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train · 219k rows

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
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	obtain ⟨z₀, hz₀⟩ := nonempty_iff_ne_empty.2 h	z z₀ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ ⊢ _root_.has_logs U	case intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U ⊢ _root_.has_logs U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	rintro f hf hfz	case intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U ⊢ _root_.has_logs U	case intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	obtain ⟨lf, hlf1, hlf2⟩ := hp (deriv f / f) ((hf.deriv hU).div hf hfz)	case intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	let g : ℂ → ℂ := λ z => lf z + (log (f z₀) - lf z₀)	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	set h : ℂ → ℂ := f / (exp ∘ g)	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	have h3 : DifferentiableOn ℂ g U := hlf1.add (differentiableOn_const _)	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	have e4 : DifferentiableOn ℂ (exp ∘ g) U := differentiable_exp.comp_differentiableOn h3	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	have e1 : DifferentiableOn ℂ h U := hf.div e4 (λ z _ => exp_ne_zero _)	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	refine ⟨g, h3, ?_⟩	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ EqOn f (cexp ∘ g) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	suffices h : EqOn h (λ _ => 1) U by exact λ z hz => eq_of_div_eq_one (h hz)	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ EqOn f (cexp ∘ g) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ EqOn h (fun x => 1) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	have : 1 = h z₀ := by unfold_let ; simp [exp_log, hfz z₀ hz₀]	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ EqOn h (fun x => 1) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ ⊢ EqOn h (fun x => 1) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	rw [this]	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ ⊢ EqOn h (fun x => 1) U	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ ⊢ EqOn h (fun x => h z₀) U
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	refine EqOn_of_deriv_eq_zero hU hU' e1 (λ z hz => ?_) hz₀	case intro.intro.intro z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ ⊢ EqOn h (fun x => h z₀) U	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U ⊢ deriv h z = 0 z
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	have f0 : U ∈ 𝓝 z := hU.mem_nhds hz	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U ⊢ deriv h z = 0 z	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv h z = 0 z
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	dsimp	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv h z = 0 z	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv h z = 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	unfold_let	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv h z = 0	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv (f / cexp ∘ fun z => lf z + (log (f z₀) - lf z₀)) z = 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	rw [Pi.div_def, deriv_div (hf.differentiableAt f0) (e4.differentiableAt f0) (exp_ne_zero _)]	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv (f / cexp ∘ fun z => lf z + (log (f z₀) - lf z₀)) z = 0	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ (deriv f z * (cexp ∘ g) z - f z * deriv (cexp ∘ g) z) / (cexp ∘ g) z ^ 2 = 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	rw [deriv.scomp z differentiableAt_exp (h3.differentiableAt f0)]	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ (deriv f z * (cexp ∘ g) z - f z * deriv (cexp ∘ g) z) / (cexp ∘ g) z ^ 2 = 0	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ (deriv f z * (cexp ∘ g) z - f z * deriv g z • deriv cexp (g z)) / (cexp ∘ g) z ^ 2 = 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	have e5 : deriv g z = deriv lf z := by unfold_let ; simp	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ (deriv f z * (cexp ∘ g) z - f z * deriv g z • deriv cexp (g z)) / (cexp ∘ g) z ^ 2 = 0	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z e5 : deriv g z = deriv lf z ⊢ (deriv f z * (cexp ∘ g) z - f z * deriv g z • deriv cexp (g z)) / (cexp ∘ g) z ^ 2 = 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	field_simp [exp_ne_zero, hlf2 hz, hfz z hz, e5]	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z e5 : deriv g z = deriv lf z ⊢ (deriv f z * (cexp ∘ g) z - f z * deriv g z • deriv cexp (g z)) / (cexp ∘ g) z ^ 2 = 0	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z e5 : deriv g z = deriv lf z ⊢ deriv f z * cexp (g z) * f z - f z * (deriv f z * cexp (g z)) = 0
