url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | . sorry | case a.left
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'(φ ∨' ψ) →' ¬'φ ∧' ¬'ψ
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'φ ∧' ¬'ψ →' ¬'(φ ∨' ψ) | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'φ ∧' ¬'ψ →' ¬'(φ ∨' ψ) | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'(φ ∨' ψ) →' ¬'φ ∧' ¬'ψ
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'φ ∧' ¬'ψ →' ¬'(φ ∨' ψ)
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | . simp [deduction];
have h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥') := by simp;
have h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ := by simp;
have h3 := (EDisj (EConj₁ h1) (EConj₂ h1));
have h4 := MP h3 h2;
assumption; | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'φ ∧' ¬'ψ →' ¬'(φ ∨' ψ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'φ ∧' ¬'ψ →' ¬'(φ ∨' ψ)
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | sorry | case a.left
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'(φ ∨' ψ) →' ¬'φ ∧' ¬'ψ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.left
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'(φ ∨' ψ) →' ¬'φ ∧' ¬'ψ
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | simp [deduction] | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'φ ∧' ¬'ψ →' ¬'(φ ∨' ψ) | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ ∅ ⊢ ¬'φ ∧' ¬'ψ →' ¬'(φ ∨' ψ)
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | have h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥') := by simp; | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥'
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | have h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ := by simp; | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥'
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | have h3 := (EDisj (EConj₁ h1) (EConj₂ h1)) | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
h3 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ →' ⊥'
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥'
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | have h4 := MP h3 h2 | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
h3 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ →' ⊥'
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
h3 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ →' ⊥'
h4 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥'
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
h3 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ →' ⊥'
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥'
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | assumption | case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
h3 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ →' ⊥'
h4 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥'
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.right
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
h2 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
h3 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ →' ⊥'
h4 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥'
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ ⊥'
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | simp | L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.duMorganConj | [161, 1] | [170, 16] | simp | L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type u
inst✝⁸ : DecidableEq L
inst✝⁷ : HasBot L
inst✝⁶ : HasArrow L
inst✝⁵ : HasLnot L
inst✝⁴ : HasLor L
inst✝³ : HasLand L
inst✝² : HasLiff L
inst✝¹ : HilbertSystem L
inst✝ : HPM L
φ ψ : L
h1 : {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ (φ →' ⊥') ∧' (ψ →' ⊥')
⊢ {(φ →' ⊥') ∧' (ψ →' ⊥')} ∪ {φ ∨' ψ} ⊢ φ ∨' ψ
TACTIC:
|
https://github.com/SnO2WMaN/lean4-propositional-logic.git | 01e7b1a79bbbe2b5aa55cc4806cff3eb45efd7cb | PropositionalLogic/HilbertSystem/HPM.lean | PropositionalLogic.HilbertSystem.HPM.strongerThanHPM₀ | [266, 1] | [268, 8] | admit | L : Type u
inst✝⁷ : DecidableEq L
inst✝⁶ : HilbertSystem L
inst✝⁵ : HasBot L
inst✝⁴ : HasArrow L
inst✝³ : HasLnot L
inst✝² : HasLor L
inst✝¹ : HasLand L
inst✝ : HasLiff L
Γ : Context L
φ : L
⊢ (HPM₀ L → (Γ ⊢ φ)) → HPM L → (Γ ⊢ φ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type u
inst✝⁷ : DecidableEq L
inst✝⁶ : HilbertSystem L
inst✝⁵ : HasBot L
inst✝⁴ : HasArrow L
inst✝³ : HasLnot L
inst✝² : HasLor L
inst✝¹ : HasLand L
inst✝ : HasLiff L
Γ : Context L
φ : L
⊢ (HPM₀ L → (Γ ⊢ φ)) → HPM L → (Γ ⊢ φ)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section06orderingsAndLattices/Sheet3.lean | Section6Sheet3solutions.foo | [31, 1] | [36, 34] | letI : DistribLattice L := { (show Lattice L by infer_instance) with le_sup_inf := fun x y z => by rw [← h] } | L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
⊢ ∀ (a b c : L), a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c | L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
this : DistribLattice L :=
let src :=
let_fun this := inferInstance;
this;
DistribLattice.mk (_ : ∀ (x y z : L), (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z)
⊢ ∀ (a b c : L), a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
⊢ ∀ (a b c : L), a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section06orderingsAndLattices/Sheet3.lean | Section6Sheet3solutions.foo | [31, 1] | [36, 34] | exact fun a b c => inf_sup_left | L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
