url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | obtain ⟨z₀, hz₀⟩ := nonempty_iff_ne_empty.2 h | z z₀ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
⊢ _root_.has_logs U | case intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
⊢ _root_.has_logs U | Please generate a tactic in lean4 to solve the state.
STATE:
z z₀ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
⊢ _root_.has_logs U
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | rintro f hf hfz | case intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
⊢ _root_.has_logs U | case intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
⊢ _root_.has_logs U
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | obtain ⟨lf, hlf1, hlf2⟩ := hp (deriv f / f) ((hf.deriv hU).div hf hfz) | case intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ∘ g) U
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | let g : ℂ → ℂ := λ z => lf z + (log (f z₀) - lf z₀) | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
⊢ ∃ g, DifferentiableOn ℂ g U ∧ EqOn f (cexp ... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | set h : ℂ → ℂ := f / (exp ∘ g) | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have h3 : DifferentiableOn ℂ g U := hlf1.add (differentiableOn_const _) | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf)... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have e4 : DifferentiableOn ℂ (exp ∘ g) U := differentiable_exp.comp_differentiableOn h3 | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf)... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have e1 : DifferentiableOn ℂ h U := hf.div e4 (λ z _ => exp_ne_zero _) | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf)... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | refine ⟨g, h3, ?_⟩ | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf)... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | suffices h : EqOn h (λ _ => 1) U
by exact λ z hz => eq_of_div_eq_one (h hz) | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf)... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have : 1 = h z₀ := by unfold_let ; simp [exp_log, hfz z₀ hz₀] | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf)... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | rw [this] | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf)... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | refine EqOn_of_deriv_eq_zero hU hU' e1 (λ z hz => ?_) hz₀ | case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - l... | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf)... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have f0 : U ∈ 𝓝 z := hU.mem_nhds hz | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | dsimp | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | unfold_let | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | rw [Pi.div_def, deriv_div (hf.differentiableAt f0) (e4.differentiableAt f0) (exp_ne_zero _)] | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | rw [deriv.scomp z differentiableAt_exp (h3.differentiableAt f0)] | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have e5 : deriv g z = deriv lf z := by unfold_let ; simp | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | field_simp [exp_ne_zero, hlf2 hz, hfz z hz, e5] | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | ring | case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | exact λ z hz => eq_of_div_eq_one (h hz) | z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝¹ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀)
h✝ : ℂ → ℂ := f ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝¹ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | unfold_let | z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀)
h : ℂ → ℂ := f / ... | z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀)
h : ℂ → ℂ := f / ... | Please generate a tactic in lean4 to solve the state.
STATE:
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | simp [exp_log, hfz z₀ hz₀] | z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀)
h : ℂ → ℂ := f / ... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | unfold_let | z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀)
h : ℂ → ℂ := f /... | z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀)
h : ℂ → ℂ := f /... | Please generate a tactic in lean4 to solve the state.
STATE:
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | simp | z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ → ℂ := fun z => lf z + (log (f z₀) - lf z₀)
h : ℂ → ℂ := f /... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
z✝ z₀✝ : ℂ
U : Set ℂ
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h✝ : ¬U = ∅
z₀ : ℂ
hz₀ : z₀ ∈ U
f : ℂ → ℂ
hf : DifferentiableOn ℂ f U
hfz : ∀ z ∈ U, f z ≠ 0
lf : ℂ → ℂ
hlf1 : DifferentiableOn ℂ lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : ℂ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | induction n generalizing f | f : ℝ → ℝ
a b : ℝ
hab : a < b
n : ℕ
h : ContDiffOn ℝ (↑n) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b) | case zero
a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b)
case succ
a b : ℝ
hab : a < b
n✝ : ℕ
n_ih✝ : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n✝) f (Icc a b) → ∃ g, ContDiff ℝ (↑n✝) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n✝)) f (... | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
a b : ℝ
hab : a < b
n : ℕ
h : ContDiffOn ℝ (↑n) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | case zero =>
simp only [CharP.cast_eq_zero, contDiff_zero, contDiffOn_zero] at h ⊢
refine ⟨IccExtend hab.le (restrict (Icc a b) f), h.restrict.Icc_extend', ?_⟩
exact λ t ht => IccExtend_of_mem _ _ ht | a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp only [CharP.cast_eq_zero, contDiff_zero, contDiffOn_zero] at h ⊢ | a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b) | a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContinuousOn f (Icc a b)
⊢ ∃ g, Continuous g ∧ EqOn g f (Icc a b) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContDiffOn ℝ (↑Nat.zero) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑Nat.zero) g ∧ EqOn g f (Icc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | refine ⟨IccExtend hab.le (restrict (Icc a b) f), h.restrict.Icc_extend', ?_⟩ | a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContinuousOn f (Icc a b)
⊢ ∃ g, Continuous g ∧ EqOn g f (Icc a b) | a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContinuousOn f (Icc a b)
