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 succ
S : Type u
inst✝ : Semiring S
a b : ℕ
⊢ eval a (ascPochhammer ℕ (Nat.succ b)) = Nat.descFactorial (a + Nat.succ b - 1) (Nat.succ b) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [Nat.add_succ, Nat.succ_sub_succ, tsub_zero] | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b := by
cases' b with b
· rw [Nat.descFactorial_zero, ascPochhammer_zero, Polynomial.eval_one]
| Mathlib.RingTheory.Polynomial.Pochhammer.162_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝ : Semiring S
a b : ℕ
⊢ eval a (ascPochhammer ℕ (Nat.succ b)) = Nat.descFactorial (a + b) (Nat.succ b) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | cases a | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b := by
cases' b with b
· rw [Nat.descFactorial_zero, ascPochhammer_zero, Polynomial.eval_one]
rw [Nat.add_succ, Nat.succ_sub_succ, tsub_zero]
| Mathlib.RingTheory.Polynomial.Pochhammer.162_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ.zero
S : Type u
inst✝ : Semiring S
b : ℕ
⊢ eval Nat.zero (ascPochhammer ℕ (Nat.succ b)) = Nat.descFactorial (Nat.zero + b) (Nat.succ b) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [Nat.zero_eq, ne_eq, Nat.succ_ne_zero, not_false_iff, ascPochhammer_ne_zero_eval_zero,
zero_add, Nat.descFactorial_succ, le_refl, tsub_eq_zero_of_le, zero_mul] | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b := by
cases' b with b
· rw [Nat.descFactorial_zero, ascPochhammer_zero, Polynomial.eval_one]
rw [Nat.add_succ, Nat.succ_sub_succ, tsub_zero]
cases a
· | Mathlib.RingTheory.Polynomial.Pochhammer.162_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ.succ
S : Type u
inst✝ : Semiring S
b n✝ : ℕ
⊢ eval (Nat.succ n✝) (ascPochhammer ℕ (Nat.succ b)) = Nat.descFactorial (Nat.succ n✝ + b) (Nat.succ b) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [Nat.succ_add, ← Nat.add_succ, Nat.add_descFactorial_eq_ascFactorial,
ascPochhammer_nat_eq_ascFactorial] | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b := by
cases' b with b
· rw [Nat.descFactorial_zero, ascPochhammer_zero, Polynomial.eval_one]
rw [Nat.add_succ, Nat.succ_sub_succ, tsub_zero]
cases a
· simp only [Nat.zero_eq, ne_eq, Nat.succ_ne_... | Mathlib.RingTheory.Polynomial.Pochhammer.162_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eq_descFactorial (a b : ℕ) :
(ascPochhammer ℕ b).eval a = (a + b - 1).descFactorial b | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝² : Semiring S
n : ℕ
inst✝¹ : NoZeroDivisors S
inst✝ : Nontrivial S
⊢ natDegree (ascPochhammer S n) = n | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n hn | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n := by
| Mathlib.RingTheory.Polynomial.Pochhammer.174_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u
inst✝² : Semiring S
inst✝¹ : NoZeroDivisors S
inst✝ : Nontrivial S
⊢ natDegree (ascPochhammer S Nat.zero) = Nat.zero | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n := by
induction' n with n hn
· | Mathlib.RingTheory.Polynomial.Pochhammer.174_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝² : Semiring S
inst✝¹ : NoZeroDivisors S
inst✝ : Nontrivial S
n : ℕ
hn : natDegree (ascPochhammer S n) = n
⊢ natDegree (ascPochhammer S (Nat.succ n)) = Nat.succ n | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | have : natDegree (X + (n : S[X])) = 1 := natDegree_X_add_C (n : S) | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n := by
induction' n with n hn
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.174_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝² : Semiring S
inst✝¹ : NoZeroDivisors S
inst✝ : Nontrivial S
n : ℕ
hn : natDegree (ascPochhammer S n) = n
this : natDegree (X + ↑n) = 1
⊢ natDegree (ascPochhammer S (Nat.succ n)) = Nat.succ n | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <| this.symm ▸ Nat.zero_lt_one), hn, this] | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n := by
induction' n with n hn
· simp
· have : natDegree (X + (n : S[X])) = 1 := natDegree_X_add_C (n : S)
| Mathlib.RingTheory.Polynomial.Pochhammer.174_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u
inst✝² : Semiring S
inst✝¹ : NoZeroDivisors S
inst✝ : Nontrivial S
n : ℕ
hn : natDegree (ascPochhammer S n) = n
this : natDegree (X + ↑n) = 1
⊢ ascPochhammer S n ≠ 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | cases n | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n := by
induction' n with n hn
· simp
· have : natDegree (X + (n : S[X])) = 1 := natDegree_X_add_C (n : S)
rw [ascPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <| t... | Mathlib.RingTheory.Polynomial.Pochhammer.174_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u
inst✝² : Semiring S
inst✝¹ : NoZeroDivisors S
inst✝ : Nontrivial S
hn : natDegree (ascPochhammer S Nat.zero) = Nat.zero
this : natDegree (X + ↑Nat.zero) = 1
⊢ ascPochhammer S Nat.zero ≠ 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n := by
induction' n with n hn
· simp
· have : natDegree (X + (n : S[X])) = 1 := natDegree_X_add_C (n : S)
rw [ascPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <| t... | Mathlib.RingTheory.Polynomial.Pochhammer.174_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u
inst✝² : Semiring S
inst✝¹ : NoZeroDivisors S
inst✝ : Nontrivial S
n✝ : ℕ
hn : natDegree (ascPochhammer S (Nat.succ n✝)) = Nat.succ n✝
this : natDegree (X + ↑(Nat.succ n✝)) = 1
⊢ ascPochhammer S (Nat.succ n✝) ≠ 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | refine' ne_zero_of_natDegree_gt <| hn.symm ▸ Nat.succ_pos _ | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n := by
induction' n with n hn
· simp
· have : natDegree (X + (n : S[X])) = 1 := natDegree_X_add_C (n : S)
rw [ascPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <| t... | Mathlib.RingTheory.Polynomial.Pochhammer.174_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_natDegree (n : ℕ) [NoZeroDivisors S] [Nontrivial S] :