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	ring	case intro.intro.intro z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z e5 : deriv g z = deriv lf z ⊢ deriv f z * cexp (g z) * f z - f z * (deriv f z * cexp (g z)) = 0	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	exact λ z hz => eq_of_div_eq_one (h hz)	z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝¹ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h✝ : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h✝ U h : EqOn h✝ (fun x => 1) U ⊢ EqOn f (cexp ∘ g) U	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	unfold_let	z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ 1 = h z₀	z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ 1 = (f / cexp ∘ fun z => lf z + (log (f z₀) - lf z₀)) z₀
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	simp [exp_log, hfz z₀ hz₀]	z z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U ⊢ 1 = (f / cexp ∘ fun z => lf z + (log (f z₀) - lf z₀)) z₀	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	unfold_let	z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv g z = deriv lf z	z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv (fun z => lf z + (log (f z₀) - lf z₀)) z = deriv lf z
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	RMT4/has_sqrt.lean	has_primitives.has_logs	[59, 1]	[85, 9]	simp	z✝ z₀✝ : ℂ U : Set ℂ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : ¬U = ∅ z₀ : ℂ hz₀ : z₀ ∈ U f : ℂ → ℂ hf : DifferentiableOn ℂ f U hfz : ∀ z ∈ U, f z ≠ 0 lf : ℂ → ℂ hlf1 : DifferentiableOn ℂ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀) h : ℂ → ℂ := f / cexp ∘ g h3 : DifferentiableOn ℂ g U e4 : DifferentiableOn ℂ (cexp ∘ g) U e1 : DifferentiableOn ℂ h U this : 1 = h z₀ z : ℂ hz : z ∈ U f0 : U ∈ 𝓝 z ⊢ deriv (fun z => lf z + (log (f z₀) - lf z₀)) z = deriv lf z	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	induction n generalizing f	f : ℝ → ℝ a b : ℝ hab : a < b n : ℕ h : ContDiffOn ℝ (↑n) f (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)	case zero a b : ℝ hab : a < b f : ℝ → ℝ h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b) case succ a b : ℝ hab : a < b n✝ : ℕ n_ih✝ : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n✝) f (Icc a b) → ∃ g, ContDiff ℝ (↑n✝) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n✝)) f (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n✝)) g ∧ EqOn g f (Icc a b)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	case zero => simp only [CharP.cast_eq_zero, contDiff_zero, contDiffOn_zero] at h ⊢ refine ⟨IccExtend hab.le (restrict (Icc a b) f), h.restrict.Icc_extend', ?_⟩ exact λ t ht => IccExtend_of_mem _ _ ht	a b : ℝ hab : a < b f : ℝ → ℝ h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	simp only [CharP.cast_eq_zero, contDiff_zero, contDiffOn_zero] at h ⊢	a b : ℝ hab : a < b f : ℝ → ℝ h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b)	a b : ℝ hab : a < b f : ℝ → ℝ h : ContinuousOn f (Icc a b) ⊢ ∃ g, Continuous g ∧ EqOn g f (Icc a b)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	refine ⟨IccExtend hab.le (restrict (Icc a b) f), h.restrict.Icc_extend', ?_⟩	a b : ℝ hab : a < b f : ℝ → ℝ h : ContinuousOn f (Icc a b) ⊢ ∃ g, Continuous g ∧ EqOn g f (Icc a b)	a b : ℝ hab : a < b f : ℝ → ℝ h : ContinuousOn f (Icc a b) ⊢ EqOn (IccExtend ⋯ (restrict (Icc a b) f)) f (Icc a b)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	exact λ t ht => IccExtend_of_mem _ _ ht	a b : ℝ hab : a < b f : ℝ → ℝ h : ContinuousOn f (Icc a b) ⊢ EqOn (IccExtend ⋯ (restrict (Icc a b) f)) f (Icc a b)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	have h1 : ContDiffOn ℝ n (derivWithin f (Icc a b)) (Icc a b) := h.derivWithin (uniqueDiffOn_Icc hab) le_rfl	a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b)	a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	obtain ⟨gg, h2, h3⟩ := ih h1	a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b)	case intro.intro a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	refine ⟨λ t => f a + ∫ u in a..t, gg u, ?_, ?