⊢ EqOn (IccExtend ⋯ (restrict (Icc a b) f)) f (Icc a b) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContinuousOn f (Icc a b)
⊢ ∃ g, Continuous g ∧ EqOn g f (Icc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | exact λ t ht => IccExtend_of_mem _ _ ht | a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContinuousOn f (Icc a b)
⊢ EqOn (IccExtend ⋯ (restrict (Icc a b) f)) f (Icc a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
f : ℝ → ℝ
h : ContinuousOn f (Icc a b)
⊢ EqOn (IccExtend ⋯ (restrict (Icc a b) f)) f (Icc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | have h1 : ContDiffOn ℝ n (derivWithin f (Icc a b)) (Icc a b) :=
h.derivWithin (uniqueDiffOn_Icc hab) le_rfl | a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b) | a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | obtain ⟨gg, h2, h3⟩ := ih h1 | a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑(Nat.succ n)) g ∧ EqOn g f (Icc a b) | case intro.intro
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWithin f (Icc... | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
⊢ ∃ g, ContDiff ℝ (↑(Na... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | refine ⟨λ t => f a + ∫ u in a..t, gg u, ?_, ?_⟩ | case intro.intro
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWithin f (Icc... | case intro.intro.refine_1
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | rw [contDiff_succ_iff_deriv] | case intro.intro.refine_1
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | case intro.intro.refine_1
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_1
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | constructor | case intro.intro.refine_1
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | case intro.intro.refine_1.left
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (deri... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_1
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | refine differentiableOn_univ.1 ((differentiableOn_integral_of_continuous ?_ h2.continuous).const_add _) | case intro.intro.refine_1.left
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (deri... | case intro.intro.refine_1.left
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (deri... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_1.left
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (I... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp [h2.continuous.intervalIntegrable] | case intro.intro.refine_1.left
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (deri... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_1.left
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (I... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | convert h2 | case intro.intro.refine_1.right
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (der... | case h.e'_10
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWithin f (Icc a b... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_1.right
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | ext t | case h.e'_10
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWithin f (Icc a b... | case h.e'_10.h
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWithin f (Icc a... | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_10
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp [deriv_const_add, h2.continuous.deriv_integral] | case h.e'_10.h
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWithin f (Icc a... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_10.h
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ →... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | intro t ht | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | dsimp | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | have l6 : Icc a t ⊆ Icc a b := Icc_subset_Icc_right ht.2 | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | have l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t) := h3.mono (by simp [uIcc, ht.1, l6]) | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | have l10 := h.one_of_succ.integral_eq_sub'' hab.le ht | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp [integral_congr l9, l10] | case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWith... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.refine_2
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a ... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp [uIcc, ht.1, l6] | a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDiff ℝ (↑n) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : ℝ
hab : a < b
n : ℕ
ih : ∀ {f : ℝ → ℝ}, ContDiffOn ℝ (↑n) f (Icc a b) → ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (Icc a b)
f : ℝ → ℝ
h : ContDiffOn ℝ (↑(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn ℝ (↑n) (derivWithin f (Icc a b)) (Icc a b)
gg : ℝ → ℝ
h2 : ContDif... |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | cases eq_or_ne a b | f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b) | case inl
f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
h✝ : a = b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)
case inr
f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
h✝ : a ≠ b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b) | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | case inl hab => exact ⟨λ _ => f a, by simp [hab, contDiff_const]⟩ | f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a = b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a = b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | exact ⟨λ _ => f a, by simp [hab, contDiff_const]⟩ | f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a = b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a = b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | simp [hab, contDiff_const] | f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a = b
⊢ (ContDiff ℝ ↑n fun x => f a) ∧ EqOn (fun x => f a) f (uIcc a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a = b
⊢ (ContDiff ℝ ↑n fun x => f a) ∧ EqOn (fun x => f a) f (uIcc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | case inr hab => exact toto (min_lt_max.2 hab) h | f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a ≠ b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a ≠ b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)
TACTIC:
