url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
---|---|---|---|---|---|---|---|---|
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | obtain β¨zβ, hzββ© := nonempty_iff_ne_empty.2 h | z zβ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
β’ _root_.has_logs U | case intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
zβ : β
hzβ : zβ β U
β’ _root_.has_logs U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | rintro f hf hfz | case intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
zβ : β
hzβ : zβ β U
β’ _root_.has_logs U | case intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | obtain β¨lf, hlf1, hlf2β© := hp (deriv f / f) ((hf.deriv hU).div hf hfz) | case intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | let g : β β β := Ξ» z => lf z + (log (f zβ) - lf zβ) | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | set h : β β β := f / (exp β g) | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
h : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have h3 : DifferentiableOn β g U := hlf1.add (differentiableOn_const _) | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have e4 : DifferentiableOn β (exp β g) U := differentiable_exp.comp_differentiableOn h3 | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have e1 : DifferentiableOn β h U := hf.div e4 (Ξ» z _ => exp_ne_zero _) | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | refine β¨g, h3, ?_β© | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ β g, DifferentiableOn β g U β§ EqOn f (cexp β g) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ EqOn f (cexp β g) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | suffices h : EqOn h (Ξ» _ => 1) U
by exact Ξ» z hz => eq_of_div_eq_one (h hz) | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ EqOn f (cexp β g) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ EqOn h (fun x => 1) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have : 1 = h zβ := by unfold_let ; simp [exp_log, hfz zβ hzβ] | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ EqOn h (fun x => 1) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
β’ EqOn h (fun x => 1) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | rw [this] | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
β’ EqOn h (fun x => 1) U | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
β’ EqOn h (fun x => h zβ) U |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | refine EqOn_of_deriv_eq_zero hU hU' e1 (Ξ» z hz => ?_) hzβ | case intro.intro.intro
z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
β’ EqOn h (fun x => h zβ) U | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
β’ deriv h z = 0 z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have f0 : U β π z := hU.mem_nhds hz | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
β’ deriv h z = 0 z | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv h z = 0 z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | dsimp | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv h z = 0 z | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv h z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | unfold_let | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv h z = 0 | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv (f / cexp β fun z => lf z + (log (f zβ) - lf zβ)) z = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | rw [Pi.div_def, deriv_div (hf.differentiableAt f0) (e4.differentiableAt f0) (exp_ne_zero _)] | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv (f / cexp β fun z => lf z + (log (f zβ) - lf zβ)) z = 0 | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ (deriv f z * (cexp β g) z - f z * deriv (cexp β g) z) / (cexp β g) z ^ 2 = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | rw [deriv.scomp z differentiableAt_exp (h3.differentiableAt f0)] | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ (deriv f z * (cexp β g) z - f z * deriv (cexp β g) z) / (cexp β g) z ^ 2 = 0 | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ (deriv f z * (cexp β g) z - f z * deriv g z β’ deriv cexp (g z)) / (cexp β g) z ^ 2 = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | have e5 : deriv g z = deriv lf z := by unfold_let ; simp | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ (deriv f z * (cexp β g) z - f z * deriv g z β’ deriv cexp (g z)) / (cexp β g) z ^ 2 = 0 | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
e5 : deriv g z = deriv lf z
