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stringclasses
147 values
file_path
stringlengths
7
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stringlengths
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stringlengths
6
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end
stringlengths
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stringlengths
6
2.09M
https://github.com/vbeffara/RMT4.git
c2a092d029d0e6d29a381ac4ad9e85b10d97391c
RMT4/has_sqrt.lean
has_primitives.has_logs
[59, 1]
[85, 9]
obtain ⟨zβ‚€, hzβ‚€βŸ© := nonempty_iff_ne_empty.2 h
z zβ‚€ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… ⊒ _root_.has_logs U
case intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ 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 zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U ⊒ _root_.has_logs U
case intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 ⊒ βˆƒ g, DifferentiableOn β„‚ 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 zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 ⊒ βˆƒ g, DifferentiableOn β„‚ g U ∧ EqOn f (cexp ∘ g) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U ⊒ βˆƒ g, DifferentiableOn β„‚ 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 : β„‚ β†’ β„‚ := Ξ» z => lf z + (log (f zβ‚€) - lf zβ‚€)
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U ⊒ βˆƒ g, DifferentiableOn β„‚ g U ∧ EqOn f (cexp ∘ g) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) ⊒ βˆƒ g, DifferentiableOn β„‚ 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 : β„‚ β†’ β„‚ := f / (exp ∘ g)
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) ⊒ βˆƒ g, DifferentiableOn β„‚ g U ∧ EqOn f (cexp ∘ g) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g ⊒ βˆƒ g, DifferentiableOn β„‚ 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 β„‚ g U := hlf1.add (differentiableOn_const _)
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g ⊒ βˆƒ g, DifferentiableOn β„‚ g U ∧ EqOn f (cexp ∘ g) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U ⊒ βˆƒ g, DifferentiableOn β„‚ 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 β„‚ (exp ∘ g) U := differentiable_exp.comp_differentiableOn h3
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U ⊒ βˆƒ g, DifferentiableOn β„‚ g U ∧ EqOn f (cexp ∘ g) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U ⊒ βˆƒ g, DifferentiableOn β„‚ 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 β„‚ h U := hf.div e4 (Ξ» z _ => exp_ne_zero _)
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U ⊒ βˆƒ g, DifferentiableOn β„‚ g U ∧ EqOn f (cexp ∘ g) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U ⊒ βˆƒ g, DifferentiableOn β„‚ 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 zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U ⊒ βˆƒ g, DifferentiableOn β„‚ g U ∧ EqOn f (cexp ∘ g) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ 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 (Ξ» _ => 1) U by exact Ξ» z hz => eq_of_div_eq_one (h hz)
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U ⊒ EqOn f (cexp ∘ g) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ 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 zβ‚€ := by unfold_let ; simp [exp_log, hfz zβ‚€ hzβ‚€]
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U ⊒ EqOn h (fun x => 1) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ ⊒ 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 zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ ⊒ EqOn h (fun x => 1) U
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ ⊒ EqOn h (fun x => h zβ‚€) 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 (Ξ» z hz => ?_) hzβ‚€
case intro.intro.intro z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ ⊒ EqOn h (fun x => h zβ‚€) U
case intro.intro.intro z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U ⊒ deriv h z = 0 z
case intro.intro.intro z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ deriv h z = 0 z
case intro.intro.intro z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ deriv h z = 0
case intro.intro.intro z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ deriv (f / cexp ∘ fun z => lf z + (log (f zβ‚€) - lf zβ‚€)) 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ deriv (f / cexp ∘ fun z => lf z + (log (f zβ‚€) - lf zβ‚€)) z = 0
case intro.intro.intro z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ (deriv f z * (cexp ∘ g) z - f z * deriv g z β€’ 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ (deriv f z * (cexp ∘ g) z - f z * deriv g z β€’ deriv cexp (g z)) / (cexp ∘ g) z ^ 2 = 0
case intro.intro.intro z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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 β€’ 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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 β€’ deriv cexp (g z)) / (cexp ∘ g) z ^ 2 = 0
