├── .gitignore ├── .gitmodules ├── COPYING ├── Makefile ├── README.md ├── config.json ├── findings.md ├── shell.nix └── src ├── dirty_lemmas.k.md ├── dss.md ├── lemmas.k.md ├── prelude.smt2.md └── storage.k.md /.gitignore: -------------------------------------------------------------------------------- 1 | *~ 2 | *.timestamp 3 | tmp/ 4 | out/* -------------------------------------------------------------------------------- /.gitmodules: -------------------------------------------------------------------------------- 1 | [submodule "dss"] 2 | path = dss 3 | url = https://github.com/makerdao/dss.git 4 | -------------------------------------------------------------------------------- /COPYING: -------------------------------------------------------------------------------- 1 | GNU AFFERO GENERAL PUBLIC LICENSE 2 | Version 3, 19 November 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU Affero General Public License is a free, copyleft license for 11 | software and other kinds of works, specifically designed to ensure 12 | cooperation with the community in the case of network server software. 13 | 14 | The licenses for most software and other practical works are designed 15 | to take away your freedom to share and change the works. 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It is safest 628 | to attach them to the start of each source file to most effectively 629 | state the exclusion of warranty; and each file should have at least 630 | the "copyright" line and a pointer to where the full notice is found. 631 | 632 | 633 | Copyright (C) 634 | 635 | This program is free software: you can redistribute it and/or modify 636 | it under the terms of the GNU Affero General Public License as published by 637 | the Free Software Foundation, either version 3 of the License, or 638 | (at your option) any later version. 639 | 640 | This program is distributed in the hope that it will be useful, 641 | but WITHOUT ANY WARRANTY; without even the implied warranty of 642 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 643 | GNU Affero General Public License for more details. 644 | 645 | You should have received a copy of the GNU Affero General Public License 646 | along with this program. If not, see . 647 | 648 | Also add information on how to contact you by electronic and paper mail. 649 | 650 | If your software can interact with users remotely through a computer 651 | network, you should also make sure that it provides a way for users to 652 | get its source. For example, if your program is a web application, its 653 | interface could display a "Source" link that leads users to an archive 654 | of the code. There are many ways you could offer source, and different 655 | solutions will be better for different programs; see section 13 for the 656 | specific requirements. 657 | 658 | You should also get your employer (if you work as a programmer) or school, 659 | if any, to sign a "copyright disclaimer" for the program, if necessary. 660 | For more information on this, and how to apply and follow the GNU AGPL, see 661 | . 662 | -------------------------------------------------------------------------------- /Makefile: -------------------------------------------------------------------------------- 1 | DAPP_DIR = $(CURDIR)/dss 2 | SRC_DIR = $(CURDIR)/src 3 | SRCS = $(addprefix $(SRC_DIR)/, dss.md lemmas.k.md storage.k.md prelude.smt2.md) 4 | DAPP_SRCS = $(wildcard $(DAPP_DIR)/src/*) 5 | # if KLAB_OUT isn't defined, default is to use out/ 6 | ifdef KLAB_OUT 7 | OUT_DIR = $(KLAB_OUT) 8 | else 9 | OUT_DIR = $(CURDIR)/out 10 | endif 11 | TMPDIR ?= $(CURDIR)/tmp 12 | ifndef KLAB_EVMS_PATH 13 | $(error $(red)Error$(reset): KLAB_EVMS_PATH must be defined and point to evm-semantics!) 14 | endif 15 | SPECS_DIR = $(OUT_DIR)/specs 16 | ACTS_DIR = $(OUT_DIR)/acts 17 | DOC_DIR = $(OUT_DIR)/doc 18 | 19 | KLAB_FLAGS = KLAB_OUT=$(OUT_DIR) 20 | 21 | SMT_PRELUDE = $(OUT_DIR)/prelude.smt2 22 | RULES = $(OUT_DIR)/rules.k 23 | 24 | SPEC_MANIFEST = $(SPECS_DIR)/specs.manifest 25 | 26 | all: dapp spec 27 | 28 | dapp: 29 | dapp --version 30 | git submodule update --init --recursive 31 | cd $(DAPP_DIR) && dapp --use solc:0.5.9 build && cd ../ 32 | 33 | dapp-clean: 34 | cd $(DAPP_DIR) && dapp clean && cd ../ 35 | 36 | $(SPEC_MANIFEST): $(SRCS) $(DAPP_SRCS) 37 | mkdir -p $(SPECS_DIR) 38 | $(KLAB_FLAGS) klab build 39 | 40 | spec: $(SPEC_MANIFEST) 41 | 42 | spec-clean: 43 | rm -f $(SPECS_DIR)/* $(ACTS_DIR)/* $(SPEC_MANIFEST) 44 | 45 | $(DOC_DIR)/dss.html: $(SRCS) 46 | $(info Generating html documentation: $@) 47 | mkdir -p $(DOC_DIR) 48 | $(KLAB_FLAGS) klab report > $@ 49 | 50 | doc: $(DOC_DIR)/dss.html 51 | 52 | doc-clean: 53 | rm -rf $(DOC_DIR)/* 54 | 55 | log-clean: 56 | rm -rf $(TMPDIR)/klab 57 | 58 | clean: dapp-clean spec-clean doc-clean 59 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | Table of Contents 2 | ================= 3 | 4 | * [Installation](#installation) 5 | * [dependencies](#dependencies) 6 | * [build](#build) 7 | * [Usage](#usage) 8 | * [Progress](#progress) 9 | * [Documentation](#documentation) 10 | * [specification format](#specification-format) 11 | * [License](#license) 12 | 13 | This repo contains the formal specification and verification of [multicollateral dai](https://github.com/makerdao/dss). 14 | 15 | The behavior of the contracts is specified in a literate format at [dss.md](src/dss.md), which generates a series of reachability claims, defining `succeeding` and `reverting` behavior for each function of each contract. These reachability claims are then tested against the [formal semantics of the EVM](https://github.com/kframework/evm-semantics) using the [klab](https://github.com/dapphub/klab) tool for interactive proof inspection and debugging. 16 | 17 | An html version of the specification, together with links to in-browser symbolic execution previews, is available at [dapp.ci/k-dss](https://dapp.ci/k-dss) 18 | 19 | # Installation 20 | ## dependencies 21 | * klab. Installation instructions can be found at [klab](https://github.com/dapphub/klab). 22 | 23 | ## build 24 | ```sh 25 | git clone git@github.com:dapphub/k-dss.git 26 | make 27 | ``` 28 | 29 | This will download and build the target contracts in `dss/`, and compile the literate specifications in `src/` to K specifications, saving the results in `out/specs`. 30 | 31 | # Usage 32 | 33 | To run a proof with [klab](https://github.com/dapphub/klab), try: 34 | 35 | ```sh 36 | klab prove --dump out/specs/proof-Vat_dai_pass_rough.k 37 | ``` 38 | 39 | After it finishes, you can open an interactive debug session to exploring the success behaviour of the `dai()` method of the contract `Vat`: 40 | 41 | ```sh 42 | klab debug $(klab hash out/specs/proof-Vat_dai_pass_rough.k) 43 | ``` 44 | 45 | If you aren't interested in exploring with the debugger, you can omit the `--dump` flag for better performance. 46 | 47 | # Progress 48 | 49 | You can inspect the current state of the proofs in the CI running at [dapp.ci](https://dapp.ci/k-dss). 50 | 51 | # Documentation 52 | 53 | To build the literate specification in HTML, run `make doc`. The output of this process is available at [dapp.ci/k-dss](https://dapp.ci/k-dss). 54 | 55 | ## specification format 56 | The format used in [dss.md](src/dss.md) provides a concise way of specifying the behavior of a contract method (see [act-mode](https://github.com/livnev/act-mode) for a simple emacs major mode for `.act` specs). 57 | 58 | Let's break down the specification of the behavior of the function `heal` in the contract `Vat`: 59 | 60 | ```act 61 | behaviour heal of Vat 62 | interface heal(bytes32 u, bytes32 v, int256 rad) 63 | 64 | types 65 | 66 | Can : uint256 67 | Dai_v : uint256 68 | Sin_u : uint256 69 | Debt : uint256 70 | Vice : uint256 71 | 72 | storage 73 | 74 | #Vat.wards(CALLER_ID) |-> Can 75 | #Vat.dai(v) |-> Dai_v => Dai_v - rad 76 | #Vat.sin(u) |-> Sin_u => Sin_u - rad 77 | #Vat.debt |-> Debt => Debt - rad 78 | #Vat.vice |-> Vice => Vice - rad 79 | 80 | iff 81 | 82 | Can == 1 83 | 84 | iff in range uint256 85 | 86 | Dai_v - rad 87 | Sin_u - rad 88 | Debt - rad 89 | Vice - rad 90 | ``` 91 | 92 | This snippet of code will generate two reachability claims, `proof-Vat_heal_succ.k` and `proof-Vat_heal_fail.k`. Both of these claims will refer to the bytecode of the contract `Vat` and use the function signature of `heal(bytes32,bytes32,int256)` as the first 4 bytes of calldata (keeping the rest of the calldata abstract). In the `success` spec, the conditions under the `iff` headers are postulated, while in the `fail` spec their negation is. 93 | 94 | The interesting part of this particular function happens under the `storage` header. The meaning of the line: 95 | `#Vat.dai(v) |-> Dai_v => Dai_v - rad` 96 | is that in the `success` case, the value at the storage location which we call `#Vat.dai(v)` will be updated from `Dai_v` to `Dai_v - rad`. 