621 lines
20 KiB
Plaintext
621 lines
20 KiB
Plaintext
(*
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* Copyright 2015, NICTA
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*
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* This software may be distributed and modified according to the terms of
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* the BSD 2-Clause license. Note that NO WARRANTY is provided.
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* See "LICENSE_BSD2.txt" for details.
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*
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* @TAG(NICTA_BSD)
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*)
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theory Etanercept imports
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"~~/src/HOL/Word/Word"
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NICTACompat
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SignedWords
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WordBitwiseSigned
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"ml-helpers/TermPatternAntiquote"
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keywords
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"word_refute" :: diag
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begin
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text {*
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This theory implements a tool for refuting word propositions. It works by constructing a C program
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that exhaustively explores the state space of your proposition.
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Usage:
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Run the "word_refute" command in a proof. The proof goal should involve only word
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and boolean expressions.
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Options:
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These are config options which can be set using "declare" or "using".
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- word_refute_debug enable verbose debugging output
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- word_refute_timeout timeout, in seconds
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It is currently a work in progress and has the following known issues:
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- Temporary files are just left for Isabelle to clean up when it exits. Should we be attempting
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to remove these ourselves?
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- The exploration strategy is completely naive, which simplifies the code, but sometimes leads to
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a failure to find trivial counter-examples.
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- There's no support for HOL binders like \<forall> and \<exists>. These could be supported relatively easily
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with GNU statement expressions, but I'm unsure if it's worth it.
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- We currently only support 8-, 16-, 32-, and 64-bit words. It would be straightforward to
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support other non-standard widths if required.
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The tool is named after etanercept, an ill-advised treatment for primary progressive aphasia, a
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condition who's symptoms include an inability to find the right words for things. Amusingly the
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word "etanercept" is also quite hard to remember.
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*}
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ML {*
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signature ETANERCEPT =
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sig
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end;
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structure Etanercept : ETANERCEPT =
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struct
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(* Toggle this to enable debugging output. *)
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val config_debug = let
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val (config, setup) = Attrib.config_bool (Binding.name "word_refute_debug") (K false)
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in
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Context.>>(Context.map_theory setup);
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config
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end
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(* Timeout, in seconds. *)
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val config_timeout = let
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val (config, setup) = Attrib.config_real (Binding.name "word_refute_timeout") (K 60.0)
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in
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Context.>>(Context.map_theory setup);
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config
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end
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(* C compiler options. Rationale:
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* -O3: search for counterexamples faster
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* -fwrapv: match HOL-Word semantics for signed overflow
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*)
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val config_cflags = "-O3 -fwrapv"
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fun debug_log ctxt str = if Config.get ctxt config_debug then tracing str else ()
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(* XXX: Clagged from metis *)
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fun dropWhile p =
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let
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fun f [] = []
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| f (l as x :: xs) = if p x then f xs else l
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in
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f
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end
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(* Exponentiation. *)
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fun exp _ 0 = 1
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| exp base n = if n mod 2 = 0 then exp (base*base) (n div 2) else base * exp base (n - 1)
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(* Strip whitespace from the end of a string. *)
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fun rstrip s =
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s
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|> String.explode
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|> List.rev
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|> dropWhile Char.isSpace
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|> List.rev
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|> String.implode
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(* Generate a string that's safe to put in a C string. In particular, we escape backslashes and
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* remove trailing underscores. The latter is not for string safety, but because focusing leads to
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* a variable named, e.g. "x__", that originated from a meta-bound "x". Stripping these
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* underscores makes the counter-example output clearer to the user.