this : DistribLattice L :=
let src :=
let_fun this := inferInstance;
this;
DistribLattice.mk (_ : ∀ (x y z : L), (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z)
⊢ ∀ (a b c : L), a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
this : DistribLattice L :=
let src :=
let_fun this := inferInstance;
this;
DistribLattice.mk (_ : ∀ (x y z : L), (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z)
⊢ ∀ (a b c : L), a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section06orderingsAndLattices/Sheet3.lean | Section6Sheet3solutions.foo | [31, 1] | [36, 34] | infer_instance | L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
⊢ Lattice L | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
⊢ Lattice L
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section06orderingsAndLattices/Sheet3.lean | Section6Sheet3solutions.foo | [31, 1] | [36, 34] | rw [← h] | L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
src✝ : Lattice L :=
let_fun this := inferInstance;
this
x y z : L
⊢ (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Type
inst✝ : Lattice L
h : ∀ (a b c : L), a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c)
src✝ : Lattice L :=
let_fun this := inferInstance;
this
x y z : L
⊢ (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_cubes | [24, 1] | [30, 9] | induction' n with d hd | n : ℕ
⊢ ∑ i in range n, ↑i ^ 3 = (↑n * (↑n - 1) / 2) ^ 2 | case zero
⊢ ∑ i in range Nat.zero, ↑i ^ 3 = (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 2
case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ ∑ i in range (Nat.succ d), ↑i ^ 3 = (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∑ i in range n, ↑i ^ 3 = (↑n * (↑n - 1) / 2) ^ 2
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_cubes | [24, 1] | [30, 9] | simp | case zero
⊢ ∑ i in range Nat.zero, ↑i ^ 3 = (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ ∑ i in range Nat.zero, ↑i ^ 3 = (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 2
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_cubes | [24, 1] | [30, 9] | rw [Finset.sum_range_succ, hd] | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ ∑ i in range (Nat.succ d), ↑i ^ 3 = (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2 | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ (↑d * (↑d - 1) / 2) ^ 2 + ↑d ^ 3 = (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ ∑ i in range (Nat.succ d), ↑i ^ 3 = (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_cubes | [24, 1] | [30, 9] | simp | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ (↑d * (↑d - 1) / 2) ^ 2 + ↑d ^ 3 = (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2 | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ (↑d * (↑d - 1)) ^ 2 / 2 ^ 2 + ↑d ^ 3 = ((↑d + 1) * ↑d) ^ 2 / 2 ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ (↑d * (↑d - 1) / 2) ^ 2 + ↑d ^ 3 = (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_cubes | [24, 1] | [30, 9] | ring | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ (↑d * (↑d - 1)) ^ 2 / 2 ^ 2 + ↑d ^ 3 = ((↑d + 1) * ↑d) ^ 2 / 2 ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 3 = (↑d * (↑d - 1) / 2) ^ 2
⊢ (↑d * (↑d - 1)) ^ 2 / 2 ^ 2 + ↑d ^ 3 = ((↑d + 1) * ↑d) ^ 2 / 2 ^ 2
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_fifths | [33, 1] | [40, 9] | induction' n with d hd | n : ℕ
⊢ ∑ i in range n, ↑i ^ 5 = (4 * (↑n * (↑n - 1) / 2) ^ 3 - (↑n * (↑n - 1) / 2) ^ 2) / 3 | case zero
⊢ ∑ i in range Nat.zero, ↑i ^ 5 =
(4 * (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 3 - (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 2) / 3
case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ ∑ i in range (Nat.succ d), ↑i ^ 5 =
(4 * (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 3 - (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2) / 3 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ ∑ i in range n, ↑i ^ 5 = (4 * (↑n * (↑n - 1) / 2) ^ 3 - (↑n * (↑n - 1) / 2) ^ 2) / 3
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_fifths | [33, 1] | [40, 9] | simp | case zero
⊢ ∑ i in range Nat.zero, ↑i ^ 5 =
(4 * (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 3 - (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 2) / 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ ∑ i in range Nat.zero, ↑i ^ 5 =
(4 * (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 3 - (↑Nat.zero * (↑Nat.zero - 1) / 2) ^ 2) / 3
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_fifths | [33, 1] | [40, 9] | rw [Finset.sum_range_succ, hd] | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ ∑ i in range (Nat.succ d), ↑i ^ 5 =
(4 * (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 3 - (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2) / 3 | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3 + ↑d ^ 5 =
(4 * (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 3 - (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2) / 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ ∑ i in range (Nat.succ d), ↑i ^ 5 =
(4 * (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 3 - (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2) / 3
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_fifths | [33, 1] | [40, 9] | simp | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3 + ↑d ^ 5 =
