RPowSimp.lean`

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40 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Mathlib.Analysis.SpecialFunctions.Pow.Real

Imported by 2 files

structure Mathlib.Tactic.RPowRing.Context The read-only state of the RPowRing monad line 15 source
Source documentation
The read-only state of the RPowRing monad.

standard axioms: Classical.choice, Quot.sound, propext what this means
```
structure Context where
…
```
structure Mathlib.Tactic.RPowRing.Config Configuration for ring_nf line 24 source
Source documentation
Configuration for ring_nf.

no axioms what this means
```
structure Config where
…
```
abbrev Mathlib.Tactic.RPowRing.M Documented declaration: M line 37 source
Source documentation
The monad for RingNF contains, in addition to the AtomM state, a simp context for the main traversal and a simp function (which has another simp context) to simplify normalized polynomials.

standard axioms: Classical.choice, Quot.sound, propext what this means
Proof uses: ReaderT, Mathlib.Tactic.RPowRing.Context, Mathlib.Tactic.AtomM
```
abbrev M := ReaderT Context AtomM
```
Mentions: Mathlib.Tactic.RPowRing.Context

inductive Mathlib.Tactic.RPowRing.ExBase A monomial, which is a product of powers of ExBase expressions, terminated by a (nonzero) constant coefficient line 43 source

Source documentation

A monomial, which is a product of powers of ExBase expressions, terminated by a (nonzero) constant coefficient.

no axioms what this means

Statement uses: Qq.Quoted, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level

inductive ExBase : (e : Q(ℝ)) → Type
  /-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast.
  If `value` is not an integer, then `hyp` should be a proof of `(value.den : α) ≠ 0`. -/
  | atom (id : ℕ) (x : Q(ℝ)) : ExBase x
  /-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase`
  and `e` is an `ExBase` representing a monomial expression in `ℕ` (it is a monomial instead of
  a polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/
  | pow (id : ℕ) (x : Q(ℝ)) (h : Q(0 < $x)) (e : Q(ℝ)) : ExBase q($x ^ $e)

def Mathlib.Tactic.RPowRing.ExBase.id line 52 source
no axioms what this means
Statement uses: Qq.Quoted, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level, Mathlib.Tactic.RPowRing.ExBase, Nat
Proof uses: Lean.Expr.app, Lean.Name.mkStr2, List.cons, Lean.Level.zero, Lean.Expr.lit, Lean.Literal.natVal
1 instances, coercions and auxiliary constants left out.

inductive Mathlib.Tactic.RPowRing.ExProd A monomial, which is a product of powers of ExBase expressions, terminated by a (nonzero) constant coefficient line 59 source

Source documentation

A monomial, which is a product of powers of ExBase expressions, terminated by a (nonzero) constant coefficient.

no axioms what this means

Statement uses: Qq.Quoted, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level

inductive ExProd : (e : Q(ℝ)) → Type
  /-- A coefficient `value`, which must not be `0`. `e` is a raw rat cast.
  If `value` is not an integer, then `hyp` should be a proof of `(value.den : α) ≠ 0`. -/
  | one : ExProd q(1)
  /-- A product `x ^ e * b` is a monomial if `b` is a monomial. Here `x` is an `ExBase`
  and `e` is an `ExProd` representing a monomial expression in `ℕ` (it is a monomial instead of
  a polynomial because we eagerly normalize `x ^ (a + b) = x ^ a * x ^ b`.) -/
  | mul {x b : Q(ℝ)} : ExBase x → ExProd b → ExProd q($x * $b)

Mentions: Mathlib.Tactic.RPowRing.ExBase

instance Mathlib.Tactic.RPowRing.(instance at line 68) line 68 source
instance Mathlib.Tactic.RPowRing.(instance at line 69) line 69 source
def Mathlib.Tactic.RPowRing.ExBase.toProd Embed an exponent (an ExBase, ExProd pair) as an ExProd by multiplying by 1 line 72 source
Source documentation
Embed an exponent (an ExBase, ExProd pair) as an ExProd by multiplying by 1.