_⟩	case intro.intro a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b)	case intro.intro.refine_1 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ ContDiff ℝ ↑(Nat.succ n) fun t => f a + ∫ (u : ℝ) in a..t, gg u case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ EqOn (fun t => f a + ∫ (u : ℝ) in a..t, gg u) f (Icc a b)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	rw [contDiff_succ_iff_deriv]	case intro.intro.refine_1 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ ContDiff ℝ ↑(Nat.succ n) fun t => f a + ∫ (u : ℝ) in a..t, gg u	case intro.intro.refine_1 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ (Differentiable ℝ fun t => f a + ∫ (u : ℝ) in a..t, gg u) ∧ ContDiff ℝ (↑n) (deriv fun t => f a + ∫ (u : ℝ) in a..t, gg u)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	constructor	case intro.intro.refine_1 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ (Differentiable ℝ fun t => f a + ∫ (u : ℝ) in a..t, gg u) ∧ ContDiff ℝ (↑n) (deriv fun t => f a + ∫ (u : ℝ) in a..t, gg u)	case intro.intro.refine_1.left a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ Differentiable ℝ fun t => f a + ∫ (u : ℝ) in a..t, gg u case intro.intro.refine_1.right a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ ContDiff ℝ (↑n) (deriv fun t => f a + ∫ (u : ℝ) in a..t, gg u)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	refine differentiableOn_univ.1 ((differentiableOn_integral_of_continuous ?_ h2.continuous).const_add _)	case intro.intro.refine_1.left a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ Differentiable ℝ fun t => f a + ∫ (u : ℝ) in a..t, gg u	case intro.intro.refine_1.left a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ ∀ x ∈ univ, IntervalIntegrable gg MeasureTheory.volume a x
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	simp [h2.continuous.intervalIntegrable]	case intro.intro.refine_1.left a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ ∀ x ∈ univ, IntervalIntegrable gg MeasureTheory.volume a x	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	convert h2	case intro.intro.refine_1.right a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ ContDiff ℝ (↑n) (deriv fun t => f a + ∫ (u : ℝ) in a..t, gg u)	case h.e'_10 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ (deriv fun t => f a + ∫ (u : ℝ) in a..t, gg u) = gg
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	ext t	case h.e'_10 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ (deriv fun t => f a + ∫ (u : ℝ) in a..t, gg u) = gg	case h.e'_10.h a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ⊢ deriv (fun t => f a + ∫ (u : ℝ) in a..t, gg u) t = gg t
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	simp [deriv_const_add, h2.continuous.deriv_integral]	case h.e'_10.h a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ⊢ deriv (fun t => f a + ∫ (u : ℝ) in a..t, gg u) t = gg t	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	intro t ht	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊢ EqOn (fun t => f a + ∫ (u : ℝ) in a..t, gg u) f (Icc a b)	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b ⊢ (fun t => f a + ∫ (u : ℝ) in a..t, gg u) t = f t
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	dsimp	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b ⊢ (fun t => f a + ∫ (u : ℝ) in a..t, gg u) t = f t	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b ⊢ f a + ∫ (u : ℝ) in a..t, gg u = f t
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	have l6 : Icc a t ⊆ Icc a b := Icc_subset_Icc_right ht.2	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b ⊢ f a + ∫ (u : ℝ) in a..t, gg u = f t	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b l6 : Icc a t ⊆ Icc a b ⊢ f a + ∫ (u : ℝ) in a..t, gg u = f t
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	have l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t) := h3.mono (by simp [uIcc, ht.1, l6])	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b l6 : Icc a t ⊆ Icc a b ⊢ f a + ∫ (u : ℝ) in a..t, gg u = f t	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b l6 : Icc a t ⊆ Icc a b l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t) ⊢ f a + ∫ (u : ℝ) in a..t, gg u = f t
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	have l10 := h.one_of_succ.integral_eq_sub'' hab.le ht	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b l6 : Icc a t ⊆ Icc a b l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t) ⊢ f a + ∫ (u : ℝ) in a..t, gg u = f t	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b l6 : Icc a t ⊆ Icc a b l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t) l10 : ∫ (y : ℝ) in a..t, derivWithin f (Icc a b) y = f t - f a ⊢ f a + ∫ (u : ℝ) in a..t, gg u = f t
