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Project-declaredLean 4.29.1 · mathlib@5e932f97

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 G

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
@[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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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...

universal algebraequational logiccombinatorics

Source project: Equational Theories

Person-level attribution pending.

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Project-declaredLean 4.29.1

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.

universal algebraequational logiccombinatorics

Source project: Equational Theories

Person-level attribution pending.

View proof record