(ascPochhammer S n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u_1
inst✝ : StrictOrderedSemiring S
n : ℕ
s : S
h : 0 < s
⊢ 0 < eval s (ascPochhammer S n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n ih | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s := by
| Mathlib.RingTheory.Polynomial.Pochhammer.192_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u_1
inst✝ : StrictOrderedSemiring S
s : S
h : 0 < s
⊢ 0 < eval s (ascPochhammer S Nat.zero) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [Nat.zero_eq, ascPochhammer_zero, eval_one] | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s := by
induction' n with n ih
· | Mathlib.RingTheory.Polynomial.Pochhammer.192_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
S : Type u_1
inst✝ : StrictOrderedSemiring S
s : S
h : 0 < s
⊢ 0 < 1 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | exact zero_lt_one | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s := by
induction' n with n ih
· simp only [Nat.zero_eq, ascPochhammer_zero, eval_one]
| Mathlib.RingTheory.Polynomial.Pochhammer.192_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u_1
inst✝ : StrictOrderedSemiring S
s : S
h : 0 < s
n : ℕ
ih : 0 < eval s (ascPochhammer S n)
⊢ 0 < eval s (ascPochhammer S (Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [ascPochhammer_succ_right, mul_add, eval_add, ← Nat.cast_comm, eval_nat_cast_mul, eval_mul_X,
Nat.cast_comm, ← mul_add] | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s := by
induction' n with n ih
· simp only [Nat.zero_eq, ascPochhammer_zero, eval_one]
exact zero_lt_one
· | Mathlib.RingTheory.Polynomial.Pochhammer.192_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
S : Type u_1
inst✝ : StrictOrderedSemiring S
s : S
h : 0 < s
n : ℕ
ih : 0 < eval s (ascPochhammer S n)
⊢ 0 < eval s (ascPochhammer S n) * (s + ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | exact mul_pos ih (lt_of_lt_of_le h ((le_add_iff_nonneg_right _).mpr (Nat.cast_nonneg n))) | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s := by
induction' n with n ih
· simp only [Nat.zero_eq, ascPochhammer_zero, eval_one]
exact zero_lt_one
· rw [ascPochhammer_succ_right, mul_add, eval_add, ← Nat.cast_comm, eval_nat_cast_mul, eval_mul_X,
Nat.cast_comm, ... | Mathlib.RingTheory.Polynomial.Pochhammer.192_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_pos (n : ℕ) (s : S) (h : 0 < s) : 0 < (ascPochhammer S n).eval s | Mathlib_RingTheory_Polynomial_Pochhammer |
S✝ : Type u_1
inst✝¹ : Semiring S✝
r n✝ : ℕ
S : Type u_2
inst✝ : Semiring S
n : ℕ
⊢ eval 1 (ascPochhammer S n) = ↑n ! | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw_mod_cast [ascPochhammer_nat_eq_ascFactorial, Nat.zero_ascFactorial] | @[simp]
theorem ascPochhammer_eval_one (S : Type*) [Semiring S] (n : ℕ) :
(ascPochhammer S n).eval (1 : S) = (n ! : S) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.209_0.yf6mY7NVFIgfXWQ | @[simp]
theorem ascPochhammer_eval_one (S : Type*) [Semiring S] (n : ℕ) :
(ascPochhammer S n).eval (1 : S) = (n ! : S) | Mathlib_RingTheory_Polynomial_Pochhammer |
S✝ : Type u_1
inst✝¹ : Semiring S✝
r✝ n✝ : ℕ
S : Type u_2
inst✝ : Semiring S
r n : ℕ
⊢ ↑r ! * eval (↑r + 1) (ascPochhammer S n) = ↑(r + n)! | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw_mod_cast [ascPochhammer_nat_eq_ascFactorial, Nat.factorial_mul_ascFactorial] | theorem factorial_mul_ascPochhammer (S : Type*) [Semiring S] (r n : ℕ) :
(r ! : S) * (ascPochhammer S n).eval (r + 1 : S) = (r + n)! := by
| Mathlib.RingTheory.Polynomial.Pochhammer.215_0.yf6mY7NVFIgfXWQ | theorem factorial_mul_ascPochhammer (S : Type*) [Semiring S] (r n : ℕ) :
(r ! : S) * (ascPochhammer S n).eval (r + 1 : S) = (r + n)! | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u_1
inst✝ : Semiring S
r✝ n r : ℕ
⊢ 0 * eval (0 + 1) (ascPochhammer ℕ r) = (0 + r) * eval 0 (ascPochhammer ℕ r) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | by_cases h : r = 0 | theorem ascPochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (ascPochhammer ℕ r).eval (n + 1) = (n + r) * (ascPochhammer ℕ r).eval n
| 0 => by
| Mathlib.RingTheory.Polynomial.Pochhammer.220_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (ascPochhammer ℕ r).eval (n + 1) = (n + r) * (ascPochhammer ℕ r).eval n
| 0 => by
by_cases h : r = 0
· simp only [h, zero_mul, zero_add]
· simp only [ascPochhammer_eval_zero, zero_mul, if_neg h, mul_zero]
| k + 1 => by simp only [ascPochhamm... | Mathlib_RingTheory_Polynomial_Pochhammer |
case pos
S : Type u_1
inst✝ : Semiring S
r✝ n r : ℕ
h : r = 0
⊢ 0 * eval (0 + 1) (ascPochhammer ℕ r) = (0 + r) * eval 0 (ascPochhammer ℕ r) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [h, zero_mul, zero_add] | theorem ascPochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (ascPochhammer ℕ r).eval (n + 1) = (n + r) * (ascPochhammer ℕ r).eval n
| 0 => by
by_cases h : r = 0
· | Mathlib.RingTheory.Polynomial.Pochhammer.220_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (ascPochhammer ℕ r).eval (n + 1) = (n + r) * (ascPochhammer ℕ r).eval n
| 0 => by
by_cases h : r = 0
· simp only [h, zero_mul, zero_add]
· simp only [ascPochhammer_eval_zero, zero_mul, if_neg h, mul_zero]
| k + 1 => by simp only [ascPochhamm... | Mathlib_RingTheory_Polynomial_Pochhammer |
case neg
S : Type u_1
inst✝ : Semiring S
r✝ n r : ℕ
h : ¬r = 0
⊢ 0 * eval (0 + 1) (ascPochhammer ℕ r) = (0 + r) * eval 0 (ascPochhammer ℕ r) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [ascPochhammer_eval_zero, zero_mul, if_neg h, mul_zero] | theorem ascPochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (ascPochhammer ℕ r).eval (n + 1) = (n + r) * (ascPochhammer ℕ r).eval n
| 0 => by
by_cases h : r = 0
· simp only [h, zero_mul, zero_add]
· | Mathlib.RingTheory.Polynomial.Pochhammer.220_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (ascPochhammer ℕ r).eval (n + 1) = (n + r) * (ascPochhammer ℕ r).eval n