|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | exact toto (min_lt_max.2 hab) h | f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a ≠ b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
a b : ℝ
n : ℕ
h : ContDiffOn ℝ (↑n) f (uIcc a b)
hab : a ≠ b
⊢ ∃ g, ContDiff ℝ (↑n) g ∧ EqOn g f (uIcc a b)
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.eta | [34, 1] | [34, 87] | cases f | R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q f : R[X;φ]
⊢ { toFinsupp := f.toFinsupp } = f | case ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := { toFinsupp := toFinsupp✝ }.toFinsupp } = { toFinsupp := toFinsupp✝ } | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q f : R[X;φ]
⊢ { toFinsupp := f.toFinsupp } = f
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.eta | [34, 1] | [34, 87] | rfl | case ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := { toFinsupp := toFinsupp✝ }.toFinsupp } = { toFinsupp := toFinsupp✝ } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := { toFinsupp := toFinsupp✝ }.toFinsupp } = { toFinsupp := toFinsupp✝ }
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_add | [102, 1] | [103, 35] | rw [add_def] | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a b : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := a + b } = SkewPolynomial.add { toFinsupp := a } { toFinsupp := b } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a b : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := a + b } = SkewPolynomial.add { toFinsupp := a } { toFinsupp := b }
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_neg | [106, 1] | [107, 33] | rw [neg_def] | R : Type u
a✝ b : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a : AddMonoidAlgebra S ℕ
⊢ { toFinsupp := -a } = SkewPolynomial.neg { toFinsupp := a } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a : AddMonoidAlgebra S ℕ
⊢ { toFinsupp := -a } = SkewPolynomial.neg { toFinsupp := a }
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_sub | [110, 1] | [113, 6] | rw [sub_eq_add_neg, ofFinsupp_add, ofFinsupp_neg] | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : AddMonoidAlgebra S ℕ
⊢ { toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b } | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : AddMonoidAlgebra S ℕ
⊢ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b } | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : AddMonoidAlgebra S ℕ
⊢ { toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b }
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_sub | [110, 1] | [113, 6] | rfl | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : AddMonoidAlgebra S ℕ
⊢ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : AddMonoidAlgebra S ℕ
⊢ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b }
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_mul | [116, 1] | [117, 35] | rw [mul_def] | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a b : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := AddMonoidAlgebra.mul' φ a b } = SkewPolynomial.mul { toFinsupp := a } { toFinsupp := b } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a b : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := AddMonoidAlgebra.mul' φ a b } = SkewPolynomial.mul { toFinsupp := a } { toFinsupp := b }
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | change _ = npowRec n _ | R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = { toFinsupp := a } ^ n | R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a } | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = { toFinsupp := a } ^ n
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | induction n with
| zero => simp [npowRec]; rfl
| succ n n_ih => simp [npowRec, pow_succ]; rw [<- n_ih, <- ofFinsupp_mul]; rfl | R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a } | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | simp [npowRec] | case zero
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ Nat.zero a } = npowRec Nat.zero { toFinsupp := a } | case zero
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ 0 a } = 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ Nat.zero a } = npowRec Nat.zero { toFinsupp := a }
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | rfl | case zero
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ 0 a } = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ 0 a } = 1
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | simp [npowRec, pow_succ] | case succ
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } = npowRec (Nat.succ n) { toFinsupp ... | case succ
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } =
{ toFinsupp := a } * npowRec ... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebr... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | rw [<- n_ih, <- ofFinsupp_mul] | case succ
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } =
{ toFinsupp := a } * npowRec ... | case succ
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } =
{ toFinsupp := AddMonoidAlgeb... | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebr... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | rfl | case succ
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ (Nat.succ n) a } =
{ toFinsupp := AddMonoidAlgeb... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
n : ℕ
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' φ n a } = npowRec n { toFinsupp := a }
⊢ { toFinsupp := SkewPolynomial.AddMonoidAlgebr... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_add | [140, 1] | [143, 23] | cases a | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b : R[X;φ]
⊢ (a + b).toFinsupp = a.toFinsupp + b.toFinsupp | case ofFinsupp
R : Type u
a b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q b : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } + b).toFinsupp = { toFinsupp := toFinsupp✝ }.toFinsupp + b.toFinsupp | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b : R[X;φ]
⊢ (a + b).toFinsupp = a.toFinsupp + b.toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_add | [140, 1] | [143, 23] | cases b | case ofFinsupp
R : Type u
a b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q b : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } + b).toFinsupp = { toFinsupp := toFinsupp✝ }.toFinsupp + b.toFinsupp | case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }).toFinsupp =
{ toFinsupp := toFinsupp✝¹ }.toFinsupp + { toFinsupp := toFinsupp✝ }.toFinsupp | Please generate a tactic in lean4 to solve the state.
