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...
Exact Lean statement
theorem AssocImpliesSgrProjFaithful {α : Type _} {G : Type _} [Magma G]
(assoc : Equation4512 G) (f : α → G) (t : FreeMagma α) :
evalInMagma f t = evalInSgr f (freeMagmaToFreeSgr t)Formal artifact
Lean source
theorem AssocImpliesSgrProjFaithful {α : Type _} {G : Type _} [Magma G] (assoc : Equation4512 G) (f : α → G) (t : FreeMagma α) : evalInMagma f t = evalInSgr f (freeMagmaToFreeSgr t) := match t with | Lf a => rfl | Lf a ⋆ tr => congrArg (f a ◇ ·) (AssocImpliesSgrProjFaithful assoc f tr) | (tl1 ⋆ tl2) ⋆ tr => by symm trans evalInSgr f (freeMagmaToFreeSgr.toFun (tl1 ⋆ tl2) ◇ freeMagmaToFreeSgr.toFun tr) · exact congrArg _ $ freeMagmaToFreeSgr.map_op' _ tr trans evalInSgr f ((freeMagmaToFreeSgr.toFun tl1 ◇ freeMagmaToFreeSgr.toFun tl2) ◇ freeMagmaToFreeSgr.toFun tr) · exact congrArg (fun s ↦ evalInSgr f (s ◇ tr ↠Sgr)) $ freeMagmaToFreeSgr.map_op' tl1 tl2 trans (evalInSgrHom assoc f).toFun (freeMagmaToFreeSgr.toFun tl1 ◇ freeMagmaToFreeSgr.toFun tl2) ◇ (evalInSgrHom assoc f).toFun (freeMagmaToFreeSgr.toFun tr) · apply (evalInSgrHom assoc f).map_op' trans ((evalInSgrHom assoc f).toFun (freeMagmaToFreeSgr.toFun tl1) ◇ (evalInSgrHom assoc f).toFun (freeMagmaToFreeSgr.toFun tl2)) ◇ evalInSgr f (freeMagmaToFreeSgr.toFun tr) · exact congrArg (· ◇ _) $ (evalInSgrHom assoc f).map_op' (tl1 ↠Sgr) (tl2 ↠Sgr) symm trans (tl1 ⋆ tl2 ⬝ f) ◇ evalInSgr f (freeMagmaToFreeSgr tr) · exact congrArg _ (AssocImpliesSgrProjFaithful assoc f tr) trans (evalInSgr f (freeMagmaToFreeSgr tl1) ◇ (tl2 ⬝ f)) ◇ evalInSgr f (freeMagmaToFreeSgr tr) · exact congrArg (fun s ↦ (s ◇ _) ◇ _) (AssocImpliesSgrProjFaithful assoc f tl1) · exact congrArg (fun s ↦ (_ ◇ s) ◇ _) (AssocImpliesSgrProjFaithful assoc f tl2)- Project
- Equational Theories
- License
- Apache-2.0
- Commit
- 7e276a2d05e8
- Source
- equational_theories/FreeSemigroup.lean:107-133
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
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.
Equation374794 not implies Equation2
InfModel.Equation374794_not_implies_Equation2
Plain-language statement
However, Equation374794 doesn't imply Equation2.
Source project: Equational Theories
Person-level attribution pending.