f8cdb03352
This PR add constructors `.intCast k` and `.natCast k` to
`CommRing.Expr`. We need them because terms such as `Nat.cast (R := α)
1` and `(1 : α)` are not definitionally equal. This is pervaise in
Mathlib for the numerals `0` and `1`.
```lean
import Mathlib
example {α : Type} [AddMonoidWithOne α] : Nat.cast (R := α) 0 = (0 : α) := rfl -- not defeq
example {α : Type} [AddMonoidWithOne α] : Nat.cast (R := α) 1 = (1 : α) := rfl -- not defeq
example {α : Type} [AddMonoidWithOne α] : Nat.cast (R := α) 2 = (2 : α) := rfl -- defeq from here
-- Similarly for everything past `AddMonoidWithOne` in the Mathlib hierarchy, e.g. `Ring`.
```
109 lines
3.9 KiB
Text
109 lines
3.9 KiB
Text
open Lean Grind
|
||
set_option grind.debug true
|
||
|
||
example [IntModule α] [Preorder α] [OrderedAdd α] (a b : α)
|
||
: a + b = b + a := by
|
||
grind
|
||
|
||
/--
|
||
trace: [grind.debug.proof] Classical.byContradiction fun h =>
|
||
let ctx := RArray.branch 1 (RArray.leaf One.one) (RArray.branch 2 (RArray.leaf a) (RArray.leaf b));
|
||
let rctx := RArray.branch 1 (RArray.leaf a) (RArray.leaf b);
|
||
let re_1 := (CommRing.Expr.var 0).add (CommRing.Expr.var 1);
|
||
let re_2 := (CommRing.Expr.var 1).add (CommRing.Expr.var 0);
|
||
let rp_1 := CommRing.Poly.num 0;
|
||
let e_1 := Expr.zero;
|
||
let e_2 := Expr.intMul 0 (Expr.var 0);
|
||
let p_1 := Poly.nil;
|
||
diseq_unsat ctx
|
||
(diseq_norm ctx e_2 e_1 p_1 (Eq.refl true) (CommRing.diseq_norm rctx re_1 re_2 rp_1 (Eq.refl true) h))
|
||
-/
|
||
#guard_msgs in
|
||
open Linarith in
|
||
set_option trace.grind.debug.proof true in
|
||
example [CommRing α] [Preorder α] [OrderedRing α] (a b : α)
|
||
: a + b = b + a := by
|
||
grind -ring
|
||
|
||
-- Temporary test. TODO: delete after we implement backtracking
|
||
/--
|
||
trace: [grind.debug.linarith.search.assign] One.one := 1
|
||
[grind.debug.linarith.search.assign] b := 1
|
||
[grind.debug.linarith.search.assign] c := 2
|
||
[grind.debug.linarith.search.assign] d := 2
|
||
[grind.debug.linarith.search.assign] d := 3
|
||
[grind.debug.linarith.search.assign] a := 7/2
|
||
-/
|
||
#guard_msgs (drop error, trace) in
|
||
set_option trace.grind.debug.linarith.search.assign true in
|
||
example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c d : α)
|
||
: b ≥ 0 → c > b → d > b → a ≠ b + c → a > b + c → a < b + d → False := by
|
||
grind
|
||
|
||
example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c d : α)
|
||
: a ≤ b → a ≥ c + d → d = 0 → b = c → a = b := by
|
||
grind
|
||
|
||
example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b c d : α)
|
||
: a ≤ b → a ≥ c + d → d ≤ 0 → d ≥ 0 → b = c → a = b := by
|
||
grind
|
||
|
||
open Linarith RArray
|
||
set_option trace.grind.debug.proof true in
|
||
example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b c d : α)
|
||
: a ≤ b → a - c ≥ 0 + d → d ≤ 0 → d ≥ 0 → b = c → a ≠ b → False := by
|
||
grind
|
||
|
||
example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c : α)
|
||
: a + 2*b = 0 → c + b = -b → a = c := by
|
||
grind
|
||
|
||
example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c : α)
|
||
: a + 2*b = 0 → a = c → c + b = -b := by
|
||
grind
|
||
|
||
example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c : α)
|
||
: c = a → a + b ≤ 3 → 3 ≤ b + c → a + b = 3 := by
|
||
grind
|
||
|
||
/--
|
||
trace: [grind.linarith.model] a := 7/2
|
||
[grind.linarith.model] b := 1
|
||
[grind.linarith.model] c := 2
|
||
[grind.linarith.model] d := 3
|
||
-/
|
||
#guard_msgs (drop error, trace) in
|
||
set_option trace.grind.linarith.model true in
|
||
example [CommRing α] [LinearOrder α] [OrderedRing α] (a b c d : α)
|
||
: b ≥ 0 → c > b → d > b → a ≠ b + c → a > b + c → a < b + d → False := by
|
||
grind
|
||
|
||
/--
|
||
trace: [grind.linarith.model] a := 0
|
||
[grind.linarith.model] b := 1
|
||
[grind.linarith.model] c := 1
|
||
[grind.linarith.model] d := -1
|
||
-/
|
||
#guard_msgs (drop error, trace) in
|
||
set_option trace.grind.linarith.model true in
|
||
example [IntModule α] [LinearOrder α] [OrderedAdd α] (a b c d : α)
|
||
: a ≤ b → a - c ≥ 0 + d → d ≤ 0 → b = c → a ≠ b → False := by
|
||
grind
|
||
|
||
/-- trace: [grind.split] x = 0, generation: 0 -/
|
||
#guard_msgs (trace) in
|
||
set_option trace.grind.split true in
|
||
example [IntModule α] [LinearOrder α] [OrderedAdd α] (f : α → α) (x : α)
|
||
: Zero.zero ≤ x → x ≤ 0 → f x = a → f 0 = a := by
|
||
grind
|
||
|
||
-- should not split on `x = 0` since `α` is not a linear order
|
||
#guard_msgs (drop error, trace) in
|
||
set_option trace.grind.split true in
|
||
example [IntModule α] [Preorder α] [OrderedAdd α] (f : α → α) (x : α)
|
||
: Zero.zero ≤ x → x ≤ 0 → f x = a → f 0 = a := by
|
||
grind
|
||
|
||
example [CommRing α] [LinearOrder α] [OrderedRing α] (f : α → α → α) (x y z : α)
|
||
: z ≤ x → x ≤ 1 → z = 1 → f x y = 2 → f 1 y = 2 := by
|
||
grind
|