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*)
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fun safe_name s =
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s
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|> String.explode
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|> map (fn c => if c = #"\\" then "\\\\" else String.implode [c])
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|> List.rev
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|> dropWhile (fn c => c = "_")
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|> List.rev
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|> String.concat
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(* Find a C compiler. Prefer Clang. *)
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fun cc () =
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let
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val (clang, r1) = Isabelle_System.bash_output "which clang";
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val (gcc, r2) = Isabelle_System.bash_output "which gcc"
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in
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if r1 = 0 then SOME (rstrip clang ^ " -Wno-tautological-compare") else
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if r2 = 0 then SOME (rstrip gcc) else
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NONE
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end
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(* Compile a C program. *)
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fun compile ctxt file =
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case cc () of
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SOME compiler =>
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let
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val serial = serial_string ();
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val tmp = File.shell_path (File.tmp_path (Path.explode ("etanercept" ^ serial ^ ".exe")));
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val cmd = compiler ^ " " ^ config_cflags ^ " -o " ^ tmp ^ " " ^ file
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val _ = debug_log ctxt ("Compiling: " ^ cmd)
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val (_, ret) = Isabelle_System.bash_output cmd
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in
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if ret = 0
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then tmp
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else error "Etanercept: failed to compile generated program (BUG)"
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end
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| NONE => error "Etanercept: no available C compiler"
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(* Mapping between Isabelle and C variables. *)
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fun var_index (vs, sz) v =
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case Termtab.lookup vs v of
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NONE => ((Termtab.update_new (v, sz) vs, sz+1), sz)
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| SOME n => ((vs, sz), n)
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val empty_var_index = (Termtab.empty, 0)
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(* The C symbol for the nth variable. *)
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fun to_var n = "v" ^ Int.toString n
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(* Create variable list from mapping *)
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val var_index_list =
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fst #> Termtab.dest
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#> sort (int_ord o apply2 snd)
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#> map (apsnd to_var)
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fun name_of (Free (name, _)) = safe_name name
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| name_of t = raise TERM ("Etanercept: unexpected variable (BUG)", [t])
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(* Types that we know about. *)
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type IntInfo = { isa_type: typ,
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c_type: string,
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c_min: string,
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c_max: string,
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c_printf: string
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}
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val type_info : IntInfo Typtab.table =
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[
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{ isa_type = @{typ "8 word"},
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c_type = "uint8_t",
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c_min = "((uint8_t)0)",
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c_max = "UINT8_MAX",
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c_printf = "PRIu8"
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},
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{ isa_type = @{typ "16 word"},
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c_type = "uint16_t",
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c_min = "((uint16_t)0)",
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c_max = "UINT16_MAX",
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c_printf = "PRIu16"
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},
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{ isa_type = @{typ "32 word"},
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c_type = "uint32_t",
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c_min = "((uint32_t)0)",
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c_max = "UINT32_MAX",
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c_printf = "PRIu32"
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},
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{ isa_type = @{typ "64 word"},
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c_type = "uint64_t",
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c_min = "((uint64_t)0)",
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c_max = "UINT64_MAX",
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c_printf = "PRIu64"
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},
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{ isa_type = @{typ "nat"},
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c_type = "uintmax_t",
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c_min = "((uintmax_t)0)",
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c_max = "UINTMAX_MAX",
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c_printf = "PRIuMAX"
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},
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{ isa_type = @{typ "8 signed word"},
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c_type = "int8_t",
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c_min = "INT8_MIN",
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c_max = "INT8_MAX",
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c_printf = "PRId8"
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},
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{ isa_type = @{typ "16 signed word"},
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c_type = "int16_t",
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c_min = "INT16_MIN",
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c_max = "INT16_MAX",
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c_printf = "PRId16"
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},
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{ isa_type = @{typ "32 signed word"},
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c_type = "int32_t",
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c_min = "INT32_MIN",
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c_max = "INT32_MAX",
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c_printf = "PRId32"
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},
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{ isa_type = @{typ "64 signed word"},
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c_type = "int64_t",
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c_min = "INT64_MIN",
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c_max = "INT64_MAX",
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c_printf = "PRId64"
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},
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{ isa_type = @{typ "int"},
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c_type = "intmax_t",
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c_min = "INTMAX_MIN",
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c_max = "INTMAX_MAX",
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c_printf = "PRIdMAX"
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}
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] |> (fn infos => Typtab.make (map (fn info => (#isa_type info, info)) infos))
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fun lookup_info f expect_success t =
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let val severity = if expect_success then " (BUG)" else ""
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in case t of
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Free (_, ty) => (case Typtab.lookup type_info ty of
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SOME info => f info
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| NONE => raise TYPE ("etanercept: unsupported type" ^ severity, [ty], [t]))
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| _ => raise TERM ("Etanercept: unsupported term" ^ severity, [t]) end
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val min_of = lookup_info #c_min true
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val max_of = lookup_info #c_max true
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val type_of = lookup_info #c_type true
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val format_of = lookup_info #c_printf true
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fun cast_to _ (Type ("fun", [from, to])) =
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(case try (lookup_info #c_type false) (Free ("_", to)) of
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SOME c_type => c_type
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| NONE => raise TYPE ("Etanercept: unsupported ucast/scast result type", [to], []))
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| cast_to _ T = raise TYPE ("Etanercept: strange ucast/scast type (BUG)", [T], [])
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(* A printf format string for printing the variables. *)
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fun format_string vs =
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var_index_list vs
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|> map (fn (var, c_var) => "\" " ^ name_of var ^ " = %\" " ^ format_of var ^ " \"\\n\" ")
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|> String.concat
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(* The variables as a list of arguments to be passed to a C function. *)
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fun as_args vs =
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var_index_list vs
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|> map (fn (var, c_var) => ", " ^ c_var)
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|> String.concat
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(* Initialisation for the variables. *)
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fun loop_header vs =
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var_index_list vs
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|> map (fn (var, c_var) =>
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type_of var ^ " " ^ c_var ^ ";\n" ^
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"for (" ^ c_var ^ " = " ^ min_of var ^ "; ; " ^ c_var ^ "++) {\n")
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|> String.concat
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(* Per-variable loop termination. *)
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fun loop_footer vs =
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var_index_list vs
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|> rev
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|> map (fn (var, c_var) =>
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"if (" ^ c_var ^ " == " ^ max_of var ^ ") { break; }\n}\n")
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|> String.concat
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(* Translate an Isabelle term into the equivalent C expression, collecting discovered variables
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* along the way.