| 0 => by
by_cases h : r = 0
· simp only [h, zero_mul, zero_add]
· simp only [ascPochhammer_eval_zero, zero_mul, if_neg h, mul_zero]
| k + 1 => by simp only [ascPochhamm... | Mathlib_RingTheory_Polynomial_Pochhammer |
S : Type u_1
inst✝ : Semiring S
r✝ n r k : ℕ
⊢ (k + 1) * eval (k + 1 + 1) (ascPochhammer ℕ r) = (k + 1 + r) * eval (k + 1) (ascPochhammer ℕ r) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [ascPochhammer_nat_eq_ascFactorial, Nat.succ_ascFactorial, add_right_comm] | theorem ascPochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (ascPochhammer ℕ r).eval (n + 1) = (n + r) * (ascPochhammer ℕ r).eval n
| 0 => by
by_cases h : r = 0
· simp only [h, zero_mul, zero_add]
· simp only [ascPochhammer_eval_zero, zero_mul, if_neg h, mul_zero]
| k + 1 => by | Mathlib.RingTheory.Polynomial.Pochhammer.220_0.yf6mY7NVFIgfXWQ | theorem ascPochhammer_nat_eval_succ (r : ℕ) :
∀ n : ℕ, n * (ascPochhammer ℕ r).eval (n + 1) = (n + r) * (ascPochhammer ℕ r).eval n
| 0 => by
by_cases h : r = 0
· simp only [h, zero_mul, zero_add]
· simp only [ascPochhammer_eval_zero, zero_mul, if_neg h, mul_zero]
| k + 1 => by simp only [ascPochhamm... | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
⊢ descPochhammer R 1 = X | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [descPochhammer] | @[simp]
theorem descPochhammer_one : descPochhammer R 1 = X := by | Mathlib.RingTheory.Polynomial.Pochhammer.252_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_one : descPochhammer R 1 = X | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ descPochhammer R (n + 1) = X * comp (descPochhammer R n) (X - 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer] | theorem descPochhammer_succ_left (n : ℕ) :
descPochhammer R (n + 1) = X * (descPochhammer R n).comp (X - 1) :=
by | Mathlib.RingTheory.Polynomial.Pochhammer.255_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_left (n : ℕ) :
descPochhammer R (n + 1) = X * (descPochhammer R n).comp (X - 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝² : Ring R
n : ℕ
inst✝¹ : Nontrivial R
inst✝ : NoZeroDivisors R
⊢ Monic (descPochhammer R n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n hn | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n := by
| Mathlib.RingTheory.Polynomial.Pochhammer.259_0.yf6mY7NVFIgfXWQ | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
R : Type u
inst✝² : Ring R
inst✝¹ : Nontrivial R
inst✝ : NoZeroDivisors R
⊢ Monic (descPochhammer R Nat.zero) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n := by
induction' n with n hn
· | Mathlib.RingTheory.Polynomial.Pochhammer.259_0.yf6mY7NVFIgfXWQ | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝² : Ring R
inst✝¹ : Nontrivial R
inst✝ : NoZeroDivisors R
n : ℕ
hn : Monic (descPochhammer R n)
⊢ Monic (descPochhammer R (Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | have h : leadingCoeff (X - 1 : R[X]) = 1 := leadingCoeff_X_sub_C 1 | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n := by
induction' n with n hn
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.259_0.yf6mY7NVFIgfXWQ | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝² : Ring R
inst✝¹ : Nontrivial R
inst✝ : NoZeroDivisors R
n : ℕ
hn : Monic (descPochhammer R n)
h : leadingCoeff (X - 1) = 1
⊢ Monic (descPochhammer R (Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | have : natDegree (X - (1 : R[X])) ≠ 0 := ne_zero_of_eq_one <| natDegree_X_sub_C (1 : R) | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n := by
induction' n with n hn
· simp
· have h : leadingCoeff (X - 1 : R[X]) = 1 := leadingCoeff_X_sub_C 1
| Mathlib.RingTheory.Polynomial.Pochhammer.259_0.yf6mY7NVFIgfXWQ | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝² : Ring R
inst✝¹ : Nontrivial R
inst✝ : NoZeroDivisors R
n : ℕ
hn : Monic (descPochhammer R n)
h : leadingCoeff (X - 1) = 1
this : natDegree (X - 1) ≠ 0
⊢ Monic (descPochhammer R (Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_succ_left, Monic.def, leadingCoeff_mul, leadingCoeff_comp this, hn, monic_X,
one_mul, one_mul, h, one_pow] | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n := by
induction' n with n hn
· simp
· have h : leadingCoeff (X - 1 : R[X]) = 1 := leadingCoeff_X_sub_C 1
have : natDegree (X - (1 : R[X])) ≠ 0 := ne_zero_of_eq_one <| natDegree_X_sub_C (1 : R)
| Mathlib.RingTheory.Polynomial.Pochhammer.259_0.yf6mY7NVFIgfXWQ | theorem monic_descPochhammer (n : ℕ) [Nontrivial R] [NoZeroDivisors R] :
Monic <| descPochhammer R n | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝¹ : Ring R
T : Type v
inst✝ : Ring T
f : R →+* T
n : ℕ
⊢ map f (descPochhammer R n) = descPochhammer T n | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n ih | @[simp]
theorem descPochhammer_map (f : R →+* T) (n : ℕ) :
(descPochhammer R n).map f = descPochhammer T n := by
| Mathlib.RingTheory.Polynomial.Pochhammer.272_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_map (f : R →+* T) (n : ℕ) :
(descPochhammer R n).map f = descPochhammer T n | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
R : Type u
inst✝¹ : Ring R
T : Type v
inst✝ : Ring T
f : R →+* T
⊢ map f (descPochhammer R Nat.zero) = descPochhammer T Nat.zero | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | @[simp]
theorem descPochhammer_map (f : R →+* T) (n : ℕ) :
(descPochhammer R n).map f = descPochhammer T n := by
induction' n with n ih
· | Mathlib.RingTheory.Polynomial.Pochhammer.272_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_map (f : R →+* T) (n : ℕ) :
(descPochhammer R n).map f = descPochhammer T n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝¹ : Ring R
T : Type v
inst✝ : Ring T
f : R →+* T
n : ℕ
ih : map f (descPochhammer R n) = descPochhammer T n
⊢ map f (descPochhammer R (Nat.succ n)) = descPochhammer T (Nat.succ n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [ih, descPochhammer_succ_left, map_comp] | @[simp]
