state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
a✝ : R
⊢ (rename k) (eval₂ C (g ∘ k) (C a✝)) = eval₂ C (⇑(rename k) ∘ g) ((rename k) (C a✝)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
⊢ ∀ (p q : MvPolynomial σ R),
(rename k) (eval₂ C (g ∘ k) p) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p) →
(r... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
p✝ q✝ : MvPolynomial σ R
a✝¹ : (rename k) (eval₂ C (g ∘ k) p✝) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p✝)
a✝ : (renam... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ),
(rename k) (eval₂ C (g ∘ k) p) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p) →
... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
g : τ → MvPolynomial σ R
p✝ : MvPolynomial σ R
n✝ : σ
a✝ : (rename k) (eval₂ C (g ∘ k) p✝) = eval₂ C (⇑(rename k) ∘ g) ((rename k) p✝)
⊢ (rename ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.199_0.3NqVCwOs1E93kvK | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
j : τ
g : σ → MvPolynomial σ R
⊢ (rename (Prod.mk j)) (eval₂ C g p) = eval₂ C (fun x => (rename (Prod.mk j)) (g x)) p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply MvPolynomial.induction_on p | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) := by
| Mathlib.Data.MvPolynomial.Rename.206_0.3NqVCwOs1E93kvK | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
j : τ
g : σ → MvPolynomial σ R
⊢ ∀ (a : R), (rename (Prod.mk j)) (eval₂ C g (C a)) = eval₂ C (fun x => (rename (Prod.mk j)) (g x)) (C a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.206_0.3NqVCwOs1E93kvK | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
j : τ
g : σ → MvPolynomial σ R
a✝ : R
⊢ (rename (Prod.mk j)) (eval₂ C g (C a✝)) = eval₂ C (fun x => (rename (Prod.mk j)) (g x)) (C a✝) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.206_0.3NqVCwOs1E93kvK | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
j : τ
g : σ → MvPolynomial σ R
⊢ ∀ (p q : MvPolynomial σ R),
(rename (Prod.mk j)) (eval₂ C g p) = eval₂ C (fun x => (rename (Prod.mk j)) (g... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.206_0.3NqVCwOs1E93kvK | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
j : τ
g : σ → MvPolynomial σ R
p✝ q✝ : MvPolynomial σ R
a✝¹ : (rename (Prod.mk j)) (eval₂ C g p✝) = eval₂ C (fun x => (rename (Prod.mk j)) (g x... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.206_0.3NqVCwOs1E93kvK | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
j : τ
g : σ → MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ),
(rename (Prod.mk j)) (eval₂ C g p) = eval₂ C (fun x => (rename (Prod.mk j)... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.206_0.3NqVCwOs1E93kvK | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p : MvPolynomial σ R
j : τ
g : σ → MvPolynomial σ R
p✝ : MvPolynomial σ R
n✝ : σ
a✝ : (rename (Prod.mk j)) (eval₂ C g p✝) = eval₂ C (fun x => (rename (Prod.mk j)) (g ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.206_0.3NqVCwOs1E93kvK | theorem rename_prod_mk_eval₂ (j : τ) (g : σ → MvPolynomial σ R) :
rename (Prod.mk j) (p.eval₂ C g) = p.eval₂ C fun x => rename (Prod.mk j) (g x) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p✝ : MvPolynomial σ R
g : σ × τ → S
i : σ
p : MvPolynomial τ R
⊢ eval₂ f g ((rename (Prod.mk i)) p) = eval₂ f (fun j => g (i, j)) p | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply MvPolynomial.induction_on p | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p := by
| Mathlib.Data.MvPolynomial.Rename.213_0.3NqVCwOs1E93kvK | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p✝ : MvPolynomial σ R
g : σ × τ → S
i : σ
p : MvPolynomial τ R
⊢ ∀ (a : R), eval₂ f g ((rename (Prod.mk i)) (C a)) = eval₂ f (fun j => g (i, j)) (C a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.213_0.3NqVCwOs1E93kvK | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p✝ : MvPolynomial σ R
g : σ × τ → S
i : σ
p : MvPolynomial τ R
a✝ : R
⊢ eval₂ f g ((rename (Prod.mk i)) (C a✝)) = eval₂ f (fun j => g (i, j)) (C a✝) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.213_0.3NqVCwOs1E93kvK | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p✝ : MvPolynomial σ R
g : σ × τ → S
i : σ
p : MvPolynomial τ R
⊢ ∀ (p q : MvPolynomial τ R),
eval₂ f g ((rename (Prod.mk i)) p) = eval₂ f (fun j => g (i, j)) p ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.213_0.3NqVCwOs1E93kvK | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p✝¹ : MvPolynomial σ R
g : σ × τ → S
i : σ
p p✝ q✝ : MvPolynomial τ R
a✝¹ : eval₂ f g ((rename (Prod.mk i)) p✝) = eval₂ f (fun j => g (i, j)) p✝
a✝ : eval₂ f g ((re... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.213_0.3NqVCwOs1E93kvK | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p✝ : MvPolynomial σ R
g : σ × τ → S
i : σ
p : MvPolynomial τ R
⊢ ∀ (p : MvPolynomial τ R) (n : τ),
eval₂ f g ((rename (Prod.mk i)) p) = eval₂ f (fun j => g (i, j)... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p := by
apply MvPolynomial.induction_on p <;>
· | Mathlib.Data.MvPolynomial.Rename.213_0.3NqVCwOs1E93kvK | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : R →+* S
k : σ → τ
g✝ : τ → S
p✝¹ : MvPolynomial σ R
g : σ × τ → S
i : σ
p p✝ : MvPolynomial τ R
n✝ : τ
a✝ : eval₂ f g ((rename (Prod.mk i)) p✝) = eval₂ f (fun j => g (i, j)) p✝
⊢ eval₂ f g ((rena... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp [*] | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p := by
