEquals',
			vals = [sp1, st1, sp2, st2])
	elif ops == set (['MemAccWrapper', 'MemWrapper']):
		[acc] = [v for v in [lhs, rhs] if v.is_op ('MemAccWrapper')]
		[addr, val] = acc.vals
		assert addr.typ == syntax.word32T
		[m] = [v for v in [lhs, rhs] if v.is_op ('MemWrapper')]
		[m] = m.vals
		assert m.typ == builtinTs['Mem']
		expr = mk_eq (mk_memacc (m, addr, val.typ), val)
		return expr
	elif ops == set (['EqSelectiveWrapper']):
		[lhs_v, _, _] = lhs.vals
		[rhs_v, _, _] = rhs.vals
		if lhs_v.typ == syntax.builtinTs['RelWrapper']:
			return apply_rel_wrapper (lhs_v, rhs_v)
		else:
			return mk_eq (lhs, rhs)
	else:
		assert not 'rel wrapper opname understood'

def inst_eq_at_visit (exp, vis):
	if not exp.is_op ('EqSelectiveWrapper'):
		return True
	[_, xs, ys] = exp.vals
	# hacks
	xs = word32_list_from_tm (xs)
	ys = word32_list_from_tm (ys)
	if vis.kind == 'Number':
		return vis.n in xs
	elif vis.kind == 'Offset':
		return vis.n in ys
	else:
		assert not 'visit kind useable', vis

def strengthen_hyp (expr, sign = 1):
	if not expr.kind == 'Op':
		return expr
	if expr.name in ['And', 'Or']:
		vals = [strengthen_hyp (v, sign) for v in expr.vals]
		return syntax.adjust_op_vals (expr, vals)
	elif expr.name == 'Implies':
		[l, r] = expr.vals
		l = strengthen_hyp (l, - sign)
		r = strengthen_hyp (r, sign)
		return syntax.mk_implies (l, r)
	elif expr.name == 'Not':
		[x] = expr.vals
		x = strengthen_hyp (x, - sign)
		return syntax.mk_not (x)
	elif expr.name == 'StackEquals':
		if sign == 1:
			return syntax.Expr ('Op', boolT,
				name = 'ImpliesStackEquals', vals = expr.vals)
		else:
			return syntax.Expr ('Op', boolT,
				name = 'StackEqualsImplies', vals = expr.vals)
	elif expr.name == 'ROData':
		if sign == 1:
			return syntax.Expr ('Op', boolT,
				name = 'ImpliesROData', vals = expr.vals)
		else:
			return expr
	elif expr.name == 'Equals' and expr.vals[0].typ == boolT:
		vals = expr.vals
		if vals[1] in [syntax.true_term, syntax.false_term]:
			vals = [vals[1], vals[0]]
		if vals[0] == syntax.true_term:
			return strengthen_hyp (vals[1], sign)
		elif vals[0] == syntax.false_term:
			return strengthen_hyp (syntax.mk_not (vals[1]), sign)
		else:
			return expr
	else:
		return expr

def weaken_assert (expr):
	return strengthen_hyp (expr, -1)

pred_logic_ops = set (['Not', 'And', 'Or', 'Implies'])

def norm_neg (expr):
	if not expr.is_op ('Not'):
		return expr
	[nexpr] = expr.vals
	if not nexpr.is_op (pred_logic_ops):
		return expr
	if nexpr.is_op ('Not'):
		[expr] = nexpr.vals
		return norm_neg (expr)
	[x, y] = nexpr.vals
	if nexpr.is_op ('And'):
		return mk_or (norm_mk_not (x), norm_mk_not (y))
	elif nexpr.is_op ('Or'):
		return mk_and (norm_mk_not (x), norm_mk_not (y))
	elif nexpr.is_op ('Implies'):
		return mk_and (x, mk_not (y))

def norm_mk_not (expr):
	return norm_neg (mk_not (expr))

def split_conjuncts (expr):
	expr = norm_neg (expr)
	if expr.is_op ('And'):
		[x, y] = expr.vals
		return split_conjuncts (x) + split_conjuncts (y)
	else:
		return [expr]

def split_disjuncts (expr):
	expr = norm_neg (expr)
	if expr.is_op ('Or'):
		[x, y] = expr.vals
		return split_disjuncts (x) + split_disjuncts (y)
	else:
		return [expr]

def binary_search_least (test, minimum, maximum):
	"""find least n, minimum <= n <= maximum, for which test (n)."""
	assert maximum >= minimum
	if test (minimum):
		return minimum
	if maximum == minimum or not test (maximum):
		return None
	while maximum > minimum + 1:
		cur = (minimum + maximum) / 2
		if test (cur):
			maximum = cur
		else:
			minimum = cur + 1
	assert minimum + 1 == maximum
	return maximum

def binary_search_greatest (test, minimum, maximum):
	"""find greatest n, minimum <= n <= maximum, for which test (n)."""
	assert maximum >= minimum
	if test (maximum):
		return maximum
	if maximum == minimum or not test (minimum):
		return None
	while maximum > minimum + 1:
		cur = (minimum + maximum) / 2
		if test (cur):
			minimum = cur
		else:
			maximum = cur - 1
	assert minimum + 1 == maximum
	return minimum