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*)
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fun translate state vs t =
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let
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val tr = translate state
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fun bin_op op1 op2 =
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let val (vs', s1) = tr vs op1;
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val (vs'', s2) = tr vs' op2
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in (vs'', s1, s2)
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end
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in
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case t of
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@{term "Trueprop"} $ t' => tr vs t'
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| @{term "True"} => (vs, "true")
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| @{term "False"} => (vs, "false")
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| @{term_pat "?a = ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " == " ^ s2 ^ ")")
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end
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| @{term_pat "\<not> ?a"} =>
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let val (vs', s) = tr vs a
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in (vs', "(!" ^ s ^ ")")
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end
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| @{term_pat "?a < ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " < " ^ s2 ^ ")")
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end
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| @{term_pat "?a \<le> ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " <= " ^ s2 ^ ")")
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end
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| @{term_pat "?a + ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " + " ^ s2 ^ ")")
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end
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| @{term_pat "?a - ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " - " ^ s2 ^ ")")
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end
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| @{term_pat "?a * ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " * " ^ s2 ^ ")")
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end
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| @{term_pat "- ?a"} =>
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let val (vs', s) = tr vs a
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in (vs', "(-" ^ s ^ ")")
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end
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| @{term_pat "?a div ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s2 ^ " == 0 ? 0 : (" ^ s1 ^ " / " ^ s2 ^ "))")
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end
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| @{term_pat "?a mod ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s2 ^ " == 0 ? " ^ s1 ^ " : (" ^ s1 ^ " % " ^ s2 ^ "))")
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end
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| @{term_pat "?a \<longrightarrow> ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "((!" ^ s1 ^ ") || (" ^ s2 ^ "))")
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end
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| @{term_pat "shiftl ?a ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " << " ^ s2 ^ ")")
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end
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| @{term_pat "shiftr ?a ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " >> " ^ s2 ^ ")")
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end
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| @{term_pat "?a && ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " & " ^ s2 ^ ")")
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end
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| @{term_pat "?a || ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " | " ^ s2 ^ ")")
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end
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| @{term_pat "?a xor ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " ^ " ^ s2 ^ ")")
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end
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| @{term_pat "NOT ?a"} =>
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let val (vs', s) = tr vs a
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in (vs', "(~" ^ s ^ ")")
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end
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| @{term_pat "test_bit ?a ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " ^ & (1ULL << " ^ s2 ^ "))")
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end
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| @{term_pat "lsb ?a"} =>
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let val (vs', s) = tr vs a
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in (vs', "(" ^ s ^ " & 1)")
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end
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| Const (@{const_name Word.ucast}, typ) $ a =>
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let val (vs', s) = tr vs a
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in (vs', "((" ^ cast_to state typ ^ ")" ^ s ^ ")")
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end
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| Const (@{const_name Word.scast}, typ) $ a =>
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let val (vs', s) = tr vs a
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in (vs', "((" ^ cast_to state typ ^ ")" ^ s ^ ")")
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end
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| Free (name, T) =>
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if Typtab.defined type_info T
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then let val (vs', n) = var_index vs t
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in (vs', to_var n)
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end
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else (case T of Type ("Word.word", _) =>
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raise TYPE ("unsupported word width of variable " ^ name, [T], [])
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| _ => raise TYPE ("unsupported type of variable " ^ name, [T], []))
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| @{term_pat "?a \<and> ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " && " ^ s2 ^ ")")
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end
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| @{term_pat "?a \<or> ?b"} =>
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let val (vs', s1, s2) = bin_op a b
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in (vs', "(" ^ s1 ^ " || " ^ s2 ^ ")")
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end
|
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| @{term_pat "0"} => (vs, "(0)")
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| @{term_pat "1"} => (vs, "(1)")
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| @{term_pat "Suc ?a"} =>
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let val (vs', s) = tr vs a
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in (vs', "(1 + " ^ s ^ ")")
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end
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| @{term_pat "numeral ?a"} =>
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let val a' = HOLogic.dest_num a
