Arithmetization.Vorspiel.Lemmata

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class LO.FirstOrder.Structure.Monotone (L : LO.FirstOrder.Language) (M : Type u_1) [LE M] [LO.FirstOrder.Structure L M] :

Type

monotone : ∀ {k : ℕ} (f : L.Func k) (v₁ v₂ : Fin k → M), (∀ (i : Fin k), v₁ i ≤ v₂ i) → LO.FirstOrder.Structure.func f v₁ ≤ LO.FirstOrder.Structure.func f v₂

Instances

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theorem LO.FirstOrder.Structure.Monotone.monotone {L : LO.FirstOrder.Language} {M : Type u_1} [LE M] [LO.FirstOrder.Structure L M] [self : LO.FirstOrder.Structure.Monotone L M] {k : ℕ} (f : L.Func k) (v₁ : Fin k → M) (v₂ : Fin k → M) :

(∀ (i : Fin k), v₁ i ≤ v₂ i) → LO.FirstOrder.Structure.func f v₁ ≤ LO.FirstOrder.Structure.func f v₂

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theorem LO.FirstOrder.Structure.Monotone.term_monotone {L : LO.FirstOrder.Language} {M : Type u_1} [LE M] [LO.FirstOrder.Structure L M] [LO.FirstOrder.Structure.Monotone L M] {ξ : Type u_2} {n : ℕ} (t : LO.FirstOrder.Semiterm L ξ n) {e₁ : Fin n → M} {e₂ : Fin n → M} {ε₁ : ξ → M} {ε₂ : ξ → M} (he : ∀ (i : Fin n), e₁ i ≤ e₂ i) (hε : ∀ (i : ξ), ε₁ i ≤ ε₂ i) :

LO.FirstOrder.Semiterm.valm M e₁ ε₁ t ≤ LO.FirstOrder.Semiterm.valm M e₂ ε₂ t

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def LO.FirstOrder.Semiterm.arithCases {ξ : Type u_1} {n : ℕ} {C : LO.FirstOrder.Semiterm ℒₒᵣ ξ n → Sort w} (hbvar : (x : Fin n) → C (LO.FirstOrder.Semiterm.bvar x)) (hfvar : (x : ξ) → C (LO.FirstOrder.Semiterm.fvar x)) (hzero : C (‘0’)) (hone : C (‘1’)) (hadd : (t u : LO.FirstOrder.Semiterm ℒₒᵣ ξ n) → C (‘(!!t + !!u)’)) (hmul : (t u : LO.FirstOrder.Semiterm ℒₒᵣ ξ n) → C (‘(!!t * !!u)’)) (t : LO.FirstOrder.Semiterm ℒₒᵣ ξ n) :

C t

Equations

One or more equations did not get rendered due to their size.

Instances For

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@[irreducible]

def LO.FirstOrder.Semiterm.arithRec {ξ : Type u_1} {n : ℕ} {C : LO.FirstOrder.Semiterm ℒₒᵣ ξ n → Sort w} (hbvar : (x : Fin n) → C (LO.FirstOrder.Semiterm.bvar x)) (hfvar : (x : ξ) → C (LO.FirstOrder.Semiterm.fvar x)) (hzero : C (‘0’)) (hone : C (‘1’)) (hadd : {t u : LO.FirstOrder.Semiterm ℒₒᵣ ξ n} → C t → C u → C (‘(!!t + !!u)’)) (hmul : {t u : LO.FirstOrder.Semiterm ℒₒᵣ ξ n} → C t → C u → C (‘(!!t * !!u)’)) (t : LO.FirstOrder.Semiterm ℒₒᵣ ξ n) :

C t

Equations

One or more equations did not get rendered due to their size.
LO.FirstOrder.Semiterm.arithRec hbvar hfvar hzero hone hadd hmul (LO.FirstOrder.Semiterm.bvar x_1) = hbvar x_1
LO.FirstOrder.Semiterm.arithRec hbvar hfvar hzero hone hadd hmul (LO.FirstOrder.Semiterm.fvar x_1) = hfvar x_1
LO.FirstOrder.Semiterm.arithRec hbvar hfvar hzero hone hadd hmul (LO.FirstOrder.Semiterm.func LO.FirstOrder.Language.ORing.Func.zero a) = ⋯.mpr (⋯.mp hzero)
LO.FirstOrder.Semiterm.arithRec hbvar hfvar hzero hone hadd hmul (LO.FirstOrder.Semiterm.func LO.FirstOrder.Language.ORing.Func.one a) = ⋯.mpr (⋯.mp hone)

Instances For

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instance LO.Arith.instNonempty_arithmetization {M : Type u_1} [LO.ORingStruc M] :

Nonempty M

Equations

⋯ = ⋯

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@[simp]

theorem LO.Arith.numeral_two_eq_two {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

LO.ORingStruc.numeral 2 = 2

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@[simp]

theorem LO.Arith.numeral_three_eq_three {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

