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
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
|
/- This file contains tactics to solve arithmetic goals -/
import Lean
import Lean.Meta.Tactic.Simp
import Init.Data.List.Basic
import Mathlib.Tactic.RunCmd
import Mathlib.Tactic.Linarith
-- TODO: there is no Omega tactic for now - it seems it hasn't been ported yet
--import Mathlib.Tactic.Omega
import Base.Primitives
import Base.Utils
import Base.Arith.Base
namespace Arith
open Primitives Utils
-- TODO: move
/- Remark: we can't write the following instance because of restrictions about
the type class parameters (`ty` doesn't appear in the return type, which is
forbidden):
```
instance Scalar.cast (ty : ScalarTy) : Coe (Scalar ty) Int where coe := λ v => v.val
```
-/
def Scalar.toInt {ty : ScalarTy} (x : Scalar ty) : Int := x.val
-- Remark: I tried a version of the shape `HasScalarProp {a : Type} (x : a)`
-- but the lookup didn't work
class HasScalarProp (a : Sort u) where
prop_ty : a → Prop
prop : ∀ x:a, prop_ty x
class HasIntProp (a : Sort u) where
prop_ty : a → Prop
prop : ∀ x:a, prop_ty x
instance (ty : ScalarTy) : HasScalarProp (Scalar ty) where
-- prop_ty is inferred
prop := λ x => And.intro x.hmin x.hmax
instance (a : Type) : HasScalarProp (Vec a) where
prop_ty := λ v => v.val.length ≤ Scalar.max ScalarTy.Usize
prop := λ ⟨ _, l ⟩ => l
class PropHasImp (x : Prop) where
concl : Prop
prop : x → concl
-- This also works for `x ≠ y` because this expression reduces to `¬ x = y`
-- and `Ne` is marked as `reducible`
instance (x y : Int) : PropHasImp (¬ x = y) where
concl := x < y ∨ x > y
prop := λ (h:x ≠ y) => ne_is_lt_or_gt h
open Lean Lean.Elab Command Term Lean.Meta
-- Small utility: print all the declarations in the context
elab "print_all_decls" : tactic => do
let ctx ← Lean.MonadLCtx.getLCtx
for decl in ← ctx.getDecls do
let ty ← Lean.Meta.inferType decl.toExpr
logInfo m!"{decl.toExpr} : {ty}"
pure ()
-- Explore a term by decomposing the applications (we explore the applied
-- functions and their arguments, but ignore lambdas, forall, etc. -
-- should we go inside?).
partial def foldTermApps (k : α → Expr → MetaM α) (s : α) (e : Expr) : MetaM α := do
-- We do it in a very simpler manner: we deconstruct applications,
-- and recursively explore the sub-expressions. Note that we do
-- not go inside foralls and abstractions (should we?).
e.withApp fun f args => do
let s ← k s f
args.foldlM (foldTermApps k) s
-- Provided a function `k` which lookups type class instances on an expression,
-- collect all the instances lookuped by applying `k` on the sub-expressions of `e`.
def collectInstances
(k : Expr → MetaM (Option Expr)) (s : HashSet Expr) (e : Expr) : MetaM (HashSet Expr) := do
let k s e := do
match ← k e with
| none => pure s
| some i => pure (s.insert i)
foldTermApps k s e
-- Similar to `collectInstances`, but explores all the local declarations in the
-- main context.
def collectInstancesFromMainCtx (k : Expr → MetaM (Option Expr)) : Tactic.TacticM (HashSet Expr) := do
Tactic.withMainContext do
-- Get the local context
let ctx ← Lean.MonadLCtx.getLCtx
-- Just a matter of precaution
let ctx ← instantiateLCtxMVars ctx
-- Initialize the hashset
let hs := HashSet.empty
-- Explore the declarations
let decls ← ctx.getDecls
decls.foldlM (fun hs d => collectInstances k hs d.toExpr) hs
-- Helper
def lookupProp (fName : String) (className : Name) (e : Expr) : MetaM (Option Expr) := do
trace[Arith] fName
-- TODO: do we need Lean.observing?