theorem descPochhammer_map (f : R →+* T) (n : ℕ) :
(descPochhammer R n).map f = descPochhammer T n := by
induction' n with n ih
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.272_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_map (f : R →+* T) (n : ℕ) :
(descPochhammer R n).map f = descPochhammer T n | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
k : ℤ
⊢ ↑(eval k (descPochhammer ℤ n)) = eval (↑k) (descPochhammer R n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [← descPochhammer_map (algebraMap ℤ R), eval_map, ← eq_intCast (algebraMap ℤ R)] | @[simp, norm_cast]
theorem descPochhammer_eval_cast (n : ℕ) (k : ℤ) :
(((descPochhammer ℤ n).eval k : ℤ) : R) = ((descPochhammer R n).eval k : R) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.280_0.yf6mY7NVFIgfXWQ | @[simp, norm_cast]
theorem descPochhammer_eval_cast (n : ℕ) (k : ℤ) :
(((descPochhammer ℤ n).eval k : ℤ) : R) = ((descPochhammer R n).eval k : R) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
k : ℤ
⊢ (algebraMap ℤ R) (eval k (descPochhammer ℤ n)) = eval₂ (algebraMap ℤ R) (↑k) (descPochhammer ℤ n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [algebraMap_int_eq, eq_intCast, eval₂_at_int_cast, Nat.cast_id, eq_natCast, Int.cast_id] | @[simp, norm_cast]
theorem descPochhammer_eval_cast (n : ℕ) (k : ℤ) :
(((descPochhammer ℤ n).eval k : ℤ) : R) = ((descPochhammer R n).eval k : R) := by
rw [← descPochhammer_map (algebraMap ℤ R), eval_map, ← eq_intCast (algebraMap ℤ R)]
| Mathlib.RingTheory.Polynomial.Pochhammer.280_0.yf6mY7NVFIgfXWQ | @[simp, norm_cast]
theorem descPochhammer_eval_cast (n : ℕ) (k : ℤ) :
(((descPochhammer ℤ n).eval k : ℤ) : R) = ((descPochhammer R n).eval k : R) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ eval 0 (descPochhammer R n) = if n = 0 then 1 else 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | cases n | theorem descPochhammer_eval_zero {n : ℕ} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 := by
| Mathlib.RingTheory.Polynomial.Pochhammer.286_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_eval_zero {n : ℕ} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
R : Type u
inst✝ : Ring R
⊢ eval 0 (descPochhammer R Nat.zero) = if Nat.zero = 0 then 1 else 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem descPochhammer_eval_zero {n : ℕ} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 := by
cases n
· | Mathlib.RingTheory.Polynomial.Pochhammer.286_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_eval_zero {n : ℕ} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝ : Ring R
n✝ : ℕ
⊢ eval 0 (descPochhammer R (Nat.succ n✝)) = if Nat.succ n✝ = 0 then 1 else 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [X_mul, Nat.succ_ne_zero, descPochhammer_succ_left] | theorem descPochhammer_eval_zero {n : ℕ} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 := by
cases n
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.286_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_eval_zero {n : ℕ} :
(descPochhammer R n).eval 0 = if n = 0 then 1 else 0 | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
⊢ eval 0 (descPochhammer R 0) = 1 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem descPochhammer_zero_eval_zero : (descPochhammer R 0).eval 0 = 1 := by | Mathlib.RingTheory.Polynomial.Pochhammer.292_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_zero_eval_zero : (descPochhammer R 0).eval 0 = 1 | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
h : n ≠ 0
⊢ eval 0 (descPochhammer R n) = 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [descPochhammer_eval_zero, h] | @[simp]
theorem descPochhammer_ne_zero_eval_zero {n : ℕ} (h : n ≠ 0) : (descPochhammer R n).eval 0 = 0 := by
| Mathlib.RingTheory.Polynomial.Pochhammer.294_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_ne_zero_eval_zero {n : ℕ} (h : n ≠ 0) : (descPochhammer R n).eval 0 = 0 | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ descPochhammer R (n + 1) = descPochhammer R n * (X - ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X])) | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n)
⊢ descPochhammer R (n + 1) = descPochhammer R n * (X - ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | apply_fun Polynomial.map (algebraMap ℤ R) at h | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
h : map (algebraMap ℤ R) (descPochhammer ℤ (n + 1)) = map (algebraMap ℤ R) (descPochhammer ℤ n * (X - ↑n))
⊢ descPochhammer R (n + 1) = descPochhammer R n * (X - ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
Polynomial.map_int_cast] using h | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
| Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
case h
R : Type u
inst✝ : Ring R
n : ℕ
⊢ descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n ih | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add,... | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
case h.zero
R : Type u
inst✝ : Ring R
⊢ descPochhammer ℤ (Nat.zero + 1) = descPochhammer ℤ Nat.zero * (X - ↑Nat.zero) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp [descPochhammer] | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add,... | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
case h.succ
R : Type u
inst✝ : Ring R
n : ℕ
ih : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n)
⊢ descPochhammer ℤ (Nat.succ n + 1) = descPochhammer ℤ (Nat.succ n) * (X - ↑(Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | conv_lhs =>
rw [descPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← descPochhammer_succ_left, sub_comp,
X_comp, nat_cast_comp] | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add,... | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
ih : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n)
| descPochhammer ℤ (Nat.succ n + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← descPochhammer_succ_left, sub_comp,
