ProofMode.md 7.96 KB
 Robbert Krebbers committed Jul 05, 2016 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 ``````Tactic overview =============== Applying hypotheses and lemmas ------------------------------ - `iExact "H"` : finish the goal if the conclusion matches the hypothesis `H` - `iAssumption` : finish the goal if the conclusion matches any hypothesis - `iApply trm` : match the conclusion of the current goal against the conclusion of `tetrmrm` and generates goals for the premises of `trm`. See proof mode terms below. Context management ------------------ - `iIntros {x1 ... xn} "ipat1 ... ipatn"` : introduce universal quantifiers using Coq introduction patterns `x1 ... xn` and implications/wands using proof mode introduction patterns `ipat1 ... ipatn`. - `iClear "H1 ... Hn"` : clear the hypothesis `H1 ... Hn`. The symbol `★` can be used to clear entire spatial context. - `iRevert {x1 ... xn} "H1 ... Hn"` : revert the proof mode hypotheses `H1 ... Hn` into wands, as well as the Coq level hypotheses/variables `x1 ... xn` into universal quantifiers. The symbol `★` can be used to revert the entire spatial context. - `iRename "H1" into "H2"` : rename the hypothesis `H1` into `H2`. - `iSpecialize trm` : instantiate universal quantifiers and eliminate implications/wands of a hypothesis `trm`. See proof mode terms below. - `iPoseProof trm as "H"` : put `trm` into the context as a new hypothesis `H`. - `iAssert P with "spat" as "ipat"` : create a new goal with conclusion `P` and put `P` in the context of the original goal. The specialization pattern `spat` specifies which hypotheses will be consumed by proving `P` and the introduction pattern `ipat` specifies how to eliminate `P`. Introduction of logical connectives ----------------------------------- - `iPureIntro` : turn a pure goal into a Coq goal. This tactic works for goals of the shape `■ φ`, `a ≡ b` on discrete COFEs, and `✓ a` on discrete CMRAs. - `iLeft` : left introduction of disjunction. - `iRight` : right introduction of disjunction. - `iSplit` : introduction of a conjunction, or separating conjunction provided one of the operands is persistent. - `iSplitL "H1 ... Hn"` : introduction of a separating conjunction. The hypotheses `H1 ... Hn` are used for the left conjunct, and the remaining ones for the right conjunct. - `iSplitR "H0 ... Hn"` : symmetric version of the above. - `iExist t1, .., tn` : introduction of an existential quantifier. Elimination of logical connectives ---------------------------------- - `iExFalso` : Ex falso sequitur quod libet. - `iDestruct trm as {x1 ... xn} "spat1 ... spatn"` : elimination of existential quantifiers using Coq introduction patterns `x1 ... xn` and elimination of object level connectives using the proof mode introduction patterns `ipat1 ... ipatn`. Separating logic specific tactics --------------------------------- - `iFrame "H0 ... Hn"` : cancel the hypotheses `H0 ... Hn` in the goal. - `iCombine "H1" "H2" as "H"` : turns `H1 : P1` and `H2 : P2` into `H : P1 ★ P2`. The later modality ------------------ - `iNext` : introduce a later by stripping laters from all hypotheses. - `iLöb {x1 ... xn} as "IH"` : perform Löb induction by generalizing over the Coq level variables `x1 ... xn` and the entire spatial context. Rewriting --------- - `iRewrite trm` : rewrite an equality in the conclusion. - `iRewrite trm in "H"` : rewrite an equality in the hypothesis `H`. Iris ---- - `iPvsIntro` : introduction of a primitive view shift. Generates a goal if the masks are not syntactically equal. - `iPvs trm as {x1 ... xn} "ipat"` : runs a primitive view shift `trm`. - `iInv N as {x1 ... xn} "ipat"` : open the invariant `N`. - `iInv> N as {x1 ... xn} "ipat"` : open the invariant `N` and establish that it is timeless so no laters have to be added. - `iTimeless "H"` : strip a later of a timeless hypotheses `H` in case the conclusion is a primitive view shifts or weakest precondition. Miscellaneous ------------- - The tactic `done` is extended so that it also performs `iAssumption` and introduces pure connectives. - The proof mode adds hints to the core `eauto` database so that `eauto` automatically introduces: conjunctions and disjunctions, universal and existential quantifiers, implications and wand, always and later modalities, primitive view shifts, and pure connectives. Introduction patterns ===================== Introduction patterns are used to perform introductions and eliminations of multiple connectives on the fly. The proof mode supports the following introduction patterns: - `H` : create a hypothesis named H. - `?` : create an anonymous hypothesis. - `_` : remove the hypothesis. - `\$` : frame the hypothesis in the goal. - `# ipat` : move the hypothesis to the persistent context. - `%` : move the hypothesis to the pure Coq context (anonymously). - `[ipat ipat]` : (separating) conjunction elimination. - `|ipat|ipat]` : disjunction elimination. - `[]` : false elimination. Apart from this, there are the following introduction patterns that can only appear at the top level: - `!