97 | 98 | To prove this reachability claim, the k prover explores all possible execution paths starting from the precondition (whats on the left hand side of a `=>`) and the claim is proven if they all end in a state satisfying the postcondition (right hand side of the `=>`). 99 | 100 | More information about how the K prover and the K Framework in general works can be found at [Semantics-Based Program Verifiers for All Languages](http://fsl.cs.illinois.edu/FSL/papers/2016/stefanescu-park-yuwen-li-rosu-2016-oopsla/stefanescu-park-yuwen-li-rosu-2016-oopsla-public.pdf) and a detailed description of the semantics of EVM defined in K is given in [KEVM: A Complete Semantics of the Ethereum Virtual Machine](https://www.ideals.illinois.edu/handle/2142/97207). 101 | 102 | # License 103 | All applicable work in this repository is licensed under AGPL-3.0. Authors: 104 | * Lev Livnev 105 | * Denis Erfurt 106 | * Martin Lundfall 107 | -------------------------------------------------------------------------------- /config.json: -------------------------------------------------------------------------------- 1 | { 2 | "name": "k-dss", 3 | "url": "https://github.com/dapphub/k-dss/", 4 | "src": { 5 | "specification": "./src/dss.md", 6 | "smt_prelude": "./src/prelude.smt2.md", 7 | "rules": [ 8 | "./src/storage.k.md", 9 | "./src/lemmas.k.md" 10 | ], 11 | "dirty_rules": [ 12 | "./src/dirty_lemmas.k.md" 13 | ] 14 | }, 15 | "implementations": { 16 | "Vat": { 17 | "src": "src/vat.sol" 18 | }, 19 | "Vow": { 20 | "src": "src/vow.sol" 21 | }, 22 | "Dai": { 23 | "src": "src/dai.sol" 24 | }, 25 | "Jug": { 26 | "src": "src/jug.sol" 27 | }, 28 | "Pot": { 29 | "src": "src/pot.sol" 30 | }, 31 | "Cat": { 32 | "src": "src/cat.sol" 33 | }, 34 | "GemJoin": { 35 | "src": "src/join.sol" 36 | }, 37 | "DaiJoin": { 38 | "src": "src/join.sol" 39 | }, 40 | "Flipper": { 41 | "src": "src/flip.sol" 42 | }, 43 | "Flopper": { 44 | "src": "src/flop.sol" 45 | }, 46 | "Flapper": { 47 | "src": "src/flap.sol" 48 | }, 49 | "DSToken": { 50 | "src": "lib/ds-token/src/token.sol" 51 | }, 52 | "End": { 53 | "src": "src/end.sol" 54 | }, 55 | "DSValue": { 56 | "src": "lib/ds-value/src/value.sol" 57 | }, 58 | "Spotter": { 59 | "src": "src/spot.sol" 60 | } 61 | }, 62 | "split_fail":[ 63 | ], 64 | "timeouts": { 65 | "Vat_grab_pass_rough": "16h", 66 | "Vat_grab_fail_rough": "16h", 67 | "Vat_frob-diff-nonzero_pass_rough": "16h", 68 | "Vat_frob-diff-nonzero_fail_rough": "2d", 69 | "Vat_frob-diff-zero-dink_pass_rough": "16h", 70 | "Vat_frob-diff-zero-dink_fail_rough": "2d", 71 | "Vat_frob-diff-zero-dink_pass": "36h", 72 | "Vat_frob-diff-zero-dart_pass_rough": "16h", 73 | "Vat_frob-diff-zero-dart_fail_rough": "2d", 74 | "Vat_frob-diff-zero_pass_rough": "16h", 75 | "Vat_frob-diff-zero_fail_rough": "2d", 76 | "Vat_frob-same-nonzero_pass_rough": "16h", 77 | "Vat_frob-same-nonzero_fail_rough": "2d", 78 | "Vat_frob-same-zero-dink_pass_rough": "16h", 79 | "Vat_frob-same-zero-dink_fail_rough": "2d", 80 | "Vat_frob-same-zero-dink_pass": "2d", 81 | "Vat_frob-same-zero-dart_pass_rough": "16h", 82 | "Vat_frob-same-zero-dart_fail_rough": "2d", 83 | "Vat_frob-same-zero_pass_rough": "16h", 84 | "Vat_frob-same-zero_fail_rough": "2d", 85 | "Vat_fork-diff_pass_rough": "16h", 86 | "Vat_fork-diff_fail_rough": "16h", 87 | "Vat_fork-same_pass_rough": "16h", 88 | "Vat_fork-same_fail_rough": "2d", 89 | "Cat_bite-full_pass_rough": "3d", 90 | "Cat_bite-full_pass": "3d", 91 | "Cat_bite-full_fail_rough": "3d", 92 | "Cat_bite-lump_pass_rough": "3d", 93 | "Cat_bite-lump_pass": "46h", 94 | "Cat_bite-lump_fail_rough": "3d", 95 | "End_free_pass_rough": "16h", 96 | "End_skim_pass_rough": "16h", 97 | "End_skim_fail_rough": "24h", 98 | "End_bail_pass_rough": "16h", 99 | "End_bail_fail_rough": "24h", 100 | "End_skip_pass_rough": "42h", 101 | "End_skip_pass": "42h", 102 | "End_skip_fail_rough": "36h", 103 | "Flipper_dent_fail_rough": "16h", 104 | "Flopper_dent_fail_rough": "16h", 105 | "Flipper_tend_fail_rough": "16h", 106 | "Flapper_tend_fail_rough": "2d", 107 | "Flapper_deal_fail_rough": "16h", 108 | "Pot_drip_fail_rough": "16h", 109 | "Jug_drip_fail_rough": "16h", 110 | "Vow_cage-surplus_fail_rough": "16h", 111 | "Vow_cage-deficit_fail_rough": "16h", 112 | "Vow_cage-balance_fail_rough": "16h", 113 | "End_cage-surplus_fail_rough": "35h", 114 | "End_cage-deficit_fail_rough": "35h", 115 | "End_cage-balance_fail_rough": "35h" 116 | }, 117 | "memory" : { 118 | "Vat_frob-diff-nonzero_fail_rough": "25G", 119 | "Vat_frob-diff-zero-dink_fail_rough": "20G", 120 | "Vat_frob-diff-zero-dart_fail_rough": "20G", 121 | "Vat_frob-diff-zero_fail_rough": "20G", 122 | "Vat_frob-same-nonzero_fail_rough": "25G", 123 | "Vat_frob-same-zero_fail_rough": "25G", 124 | "Flapper_tend_fail_rough": "20G", 125 | "Cat_bite-full_pass_rough": "22G", 126 | "Cat_bite-full_pass": "22G", 127 | "Cat_bite-full_fail_rough": "30G", 128 | "Cat_bite-lump_pass_rough": "22G", 129 | "Cat_bite-lump_pass": "22G", 130 | "Cat_bite-lump_fail_rough": "35G", 131 | "End_skip_pass_rough": "20G", 132 | "End_skip_fail_rough": "20G", 133 | "End_skim_fail_rough": "20G", 134 | "End_bail_fail_rough": "20G" 135 | }, 136 | "dapp_root": "./dss", 137 | "host": "127.0.0.1:8080" 138 | } 139 | -------------------------------------------------------------------------------- /findings.md: -------------------------------------------------------------------------------- 1 | frob: 2 | 3 | callData is maybe malformed: ABI_dart is not wrapped in #unsigned 4 | 5 | in general: ***ui proofs don't apply if the i value is not wrapped in unsigned 6 | -------------------------------------------------------------------------------- /shell.nix: -------------------------------------------------------------------------------- 1 | with import {}; 2 | stdenv.mkDerivation { 3 | name = "k-dss"; 4 | buildInputs = [ 5 | flex 6 | getopt 7 | utillinux 8 | git 9 | gnumake 10 | jq 11 | nodejs 12 | openjdk8 13 | parallel 14 | wget 15 | zip 16 | z3 17 | ]; 18 | shellHook = '' 19 | if [ -z "$KLAB_PATH" ]; then echo "WARNING: The environment variable KLAB_PATH should be set to point to the klab repo. Please fix and reënter the nix shell."; fi 20 | export PATH=$KLAB_PATH/node_modules/.bin/:$KLAB_PATH/bin:$PATH 21 | export KLAB_EVMS_PATH=$KLAB_PATH/evm-semantics 22 | ''; 23 | } 24 | -------------------------------------------------------------------------------- /src/dirty_lemmas.k.md: -------------------------------------------------------------------------------- 1 | This is an example for rules that won't affect the proof hashes 2 | this should be "flushed" once in a while to the real lemmas.k file 3 | 4 | ```k 5 | rule WM[ N := #take(X, WS) ] => WM [ N := #asByteStackInWidth(#asWord(#take(X, WS)), X) ] 6 | 7 | rule 0 -Word X => #unsigned(0 -Int X) 8 | requires 0 <=Int X andBool X <=Int pow255 9 | /* 10 | proof: 11 | 12 | 1) rule W0 -Word W1 => chop( (W0 +Int pow256) -Int W1 ) requires W0 I modInt pow256 [concrete, smt-lemma] 14 | 3) rule W0 -Word W1 => chop( W0 -Int W1 ) requires W0 >=Int W1 15 | 16 | Assume X != 0: 17 | 18 | 0 < X : 0 -W X =(1)=> chop( pow256 - X ) 19 | 0 < pow256 - X < pow256 : chop( pow256 - X ) =(2)=> pow256 - X 20 | 21 | Assume X == 0: 22 | 23 | 0 == X : 0 -W 0 =(3)=> chop( 0 - 0 ) 24 | */ 25 | 26 | rule #range(WS [ X := #padToWidth(32, Y) ], Z, 32, WSS) => #range(WS, Z, 32, WSS) 27 | requires Z +Int 32 Z 31 | 32 | //assume ecrec returns an address 33 | rule maxUInt160 &Int #symEcrec(MSG, V, R, S) => #symEcrec(MSG, V, R, S) 34 | 35 | rule 0 <=Int #symEcrec(MSG, V, R, S) => true 36 | rule #symEcrec(MSG, V, R, S) true 37 | 38 | rule A -Word B <=Int A => 0 <=Int A -Int B 39 | requires #rangeUInt(256, A) 40 | andBool #rangeUInt(256, B) 41 | 42 | rule A <=Int chop(A +Int B) => A +Int B <=Int maxUInt256 43 | requires #rangeUInt(256, A) 44 | andBool #rangeUInt(256, B) 45 | 46 | rule #sgnInterp(sgn(chop(A *Int #unsigned(B))) *Int -1, abs(chop(A *Int #unsigned(B))) /Int (pow256 -Int #unsigned(B))) ==K A => #rangeSInt(256, A *Int B) 47 | requires #rangeUInt(256, A) 48 | andBool #rangeSInt(256, B) 49 | andBool B #rangeSInt(256, A *Int B) 52 | requires #rangeUInt(256, A) 53 | andBool #rangeSInt(256, B) 54 | andBool 0 A +Int B 58 | requires #rangeUInt(256, A) 59 | andBool #rangeUInt(256, B) 60 | andBool #rangeUInt(256, A +Int B) 61 | 62 | rule A +Int #unsigned(B) => A 63 | requires B ==K 0 64 | 