X_comp, nat_cast_comp] | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add,... | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
ih : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n)
| descPochhammer ℤ (Nat.succ n + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← descPochhammer_succ_left, sub_comp,
X_comp, nat_cast_comp] | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add,... | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
ih : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n)
| descPochhammer ℤ (Nat.succ n + 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← descPochhammer_succ_left, sub_comp,
X_comp, nat_cast_comp] | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add,... | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
case h.succ
R : Type u
inst✝ : Ring R
n : ℕ
ih : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n)
⊢ descPochhammer ℤ (n + 1) * (X - 1 - ↑n) = descPochhammer ℤ (Nat.succ n) * (X - ↑(Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | nth_rw 1 [Nat.succ_eq_add_one] | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add,... | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
case h.succ
R : Type u
inst✝ : Ring R
n : ℕ
ih : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - ↑n)
⊢ descPochhammer ℤ (n + 1) * (X - 1 - ↑n) = descPochhammer ℤ (n + 1) * (X - ↑(Nat.succ n)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [Nat.succ_eq_one_add, Nat.cast_add, Nat.cast_one, sub_add_eq_sub_sub] | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X]))
· apply_fun Polynomial.map (algebraMap ℤ R) at h
simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add,... | Mathlib.RingTheory.Polynomial.Pochhammer.298_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝² : Ring R
n : ℕ
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ natDegree (descPochhammer R n) = n | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' n with n hn | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by
| Mathlib.RingTheory.Polynomial.Pochhammer.312_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
R : Type u
inst✝² : Ring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
⊢ natDegree (descPochhammer R Nat.zero) = Nat.zero | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by
induction' n with n hn
· | Mathlib.RingTheory.Polynomial.Pochhammer.312_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝² : Ring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
n : ℕ
hn : natDegree (descPochhammer R n) = n
⊢ natDegree (descPochhammer R (Nat.succ n)) = Nat.succ n | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | have : natDegree (X - (n : R[X])) = 1 := natDegree_X_sub_C (n : R) | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by
induction' n with n hn
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.312_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝² : Ring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
n : ℕ
hn : natDegree (descPochhammer R n) = n
this : natDegree (X - ↑n) = 1
⊢ natDegree (descPochhammer R (Nat.succ n)) = Nat.succ n | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <| this.symm ▸ Nat.zero_lt_one), hn, this] | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by
induction' n with n hn
· simp
· have : natDegree (X - (n : R[X])) = 1 := natDegree_X_sub_C (n : R)
| Mathlib.RingTheory.Polynomial.Pochhammer.312_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝² : Ring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
n : ℕ
hn : natDegree (descPochhammer R n) = n
this : natDegree (X - ↑n) = 1
⊢ descPochhammer R n ≠ 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | cases n | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by
induction' n with n hn
· simp
· have : natDegree (X - (n : R[X])) = 1 := natDegree_X_sub_C (n : R)
rw [descPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <... | Mathlib.RingTheory.Polynomial.Pochhammer.312_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
R : Type u
inst✝² : Ring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
hn : natDegree (descPochhammer R Nat.zero) = Nat.zero
this : natDegree (X - ↑Nat.zero) = 1
⊢ descPochhammer R Nat.zero ≠ 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by
induction' n with n hn
· simp
· have : natDegree (X - (n : R[X])) = 1 := natDegree_X_sub_C (n : R)
rw [descPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <... | Mathlib.RingTheory.Polynomial.Pochhammer.312_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝² : Ring R
inst✝¹ : NoZeroDivisors R
inst✝ : Nontrivial R
n✝ : ℕ
hn : natDegree (descPochhammer R (Nat.succ n✝)) = Nat.succ n✝
this : natDegree (X - ↑(Nat.succ n✝)) = 1
⊢ descPochhammer R (Nat.succ n✝) ≠ 0 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | refine' ne_zero_of_natDegree_gt <| hn.symm ▸ Nat.succ_pos _ | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n := by
induction' n with n hn
· simp
· have : natDegree (X - (n : R[X])) = 1 := natDegree_X_sub_C (n : R)
rw [descPochhammer_succ_right,
natDegree_mul _ (ne_zero_of_natDegree_gt <... | Mathlib.RingTheory.Polynomial.Pochhammer.312_0.yf6mY7NVFIgfXWQ | @[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝¹ : Ring R
S : Type u_1
inst✝ : Ring S
n : ℕ
k : S
⊢ eval k (descPochhammer S (n + 1)) = eval k (descPochhammer S n) * (k - ↑n) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_succ_right, mul_sub, eval_sub, eval_mul_X, ← Nat.cast_comm, ← C_eq_nat_cast,
eval_C_mul, Nat.cast_comm, ← mul_sub] | theorem descPochhammer_succ_eval {S : Type*} [Ring S] (n : ℕ) (k : S) :
(descPochhammer S (n + 1)).eval k = (descPochhammer S n).eval k * (k - n) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.324_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_eval {S : Type*} [Ring S] (n : ℕ) (k : S) :