LO.ORingStruc.numeral 3 = 3

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@[simp]

theorem LO.Arith.numeral_four_eq_four {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

LO.ORingStruc.numeral 4 = 4

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theorem LO.Arith.lt_succ_iff_le {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {x : M} {y : M} :

x < y + 1 ↔ x ≤ y

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theorem LO.Arith.lt_iff_succ_le {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} :

a < b ↔ a + 1 ≤ b

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theorem LO.Arith.succ_le_iff_lt {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} :

a + 1 ≤ b ↔ a < b

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theorem LO.Arith.pos_iff_one_le {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

0 < a ↔ 1 ≤ a

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theorem LO.Arith.one_lt_iff_two_le {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

1 < a ↔ 2 ≤ a

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@[simp]

theorem LO.Arith.not_nonpos {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (a : M) :

¬a < 0

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theorem LO.Arith.lt_two_iff_le_one {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

a < 2 ↔ a ≤ 1

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@[simp]

theorem LO.Arith.lt_one_iff_eq_zero {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

a < 1 ↔ a = 0

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theorem LO.Arith.le_one_iff_eq_zero_or_one {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

a ≤ 1 ↔ a = 0 ∨ a = 1

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theorem LO.Arith.le_two_iff_eq_zero_or_one_or_two {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

a ≤ 2 ↔ a = 0 ∨ a = 1 ∨ a = 2

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theorem LO.Arith.le_three_iff_eq_zero_or_one_or_two_or_three {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

a ≤ 3 ↔ a = 0 ∨ a = 1 ∨ a = 2 ∨ a = 3

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theorem LO.Arith.two_mul_two_eq_four {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

2 * 2 = 4

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theorem LO.Arith.two_pow_two_eq_four {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

2 ^ 2 = 4

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theorem LO.Arith.two_pos {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

0 < 2

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@[simp]

theorem LO.Arith.le_mul_self {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (a : M) :

a ≤ a * a

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@[simp]

theorem LO.Arith.le_sq {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (a : M) :

a ≤ a ^ 2

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@[simp]

theorem LO.Arith.sq_le_sq {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} :

a ^ 2 ≤ b ^ 2 ↔ a ≤ b

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@[simp]

theorem LO.Arith.sq_lt_sq {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} :

a ^ 2 < b ^ 2 ↔ a < b

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theorem LO.Arith.le_mul_of_pos_right {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} (h : 0 < b) :

a ≤ a * b

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theorem LO.Arith.le_mul_of_pos_left {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} (h : 0 < b) :

a ≤ b * a

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@[simp]

theorem LO.Arith.le_two_mul_left {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

a ≤ 2 * a

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theorem LO.Arith.lt_mul_of_pos_of_one_lt_right {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} (pos : 0 < a) (h : 1 < b) :

a < a * b

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theorem LO.Arith.lt_mul_of_pos_of_one_lt_left {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} (pos : 0 < a) (h : 1 < b) :

a < b * a

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theorem LO.Arith.mul_le_mul_left {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} {c : M} (h : b ≤ c) :

a * b ≤ a * c

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theorem LO.Arith.mul_le_mul_right {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} {c : M} (h : b ≤ c) :

b * a ≤ c * a

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theorem LO.Arith.lt_of_mul_lt_mul_left {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} {c : M} (h : a * b < a * c) :

b < c

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theorem LO.Arith.lt_of_mul_lt_mul_right {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} {c : M} (h : b * a < c * a) :

b < c

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theorem LO.Arith.pow_three {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (x : M) :

x ^ 3 = x * x * x

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theorem LO.Arith.pow_four {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (x : M) :

x ^ 4 = x * x * x * x

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theorem LO.Arith.pow_four_eq_sq_sq {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (x : M) :

x ^ 4 = (x ^ 2) ^ 2

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def LO.Arith.instCovariantClassHMulLe_arithmetization {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

CovariantClass M M (fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1

Equations

⋯ = ⋯

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def LO.Arith.instCovariantClassHAddLe_arithmetization {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

CovariantClass M M (fun (x x_1 : M) => x + x_1) fun (x x_1 : M) => x ≤ x_1

Equations

⋯ = ⋯

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def LO.Arith.instCovariantClassSwapHMulLe_arithmetization {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

CovariantClass M M (Function.swap fun (x x_1 : M) => x * x_1) fun (x x_1 : M) => x ≤ x_1

Equations

⋯ = ⋯

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@[simp]

theorem LO.Arith.one_lt_mul_self_iff {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

1 < a * a ↔ 1 < a

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@[simp]

theorem LO.Arith.opos_lt_sq_pos_iff {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

0 < a ^ 2 ↔ 0 < a

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@[simp]

theorem LO.Arith.one_lt_sq_iff {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

