Documentation

Mathlib.Tactic.Ring.RingNF

ring_nf tactic #

A tactic which uses ring to rewrite expressions. This can be used non-terminally to normalize ring expressions in the goal such as ⊢ P (x + x + x) ~> ⊢ P (x * 3), as well as being able to prove some equations that ring cannot because they involve ring reasoning inside a subterm, such as sin (x + y) + sin (y + x) = 2 * sin (x + y).

def Mathlib.Tactic.Ring.ExBase.isAtom :
{a : Lean.Level} → {arg : Q(Type a)} → { : Q(CommSemiring «$arg»)} → {a_1 : Q(«$arg»)} → Mathlib.Tactic.Ring.ExBase a_1Bool

True if this represents an atomic expression.

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    def Mathlib.Tactic.Ring.ExProd.isAtom :
    {a : Lean.Level} → {arg : Q(Type a)} → { : Q(CommSemiring «$arg»)} → {a_1 : Q(«$arg»)} → Mathlib.Tactic.Ring.ExProd a_1Bool

    True if this represents an atomic expression.

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      def Mathlib.Tactic.Ring.ExSum.isAtom :
      {a : Lean.Level} → {arg : Q(Type a)} → { : Q(CommSemiring «$arg»)} → {a_1 : Q(«$arg»)} → Mathlib.Tactic.Ring.ExSum a_1Bool

      True if this represents an atomic expression.

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        The normalization style for ring_nf.

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          Configuration for ring_nf.

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            The read-only state of the RingNF monad.

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              @[inline, reducible]

              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.

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                A tactic in the RingNF.M monad which will simplify expression parent to a normal form.

                • root: true if this is a direct call to the function. RingNF.M.run sets this to false in recursive mode.
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                  theorem Mathlib.Tactic.RingNF.add_assoc_rev {R : Type u_1} [CommSemiring R] (a : R) (b : R) (c : R) :
                  a + (b + c) = a + b + c
                  theorem Mathlib.Tactic.RingNF.mul_assoc_rev {R : Type u_1} [CommSemiring R] (a : R) (b : R) (c : R) :
                  a * (b * c) = a * b * c
                  theorem Mathlib.Tactic.RingNF.mul_neg {R : Type u_1} [Ring R] (a : R) (b : R) :
                  a * -b = -(a * b)
                  theorem Mathlib.Tactic.RingNF.add_neg {R : Type u_1} [Ring R] (a : R) (b : R) :
                  a + -b = a - b
                  theorem Mathlib.Tactic.RingNF.rat_rawCast_2 {n : } {d : } {R : Type u_1} [DivisionRing R] :
                  Rat.rawCast n d = n / d

                  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
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                    Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

                    • ring_nf! will use a more aggressive reducibility setting to identify atoms.
                    • ring_nf (config := cfg) allows for additional configuration:
                      • red: the reducibility setting (overridden by !)
                      • recursive: if true, ring_nf will also recurse into atoms
                    • ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.
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                      Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

                      • ring_nf! will use a more aggressive reducibility setting to identify atoms.
                      • ring_nf (config := cfg) allows for additional configuration:
                        • red: the reducibility setting (overridden by !)
                        • recursive: if true, ring_nf will also recurse into atoms
                      • ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.
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                        Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

                        • ring_nf! will use a more aggressive reducibility setting to identify atoms.
                        • ring_nf (config := cfg) allows for additional configuration:
                          • red: the reducibility setting (overridden by !)
                          • recursive: if true, ring_nf will also recurse into atoms
                        • ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.
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                          Tactic for solving equations of commutative (semi)rings, allowing variables in the exponent.

                          • This version of ring1 uses ring_nf to simplify in atoms.
                          • The variant ring1_nf! will use a more aggressive reducibility setting to determine equality of atoms.
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                            Tactic for solving equations of commutative (semi)rings, allowing variables in the exponent.

                            • This version of ring1 uses ring_nf to simplify in atoms.
                            • The variant ring1_nf! will use a more aggressive reducibility setting to determine equality of atoms.
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                              Elaborator for the ring_nf tactic.

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                                Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

                                • ring_nf! will use a more aggressive reducibility setting to identify atoms.
                                • ring_nf (config := cfg) allows for additional configuration:
                                  • red: the reducibility setting (overridden by !)
                                  • recursive: if true, ring_nf will also recurse into atoms
                                • ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.
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                                  Tactic for evaluating expressions in commutative (semi)rings, allowing for variables in the exponent.

                                  • ring! will use a more aggressive reducibility setting to determine equality of atoms.
                                  • ring1 fails if the target is not an equality.

                                  For example:

                                  example (n : ℕ) (m : ℤ) : 2^(n+1) * m = 2 * 2^n * m := by ring
                                  example (a b : ℤ) (n : ℕ) : (a + b)^(n + 2) = (a^2 + b^2 + a * b + b * a) * (a + b)^n := by ring
                                  example (x y : ℕ) : x + id y = y + id x := by ring!
                                  
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                                    Tactic for evaluating expressions in commutative (semi)rings, allowing for variables in the exponent.

                                    • ring! will use a more aggressive reducibility setting to determine equality of atoms.
                                    • ring1 fails if the target is not an equality.

                                    For example:

                                    example (n : ℕ) (m : ℤ) : 2^(n+1) * m = 2 * 2^n * m := by ring
                                    example (a b : ℤ) (n : ℕ) : (a + b)^(n + 2) = (a^2 + b^2 + a * b + b * a) * (a + b)^n := by ring
                                    example (x y : ℕ) : x + id y = y + id x := by ring!
                                    
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                                      The tactic ring evaluates expressions in commutative (semi)rings. This is the conv tactic version, which rewrites a target which is a ring equality to True.

                                      See also the ring tactic.

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                                        The tactic ring evaluates expressions in commutative (semi)rings. This is the conv tactic version, which rewrites a target which is a ring equality to True.

                                        See also the ring tactic.

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