(descPochhammer S (n + 1)).eval k = (descPochhammer S n).eval k * (k - n) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ comp (descPochhammer R (n + 1)) (X - 1) = descPochhammer R (n + 1) - (↑n + 1) • comp (descPochhammer R n) (X - 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | suffices (descPochhammer ℤ (n + 1)).comp (X - 1) =
descPochhammer ℤ (n + 1) - (n + 1) * (descPochhammer ℤ n).comp (X - 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Int.castRingHom R)) this | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.329_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
this : comp (descPochhammer ℤ (n + 1)) (X - 1) = descPochhammer ℤ (n + 1) - (↑n + 1) * comp (descPochhammer ℤ n) (X - 1)
⊢ comp (descPochhammer R (n + 1)) (X - 1) = descPochhammer R (n + 1) - (↑n + 1) • comp (descPochhammer R n) (X - 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simpa [map_comp] using congr_arg (Polynomial.map (Int.castRingHom R)) this | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) := by
suffices (descPochhammer ℤ (n + 1)).comp (X - 1) =
descPochhammer ℤ (n + 1) - (n + 1) * (descPochhammer ℤ n).comp (X - 1)... | Mathlib.RingTheory.Polynomial.Pochhammer.329_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ comp (descPochhammer ℤ (n + 1)) (X - 1) = descPochhammer ℤ (n + 1) - (↑n + 1) * comp (descPochhammer ℤ n) (X - 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | nth_rw 2 [descPochhammer_succ_left] | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) := by
suffices (descPochhammer ℤ (n + 1)).comp (X - 1) =
descPochhammer ℤ (n + 1) - (n + 1) * (descPochhammer ℤ n).comp (X - 1)... | Mathlib.RingTheory.Polynomial.Pochhammer.329_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ comp (descPochhammer ℤ (n + 1)) (X - 1) =
X * comp (descPochhammer ℤ n) (X - 1) - (↑n + 1) * comp (descPochhammer ℤ n) (X - 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [← sub_mul, descPochhammer_succ_right ℤ n, mul_comp, mul_comm, sub_comp, X_comp, nat_cast_comp] | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) := by
suffices (descPochhammer ℤ (n + 1)).comp (X - 1) =
descPochhammer ℤ (n + 1) - (n + 1) * (descPochhammer ℤ n).comp (X - 1)... | Mathlib.RingTheory.Polynomial.Pochhammer.329_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ (X - 1 - ↑n) * comp (descPochhammer ℤ n) (X - 1) = (X - (↑n + 1)) * comp (descPochhammer ℤ n) (X - 1) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | ring | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) := by
suffices (descPochhammer ℤ (n + 1)).comp (X - 1) =
descPochhammer ℤ (n + 1) - (n + 1) * (descPochhammer ℤ n).comp (X - 1)... | Mathlib.RingTheory.Polynomial.Pochhammer.329_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n m : ℕ
⊢ descPochhammer R n * comp (descPochhammer R m) (X - ↑n) = descPochhammer R (n + m) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | induction' m with m ih | theorem descPochhammer_mul (n m : ℕ) :
descPochhammer R n * (descPochhammer R m).comp (X - (n : R[X])) = descPochhammer R (n + m) := by
| Mathlib.RingTheory.Polynomial.Pochhammer.339_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_mul (n m : ℕ) :
descPochhammer R n * (descPochhammer R m).comp (X - (n : R[X])) = descPochhammer R (n + m) | Mathlib_RingTheory_Polynomial_Pochhammer |
case zero
R : Type u
inst✝ : Ring R
n : ℕ
⊢ descPochhammer R n * comp (descPochhammer R Nat.zero) (X - ↑n) = descPochhammer R (n + Nat.zero) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp | theorem descPochhammer_mul (n m : ℕ) :
descPochhammer R n * (descPochhammer R m).comp (X - (n : R[X])) = descPochhammer R (n + m) := by
induction' m with m ih
· | Mathlib.RingTheory.Polynomial.Pochhammer.339_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_mul (n m : ℕ) :
descPochhammer R n * (descPochhammer R m).comp (X - (n : R[X])) = descPochhammer R (n + m) | Mathlib_RingTheory_Polynomial_Pochhammer |
case succ
R : Type u
inst✝ : Ring R
n m : ℕ
ih : descPochhammer R n * comp (descPochhammer R m) (X - ↑n) = descPochhammer R (n + m)
⊢ descPochhammer R n * comp (descPochhammer R (Nat.succ m)) (X - ↑n) = descPochhammer R (n + Nat.succ m) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_succ_right, Polynomial.mul_X_sub_int_cast_comp, ← mul_assoc, ih,
Nat.succ_eq_add_one, ← add_assoc, descPochhammer_succ_right, Nat.cast_add, sub_add_eq_sub_sub] | theorem descPochhammer_mul (n m : ℕ) :
descPochhammer R n * (descPochhammer R m).comp (X - (n : R[X])) = descPochhammer R (n + m) := by
induction' m with m ih
· simp
· | Mathlib.RingTheory.Polynomial.Pochhammer.339_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_mul (n m : ℕ) :
descPochhammer R n * (descPochhammer R m).comp (X - (n : R[X])) = descPochhammer R (n + m) | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ eval (↑n) (descPochhammer ℤ 0) = ↑(Nat.descFactorial n 0) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero] | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
| Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n : ℕ
⊢ 1 = ↑1 | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rfl | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
| Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n t : ℕ
⊢ eval (↑n) (descPochhammer ℤ (t + 1)) = ↑(Nat.descFactorial n (t + 1)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t] | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
| Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n t : ℕ
⊢ ↑(Nat.descFactorial n t) * eval (↑n) (X - ↑t) = ↑(Nat.descFactorial n (t + 1)) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [eval_sub, eval_X, eval_nat_cast, Nat.descFactorial_succ, Nat.cast_mul,
Nat.descFactorial_eq_zero_iff_lt] | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
| Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n t : ℕ
⊢ ↑(Nat.descFactorial n t) * (↑n - ↑t) = ↑(n - t) * ↑(Nat.descFactorial n t) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [mul_comm] | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n t : ℕ