β’ (deriv f z * (cexp β g) z - f z * deriv g z β’ deriv cexp (g z)) / (cexp β g) z ^ 2 = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | field_simp [exp_ne_zero, hlf2 hz, hfz z hz, e5] | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
e5 : deriv g z = deriv lf z
β’ (deriv f z * (cexp β g) z - f z * deriv g z β’ deriv cexp (g z)) / (cexp β g) z ^ 2 = 0 | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
e5 : deriv g z = deriv lf z
β’ deriv f z * cexp (g z) * f z - f z * (deriv f z * cexp (g z)) = 0 |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | ring | case intro.intro.intro
zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
e5 : deriv g z = deriv lf z
β’ deriv f z * cexp (g z) * f z - f z * (deriv f z * cexp (g z)) = 0 | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | exact Ξ» z hz => eq_of_div_eq_one (h hz) | z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβΒΉ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
hβ : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β hβ U
h : EqOn hβ (fun x => 1) U
β’ EqOn f (cexp β g) U | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | unfold_let | z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ 1 = h zβ | z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ 1 = (f / cexp β fun z => lf z + (log (f zβ) - lf zβ)) zβ |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | simp [exp_log, hfz zβ hzβ] | z zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
β’ 1 = (f / cexp β fun z => lf z + (log (f zβ) - lf zβ)) zβ | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | unfold_let | zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv g z = deriv lf z | zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv (fun z => lf z + (log (f zβ) - lf zβ)) z = deriv lf z |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | RMT4/has_sqrt.lean | has_primitives.has_logs | [59, 1] | [85, 9] | simp | zβ zββ : β
U : Set β
hp : has_primitives U
hU : IsOpen U
hU' : IsPreconnected U
hβ : Β¬U = β
zβ : β
hzβ : zβ β U
f : β β β
hf : DifferentiableOn β f U
hfz : β z β U, f z β 0
lf : β β β
hlf1 : DifferentiableOn β lf U
hlf2 : EqOn (deriv lf) (deriv f / f) U
g : β β β := fun z => lf z + (log (f zβ) - lf zβ)
h : β β β := f / cexp β g
h3 : DifferentiableOn β g U
e4 : DifferentiableOn β (cexp β g) U
e1 : DifferentiableOn β h U
this : 1 = h zβ
z : β
hz : z β U
f0 : U β π z
β’ deriv (fun z => lf z + (log (f zβ) - lf zβ)) z = deriv lf z | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | induction n generalizing f | f : β β β
a b : β
hab : a < b
n : β
h : ContDiffOn β (βn) f (Icc a b)
β’ β g, ContDiff β (βn) g β§ EqOn g f (Icc a b) | case zero
a b : β
hab : a < b
f : β β β
h : ContDiffOn β (βNat.zero) f (Icc a b)
β’ β g, ContDiff β (βNat.zero) g β§ EqOn g f (Icc a b)
case succ
a b : β
hab : a < b
nβ : β
n_ihβ : β {f : β β β}, ContDiffOn β (βnβ) f (Icc a b) β β g, ContDiff β (βnβ) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ nβ)) f (Icc a b)
β’ β g, ContDiff β (β(Nat.succ nβ)) g β§ EqOn g f (Icc a b) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | case zero =>
simp only [CharP.cast_eq_zero, contDiff_zero, contDiffOn_zero] at h β’
refine β¨IccExtend hab.le (restrict (Icc a b) f), h.restrict.Icc_extend', ?_β©
exact Ξ» t ht => IccExtend_of_mem _ _ ht | a b : β
hab : a < b
f : β β β
h : ContDiffOn β (βNat.zero) f (Icc a b)
β’ β g, ContDiff β (βNat.zero) g β§ EqOn g f (Icc a b) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp only [CharP.cast_eq_zero, contDiff_zero, contDiffOn_zero] at h β’ | a b : β
hab : a < b
f : β β β
h : ContDiffOn β (βNat.zero) f (Icc a b)
β’ β g, ContDiff β (βNat.zero) g β§ EqOn g f (Icc a b) | a b : β
hab : a < b
f : β β β
h : ContinuousOn f (Icc a b)
β’ β g, Continuous g β§ EqOn g f (Icc a b) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | refine β¨IccExtend hab.le (restrict (Icc a b) f), h.restrict.Icc_extend', ?_β© | a b : β
hab : a < b
f : β β β
h : ContinuousOn f (Icc a b)
β’ β g, Continuous g β§ EqOn g f (Icc a b) | a b : β
hab : a < b
f : β β β
h : ContinuousOn f (Icc a b)
β’ EqOn (IccExtend β― (restrict (Icc a b) f)) f (Icc a b) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | exact Ξ» t ht => IccExtend_of_mem _ _ ht | a b : β
hab : a < b
f : β β β
h : ContinuousOn f (Icc a b)
β’ EqOn (IccExtend β― (restrict (Icc a b) f)) f (Icc a b) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | have h1 : ContDiffOn β n (derivWithin f (Icc a b)) (Icc a b) :=
h.derivWithin (uniqueDiffOn_Icc hab) le_rfl | a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
