This PR introduces the basic theory of ordered modules over Nat (i.e.
without subtraction), for `grind`. We'll solve problems here by
embedding them in the `IntModule` envelope.
This PR adds the following instance
```
instance [Field α] [LinearOrder α] [Ring.IsOrdered α] : IsCharP α 0
```
The goal is to ensure we do not perform unnecessary case-splits in our
test suite.
This PR adds `have` forms of simp lemmas that will be used in a future
`have` simplifier. This depends on #8751 and future elaboration changes,
since these are meant to elaborate using `Expr.letE (nondep := true) ..`
expressions; for now they are duplicates of the `letFun_*` lemmas.
This PR implements the Rabinowitsch transformation for `Field`
disequalities in `grind`. For example, this transformation is necessary
for solving:
```lean
example [Field α] (a : α) : a^2 = 0 → a = 0 := by
grind
```
This PR improves the support for fields in `grind`. New supported
examples:
```lean
example [Field α] [IsCharP α 0] (x : α) : x ≠ 0 → (4 / x)⁻¹ * ((3 * x^3) / x)^2 * ((1 / (2 * x))⁻¹)^3 = 18 * x^8 := by grind
example [Field α] (a : α) : 2 * a ≠ 0 → 1 / a + 1 / (2 * a) = 3 / (2 * a) := by grind
example [Field α] [IsCharP α 0] (a : α) : 1 / a + 1 / (2 * a) = 3 / (2 * a) := by grind
example [Field α] [IsCharP α 0] (a b : α) : 2*b - a = a + b → 1 / a + 1 / (2 * a) = 3 / b := by grind
example [Field α] [NoNatZeroDivisors α] (a : α) : 1 / a + 1 / (2 * a) = 3 / (2 * a) := by grind
example [Field α] {x y z w : α} : x / y = z / w → y ≠ 0 → w ≠ 0 → x * w = z * y := by grind
example [Field α] (a : α) : a = 0 → a ≠ 1 := by grind
example [Field α] (a : α) : a = 0 → a ≠ 1 - a := by grind
```
This PR implements basic `Field` support in the commutative ring module
in `grind`. It is just division by numerals for now. Examples:
```lean
open Lean Grind
example [Field α] [IsCharP α 0] (a b c : α) : a/3 = b → c = a/3 → a/2 + a/2 = b + 2*c := by
grind
example [Field α] (a b : α) : b = 0 → (a + a) / 0 = b := by
grind
example [Field α] [IsCharP α 3] (a b : α) : a/3 = b → b = 0 := by
grind
example [Field α] [IsCharP α 7] (a b c : α) : a/3 = b → c = a/3 → a/2 + a/2 = b + 2*c + 7 := by
grind
example [Field R] [IsCharP R 0] (x : R) (cos : R → R) :
(cos x ^ 2 + (2 * cos x ^ 2 - 1) ^ 2 + (4 * cos x ^ 3 - 3 * cos x) ^ 2 - 1) / 4 =
cos x * (cos x ^ 2 - 1 / 2) * (4 * cos x ^ 3 - 3 * cos x) := by
grind
```
This PR adds an option for disabling the cutsat procedure in `grind`.
The linarith module takes over linear integer/nat constraints. Example:
```lean
set_option trace.grind.cutsat.assert true in -- cutsat should **not** process the following constraints
example (x y z : Int) (h1 : 2 * x < 3 * y) (h2 : -4 * x + 2 * z < 0) : ¬ 12*y - 4* z < 0 := by
grind -cutsat -- `linarith` module solves it
```
This PR fixes a bug where the single-quote character `Char.ofNat 39`
would delaborate as `'''`, which causes a parse error if pasted back in
to the source code.
---------
Co-authored-by: Kyle Miller <kmill31415@gmail.com>
This PR changes the `show t` tactic to match its documentation.
Previously it was a synonym for `change t`, but now it finds the first
goal that unifies with the term `t` and moves it to the front of the
goal list.
This PR adds grind annotations for `List/Array/Vector.ofFn` theorems and
additional `List.Impl` find operations.
The annotations are added to theorems that correspond to those already
annotated in the List implementation, ensuring consistency across all
three container types (List, Array, Vector) for ofFn operations and
related functionality.
Key theorems annotated include:
- Element access theorems (`getElem_ofFn`, `getElem?_ofFn`)
- Construction and conversion theorems (`ofFn_zero`, `toList_ofFn`,
`toArray_ofFn`)
- Membership theorems (`mem_ofFn`)
- Head/tail operations (`back_ofFn`)
- Monadic operations (`ofFnM_zero`, `toList_ofFnM`, `toArray_ofFnM`,
`idRun_ofFnM`)
- List.Impl find operations (`find?_singleton`, `find?_append`,
`findSome?_singleton`, `findSome?_append`)
This PR adds grind annotations for `Array/Vector.mapIdx` and `mapFinIdx`
theorems.
The annotations are added to theorems that correspond to those already
annotated in the List implementation, ensuring consistency across all
three container types (List, Array, Vector) for indexed mapping
operations.
Key theorems annotated include:
- Size and element access theorems (`size_mapIdx`, `getElem_mapIdx`,
`getElem?_mapIdx`)
- Construction theorems (`mapIdx_empty`, `mapIdx_push`, `mapIdx_append`)
- Membership and equality theorems (`mem_mapIdx`, `mapIdx_mapIdx`)
- Conversion theorems (`toList_mapIdx`, `mapIdx_toArray`, etc.)
