ProofMode.md 8.06 KB
 Robbert Krebbers committed Jul 05, 2016 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ``````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 ------------------ `````` Robbert Krebbers committed Jul 13, 2016 16 ``````- `iIntros (x1 ... xn) "ipat1 ... ipatn"` : introduce universal quantifiers `````` Robbert Krebbers committed Jul 05, 2016 17 18 19 `````` 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 `````` Robbert Krebbers committed Jul 13, 2016 20 21 `````` be used to clear entire spatial context. - `iRevert (x1 ... xn) "H1 ... Hn"` : revert the proof mode hypotheses `````` Robbert Krebbers committed Jul 05, 2016 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 `````` `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. `````` Robbert Krebbers committed Jul 13, 2016 55 ``````- `iDestruct trm as (x1 ... xn) "spat1 ... spatn"` : elimination of existential `````` Robbert Krebbers committed Jul 05, 2016 56 57 58 `````` quantifiers using Coq introduction patterns `x1 ... xn` and elimination of object level connectives using the proof mode introduction patterns `ipat1 ... ipatn`. `````` Robbert Krebbers committed Jul 13, 2016 59 60 ``````- `iDestruct trm as %cpat` : elimination of a pure hypothesis using the Coq introduction pattern `cpat`. `````` Robbert Krebbers committed Jul 05, 2016 61 62 63 64 65 66 67 68 69 70 71 `````` 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. `````` Robbert Krebbers committed Jul 13, 2016 72 ``````- `iLöb (x1 ... xn) as "IH"` : perform Löb induction by generalizing over the `````` Robbert Krebbers committed Jul 05, 2016 73 74 75 76 77 78 79 80 81 82 83 84 85 `````` 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. `````` Robbert Krebbers committed Jul 13, 2016 86 87 88 ``````- `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 `````` Robbert Krebbers committed Jul 05, 2016 89 90 `````` it is timeless so no laters have to be added. - `iTimeless "H"` : strip a later of a timeless hypotheses `H` in case the `````` Robbert Krebbers committed Jul 13, 2016 91 `````` conclusion is a primitive view shifts or weakest precondition. `````` Robbert Krebbers committed Jul 05, 2016 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 `````` 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. `````` Ralf Jung committed Jul 11, 2016 117 ``````- `[ipat|ipat]` : disjunction elimination. `````` Robbert Krebbers committed Jul 05, 2016 118 119 120 121 122 123 124 125 126 ``````- `[]` : 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 `````` Robbert Krebbers committed Jul 05, 2016 127 `````` previous pattern, e.g., `{\$H1 H2 \$H3}`). `````` Robbert Krebbers committed Jul 05, 2016 128 129 130 131 132 ``````- `/=` : 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 133 `````` `````` Robbert Krebbers committed Jul 05, 2016 134 `````` ∀ x, x = 0 ⊢ □ (P → False ∨ □ (Q ∧ ▷ R) -★ P ★ ▷ (R ★ Q ∧ x = pred 2)). `````` Ralf Jung committed Apr 19, 2016 135 `````` `````` Robbert Krebbers committed Jul 05, 2016 136 ``````You can write `````` Ralf Jung committed Apr 19, 2016 137 `````` `````` Robbert Krebbers committed Jul 13, 2016 138 `````` iIntros (x) "% ! \$ [[] | #[HQ HR]] /= >". `````` Robbert Krebbers committed Jul 05, 2016 139 140 `````` which results in: `````` Ralf Jung committed Apr 19, 2016 141 `````` `````` Robbert Krebbers committed Jul 05, 2016 142 143 144 `````` x : nat H : x = 0 ______________________________________(1/1) `````` Robbert Krebbers committed Jul 05, 2016 145 `````` "HQ" : Q `````` Robbert Krebbers committed Jul 05, 2016 146 147 148 `````` "HR" : R --------------------------------------□ R ★ Q ∧ x = 1 `````` Ralf Jung committed Apr 19, 2016 149 150 `````` `````` Robbert Krebbers committed Jul 05, 2016 151 152 ``````Specialization patterns ======================= `````` Ralf Jung committed Apr 19, 2016 153 `````` `````` Robbert Krebbers committed Jul 05, 2016 154 155 156 157 ``````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 158 `````` `````` Robbert Krebbers committed Jul 05, 2016 159 160 161 162 163 164 ``````- `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 `````` Robbert Krebbers committed Jul 13, 2016 165 166 `````` is a primitive view shift, in which case the view shift will be kept in the goal of the premise too. `````` Robbert Krebbers committed Jul 05, 2016 167 168 169 170 171 172 ``````- `[#]` : 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 173 `````` `````` Robbert Krebbers committed Jul 05, 2016 174 ``````For example, given: `````` Ralf Jung committed Apr 19, 2016 175 `````` `````` Robbert Krebbers committed Jul 05, 2016 176 `````` H : □ P -★ P 2 -★ x = 0 -★ Q1 ∗ Q2 `````` Ralf Jung committed Apr 19, 2016 177 `````` `````` Robbert Krebbers committed Jul 05, 2016 178 ``````You can write: `````` Ralf Jung committed Apr 19, 2016 179 `````` `````` Robbert Krebbers committed Jul 05, 2016 180 `````` iDestruct ("H" with "[#] [H1 H2] [%]") as "[H4 H5]". `````` Ralf Jung committed Apr 19, 2016 181 `````` `````` Robbert Krebbers committed Jul 05, 2016 182 183 184 185 186 187 ``````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 188 `````` `````` Ralf Jung committed Jul 11, 2016 189 ``````The syntax for the arguments, called _proof mode terms_, of these tactics is: `````` Ralf Jung committed Apr 19, 2016 190 `````` `````` Robbert Krebbers committed Jul 05, 2016 191 `````` (H \$! t1 ... tn with "spat1 .. spatn") `````` Ralf Jung committed Apr 19, 2016 192 `````` `````` Robbert Krebbers committed Jul 05, 2016 193 194 195 196 ``````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 197 `````` `````` Robbert Krebbers committed Jul 05, 2016 198 ``````Proof mode terms can be written down using the following short hands too: `````` Ralf Jung committed Apr 19, 2016 199 `````` `````` Robbert Krebbers committed Jul 05, 2016 200 201 202 `````` (H with "spat1 .. spatn") (H \$! t1 ... tn) H `````` Ralf Jung committed Apr 19, 2016 203