case intro.intro.intro z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ 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 Ξ» z hz => eq_of_div_eq_one (h hz)
z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝¹ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h✝ : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ 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 zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U ⊒ 1 = h zβ‚€
z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U ⊒ 1 = (f / cexp ∘ fun z => lf z + (log (f zβ‚€) - lf zβ‚€)) zβ‚€
https://github.com/vbeffara/RMT4.git
c2a092d029d0e6d29a381ac4ad9e85b10d97391c
RMT4/has_sqrt.lean
has_primitives.has_logs
[59, 1]
[85, 9]
simp [exp_log, hfz zβ‚€ hzβ‚€]
z zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U ⊒ 1 = (f / cexp ∘ fun z => lf z + (log (f zβ‚€) - lf zβ‚€)) zβ‚€
no goals
https://github.com/vbeffara/RMT4.git
c2a092d029d0e6d29a381ac4ad9e85b10d97391c
RMT4/has_sqrt.lean
has_primitives.has_logs
[59, 1]
[85, 9]
unfold_let
z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ deriv g z = deriv lf z
z✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ deriv (fun z => lf z + (log (f zβ‚€) - lf zβ‚€)) 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✝ zβ‚€βœ : β„‚ U : Set β„‚ hp : has_primitives U hU : IsOpen U hU' : IsPreconnected U h✝ : Β¬U = βˆ… zβ‚€ : β„‚ hzβ‚€ : zβ‚€ ∈ U f : β„‚ β†’ β„‚ hf : DifferentiableOn β„‚ f U hfz : βˆ€ z ∈ U, f z β‰  0 lf : β„‚ β†’ β„‚ hlf1 : DifferentiableOn β„‚ lf U hlf2 : EqOn (deriv lf) (deriv f / f) U g : β„‚ β†’ β„‚ := fun z => lf z + (log (f zβ‚€) - lf zβ‚€) h : β„‚ β†’ β„‚ := f / cexp ∘ g h3 : DifferentiableOn β„‚ g U e4 : DifferentiableOn β„‚ (cexp ∘ g) U e1 : DifferentiableOn β„‚ h U this : 1 = h zβ‚€ z : β„‚ hz : z ∈ U f0 : U ∈ 𝓝 z ⊒ deriv (fun z => lf z + (log (f zβ‚€) - lf zβ‚€)) 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 β†’ ℝ a b : ℝ hab : a < b n : β„• h : ContDiffOn ℝ (↑n) f (Icc a b) ⊒ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
case zero a b : ℝ hab : a < b f : ℝ β†’ ℝ h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b) ⊒ βˆƒ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b) case succ a b : ℝ hab : a < b n✝ : β„• n_ih✝ : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n✝) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n✝) g ∧ EqOn g f (Icc a b) f : ℝ β†’ ℝ h : ContDiffOn ℝ (↑(Nat.succ n✝)) f (Icc a b) ⊒ βˆƒ 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 β†’ ℝ h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b) ⊒ βˆƒ 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 β†’ ℝ h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b) ⊒ βˆƒ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b)
a b : ℝ hab : a < b f : ℝ β†’ ℝ h : ContinuousOn f (Icc a b) ⊒ βˆƒ 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 β†’ ℝ h : ContinuousOn f (Icc a b) ⊒ βˆƒ g, Continuous g ∧ EqOn g f (Icc a b)
a b : ℝ hab : a < b f : ℝ β†’ ℝ h : ContinuousOn f (Icc a b) ⊒ EqOn (IccExtend β‹― (restrict (Icc a b) f)) f (Icc a b)
https://github.com/vbeffara/RMT4.git
c2a092d029d0e6d29a381ac4ad9e85b10d97391c
extend.lean
toto
[5, 1]
[29, 36]
exact Ξ» t ht => IccExtend_of_mem _ _ ht
a b : ℝ hab : a < b f : ℝ β†’ ℝ h : ContinuousOn f (Icc a b) ⊒ EqOn (IccExtend β‹― (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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) ⊒ βˆƒ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b)
a b : ℝ hab : a < b n : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) ⊒ βˆƒ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ ℝ h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b) h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b) ⊒ βˆƒ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b)
case intro.intro a b : ℝ hab : a < b n : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊒ βˆƒ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊒ βˆƒ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊒ βˆ€ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ ℝ h2 : ContDiff ℝ (↑n) gg h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b) ⊒ βˆ€ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 βŠ† Icc a b := Icc_subset_Icc_right ht.2