(4 * (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 3 - (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2) / 3 | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ (4 * ((↑d * (↑d - 1)) ^ 3 / 2 ^ 3) - (↑d * (↑d - 1)) ^ 2 / 2 ^ 2) / 3 + ↑d ^ 5 =
(4 * (((↑d + 1) * ↑d) ^ 3 / 2 ^ 3) - ((↑d + 1) * ↑d) ^ 2 / 2 ^ 2) / 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3 + ↑d ^ 5 =
(4 * (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 3 - (↑(Nat.succ d) * (↑(Nat.succ d) - 1) / 2) ^ 2) / 3
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet7.lean | Section15Sheet7Solutions.sum_fifths | [33, 1] | [40, 9] | ring | case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ (4 * ((↑d * (↑d - 1)) ^ 3 / 2 ^ 3) - (↑d * (↑d - 1)) ^ 2 / 2 ^ 2) / 3 + ↑d ^ 5 =
(4 * (((↑d + 1) * ↑d) ^ 3 / 2 ^ 3) - ((↑d + 1) * ↑d) ^ 2 / 2 ^ 2) / 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : ∑ i in range d, ↑i ^ 5 = (4 * (↑d * (↑d - 1) / 2) ^ 3 - (↑d * (↑d - 1) / 2) ^ 2) / 3
⊢ (4 * ((↑d * (↑d - 1)) ^ 3 / 2 ^ 3) - (↑d * (↑d - 1)) ^ 2 / 2 ^ 2) / 3 + ↑d ^ 5 =
(4 * (((↑d + 1) * ↑d) ^ 3 / 2 ^ 3) - ((↑d + 1) * ↑d) ^ 2 / 2 ^ 2) / 3
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section20representationTheory/Sheet2.lean | Section20Sheet2Solutions.RepMap.comp_id | [94, 1] | [95, 11] | ext | k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
⊢ comp φ (id ρ) = φ | case toLinearMap.h
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
x✝ : V
⊢ (comp φ (id ρ)).toLinearMap x✝ = φ.toLinearMap x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
⊢ comp φ (id ρ) = φ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section20representationTheory/Sheet2.lean | Section20Sheet2Solutions.RepMap.comp_id | [94, 1] | [95, 11] | rfl | case toLinearMap.h
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
x✝ : V
⊢ (comp φ (id ρ)).toLinearMap x✝ = φ.toLinearMap x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case toLinearMap.h
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
x✝ : V
⊢ (comp φ (id ρ)).toLinearMap x✝ = φ.toLinearMap x✝
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section20representationTheory/Sheet2.lean | Section20Sheet2Solutions.RepMap.id_comp | [97, 1] | [98, 11] | ext | k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
⊢ comp (id σ) φ = φ | case toLinearMap.h
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
x✝ : V
⊢ (comp (id σ) φ).toLinearMap x✝ = φ.toLinearMap x✝ | Please generate a tactic in lean4 to solve the state.
STATE:
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
⊢ comp (id σ) φ = φ
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section20representationTheory/Sheet2.lean | Section20Sheet2Solutions.RepMap.id_comp | [97, 1] | [98, 11] | rfl | case toLinearMap.h
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
x✝ : V
⊢ (comp (id σ) φ).toLinearMap x✝ = φ.toLinearMap x✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case toLinearMap.h
k : Type
inst✝⁷ : Field k
G : Type
inst✝⁶ : Group G
V : Type
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module k V
W : Type
inst✝³ : AddCommGroup W
inst✝² : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝¹ : AddCommGroup X
inst✝ : Module k X
φ : RepMap ρ σ
x✝ : V
⊢ (comp (id σ) φ).toLinearMap x✝ = φ.toLinearMap x✝
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section20representationTheory/Sheet2.lean | Section20Sheet2Solutions.RepMap.comp_assoc | [100, 1] | [105, 6] | rfl | k : Type
inst✝⁹ : Field k
G : Type
inst✝⁸ : Group G
V : Type
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
W : Type
inst✝⁵ : AddCommGroup W
inst✝⁴ : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝³ : AddCommGroup X
inst✝² : Module k X
τ : Representation k G X
Y : Type
inst✝¹ : AddCommGroup Y
inst✝ : Module k Y
υ : Representation k G Y
ξ : RepMap τ υ
ψ : RepMap σ τ
φ : RepMap ρ σ
⊢ comp (comp ξ ψ) φ = comp ξ (comp ψ φ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : Type
inst✝⁹ : Field k
G : Type
inst✝⁸ : Group G
V : Type
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
W : Type
inst✝⁵ : AddCommGroup W
inst✝⁴ : Module k W
ρ : Representation k G V
σ : Representation k G W
X : Type
inst✝³ : AddCommGroup X
inst✝² : Module k X
τ : Representation k G X
Y : Type
inst✝¹ : AddCommGroup Y
inst✝ : Module k Y
υ : Representation k G Y
ξ : RepMap τ υ
ψ : RepMap σ τ
φ : RepMap ρ σ
⊢ comp (comp ξ ψ) φ = comp ξ (comp ψ φ)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.factorial_eq_prod | [14, 1] | [21, 10] | induction' n with d hd | n : ℕ
⊢ Nat.factorial n = ∏ i in Finset.Icc 1 n, i | case zero
⊢ Nat.factorial Nat.zero = ∏ i in Finset.Icc 1 Nat.zero, i
case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ Nat.factorial (Nat.succ d) = ∏ i in Finset.Icc 1 (Nat.succ d), i | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ Nat.factorial n = ∏ i in Finset.Icc 1 n, i
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.factorial_eq_prod | [14, 1] | [21, 10] | rfl | case zero
⊢ Nat.factorial Nat.zero = ∏ i in Finset.Icc 1 Nat.zero, i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ Nat.factorial Nat.zero = ∏ i in Finset.Icc 1 Nat.zero, i