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val suffix = if a' > exp 2 32 - 1 then "ull" else ""
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in (vs, "(" ^ string_of_int a' ^ suffix ^ ")")
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end
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| _ => raise TERM ("unsupported subterm ", [t])
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end
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(* Construct a C program to match the current goal state. *)
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fun make_program st =
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let
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val state = Toplevel.proof_of st;
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val {goal = g, ...} = Proof.raw_goal state;
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val (_, g') = Subgoal.focus (Proof.context_of state) 1 g
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val (gi, _) = Logic.goal_params (Thm.prop_of g') 1
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val (vars, expr) = translate state empty_var_index gi
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in
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"#include <inttypes.h>\n" ^
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"#include <limits.h>\n" ^
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"#include <stdbool.h>\n" ^
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"#include <stdint.h>\n" ^
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"#include <stdio.h>\n" ^
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"int main(void) {\n" ^
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loop_header vars ^
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"if (!" ^ expr ^ ") {\n" ^
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"printf(\"Found counter-example:\\n\"" ^ format_string vars ^ as_args vars ^ ");\n" ^
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"return 0;\n" ^
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"}\n" ^
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loop_footer vars ^
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"return -1;\n" ^
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"}"
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end
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(* Try to refute the current goal by using a C program to find a counter example. *)
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fun refute st =
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let
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val ctxt = Proof.context_of (Toplevel.proof_of st);
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val program = make_program st;
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val serial = serial_string ();
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val tmp = File.tmp_path (Path.explode ("etanercept" ^ serial ^ ".c"));
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val _ = File.write tmp program;
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val aout = compile ctxt (File.shell_path tmp);
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val _ = debug_log ctxt
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("Program:\n" ^ program ^ "\nWritten to: " ^ File.shell_path tmp ^ "\nCompiled to: " ^ aout)
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val (msg, ret) = TimeLimit.timeLimit (Config.get ctxt config_timeout |> Time.fromReal)
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Isabelle_System.bash_output aout
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in
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(if ret = 0
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then msg
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else "Etanercept found no counter example")
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|> writeln
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end
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handle TimeLimit.TimeOut =>
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warning "Etanercept: timed out"
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|
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(* Install the command itself. *)
|
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val _ = Outer_Syntax.command @{command_keyword word_refute}
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"Construct a C program to find counter-examples to word propositions"
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(Scan.succeed [] >> (fn _ => Toplevel.keep_proof refute))
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end
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*}
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text {* Basic examples *}
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lemma "True \<and> False"
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word_refute
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oops
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|
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lemma "(x::32 word) = 0"
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word_refute
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|
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using [[word_refute_debug]]
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word_refute
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oops
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|
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word_refute -- "requires a proof state"
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|
|
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text {* Can deal with top-level quantified vars *}
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lemma "\<And>x. (x::32 word) \<noteq> y\<^sub>1 \<and> y \<ge> x"
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word_refute
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|
oops
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|
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lemma "(x::8 word) = y"
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word_refute
|
|
oops
|
|
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|
lemma "(x::64 word) < y"
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word_refute
|
|
oops
|
|
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lemma "\<And>(x::32 word) y. x = y"
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word_refute
|
|
oops
|
|
|
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text {* Previously, this example would give us a tautological comparison warning from Clang. *}
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lemma "y = y \<and> (x::32 word) << y = x"
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word_refute
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|
oops
|
|
|
|
text {* Also works for nats (approximated with uint64) *}
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|
lemma "(x :: nat) = 0"
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|
word_refute
|
|
oops
|
|
|
|
text {* Example that partially demonstrates Etanercept's utility. *}
|
|
lemma "(x::8 word) > y \<or> x < y + y + y + y"
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|
(* quickcheck cannot handle this *)
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|
(* quickcheck *)
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|
(* quickcheck[random] takes a little while *)
|
|
quickcheck[random]
|
|
(* Etanercept immediately finds the trivial counterexample *)
|
|
word_refute
|
|
oops
|
|
|
|
text {* Example that demonstrates one of Etanercept's weaknesses. *}
|
|
lemma "(x::32 word) div y = x"
|
|
(* The naive exploration strategy means we wait for Etanercept to try every value of y before
|
|
* moving x beyond 0.