65 | // lemma for Jug_drip 66 | rule A -Word B => #unsigned(A -Int B) 67 | requires #rangeSInt(256, A) 68 | andBool #rangeSInt(256, B) 69 | andBool 0 <=Int B 70 | andBool 0 <=Int A 71 | 72 | // lemmas for End_skim 73 | rule (A +Int (0 -Int B)) => A -Int B 74 | rule (A *Int (0 -Int B)) => (0 -Int (A *Int B)) 75 | rule (A -Int (0 -Int B)) => A +Int B 76 | //lemmas for End_bail 77 | rule (0 -Int A) (0 -Int B) (0 -Int B) <=Int A 81 | requires (notBool #isConcrete(A)) 82 | andBool #isConcrete(B) 83 | rule A <=Int (0 -Int B) => B <=Int 0 -Int A 84 | requires (notBool #isConcrete(B)) 85 | andBool #isConcrete(A) 86 | rule A B 0 -Int (X *Int Y) 92 | rule (0 -Int X) /Int Y => 0 -Int (X /Int Y) 93 | 94 | rule #unsigned( X *Int Y ) /Int #unsigned( Y ) => X 95 | requires #rangeSInt(256, X *Int Y) 96 | andBool #rangeSInt(256, X) 97 | andBool #rangeSInt(256, Y) 98 | andBool 0 <=Int X 99 | andBool 0 A 102 | requires B ==K 0 103 | andBool #isVariable(A) 104 | andBool #isVariable(B) 105 | 106 | rule A +Int B => B 107 | requires A ==K 0 108 | andBool #isVariable(A) 109 | andBool #isVariable(B) 110 | 111 | // lemma for Cat_bite-full to prevent unsigned(0 - X) devision 112 | rule pow256 -Int #unsigned(0 -Int X) => X 113 | requires X >Int 0 114 | 115 | 116 | // lemma to deal with deep nested calls - gas stuff 117 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (A14 +Int (A15 +Int (A16 +Int (A17 +Int (X +Int AS)))))))))))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (A14 +Int (A15 +Int (A16 +Int (A17 +Int AS))))))))))))))))) 118 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (A14 +Int (A15 +Int (A16 +Int (X +Int AS))))))))))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (A14 +Int (A15 +Int (A16 +Int AS)))))))))))))))) 119 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (A14 +Int (A15 +Int (X +Int AS)))))))))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (A14 +Int (A15 +Int AS))))))))))))))) 120 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (A14 +Int (X +Int AS))))))))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (A14 +Int AS)))))))))))))) 121 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int (X +Int AS)))))))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (A13 +Int AS))))))))))))) 122 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int (X +Int AS))))))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (A12 +Int AS)))))))))))) 123 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int (X +Int AS)))))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (A11 +Int AS))))))))))) 124 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int (X +Int AS))))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (A10 +Int AS)))))))))) 125 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int (X +Int AS)))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (A9 +Int AS))))))))) 126 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int (X +Int AS))))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (A8 +Int AS)))))))) 127 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int (X +Int AS)))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (A7 +Int AS))))))) 128 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int (X +Int AS))))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (A6 +Int AS)))))) 129 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int (X +Int AS)))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (A5 +Int AS))))) 130 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int (X +Int AS))))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int (A4 +Int AS)))) 131 | rule X -Int (A1 +Int (A2 +Int (A3 +Int (X +Int AS)))) => 0 -Int (A1 +Int (A2 +Int (A3 +Int AS))) 132 | rule X -Int (A1 +Int (A2 +Int (X +Int AS))) => 0 -Int (A1 +Int (A2 +Int AS)) 133 | 134 | // Vat_fork-same_fail lemma 135 | rule X +Int (pow256 -Int #unsigned(Y)) => X -Int Y 136 | requires Y 0 140 | requires X #rangeSInt(256, A *Int (0 -Int B)) 144 | // requires #rangeUInt(256, A) 145 | // andBool 0 #rangeSInt(256, 0 -Int (A *Int B)) 150 | requires #rangeUInt(256, A) 151 | andBool 0 #rangeSInt(256, A *Int (0 -Int B)) 155 | requires #rangeUInt256(A) 156 | andBool #rangeUInt256(A *Int B) 157 | andBool 0 =Int 0 => num0(N) >=Int 1 6 | requires N >=Int 1 7 | andBool N modInt 2 ==Int 0 8 | 9 | rule num1(N) >=Int 0 => num1(N) >=Int 1 10 | requires N >Int 1 11 | andBool N modInt 2 =/=Int 0 12 | 13 | rule num0(N /Int 2) => num0(N) -Int 1 14 | requires N >=Int 1 15 | andBool N modInt 2 ==Int 0 16 | 17 | rule num0(N /Int 2) => num0(N) 18 | requires N >=Int 1 19 | andBool N modInt 2 ==Int 1 20 | 21 | rule num1(N /Int 2) => num1(N) -Int 1 22 | requires N >Int 1 23 | andBool N modInt 2 ==Int 1 24 | 25 | rule num1(N /Int 2) => num1(N) 26 | requires N >Int 1 27 | andBool N modInt 2 ==Int 0 28 | 29 | rule num0(N) => 0 30 | requires N ==Int 1 31 | 32 | rule num1(N) => 0 33 | requires N ==Int 1 34 | 35 | rule (#if C #then A #else B #fi *Int X) <=Int maxUInt256 => true 36 | requires A *Int X <=Int maxUInt256 37 | andBool B *Int X <=Int maxUInt256 38 | ``` 39 | 40 | ### Solidity Word Packing 41 | We will have to use some of these tricks when reasoning about solidity implementations of `Flip`, `Flap`, and `Flop`: 42 | 43 | ```k 44 | syntax Int ::= "pow48" [function] 45 | syntax Int ::= "pow208" [function] 46 | rule pow48 => 281474976710656 [macro] 47 | rule pow208 => 411376139330301510538742295639337626245683966408394965837152256 [macro] 48 | 49 | syntax Int ::= "#WordPackUInt48UInt48" "(" Int "," Int ")" [function] 50 | // ---------------------------------------------------------- 51 | rule #WordPackUInt48UInt48(X, Y) => Y *Int pow48 +Int X 52 | requires #rangeUInt(48, X) 53 | andBool #rangeUInt(48, Y) 54 | 55 | syntax Int ::= "#WordPackAddrUInt48UInt48" "(" Int "," Int "," Int ")" [function] 56 | // ---------------------------------------------------------------------- 57 | rule #WordPackAddrUInt48UInt48(A, X, Y) => Y *Int pow208 +Int X *Int pow160 +Int A 58 | requires #rangeAddress(A) 59 | andBool #rangeUInt(48, X) 60 | andBool #rangeUInt(48, Y) 61 | 62 | syntax Int ::= "#WordPackAddrUInt8" "(" Int "," Int ")" [function] 63 | // ---------------------------------------------------------- 64 | rule #WordPackAddrUInt8(X, Y) => Y *Int pow160 +Int X 65 | requires #rangeAddress(X) 66 | andBool #rangeUInt(8, Y) 67 | ``` 68 | 69 | # dss lemmas 70 | 71 | ### string literal syntax 72 | 73 | ```k 74 | 75 | syntax Int ::= "#string2Word" "(" String ")" [function] 76 | // ---------------------------------------------------- 77 | rule #string2Word(S) => #asWord(#padRightToWidth(32, #parseByteStackRaw(S))) 78 | ``` 79 | 80 | ### special fixed-point arithmetic 81 | 82 | ```k 83 | syntax Int ::= "#Wad" [function] 84 | // ----------------------------- 85 | rule #Wad => 1000000000000000000 [macro] 86 | 87 | syntax Int ::= "#Ray" [function] 88 | // ----------------------------- 89 | rule #Ray => 1000000000000000000000000000 [macro] 90 | 91 | syntax Int ::= "#rmul" "(" Int "," Int ")" [function] 92 | rule #rmul(X, Y) => (X *Int Y) /Int #Ray 93 | ``` 94 | 95 | We leave these symbolic for now: 96 | 97 | ```k 98 | syntax Int ::= "#rpow" "(" Int "," Int "," Int "," Int ")" [function, smtlib(smt_rpow), smt-prelude] 99 | 100 | syntax Int ::= "#ifInt" Bool "#then" Int "#else" Int "#fi" [function, smtlib(ite), hook(KEQUAL.ite)] 101 | 102 | rule A *Int C /Int C => A 103 | requires A *Int C #ifInt Y ==Int 0 #then 0 #else chop(X) #fi 106 | 107 | rule A /Int B true 108 | requires A Z 111 | requires N ==Int 0 112 | 113 | rule Z *Int (X ^Int (N %Int 2)) => Z 114 | requires N ==Int 0 115 | 116 | rule 0 ^Int N => 1 117 | requires N ==Int 0 118 | 119 | rule 0 ^Int N => 0 120 | requires N >Int 0 121 | 122 | rule 0 <=Int (N /Int 2) => true 123 | requires 0 <=Int N 124 | 125 | rule N /Int 2 true 126 | requires N X *Int X 130 | requires #rpow(Z, X, N, B) *Int B =Int 2 132 | 133 | 134 | rule #rpow(Z, X, 0, Base) => Z 135 | 136 | rule #rpow(Z, X, N, Base) => Z 137 | requires N modInt 2 ==Int 0 138 | andBool N /Int 2 ==Int 0 139 | 140 | rule #rpow(Z, 0, N, Base) => 0 141 | 142 | rule #rpow(Base, X, N, Base) => X 143 | requires N ==Int 1 144 | 145 | rule #rpow(((Z *Int X) +Int Half) /Int Base, X, N /Int 2, Base) => 146 | #rpow(Z, X, N, Base) 147 | requires Half ==Int Base /Int 2 148 | andBool N ==Int 1 149 | 150 | rule #rpow(((Z *Int X) +Int Half) /Int Base, ((X *Int X) +Int Half) /Int Base, N /Int 2, Base) => 151 | #rpow( Z , X , N , Base ) 152 | requires N modInt 2 =/=Int 0 153 | andBool N >=Int 2 154 | andBool Half ==Int Base /Int 2 155 | 156 | rule #rpow( Z , ((X *Int X) +Int Half) /Int Base, N /Int 2, Base) => 157 | #rpow( Z , X , N , Base ) 158 | requires N modInt 2 ==Int 0 159 | andBool N >=Int 2 160 | andBool Half ==Int Base /Int 2 161 | 162 | rule #rpow( X , ((X *Int X) +Int Half) /Int Base, N /Int 2, Base) => 163 | #rpow( Base , X , N , Base ) 164 | requires N modInt 2 =/=Int 0 165 | andBool N /Int 2 =/=Int 0 166 | andBool Half ==Int Base /Int 2 167 | 168 | rule Z *Int X true 169 | requires #rpow(Z, X, N, Base) keccakIntList(L) +Int N 177 | requires N <=Int 100 178 | 179 | // solidity also needs: 180 | rule chop(keccakIntList(L)) => keccakIntList(L) 181 | // and 182 | rule chop(N +Int keccakIntList(L)) => keccakIntList(L) +Int N 183 | requires N <=Int 100 184 | ``` 185 | 186 | ### solidity masking 187 | 188 | **TODO**: refactor and tidy these. 