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.factorial_eq_prod | [14, 1] | [21, 10] | rw [Nat.factorial_succ, hd] | case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ Nat.factorial (Nat.succ d) = ∏ i in Finset.Icc 1 (Nat.succ d), i | case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i = ∏ i in Finset.Icc 1 (Nat.succ d), i | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ Nat.factorial (Nat.succ d) = ∏ i in Finset.Icc 1 (Nat.succ d), i
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.factorial_eq_prod | [14, 1] | [21, 10] | rw [Finset.Icc_eq_cons_Ico (show 1 ≤ d + 1 by linarith)] | case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i = ∏ i in Finset.Icc 1 (Nat.succ d), i | case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i =
∏ i in Finset.cons (d + 1) (Finset.Ico 1 (d + 1)) (_ : d + 1 ∉ Finset.Ico 1 (d + 1)), i | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i = ∏ i in Finset.Icc 1 (Nat.succ d), i
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.factorial_eq_prod | [14, 1] | [21, 10] | rw [Finset.prod_cons] | case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i =
∏ i in Finset.cons (d + 1) (Finset.Ico 1 (d + 1)) (_ : d + 1 ∉ Finset.Ico 1 (d + 1)), i | case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i = (d + 1) * ∏ x in Finset.Ico 1 (d + 1), x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i =
∏ i in Finset.cons (d + 1) (Finset.Ico 1 (d + 1)) (_ : d + 1 ∉ Finset.Ico 1 (d + 1)), i
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.factorial_eq_prod | [14, 1] | [21, 10] | congr | case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i = (d + 1) * ∏ x in Finset.Ico 1 (d + 1), x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ (d + 1) * ∏ i in Finset.Icc 1 d, i = (d + 1) * ∏ x in Finset.Ico 1 (d + 1), x
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.factorial_eq_prod | [14, 1] | [21, 10] | linarith | d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ 1 ≤ d + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
d : ℕ
hd : Nat.factorial d = ∏ i in Finset.Icc 1 d, i
⊢ 1 ≤ d + 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.wilson_theorem | [23, 1] | [32, 30] | have := (Nat.prime_iff_fac_equiv_neg_one (?_ : p ≠ 1)).1 hp | p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = -1 | case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = -1
case refine_1
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
⊢ p ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.wilson_theorem | [23, 1] | [32, 30] | rw [← this, hn] | case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = -1 | case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = ↑(Nat.factorial (4 * n + 1 - 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.wilson_theorem | [23, 1] | [32, 30] | simp | case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = ↑(Nat.factorial (4 * n + 1 - 1)) | case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = ↑(Nat.factorial (4 * n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = ↑(Nat.factorial (4 * n + 1 - 1))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.wilson_theorem | [23, 1] | [32, 30] | rw [factorial_eq_prod, Nat.cast_prod] | case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = ↑(Nat.factorial (4 * n)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
this : ↑(Nat.factorial (p - 1)) = -1
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = ↑(Nat.factorial (4 * n))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.wilson_theorem | [23, 1] | [32, 30] | exact Nat.Prime.ne_one hp | case refine_1
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
⊢ p ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
p n : ℕ
hp : Nat.Prime p
hn : p = 4 * n + 1
⊢ p ≠ 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | cases' hp2 with n hn | p : ℕ
hp : Nat.Prime p
hp2 : ∃ n, p = 4 * n + 1
⊢ ∃ i, i ^ 2 = -1 | case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
⊢ ∃ i, i ^ 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
p : ℕ
hp : Nat.Prime p
hp2 : ∃ n, p = 4 * n + 1
⊢ ∃ i, i ^ 2 = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | set i := ∏ j in Finset.Icc 1 (2 * n), (j : ZMod p) with hi | case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
⊢ ∃ i, i ^ 2 = -1 | case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ ∃ i, i ^ 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
⊢ ∃ i, i ^ 2 = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | have h1 : ∏ j in Finset.Icc 1 (2 * n), (-1 : ZMod p) = 1 | case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ ∃ i, i ^ 2 = -1 | case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ ∏ j in Finset.Icc 1 (2 * n), -1 = 1
case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∃ i, i ^ 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ ∃ i, i ^ 2 = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | have h2 : ∏ j in Finset.Icc 1 (2 * n), (-j : ZMod p) = i | case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∃ i, i ^ 2 = -1 | case h2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∏ j in Finset.Icc 1 (2 * n), -↑j = i
case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∃ i, i ^ 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∃ i, i ^ 2 = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | have h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), (j : ZMod p) = i | case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∃ i, i ^ 2 = -1 | case h3