β’ β g, ContDiff β (β(Nat.succ n)) g β§ EqOn g f (Icc a b) | a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
β’ β g, ContDiff β (β(Nat.succ n)) g β§ EqOn g f (Icc a b) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | obtain β¨gg, h2, h3β© := ih h1 | a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
β’ β g, ContDiff β (β(Nat.succ n)) g β§ EqOn g f (Icc a b) | case intro.intro
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ β g, ContDiff β (β(Nat.succ n)) g β§ EqOn g f (Icc a b) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | refine β¨Ξ» t => f a + β« u in a..t, gg u, ?_, ?_β© | case intro.intro
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ β g, ContDiff β (β(Nat.succ n)) g β§ EqOn g f (Icc a b) | case intro.intro.refine_1
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ ContDiff β β(Nat.succ n) fun t => f a + β« (u : β) in a..t, gg u
case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ EqOn (fun t => f a + β« (u : β) in a..t, gg u) f (Icc a b) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | rw [contDiff_succ_iff_deriv] | case intro.intro.refine_1
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ ContDiff β β(Nat.succ n) fun t => f a + β« (u : β) in a..t, gg u | case intro.intro.refine_1
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ (Differentiable β fun t => f a + β« (u : β) in a..t, gg u) β§
ContDiff β (βn) (deriv fun t => f a + β« (u : β) in a..t, gg u) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | constructor | case intro.intro.refine_1
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ (Differentiable β fun t => f a + β« (u : β) in a..t, gg u) β§
ContDiff β (βn) (deriv fun t => f a + β« (u : β) in a..t, gg u) | case intro.intro.refine_1.left
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ Differentiable β fun t => f a + β« (u : β) in a..t, gg u
case intro.intro.refine_1.right
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ ContDiff β (βn) (deriv fun t => f a + β« (u : β) in a..t, gg u) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | refine differentiableOn_univ.1 ((differentiableOn_integral_of_continuous ?_ h2.continuous).const_add _) | case intro.intro.refine_1.left
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ Differentiable β fun t => f a + β« (u : β) in a..t, gg u | case intro.intro.refine_1.left
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ β x β univ, IntervalIntegrable gg MeasureTheory.volume a x |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp [h2.continuous.intervalIntegrable] | case intro.intro.refine_1.left
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ β x β univ, IntervalIntegrable gg MeasureTheory.volume a x | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | convert h2 | case intro.intro.refine_1.right
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ ContDiff β (βn) (deriv fun t => f a + β« (u : β) in a..t, gg u) | case h.e'_10
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ (deriv fun t => f a + β« (u : β) in a..t, gg u) = gg |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | ext t | case h.e'_10
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ (deriv fun t => f a + β« (u : β) in a..t, gg u) = gg | case h.e'_10.h
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
β’ deriv (fun t => f a + β« (u : β) in a..t, gg u) t = gg t |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp [deriv_const_add, h2.continuous.deriv_integral] | case h.e'_10.h
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
β’ deriv (fun t => f a + β« (u : β) in a..t, gg u) t = gg t | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | intro t ht | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
β’ EqOn (fun t => f a + β« (u : β) in a..t, gg u) f (Icc a b) | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
β’ (fun t => f a + β« (u : β) in a..t, gg u) t = f t |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | dsimp | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
β’ (fun t => f a + β« (u : β) in a..t, gg u) t = f t | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
β’ f a + β« (u : β) in a..t, gg u = f t |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | have l6 : Icc a t β Icc a b := Icc_subset_Icc_right ht.2 | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
β’ f a + β« (u : β) in a..t, gg u = f t | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
l6 : Icc a t β Icc a b