189 | 190 | ```k 191 | syntax Int ::= "Mask12_32" [function] 192 | syntax Int ::= "Mask0_6" [function] 193 | syntax Int ::= "Mask6_12" [function] 194 | syntax Int ::= "Mask0_12" [function] 195 | syntax Int ::= "Mask0_26" [function] 196 | syntax Int ::= "Mask26_32" [function] 197 | syntax Int ::= "Mask20_26" [function] 198 | // ----------------------------------- 199 | // 0x000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffff 200 | rule Mask0_6 => 411376139330301510538742295639337626245683966408394965837152255 [macro] 201 | // 0xffffffffffff000000000000ffffffffffffffffffffffffffffffffffffffff 202 | rule Mask6_12 => 115792089237315784047431654708638870748305248246218003188207458632603225030655 [macro] 203 | // 0x000000000000000000000000ffffffffffffffffffffffffffffffffffffffff 204 | rule Mask0_12 => 1461501637330902918203684832716283019655932542975 [macro] 205 | // 0xffffffffffffffffffffffff0000000000000000000000000000000000000000 206 | rule Mask12_32 => 115792089237316195423570985007226406215939081747436879206741300988257197096960 [macro] 207 | // 0x0000000000000000000000000000000000000000000000000000ffffffffffff 208 | rule Mask0_26 => 281474976710655 [macro] 209 | // 0xffffffffffffffffffffffffffffffffffffffff000000000000ffffffffffff 210 | rule Mask20_26 => 115792089237316195423570985008687907853269984665561335876943319951794562400255 [macro] 211 | // 0xffffffffffffffffffffffffffffffffffffffffffffffffffff000000000000 212 | rule Mask26_32 => 115792089237316195423570985008687907853269984665640564039457583726438152929280 [macro] 213 | 214 | syntax Int ::= "maxUInt208" [function] 215 | rule maxUInt208 => 411376139330301510538742295639337626245683966408394965837152255 [macro] 216 | 217 | rule maxUInt208 &Int ((X *Int pow208) +Int A ) => A 218 | requires #rangeAddress(A) 219 | andBool #rangeUInt(48, X) 220 | 221 | rule (X *Int pow208) |Int A => (X *Int pow208 +Int A) 222 | requires #rangeUInt(48, X) 223 | andBool #rangeAddress(A) 224 | 225 | rule Mask26_32 &Int (Y *Int pow48 +Int X) => Y *Int pow48 226 | requires #rangeUInt(48, X) 227 | andBool #rangeUInt(48, Y) 228 | 229 | rule Mask20_26 &Int (Y *Int pow48 +Int X) => X 230 | requires #rangeUInt(48, X) 231 | andBool #rangeUInt(48, Y) 232 | 233 | rule X |Int Y *Int pow48 => Y *Int pow48 +Int X 234 | requires #rangeUInt(48, Y) 235 | andBool #rangeUInt(48, X) 236 | 237 | rule (X *Int pow48) |Int Y => (X *Int pow48) +Int Y 238 | requires #rangeUInt(48, Y) 239 | andBool #rangeUInt(48, X) 240 | 241 | rule Mask12_32 &Int A => 0 242 | requires #rangeAddress(A) 243 | 244 | rule X |Int 0 => X 245 | 246 | rule chop(A &Int B) => A &Int B 247 | requires #rangeUInt(256, A) 248 | andBool #rangeUInt(256, B) 249 | 250 | rule chop(A |Int B) => A |Int B 251 | requires #rangeUInt(256, A) 252 | andBool #rangeUInt(256, B) 253 | 254 | // Masking for packed words 255 | rule Mask12_32 &Int (Y *Int pow208 +Int (X *Int pow160 +Int A)) => Y *Int pow208 +Int X *Int pow160 256 | requires #rangeAddress(A) 257 | andBool #rangeUInt(48, X) 258 | andBool #rangeUInt(48, Y) 259 | 260 | rule B |Int (Y *Int pow208 +Int X *Int pow160) => Y *Int pow208 +Int X *Int pow160 +Int B 261 | requires #rangeAddress(B) 262 | andBool #rangeUInt(48, X) 263 | andBool #rangeUInt(48, Y) 264 | 265 | rule (Y *Int pow208 +Int ( X *Int pow160 +Int A ) ) /Int pow208 => Y 266 | requires #rangeAddress(A) 267 | andBool #rangeUInt(48, X) 268 | andBool #rangeUInt(48, Y) 269 | 270 | rule (Y *Int pow48 +Int X) /Int pow48 => Y 271 | requires #rangeUInt(48, X) 272 | andBool #rangeUInt(48, Y) 273 | 274 | rule Mask0_6 &Int (X *Int pow208 +Int ( Y *Int pow160 +Int A ) ) => Y *Int pow160 +Int A 275 | requires #rangeUInt(48, X) 276 | andBool #rangeUInt(48, Y) 277 | andBool #rangeAddress(A) 278 | 279 | rule Mask6_12 &Int (Y *Int pow208 +Int ( X *Int pow160 +Int A) ) => Y *Int pow208 +Int A 280 | requires #rangeAddress(A) 281 | andBool #rangeUInt(48, X) 282 | andBool #rangeUInt(48, Y) 283 | 284 | rule (Y *Int pow208) |Int (X *Int pow160 +Int A) => Y *Int pow208 +Int X *Int pow160 +Int A 285 | requires #rangeAddress(A) 286 | andBool #rangeUInt(48, X) 287 | andBool #rangeUInt(48, Y) 288 | 289 | rule (X *Int pow160) |Int (Y *Int pow208 +Int A) => Y *Int pow208 +Int X *Int pow160 +Int A 290 | requires #rangeAddress(A) 291 | andBool #rangeUInt(48, X) 292 | andBool #rangeUInt(48, Y) 293 | 294 | rule maxUInt160 &Int ((X *Int pow208) +Int ((Y *Int pow160) +Int A)) => A 295 | requires #rangeAddress(A) 296 | andBool #rangeUInt(48, X) 297 | andBool #rangeUInt(48, Y) 298 | 299 | rule maxUInt160 &Int ((X *Int pow208) +Int (Y *Int pow160)) => 0 300 | requires #rangeUInt(48, X) 301 | andBool #rangeUInt(48, Y) 302 | 303 | rule maxUInt160 &Int ((X *Int pow208) +Int A) => A 304 | requires #rangeAddress(A) 305 | andBool #rangeUInt(48, X) 306 | 307 | rule maxUInt160 &Int ((X *Int pow160) +Int A) => A 308 | requires #rangeAddress(A) 309 | andBool #rangeUInt(48, X) 310 | 311 | rule maxUInt160 &Int (X *Int pow208) => 0 312 | requires #rangeUInt(48, X) 313 | 314 | rule maxUInt160 &Int (X *Int pow160) => 0 315 | requires #rangeUInt(48, X) 316 | 317 | rule (((X *Int pow208) +Int ( (Y *Int pow160) +Int A)) /Int pow160) => (X *Int pow48) +Int Y 318 | requires #rangeAddress(A) 319 | andBool #rangeUInt(48, X) 320 | andBool #rangeUInt(48, Y) 321 | 322 | rule maxUInt48 &Int (X *Int pow48) +Int Y => Y 323 | requires #rangeUInt(48, X) 324 | andBool #rangeUInt(48, Y) 325 | 326 | rule ((X *Int pow208) +Int A) /Int pow160 => X *Int pow48 327 | requires #rangeAddress(A) 328 | andBool #rangeUInt(48, X) 329 | 330 | rule ((X *Int pow160) +Int A) /Int pow160 => X 331 | requires #rangeAddress(A) 332 | 333 | rule (Y *Int pow208 +Int X *Int pow160) /Int pow208 => Y 334 | requires #rangeUInt(48, X) 335 | 336 | rule (Y *Int pow208 +Int A) /Int pow208 => Y 337 | requires #rangeAddress(A) 338 | 339 | rule maxUInt48 &Int (X *Int pow48) => 0 340 | requires #rangeUInt(48, X) 341 | ``` 342 | 343 | ### miscellaneous 344 | 345 | ```k 346 | rule WS ++ .WordStack => WS 347 | 348 | rule #sizeWordStack ( #padToWidth ( 32 , #asByteStack ( #unsigned ( W ) ) ) , 0) => 32 349 | requires #rangeSInt(256, W) 350 | 351 | // custom ones: 352 | rule #asWord(#padToWidth(32, #asByteStack(#unsigned(X)))) => #unsigned(X) 353 | requires #rangeSInt(256, X) 354 | 355 | // rule #take(N, #padToWidth(N, WS) ++ WS' ) => #padToWidth(N, WS) 356 | 357 | // potentially useful 358 | // rule #padToWidth(N, WS) ++ WS' => #padToWidth(N + #sizeWordStack(WS'), WS ++ WS') 359 | // and the N, M versions 360 | 361 | rule #take(N, #padToWidth(N, WS) ) => #padToWidth(N, WS) 362 | ``` 363 | 364 | ### 48-bit integer arithmetic 365 | 366 | ```k 367 | rule notBool((Mask0_26 &Int (A +Int B)) A +Int B <=Int maxUInt48 368 | requires #rangeUInt(48, A) 369 | andBool #rangeUInt(48, B) 370 | ``` 371 | 372 | ### signed 256-bit integer arithmetic 373 | 374 | ```k 375 | rule #unsigned(X) ==K 0 => X ==Int 0 376 | requires #rangeSInt(256, X) 377 | 378 | // rule 0 0 0 <=Int A -Int B 384 | // requires #rangeUInt(256, A) 385 | // andBool #rangeUInt(256, B) 386 | 387 | // addui 388 | // lemmas for sufficiency 389 | rule chop(A +Int #unsigned(B)) => A +Int B 390 | requires #rangeUInt(256, A) 391 | andBool #rangeSInt(256, B) 392 | andBool #rangeUInt(256, A +Int B) 393 | 394 | rule A <=Int chop(A +Int #unsigned(B)) => A +Int B <=Int maxUInt256 395 | requires #rangeUInt(256, A) 396 | andBool #rangeSInt(256, B) 397 | andBool 0 0 <=Int A +Int B 400 | requires #rangeUInt(256, A) 401 | andBool #rangeSInt(256, B) 402 | andBool B (A +Int B <=Int maxUInt256) 405 | // requires #rangeUInt(256, A) 406 | // andBool #rangeSInt(256, B) 407 | // andBool B >=Int 0 408 | 409 | // rule A <=Int chop(A +Int #unsigned(B)) => A +Int B A -Int B 417 | requires #rangeUInt(256, A) 418 | andBool #rangeSInt(256, B) 419 | andBool #rangeUInt(256, A -Int B) 420 | 421 | rule A -Word #unsigned(B) <=Int