apply MvPolynomial.induction_on p <;>
· intros
| Mathlib.Data.MvPolynomial.Rename.213_0.3NqVCwOs1E93kvK | theorem eval₂_rename_prod_mk (g : σ × τ → S) (i : σ) (p : MvPolynomial τ R) :
(rename (Prod.mk i) p).eval₂ f g = eval₂ f (fun j => g (i, j)) p | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p : MvPolynomial σ R
⊢ ∃ s q, p = (rename Subtype.val) q | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q, rfl⟩
refine' ⟨s ∪ t, ⟨_, _⟩⟩
· refine' rename (Subtype.map id _) p + rename (Subtype.map id _) q <;>
simp (config := { contextual := true }) only [id.def, true_or_iff, or_true_iff,
... | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
| Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p : MvPolynomial σ R
⊢ ∃ s q, p = (rename Subtype.val) q | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply induction_on p | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
| Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p : MvPolynomial σ R
⊢ ∀ (a : R), ∃ s q, C a = (rename Subtype.val) q | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intro r | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p : MvPolynomial σ R
r : R
⊢ ∃ s q, C r = (rename Subtype.val) q | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | exact ⟨∅, C r, by rw [rename_C]⟩ | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
| Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p : MvPolynomial σ R
r : R
⊢ C r = (rename Subtype.val) (C r) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [rename_C] | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p : MvPolynomial σ R
⊢ ∀ (p q : MvPolynomial σ R),
(∃ s q, p = (rename Subtype.val) q) →
(∃ s q_1, q = (rename Subtype.val) q_1) → ∃ s q_1, p + q = (rename Subtype.val) q_1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rintro p q ⟨s, p, rfl⟩ ⟨t, q, rfl⟩ | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_add.intro.intro.intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
t : Finset σ
q : MvPolynomial { x // x ∈ t } R
⊢ ∃ s_1 q_1, (rename Subtype.val) p + (rename Subtype.val... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | refine' ⟨s ∪ t, ⟨_, _⟩⟩ | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_add.intro.intro.intro.intro.refine'_1
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
t : Finset σ
q : MvPolynomial { x // x ∈ t } R
⊢ MvPolynomial { x // x ∈ s ∪ t } R | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | refine' rename (Subtype.map id _) p + rename (Subtype.map id _) q | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_add.intro.intro.intro.intro.refine'_1.refine'_1
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
t : Finset σ
q : MvPolynomial { x // x ∈ t } R
⊢ ∀ a ∈ s, id a ∈ s ∪ t | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp (config := { contextual := true }) only [id.def, true_or_iff, or_true_iff,
Finset.mem_union, forall_true_iff] | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_add.intro.intro.intro.intro.refine'_1.refine'_2
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
t : Finset σ
q : MvPolynomial { x // x ∈ t } R
⊢ ∀ a ∈ t, id a ∈ s ∪ t | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp (config := { contextual := true }) only [id.def, true_or_iff, or_true_iff,
Finset.mem_union, forall_true_iff] | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_add.intro.intro.intro.intro.refine'_2
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
t : Finset σ
q : MvPolynomial { x // x ∈ t } R
⊢ (rename Subtype.val) p + (rename Subtype.val)... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [rename_rename, AlgHom.map_add] | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_add.intro.intro.intro.intro.refine'_2
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
t : Finset σ
q : MvPolynomial { x // x ∈ t } R
⊢ (rename Subtype.val) p + (rename Subtype.val)... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rfl | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p : MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ), (∃ s q, p = (rename Subtype.val) q) → ∃ s q, p * X n = (rename Subtype.val) q | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rintro p n ⟨s, p, rfl⟩ | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_X.intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
n : σ
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
⊢ ∃ s_1 q, (rename Subtype.val) p * X n = (rename Subtype.val) q | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | refine' ⟨insert n s, ⟨_, _⟩⟩ | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_X.intro.intro.refine'_1
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
n : σ
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
⊢ MvPolynomial { x // x ∈ insert n s } R | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | refine' rename (Subtype.map id _) p * X ⟨n, s.mem_insert_self n⟩ | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_X.intro.intro.refine'_1
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
n : σ
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
⊢ ∀ a ∈ s, id a ∈ insert n s | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp (config := { contextual := true }) only [id.def, or_true_iff, Finset.mem_insert,