no axioms what this means
Statement uses: Qq.Quoted, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level, Mathlib.Tactic.RPowRing.ExBase, Mathlib.Tactic.RPowRing.ExProd, Qq.Quoted.unsafeMk, Lean.Expr.app, Lean.Name.mkStr2, List.cons, Lean.Level.zero, Lean.Expr.lit, Lean.Literal.natVal
Proof uses: Mathlib.Tactic.RPowRing.ExProd.mul, Mathlib.Tactic.RPowRing.ExProd.one
```
def ExBase.toProd {a : Q(ℝ)} (va : ExBase a) : ExProd q($a * 1) := .mul va .one
```
Mentions: Mathlib.Tactic.RPowRing.ExBase, Mathlib.Tactic.RPowRing.ExProd
abbrev Mathlib.Tactic.RPowRing.Result line 74 source
no axioms what this means
Statement uses: Qq.Quoted, Qq.Quoted.unsafeMk, Lean.Expr.sort, Lean.Level.succ, Lean.Level.zero, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level
Proof uses: Mathlib.Tactic.Ring.Common.Result
instance Mathlib.Tactic.RPowRing.(instance at line 76) line 76 source
theorem Mathlib.Tactic.RPowRing.atom_pf line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HMul.hMul, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrArg, mul_one, eq_self
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Mathlib.Tactic.RPowRing.atom_pf' line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HMul.hMul, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congr, congrArg, congrFun', mul_one, eq_self
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Mathlib.Tactic.RPowRing.atom_pow_pf line 81 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HMul.hMul, HPow.hPow, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrArg, congrFun', Real.rpow_one, mul_one, eq_self
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Mathlib.Tactic.RPowRing.atom_pow_pf' line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HMul.hMul, HPow.hPow, Nat, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congr, congrArg, congrFun', pow_one, mul_one, eq_self
21 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def Mathlib.Tactic.RPowRing.evalAtom Evaluates an atom, an expression where ring can find no additional structure line 89 source
Source documentation
Evaluates an atom, an expression where ring can find no additional structure.
- a = a ^ 1 * 1 + 0
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Qq.Quoted, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level, Mathlib.Tactic.AtomM, Mathlib.Tactic.RPowRing.Result, Mathlib.Tactic.RPowRing.ExProd
Proof uses: Bind.bind, Mathlib.Tactic.AtomM.Context, StateRefT', IO.RealWorld, Mathlib.Tactic.AtomM.State, Lean.Meta.MetaM, MonadReader.read, Lean.Meta.Simp.Result, liftM, ReaderT, Lean.Meta.Context, Lean.Meta.State, Lean.Core.CoreM, Mathlib.Tactic.AtomM.Context.evalAtom, Lean.Meta.Simp.Result.expr, Prod, Nat, Lean.Expr, Mathlib.Tactic.AtomM.addAtom, Mathlib.Meta.Positivity.Strictness and 20 more
14 instances, coercions and auxiliary constants left out.
```
def evalAtom (e : Q(ℝ)) : AtomM (Result ExProd e) := do
…
```
Mentions: Mathlib.Tactic.RPowRing.Result, Mathlib.Tactic.RPowRing.ExProd
theorem Mathlib.Tactic.RPowRing.mul_pf_left line 102 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HMul.hMul
Proof uses: Eq.mpr, id, congrArg, mul_assoc, Eq.refl
7 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Mathlib.Tactic.RPowRing.mul_pf_right line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HMul.hMul