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	simp [integral_congr l9, l10]	case intro.intro.refine_2 a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b l6 : Icc a t ⊆ Icc a b l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t) l10 : ∫ (y : ℝ) in a..t, derivWithin f (Icc a b) y = f t - f a ⊢ f a + ∫ (u : ℝ) in a..t, gg u = f t	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto	[5, 1]	[29, 36]	simp [uIcc, ht.1, l6]	a b : ℝ hab : a < b n : ℕ ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ → ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) gg : ℝ → ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) t : ℝ ht : t ∈ Icc a b l6 : Icc a t ⊆ Icc a b ⊢ uIcc a t ⊆ Icc a b	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto'	[31, 1]	[35, 52]	cases eq_or_ne a b	f : ℝ → ℝ a b : ℝ n : ℕ h : ContDiffOn ℝ (↑n) f (uIcc a b) ⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)	case inl f : ℝ → ℝ a b : ℝ n : ℕ h : ContDiffOn ℝ (↑n) f (uIcc a b) h✝ : a = b ⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b) case inr f : ℝ → ℝ a b : ℝ n : ℕ h : ContDiffOn ℝ (↑n) f (uIcc a b) h✝ : a ≠ b ⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto'	[31, 1]	[35, 52]	case inl hab => exact ⟨λ _ => f a, by simp [hab, contDiff_const]⟩	f : ℝ → ℝ a b : ℝ n : ℕ h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a = b ⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto'	[31, 1]	[35, 52]	exact ⟨λ _ => f a, by simp [hab, contDiff_const]⟩	f : ℝ → ℝ a b : ℝ n : ℕ h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a = b ⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto'	[31, 1]	[35, 52]	simp [hab, contDiff_const]	f : ℝ → ℝ a b : ℝ n : ℕ h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a = b ⊢ (ContDiff ℝ ↑n fun x => f a) ∧ EqOn (fun x => f a) f (uIcc a b)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto'	[31, 1]	[35, 52]	case inr hab => exact toto (min_lt_max.2 hab) h	f : ℝ → ℝ a b : ℝ n : ℕ h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a ≠ b ⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)	no goals
https://github.com/vbeffara/RMT4.git	c2a092d029d0e6d29a381ac4ad9e85b10d97391c	extend.lean	toto'	[31, 1]	[35, 52]	exact toto (min_lt_max.2 hab) h	f : ℝ → ℝ a b : ℝ n : ℕ h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a ≠ b ⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.eta	[34, 1]	[34, 87]	cases f	R : Type u a b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q f : R[X;φ] ⊢ { toFinsupp := f.toFinsupp } = f	case ofFinsupp R : Type u a b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := { toFinsupp := toFinsupp✝ }.toFinsupp } = { toFinsupp := toFinsupp✝ }
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.eta	[34, 1]	[34, 87]	rfl	case ofFinsupp R : Type u a b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := { toFinsupp := toFinsupp✝ }.toFinsupp } = { toFinsupp := toFinsupp✝ }	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_add	[102, 1]	[103, 35]	rw [add_def]	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a b : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := a + b } = SkewPolynomial.add { toFinsupp := a } { toFinsupp := b }	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_neg	[106, 1]	[107, 33]	rw [neg_def]	R : Type u a✝ b : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S a : AddMonoidAlgebra S ℕ ⊢ { toFinsupp := -a } = SkewPolynomial.neg { toFinsupp := a }	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_sub	[110, 1]	[113, 6]	rw [sub_eq_add_neg, ofFinsupp_add, ofFinsupp_neg]	R : Type u a✝ b✝ : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S a b : AddMonoidAlgebra S ℕ ⊢ { toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b }	R : Type u a✝ b✝ : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S a b : AddMonoidAlgebra S ℕ ⊢ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b }
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_sub	[110, 1]	[113, 6]	rfl	R : Type u a✝ b✝ : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S a b : AddMonoidAlgebra S ℕ ⊢ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b }	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_mul	[116, 1]	[117, 35]	rw [mul_def]	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a b : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := AddMonoidAlgebra.mul' φ a b } = SkewPolynomial.mul { toFinsupp := a } { toFinsupp := b }	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_pow	[125, 1]	[129, 81]	change _ = npowRec n _	R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ n : ℕ ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = { toFinsupp := a } ^ n	R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ n : ℕ ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_pow	[125, 1]	[129, 81]	induction n with \| zero => simp [npowRec]; rfl \| succ n n_ih => simp [npowRec, pow_succ]; rw [<- n_ih, <- ofFinsupp_mul]; rfl	R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ n : ℕ ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_pow	[125, 1]	[129, 81]	simp [npowRec]	case zero R : Type u a✝ b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ Nat.zero a } = npowRec Nat.zero { toFinsupp := a }	case zero R : Type u a✝ b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ 0 a } = 1