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∃ i, i ^ 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∃ i, i ^ 2 = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | use i | case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∃ i, i ^ 2 = -1 | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ i ^ 2 = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∃ i, i ^ 2 = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [pow_two] | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ i ^ 2 = -1 | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ i * i = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ i ^ 2 = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | nth_rw 1 [hi] | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ i * i = -1 | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ (∏ j in Finset.Icc 1 (2 * n), ↑j) * i = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ i * i = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [← h3] | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ (∏ j in Finset.Icc 1 (2 * n), ↑j) * i = -1 | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ (∏ j in Finset.Icc 1 (2 * n), ↑j) * ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ (∏ j in Finset.Icc 1 (2 * n), ↑j) * i = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [← Finset.prod_union] | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ (∏ j in Finset.Icc 1 (2 * n), ↑j) * ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = -1 | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ x in Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n), ↑x = -1
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Disjoint (Finset.Icc 1 (2 * n)) (Finset.Icc (2 * n + 1) (4 * n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ (∏ j in Finset.Icc 1 (2 * n), ↑j) * ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [Finset.prod_const] | case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ ∏ j in Finset.Icc 1 (2 * n), -1 = 1 | case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ (-1) ^ Finset.card (Finset.Icc 1 (2 * n)) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ ∏ j in Finset.Icc 1 (2 * n), -1 = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | simp only [Nat.add_succ_sub_one, add_zero, Nat.card_Icc] | case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ (-1) ^ Finset.card (Finset.Icc 1 (2 * n)) = 1 | case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ (-1) ^ (2 * n) = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ (-1) ^ Finset.card (Finset.Icc 1 (2 * n)) = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [pow_mul, neg_one_pow_two, one_pow] | case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ (-1) ^ (2 * n) = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h1
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
⊢ (-1) ^ (2 * n) = 1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | conv_lhs => rhs; ext; rw [neg_eq_neg_one_mul] | case h2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∏ j in Finset.Icc 1 (2 * n), -↑j = i | case h2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∏ x in Finset.Icc 1 (2 * n), -1 * ↑x = i | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∏ j in Finset.Icc 1 (2 * n), -↑j = i
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [Finset.prod_mul_distrib, h1, one_mul] | case h2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∏ x in Finset.Icc 1 (2 * n), -1 * ↑x = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
⊢ ∏ x in Finset.Icc 1 (2 * n), -1 * ↑x = i
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [← h2] | case h3
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i | case h3
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = ∏ j in Finset.Icc 1 (2 * n), -↑j | Please generate a tactic in lean4 to solve the state.
STATE:
case h3
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | apply Finset.prod_bij' (fun j _ => p - j) (fun j _ => p - j) <;>
rintro a ha <;>
rw [Finset.mem_Icc] at * | case h3
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = ∏ j in Finset.Icc 1 (2 * n), -↑j | case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a
case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ ↑a = -↑(p - a) | Please generate a tactic in lean4 to solve the state.
STATE:
case h3
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
⊢ ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = ∏ j in Finset.Icc 1 (2 * n), -↑j
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rotate_right | case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a
case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ ↑a = -↑(p - a) | case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ ↑a = -↑(p - a)
case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a
case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ ↑a = -↑(p - a)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | suffices : a + (p - a) = p | case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ ↑a = -↑(p - a)
case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a | case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑a = -↑(p - a)
case this
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ a + (p - a) = p
case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a | Please generate a tactic in lean4 to solve the state.