forall_true_iff] | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_X.intro.intro.refine'_2
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
n : σ
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
⊢ (rename Subtype.val) p * X n =
(rename Subtype.val)
((rename (Subtype.map id (_ : ∀ a ∈ s,... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [rename_rename, rename_X, Subtype.coe_mk, AlgHom.map_mul] | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
case h_X.intro.intro.refine'_2
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p✝ : MvPolynomial σ R
n : σ
s : Finset σ
p : MvPolynomial { x // x ∈ s } R
⊢ (rename Subtype.val) p * X n = (rename (Subtype.val ∘ Subtype.map id (_ : ∀ a ∈ s, a ∈ insert n s)))... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rfl | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q := by
classical
apply induction_on p
· intro r
exact ⟨∅, C r, by rw [rename_C]⟩
· rintro p q ⟨s, p, rfl⟩ ⟨t, q,... | Mathlib.Data.MvPolynomial.Rename.227_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_finset_rename (p : MvPolynomial σ R) :
∃ (s : Finset σ) (q : MvPolynomial { x // x ∈ s } R), p = rename (↑) q | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p₁ p₂ : MvPolynomial σ R
⊢ ∃ s q₁ q₂, p₁ = (rename Subtype.val) q₁ ∧ p₂ = (rename Subtype.val) q₂ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | obtain ⟨s₁, q₁, rfl⟩ := exists_finset_rename p₁ | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p₂ : MvPolynomial σ R
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
⊢ ∃ s q₁_1 q₂, (rename Subtype.val) q₁ = (rename Subtype.val) q₁_1 ∧ p₂ = (rename Subtype.val) q₂ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | obtain ⟨s₂, q₂, rfl⟩ := exists_finset_rename p₂ | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case intro.intro.intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ ∃ s q₁_1 q₂_1,
(rename Subtype.val) q₁ = (rename Subtype.val) q₁_1 ∧ (ren... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | classical
use s₁ ∪ s₂
use rename (Set.inclusion <| s₁.subset_union_left s₂) q₁
use rename (Set.inclusion <| s₁.subset_union_right s₂) q₂
constructor -- porting note: was `<;> simp <;> rfl` but Lean couldn't infer the arguments
· -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
... | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case intro.intro.intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ ∃ s q₁_1 q₂_1,
(rename Subtype.val) q₁ = (rename Subtype.val) q₁_1 ∧ (ren... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | use s₁ ∪ s₂ | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case h
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ ∃ q₁_1 q₂_1, (rename Subtype.val) q₁ = (rename Subtype.val) q₁_1 ∧ (rename Subtype.val) q₂ = (renam... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | use rename (Set.inclusion <| s₁.subset_union_left s₂) q₁ | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case h
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ ∃ q₂_1,
(rename Subtype.val) q₁ = (rename Subtype.val) ((rename (inclusion (_ : s₁ ⊆ s₁ ∪ s₂)))... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | use rename (Set.inclusion <| s₁.subset_union_right s₂) q₂ | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case h
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ (rename Subtype.val) q₁ = (rename Subtype.val) ((rename (inclusion (_ : s₁ ⊆ s₁ ∪ s₂))) q₁) ∧
(... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | constructor | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case h.left
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ (rename Subtype.val) q₁ = (rename Subtype.val) ((rename (inclusion (_ : s₁ ⊆ s₁ ∪ s₂))) q₁) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | erw [rename_rename (Set.inclusion <| s₁.subset_union_left s₂)] | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case h.left
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ (rename Subtype.val) q₁ = (rename (Subtype.val ∘ inclusion (_ : s₁ ⊆ s₁ ∪ s₂))) q₁ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rfl | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case h.right
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ (rename Subtype.val) q₂ = (rename Subtype.val) ((rename (inclusion (_ : s₂ ⊆ s₁ ∪ s₂))) q₂) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | erw [rename_rename (Set.inclusion <| s₁.subset_union_right s₂)] | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
case h.right
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s₁ : Finset σ
q₁ : MvPolynomial { x // x ∈ s₁ } R
s₂ : Finset σ
q₂ : MvPolynomial { x // x ∈ s₂ } R
⊢ (rename Subtype.val) q₂ = (rename (Subtype.val ∘ inclusion (_ : s₂ ⊆ s₁ ∪ s₂))) q₂ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rfl | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib.Data.MvPolynomial.Rename.250_0.3NqVCwOs1E93kvK | /-- `exists_finset_rename` for two polynomials at once: for any two polynomials `p₁`, `p₂` in a
polynomial semiring `R[σ]` of possibly infinitely many variables, `exists_finset_rename₂` yields
a finite subset `s` of `σ` such that both `p₁` and `p₂` are contained in the polynomial semiring
`R[s]` of finitely many ... | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
p : MvPolynomial σ R