STATE:
case ofFinsupp
R : Type u
a b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q b : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } + b).toFinsupp = { toFinsupp := toFinsupp✝ }.toFinsupp + b.toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_add | [140, 1] | [143, 23] | rw [← ofFinsupp_add] | case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }).toFinsupp =
{ toFinsupp := toFinsupp✝¹ }.toFinsupp + { toFinsupp := toFinsupp✝ }.toFinsupp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }).toFinsupp =
{ toFinsupp := toFinsupp✝¹ }.toFinsupp... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_neg | [146, 1] | [149, 23] | cases a | R : Type u
a✝ b : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a : S[X;φ]
⊢ (-a).toFinsupp = -a.toFinsupp | case ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
toFinsupp✝ : AddMonoidAlgebra S ℕ
⊢ (-{ toFinsupp := toFinsupp✝ }).toFinsupp = -{ toFinsupp := toFinsupp✝ }.toFinsupp | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a : S[X;φ]
⊢ (-a).toFinsupp = -a.toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_neg | [146, 1] | [149, 23] | rw [← ofFinsupp_neg] | case ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
toFinsupp✝ : AddMonoidAlgebra S ℕ
⊢ (-{ toFinsupp := toFinsupp✝ }).toFinsupp = -{ toFinsupp := toFinsupp✝ }.toFinsupp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
toFinsupp✝ : AddMonoidAlgebra S ℕ
⊢ (-{ toFinsupp := toFinsupp✝ }).toFinsupp = -{ toFinsupp := toFinsupp✝ }.toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_sub | [152, 1] | [155, 6] | rw [sub_eq_add_neg, ← toFinsupp_neg, ← toFinsupp_add] | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : S[X;φ]
⊢ (a - b).toFinsupp = a.toFinsupp - b.toFinsupp | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : S[X;φ]
⊢ (a - b).toFinsupp = (a + -b).toFinsupp | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : S[X;φ]
⊢ (a - b).toFinsupp = a.toFinsupp - b.toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_sub | [152, 1] | [155, 6] | rfl | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : S[X;φ]
⊢ (a - b).toFinsupp = (a + -b).toFinsupp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝¹ : Semiring R
φ✝ : R →+* R
p q : R[X;φ✝]
S : Type u
inst✝ : Ring S
φ : S →+* S
a b : S[X;φ]
⊢ (a - b).toFinsupp = (a + -b).toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_mul | [158, 1] | [162, 23] | cases a | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b : R[X;φ]
⊢ (a * b).toFinsupp = AddMonoidAlgebra.mul' φ a.toFinsupp b.toFinsupp | case ofFinsupp
R : Type u
a b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q b : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } * b).toFinsupp =
AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝ }.toFinsupp b.toFinsupp | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b : R[X;φ]
⊢ (a * b).toFinsupp = AddMonoidAlgebra.mul' φ a.toFinsupp b.toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_mul | [158, 1] | [162, 23] | cases b | case ofFinsupp
R : Type u
a b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q b : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } * b).toFinsupp =
AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝ }.toFinsupp b.toFinsupp | case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp =
AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝¹ }.toFinsupp { toFinsupp := toFinsupp✝ }.toFinsup... | Please generate a tactic in lean4 to solve the state.