STATE:
case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ ↑a = -↑(p - a)
case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | all_goals omega | case this
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ a + (p - a) = p
case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ a + (p - a) = p
case h3.hi
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ 1 ≤ p - a ∧ p - a ≤ 2 * n
case h3.hj
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ 2 * n + 1 ≤ p - a ∧ p - a ≤ 4 * n
case h3.left_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
⊢ p - (p - a) = a
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [eq_neg_iff_add_eq_zero] | case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑a = -↑(p - a) | case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑a + ↑(p - a) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑a = -↑(p - a)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | norm_cast | case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑a + ↑(p - a) = 0 | case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑(a + (p - a)) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑a + ↑(p - a) = 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | simp [this] | case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑(a + (p - a)) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h3.h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 2 * n + 1 ≤ a ∧ a ≤ 4 * n
this : a + (p - a) = p
⊢ ↑(a + (p - a)) = 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | omega | case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h3.right_inv
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
a : ℕ
ha : 1 ≤ a ∧ a ≤ 2 * n
⊢ p - (p - a) = a
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | convert_to ∏ j in Finset.Icc 1 (4 * n), (j : ZMod p) = -1 | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ x in Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n), ↑x = -1 | case h.e'_2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ x in Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n), ↑x = ∏ j in Finset.Icc 1 (4 * n), ↑j
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = -1 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ x in Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n), ↑x = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | congr | case h.e'_2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ x in Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n), ↑x = ∏ j in Finset.Icc 1 (4 * n), ↑j | case h.e'_2.e_s
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n) = Finset.Icc 1 (4 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ x in Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n), ↑x = ∏ j in Finset.Icc 1 (4 * n), ↑j
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | ext x | case h.e'_2.e_s
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n) = Finset.Icc 1 (4 * n) | case h.e'_2.e_s.a
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ x ∈ Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n) ↔ x ∈ Finset.Icc 1 (4 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.e_s
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n) = Finset.Icc 1 (4 * n)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [Finset.mem_union] | case h.e'_2.e_s.a
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ x ∈ Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n) ↔ x ∈ Finset.Icc 1 (4 * n) | case h.e'_2.e_s.a
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ x ∈ Finset.Icc 1 (2 * n) ∨ x ∈ Finset.Icc (2 * n + 1) (4 * n) ↔ x ∈ Finset.Icc 1 (4 * n) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.e_s.a
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ x ∈ Finset.Icc 1 (2 * n) ∪ Finset.Icc (2 * n + 1) (4 * n) ↔ x ∈ Finset.Icc 1 (4 * n)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | simp only [Finset.mem_Icc] | case h.e'_2.e_s.a
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ x ∈ Finset.Icc 1 (2 * n) ∨ x ∈ Finset.Icc (2 * n + 1) (4 * n) ↔ x ∈ Finset.Icc 1 (4 * n) | case h.e'_2.e_s.a
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ 1 ≤ x ∧ x ≤ 2 * n ∨ 2 * n + 1 ≤ x ∧ x ≤ 4 * n ↔ 1 ≤ x ∧ x ≤ 4 * n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.e_s.a
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ x ∈ Finset.Icc 1 (2 * n) ∨ x ∈ Finset.Icc (2 * n + 1) (4 * n) ↔ x ∈ Finset.Icc 1 (4 * n)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | intros | case h.e'_2.e_s.a.mpr
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ 1 ≤ x ∧ x ≤ 4 * n → 1 ≤ x ∧ x ≤ 2 * n ∨ 2 * n + 1 ≤ x ∧ x ≤ 4 * n | case h.e'_2.e_s.a.mpr
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
a✝ : 1 ≤ x ∧ x ≤ 4 * n
⊢ 1 ≤ x ∧ x ≤ 2 * n ∨ 2 * n + 1 ≤ x ∧ x ≤ 4 * n | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.e_s.a.mpr
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
⊢ 1 ≤ x ∧ x ≤ 4 * n → 1 ≤ x ∧ x ≤ 2 * n ∨ 2 * n + 1 ≤ x ∧ x ≤ 4 * n