case intro.intro.refine_2 a b : ℝ hab : a < b n : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 βŠ† 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 βŠ† 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 βŠ† 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 βŠ† 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 βŠ† 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 βŠ† 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 : β„• ih : βˆ€ {f : ℝ β†’ ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) β†’ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) f : ℝ β†’ 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 β†’ 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 βŠ† Icc a b ⊒ uIcc a t βŠ† 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 β†’ ℝ a b : ℝ n : β„• h : ContDiffOn ℝ (↑n) f (uIcc a b) ⊒ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)
case inl f : ℝ β†’ ℝ a b : ℝ n : β„• h : ContDiffOn ℝ (↑n) f (uIcc a b) h✝ : a = b ⊒ βˆƒ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b) case inr f : ℝ β†’ ℝ a b : ℝ n : β„• h : ContDiffOn ℝ (↑n) f (uIcc a b) h✝ : a β‰  b ⊒ βˆƒ 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 β†’ ℝ a b : ℝ n : β„• h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a = b ⊒ βˆƒ 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 β†’ ℝ a b : ℝ n : β„• h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a = b ⊒ βˆƒ 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 β†’ ℝ a b : ℝ n : β„• 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 β†’ ℝ a b : ℝ n : β„• h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a β‰  b ⊒ βˆƒ 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 β†’ ℝ a b : ℝ n : β„• h : ContDiffOn ℝ (↑n) f (uIcc a b) hab : a β‰  b ⊒ βˆƒ 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q f : R[X;Ο†] ⊒ { toFinsupp := f.toFinsupp } = f
case ofFinsupp R : Type u a b : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ { 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ { 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a b : AddMonoidAlgebra R β„• ⊒ { 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 : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S a : AddMonoidAlgebra S β„• ⊒ { 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 : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S a b : AddMonoidAlgebra S β„• ⊒ { toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b }
R : Type u a✝ b✝ : R m n : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S a b : AddMonoidAlgebra S β„• ⊒ { 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 : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S a b : AddMonoidAlgebra S β„• ⊒ { 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a b : AddMonoidAlgebra R β„• ⊒ { toFinsupp := AddMonoidAlgebra.mul' Ο† 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✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• n : β„• ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† n a } = { toFinsupp := a } ^ n
R : Type u a✝ b : R m n✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• n : β„• ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† 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✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• n : β„• ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† Nat.zero a } = npowRec Nat.zero { toFinsupp := a }
case zero R : Type u a✝ b : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† 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✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• n : β„• n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† n a } = npowRec n { toFinsupp := a } ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† (Nat.succ n) a } = npowRec (Nat.succ n) { toFinsupp := a }
case succ R : Type u a✝ b : R m n✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• n : β„• n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† n a } = npowRec n { toFinsupp := a } ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† (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✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• n : β„• n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† n a } = npowRec n { toFinsupp := a } ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† (Nat.succ n) a } = { toFinsupp := a } * npowRec n { toFinsupp := a }
case succ R : Type u a✝ b : R m n✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• n : β„• n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† n a } = npowRec n { toFinsupp := a } ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† (Nat.succ n) a } = { toFinsupp := AddMonoidAlgebra.mul' Ο† a (SkewPolynomial.AddMonoidAlgebra.pow' Ο† 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✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• n : β„• n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† n a } = npowRec n { toFinsupp := a } ⊒ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο† (Nat.succ n) a } = { toFinsupp := AddMonoidAlgebra.mul' Ο† a (SkewPolynomial.AddMonoidAlgebra.pow' Ο† 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b : R[X;Ο†] ⊒ (a + b).toFinsupp = a.toFinsupp + b.toFinsupp