STATE:
case ofFinsupp
R : Type u
a b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q b : R[X;φ]
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } * b).toFinsupp =
AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝ }.toFinsupp b.toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_mul | [158, 1] | [162, 23] | rw [← ofFinsupp_mul] | case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp =
AddMonoidAlgebra.mul' φ { toFinsupp := toFinsupp✝¹ }.toFinsupp { toFinsupp := toFinsupp✝ }.toFinsup... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
toFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝¹ } * { toFinsupp := toFinsupp✝ }).toFinsupp =
AddMonoidAlgebra.mul' φ { toFinsupp :=... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_pow | [171, 1] | [174, 23] | cases a | R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a : R[X;φ]
n : ℕ
⊢ (a ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' φ n a.toFinsupp | case ofFinsupp
R : Type u
a b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
n : ℕ
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } ^ n).toFinsupp =
SkewPolynomial.AddMonoidAlgebra.pow' φ n { toFinsupp := toFinsupp✝ }.toFinsupp | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a : R[X;φ]
n : ℕ
⊢ (a ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' φ n a.toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_pow | [171, 1] | [174, 23] | rw [← ofFinsupp_pow] | case ofFinsupp
R : Type u
a b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
n : ℕ
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } ^ n).toFinsupp =
SkewPolynomial.AddMonoidAlgebra.pow' φ n { toFinsupp := toFinsupp✝ }.toFinsupp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ofFinsupp
R : Type u
a b : R
m n✝ : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
n : ℕ
toFinsupp✝ : AddMonoidAlgebra R ℕ
⊢ ({ toFinsupp := toFinsupp✝ } ^ n).toFinsupp =
SkewPolynomial.AddMonoidAlgebra.pow' φ n { toFinsupp := toFinsupp✝ }.toFinsupp
T... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_eq_zero | [188, 1] | [189, 39] | rw [← toFinsupp_zero, toFinsupp_inj] | R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a : R[X;φ]
⊢ a.toFinsupp = 0 ↔ a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a : R[X;φ]
⊢ a.toFinsupp = 0 ↔ a = 0
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_eq_one | [192, 1] | [193, 38] | rw [← toFinsupp_one, toFinsupp_inj] | R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a : R[X;φ]
⊢ a.toFinsupp = 1 ↔ a = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a : R[X;φ]
⊢ a.toFinsupp = 1 ↔ a = 1
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_eq_zero | [200, 1] | [201, 39] | rw [← ofFinsupp_zero, ofFinsupp_inj] | R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := a } = 0 ↔ a = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := a } = 0 ↔ a = 0
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_eq_one | [204, 1] | [204, 100] | rw [← ofFinsupp_one, ofFinsupp_inj] | R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := a } = 1 ↔ a = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q : R[X;φ]
a : AddMonoidAlgebra R ℕ
⊢ { toFinsupp := a } = 1 ↔ a = 1
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [←toFinsupp_inj] | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ a * b * c = a * (b * c) | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (a * b * c).toFinsupp = (a * (b * c)).toFinsupp | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ a * b * c = a * (b * c)
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | simp only [toFinsupp_mul, AddMonoidAlgebra.mul'_def] | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (a * b * c).toFinsupp = (a * (b * c)).toFinsupp | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (sum (sum a.toFinsupp fun a₁ b₁ => sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum a.toFinsupp fun a₁ b₁ =>
sum (sum b... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (a * b * c).toFinsupp = (a * (b * c)).toFinsupp
TACTIC:
|
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [sum_sum_index] | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (sum (sum a.toFinsupp fun a₁ b₁ => sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum a.toFinsupp fun a₁ b₁ =>
sum (sum b... | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (sum a.toFinsupp fun a b_1 =>
sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum a.toFinsupp fun a₁ b₁ =>
sum... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (sum (sum a.toFinsupp fun a₁ b₁ => sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | congr | R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (sum a.toFinsupp fun a b_1 =>
sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum a.toFinsupp fun a₁ b₁ =>
sum... | case e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (fun a b_1 =>
sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
fun a₁ b₁ =>
sum (sum b.toFinsupp fun a₁ ... | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (sum a.toFinsupp fun a b_1 =>
sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ ... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | ext a₁ b₁ | case e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (fun a b_1 =>
sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
fun a₁ b₁ =>
sum (sum b.toFinsupp fun a₁ ... | case e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (sum (sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsu... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
⊢ (fun a b_1 =>
sum (sum b.toFinsupp fun a₂ b₂ => single (a + a₂) (b_1 * (↑φ)^[a] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [sum_sum_index, sum_sum_index] | case e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (sum (sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum (sum b.toFinsupp fun a₁ b₁ => sum c.toFinsu... | case e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (sum b.toFinsupp fun a b =>
sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum b.toFinsupp fun a b =>