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | omega | case h.e'_2.e_s.a.mpr
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
a✝ : 1 ≤ x ∧ x ≤ 4 * n
⊢ 1 ≤ x ∧ x ≤ 2 * n ∨ 2 * n + 1 ≤ x ∧ x ≤ 4 * n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_2.e_s.a.mpr
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
a✝ : 1 ≤ x ∧ x ≤ 4 * n
⊢ 1 ≤ x ∧ x ≤ 2 * n ∨ 2 * n + 1 ≤ x ∧ x ≤ 4 * n
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | apply wilson_theorem hp hn | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = -1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ ∏ j in Finset.Icc 1 (4 * n), ↑j = -1
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [disjoint_iff_inf_le] | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Disjoint (Finset.Icc 1 (2 * n)) (Finset.Icc (2 * n + 1) (4 * n)) | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Finset.Icc 1 (2 * n) ⊓ Finset.Icc (2 * n + 1) (4 * n) ≤ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Disjoint (Finset.Icc 1 (2 * n)) (Finset.Icc (2 * n + 1) (4 * n))
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rintro x (hx : x ∈ _ ∩ _) | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Finset.Icc 1 (2 * n) ⊓ Finset.Icc (2 * n + 1) (4 * n) ≤ ⊥ | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
hx : x ∈ Finset.Icc 1 (2 * n) ∩ Finset.Icc (2 * n + 1) (4 * n)
⊢ x ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
⊢ Finset.Icc 1 (2 * n) ⊓ Finset.Icc (2 * n + 1) (4 * n) ≤ ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rw [Finset.mem_inter, Finset.mem_Icc, Finset.mem_Icc] at hx | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
hx : x ∈ Finset.Icc 1 (2 * n) ∩ Finset.Icc (2 * n + 1) (4 * n)
⊢ x ∈ ⊥ | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
hx : (1 ≤ x ∧ x ≤ 2 * n) ∧ 2 * n + 1 ≤ x ∧ x ≤ 4 * n
⊢ x ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
hx : x ∈ Finset.Icc 1 (2 * n) ∩ Finset.Icc (2 * n + 1) (4 * n)
⊢ x ∈ ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | rcases hx with ⟨⟨_, _⟩, _, _⟩ | case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
hx : (1 ≤ x ∧ x ≤ 2 * n) ∧ 2 * n + 1 ≤ x ∧ x ≤ 4 * n
⊢ x ∈ ⊥ | case h.intro.intro.intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
left✝¹ : 1 ≤ x
right✝¹ : x ≤ 2 * n
left✝ : 2 * n + 1 ≤ x
right✝ : x ≤ 4 * n
⊢ x ∈ ⊥ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
hx : (1 ≤ x ∧ x ≤ 2 * n) ∧ 2 * n + 1 ≤ x ∧ x ≤ 4 * n
⊢ x ∈ ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet8.lean | Section15Sheet7Solutions.exists_sqrt_neg_one_of_one_mod_four | [34, 1] | [75, 13] | linarith | case h.intro.intro.intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
left✝¹ : 1 ≤ x
right✝¹ : x ≤ 2 * n
left✝ : 2 * n + 1 ≤ x
right✝ : x ≤ 4 * n
⊢ x ∈ ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.intro.intro.intro
p : ℕ
hp : Nat.Prime p
n : ℕ
hn : p = 4 * n + 1
i : ZMod p := ∏ j in Finset.Icc 1 (2 * n), ↑j
hi : i = ∏ j in Finset.Icc 1 (2 * n), ↑j
h1 : ∏ j in Finset.Icc 1 (2 * n), -1 = 1
h2 : ∏ j in Finset.Icc 1 (2 * n), -↑j = i
h3 : ∏ j in Finset.Icc (2 * n + 1) (4 * n), ↑j = i
x : ℕ
left✝¹ : 1 ≤ x
right✝¹ : x ≤ 2 * n
left✝ : 2 * n + 1 ≤ x
right✝ : x ≤ 4 * n
⊢ x ∈ ⊥
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section15numberTheory/Sheet2.lean | Section15sheet2.int_dvd_iff | [31, 1] | [32, 12] | simp [hn] | x n : ℤ
hn : n ≠ 0
⊢ x ∣ n ↔ Int.natAbs x ∈ Nat.divisors (Int.natAbs n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x n : ℤ
hn : n ≠ 0
⊢ x ∣ n ↔ Int.natAbs x ∈ Nat.divisors (Int.natAbs n)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section15numberTheory/Sheet5.lean | Section15Sheet5Solutions.sixteen_pow_sixtyfour_mod_nineteen | [30, 1] | [30, 80] | rfl | ⊢ 16 ^ 64 = 16 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ 16 ^ 64 = 16
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_thirtyseven_mul | [31, 1] | [33, 8] | sorry | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => 37 * a n) (37 * t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => 37 * a n) (37 * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_pos_const_mul | [37, 1] | [39, 8] | sorry | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
⊢ TendsTo (fun n => c * a n) (c * t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : 0 < c
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_neg_const_mul | [43, 1] | [45, 8] | sorry | a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
⊢ TendsTo (fun n => c * a n) (c * t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
h : TendsTo a t
c : ℝ
hc : c < 0
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_const_mul | [49, 1] | [51, 8] | sorry | a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => c * a n) (c * t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => c * a n) (c * t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_mul_const | [55, 1] | [57, 6] | sorry | a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n * c) (t * c) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t c : ℝ