β’ f a + β« (u : β) in a..t, gg u = f t |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | have l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t) := h3.mono (by simp [uIcc, ht.1, l6]) | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
l6 : Icc a t β Icc a b
β’ f a + β« (u : β) in a..t, gg u = f t | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
l6 : Icc a t β Icc a b
l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t)
β’ f a + β« (u : β) in a..t, gg u = f t |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | have l10 := h.one_of_succ.integral_eq_sub'' hab.le ht | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
l6 : Icc a t β Icc a b
l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t)
β’ f a + β« (u : β) in a..t, gg u = f t | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
l6 : Icc a t β Icc a b
l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t)
l10 : β« (y : β) in a..t, derivWithin f (Icc a b) y = f t - f a
β’ f a + β« (u : β) in a..t, gg u = f t |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp [integral_congr l9, l10] | case intro.intro.refine_2
a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
l6 : Icc a t β Icc a b
l9 : EqOn gg (derivWithin f (Icc a b)) (uIcc a t)
l10 : β« (y : β) in a..t, derivWithin f (Icc a b) y = f t - f a
β’ f a + β« (u : β) in a..t, gg u = f t | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto | [5, 1] | [29, 36] | simp [uIcc, ht.1, l6] | a b : β
hab : a < b
n : β
ih : β {f : β β β}, ContDiffOn β (βn) f (Icc a b) β β g, ContDiff β (βn) g β§ EqOn g f (Icc a b)
f : β β β
h : ContDiffOn β (β(Nat.succ n)) f (Icc a b)
h1 : ContDiffOn β (βn) (derivWithin f (Icc a b)) (Icc a b)
gg : β β β
h2 : ContDiff β (βn) gg
h3 : EqOn gg (derivWithin f (Icc a b)) (Icc a b)
t : β
ht : t β Icc a b
l6 : Icc a t β Icc a b
β’ uIcc a t β Icc a b | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | cases eq_or_ne a b | f : β β β
a b : β
n : β
h : ContDiffOn β (βn) f (uIcc a b)
β’ β g, ContDiff β (βn) g β§ EqOn g f (uIcc a b) | case inl
f : β β β
a b : β
n : β
h : ContDiffOn β (βn) f (uIcc a b)
hβ : a = b
β’ β g, ContDiff β (βn) g β§ EqOn g f (uIcc a b)
case inr
f : β β β
a b : β
n : β
h : ContDiffOn β (βn) f (uIcc a b)
hβ : a β b
β’ β g, ContDiff β (βn) g β§ EqOn g f (uIcc a b) |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | case inl hab => exact β¨Ξ» _ => f a, by simp [hab, contDiff_const]β© | f : β β β
a b : β
n : β
h : ContDiffOn β (βn) f (uIcc a b)
hab : a = b
β’ β g, ContDiff β (βn) g β§ EqOn g f (uIcc a b) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | exact β¨Ξ» _ => f a, by simp [hab, contDiff_const]β© | f : β β β
a b : β
n : β
h : ContDiffOn β (βn) f (uIcc a b)
hab : a = b
β’ β g, ContDiff β (βn) g β§ EqOn g f (uIcc a b) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | simp [hab, contDiff_const] | f : β β β
a b : β
n : β
h : ContDiffOn β (βn) f (uIcc a b)
hab : a = b
β’ (ContDiff β βn fun x => f a) β§ EqOn (fun x => f a) f (uIcc a b) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | case inr hab => exact toto (min_lt_max.2 hab) h | f : β β β
a b : β
n : β
h : ContDiffOn β (βn) f (uIcc a b)
hab : a β b
β’ β g, ContDiff β (βn) g β§ EqOn g f (uIcc a b) | no goals |
https://github.com/vbeffara/RMT4.git | c2a092d029d0e6d29a381ac4ad9e85b10d97391c | extend.lean | toto' | [31, 1] | [35, 52] | exact toto (min_lt_max.2 hab) h | f : β β β
a b : β
n : β
h : ContDiffOn β (βn) f (uIcc a b)
hab : a β b
β’ β g, ContDiff β (βn) g β§ EqOn g f (uIcc a b) | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.eta | [34, 1] | [34, 87] | cases f | R : Type u
a b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q f : R[X;Ο]
β’ { toFinsupp := f.toFinsupp } = f | case ofFinsupp
R : Type u
a b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
toFinsuppβ : AddMonoidAlgebra R β
β’ { toFinsupp := { toFinsupp := toFinsuppβ }.toFinsupp } = { toFinsupp := toFinsuppβ } |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.eta | [34, 1] | [34, 87] | rfl | case ofFinsupp
R : Type u
a b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
toFinsuppβ : AddMonoidAlgebra R β
β’ { toFinsupp := { toFinsupp := toFinsuppβ }.toFinsupp } = { toFinsupp := toFinsuppβ } | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_add | [102, 1] | [103, 35] | rw [add_def] | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a b : AddMonoidAlgebra R β
β’ { toFinsupp := a + b } = SkewPolynomial.add { toFinsupp := a } { toFinsupp := b } | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_neg | [106, 1] | [107, 33] | rw [neg_def] | R : Type u