` : introduce a box (provided that the spatial context is empty). - `>` : introduce a later (which strips laters from all hypotheses). - `{H1 ... Hn}` : clear `H1 ... Hn`. - `{\$H1 ... \$Hn}` : frame `H1 ... Hn` (this pattern can be mixed with the previous pattern, e.g., `{\$H1 H2 #H3}`). - `/=` : perform `simpl`. - `*` : introduce all universal quantifiers. - `**` : introduce all universal quantifiers, as well as all arrows and wands. For example, given: `````` Ralf Jung committed Apr 19, 2016 131 `````` `````` Robbert Krebbers committed Jul 05, 2016 132 `````` ∀ x, x = 0 ⊢ □ (P → False ∨ □ (Q ∧ ▷ R) -★ P ★ ▷ (R ★ Q ∧ x = pred 2)). `````` Ralf Jung committed Apr 19, 2016 133 `````` `````` Robbert Krebbers committed Jul 05, 2016 134 ``````You can write `````` Ralf Jung committed Apr 19, 2016 135 `````` `````` Robbert Krebbers committed Jul 05, 2016 136 137 138 `````` iIntros {x} "% ! \$ [[] | #[HQ HR]] /= >". which results in: `````` Ralf Jung committed Apr 19, 2016 139 `````` `````` Robbert Krebbers committed Jul 05, 2016 140 141 142 143 144 145 146 `````` x : nat H : x = 0 ______________________________________(1/1) "HQ" : Q "HR" : R --------------------------------------□ R ★ Q ∧ x = 1 `````` Ralf Jung committed Apr 19, 2016 147 148 `````` `````` Robbert Krebbers committed Jul 05, 2016 149 150 ``````Specialization patterns ======================= `````` Ralf Jung committed Apr 19, 2016 151 `````` `````` Robbert Krebbers committed Jul 05, 2016 152 153 154 155 ``````Since we are reasoning in a spatial logic, when eliminating a lemma or hypotheses of type ``P_0 -★ ... -★ P_n -★ R`` one has to specify how the hypotheses are split between the premises. The proof mode supports the following so called specification patterns to express this splitting: `````` Ralf Jung committed Apr 19, 2016 156 `````` `````` Robbert Krebbers committed Jul 05, 2016 157 158 159 160 161 162 163 164 165 166 167 168 169 170 ``````- `H` : use the hypothesis `H` (it should match the premise exactly). If `H` is spatial, it will be consumed. - `[H1 ... Hn]` : generate a goal with the spatial hypotheses `H1 ... Hn` and all persistent hypotheses. The hypotheses `H1 ... Hn` will be consumed. - `[-H1 ... Hn]` : negated form of the above pattern - `=>[H1 ... Hn]` : same as the above pattern, but can only be used if the goal is a primitive view shift, in which case the view shift will be kept in the goal of the premise too. - `[#]` : This pattern can be used when eliminating `P -★ Q` when either `P` or `Q` is persistent. In this case, all hypotheses are available in the goal for the premise as none will be consumed. - `[%]` : This pattern can be used when eliminating `P -★ Q` when `P` is pure. It will generate a Coq goal for `P` and does not consume any hypotheses. - `*` : instantiate all top-level universal quantifiers with meta variables. `````` Ralf Jung committed Apr 19, 2016 171 `````` `````` Robbert Krebbers committed Jul 05, 2016 172 ``````For example, given: `````` Ralf Jung committed Apr 19, 2016 173 `````` `````` Robbert Krebbers committed Jul 05, 2016 174 `````` H : □ P -★ P 2 -★ x = 0 -★ Q1 ∗ Q2 `````` Ralf Jung committed Apr 19, 2016 175 `````` `````` Robbert Krebbers committed Jul 05, 2016 176 ``````You can write: `````` Ralf Jung committed Apr 19, 2016 177 `````` `````` Robbert Krebbers committed Jul 05, 2016 178 `````` iDestruct ("H" with "[#] [H1 H2] [%]") as "[H4 H5]". `````` Ralf Jung committed Apr 19, 2016 179 `````` `````` Robbert Krebbers committed Jul 05, 2016 180 181 182 183 184 185 ``````Proof mode terms ================ Many of the proof mode tactics (such as `iDestruct`, `iApply`, `iRewrite`) can take both hypotheses and lemmas, and allow one to instantiate universal quantifiers and implications/wands of these hypotheses/lemmas on the fly. `````` Ralf Jung committed Apr 19, 2016 186 `````` `````` Robbert Krebbers committed Jul 05, 2016 187 ``````The syntax for the arguments, called _proof mode terms_ of these tactics is: `````` Ralf Jung committed Apr 19, 2016 188 `````` `````` Robbert Krebbers committed Jul 05, 2016 189 `````` (H \$! t1 ... tn with "spat1 .. spatn") `````` Ralf Jung committed Apr 19, 2016 190 `````` `````` Robbert Krebbers committed Jul 05, 2016 191 192 193 194 ``````Here, `H` can be both a hypothesis, as well as a Coq lemma whose conclusion is of the shape `P ⊢ Q`. In the above, `t1 ... tn` are arbitrary Coq terms used for instantiation of universal quantifiers, and `spat1 .. spatn` are specialization patterns to eliminate implications and wands. `````` Ralf Jung committed Apr 19, 2016 195 `````` `````` Robbert Krebbers committed Jul 05, 2016 196 ``````Proof mode terms can be written down using the following short hands too: `````` Ralf Jung committed Apr 19, 2016 197 `````` `````` Robbert Krebbers committed Jul 05, 2016 198 199 200 `````` (H with "spat1 .. spatn") (H \$! t1 ... tn) H `````` Ralf Jung committed Apr 19, 2016 201