1 < a ^ 2 ↔ 1 < a

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@[simp]

theorem LO.Arith.mul_self_eq_one_iff {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

a * a = 1 ↔ a = 1

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@[simp]

theorem LO.Arith.sq_eq_one_iff {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} :

a ^ 2 = 1 ↔ a = 1

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theorem LO.Arith.lt_square_of_lt {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} (pos : 1 < a) :

a < a ^ 2

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theorem LO.Arith.two_mul_le_sq {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {i : M} (h : 2 ≤ i) :

2 * i ≤ i ^ 2

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theorem LO.Arith.two_mul_le_sq_add_one {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (i : M) :

2 * i ≤ i ^ 2 + 1

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theorem LO.Arith.two_mul_lt_sq {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {i : M} (h : 2 < i) :

2 * i < i ^ 2

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theorem LO.Arith.succ_le_double_of_pos {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} (h : 0 < a) :

a + 1 ≤ 2 * a

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theorem LO.Arith.two_mul_add_one_lt_two_mul_of_lt {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {a : M} {b : M} (h : a < b) :

2 * a + 1 < 2 * b

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@[simp]

theorem LO.Arith.le_add_add_left {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (a : M) (b : M) (c : M) :

a ≤ a + b + c

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@[simp]

theorem LO.Arith.le_add_add_right {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (a : M) (b : M) (c : M) :

b ≤ a + b + c

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theorem LO.Arith.add_le_cancel {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (a : M) :

AddLECancellable a

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@[simp]

theorem LO.Arith.val_npow {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (k : ℕ) (a : M) :

(LO.FirstOrder.Semiterm.Operator.npow ℒₒᵣ k).val ![a] = a ^ k

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instance LO.Arith.instMonotoneORing {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

LO.FirstOrder.Structure.Monotone ℒₒᵣ M

Equations

LO.Arith.instMonotoneORing = { monotone := ⋯ }

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@[simp]

theorem LO.Arith.zero_ne_add_one {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (x : M) :

0 ≠ x + 1

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@[simp]

theorem LO.Arith.nat_cast_inj {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {n : ℕ} {m : ℕ} :

↑n = ↑m ↔ n = m

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@[simp]

theorem LO.Arith.coe_coe_lt {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] {n : ℕ} {m : ℕ} :

↑n < ↑m ↔ n < m

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theorem LO.Arith.coe_succ {M : Type u_1} [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] (x : ℕ) :

↑(x + 1) = ↑x + 1

TODO: move

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@[reducible, inline]

abbrev LO.Arith.natCast (M : Type u_1) [LO.ORingStruc M] [M ⊧ₘ* 𝐏𝐀⁻] :

NatCast M

Equations

LO.Arith.natCast M = inferInstance

Instances For

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@[simp]

theorem LO.Arith.natCast_nat (n : ℕ) :

↑n = n

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@[simp]

theorem LO.FirstOrder.Semiformula.eval_ballLTSucc' {M : Type u_1} [Zero M] [One M] [Add M] [Mul M] [LT M] [M ⊧ₘ* 𝐏𝐀⁻] {L : LO.FirstOrder.Language} [L.LT] [L.Zero] [L.One] [L.Add] [LO.FirstOrder.Structure L M] [LO.FirstOrder.Structure.LT L M] [LO.FirstOrder.Structure.One L M] [LO.FirstOrder.Structure.Add L M] {ξ : Type u_2} {n : ℕ} {e : Fin n → M} {ε : ξ → M} {t : LO.FirstOrder.Semiterm L ξ n} {p : LO.FirstOrder.Semiformula L ξ (n + 1)} :

(LO.FirstOrder.Semiformula.Evalm M e ε) (LO.FirstOrder.Semiformula.ballLTSucc t p) ↔ ∀ x ≤ LO.FirstOrder.Semiterm.valm M e ε t, (LO.FirstOrder.Semiformula.Evalm M (x :> e) ε) p

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@[simp]

theorem LO.FirstOrder.Semiformula.eval_bexLTSucc' {M : Type u_1} [Zero M] [One M] [Add M] [Mul M] [LT M] [M ⊧ₘ* 𝐏𝐀⁻] {L : LO.FirstOrder.Language} [L.LT] [L.Zero] [L.One] [L.Add] [LO.FirstOrder.Structure L M] [LO.FirstOrder.Structure.LT L M] [LO.FirstOrder.Structure.One L M] [LO.FirstOrder.Structure.Add L M] {ξ : Type u_2} {n : ℕ} {e : Fin n → M} {ε : ξ → M} {t : LO.FirstOrder.Semiterm L ξ n} {p : LO.FirstOrder.Semiformula L ξ (n + 1)} :

(LO.FirstOrder.Semiformula.Evalm M e ε) (LO.FirstOrder.Semiformula.bexLTSucc t p) ↔ ∃ x ≤ LO.FirstOrder.Semiterm.valm M e ε t, (LO.FirstOrder.Semiformula.Evalm M (x :> e) ε) p