aβ b : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
a : AddMonoidAlgebra S β
β’ { toFinsupp := -a } = SkewPolynomial.neg { toFinsupp := a } | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_sub | [110, 1] | [113, 6] | rw [sub_eq_add_neg, ofFinsupp_add, ofFinsupp_neg] | R : Type u
aβ bβ : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
a b : AddMonoidAlgebra S β
β’ { toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b } | R : Type u
aβ bβ : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
a b : AddMonoidAlgebra S β
β’ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b } |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_sub | [110, 1] | [113, 6] | rfl | R : Type u
aβ bβ : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
a b : AddMonoidAlgebra S β
β’ { toFinsupp := a } + -{ toFinsupp := b } = { toFinsupp := a } - { toFinsupp := b } | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_mul | [116, 1] | [117, 35] | rw [mul_def] | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a b : AddMonoidAlgebra R β
β’ { toFinsupp := AddMonoidAlgebra.mul' Ο a b } = SkewPolynomial.mul { toFinsupp := a } { toFinsupp := b } | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | change _ = npowRec n _ | R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
n : β
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο n a } = { toFinsupp := a } ^ n | R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
n : β
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο n a } = npowRec n { toFinsupp := a } |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | induction n with
| zero => simp [npowRec]; rfl
| succ n n_ih => simp [npowRec, pow_succ]; rw [<- n_ih, <- ofFinsupp_mul]; rfl | R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
n : β
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο n a } = npowRec n { toFinsupp := a } | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | simp [npowRec] | case zero
R : Type u
aβ b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο Nat.zero a } = npowRec Nat.zero { toFinsupp := a } | case zero
R : Type u
aβ b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο 0 a } = 1 |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | rfl | case zero
R : Type u
aβ b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο 0 a } = 1 | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | simp [npowRec, pow_succ] | case succ
R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
n : β
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο n a } = npowRec n { toFinsupp := a }
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο (Nat.succ n) a } = npowRec (Nat.succ n) { toFinsupp := a } | case succ
R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
n : β
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο n a } = npowRec n { toFinsupp := a }
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο (Nat.succ n) a } =
{ toFinsupp := a } * npowRec n { toFinsupp := a } |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | rw [<- n_ih, <- ofFinsupp_mul] | case succ
R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
n : β
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο n a } = npowRec n { toFinsupp := a }
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο (Nat.succ n) a } =
{ toFinsupp := a } * npowRec n { toFinsupp := a } | case succ
R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
n : β
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο n a } = npowRec n { toFinsupp := a }
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο (Nat.succ n) a } =
{ toFinsupp := AddMonoidAlgebra.mul' Ο a (SkewPolynomial.AddMonoidAlgebra.pow' Ο n a) } |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_pow | [125, 1] | [129, 81] | rfl | case succ
R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
n : β
n_ih : { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο n a } = npowRec n { toFinsupp := a }
β’ { toFinsupp := SkewPolynomial.AddMonoidAlgebra.pow' Ο (Nat.succ n) a } =
{ toFinsupp := AddMonoidAlgebra.mul' Ο a (SkewPolynomial.AddMonoidAlgebra.pow' Ο n a) } | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_add | [140, 1] | [143, 23] | cases a | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b : R[X;Ο]
β’ (a + b).toFinsupp = a.toFinsupp + b.toFinsupp | case ofFinsupp
R : Type u
a bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q b : R[X;Ο]
toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβ } + b).toFinsupp = { toFinsupp := toFinsuppβ }.toFinsupp + b.toFinsupp |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_add | [140, 1] | [143, 23] | cases b | case ofFinsupp
R : Type u
a bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q b : R[X;Ο]
toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβ } + b).toFinsupp = { toFinsupp := toFinsuppβ }.toFinsupp + b.toFinsupp | case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
toFinsuppβΒΉ toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβΒΉ } + { toFinsupp := toFinsuppβ }).toFinsupp =
{ toFinsupp := toFinsuppβΒΉ }.toFinsupp + { toFinsupp := toFinsuppβ }.toFinsupp |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_add | [140, 1] | [143, 23] | rw [β ofFinsupp_add] | case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
toFinsuppβΒΉ toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβΒΉ } + { toFinsupp := toFinsuppβ }).toFinsupp =
{ toFinsupp := toFinsuppβΒΉ }.toFinsupp + { toFinsupp := toFinsuppβ }.toFinsupp | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_neg | [146, 1] | [149, 23] | cases a | R : Type u
aβ b : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
a : S[X;Ο]
β’ (-a).toFinsupp = -a.toFinsupp | case ofFinsupp
R : Type u
a b : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
toFinsuppβ : AddMonoidAlgebra S β
β’ (-{ toFinsupp := toFinsuppβ }).toFinsupp = -{ toFinsupp := toFinsuppβ }.toFinsupp |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_neg | [146, 1] | [149, 23] | rw [β ofFinsupp_neg] | case ofFinsupp
R : Type u
a b : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
toFinsuppβ : AddMonoidAlgebra S β
β’ (-{ toFinsupp := toFinsuppβ }).toFinsupp = -{ toFinsupp := toFinsuppβ }.toFinsupp | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_sub | [152, 1] | [155, 6] | rw [sub_eq_add_neg, β toFinsupp_neg, β toFinsupp_add] | R : Type u
aβ bβ : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
a b : S[X;Ο]
β’ (a - b).toFinsupp = a.toFinsupp - b.toFinsupp | R : Type u
aβ bβ : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
a b : S[X;Ο]
β’ (a - b).toFinsupp = (a + -b).toFinsupp |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_sub | [152, 1] | [155, 6] | rfl | R : Type u
aβ bβ : R
m n : β
instβΒΉ : Semiring R
Οβ : R β+* R
p q : R[X;Οβ]
S : Type u
instβ : Ring S
Ο : S β+* S
a b : S[X;Ο]
β’ (a - b).toFinsupp = (a + -b).toFinsupp | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_mul | [158, 1] | [162, 23] | cases a | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b : R[X;Ο]
β’ (a * b).toFinsupp = AddMonoidAlgebra.mul' Ο a.toFinsupp b.toFinsupp | case ofFinsupp
R : Type u
a bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q b : R[X;Ο]
toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβ } * b).toFinsupp =
AddMonoidAlgebra.mul' Ο { toFinsupp := toFinsuppβ }.toFinsupp b.toFinsupp |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_mul | [158, 1] | [162, 23] | cases b | case ofFinsupp
R : Type u
a bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q b : R[X;Ο]
toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβ } * b).toFinsupp =
AddMonoidAlgebra.mul' Ο { toFinsupp := toFinsuppβ }.toFinsupp b.toFinsupp | case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
toFinsuppβΒΉ toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβΒΉ } * { toFinsupp := toFinsuppβ }).toFinsupp =
AddMonoidAlgebra.mul' Ο { toFinsupp := toFinsuppβΒΉ }.toFinsupp { toFinsupp := toFinsuppβ }.toFinsupp |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_mul | [158, 1] | [162, 23] | rw [β ofFinsupp_mul] | case ofFinsupp.ofFinsupp
R : Type u
a b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
toFinsuppβΒΉ toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβΒΉ } * { toFinsupp := toFinsuppβ }).toFinsupp =
AddMonoidAlgebra.mul' Ο { toFinsupp := toFinsuppβΒΉ }.toFinsupp { toFinsupp := toFinsuppβ }.toFinsupp | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_pow | [171, 1] | [174, 23] | cases a | R : Type u
aβ b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q a : R[X;Ο]
n : β
β’ (a ^ n).toFinsupp = SkewPolynomial.AddMonoidAlgebra.pow' Ο n a.toFinsupp | case ofFinsupp
R : Type u
a b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
n : β
toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβ } ^ n).toFinsupp =
SkewPolynomial.AddMonoidAlgebra.pow' Ο n { toFinsupp := toFinsuppβ }.toFinsupp |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_pow | [171, 1] | [174, 23] | rw [β ofFinsupp_pow] | case ofFinsupp
R : Type u
a b : R