-- This actually eliminates the error messages
Lean.observing? do
trace[Arith] m!"{fName}: observing"
let ty ← Lean.Meta.inferType e
let hasProp ← mkAppM className #[ty]
let hasPropInst ← trySynthInstance hasProp
match hasPropInst with
| LOption.some i =>
trace[Arith] "Found HasScalarProp instance"
let i_prop ← mkProjection i (Name.mkSimple "prop")
some (← mkAppM' i_prop #[e])
| _ => none
-- Return an instance of `HasIntProp` for `e` if it has some
def lookupHasIntProp (e : Expr) : MetaM (Option Expr) :=
lookupProp "lookupHasScalarProp" ``HasIntProp e
-- Return an instance of `HasScalarProp` for `e` if it has some
def lookupHasScalarProp (e : Expr) : MetaM (Option Expr) :=
lookupProp "lookupHasScalarProp" ``HasScalarProp e
-- Collect the instances of `HasIntProp` for the subexpressions in the context
def collectHasIntPropInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
collectInstancesFromMainCtx lookupHasIntProp
-- Collect the instances of `HasScalarProp` for the subexpressions in the context
def collectHasScalarPropInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
collectInstancesFromMainCtx lookupHasScalarProp
elab "display_has_prop_instances" : tactic => do
trace[Arith] "Displaying the HasScalarProp instances"
let hs ← collectHasScalarPropInstancesFromMainCtx
hs.forM fun e => do
trace[Arith] "+ HasScalarProp instance: {e}"
example (x : U32) : True := by
let i : HasScalarProp U32 := inferInstance
have p := @HasScalarProp.prop _ i x
simp only [HasScalarProp.prop_ty] at p
display_has_prop_instances
simp
-- Return an instance of `PropHasImp` for `e` if it has some
def lookupPropHasImp (e : Expr) : MetaM (Option Expr) := do
trace[Arith] "lookupPropHasImp"
-- TODO: do we need Lean.observing?
-- This actually eliminates the error messages
Lean.observing? do
trace[Arith] "lookupPropHasImp: observing"
let ty ← Lean.Meta.inferType e
trace[Arith] "lookupPropHasImp: ty: {ty}"
let cl ← mkAppM ``PropHasImp #[ty]
let inst ← trySynthInstance cl
match inst with
| LOption.some i =>
trace[Arith] "Found PropHasImp instance"
let i_prop ← mkProjection i (Name.mkSimple "prop")
some (← mkAppM' i_prop #[e])
| _ => none
-- Collect the instances of `PropHasImp` for the subexpressions in the context
def collectPropHasImpInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
collectInstancesFromMainCtx lookupPropHasImp
elab "display_prop_has_imp_instances" : tactic => do
trace[Arith] "Displaying the PropHasImp instances"
let hs ← collectPropHasImpInstancesFromMainCtx
hs.forM fun e => do
trace[Arith] "+ PropHasImp instance: {e}"
example (x y : Int) (_ : x ≠ y) (_ : ¬ x = y) : True := by
display_prop_has_imp_instances
simp
-- Lookup instances in a context and introduce them with additional declarations.
def introInstances (declToUnfold : Name) (lookup : Expr → MetaM (Option Expr)) : Tactic.TacticM (Array Expr) := do
let hs ← collectInstancesFromMainCtx lookup
hs.toArray.mapM fun e => do
let type ← inferType e
let name ← mkFreshUserName `h
-- Add a declaration
let nval ← Utils.addDeclTac name e type (asLet := false)
-- Simplify to unfold the declaration to unfold (i.e., the projector)
Utils.simpAt [declToUnfold] [] [] (Tactic.Location.targets #[mkIdent name] false)
-- Return the new value
pure nval
def introHasIntPropInstances : Tactic.TacticM (Array Expr) := do
trace[Arith] "Introducing the HasIntProp instances"
introInstances ``HasIntProp.prop_ty lookupHasIntProp
def introHasScalarPropInstances : Tactic.TacticM (Array Expr) := do
trace[Arith] "Introducing the HasScalarProp instances"
introInstances ``HasScalarProp.prop_ty lookupHasScalarProp
-- Lookup the instances of `HasScalarProp for all the sub-expressions in the context,
-- and introduce the corresponding assumptions
elab "intro_has_prop_instances" : tactic => do
let _ ← introHasScalarPropInstances
example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
intro_has_prop_instances
simp [*]
example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
intro_has_prop_instances
simp_all [Scalar.max, Scalar.min]
-- Lookup the instances of `PropHasImp for all the sub-expressions in the context,
-- and introduce the corresponding assumptions
elab "intro_prop_has_imp_instances" : tactic => do
trace[Arith] "Introducing the PropHasImp instances"
let _ ← introInstances ``PropHasImp.concl lookupPropHasImp
example (x y : Int) (h0 : x ≤ y) (h1 : x ≠ y) : x < y := by
intro_prop_has_imp_instances
rename_i h
split_disj h
. linarith
. linarith
/- Boosting a bit the linarith tac.