⊢ ∃ n f, ∃ (_ : Injective f), ∃ q, p = (rename f) q | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | obtain ⟨s, q, rfl⟩ := exists_finset_rename p | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q := by
| Mathlib.Data.MvPolynomial.Rename.271_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s : Finset σ
q : MvPolynomial { x // x ∈ s } R
⊢ ∃ n f, ∃ (_ : Injective f), ∃ q_1, (rename Subtype.val) q = (rename f) q_1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | let n := Fintype.card { x // x ∈ s } | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q := by
obtain ⟨s, q, rfl⟩ := exists_finset_rename p
| Mathlib.Data.MvPolynomial.Rename.271_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s : Finset σ
q : MvPolynomial { x // x ∈ s } R
n : ℕ := Fintype.card { x // x ∈ s }
⊢ ∃ n f, ∃ (_ : Injective f), ∃ q_1, (rename Subtype.val) q = (rename f) q_1 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | let e := Fintype.equivFin { x // x ∈ s } | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q := by
obtain ⟨s, q, rfl⟩ := exists_finset_rename p
let n := Fintype.card { x // x ∈ s }
| Mathlib.Data.MvPolynomial.Rename.271_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s : Finset σ
q : MvPolynomial { x // x ∈ s } R
n : ℕ := Fintype.card { x // x ∈ s }
e : { x // x ∈ s } ≃ Fin (Fintype.card { x // x ∈ s }) := Fintype.equivFin { x // x ∈ s }
⊢ ∃ n f, ∃ (_ : I... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | refine' ⟨n, (↑) ∘ e.symm, Subtype.val_injective.comp e.symm.injective, rename e q, _⟩ | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q := by
obtain ⟨s, q, rfl⟩ := exists_finset_rename p
let n := Fintype.card { x // x ∈ s }
let e := Fin... | Mathlib.Data.MvPolynomial.Rename.271_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s : Finset σ
q : MvPolynomial { x // x ∈ s } R
n : ℕ := Fintype.card { x // x ∈ s }
e : { x // x ∈ s } ≃ Fin (Fintype.card { x // x ∈ s }) := Fintype.equivFin { x // x ∈ s }
⊢ (rename Subtype... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [← rename_rename, rename_rename e] | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q := by
obtain ⟨s, q, rfl⟩ := exists_finset_rename p
let n := Fintype.card { x // x ∈ s }
let e := Fin... | Mathlib.Data.MvPolynomial.Rename.271_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
s : Finset σ
q : MvPolynomial { x // x ∈ s } R
n : ℕ := Fintype.card { x // x ∈ s }
e : { x // x ∈ s } ≃ Fin (Fintype.card { x // x ∈ s }) := Fintype.equivFin { x // x ∈ s }
⊢ (rename Subtype... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [Function.comp, Equiv.symm_apply_apply, rename_rename] | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q := by
obtain ⟨s, q, rfl⟩ := exists_finset_rename p
let n := Fintype.card { x // x ∈ s }
let e := Fin... | Mathlib.Data.MvPolynomial.Rename.271_0.3NqVCwOs1E93kvK | /-- Every polynomial is a polynomial in finitely many variables. -/
theorem exists_fin_rename (p : MvPolynomial σ R) :
∃ (n : ℕ) (f : Fin n → σ) (_hf : Injective f) (q : MvPolynomial (Fin n) R), p = rename f q | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
c : ℤ →+* R
g : τ → R
p : MvPolynomial σ ℤ
⊢ eval₂ c (g ∘ f) p = eval₂ c g ((rename f) p) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply MvPolynomial.induction_on p (fun n => by simp only [eval₂_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, rename, eval₂_add, AlgHom.map_add])
fun p n hp => by simp only [eval₂_mul, hp, eval₂_X, comp_apply, map_mul, rename_X, eval₂_mul] | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) := by
| Mathlib.Data.MvPolynomial.Rename.284_0.3NqVCwOs1E93kvK | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
c : ℤ →+* R
g : τ → R
p : MvPolynomial σ ℤ
n : ℤ
⊢ eval₂ c (g ∘ f) (C n) = eval₂ c g ((rename f) (C n)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [eval₂_C, rename_C] | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) := by
apply MvPolynomial.induction_on p (fun n => by | Mathlib.Data.MvPolynomial.Rename.284_0.3NqVCwOs1E93kvK | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
c : ℤ →+* R
g : τ → R
p✝ p q : MvPolynomial σ ℤ
hp : eval₂ c (g ∘ f) p = eval₂ c g ((rename f) p)
hq : eval₂ c (g ∘ f) q = eval₂ c g ((rename f) q)
⊢ eval₂ c (g ∘ f) (p + q) = eval₂ c g ((rename f) ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [hp, hq, rename, eval₂_add, AlgHom.map_add] | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) := by
apply MvPolynomial.induction_on p (fun n => by simp only [eval₂_C, rename_C])
(fun p q hp hq => by | Mathlib.Data.MvPolynomial.Rename.284_0.3NqVCwOs1E93kvK | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
c : ℤ →+* R
g : τ → R
p✝ p : MvPolynomial σ ℤ
n : σ
hp : eval₂ c (g ∘ f) p = eval₂ c g ((rename f) p)
⊢ eval₂ c (g ∘ f) (p * X n) = eval₂ c g ((rename f) (p * X n)) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [eval₂_mul, hp, eval₂_X, comp_apply, map_mul, rename_X, eval₂_mul] | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) := by
apply MvPolynomial.induction_on p (fun n => by simp only [eval₂_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, rename, eval₂_add, AlgHom.map_add])
fun p n hp => by... | Mathlib.Data.MvPolynomial.Rename.284_0.3NqVCwOs1E93kvK | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
φ : MvPolynomial σ R
d : σ →₀ ℕ
⊢ coeff (Finsupp.mapDomain f d) ((rename f) φ) = coeff d φ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | classical