Proof uses: Eq.mpr, id, congrArg, mul_left_comm, Eq.refl
8 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Mathlib.Tactic.RPowRing.mul_pp_pf_overlap line 108 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Eq, HMul.hMul, HPow.hPow, HAdd.hAdd
Proof uses: Eq.mpr, id, congrArg, Real.rpow_add, mul_mul_mul_comm, Eq.refl
22 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def Mathlib.Tactic.RPowRing.evalMul Multiplies two monomials va, vb together to get a normalized result monomial line 122 source
Source documentation
Multiplies two monomials va, vb together to get a normalized result monomial.
- x * y = (x * y) (for x, y coefficients)
- x * (b₁ * b₂) = b₁ * (b₂ * x) (for x coefficient)
- (a₁ * a₂) * y = a₁ * (a₂ * y) (for y coefficient)
- (x ^ ea * a₂) * (x ^ eb * b₂) = x ^ (ea + eb) * (a₂ * b₂) (if ea and eb are identical except coefficient)
- (a₁ * a₂) * (b₁ * b₂) = a₁ * (a₂ * (b₁ * b₂)) (if a₁.lt b₁)
- (a₁ * a₂) * (b₁ * b₂) = b₁ * ((a₁ * a₂) * b₂) (if not a₁.lt b₁)
no axioms what this means
Statement uses: Qq.Quoted, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level, Mathlib.Tactic.RPowRing.ExProd, Mathlib.Tactic.RPowRing.Result, Qq.Quoted.unsafeMk, Lean.Expr.sort, Lean.Level.succ, Lean.Level.zero, Lean.Expr.app, Lean.Name.mkStr2, List.cons
Proof uses: Inhabited.default
1 instances, coercions and auxiliary constants left out.
```
partial def evalMul {a b : Q(ℝ)} (va : ExProd a) (vb : ExProd b) : Result ExProd q($a * $b) :=
…
```
Mentions: Mathlib.Tactic.RPowRing.ExProd, Mathlib.Tactic.RPowRing.Result
theorem Mathlib.Tactic.RPowRing.pow_pos line 146 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LT.lt, OfNat.ofNat, HMul.hMul, HPow.hPow
Proof uses: mul_pos, Real.rpow_pos_of_pos
18 instances, coercions and auxiliary constants left out.
theorem Mathlib.Tactic.RPowRing.pow_pf line 149 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Eq, HPow.hPow, HMul.hMul
Proof uses: Eq.mpr, id, congrArg, Real.mul_rpow, LT.lt.le, Real.rpow_pos_of_pos, Real.rpow_mul, Eq.refl
16 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def Mathlib.Tactic.RPowRing.evalPow line 153 source
no axioms what this means
Statement uses: Qq.Quoted, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level, Mathlib.Tactic.RPowRing.ExProd, Option, Prod, Lean.Expr.app, Lean.Name.mkStr2, List.cons, Lean.Level.zero, Lean.Expr.lit, Lean.Literal.natVal, Mathlib.Tactic.RPowRing.Result, Qq.Quoted.unsafeMk, Lean.Expr.sort, Lean.Level.succ
2 instances, coercions and auxiliary constants left out.
theorem Mathlib.Tactic.RPowRing.pow_congr line 163 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HPow.hPow
Proof uses: Eq.refl, Eq.symm
5 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Mathlib.Tactic.RPowRing.inv_congr line 165 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Eq, HPow.hPow, Neg.neg, OfNat.ofNat, Inv.inv
Proof uses: of_eq_true, Eq.trans, True, congrArg, Real.rpow_neg_one, eq_self, Eq.symm
13 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Mathlib.Tactic.RPowRing.npow_congr line 168 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, Eq, HPow.hPow, NPow.npow
Proof uses: of_eq_true, Eq.trans, True, congrArg, Real.rpow_natCast, eq_self, Eq.symm