m nβ : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
n : β
toFinsuppβ : AddMonoidAlgebra R β
β’ ({ toFinsupp := toFinsuppβ } ^ n).toFinsupp =
SkewPolynomial.AddMonoidAlgebra.pow' Ο n { toFinsupp := toFinsuppβ }.toFinsupp | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_eq_zero | [188, 1] | [189, 39] | rw [β toFinsupp_zero, toFinsupp_inj] | R : Type u
aβ b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a : R[X;Ο]
β’ a.toFinsupp = 0 β a = 0 | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.toFinsupp_eq_one | [192, 1] | [193, 38] | rw [β toFinsupp_one, toFinsupp_inj] | R : Type u
aβ b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a : R[X;Ο]
β’ a.toFinsupp = 1 β a = 1 | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_eq_zero | [200, 1] | [201, 39] | rw [β ofFinsupp_zero, ofFinsupp_inj] | R : Type u
aβ b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
β’ { toFinsupp := a } = 0 β a = 0 | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.ofFinsupp_eq_one | [204, 1] | [204, 100] | rw [β ofFinsupp_one, ofFinsupp_inj] | R : Type u
aβ b : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q : R[X;Ο]
a : AddMonoidAlgebra R β
β’ { toFinsupp := a } = 1 β a = 1 | no goals |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [βtoFinsupp_inj] | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ a * b * c = a * (b * c) | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ (a * b * c).toFinsupp = (a * (b * c)).toFinsupp |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | simp only [toFinsupp_mul, AddMonoidAlgebra.mul'_def] | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ (a * b * c).toFinsupp = (a * (b * c)).toFinsupp | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ (sum (sum a.toFinsupp fun aβ bβ => sum b.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum a.toFinsupp fun aβ bβ =>
sum (sum b.toFinsupp fun aβ bβ => sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [sum_sum_index] | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ (sum (sum a.toFinsupp fun aβ bβ => sum b.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum a.toFinsupp fun aβ bβ =>
sum (sum b.toFinsupp fun aβ bβ => sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ) | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ (sum a.toFinsupp fun a b_1 =>
sum (sum b.toFinsupp fun aβ bβ => single (a + aβ) (b_1 * (βΟ)^[a] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum a.toFinsupp fun aβ bβ =>
sum (sum b.toFinsupp fun aβ bβ => sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | congr | R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ (sum a.toFinsupp fun a b_1 =>
sum (sum b.toFinsupp fun aβ bβ => single (a + aβ) (b_1 * (βΟ)^[a] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum a.toFinsupp fun aβ bβ =>
sum (sum b.toFinsupp fun aβ bβ => sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ (fun a b_1 =>
sum (sum b.toFinsupp fun aβ bβ => single (a + aβ) (b_1 * (βΟ)^[a] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
fun aβ bβ =>
sum (sum b.toFinsupp fun aβ bβ => sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | ext aβ bβ | case e_g
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ (fun a b_1 =>
sum (sum b.toFinsupp fun aβ bβ => single (a + aβ) (b_1 * (βΟ)^[a] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
fun aβ bβ =>
sum (sum b.toFinsupp fun aβ bβ => sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ (sum (sum b.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum (sum b.toFinsupp fun aβ bβ => sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [sum_sum_index, sum_sum_index] | case e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ (sum (sum b.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum (sum b.toFinsupp fun aβ bβ => sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ (sum b.toFinsupp fun a b =>
sum (single (aβ + a) (bβ * (βΟ)^[aβ] b)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum b.toFinsupp fun a b =>
sum (sum c.toFinsupp fun aβ bβ => single (a + aβ) (b * (βΟ)^[a] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | congr | case e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ (sum b.toFinsupp fun a b =>
sum (single (aβ + a) (bβ * (βΟ)^[aβ] b)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum b.toFinsupp fun a b =>
sum (sum c.toFinsupp fun aβ bβ => single (a + aβ) (b * (βΟ)^[a] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ (fun a b =>
sum (single (aβ + a) (bβ * (βΟ)^[aβ] b)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