case ofFinsupp R : Type u a b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q b : R[X;Ο†] toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q b : R[X;Ο†] toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ toFinsupp := toFinsupp✝ } + b).toFinsupp = { toFinsupp := toFinsupp✝ }.toFinsupp + b.toFinsupp
case ofFinsupp.ofFinsupp R : Type u a b : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ 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 : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S a : S[X;Ο†] ⊒ (-a).toFinsupp = -a.toFinsupp
case ofFinsupp R : Type u a b : R m n : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S toFinsupp✝ : AddMonoidAlgebra S β„• ⊒ (-{ 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 : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S toFinsupp✝ : AddMonoidAlgebra S β„• ⊒ (-{ 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 : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S a b : S[X;Ο†] ⊒ (a - b).toFinsupp = a.toFinsupp - b.toFinsupp
R : Type u a✝ b✝ : R m n : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S a b : S[X;Ο†] ⊒ (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 : β„• inst✝¹ : Semiring R Ο†βœ : R β†’+* R p q : R[X;Ο†βœ] S : Type u inst✝ : Ring S Ο† : S β†’+* S a b : S[X;Ο†] ⊒ (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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b : R[X;Ο†] ⊒ (a * b).toFinsupp = AddMonoidAlgebra.mul' Ο† a.toFinsupp b.toFinsupp
case ofFinsupp R : Type u a b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q b : R[X;Ο†] toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ toFinsupp := toFinsupp✝ } * b).toFinsupp = AddMonoidAlgebra.mul' Ο† { 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q b : R[X;Ο†] toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ toFinsupp := toFinsupp✝ } * b).toFinsupp = AddMonoidAlgebra.mul' Ο† { toFinsupp := toFinsupp✝ }.toFinsupp b.toFinsupp
case ofFinsupp.ofFinsupp R : Type u a b : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp = AddMonoidAlgebra.mul' Ο† { 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp = AddMonoidAlgebra.mul' Ο† { 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✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a : R[X;Ο†] n : β„• ⊒ (a ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' Ο† n a.toFinsupp
case ofFinsupp R : Type u a b : R m n✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] n : β„• toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ toFinsupp := toFinsupp✝ } ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' Ο† 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✝ : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] n : β„• toFinsupp✝ : AddMonoidAlgebra R β„• ⊒ ({ toFinsupp := toFinsupp✝ } ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' Ο† 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a : R[X;Ο†] ⊒ 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a : R[X;Ο†] ⊒ 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• ⊒ { 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q : R[X;Ο†] a : AddMonoidAlgebra R β„• ⊒ { 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ a * b * c = a * (b * c)
R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ (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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ (a * b * c).toFinsupp = (a * (b * c)).toFinsupp
R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ (sum (sum a.toFinsupp fun a₁ b₁ => sum b.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum a.toFinsupp fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)
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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ (sum (sum a.toFinsupp fun a₁ b₁ => sum b.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum a.toFinsupp fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)
R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ (sum a.toFinsupp fun a b_1 => sum (sum b.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b_1 * (↑φ)^[a] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum a.toFinsupp fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ (sum a.toFinsupp fun a b_1 => sum (sum b.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b_1 * (↑φ)^[a] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum a.toFinsupp fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ (fun a b_1 => sum (sum b.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b_1 * (↑φ)^[a] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ (fun a b_1 => sum (sum b.