14 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
def Mathlib.Tactic.RPowRing.eval line 171 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Qq.Quoted, Lean.Expr.const, Lean.Name.mkStr1, List.nil, Lean.Level, Mathlib.Tactic.AtomM, Mathlib.Tactic.RPowRing.Result, Mathlib.Tactic.RPowRing.ExProd
Proof uses: Inhabited.default, Mathlib.Tactic.AtomM.Context, StateRefT', IO.RealWorld, Mathlib.Tactic.AtomM.State, Lean.Meta.MetaM
5 instances, coercions and auxiliary constants left out.
def Mathlib.Tactic.RPowRing.rewrite line 200 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Lean.Expr, optParam, Bool, Bool.true, Mathlib.Tactic.RPowRing.M, Lean.Meta.Simp.Result
Proof uses: Mathlib.Tactic.RPowRing.Context, Mathlib.Tactic.AtomM.Context, ST.Ref, IO.RealWorld, Mathlib.Tactic.AtomM.State, Lean.Meta.Simp.Simproc, Bind.bind, Lean.Meta.Simp.SimpM, Lean.Meta.Simp.MethodsRef, ReaderT, Lean.Meta.Simp.Context, StateRefT', Lean.Meta.Simp.State, Lean.Meta.MetaM, Except, Lean.Meta.Simp.Step, MonadExcept.tryCatch, Lean.Exception, Lean.Meta.Context, Lean.Meta.State and 71 more
43 instances, coercions and auxiliary constants left out.
def Mathlib.Tactic.RPowRing.M.run Runs a tactic in the RingNF.M monad, given initial data: * s: a reference to the mutable state of ring, for persisting across calls line 224 source
Source documentation
Runs a tactic in the RingNF.M monad, given initial data:
- s: a reference to the mutable state of ring, for persisting across calls. This ensures that atom ordering is used consistently.
- cfg: the configuration options
- x: the tactic to run
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IO.Ref, Mathlib.Tactic.AtomM.State, Mathlib.Tactic.RPowRing.Config, Mathlib.Tactic.RPowRing.M, Lean.Meta.MetaM
Proof uses: Bind.bind, Lean.Meta.SimpTheorems, List.foldlM, Lean.Name, Lean.Meta.SimpTheorems.addConst, Bool.true, Bool.false, OfNat.ofNat, Nat, Lean.Meta.SimpTheorems.mk, Lean.Meta.DiscrTree.empty, Lean.Meta.SimpTheorem, EmptyCollection.emptyCollection, Lean.PHashSet, Lean.Meta.Origin, Lean.PersistentHashMap.mk, Array, Lean.PersistentHashMap.Node.entries, Lean.PersistentHashMap.mkEmptyEntriesArray, Lean.Elab.Tactic.simpOnlyBuiltins and 87 more
18 instances, coercions and auxiliary constants left out.
```
def M.run {α : Type} (s : IO.Ref AtomM.State) (cfg : RPowRing.Config) (x : M α) : MetaM α := do
…
```
Mentions: Mathlib.Tactic.RPowRing.Config, Mathlib.Tactic.RPowRing.M
def Mathlib.Tactic.RPowRing.rpowRingTarget Use rpow_ring to rewrite the main goal line 241 source
Source documentation
Use rpow_ring to rewrite the main goal.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IO.Ref, Mathlib.Tactic.AtomM.State, Mathlib.Tactic.RPowRing.Config, Lean.Elab.Tactic.TacticM, Unit
Proof uses: Lean.Elab.Tactic.withMainContext, Bind.bind, Lean.MVarId, Lean.Elab.Tactic.getMainGoal, Lean.Expr, liftM, ReaderT, Lean.Meta.Context, StateRefT', IO.RealWorld, Lean.Meta.State, Lean.Core.CoreM, Lean.Elab.Tactic.Context, Lean.Elab.Tactic.State, Lean.Elab.Term.TermElabM, Lean.Elab.Term.State, Lean.Meta.MetaM, Lean.Elab.Term.Context, Lean.MVarId.getType, Lean.instantiateMVars and 18 more
9 instances, coercions and auxiliary constants left out.