- Reverse and composition operations
- Similar annotations for `mapFinIdx` variants
This PR adds an equivalence relation to tree maps akin to the existing
one for hash maps. In order to get many congruence lemmas to eventually
use for defining functions on extensional tree maps, almost all of the
remaining tree map functions have also been given lemmas to relate them
to list functions, although these aren't currently used to prove lemmas
other than congruence lemmas.
This PR implements support for inequalities in the `grind` linear
arithmetic procedure and simplifies its design. Some examples that can
already be solved:
```lean
open Lean.Grind
example [IntModule α] [Preorder α] [IntModule.IsOrdered α] (a b c d : α)
: a + d < c → b = a + (2:Int)*d → b - d > c → False := by
grind
example [CommRing α] [LinearOrder α] [Ring.IsOrdered α] (a b : α)
: a = 0 → b = 1 → a + b ≤ 2 := by
grind
example [CommRing α] [Preorder α] [Ring.IsOrdered α] (a b c d e : α) :
2*a + b ≥ 1 → b ≥ 0 → c ≥ 0 → d ≥ 0 → e ≥ 0
→ a ≥ 3*c → c ≥ 6*e → d - e*5 ≥ 0
→ a + b + 3*c + d + 2*e < 0 → False := by
grind
```
This PR makes `unsafeBaseIO` `noinline`. The new compiler is better at
optimizing `Result`-like types, which can cause the final operation in
an `unsafeBaseIO` block to be dropped, since `unsafeBaseIO` is
discarding the state.
This PR implements the main framework of the model search procedure for
the linarith component in grind. It currently handles only inequalities.
It can already solve simple goals such as
```lean
example [IntModule α] [Preorder α] [IntModule.IsOrdered α] (a b c : α)
: a < b → b < c → c < a → False := by
grind
example [IntModule α] [LinearOrder α] [IntModule.IsOrdered α] (a b c : α)
: a < b → b < c + d → a - d < c := by
grind
```
This PR implements the infrastructure for constructing proof terms in
the linarith procedure in `grind`. It also adds the `ToExpr` instances
for the reified objects.
This PR introduces an explicit `defeq` attribute to mark theorems that
can be used by `dsimp`. The benefit of an explicit attribute over the
prior logic of looking at the proof body is that we can reliably omit
theorem bodies across module boundaries. It also helps with intra-file
parallelism.
If a theorem is syntactically defined by `:= rfl`, then the attribute is
assumed and need not given explicitly. This is a purely syntactic check
and can be fooled, e.g. if in the current namespace, `rfl` is not
actually “the” `rfl` of `Eq`. In that case, some other syntax has be
used, such as `:= (rfl)`. This is also the way to go if a theorem can be
proved by `defeq`, but one does not actually want `dsimp` to use this
fact.
The `defeq` attribute will look at the *type* of the declaration, not
the body, to check if it really holds definitionally. Because of
different reduction settings, this can sometimes go wrong. Then one
should also write `:= (rfl)`, if one does not want this to be a defeq
theorem. (If one does then this is currently not possible, but it’s
probably a bad idea anyways).
The `set_option debug.tactic.simp.checkDefEqAttr true`, `dsimp` will
warn if could not apply a lemma due to a missing `defeq` attribute.
With `set_option backward.dsimp.useDefEqAttr.get false` one can revert
to the old behavior of inferring rfl-ness based on the theorem body.
Both options will go away eventually (too bad we can’t mark them as
deprecated right away, see #7969)
Meta programs that generate theorems (e.g. equational theorems) can use
`inferDefEqAttr` to set the attribute based on the theorem body of the
just created declaration.
This builds on #8501 to update Init to `@[expose]` a fair amount of
definitions that, if not exposed, would prevent some existing `:= rfl`
theorems from being `defeq` theorems. In the interest of starting
backwards compatible, I exposed these function. Hopefully many can be
un-exposed later again.
A mathlib adaption branch exists that includes both the meta programming
fixes and changes to the theorems (e.g. changing `:= by rfl` to `:=
rfl`).
With the module system there is now no special handling for `defeq`
theorem bodies, because we don’t look at the body anymore. The previous
hack is removed. The `defeq`-ness of the theorem needs to be checked in
the context of the theorem’s *type*; the error message contains a hint
if the defeq check fails because of the exported context.
This PR adds many helper theorems for the future `IntModule` linear
arithmetic procedure in `grind`.
It also adds helper theorems for normalizing input atoms and support for
disequality in the new linear arithmetic procedure in `grind`.
This PR completes the `ToInt` family of typeclasses which `grind` will
use to embed types into the integers for `cutsat`. It contains instances
for the usual concrete data types (`Fin`, `UIntX`, `IntX`, `BitVec`),
and is extensible (e.g. for Mathlib's `PNat`).
This PR adds a simp lemma that simplifies T-division where the numerator
is a `Nat` into an E-division:
```lean
@[simp] theorem ofNat_tdiv_eq_ediv {a : Nat} {b : Int} : (a : Int).tdiv b = a / b :=
tdiv_eq_ediv_of_nonneg (by simp)
```
---------
Co-authored-by: Tobias Grosser <tobias@grosser.es>
This PR adds trichotomy lemmas for unsigned and signed comparisons,
stating that only one of three cases may happen: either `x < y`, `x =
y`, or `x > y` (for both signed and unsigned comparsions). We use
explicit arguments so that users can write `rcases slt_trichotomy x y
with hlt | heq | hgt`.