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b_1 * (↑φ)^[a] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = fun a₁ b₁ => sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ (sum (sum b.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ (sum (sum b.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ (sum b.toFinsupp fun a b => sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum b.toFinsupp fun a b => sum (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b * (↑φ)^[a] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ (sum b.toFinsupp fun a b => sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum b.toFinsupp fun a b => sum (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b * (↑φ)^[a] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ (fun a b => sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = fun a b => sum (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b * (↑φ)^[a] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
https://github.com/mariainesdff/skew_polynomials.git
16371a025f5c867f83ff258a22df5c0341793888
SkewPolynomials.lean
SkewPolynomial.mul_assoc
[221, 1]
[241, 84]
ext aβ‚‚ bβ‚‚
case e_g.h.h.e_g R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ (fun a b => sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = fun a b => sum (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b * (↑φ)^[a] bβ‚‚)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum (single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚_1 => single (aβ‚‚ + aβ‚‚_1) (bβ‚‚ * (↑φ)^[aβ‚‚] bβ‚‚_1)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum (single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) fun a₁ b₁ => sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚)) = sum (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚_1 => single (aβ‚‚ + aβ‚‚_1) (bβ‚‚ * (↑φ)^[aβ‚‚] bβ‚‚_1)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚_1 => single (a₁ + aβ‚‚ + aβ‚‚_1) (b₁ * (↑φ)^[a₁] bβ‚‚ * (↑φ)^[a₁ + aβ‚‚] bβ‚‚_1)) = sum c.toFinsupp fun a b => sum (single (aβ‚‚ + a) (bβ‚‚ * (↑φ)^[aβ‚‚] b)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚_1 => single (a₁ + aβ‚‚ + aβ‚‚_1) (b₁ * (↑φ)^[a₁] bβ‚‚ * (↑φ)^[a₁ + aβ‚‚] bβ‚‚_1)) = sum c.toFinsupp fun a b => sum (single (aβ‚‚ + a) (bβ‚‚ * (↑φ)^[aβ‚‚] b)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g.h.h.e_g R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (fun aβ‚‚_1 bβ‚‚_1 => single (a₁ + aβ‚‚ + aβ‚‚_1) (b₁ * (↑φ)^[a₁] bβ‚‚ * (↑φ)^[a₁ + aβ‚‚] bβ‚‚_1)) = fun a b => sum (single (aβ‚‚ + a) (bβ‚‚ * (↑φ)^[aβ‚‚] b)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (fun aβ‚‚_1 bβ‚‚_1 => single (a₁ + aβ‚‚ + aβ‚‚_1) (b₁ * (↑φ)^[a₁] bβ‚‚ * (↑φ)^[a₁ + aβ‚‚] bβ‚‚_1)) = fun a b => sum (single (aβ‚‚ + a) (bβ‚‚ * (↑φ)^[aβ‚‚] b)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + aβ‚‚ + a₃) (b₁ * (↑φ)^[a₁] bβ‚‚ * (↑φ)^[a₁ + aβ‚‚] b₃) = sum (single (aβ‚‚ + a₃) (bβ‚‚ * (↑φ)^[aβ‚‚] b₃)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + aβ‚‚ + a₃) (b₁ * (↑φ)^[a₁] bβ‚‚ * (↑φ)^[a₁ + aβ‚‚] b₃) = sum (single (aβ‚‚ + a₃) (bβ‚‚ * (↑φ)^[aβ‚‚] b₃)) fun aβ‚‚ bβ‚‚ => single (a₁ + aβ‚‚) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), 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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_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 : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁_1 bβ‚‚ : R), single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + bβ‚‚)) = single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] bβ‚‚) case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)
case e_g.h.h.e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R a₃ : β„• b₃ : R ⊒ single (a₁ + (aβ‚‚ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.e_g.h.h R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ (sum c.toFinsupp fun aβ‚‚_1 bβ‚‚ => single (a₁ + aβ‚‚ + aβ‚‚_1) (0 * (↑φ)^[a₁ + aβ‚‚] bβ‚‚)) = 0 case e_g.h.h.e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R aβ‚‚ : β„• bβ‚‚ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), single (a₁ + a) (b₁ * (↑φ)^[a₁] 0) = 0 case e_g.h.h.h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case e_g.h.h.h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] a₁ : β„• b₁ : R ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1) case h_zero R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•), (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (0 * (↑φ)^[a] bβ‚‚)) = 0 case h_add R : Type u a✝ b✝ : R m n : β„• inst✝ : Semiring R Ο† : R β†’+* R p q a b c : R[X;Ο†] ⊒ βˆ€ (a : β„•) (b₁ bβ‚‚ : R), (sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) ((b₁ + bβ‚‚) * (↑φ)^[a] bβ‚‚_1)) = (sum c.toFinsupp fun aβ‚‚ bβ‚‚ => single (a + aβ‚‚) (b₁ * (↑φ)^[a] bβ‚‚)) + sum c.toFinsupp fun aβ‚‚ bβ‚‚_1 => single (a + aβ‚‚) (bβ‚‚ * (↑φ)^[a] bβ‚‚_1)