⊢ (↑n - ↑t) * ↑(Nat.descFactorial n t) = ↑(n - t) * ↑(Nat.descFactorial n t) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | simp only [mul_eq_mul_right_iff, Nat.cast_eq_zero, Nat.descFactorial_eq_zero_iff_lt] | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
n t : ℕ
⊢ ↑n - ↑t = ↑(n - t) ∨ n < t | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | by_cases h : n < t | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
case pos
R : Type u
inst✝ : Ring R
n t : ℕ
h : n < t
⊢ ↑n - ↑t = ↑(n - t) ∨ n < t | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | tauto | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
case neg
R : Type u
inst✝ : Ring R
n t : ℕ
h : ¬n < t
⊢ ↑n - ↑t = ↑(n - t) ∨ n < t | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | left | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
case neg.h
R : Type u
inst✝ : Ring R
n t : ℕ
h : ¬n < t
⊢ ↑n - ↑t = ↑(n - t) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | exact (Int.ofNat_sub <| not_lt.mp h).symm | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib.RingTheory.Polynomial.Pochhammer.346_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_descFactorial (n : ℕ) :
∀ k, (descPochhammer ℤ k).eval (n : ℤ) = n.descFactorial k
| 0 => by
rw [descPochhammer_zero, eval_one, Nat.descFactorial_zero]
rfl
| t + 1 => by
rw [descPochhammer_succ_right, eval_mul, descPochhammer_int_eq_descFactorial n t]
simp only [eva... | Mathlib_RingTheory_Polynomial_Pochhammer |
R : Type u
inst✝ : Ring R
a b : ℕ
⊢ eval (↑a + ↑b) (descPochhammer ℤ b) = ↑(Nat.ascFactorial a b) | /-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Tactic.Abel
import Mathlib.Data.Polynomial.Degree.Definitions
import Mathlib.Data.Polynomial.Eval
import Mathlib.Data.Polynomial.Monic
import Mathlib.D... | rw [← Nat.cast_add, descPochhammer_int_eq_descFactorial (a + b) b,
Nat.add_descFactorial_eq_ascFactorial] | theorem descPochhammer_int_eq_ascFactorial (a b : ℕ) :
(descPochhammer ℤ b).eval (a + b : ℤ) = a.ascFactorial b := by
| Mathlib.RingTheory.Polynomial.Pochhammer.362_0.yf6mY7NVFIgfXWQ | theorem descPochhammer_int_eq_ascFactorial (a b : ℕ) :
(descPochhammer ℤ b).eval (a + b : ℤ) = a.ascFactorial b | Mathlib_RingTheory_Polynomial_Pochhammer |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : Preorder ι
inst✝¹ : OrderBot ι
inst✝ : InfSet ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
⊢ upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [upperCrossingTime] | theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω := by
| Mathlib.Probability.Martingale.Upcrossing.168_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : Preorder ι
inst✝¹ : OrderBot ι
inst✝ : InfSet ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω✝ ω : Ω
⊢ upperCrossingTime a b f N (n + 1) ω = hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [upperCrossingTime_succ] | theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
| Mathlib.Probability.Martingale.Upcrossing.173_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝² : Preorder ι
inst✝¹ : OrderBot ι
inst✝ : InfSet ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω✝ ω : Ω
⊢ hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rfl | theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by
simp only [upperCrossingTime_succ]
| Mathlib.Probability.Martingale.Upcrossing.173_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
⊢ upperCrossingTime a b f N n ω ≤ N | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | cases n | theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
| Mathlib.Probability.Martingale.Upcrossing.187_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N | Mathlib_Probability_Martingale_Upcrossing |
case zero
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
m : ℕ
ω : Ω
⊢ upperCrossingTime a b f N Nat.zero ω ≤ N | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq] | theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· | Mathlib.Probability.Martingale.Upcrossing.187_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N | Mathlib_Probability_Martingale_Upcrossing |
case succ
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
m : ℕ
ω : Ω
n✝ : ℕ
⊢ upperCrossingTime a b f N (Nat.succ n✝) ω ≤ N | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [upperCrossingTime_succ, hitting_le] | theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N := by
cases n
· simp only [upperCrossingTime_zero, Pi.bot_apply, bot_le, Nat.zero_eq]
· | Mathlib.Probability.Martingale.Upcrossing.187_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
⊢ lowerCrossingTime a b f N n ω ≤ N | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [lowerCrossingTime, hitting_le ω] | theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N := by
| Mathlib.Probability.Martingale.Upcrossing.198_0.80Cpy4Qgm9i1y9y | theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
⊢ upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [lowerCrossingTime, le_hitting upperCrossingTime_le ω] | theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω := by
| Mathlib.Probability.Martingale.Upcrossing.202_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_le_lowerCrossingTime :
upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
⊢ lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [upperCrossingTime_succ] | theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
| Mathlib.Probability.Martingale.Upcrossing.207_0.80Cpy4Qgm9i1y9y | theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
⊢ lowerCrossingTime a b f N n ω ≤
hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact le_hitting lowerCrossingTime_le ω | theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω := by
rw [upperCrossingTime_succ]
| Mathlib.Probability.Martingale.Upcrossing.207_0.80Cpy4Qgm9i1y9y | theorem lowerCrossingTime_le_upperCrossingTime_succ :
lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