A => minUInt256 <=Int A -Int B 422 | requires #rangeUInt(256, A) 423 | andBool #rangeSInt(256, B) 424 | andBool 0 <=Int B 425 | 426 | rule A A -Int B <=Int maxUInt256 427 | requires #rangeUInt(256, A) 428 | andBool #rangeSInt(256, B) 429 | andBool B maxUInt256 A -Int B 437 | requires #rangeUInt(256, A) 438 | andBool #rangeSInt(256, B) 439 | andBool #rangeUInt(256, A -Int B) 440 | andBool B #unsigned(A *Int B) 445 | requires #rangeUInt(256, A) 446 | andBool #rangeSInt(256, B) 447 | andBool #rangeSInt(256, A *Int B) 448 | 449 | rule abs(#unsigned(A *Int B)) /Int abs(#unsigned(B)) => A 450 | requires #rangeUInt(256, A) 451 | andBool #rangeSInt(256, B) 452 | andBool #rangeSInt(256, A *Int B) 453 | andBool notBool (#unsigned(B) ==Int 0) 454 | 455 | 456 | rule abs(B) ==K 0 => B ==K 0 457 | 458 | rule #sgnInterp(sgn(#unsigned(A *Int B)) *Int sgn(#unsigned(B)), A) => A 459 | requires #rangeSInt(256, A *Int B) 460 | andBool #rangeUInt(256, A) 461 | andBool #rangeSInt(256, B) 462 | 463 | // lemmas for necessity 464 | rule #signed(X) notBool #rangeSInt(256, X) 465 | requires #rangeUInt(256, X) 466 | 467 | // rule #sgnInterp(sgn(chop(A *Int B)), abs(chop(A *Int B)) /Int B) ==K A => #rangeSInt(256, A *Int B) 468 | // requires #rangeUInt(256, A) 469 | // andBool #rangeSInt(256, B) 470 | 471 | // rule #sgnInterp(sgn(chop(A *Int (pow256 +Int B))) *Int -1, chop(abs(chop(A *Int (pow256 +Int B))) /Int (pow256 -Int (pow256 +Int B)))) ==K A => #rangeSInt(256, A *Int B) 472 | // requires #rangeUInt(256, A) 473 | // andBool #rangeSInt(256, B) 474 | 475 | rule (chop(A *Int B) /Int B ==K A) => #rangeUInt(256, A *Int B) 476 | requires #rangeUInt(256, A) 477 | andBool #rangeUInt(256, B) 478 | 479 | rule (#sgnInterp(sgn(chop(A *Int #unsigned(B))) *Int sgn(#unsigned(B)), abs(chop(A *Int #unsigned(B))) /Int abs(#unsigned(B))) ==K A) => #rangeSInt(256, A *Int B) 480 | requires #rangeUInt(256, A) 481 | andBool #rangeSInt(256, B) 482 | andBool B =/=Int 0 483 | ``` 484 | -------------------------------------------------------------------------------- /src/prelude.smt2.md: -------------------------------------------------------------------------------- 1 | ```smt2 2 | 3 | (set-option :auto-config false) 4 | (set-option :smt.mbqi false) 5 | ;(set-option :smt.mbqi.max_iterations 15) 6 | 7 | ; (set-option :auto-config false) 8 | ; (set-option :smt.mbqi false) 9 | ; (set-option :smt.array.extensional false) 10 | 11 | ; int extra 12 | (define-fun int_max ((x Int) (y Int)) Int (ite (< x y) y x)) 13 | (define-fun int_min ((x Int) (y Int)) Int (ite (< x y) x y)) 14 | (define-fun int_abs ((x Int)) Int (ite (< x 0) (- 0 x) x)) 15 | 16 | ; bool to int 17 | (define-fun smt_bool2int ((b Bool)) Int (ite b 1 0)) 18 | 19 | ; ceil32 20 | (define-fun ceil32 ((x Int)) Int ( * ( div ( + x 31 ) 32 ) 32 ) ) 21 | 22 | ;; pow256 and pow255 23 | (define-fun pow256 () Int 24 | 115792089237316195423570985008687907853269984665640564039457584007913129639936) 25 | (define-fun pow255 () Int 26 | 57896044618658097711785492504343953926634992332820282019728792003956564819968) 27 | ;; weird declaration trick (doesn't seem to be needed currently) 28 | ;; (declare-fun pow256 () Int) 29 | ;; (assert (>= pow256 115792089237316195423570985008687907853269984665640564039457584007913129639936)) 30 | ;; (assert (<= pow256 115792089237316195423570985008687907853269984665640564039457584007913129639936)) 31 | 32 | ;; signed word arithmetic definitions: 33 | ;; integer to word: 34 | (define-fun unsigned ((x Int)) Int 35 | (if (>= x 0) 36 | x 37 | (+ x pow256))) 38 | 39 | ;; word to integer 40 | (define-fun signed ((x Int)) Int 41 | (if (< x pow255) 42 | x 43 | (- x pow256))) 44 | 45 | ;; signed_abs 46 | (define-fun signed_abs ((x Int)) Int 47 | (if (< x pow255) 48 | x 49 | (- pow256 x))) 50 | 51 | ;; signed_sgn 52 | (define-fun signed_sgn ((x Int)) Int 53 | (if (< x pow255) 54 | 1 55 | -1)) 56 | 57 | ;; smt_rpow 58 | (declare-fun smt_rpow (Int Int Int Int) Int) 59 | (assert (forall ((z Int) (x Int) (y Int) (b Int)) (! (=> (= y 1) (= (smt_rpow z x y b) (div (+ (* z x) (div b 2)) b))) :pattern ((smt_rpow z x y b))))) 60 | 61 | (assert (forall ((z1 Int) (z2 Int) (x1 Int) (y1 Int) (x2 Int) (y2 Int) (b1 Int) (b2 Int)) 62 | (! 63 | (=> (and (<= z1 z2) (<= x1 x2) (<= y1 y2) (<= b1 b2)) 64 | (<= (smt_rpow z1 x1 y1 b1) (smt_rpow z2 x2 y2 b2)) 65 | ) 66 | :pattern ((smt_rpow z1 x1 y1 b1) (smt_rpow z2 x2 y2 b2)) 67 | ) 68 | )) 69 | 70 | (assert (forall ((z Int) (x Int) (y Int) (b Int)) (! (=> (>= y 1) (>= (* (smt_rpow z x y b) b) (* z x))) :pattern ((smt_rpow z x y b))))) 71 | (assert (forall ((z Int) (x Int) (y Int) (b Int)) (! (=> (>= y 2) (>= (* (smt_rpow z x y b) b) (* x x))) :pattern ((smt_rpow z x y b))))) 72 | 73 | (assert (forall ((z Int) (x Int) (y Int) (b Int)) (! (=> (>= y 1) (>= (* (smt_rpow z x y b) b) (+ (* z x) (div b 2)))) :pattern ((smt_rpow z x y b))))) 74 | (assert (forall ((z Int) (x Int) (y Int) (b Int)) (! (=> (>= y 2) (>= (* (smt_rpow z x y b) b) (+ (* x x) (div b 2)))) :pattern ((smt_rpow z x y b))))) 75 | ; 76 | ; end of prelude 77 | ; 78 | ``` 79 | -------------------------------------------------------------------------------- /src/storage.k.md: -------------------------------------------------------------------------------- 1 | # dss storage model 2 | 3 | ### Vat 4 | 5 | ```k 6 | // act: public 7 | syntax Int ::= "#Vat.wards" "[" Int "]" [function] 8 | // ----------------------------------------------- 9 | // doc: whether `$0` is an owner of `Vat` 10 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 11 | rule #Vat.wards[A] => #hashedLocation("Solidity", 0, A) 12 | 13 | syntax Int ::= "#Vat.can" "[" Int "][" Int "]" [function] 14 | // ----------------------------------------------- 15 | // doc: whether `$1` can spend the resources of `$0` 16 | // act: address `$0` has authorized `$1` 17 | rule #Vat.can[A][B] => #hashedLocation("Solidity", 1, A B) 18 | 19 | syntax Int ::= "#Vat.ilks" "[" Int "].Art" [function] 20 | // ---------------------------------------------------- 21 | // doc: total debt units issued from `$0` 22 | // act: `$0` has debt issuance `.` 23 | rule #Vat.ilks[Ilk].Art => #hashedLocation("Solidity", 2, Ilk) +Int 0 24 | 25 | syntax Int ::= "#Vat.ilks" "[" Int "].rate" [function] 26 | // ---------------------------------------------------- 27 | // doc: debt unit rate of `$0` 28 | // act: `$0` has debt unit rate `.` 29 | rule #Vat.ilks[Ilk].rate => #hashedLocation("Solidity", 2, Ilk) +Int 1 30 | 31 | syntax Int ::= "#Vat.ilks" "[" Int "].spot" [function] 32 | // ----------------------------------------------- 33 | // doc: price with safety margin for `$0` 34 | // act: `$0` has safety margin `.` 35 | rule #Vat.ilks[Ilk].spot => #hashedLocation("Solidity", 2, Ilk) +Int 2 36 | 37 | syntax Int ::= "#Vat.ilks" "[" Int "].line" [function] 38 | // ----------------------------------------------- 39 | // doc: debt ceiling for `$0` 40 | // act: `$0` has debt ceiling `.` 41 | rule #Vat.ilks[Ilk].line => #hashedLocation("Solidity", 2, Ilk) +Int 3 42 | 43 | syntax Int ::= "#Vat.ilks" "[" Int "].dust" [function] 44 | // ----------------------------------------------- 45 | // doc: urn debt floor for `$0` 46 | // act: `$0` has debt floor `.` 47 | rule #Vat.ilks[Ilk].dust => #hashedLocation("Solidity", 2, Ilk) +Int 4 48 | 49 | syntax Int ::= "#Vat.urns" "[" Int "][" Int "].ink" [function] 50 | // ---------------------------------------------------------- 51 | // doc: locked collateral units in `$0` assigned to `$1` 52 | // act: agent `$1` has `.` collateral units in `$0` 53 | rule #Vat.urns[Ilk][Usr].ink => #hashedLocation("Solidity", 3, Ilk Usr) 54 | 55 | syntax Int ::= "#Vat.urns" "[" Int "][" Int "].art" [function] 56 | // ---------------------------------------------------------- 57 | // doc: debt units in `$0` assigned to `$1` 58 | // act: agent `$1` has `.` debt units in `$0` 59 | rule #Vat.urns[Ilk][Usr].art => #hashedLocation("Solidity", 3, Ilk Usr) +Int 1 60 | 61 | syntax Int ::= "#Vat.gem" "[" Int "][" Int "]" [function] 62 | // --------------------------------------------- 63 | // doc: unlocked collateral in `$0` assigned to `$1` 64 | // act: agent `$1` has `.` unlocked collateral in `$0` 65 | rule #Vat.gem[Ilk][Usr] => #hashedLocation("Solidity", 4, Ilk Usr) 66 | 67 | syntax Int ::= "#Vat.dai" "[" Int "]" [function] 68 | // --------------------------------------------- 69 | // doc: dai assigned to `$0` 70 | // act: agent `$0` has `.` dai 71 | rule #Vat.dai[A] => #hashedLocation("Solidity", 5, A) 72 | 73 | syntax Int ::= "#Vat.sin" "[" Int "]" [function] 74 | // --------------------------------------------- 75 | // doc: system debt assigned to `$0` 76 | // act: agent `$0` has `.