sum (sum c... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (sum (sum b.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | congr | case e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (sum b.toFinsupp fun a b =>
sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum b.toFinsupp fun a b =>
sum (sum c... | case e_g.h.h.e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (fun a b =>
sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
fun a b =>
sum (sum c.toFinsupp fun a₂ b₂ => single... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (sum b.toFinsupp fun a b =>
sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | ext a₂ b₂ | case e_g.h.h.e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (fun a b =>
sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
fun a b =>
sum (sum c.toFinsupp fun a₂ b₂ => single... | case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (sum (single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₂ + a₂_... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ (fun a b =>
sum (single (a₁ + a) (b₁ * (↑φ)^[a₁] b)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) ... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [sum_sum_index, AddMonoidAlgebra.sum_single_index] | case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (sum (single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) =
sum (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₂ + a₂_... | case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) =
sum c.toFinsupp fun a b =>
sum (single (a₂ + a) (b₂ * (↑φ)^[a₂] b)) fun a₂ b₂... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (sum (single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)) fun a₁ b₁ =>
sum c.toFinsupp fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂))... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | congr | case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) =
sum c.toFinsupp fun a b =>
sum (single (a₂ + a) (b₂ * (↑φ)^[a₂] b)) fun a₂ b₂... | case e_g.h.h.e_g.h.h.e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) = fun a b =>
sum (single (a₂ + a) (b₂ * (↑φ)^[a₂] b)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (sum c.toFinsupp fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) =
sum c.toFinsupp fun a ... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | ext a₃ b₃ | case e_g.h.h.e_g.h.h.e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) = fun a b =>
sum (single (a₂ + a) (b₂ * (↑φ)^[a₂] b)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a... | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + a₂ + a₃) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₃) =
sum (single (a₂ + a₃) (b₂ * (↑φ)^[a₂] b₃)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)
case e... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.e_g.h.h.e_g
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ (fun a₂_1 b₂_1 => single (a₁ + a₂ + a₂_1) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₂_1)) = fun a b =>
sum (single (a₂ + a) (b... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [AddMonoidAlgebra.sum_single_index, _root_.mul_assoc, RingHom.iterate_map_mul,
← Function.iterate_add_apply, add_assoc] | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + a₂ + a₃) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₃) =
sum (single (a₂ + a₃) (b₂ * (↑φ)^[a₂] b₃)) fun a₂ b₂ => single (a₁ + a₂) (b₁ * (↑φ)^[a₁] b₂)
case e... | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + a₂ + a₃) (b₁ * (↑φ)^[a₁] b₂ * (↑φ)^[a₁ + a₂] b₃) =
sum (single (a₂ + a₃) (b₂ * (↑φ)^[a₂] b... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | pick_goal 4 | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ... | case e_g.h.h.e_g.h.h.h_add
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ ∀ (a : ℕ) (b₁_1 b₂ : R),
single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) =
single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂)
case e_g.h.h.e_g.h.h.e_g.... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : S... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | . intros n r1 r2
rw [RingHom.iterate_map_add, mul_add, AddMonoidAlgebra.single_add] | case e_g.h.h.e_g.h.h.h_add
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ ∀ (a : ℕ) (b₁_1 b₂ : R),
single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) =
single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂)
case e_g.h.h.e_g.h.h.e_g.... | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.e_g.h.h.h_add
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
⊢ ∀ (a : ℕ) (b₁_1 b₂ : R),
single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) =
single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + s... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | pick_goal 5 | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ... | case e_g.h.h.h_add
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ ∀ (a : ℕ) (b₁_1 b₂ : R),
single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) =
single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂)
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ :... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : S... |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | . intros n r1 r2
rw [RingHom.iterate_map_add, mul_add, AddMonoidAlgebra.single_add] | case e_g.h.h.h_add
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ ∀ (a : ℕ) (b₁_1 b₂ : R),
single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) =
single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (↑φ)^[a₁] b₂)
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ :... | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ
b₂ : R
a₃ : ℕ
b₃ : R
⊢ single (a₁ + (a₂ + a₃)) (b₁ * (↑φ)^[a₁] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
a₂ : ℕ... | Please generate a tactic in lean4 to solve the state.
STATE:
case e_g.h.h.h_add
R : Type u
a✝ b✝ : R
m n : ℕ
inst✝ : Semiring R
φ : R →+* R
p q a b c : R[X;φ]
a₁ : ℕ
b₁ : R
⊢ ∀ (a : ℕ) (b₁_1 b₂ : R),
single (a₁ + a) (b₁ * (↑φ)^[a₁] (b₁_1 + b₂)) =
single (a₁ + a) (b₁ * (↑φ)^[a₁] b₁_1) + single (a₁ + a) (b₁ * (... |
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