hnm : n ≤ m
⊢ lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm | theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
| Mathlib.Probability.Martingale.Upcrossing.213_0.80Cpy4Qgm9i1y9y | theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
hnm : n ≤ m
this : Monotone fun n => lowerCrossingTime a b f N n ω
⊢ lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact this hnm | theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by | Mathlib.Probability.Martingale.Upcrossing.213_0.80Cpy4Qgm9i1y9y | theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
hnm : n ≤ m
⊢ Monotone fun n => lowerCrossingTime a b f N n ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact monotone_nat_of_le_succ fun n =>
le_trans lowerCrossingTime_le_upperCrossingTime_succ upperCrossingTime_le_lowerCrossingTime | theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω := by
suffices Monotone fun n => lowerCrossingTime a b f N n ω by exact this hnm
| Mathlib.Probability.Martingale.Upcrossing.213_0.80Cpy4Qgm9i1y9y | theorem lowerCrossingTime_mono (hnm : n ≤ m) :
lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
hnm : n ≤ m
⊢ upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm | theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
| Mathlib.Probability.Martingale.Upcrossing.220_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
hnm : n ≤ m
this : Monotone fun n => upperCrossingTime a b f N n ω
⊢ upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact this hnm | theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by | Mathlib.Probability.Martingale.Upcrossing.220_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
inst✝ : ConditionallyCompleteLinearOrderBot ι
a b : ℝ
f : ι → Ω → ℝ
N : ι
n m : ℕ
ω : Ω
hnm : n ≤ m
⊢ Monotone fun n => upperCrossingTime a b f N n ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact monotone_nat_of_le_succ fun n =>
le_trans upperCrossingTime_le_lowerCrossingTime lowerCrossingTime_le_upperCrossingTime_succ | theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω := by
suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm
| Mathlib.Probability.Martingale.Upcrossing.220_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_mono (hnm : n ≤ m) :
upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
h : lowerCrossingTime a b f N n ω ≠ N
⊢ stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl | theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
| Mathlib.Probability.Martingale.Upcrossing.231_0.80Cpy4Qgm9i1y9y | theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a | Mathlib_Probability_Martingale_Upcrossing |
case intro.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
h : lowerCrossingTime a b f N n ω ≠ N
j : ℕ
hj₁ : j ∈ Set.Icc (upperCrossingTime a b f N n ω) (lowerCrossingTime a b f N n ω)
hj₂ : f j ω ∈ Set.Iic a
⊢ stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 lowerCrossingTime_le⟩, hj₂⟩ | theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne lowerCrossingTime_le h)).1 le_rfl
| Mathlib.Probability.Martingale.Upcrossing.231_0.80Cpy4Qgm9i1y9y | theorem stoppedValue_lowerCrossingTime (h : lowerCrossingTime a b f N n ω ≠ N) :
stoppedValue f (lowerCrossingTime a b f N n) ω ≤ a | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
h : upperCrossingTime a b f N (n + 1) ω ≠ N
⊢ b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl | theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
| Mathlib.Probability.Martingale.Upcrossing.237_0.80Cpy4Qgm9i1y9y | theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω | Mathlib_Probability_Martingale_Upcrossing |
case intro.intro
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
h : upperCrossingTime a b f N (n + 1) ω ≠ N
j : ℕ
hj₁ :
j ∈
Set.Icc (lowerCrossingTimeAux a f (upperCrossingTime a b f N (Nat.add n 0) ω) N ω)
(upperCrossingTime a b f N (n + 1) ω)
hj₂ : f j... | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩ | theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
| Mathlib.Probability.Martingale.Upcrossing.237_0.80Cpy4Qgm9i1y9y | theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
hab : a < b
hn : lowerCrossingTime a b f N (n + 1) ω ≠ N
⊢ upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowerCrossingTime hn) | theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
| Mathlib.Probability.Martingale.Upcrossing.243_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
hab : a < b
hn : lowerCrossingTime a b f N (n + 1) ω ≠ N
h : upperCrossingTime a b f N (n + 1) ω = lowerCrossingTime a b f N (n + 1) ω
⊢ b ≤ stoppedValue f (lowerCrossingTime a b f N (n + 1)) ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | simp only [stoppedValue] | theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowe... | Mathlib.Probability.Martingale.Upcrossing.243_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω | Mathlib_Probability_Martingale_Upcrossing |
Ω : Type u_1
ι : Type u_2
m0 : MeasurableSpace Ω
μ : Measure Ω
a b : ℝ
f : ℕ → Ω → ℝ
N n m : ℕ
ω : Ω
hab : a < b
hn : lowerCrossingTime a b f N (n + 1) ω ≠ N
h : upperCrossingTime a b f N (n + 1) ω = lowerCrossingTime a b f N (n + 1) ω
⊢ b ≤ f (lowerCrossingTime a b f N (n + 1) ω) ω | /-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Data.Set.Intervals.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
#align_import probability.martingale.... | rw [← h] | theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω := by
refine' lt_of_le_of_ne upperCrossingTime_le_lowerCrossingTime fun h =>
not_le.2 hab <| le_trans _ (stoppedValue_lowe... | Mathlib.Probability.Martingale.Upcrossing.243_0.80Cpy4Qgm9i1y9y | theorem upperCrossingTime_lt_lowerCrossingTime (hab : a < b)
(hn : lowerCrossingTime a b f N (n + 1) ω ≠ N) :
upperCrossingTime a b f N (n + 1) ω < lowerCrossingTime a b f N (n + 1) ω | Mathlib_Probability_Martingale_Upcrossing |
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