Finite Equation374794 implies Equation2
InfModel.Finite.Equation374794_implies_Equation2
Plain-language statement
In a finite model Equation374794 implies Equation2, that the model is a subsingleton.
Exact Lean statement
@[equational_result]
theorem Finite.Equation374794_implies_Equation2 (G : Type*) [Magma G] [Finite G] (h : Equation374794 G) :
Equation2 GFormal artifact
Lean source
@[equational_result]theorem Finite.Equation374794_implies_Equation2 (G : Type*) [Magma G] [Finite G] (h : Equation374794 G) : Equation2 G := by have : ∀ (y z u : G), (y ◇ y) ◇ z = (y ◇ y) ◇ u := by intro y let f (x : G) := ((y ◇ y) ◇ y) ◇ x let g (x : G) := x ◇ ((y ◇ y) ◇ y) have : Function.RightInverse f g := fun x ↦ by simp [f, g, ← h] refine fun z u ↦ this.injective ?_ obtain ⟨finv, hf⟩ := (Finite.surjective_of_injective this.injective).hasRightInverse let fy := finv ((y ◇ y) ◇ y) replace hf : ((y ◇ y) ◇ y) ◇ fy = (y ◇ y) ◇ y := hf _ have := h fy y simp only [hf] at this simp [f, ← this] intro x u have y := x have z := x rw [h x y z, this y y (y ◇ y), this (y ◇ y) x u, ← this y y (y ◇ y), ← h]- Project
- Equational Theories
- License
- Apache-2.0
- Commit
- 7e276a2d05e8
- Source
- equational_theories/InfModel.lean:16-34
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Assoc Implies Sgr Proj Faithful
AssocImpliesSgrProjFaithful
Project documentation
Example usage of AssocFullyRightAssociate -/ theorem Assoc4 {G : Type _} [Magma G] (assoc : Equation4512 G) : ∀ x y z w : G, ((x ◇ y) ◇ z) ◇ w = x ◇ (y ◇ (z ◇ w)) := fun x y z w ↦ AssocFullyRightAssociate assoc (fun | 0 => x | 1 => y | 2 => z | 3 => w : Fin 4 → G) (((Lf 0 ⋆ Lf 1) ⋆ Lf 2) ⋆ Lf 3) inductive FreeSemigroup (α : Type _) | Singleton : α → FreeS...
Source project: Equational Theories
Person-level attribution pending.
Equation1323 not implies Equation2744
Eq1323.Equation1323_not_implies_Equation2744
Plain-language statement
Source project: Equational Theories
Person-level attribution pending.
Eq255 equiv Lx Rx
Eq677.eq255_equiv_LxRx
Plain-language statement
Blueprint Lemma 13.2(v). E255 at x ↔ L_x ∘ R_x has a fixed point.
Source project: Equational Theories
Person-level attribution pending.