```
def rpowRingTarget (s : IO.Ref AtomM.State) (cfg : Config) : TacticM Unit := withMainContext do
…
```
Mentions: Mathlib.Tactic.RPowRing.Config
def Mathlib.Tactic.RPowRing.rpowRingLocalDecl Use rpow_ring to rewrite hypothesis h line 252 source
Source documentation
Use rpow_ring to rewrite hypothesis h.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IO.Ref, Mathlib.Tactic.AtomM.State, Mathlib.Tactic.RPowRing.Config, Lean.FVarId, Lean.Elab.Tactic.TacticM, Unit
Proof uses: Lean.Elab.Tactic.withMainContext, Bind.bind, Lean.Expr, liftM, ReaderT, Lean.Meta.Context, StateRefT', IO.RealWorld, Lean.Meta.State, Lean.Core.CoreM, Lean.Elab.Tactic.Context, Lean.Elab.Tactic.State, Lean.Elab.Term.TermElabM, Lean.Elab.Term.State, Lean.Meta.MetaM, Lean.Elab.Term.Context, Lean.FVarId.getType, Lean.instantiateMVars, Lean.MVarId, Lean.Elab.Tactic.getMainGoal and 11 more
9 instances, coercions and auxiliary constants left out.
```
def rpowRingLocalDecl (s : IO.Ref AtomM.State) (cfg : Config) (fvarId : FVarId) :
    TacticM Unit := withMainContext do
…
```
Mentions: Mathlib.Tactic.RPowRing.Config
def Mathlib.Tactic.RPowRing.elabRPowRingConv Elaborator for the rpow_ring tactic line 285 source
Source documentation
Elaborator for the rpow_ring tactic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Lean.Elab.Tactic.Tactic
Proof uses: Lean.Syntax, ite, Lean.Elab.Tactic.TacticM, Unit, Eq, Bool, Lean.Syntax.isOfKind, Lean.Name.mkStr4, Bool.true, Lean.Syntax.getArg, OfNat.ofNat, Nat, Option, Lean.TSyntax, List.cons, Lean.SyntaxNodeKind, List.nil, Option.map, Lean.TSyntax.mk, Lean.Syntax.getOptional? and 46 more
16 instances, coercions and auxiliary constants left out.
```
@[tactic rpowRingConv] def elabRPowRingConv : Tactic := fun stx ↦ match stx with
…
```
theorem Mathlib.Tactic.RPowRing._root_.Real.pow_neg line 293 source
theorem Mathlib.Tactic.RPowRing._root_.Real.inv_rpow' line 296 source
theorem Mathlib.Tactic.RPowRing._root_.Real.rpow_inv line 299 source
lemma Mathlib.Tactic.RPowRing.fix_cast₁ line 302 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, Int.ofNat, OfNat.ofNat, Nat
Proof uses: Iff.mpr, Int, Int.cast_eq_one, rfl
20 instances, coercions and auxiliary constants left out.
lemma Mathlib.Tactic.RPowRing.fix_cast₂ line 304 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Real, Int.ofNat
Proof uses: rfl
6 instances, coercions and auxiliary constants left out.
lemma Mathlib.Tactic.RPowRing.fix_cast₃ line 306 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.AtLeastTwo, Eq, Real, OfNat.ofNat
Proof uses: Eq.refl
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
def Mathlib.Tactic.RPowRing.delab_ofNat line 309 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Lean.PrettyPrinter.Delaborator.Delab
Proof uses: Lean.PrettyPrinter.Delaborator.whenPPOption, Lean.getPPNotation, Lean.PrettyPrinter.Delaborator.SubExpr.withNaryArg, Lean.Syntax.Term, Lean.PrettyPrinter.Delaborator.DelabM, Lean.PrettyPrinter.Delaborator.Context, StateRefT', IO.RealWorld, Lean.PrettyPrinter.Delaborator.State, Lean.Meta.MetaM, OfNat.ofNat, Nat, Lean.PrettyPrinter.Delaborator.delab
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
example Mathlib.Tactic.RPowRing.(example at line 323) line 323 source