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_pow	[125, 1]	[129, 81]	rfl	case zero R : Type u a✝ b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ 0 a } = 1	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_pow	[125, 1]	[129, 81]	simp [npowRec, pow_succ]	case succ R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ n : ℕ n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a } ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } = npowRec (Nat.succ n) { toFinsupp := a }	case succ R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ n : ℕ n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a } ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } = { toFinsupp := a } * npowRec n { toFinsupp := a }
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_pow	[125, 1]	[129, 81]	rw [<- n_ih, <- ofFinsupp_mul]	case succ R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ n : ℕ n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a } ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } = { toFinsupp := a } * npowRec n { toFinsupp := a }	case succ R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ n : ℕ n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a } ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } = { toFinsupp := AddMonoidAlgebra.mul' φ a (SkewPolynomial.AddMonoidAlgebra.pow' φ n a) }
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_pow	[125, 1]	[129, 81]	rfl	case succ R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ n : ℕ n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a } ⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } = { toFinsupp := AddMonoidAlgebra.mul' φ a (SkewPolynomial.AddMonoidAlgebra.pow' φ n a) }	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_add	[140, 1]	[143, 23]	cases a	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b : R[X;φ] ⊢ (a + b).toFinsupp = a.toFinsupp + b.toFinsupp	case ofFinsupp R : Type u a b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q b : R[X;φ] toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝ } + b).toFinsupp = { toFinsupp := toFinsupp✝ }.toFinsupp + b.toFinsupp
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_add	[140, 1]	[143, 23]	cases b	case ofFinsupp R : Type u a b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q b : R[X;φ] toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝ } + b).toFinsupp = { toFinsupp := toFinsupp✝ }.toFinsupp + b.toFinsupp	case ofFinsupp.ofFinsupp R : Type u a b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }).toFinsupp = { toFinsupp := toFinsupp✝¹ }.toFinsupp + { toFinsupp := toFinsupp✝ }.toFinsupp
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_add	[140, 1]	[143, 23]	rw [← ofFinsupp_add]	case ofFinsupp.ofFinsupp R : Type u a b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }).toFinsupp = { toFinsupp := toFinsupp✝¹ }.toFinsupp + { toFinsupp := toFinsupp✝ }.toFinsupp	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_neg	[146, 1]	[149, 23]	cases a	R : Type u a✝ b : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S a : S[X;φ] ⊢ (-a).toFinsupp = -a.toFinsupp	case ofFinsupp R : Type u a b : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S toFinsupp✝ : AddMonoidAlgebra S ℕ ⊢ (-{ toFinsupp := toFinsupp✝ }).toFinsupp = -{ toFinsupp := toFinsupp✝ }.toFinsupp
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_neg	[146, 1]	[149, 23]	rw [← ofFinsupp_neg]	case ofFinsupp R : Type u a b : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S toFinsupp✝ : AddMonoidAlgebra S ℕ ⊢ (-{ toFinsupp := toFinsupp✝ }).toFinsupp = -{ toFinsupp := toFinsupp✝ }.toFinsupp	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_sub	[152, 1]	[155, 6]	rw [sub_eq_add_neg, ← toFinsupp_neg, ← toFinsupp_add]	R : Type u a✝ b✝ : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S a b : S[X;φ] ⊢ (a - b).toFinsupp = a.toFinsupp - b.toFinsupp	R : Type u a✝ b✝ : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S a b : S[X;φ] ⊢ (a - b).toFinsupp = (a + -b).toFinsupp