apply φ.induction_on' (P := fun ψ => coeff (Finsupp.mapDomain f d) ((rename f) ψ) = coeff d ψ)
-- Lean could no longer infer the motive
· intro u r
rw [rename_monomial, coeff_monomial, coeff_monomial]
simp only [(Finsupp.mapDomain_injective hf).eq_iff]
· intros
simp only [*, AlgHom.map_add... | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d := by
| Mathlib.Data.MvPolynomial.Rename.293_0.3NqVCwOs1E93kvK | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
φ : MvPolynomial σ R
d : σ →₀ ℕ
⊢ coeff (Finsupp.mapDomain f d) ((rename f) φ) = coeff d φ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply φ.induction_on' (P := fun ψ => coeff (Finsupp.mapDomain f d) ((rename f) ψ) = coeff d ψ) | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d := by
classical
| Mathlib.Data.MvPolynomial.Rename.293_0.3NqVCwOs1E93kvK | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d | Mathlib_Data_MvPolynomial_Rename |
case h1
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
φ : MvPolynomial σ R
d : σ →₀ ℕ
⊢ ∀ (u : σ →₀ ℕ) (a : R), coeff (Finsupp.mapDomain f d) ((rename f) ((monomial u) a)) = coeff d ((monomial u) a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intro u r | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d := by
classical
apply φ.induction_on' (P := fun ψ => coeff (Finsupp.mapDomain f d) ((rename f) ψ) = coeff d ψ)
-- Lean could no longer infer the motive
·... | Mathlib.Data.MvPolynomial.Rename.293_0.3NqVCwOs1E93kvK | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d | Mathlib_Data_MvPolynomial_Rename |
case h1
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
φ : MvPolynomial σ R
d u : σ →₀ ℕ
r : R
⊢ coeff (Finsupp.mapDomain f d) ((rename f) ((monomial u) r)) = coeff d ((monomial u) r) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [rename_monomial, coeff_monomial, coeff_monomial] | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d := by
classical
apply φ.induction_on' (P := fun ψ => coeff (Finsupp.mapDomain f d) ((rename f) ψ) = coeff d ψ)
-- Lean could no longer infer the motive
·... | Mathlib.Data.MvPolynomial.Rename.293_0.3NqVCwOs1E93kvK | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d | Mathlib_Data_MvPolynomial_Rename |
case h1
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
φ : MvPolynomial σ R
d u : σ →₀ ℕ
r : R
⊢ (if Finsupp.mapDomain f u = Finsupp.mapDomain f d then r else 0) = if u = d then r else 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [(Finsupp.mapDomain_injective hf).eq_iff] | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d := by
classical
apply φ.induction_on' (P := fun ψ => coeff (Finsupp.mapDomain f d) ((rename f) ψ) = coeff d ψ)
-- Lean could no longer infer the motive
·... | Mathlib.Data.MvPolynomial.Rename.293_0.3NqVCwOs1E93kvK | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d | Mathlib_Data_MvPolynomial_Rename |
case h2
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
φ : MvPolynomial σ R
d : σ →₀ ℕ
⊢ ∀ (p q : MvPolynomial σ R),
coeff (Finsupp.mapDomain f d) ((rename f) p) = coeff d p →
coeff (Finsupp.mapDomain f d) ((rename f) ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intros | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d := by
classical
apply φ.induction_on' (P := fun ψ => coeff (Finsupp.mapDomain f d) ((rename f) ψ) = coeff d ψ)
-- Lean could no longer infer the motive
·... | Mathlib.Data.MvPolynomial.Rename.293_0.3NqVCwOs1E93kvK | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d | Mathlib_Data_MvPolynomial_Rename |
case h2
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
hf : Injective f
φ : MvPolynomial σ R
d : σ →₀ ℕ
p✝ q✝ : MvPolynomial σ R
a✝¹ : coeff (Finsupp.mapDomain f d) ((rename f) p✝) = coeff d p✝
a✝ : coeff (Finsupp.mapDomain f d) ((rename f) q✝) ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [*, AlgHom.map_add, coeff_add] | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d := by
classical
apply φ.induction_on' (P := fun ψ => coeff (Finsupp.mapDomain f d) ((rename f) ψ) = coeff d ψ)
-- Lean could no longer infer the motive
·... | Mathlib.Data.MvPolynomial.Rename.293_0.3NqVCwOs1E93kvK | @[simp]
theorem coeff_rename_mapDomain (f : σ → τ) (hf : Injective f) (φ : MvPolynomial σ R) (d : σ →₀ ℕ) :
(rename f φ).coeff (d.mapDomain f) = φ.coeff d | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
⊢ coeff d ((rename f) φ) = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | classical
rw [rename_eq, ← not_mem_support_iff]
intro H
replace H := mapDomain_support H
rw [Finset.mem_image] at H
obtain ⟨u, hu, rfl⟩ := H
specialize h u rfl
simp? at h hu says simp only [Finsupp.mem_support_iff, ne_eq] at h hu
contradiction | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
| Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
⊢ coeff d ((rename f) φ) = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [rename_eq, ← not_mem_support_iff] | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
| Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
⊢ d ∉ support (Finsupp.mapDomain (Finsupp.mapDomain f) φ) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intro H | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
rw [rename_eq, ← not_mem_support_iff]
| Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
H : d ∈ support (Finsupp.mapDomain (Finsupp.mapDomain f) φ)