h : TendsTo a t
⊢ TendsTo (fun n => a n * c) (t * c)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_neg' | [60, 1] | [61, 40] | simpa using tendsTo_const_mul (-1) ha | a : ℕ → ℝ
t : ℝ
ha : TendsTo a t
⊢ TendsTo (fun n => -a n) (-t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
ha : TendsTo a t
⊢ TendsTo (fun n => -a n) (-t)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_of_tendsTo_sub | [65, 1] | [67, 8] | sorry | a b : ℕ → ℝ
t u : ℝ
h1 : TendsTo (fun n => a n - b n) t
h2 : TendsTo b u
⊢ TendsTo a (t + u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
h1 : TendsTo (fun n => a n - b n) t
h2 : TendsTo b u
⊢ TendsTo a (t + u)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_sub_lim_iff | [70, 1] | [71, 8] | sorry | a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t ↔ TendsTo (fun n => a n - t) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
t : ℝ
⊢ TendsTo a t ↔ TendsTo (fun n => a n - t) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_zero_mul_tendsTo_zero | [75, 1] | [77, 8] | sorry | a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
⊢ TendsTo (fun n => a n * b n) 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
ha : TendsTo a 0
hb : TendsTo b 0
⊢ TendsTo (fun n => a n * b n) 0
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_mul | [81, 1] | [83, 6] | sorry | a b : ℕ → ℝ
t u : ℝ
ha : TendsTo a t
hb : TendsTo b u
⊢ TendsTo (fun n => a n * b n) (t * u) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℕ → ℝ
t u : ℝ
ha : TendsTo a t
hb : TendsTo b u
⊢ TendsTo (fun n => a n * b n) (t * u)
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Section02reals/Sheet6.lean | Section2sheet6.tendsTo_unique | [87, 1] | [88, 8] | sorry | a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
⊢ s = t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : ℕ → ℝ
s t : ℝ
hs : TendsTo a s
ht : TendsTo a t
⊢ s = t
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean | Ideal.mem_iff_associated | [106, 1] | [112, 7] | rcases hab with ⟨u, rfl⟩ | R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a b : R
hab : Associated a b
⊢ a ∈ I ↔ b ∈ I | case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
⊢ a ∈ I ↔ a * ↑u ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a b : R
hab : Associated a b
⊢ a ∈ I ↔ b ∈ I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean | Ideal.mem_iff_associated | [106, 1] | [112, 7] | refine' ⟨I.mul_mem_right _, _⟩ | case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
⊢ a ∈ I ↔ a * ↑u ∈ I | case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
⊢ a * ↑u ∈ I → a ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
⊢ a ∈ I ↔ a * ↑u ∈ I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean | Ideal.mem_iff_associated | [106, 1] | [112, 7] | intro h | case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
⊢ a * ↑u ∈ I → a ∈ I | case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
h : a * ↑u ∈ I
⊢ a ∈ I | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
⊢ a * ↑u ∈ I → a ∈ I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean | Ideal.mem_iff_associated | [106, 1] | [112, 7] | convert I.mul_mem_right ((u⁻¹ : Rˣ) : R) h | case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
h : a * ↑u ∈ I
⊢ a ∈ I | case h.e'_4
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
h : a * ↑u ∈ I
⊢ a = a * ↑u * ↑u⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
h : a * ↑u ∈ I
⊢ a ∈ I
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean | Ideal.mem_iff_associated | [106, 1] | [112, 7] | simp | case h.e'_4
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
h : a * ↑u ∈ I
⊢ a = a * ↑u * ↑u⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
I : Ideal R
a : R
u : Rˣ
h : a * ↑u ∈ I
⊢ a = a * ↑u * ↑u⁻¹
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean | Ideal.IsPrime.not_one_mem | [114, 1] | [119, 29] | intro h | R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
P : Ideal R
hI : IsPrime P
⊢ 1 ∉ P | R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
P : Ideal R
hI : IsPrime P
h : 1 ∈ P
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
P : Ideal R
hI : IsPrime P
⊢ 1 ∉ P
TACTIC:
|
https://github.com/ImperialCollegeLondon/formalising-mathematics-2024.git | b732ed1352e87b4474b0520d1383994e069f8057 | FormalisingMathematics2024/Solutions/Section14UFDsAndPIDsEtc/Sheet3.lean | Ideal.IsPrime.not_one_mem | [114, 1] | [119, 29] | apply hI.ne_top | R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
P : Ideal R
hI : IsPrime P
h : 1 ∈ P
⊢ False | R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
P : Ideal R
hI : IsPrime P
h : 1 ∈ P
⊢ P = ⊤ | Please generate a tactic in lean4 to solve the state.
STATE:
R✝ : Type
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
inst✝¹ : UniqueFactorizationMonoid R✝
R : Type
inst✝ : CommRing R
P : Ideal R
hI : IsPrime P
h : 1 ∈ P
⊢ False
TACTIC:
|
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