We do the following:
- for all the assumptions of the shape `(x : Int) ≠ y` or `¬ (x = y), we
introduce two goals with the assumptions `x < y` and `x > y`
TODO: we could create a PR for mathlib.
-/
def intTacPreprocess : Tactic.TacticM Unit := do
Tactic.withMainContext do
-- Lookup the instances of PropHasImp (this is how we detect assumptions
-- of the proper shape), introduce assumptions in the context and split
-- on those
-- TODO: get rid of the assumptions that we split
let rec splitOnAsms (asms : List Expr) : Tactic.TacticM Unit :=
match asms with
| [] => pure ()
| asm :: asms =>
let k := splitOnAsms asms
Utils.splitDisjTac asm k k
-- Introduce
let _ ← introHasIntPropInstances
let asms ← introInstances ``PropHasImp.concl lookupPropHasImp
-- Split
splitOnAsms asms.toList
elab "int_tac_preprocess" : tactic =>
intTacPreprocess
def intTac : Tactic.TacticM Unit := do
Tactic.withMainContext do
Tactic.focus do
-- Preprocess - wondering if we should do this before or after splitting
-- the goal. I think before leads to a smaller proof term?
Tactic.allGoals intTacPreprocess
-- Split the conjunctions in the goal
Utils.repeatTac Utils.splitConjTarget
-- Call linarith
let linarith :=
let cfg : Linarith.LinarithConfig := {
-- We do this with our custom preprocessing
splitNe := false
}
Tactic.liftMetaFinishingTactic <| Linarith.linarith false [] cfg
Tactic.allGoals linarith
elab "int_tac" : tactic =>
intTac
example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
int_tac_preprocess
linarith
linarith
example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
int_tac
-- Checking that things append correctly when there are several disjunctions
example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y := by
int_tac
-- Checking that things append correctly when there are several disjunctions
example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y ∧ x + y ≥ 2 := by
int_tac
def scalarTacPreprocess (tac : Tactic.TacticM Unit) : Tactic.TacticM Unit := do
Tactic.withMainContext do
-- Introduce the scalar bounds
let _ ← introHasScalarPropInstances
Tactic.allGoals do
-- Inroduce the bounds for the isize/usize types
let add (e : Expr) : Tactic.TacticM Unit := do
let ty ← inferType e
let _ ← Utils.addDeclTac (← mkFreshUserName `h) e ty (asLet := false)
add (← mkAppM ``Scalar.cMin_bound #[.const ``ScalarTy.Isize []])
add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Usize []])
add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Isize []])
-- Reveal the concrete bounds
Utils.simpAt [``Scalar.min, ``Scalar.max, ``Scalar.cMin, ``Scalar.cMax,
``I8.min, ``I16.min, ``I32.min, ``I64.min, ``I128.min,
``I8.max, ``I16.max, ``I32.max, ``I64.max, ``I128.max,
``U8.min, ``U16.min, ``U32.min, ``U64.min, ``U128.min,
``U8.max, ``U16.max, ``U32.max, ``U64.max, ``U128.max
] [] [] .wildcard
-- Finish the proof
tac
elab "scalar_tac_preprocess" : tactic =>
scalarTacPreprocess intTacPreprocess
-- A tactic to solve linear arithmetic goals in the presence of scalars
def scalarTac : Tactic.TacticM Unit := do
scalarTacPreprocess intTac
elab "scalar_tac" : tactic =>
scalarTac
example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
scalar_tac
example {a: Type} (v : Vec a) : v.val.length ≤ Scalar.max ScalarTy.Usize := by
scalar_tac
end Arith
|