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | replace H := mapDomain_support H | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
rw [rename_eq, ← not_mem_support_iff]
intro H
| Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
H : d ∈ Finset.image (Finsupp.mapDomain f) φ.support
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [Finset.mem_image] at H | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
rw [rename_eq, ← not_mem_support_iff]
intro H
replace H := mapDomain_support H
| Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
H : ∃ a ∈ φ.support, Finsupp.mapDomain f a = d
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | obtain ⟨u, hu, rfl⟩ := H | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
rw [rename_eq, ← not_mem_support_iff]
intro H
replace H := mapDomain_support H
rw [Finset.mem_image] at H
| Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
u : σ →₀ ℕ
hu : u ∈ φ.support
h : ∀ (u_1 : σ →₀ ℕ), Finsupp.mapDomain f u_1 = Finsupp.mapDomain f u → coeff u_1 φ = 0
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | specialize h u rfl | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
rw [rename_eq, ← not_mem_support_iff]
intro H
replace H := mapDomain_support H
rw [Finset.mem_image] at H
obtain ⟨u, hu, rfl⟩ := H... | Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
u : σ →₀ ℕ
hu : u ∈ φ.support
h : coeff u φ = 0
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp? at h hu says simp only [Finsupp.mem_support_iff, ne_eq] at h hu | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
rw [rename_eq, ← not_mem_support_iff]
intro H
replace H := mapDomain_support H
rw [Finset.mem_image] at H
obtain ⟨u, hu, rfl⟩ := H... | Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
u : σ →₀ ℕ
hu : u ∈ φ.support
h : coeff u φ = 0
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [Finsupp.mem_support_iff, ne_eq] at h hu | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
rw [rename_eq, ← not_mem_support_iff]
intro H
replace H := mapDomain_support H
rw [Finset.mem_image] at H
obtain ⟨u, hu, rfl⟩ := H... | Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
case intro.intro
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
u : σ →₀ ℕ
h : coeff u φ = 0
hu : ¬φ u = 0
⊢ False | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | contradiction | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 := by
classical
rw [rename_eq, ← not_mem_support_iff]
intro H
replace H := mapDomain_support H
rw [Finset.mem_image] at H
obtain ⟨u, hu, rfl⟩ := H... | Mathlib.Data.MvPolynomial.Rename.306_0.3NqVCwOs1E93kvK | theorem coeff_rename_eq_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : ∀ u : σ →₀ ℕ, u.mapDomain f = d → φ.coeff u = 0) : (rename f φ).coeff d = 0 | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : coeff d ((rename f) φ) ≠ 0
⊢ ∃ u, Finsupp.mapDomain f u = d ∧ coeff u φ ≠ 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | contrapose! h | theorem coeff_rename_ne_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : (rename f φ).coeff d ≠ 0) : ∃ u : σ →₀ ℕ, u.mapDomain f = d ∧ φ.coeff u ≠ 0 := by
| Mathlib.Data.MvPolynomial.Rename.319_0.3NqVCwOs1E93kvK | theorem coeff_rename_ne_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : (rename f φ).coeff d ≠ 0) : ∃ u : σ →₀ ℕ, u.mapDomain f = d ∧ φ.coeff u ≠ 0 | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
f : σ → τ
φ : MvPolynomial σ R
d : τ →₀ ℕ
h : ∀ (u : σ →₀ ℕ), Finsupp.mapDomain f u = d → coeff u φ = 0
⊢ coeff d ((rename f) φ) = 0 | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply coeff_rename_eq_zero _ _ _ h | theorem coeff_rename_ne_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : (rename f φ).coeff d ≠ 0) : ∃ u : σ →₀ ℕ, u.mapDomain f = d ∧ φ.coeff u ≠ 0 := by
contrapose! h
| Mathlib.Data.MvPolynomial.Rename.319_0.3NqVCwOs1E93kvK | theorem coeff_rename_ne_zero (f : σ → τ) (φ : MvPolynomial σ R) (d : τ →₀ ℕ)
(h : (rename f φ).coeff d ≠ 0) : ∃ u : σ →₀ ℕ, u.mapDomain f = d ∧ φ.coeff u ≠ 0 | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ✝ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
τ : Type u_6
f : σ → τ
φ : MvPolynomial σ R
⊢ constantCoeff ((rename f) φ) = constantCoeff φ | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | apply φ.induction_on | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ := by
| Mathlib.Data.MvPolynomial.Rename.325_0.3NqVCwOs1E93kvK | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ✝ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
τ : Type u_6
f : σ → τ
φ : MvPolynomial σ R
⊢ ∀ (a : R), constantCoeff ((rename f) (C a)) = constantCoeff (C a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intro a | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ := by
apply φ.induction_on
· | Mathlib.Data.MvPolynomial.Rename.325_0.3NqVCwOs1E93kvK | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ | Mathlib_Data_MvPolynomial_Rename |
case h_C
σ : Type u_1
τ✝ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
τ : Type u_6
f : σ → τ
φ : MvPolynomial σ R
a : R
⊢ constantCoeff ((rename f) (C a)) = constantCoeff (C a) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [constantCoeff_C, rename_C] | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ := by
apply φ.induction_on
· intro a