` dai 77 | rule #Vat.sin[A] => #hashedLocation("Solidity", 6, A) 78 | 79 | syntax Int ::= "#Vat.debt" [function] 80 | // --------------------------------- 81 | // doc: total dai issued from the system 82 | // act: there is `.` dai in total 83 | rule #Vat.debt => 7 84 | 85 | syntax Int ::= "#Vat.vice" [function] 86 | // ---------------------------------- 87 | // doc: total system debt 88 | // act: there is `.` system debt 89 | rule #Vat.vice => 8 90 | 91 | syntax Int ::= "#Vat.Line" [function] 92 | // ---------------------------------- 93 | // doc: global debt ceiling 94 | // act: the global debt ceiling is `.` 95 | rule #Vat.Line => 9 96 | 97 | syntax Int ::= "#Vat.live" [function] 98 | // ---------------------------------- 99 | // doc: system status 100 | // act: the system is `. == 1 ? : not` live 101 | rule #Vat.live => 10 102 | ``` 103 | 104 | ### Dai 105 | 106 | ```k 107 | syntax Int ::= "#Dai.wards" "[" Int "]" [function] 108 | // ----------------------------------------------- 109 | // doc: whether `$0` is an owner of `Vat` 110 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 111 | rule #Dai.wards[A] => #hashedLocation("Solidity", 0, A) 112 | 113 | syntax Int ::= "#Dai.totalSupply" [function] 114 | // ----------------------------------------------- 115 | // doc: the total supply of this token 116 | // act: the total supply is .` 117 | rule #Dai.totalSupply => 1 118 | 119 | syntax Int ::= "#Dai.balanceOf" "[" Int "]" [function] 120 | // ----------------------------------------------- 121 | // doc: the balance of a user 122 | // act: the balance of `$0 is .` us , 123 | rule #Dai.balanceOf[A] => #hashedLocation("Solidity", 2, A) 124 | 125 | syntax Int ::= "#Dai.allowance" "[" Int "][" Int "]" [function] 126 | // ----------------------------------------------- 127 | // doc: the amount that can be spent on someones behalf 128 | // act: `$1 can spend `.` tokens belonging to `$0` 129 | rule #Dai.allowance[A][B] => #hashedLocation("Solidity", 3, A B) 130 | 131 | syntax Int ::= "#Dai.nonces" "[" Int "]" [function] 132 | // ----------------------------------------------- 133 | // doc: the amount that can be spent on someones behalf 134 | // act: `$1 can spend `.` tokens belonging to `$0` 135 | rule #Dai.nonces[A] => #hashedLocation("Solidity", 4, A) 136 | 137 | syntax Int ::= "#Dai.DOMAIN_SEPARATOR" [function] 138 | // ----------------------------------------------- 139 | // doc: the amount that can be spent on someones behalf 140 | // act: `$1 can spend `.` tokens belonging to `$0` 141 | rule #Dai.DOMAIN_SEPARATOR => 5 142 | ``` 143 | 144 | ### Jug 145 | 146 | ```k 147 | syntax Int ::= "#Jug.wards" "[" Int "]" [function] 148 | // ----------------------------------------------- 149 | // doc: whether `$0` is an owner of `Jug` 150 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 151 | rule #Jug.wards[A] => #hashedLocation("Solidity", 0, A) 152 | 153 | syntax Int ::= "#Jug.ilks" "[" Int "].duty" [function] 154 | // ---------------------------------------------------- 155 | // doc: 156 | // act: 157 | rule #Jug.ilks[Ilk].duty => #hashedLocation("Solidity", 1, Ilk) +Int 0 158 | 159 | syntax Int ::= "#Jug.ilks" "[" Int "].rho" [function] 160 | // ---------------------------------------------------- 161 | // doc: 162 | // act: 163 | rule #Jug.ilks[Ilk].rho => #hashedLocation("Solidity", 1, Ilk) +Int 1 164 | 165 | syntax Int ::= "#Jug.vat" [function] 166 | // ---------------------------------- 167 | // doc: `Vat` that this `Jug` points to 168 | // act: this Jug points to Vat `.` 169 | rule #Jug.vat => 2 170 | 171 | syntax Int ::= "#Jug.vow" [function] 172 | // ---------------------------------- 173 | // doc: `Vow` that this `Jug` points to 174 | // act: this Jug points to Vow `.` 175 | rule #Jug.vow => 3 176 | 177 | syntax Int ::= "#Jug.base" [function] 178 | // ---------------------------------- 179 | // doc: 180 | // act: 181 | rule #Jug.base => 4 182 | ``` 183 | 184 | ### Drip 185 | 186 | ```k 187 | syntax Int ::= "#Drip.wards" "[" Int "]" [function] 188 | // ----------------------------------------------- 189 | // doc: whether `$0` is an owner of `Drip` 190 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 191 | rule #Drip.wards[A] => #hashedLocation("Solidity", 0, A) 192 | 193 | syntax Int ::= "#Drip.ilks" "[" Int "].tax" [function] 194 | // ---------------------------------------------------- 195 | // doc: stability fee of `$0` 196 | // act: `$0` has stability fee `.` 197 | rule #Drip.ilks[Ilk].tax => #hashedLocation("Solidity", 1, Ilk) +Int 0 198 | 199 | syntax Int ::= "#Drip.ilks" "[" Int "].rho" [function] 200 | // ---------------------------------------------------- 201 | // doc: last drip time of `$0` 202 | // act: `$0` was dripped at `.` 203 | rule #Drip.ilks[Ilk].rho => #hashedLocation("Solidity", 1, Ilk) +Int 1 204 | 205 | syntax Int ::= "#Drip.vat" [function] 206 | // ---------------------------------- 207 | // doc: `Vat` that this `Drip` points to 208 | // act: this Drip points to Vat `.` 209 | rule #Drip.vat => 2 210 | 211 | syntax Int ::= "#Drip.vow" [function] 212 | // ---------------------------------- 213 | // doc: `Vow` that this `Drip` points to 214 | // act: this Drip points to Vow `.` 215 | rule #Drip.vow => 3 216 | 217 | syntax Int ::= "#Drip.repo" [function] 218 | // ----------------------------------- 219 | // doc: base interest rate 220 | // act: the base interest rate is `.` 221 | rule #Drip.repo => 4 222 | ``` 223 | 224 | ### Vow 225 | 226 | ```k 227 | syntax Int ::= "#Vow.wards" "[" Int "]" [function] 228 | // --------------------------------- 229 | // doc: whether `$0` is an owner of `Vow` 230 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 231 | rule #Vow.wards[A] => #hashedLocation("Solidity", 0, A) 232 | 233 | syntax Int ::= "#Vow.vat" [function] 234 | // --------------------------------- 235 | // doc: `Vat` that this `Vow` points to 236 | // act: this Vow points to Vat `.` 237 | rule #Vow.vat => 1 238 | 239 | syntax Int ::= "#Vow.flapper" [function] 240 | // --------------------------------- 241 | // doc: `Flapper` that this `Vow` points to 242 | // act: this Vow points to Flapper `.` 243 | rule #Vow.flapper => 2 244 | 245 | syntax Int ::= "#Vow.flopper" [function] 246 | // --------------------------------- 247 | // doc: `Flopper` that this `Vow` points to 248 | // act: this Vow points to Flopper `.` 249 | rule #Vow.flopper => 3 250 | 251 | syntax Int ::= "#Vow.sin" "[" Int "]" [function] 252 | // --------------------------------------------- 253 | // doc: sin queued at timestamp `$0` 254 | // act: `.` sin queued at timestamp `$0` 255 | rule #Vow.sin[A] => #hashedLocation("Solidity", 4, A) 256 | 257 | syntax Int ::= "#Vow.Sin" [function] 258 | // --------------------------------- 259 | // doc: total queued sin 260 | // act: the total queued sin is `.` 261 | rule #Vow.Sin => 5 262 | 263 | syntax Int ::= "#Vow.Ash" [function] 264 | // --------------------------------- 265 | // doc: total sin in debt auctions 266 | // act: the total sin in debt auctions is `.` 267 | rule #Vow.Ash => 6 268 | 269 | syntax Int ::= "#Vow.wait" [function] 270 | // ---------------------------------- 271 | // doc: sin maturation time 272 | // act: the sin maturation time is `.` 273 | rule #Vow.wait => 7 274 | 275 | syntax Int ::= "#Vow.sump" [function] 276 | // ---------------------------------- 277 | // doc: debt auction lot size 278 | // act: the debt auction lot size is `.` 279 | rule #Vow.sump => 8 280 | 281 | syntax Int ::= "#Vow.bump" [function] 282 | // ---------------------------------- 283 | // doc: surplus auction lot size 284 | // act: the surplus auction lot size is `.` 285 | rule #Vow.bump => 9 286 | 287 | syntax Int ::= "#Vow.hump" [function] 288 | // --------------------------------- 289 | // doc: surplus dai cushion 290 | // act: the surplus dai cushion is `.` 291 | rule #Vow.hump => 10 292 | 293 | syntax Int ::= "#Vow.live" [function] 294 | // --------------------------------- 295 | // doc: liveness flag 296 | // act: the system is active/inactive `.