|
|
*)
|
|
word_refute
|
|
oops
|
|
|
|
text {*
|
|
This is an interesting example that, due to Etanercept's exploration strategy, *should* be out of
|
|
its reach with a reasonable time out. Instead, the C compiler folds the entire loop into immediate
|
|
discovery of a counter-example.
|
|
*}
|
|
lemma "(x::64 word) \<ge> y \<and> x \<ge> y + y"
|
|
word_refute
|
|
oops
|
|
|
|
text {* Various cases that test our handling of numeric literals. *}
|
|
lemma "(x::32 word) && 45 = 0"
|
|
word_refute
|
|
oops
|
|
lemma "(x::32 word) < 45"
|
|
word_refute
|
|
oops
|
|
lemma "(x::32 word) < 0"
|
|
word_refute
|
|
oops
|
|
lemma "(x::32 word) < 1"
|
|
word_refute
|
|
oops
|
|
lemma "(x::32 word) < 0x45"
|
|
word_refute
|
|
oops
|
|
|
|
text {* Test something non-trivial that we shouldn't be able to refute. *}
|
|
lemma "(x::32 word) && 1 = 1 \<and> x && (~~ 1) = 0 \<longrightarrow> x = 1"
|
|
word_refute
|
|
apply word_bitwise
|
|
done
|
|
lemma "(x::32 signed word) && 1 = 1 \<and> x && (~~ 1) = 0 \<longrightarrow> x = 1"
|
|
word_refute
|
|
apply word_bitwise_signed
|
|
done
|
|
|
|
text {* Test that division by zero is correctly modelled with Isabelle's semantics. *}
|
|
lemma "(x::64 word) div 0 = 0"
|
|
word_refute
|
|
by (simp add: word_arith_nat_defs(6))
|
|
|
|
text {* Test we can handle large literals. *}
|
|
lemma "0xdeadbeeffacecafe > 0xdeadbeeffacade00 + (x::64 word)"
|
|
word_refute
|
|
oops
|
|
|
|
text {* Test some casting operations. *}
|
|
lemma "(ucast::32 signed word \<Rightarrow> 32 word) (x::32 signed word) = (y::32 word)"
|
|
word_refute
|
|
oops
|
|
lemma "ucast (x::32 word) = (y:: 8 word)"
|
|
word_refute
|
|
oops
|
|
lemma "ucast (x::32 word) = (y::32 word)"
|
|
word_refute
|
|
oops
|
|
lemma "scast (x::32 signed word) = (y::8 word)"
|
|
word_refute
|
|
oops
|
|
|
|
text {* Try some things we shouldn't be able to refute. *}
|
|
lemma "(x::64 word) >> 0 = x"
|
|
word_refute
|
|
by simp
|
|
lemma "(x::64 word) >> 1 = x div 2"
|
|
word_refute
|
|
apply simp
|
|
apply (rule shiftr_div_2n_w[where n=1, simplified])
|
|
apply (simp add:word_size)
|
|
done
|
|
lemma "(x::64 word) << 1 = x * 2"
|
|
word_refute
|
|
apply (subst shiftl_t2n)
|
|
apply simp
|
|
done
|
|
|
|
text {* Test that our compiler setup permits signed overflow *}
|
|
lemma "(x :: 32 signed word) < x + 1"
|
|
word_refute -- "should find INT_MAX"
|
|
oops
|
|
|
|
text {* C translation pitfalls *}
|
|
lemma "(ucast (x * y :: 16 word) :: 32 signed word) \<ge> 0"
|
|
text {* A naive translation fails due to semantic mismatch *}
|
|
word_refute
|
|
by simp
|
|
|
|
|
|
text {* Unsupported constructs *}
|
|
lemma "(x :: 1 word) = 0" -- "bad word size"
|
|
(* word_refute *)
|
|
oops
|
|
|
|
end
|