| Mathlib.Data.MvPolynomial.Rename.325_0.3NqVCwOs1E93kvK | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ✝ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
τ : Type u_6
f : σ → τ
φ : MvPolynomial σ R
⊢ ∀ (p q : MvPolynomial σ R),
constantCoeff ((rename f) p) = constantCoeff p →
constantCoeff ((rename f) q) = constantCoeff q → constantCoeff ... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intro p q hp hq | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ := by
apply φ.induction_on
· intro a
simp only [constantCoeff_C, rename_C]
· | Mathlib.Data.MvPolynomial.Rename.325_0.3NqVCwOs1E93kvK | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ | Mathlib_Data_MvPolynomial_Rename |
case h_add
σ : Type u_1
τ✝ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
τ : Type u_6
f : σ → τ
φ p q : MvPolynomial σ R
hp : constantCoeff ((rename f) p) = constantCoeff p
hq : constantCoeff ((rename f) q) = constantCoeff q
⊢ constantCoeff ((rename f) (p + q)) = const... | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [hp, hq, RingHom.map_add, AlgHom.map_add] | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ := by
apply φ.induction_on
· intro a
simp only [constantCoeff_C, rename_C]
· intro p q hp hq
| Mathlib.Data.MvPolynomial.Rename.325_0.3NqVCwOs1E93kvK | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ✝ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
τ : Type u_6
f : σ → τ
φ : MvPolynomial σ R
⊢ ∀ (p : MvPolynomial σ R) (n : σ),
constantCoeff ((rename f) p) = constantCoeff p → constantCoeff ((rename f) (p * X n)) = constantCoeff (p * X n) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | intro p n hp | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ := by
apply φ.induction_on
· intro a
simp only [constantCoeff_C, rename_C]
· intro p q hp hq
simp only [hp, hq, RingHom.map_add, AlgHom.map_add]
· | Mathlib.Data.MvPolynomial.Rename.325_0.3NqVCwOs1E93kvK | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ | Mathlib_Data_MvPolynomial_Rename |
case h_X
σ : Type u_1
τ✝ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S
τ : Type u_6
f : σ → τ
φ p : MvPolynomial σ R
n : σ
hp : constantCoeff ((rename f) p) = constantCoeff p
⊢ constantCoeff ((rename f) (p * X n)) = constantCoeff (p * X n) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | simp only [hp, rename_X, constantCoeff_X, RingHom.map_mul, AlgHom.map_mul] | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ := by
apply φ.induction_on
· intro a
simp only [constantCoeff_C, rename_C]
· intro p q hp hq
simp only [hp, hq, RingHom.map_add, AlgHom.map_add]
· intro p n hp
| Mathlib.Data.MvPolynomial.Rename.325_0.3NqVCwOs1E93kvK | @[simp]
theorem constantCoeff_rename {τ : Type*} (f : σ → τ) (φ : MvPolynomial σ R) :
constantCoeff (rename f φ) = constantCoeff φ | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
p : MvPolynomial σ R
f : σ → τ
inst✝ : DecidableEq τ
h : Injective f
⊢ support ((rename f) p) = Finset.image (Finsupp.mapDomain f) (support p) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | rw [rename_eq] | theorem support_rename_of_injective {p : MvPolynomial σ R} {f : σ → τ} [DecidableEq τ]
(h : Function.Injective f) :
(rename f p).support = Finset.image (Finsupp.mapDomain f) p.support := by
| Mathlib.Data.MvPolynomial.Rename.341_0.3NqVCwOs1E93kvK | theorem support_rename_of_injective {p : MvPolynomial σ R} {f : σ → τ} [DecidableEq τ]
(h : Function.Injective f) :
(rename f p).support = Finset.image (Finsupp.mapDomain f) p.support | Mathlib_Data_MvPolynomial_Rename |
σ : Type u_1
τ : Type u_2
α : Type u_3
R : Type u_4
S : Type u_5
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
p : MvPolynomial σ R
f : σ → τ
inst✝ : DecidableEq τ
h : Injective f
⊢ support (Finsupp.mapDomain (Finsupp.mapDomain f) p) = Finset.image (Finsupp.mapDomain f) (support p) | /-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Data.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de8... | exact Finsupp.mapDomain_support_of_injective (mapDomain_injective h) _ | theorem support_rename_of_injective {p : MvPolynomial σ R} {f : σ → τ} [DecidableEq τ]
(h : Function.Injective f) :
(rename f p).support = Finset.image (Finsupp.mapDomain f) p.support := by
rw [rename_eq]
| Mathlib.Data.MvPolynomial.Rename.341_0.3NqVCwOs1E93kvK | theorem support_rename_of_injective {p : MvPolynomial σ R} {f : σ → τ} [DecidableEq τ]
(h : Function.Injective f) :
(rename f p).support = Finset.image (Finsupp.mapDomain f) p.support | Mathlib_Data_MvPolynomial_Rename |
n : ℕ
⊢ ∀ {v w : Fin n}, Adj (pathGraph n) v w → (fun u => decide (↑u % 2 = 0)) v ≠ (fun u => decide (↑u % 2 = 0)) w | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | intro u v | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool :=
Coloring.mk (fun u ↦ u.val % 2 = 0) <| by
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.24_0.jXeFS7nTQciTQGN | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
u v : Fin n
⊢ Adj (pathGraph n) u v → (fun u => decide (↑u % 2 = 0)) u ≠ (fun u => decide (↑u % 2 = 0)) v | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | rw [pathGraph_adj] | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool :=
Coloring.mk (fun u ↦ u.val % 2 = 0) <| by
intro u v
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.24_0.jXeFS7nTQciTQGN | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
u v : Fin n
