`norm_num` extensions for GCD-adjacent functions #

This module defines some norm_num extensions for functions such as Nat.gcd, Nat.lcm, Int.gcd, and Int.lcm.

Note that Nat.coprime is reducible and defined in terms of Nat.gcd, so the Nat.gcd extension also indirectly provides a Nat.coprime extension.

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theorem Tactic.NormNum.int_gcd_helper' {d : ℕ} {x : ℤ} {y : ℤ} (a : ℤ) (b : ℤ) (h₁ : ↑d ∣ x) (h₂ : ↑d ∣ y) (h₃ : x * a + y * b = ↑d) :

x.gcd y = d

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theorem Tactic.NormNum.nat_gcd_helper_dvd_left (x : ℕ) (y : ℕ) (h : y % x = 0) :

x.gcd y = x

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theorem Tactic.NormNum.nat_gcd_helper_dvd_right (x : ℕ) (y : ℕ) (h : x % y = 0) :

x.gcd y = y

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theorem Tactic.NormNum.nat_gcd_helper_2 (d : ℕ) (x : ℕ) (y : ℕ) (a : ℕ) (b : ℕ) (hu : x % d = 0) (hv : y % d = 0) (h : x * a = y * b + d) :

x.gcd y = d

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theorem Tactic.NormNum.nat_gcd_helper_1 (d : ℕ) (x : ℕ) (y : ℕ) (a : ℕ) (b : ℕ) (hu : x % d = 0) (hv : y % d = 0) (h : y * b = x * a + d) :

x.gcd y = d

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theorem Tactic.NormNum.nat_gcd_helper_1' (x : ℕ) (y : ℕ) (a : ℕ) (b : ℕ) (h : y * b = x * a + 1) :

x.gcd y = 1

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theorem Tactic.NormNum.nat_gcd_helper_2' (x : ℕ) (y : ℕ) (a : ℕ) (b : ℕ) (h : x * a = y * b + 1) :

x.gcd y = 1

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theorem Tactic.NormNum.nat_lcm_helper (x : ℕ) (y : ℕ) (d : ℕ) (m : ℕ) (hd : x.gcd y = d) (d0 : d.beq 0 = false) (dm : x * y = d * m) :

x.lcm y = m

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theorem Tactic.NormNum.int_gcd_helper {x : ℤ} {y : ℤ} {x' : ℕ} {y' : ℕ} {d : ℕ} (hx : x.natAbs = x') (hy : y.natAbs = y') (h : x'.gcd y' = d) :

x.gcd y = d

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theorem Tactic.NormNum.int_lcm_helper {x : ℤ} {y : ℤ} {x' : ℕ} {y' : ℕ} {d : ℕ} (hx : x.natAbs = x') (hy : y.natAbs = y') (h : x'.lcm y' = d) :

x.lcm y = d

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theorem Tactic.NormNum.isNat_gcd {x : ℕ} {y : ℕ} {nx : ℕ} {ny : ℕ} {z : ℕ} :

Mathlib.Meta.NormNum.IsNat x nx → Mathlib.Meta.NormNum.IsNat y ny → nx.gcd ny = z → Mathlib.Meta.NormNum.IsNat (x.gcd y) z

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theorem Tactic.NormNum.isNat_lcm {x : ℕ} {y : ℕ} {nx : ℕ} {ny : ℕ} {z : ℕ} :

Mathlib.Meta.NormNum.IsNat x nx → Mathlib.Meta.NormNum.IsNat y ny → nx.lcm ny = z → Mathlib.Meta.NormNum.IsNat (x.lcm y) z

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theorem Tactic.NormNum.isInt_gcd {x : ℤ} {y : ℤ} {nx : ℤ} {ny : ℤ} {z : ℕ} :

Mathlib.Meta.NormNum.IsInt x nx → Mathlib.Meta.NormNum.IsInt y ny → nx.gcd ny = z → Mathlib.Meta.NormNum.IsNat (x.gcd y) z

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theorem Tactic.NormNum.isInt_lcm {x : ℤ} {y : ℤ} {nx : ℤ} {ny : ℤ} {z : ℕ} :

Mathlib.Meta.NormNum.IsInt x nx → Mathlib.Meta.NormNum.IsInt y ny → nx.lcm ny = z → Mathlib.Meta.NormNum.IsNat (x.lcm y) z

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def Tactic.NormNum.proveNatGCD (ex : Q(ℕ)) (ey : Q(ℕ)) :

(ed : Q(ℕ)) × Q(«$ex».gcd «$ey» = «$ed»)

Given natural number literals ex and ey, return their GCD as a natural number literal and an equality proof. Panics if ex or ey aren't natural number literals.

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def Tactic.NormNum.evalNatGCD :

Mathlib.Meta.NormNum.NormNumExt

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def Tactic.NormNum.proveNatLCM (ex : Q(ℕ)) (ey : Q(ℕ)) :

(ed : Q(ℕ)) × Q(«$ex».lcm «$ey» = «$ed»)

Given natural number literals ex and ey, return their LCM as a natural number literal and an equality proof. Panics if ex or ey aren't natural number literals.

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