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_sub	[152, 1]	[155, 6]	rfl	R : Type u a✝ b✝ : R m n : ℕ inst✝¹ : Semiring R φ✝ : R →+* R p q : R[X;φ✝] S : Type u inst✝ : Ring S φ : S →+* S a b : S[X;φ] ⊢ (a - b).toFinsupp = (a + -b).toFinsupp	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_mul	[158, 1]	[162, 23]	cases a	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b : R[X;φ] ⊢ (a * b).toFinsupp = AddMonoidAlgebra.mul' φ a.toFinsupp b.toFinsupp	case ofFinsupp R : Type u a b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q b : R[X;φ] toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝ } * b).toFinsupp = AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝ }.toFinsupp b.toFinsupp
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_mul	[158, 1]	[162, 23]	cases b	case ofFinsupp R : Type u a b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q b : R[X;φ] toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝ } * b).toFinsupp = AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝ }.toFinsupp b.toFinsupp	case ofFinsupp.ofFinsupp R : Type u a b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp = AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝¹ }.toFinsupp { toFinsupp := toFinsupp✝ }.toFinsupp
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_mul	[158, 1]	[162, 23]	rw [← ofFinsupp_mul]	case ofFinsupp.ofFinsupp R : Type u a b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp = AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝¹ }.toFinsupp { toFinsupp := toFinsupp✝ }.toFinsupp	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_pow	[171, 1]	[174, 23]	cases a	R : Type u a✝ b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q a : R[X;φ] n : ℕ ⊢ (a ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' φ n a.toFinsupp	case ofFinsupp R : Type u a b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] n : ℕ toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝ } ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' φ n { toFinsupp := toFinsupp✝ }.toFinsupp
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_pow	[171, 1]	[174, 23]	rw [← ofFinsupp_pow]	case ofFinsupp R : Type u a b : R m n✝ : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] n : ℕ toFinsupp✝ : AddMonoidAlgebra R ℕ ⊢ ({ toFinsupp := toFinsupp✝ } ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' φ n { toFinsupp := toFinsupp✝ }.toFinsupp	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_eq_zero	[188, 1]	[189, 39]	rw [← toFinsupp_zero, toFinsupp_inj]	R : Type u a✝ b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a : R[X;φ] ⊢ a.toFinsupp = 0 ↔ a = 0	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.toFinsupp_eq_one	[192, 1]	[193, 38]	rw [← toFinsupp_one, toFinsupp_inj]	R : Type u a✝ b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a : R[X;φ] ⊢ a.toFinsupp = 1 ↔ a = 1	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_eq_zero	[200, 1]	[201, 39]	rw [← ofFinsupp_zero, ofFinsupp_inj]	R : Type u a✝ b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := a } = 0 ↔ a = 0	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.ofFinsupp_eq_one	[204, 1]	[204, 100]	rw [← ofFinsupp_one, ofFinsupp_inj]	R : Type u a✝ b : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q : R[X;φ] a : AddMonoidAlgebra R ℕ ⊢ { toFinsupp := a } = 1 ↔ a = 1	no goals
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	rw [←toFinsupp_inj]	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ a * b * c = a * (b * c)	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ (a * b * c).toFinsupp = (a * (b * c)).toFinsupp
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	simp only [toFinsupp_mul, AddMonoidAlgebra.mul'_def]	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ (a * b * c).toFinsupp = (a * (b * c)).toFinsupp	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ (sum (sum a.toFinsupp fun a₁ b₁ => sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum a.toFinsupp fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	rw [sum_sum_index]	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ (sum (sum a.toFinsupp fun a₁ b₁ => sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum a.toFinsupp fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ (sum a.toFinsupp fun a b_1 => sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum a.toFinsupp fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	congr	R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ (sum a.toFinsupp fun a b_1 => sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum a.toFinsupp fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ (fun a b_1 => sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	ext a₁ b₁	case e_g R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ (fun a b_1 => sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ (sum (sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	rw [sum_sum_index, sum_sum_index]	case e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ (sum (sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ (sum b.toFinsupp fun a b => sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum b.toFinsupp fun a b => sum (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b * (↑φ)^[a] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	congr	case e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ (sum b.toFinsupp fun a b => sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum b.toFinsupp fun a b => sum (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b * (↑φ)^[a] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ (fun a b => sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = fun a b => sum (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b * (↑φ)^[a] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	ext a₂ b₂	case e_g.h.h.e_g R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ (fun a b => sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = fun a b => sum (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b * (↑φ)^[a] b₂)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum (single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₂ + a₂_1) (b₂ * (↑φ)^[a₂] b₂_1)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	rw [sum_sum_index, AddMonoidAlgebra.sum_single_index]	case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum (single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ => sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) = sum (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₂ + a₂_1) (b₂ * (↑φ)^[a₂] b₂_1)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) = sum c.toFinsupp fun a b => sum (single (a₂ + a) (b₂ * (↑φ)^[a₂] b)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	congr	case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) = sum c.toFinsupp fun a b => sum (single (a₂ + a) (b₂ * (↑φ)^[a₂] b)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g.h.h.e_g R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) = fun a b => sum (single (a₂ + a) (b₂ * (↑φ)^[a₂] b)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	ext a₃ b₃	case e_g.h.h.e_g.h.h.e_g R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) = fun a b => sum (single (a₂ + a) (b₂ * (↑φ)^[a₂] b)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + a₂ + a₃) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₃) = sum (single (a₂ + a₃) (b₂ * (↑φ)^[a₂] b₃)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	rw [AddMonoidAlgebra.sum_single_index, _root_.mul_assoc, RingHom.iterate_map_mul, ← Function.iterate_add_apply, add_assoc]	case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + a₂ + a₃) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₃) = sum (single (a₂ + a₃) (b₂ * (↑φ)^[a₂] b₃)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	pick_goal 4	case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	. intros n r1 r2 rw [RingHom.iterate_map_add, mul_add, AddMonoidAlgebra.single_add]	case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	pick_goal 5	case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)
https://github.com/mariainesdff/skew_polynomials.git	16371a025f5c867f83ff258a22df5c0341793888	SkewPolynomials.lean	SkewPolynomial.mul_assoc	[221, 1]	[241, 84]	. intros n r1 r2 rw [RingHom.iterate_map_add, mul_add, AddMonoidAlgebra.single_add]	case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁_1 b₂ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂) case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)	case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R a₃ : ℕ b₃ : R ⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ (sum c.toFinsupp fun a₂_1 b₂ => single (a₁ + a₂ + a₂_1) (0 * (↑φ)^[a₁ + a₂] b₂)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R a₂ : ℕ b₂ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] a₁ : ℕ b₁ : R ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1) case h_zero R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ), (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (0 * (↑φ)^[a] b₂)) = 0 case h_add R : Type u a✝ b✝ : R m n : ℕ inst✝ : Semiring R φ : R →+* R p q a b c : R[X;φ] ⊢ ∀ (a : ℕ) (b₁ b₂ : R), (sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) ((b₁ + b₂) * (↑φ)^[a] b₂_1)) = (sum c.toFinsupp fun a₂ b₂ => single (a + a₂) (b₁ * (↑φ)^[a] b₂)) + sum c.toFinsupp fun a₂ b₂_1 => single (a + a₂) (b₂ * (↑φ)^[a] b₂_1)

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