fun a b =>
sum (sum c.toFinsupp fun aβ bβ => single (a + aβ) (b * (βΟ)^[a] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | ext aβ bβ | case e_g.h.h.e_g
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ (fun a b =>
sum (single (aβ + a) (bβ * (βΟ)^[aβ] b)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
fun a b =>
sum (sum c.toFinsupp fun aβ bβ => single (a + aβ) (b * (βΟ)^[a] bβ)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum (single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum (sum c.toFinsupp fun aβ_1 bβ_1 => single (aβ + aβ_1) (bβ * (βΟ)^[aβ] bβ_1)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [sum_sum_index, AddMonoidAlgebra.sum_single_index] | case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum (single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ =>
sum c.toFinsupp fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) =
sum (sum c.toFinsupp fun aβ_1 bβ_1 => single (aβ + aβ_1) (bβ * (βΟ)^[aβ] bβ_1)) fun aβ bβ =>
single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ_1 => single (aβ + aβ + aβ_1) (bβ * (βΟ)^[aβ] bβ * (βΟ)^[aβ + aβ] bβ_1)) =
sum c.toFinsupp fun a b =>
sum (single (aβ + a) (bβ * (βΟ)^[aβ] b)) fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | congr | case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ_1 => single (aβ + aβ + aβ_1) (bβ * (βΟ)^[aβ] bβ * (βΟ)^[aβ + aβ] bβ_1)) =
sum c.toFinsupp fun a b =>
sum (single (aβ + a) (bβ * (βΟ)^[aβ] b)) fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g.h.h.e_g
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (fun aβ_1 bβ_1 => single (aβ + aβ + aβ_1) (bβ * (βΟ)^[aβ] bβ * (βΟ)^[aβ + aβ] bβ_1)) = fun a b =>
sum (single (aβ + a) (bβ * (βΟ)^[aβ] b)) fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | ext aβ bβ | case e_g.h.h.e_g.h.h.e_g
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (fun aβ_1 bβ_1 => single (aβ + aβ + aβ_1) (bβ * (βΟ)^[aβ] bβ * (βΟ)^[aβ + aβ] bβ_1)) = fun a b =>
sum (single (aβ + a) (bβ * (βΟ)^[aβ] b)) fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + aβ + aβ) (bβ * (βΟ)^[aβ] bβ * (βΟ)^[aβ + aβ] bβ) =
sum (single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | rw [AddMonoidAlgebra.sum_single_index, _root_.mul_assoc, RingHom.iterate_map_mul,
β Function.iterate_add_apply, add_assoc] | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + aβ + aβ) (bβ * (βΟ)^[aβ] bβ * (βΟ)^[aβ + aβ] bβ) =
sum (single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)) fun aβ bβ => single (aβ + aβ) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | pick_goal 4 | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | . intros n r1 r2
rw [RingHom.iterate_map_add, mul_add, AddMonoidAlgebra.single_add] | case e_g.h.h.e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | pick_goal 5 | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
https://github.com/mariainesdff/skew_polynomials.git | 16371a025f5c867f83ff258a22df5c0341793888 | SkewPolynomials.lean | SkewPolynomial.mul_assoc | [221, 1] | [241, 84] | . intros n r1 r2
rw [RingHom.iterate_map_add, mul_add, AddMonoidAlgebra.single_add] | case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ_1 bβ : R),
single (aβ + a) (bβ * (βΟ)^[aβ] (bβ_1 + bβ)) =
single (aβ + a) (bβ * (βΟ)^[aβ] bβ_1) + single (aβ + a) (bβ * (βΟ)^[aβ] bβ)
case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) | case e_g.h.h.e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
aβ : β
bβ : R
β’ single (aβ + (aβ + aβ)) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.e_g.h.h
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ (sum c.toFinsupp fun aβ_1 bβ => single (aβ + aβ + aβ_1) (0 * (βΟ)^[aβ + aβ] bβ)) = 0
case e_g.h.h.e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), single (aβ + a) (bβ * (βΟ)^[aβ] 0) = 0
case e_g.h.h.h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case e_g.h.h.h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
aβ : β
bβ : R
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1)
case h_zero
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β), (sum c.toFinsupp fun aβ bβ => single (a + aβ) (0 * (βΟ)^[a] bβ)) = 0
case h_add
R : Type u
aβ bβ : R
m n : β
instβ : Semiring R
Ο : R β+* R
p q a b c : R[X;Ο]
β’ β (a : β) (bβ bβ : R),
(sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) ((bβ + bβ) * (βΟ)^[a] bβ_1)) =
(sum c.toFinsupp fun aβ bβ => single (a + aβ) (bβ * (βΟ)^[a] bβ)) +
sum c.toFinsupp fun aβ bβ_1 => single (a + aβ) (bβ * (βΟ)^[a] bβ_1) |
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