⊢ ↑u + 1 = ↑v ∨ ↑v + 1 = ↑u → (fun u => decide (↑u % 2 = 0)) u ≠ (fun u => decide (↑u % 2 = 0)) v | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | rintro (h | h) | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool :=
Coloring.mk (fun u ↦ u.val % 2 = 0) <| by
intro u v
rw [pathGraph_adj]
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.24_0.jXeFS7nTQciTQGN | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case inl
n : ℕ
u v : Fin n
h : ↑u + 1 = ↑v
⊢ (fun u => decide (↑u % 2 = 0)) u ≠ (fun u => decide (↑u % 2 = 0)) v | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | simp [← h, not_iff, Nat.succ_mod_two_eq_zero_iff] | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool :=
Coloring.mk (fun u ↦ u.val % 2 = 0) <| by
intro u v
rw [pathGraph_adj]
rintro (h | h) <;> | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.24_0.jXeFS7nTQciTQGN | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case inr
n : ℕ
u v : Fin n
h : ↑v + 1 = ↑u
⊢ (fun u => decide (↑u % 2 = 0)) u ≠ (fun u => decide (↑u % 2 = 0)) v | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | simp [← h, not_iff, Nat.succ_mod_two_eq_zero_iff] | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool :=
Coloring.mk (fun u ↦ u.val % 2 = 0) <| by
intro u v
rw [pathGraph_adj]
rintro (h | h) <;> | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.24_0.jXeFS7nTQciTQGN | /-- Bicoloring of a path graph -/
def pathGraph.bicoloring (n : ℕ) :
Coloring (pathGraph n) Bool | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
h : 2 ≤ n
⊢ Function.Injective fun v => { val := ↑v, isLt := (_ : ↑v < n) } | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | rintro v w | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
h : 2 ≤ n
v w : Fin 2
⊢ (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w → v = w | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | rw [Fin.mk.injEq] | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
h : 2 ≤ n
v w : Fin 2
⊢ ↑v = ↑w → v = w | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | exact Fin.ext | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
h : 2 ≤ n
⊢ ∀ {a b : Fin 2},
Adj (pathGraph n)
({ toFun := fun v => { val := ↑v, isLt := (_ : ↑v < n) },
inj' :=
(_ :
∀ ⦃v w : Fin 2⦄,
(fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w →
... | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | intro v w | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
exact Fin.ext
map_rel_iff' := by
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
n : ℕ
h : 2 ≤ n
v w : Fin 2
⊢ Adj (pathGraph n)
({ toFun := fun v => { val := ↑v, isLt := (_ : ↑v < n) },
inj' :=
(_ :
∀ ⦃v w : Fin 2⦄,
(fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w →
v = w)... | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | fin_cases v | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
exact Fin.ext
map_rel_iff' := by
intro v w
| Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case head
n : ℕ
h : 2 ≤ n
w : Fin 2
⊢ Adj (pathGraph n)
({ toFun := fun v => { val := ↑v, isLt := (_ : ↑v < n) },
inj' :=
(_ :
∀ ⦃v w : Fin 2⦄,
(fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w →
... | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | fin_cases w | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
exact Fin.ext
map_rel_iff' := by
intro v w
fin_cases... | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case tail.head
n : ℕ
h : 2 ≤ n
w : Fin 2
⊢ Adj (pathGraph n)
({ toFun := fun v => { val := ↑v, isLt := (_ : ↑v < n) },
inj' :=
(_ :
∀ ⦃v w : Fin 2⦄,
(fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w →
... | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | fin_cases w | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
exact Fin.ext
map_rel_iff' := by
intro v w
fin_cases... | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case head.head
n : ℕ
h : 2 ≤ n
⊢ Adj (pathGraph n)
({ toFun := fun v => { val := ↑v, isLt := (_ : ↑v < n) },
inj' :=
(_ :
∀ ⦃v w : Fin 2⦄,
(fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w →
v =... | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | simp [pathGraph, ← Fin.coe_covby_iff] | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
exact Fin.ext
map_rel_iff' := by
intro v w
fin_cases... | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
case head.tail.head
n : ℕ
h : 2 ≤ n
⊢ Adj (pathGraph n)
({ toFun := fun v => { val := ↑v, isLt := (_ : ↑v < n) },
inj' :=
(_ :
∀ ⦃v w : Fin 2⦄,
(fun v => { val := ↑v, isLt := (_ : ↑v < n) }) v = (fun v => { val := ↑v, isLt := (_ : ↑v < n) }) w →
... | /-
Copyright (c) 2023 Iván Renison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Iván Renison
-/
import Mathlib.Combinatorics.SimpleGraph.Coloring
import Mathlib.Combinatorics.SimpleGraph.Hasse
import Mathlib.Data.Nat.Parity
import Mathlib.Data.ZMod.Basic
/-!
# Conc... | simp [pathGraph, ← Fin.coe_covby_iff] | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v := ⟨v, trans v.2 h⟩
inj' := by
rintro v w
rw [Fin.mk.injEq]
exact Fin.ext
map_rel_iff' := by
intro v w
fin_cases... | Mathlib.Combinatorics.SimpleGraph.ConcreteColorings.32_0.jXeFS7nTQciTQGN | /-- Embedding of `pathGraph 2` into the first two elements of `pathGraph n` for `2 ≤ n` -/
def pathGraph_two_embedding (n : ℕ) (h : 2 ≤ n) : pathGraph 2 ↪g pathGraph n where
toFun v | Mathlib_Combinatorics_SimpleGraph_ConcreteColorings |
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