` 297 | rule #Vow.live => 11 298 | 299 | ``` 300 | 301 | ### Cat 302 | 303 | ```k 304 | syntax Int ::= "#Cat.wards" "[" Int "]" [function] 305 | // --------------------------------- 306 | // doc: whether `$0` is an owner of `Cat` 307 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 308 | rule #Cat.wards[A] => #hashedLocation("Solidity", 0, A) 309 | 310 | syntax Int ::= "#Cat.ilks" "[" Int "].flip" [function] 311 | // --------------------------------------------------- 312 | // doc: `Flipper` for `$0` 313 | // act: 314 | rule #Cat.ilks[Ilk].flip => #hashedLocation("Solidity", 1, Ilk) +Int 0 315 | 316 | syntax Int ::= "#Cat.ilks" "[" Int "].chop" [function] 317 | // --------------------------------------------------- 318 | // doc: liquidation penalty for `$0` 319 | // act: 320 | rule #Cat.ilks[Ilk].chop => #hashedLocation("Solidity", 1, Ilk) +Int 1 321 | 322 | syntax Int ::= "#Cat.ilks" "[" Int "].lump" [function] 323 | // --------------------------------------------------- 324 | // doc: liquidation lot size for `$0` 325 | // act: 326 | rule #Cat.ilks[Ilk].lump => #hashedLocation("Solidity", 1, Ilk) +Int 2 327 | 328 | syntax Int ::= "#Cat.live" [function] 329 | // ---------------------------------- 330 | // doc: system liveness 331 | // act: 332 | rule #Cat.live => 2 333 | 334 | syntax Int ::= "#Cat.vat" [function] 335 | // --------------------------------- 336 | // doc: `Vat` that this `Cat` points to 337 | // act: 338 | rule #Cat.vat => 3 339 | 340 | syntax Int ::= "#Cat.vow" [function] 341 | // --------------------------------- 342 | // doc: `Vow` that this `Cat` points to 343 | // act: 344 | rule #Cat.vow => 4 345 | ``` 346 | 347 | ### GemJoin 348 | 349 | ```k 350 | syntax Int ::= "#GemJoin.vat" [function] 351 | // ------------------------------------- 352 | // doc: `Vat` that this adapter points to 353 | // act: 354 | rule #GemJoin.vat => 0 355 | 356 | syntax Int ::= "#GemJoin.ilk" [function] 357 | // ------------------------------------- 358 | // doc: collateral type of this adapter 359 | // act: 360 | rule #GemJoin.ilk => 1 361 | 362 | syntax Int ::= "#GemJoin.gem" [function] 363 | // ------------------------------------- 364 | // doc: underlying token of this adapter 365 | // act: 366 | rule #GemJoin.gem => 2 367 | ``` 368 | 369 | ### DaiJoin 370 | 371 | ```k 372 | syntax Int ::= "#DaiJoin.vat" [function] 373 | // ------------------------------------- 374 | // doc: `Vat` that this adapter points to 375 | // act: 376 | rule #DaiJoin.vat => 0 377 | 378 | syntax Int ::= "#DaiJoin.dai" [function] 379 | // ------------------------------------- 380 | // doc: underlying dai token of this adapter 381 | // act: 382 | rule #DaiJoin.dai => 1 383 | ``` 384 | 385 | ### Flip 386 | 387 | ```k 388 | syntax Int ::= "#Flipper.wards" "[" Int "]" [function] 389 | // --------------------------------------- 390 | // doc: whether `$0` is an owner of `Flip` 391 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 392 | rule #Flipper.wards[A] => #hashedLocation("Solidity", 0, A) 393 | 394 | syntax Int ::= "#Flipper.bids" "[" Int "].bid" [function] 395 | // ------------------------------------------------------ 396 | // doc: current bid (dai) 397 | // act: 398 | rule #Flipper.bids[N].bid => #hashedLocation("Solidity", 1, N) +Int 0 399 | 400 | syntax Int ::= "#Flipper.bids" "[" Int "].lot" [function] 401 | // ------------------------------------------------------ 402 | // doc: current lot (gem) 403 | // act: 404 | rule #Flipper.bids[N].lot => #hashedLocation("Solidity", 1, N) +Int 1 405 | 406 | // packed, use #WordPackAddrUInt48UInt48 to unpack this 407 | syntax Int ::= "#Flipper.bids" "[" Int "].guy_tic_end" [function] 408 | // -------------------------------------------------------------- 409 | // doc: 410 | // act: 411 | rule #Flipper.bids[N].guy_tic_end => #hashedLocation("Solidity", 1, N) +Int 2 412 | 413 | syntax Int ::= "#Flipper.bids" "[" Int "].usr" [function] 414 | // ------------------------------------------------------ 415 | // doc: CDP owner 416 | // act: 417 | rule #Flipper.bids[N].usr => #hashedLocation("Solidity", 1, N) +Int 3 418 | 419 | syntax Int ::= "#Flipper.bids" "[" Int "].gal" [function] 420 | // ------------------------------------------------------ 421 | // doc: beneficiary of the auction 422 | // act: 423 | rule #Flipper.bids[N].gal => #hashedLocation("Solidity", 1, N) +Int 4 424 | 425 | syntax Int ::= "#Flipper.bids" "[" Int "].tab" [function] 426 | // ------------------------------------------------------ 427 | // doc: beneficiary of the auction 428 | // act: 429 | rule #Flipper.bids[N].tab => #hashedLocation("Solidity", 1, N) +Int 5 430 | 431 | syntax Int ::= "#Flipper.vat" [function] 432 | // --------------------------------------- 433 | // doc: CDP engine 434 | // act: 435 | rule #Flipper.vat => 2 436 | 437 | syntax Int ::= "#Flipper.ilk" [function] 438 | // --------------------------------------- 439 | // doc: collateral type 440 | // act: 441 | rule #Flipper.ilk => 3 442 | 443 | syntax Int ::= "#Flipper.beg" [function] 444 | // --------------------------------------- 445 | // doc: minimum bid increment 446 | // act: 447 | rule #Flipper.beg => 4 448 | 449 | // packed, use #WordPackUInt48UInt48 to unpack this 450 | syntax Int ::= "#Flipper.ttl_tau" [function] 451 | // ----------------------------------------- 452 | // doc: 453 | // act: 454 | rule #Flipper.ttl_tau => 5 455 | 456 | syntax Int ::= "#Flipper.kicks" [function] 457 | // --------------------------------------- 458 | // doc: auction counter 459 | // act: 460 | rule #Flipper.kicks => 6 461 | ``` 462 | 463 | 464 | ### Flop 465 | 466 | ```k 467 | syntax Int ::= "#Flopper.wards" "[" Int "]" [function] 468 | // --------------------------------------- 469 | // doc: whether `$0` is an owner of `Flop` 470 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 471 | rule #Flopper.wards[A] => #hashedLocation("Solidity", 0, A) 472 | 473 | syntax Int ::= "#Flopper.bids" "[" Int "].bid" [function] 474 | // ------------------------------------------------------ 475 | // doc: current bid (dai) 476 | // act: 477 | rule #Flopper.bids[N].bid => #hashedLocation("Solidity", 1, N) +Int 0 478 | 479 | syntax Int ::= "#Flopper.bids" "[" Int "].lot" [function] 480 | // ------------------------------------------------------ 481 | // doc: current lot (gem) 482 | // act: 483 | rule #Flopper.bids[N].lot => #hashedLocation("Solidity", 1, N) +Int 1 484 | 485 | // packed, use #WordPackAddrUInt48UInt48 to unpack this 486 | syntax Int ::= "#Flopper.bids" "[" Int "].guy_tic_end" [function] 487 | // -------------------------------------------------------------- 488 | // doc: 489 | // act: 490 | rule #Flopper.bids[N].guy_tic_end => #hashedLocation("Solidity", 1, N) +Int 2 491 | 492 | syntax Int ::= "#Flopper.vat" [function] 493 | // --------------------------------------- 494 | // doc: dai token 495 | // act: 496 | rule #Flopper.vat => 2 497 | 498 | syntax Int ::= "#Flopper.gem" [function] 499 | // --------------------------------------- 500 | // doc: mkr token 501 | // act: 502 | rule #Flopper.gem => 3 503 | 504 | syntax Int ::= "#Flopper.beg" [function] 505 | // --------------------------------------- 506 | // doc: minimum bid increment 507 | // act: 508 | rule #Flopper.beg => 4 509 | 510 | // packed, use #WordPackUInt48UInt48 to unpack this 511 | syntax Int ::= "#Flopper.ttl_tau" [function] 512 | // ----------------------------------------- 513 | // doc: 514 | // act: 515 | rule #Flopper.ttl_tau => 5 516 | 517 | syntax Int ::= "#Flopper.kicks" [function] 518 | // --------------------------------------- 519 | // doc: auction counter 520 | // act: 521 | rule #Flopper.kicks => 6 522 | 523 | syntax Int ::= "#Flopper.live" [function] 524 | // --------------------------------------- 525 | // doc: liveness flag 526 | // act: 527 | rule #Flopper.live => 7 528 | ``` 529 | 530 | ### Flap 531 | 532 | ```k 533 | syntax Int ::= "#Flapper.wards" "[" Int "]" [function] 534 | // --------------------------------------- 535 | // doc: whether `$0` is an owner of `Flop` 536 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 537 | rule #Flapper.wards[A] => #hashedLocation("Solidity", 0, A) 538 | 539 | syntax Int ::= "#Flapper.bids" "[" Int "].bid" [function] 540 | // ------------------------------------------------------ 541 | // doc: current bid (dai) 542 | // act: 543 | rule #Flapper.bids[N].bid => #hashedLocation("Solidity", 1, N) +Int 0 544 | 545 | syntax Int ::= "#Flapper.bids" "[" Int "].lot" [function] 546 | // ------------------------------------------------------ 547 | // doc: current lot (gem) 548 | // act: 549 | rule #Flapper.bids[N].lot => #hashedLocation("Solidity", 1, N) +Int 1 550 | 551 | // packed, use #WordPackAddrUInt48UInt48 to unpack this 552 | syntax Int ::= "#Flapper.bids" "[" Int "].guy_tic_end" [function] 553 | // -------------------------------------------------------------- 554 | // doc: 555 | // act: 556 | rule #Flapper.bids[N].guy_tic_end => #hashedLocation("Solidity", 1, N) +Int 2 557 | 558 | syntax Int ::= "#Flapper.vat" [function] 559 | // --------------------------------------- 560 | // doc: dai token 561 | // act: 562 | rule #Flapper.vat => 2 563 | 564 | syntax Int ::= "#Flapper.gem" [function] 565 | // --------------------------------------- 566 | // doc: mkr token 567 | // act: 568 | rule #Flapper.gem => 3 569 | 570 | syntax Int ::= "#Flapper.beg" [function] 571 | // --------------------------------------- 572 | // doc: minimum bid increment 573 | // act: 574 | rule #Flapper.beg => 4 575 | 576 | // packed, use #WordPackUInt48UInt48 to unpack this 577 | syntax Int ::= "#Flapper.ttl_tau" [function] 578 | // ----------------------------------------- 579 | // doc: 580 | // act: 581 | rule #Flapper.ttl_tau => 5 582 | 583 | syntax Int ::= "#Flapper.kicks" [function] 584 | // --------------------------------------- 585 | // doc: auction counter 586 | // act: 587 | rule #Flapper.kicks => 6 588 | 589 | syntax Int ::= "#Flapper.live" [function] 590 | // --------------------------------------- 591 | // doc: liveness flag 592 | // act: 593 | rule #Flapper.live => 7 594 | ``` 595 | 596 | ### GemLike 597 | 598 | A hypothetical token contract, based on `ds-token`: 599 | 600 | ```k 601 | syntax Int ::= "#Gem.balances" "[" Int "]" [function] 602 | // -------------------------------------------------- 603 | // doc: `gem` balance of `$0` 604 | // act: 605 | rule #Gem.balances[A] => #hashedLocation("Solidity", 3, A) 606 | 607 | syntax Int ::= "#Gem.stopped" [function] 608 | // -------------------------------------------------- 609 | // doc: `gem` balance of `$0` 610 | // act: 611 | rule #Gem.stopped => 4 612 | 613 | syntax Int ::= "#Gem.allowance" "[" Int "][" Int "]" [function] 614 | // ----------------------------------------------- 615 | // doc: the amount that can be spent on someones behalf 616 | // act: `$1 can spend `.` tokens belonging to `$0` 617 | rule #Gem.allowance[A][B] => #hashedLocation("Solidity", 8, A B) 618 | ``` 619 | 620 | ### End 621 | 622 | ```k 623 | // act: public 624 | syntax Int ::= "#End.wards" "[" Int "]" [function] 625 | // ----------------------------------------------- 626 | // doc: whether `$0` is an owner of `End` 627 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 628 | rule #End.wards[A] => #hashedLocation("Solidity", 0, A) 629 | 630 | syntax Int ::= "#End.vat" [function] 631 | // --------------------------------- 632 | // doc: `Vat` that this `End` points to 633 | // act: 634 | rule #End.vat => 1 635 | 636 | syntax Int ::= "#End.cat" [function] 637 | // --------------------------------- 638 | // doc: `cat` that this `End` points to 639 | // act: 640 | rule #End.cat => 2 641 | 642 | syntax Int ::= "#End.vow" [function] 643 | // --------------------------------- 644 | // doc: `Vow` that this `End` points to 645 | // act: 646 | rule #End.vow => 3 647 | 648 | syntax Int ::= "#End.spot" [function] 649 | // --------------------------------- 650 | // doc: `Spot` that this `End` points to 651 | // act: 652 | rule #End.spot => 4 653 | 654 | syntax Int ::= "#End.live" [function] 655 | // ---------------------------------- 656 | // doc: system liveness 657 | // act: 658 | rule #End.live => 5 659 | 660 | syntax Int ::= "#End.when" [function] 661 | // ---------------------------------- 662 | // doc: time of cage 663 | // act: 664 | rule #End.when => 6 665 | 666 | syntax Int ::= "#End.wait" [function] 667 | // ---------------------------------- 668 | // doc: processing period 669 | // act: 670 | rule #End.wait => 7 671 | 672 | syntax Int ::= "#End.debt" [function] 673 | // ---------------------------------- 674 | // doc: total outstanding debt following processing 675 | // act: 676 | rule #End.debt => 8 677 | 678 | syntax Int ::= "#End.tag" "[" Int "]" [function] 679 | // ----------------------------------------------- 680 | // doc: the cage price of ilk `$0` 681 | // act: 682 | rule #End.tag[Ilk] => #hashedLocation("Solidity", 9, Ilk) 683 | 684 | syntax Int ::= "#End.gap" "[" Int "]" [function] 685 | // ----------------------------------------------- 686 | // doc: the collateral shortfall of ilk `$0` 687 | // act: 688 | rule #End.gap[Ilk] => #hashedLocation("Solidity", 10, Ilk) 689 | 690 | syntax Int ::= "#End.Art" "[" Int "]" [function] 691 | // ----------------------------------------------- 692 | // doc: the total debt of ilk `$0` 693 | // act: 694 | rule #End.Art[Ilk] => #hashedLocation("Solidity", 11, Ilk) 695 | 696 | syntax Int ::= "#End.fix" "[" Int "]" [function] 697 | // ----------------------------------------------- 698 | // doc: the final cash price of ilk `$0` 699 | // act: 700 | rule #End.fix[Ilk] => #hashedLocation("Solidity", 12, Ilk) 701 | 702 | syntax Int ::= "#End.bag" "[" Int "]" [function] 703 | // ----------------------------------------------- 704 | // doc: the packed dai of user `$0` 705 | // act: 706 | rule #End.bag[Usr] => #hashedLocation("Solidity", 13, Usr) 707 | 708 | syntax Int ::= "#End.out" "[" Int "][" Int "]" [function] 709 | // --------------------------------------------- 710 | // doc: cashed collateral of ilk `$0` assigned to `$1` 711 | // act: 712 | rule #End.out[Ilk][Usr] => #hashedLocation("Solidity", 14, Ilk Usr) 713 | ``` 714 | 715 | ### Pot 716 | 717 | ```k 718 | syntax Int ::= "#Pot.wards" "[" Int "]" [function] 719 | // ----------------------------------------------- 720 | // doc: whether `$0` is an owner of `Pot` 721 | // act: address `$0` is `. == 1 ? authorised : unauthorised` 722 | rule #Pot.wards[A] => #hashedLocation("Solidity", 0, A) 723 | 724 | syntax Int ::= "#Pot.pie" "[" Int "]" [function] 725 | // ---------------------------------------------------- 726 | // doc: balance that `$0` has locked in this pot 727 | // act: 728 | rule #Pot.pie[Usr] => #hashedLocation("Solidity", 1, Usr) 729 | 730 | syntax Int ::= "#Pot.Pie" [function] 731 | // ---------------------------------- 732 | // doc: total amount of dai locked in this `Pot` 733 | // act: this Pot points to Vat `.` 734 | rule #Pot.Pie => 2 735 | 736 | syntax Int ::= "#Pot.dsr" [function] 737 | // ---------------------------------- 738 | // doc: the current deposit interest rate of this `Pot` 739 | // act: 740 | rule #Pot.dsr => 3 741 | 742 | syntax Int ::= "#Pot.chi" [function] 743 | // ---------------------------------- 744 | // doc: `Vat` that this `Pot` points to 745 | // act: this Pot points to Vat `.` 746 | rule #Pot.chi => 4 747 | 748 | syntax Int ::= "#Pot.vat" [function] 749 | // ---------------------------------- 750 | // doc: `Vat` that this `Pot` points to 751 | // act: this Pot points to Vat `.` 752 | rule #Pot.vat => 5 753 | 754 | syntax Int ::= "#Pot.vow" [function] 755 | // ---------------------------------- 756 | // doc: `Vow` that this `Pot` points to 757 | // act: this Pot points to Vow `.` 758 | rule #Pot.vow => 6 759 | 760 | syntax Int ::= "#Pot.rho" [function] 761 | // ---------------------------------- 762 | // doc: 763 | // act: 764 | rule #Pot.rho => 7 765 | ``` 766 | ### DSToken 767 | 768 | ```k 769 | syntax Int ::= "#DSToken.supply" [function] 770 | rule #DSToken.supply => 0 771 | 772 | syntax Int ::= "#DSToken.balances" "[" Int "]" [function] 773 | rule #DSToken.balances[A] => #hashedLocation("Solidity", 1, A) 774 | 775 | syntax Int ::= "#DSToken.allowance" "[" Int "][" Int "]" [function] 776 | rule #DSToken.allowance[A][B] => #hashedLocation("Solidity", 2, A B) 777 | 778 | syntax Int ::= "#DSToken.authority" [function] 779 | rule #DSToken.authority => 3 780 | 781 | syntax Int ::= "#DSToken.owner_stopped" [function] 782 | rule #DSToken.owner_stopped => 4 783 | 784 | syntax Int ::= "#DSToken.symbol" [function] 785 | rule #DSToken.symbol => 5 786 | 787 | syntax Int ::= "#DSToken.decimals" [function] 788 | rule #DSToken.decimals => 6 789 | ``` 790 | 791 | ### DSValue 792 | 793 | ```k 794 | syntax Int ::= "#DSValue.authority" [function] 795 | rule #DSValue.authority => 0 796 | 797 | syntax Int ::= "#DSValue.owner_has" [function] 798 | rule #DSValue.owner_has => 1 799 | 800 | syntax Int ::= "#DSValue.val" [function] 801 | rule #DSValue.val => 2 802 | 803 | ``` 804 | 805 | ### Spotter 806 | 807 | ```act 808 | syntax Int ::= "#Spotter.wards" "[" Int "]" [function] 809 | rule #Spotter.wards[A] => #hashedLocation("Solidity", 0, A) 810 | 811 | syntax Int ::= "#Spotter.ilks" "[" Int "].pip" [function] 812 | rule #Spotter.ilks[Ilk].pip => #hashedLocation("Solidity", 1, Ilk) +Int 0 813 | 814 | syntax Int ::= "#Spotter.ilks" "[" Int "].mat" [function] 815 | rule #Spotter.ilks[Ilk].mat => #hashedLocation("Solidity", 1, Ilk) +Int 1 816 | 817 | syntax Int ::= "#Spotter.vat" [function] 818 | rule #Spotter.vat => 2 819 | 820 | syntax Int ::= "#Spotter.par" [function] 821 | rule #Spotter.par => 3 822 | ``` 823 | --------------------------------------------------------------------------------