url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [h, add_one_mul s, add_eq_zero_iff_eq] at hr | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [hr, add_add_cancel_right] at hs | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s * (s + 1) + 1 = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rwa [← h, hs, zero_mul, eq_comm] at hr | case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : s * (s + 1) = s + 1
hs : s = 0
h✝ : (r + s + 1) * (r + s + 1) = 0
h : r = s + 1
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [add_assoc, self_eq_add_right] at h0 | case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : r = r + s + 1
⊢ r = 0 | case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : s + 1 = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : r = r + s + 1
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [h0, mul_zero, zero_add] at hs | case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : s + 1 = 0
⊢ r = 0 | case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : s + 1 = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : s + 1 = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [← mul_one r, hs, mul_zero] | case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : s + 1 = 0
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : 1 = 0
h : (r + s + 1) * (r + s + 1) = 0
h0 : s + 1 = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rcases R_elts_claim1 hS hf hr h with hr | h | h0 | case inl.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
⊢ r = 0 | case inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
hr : r = 0
⊢ r = 0
case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0
case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
h0 : r = r + s
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | exact hr | case inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
hr : r = 0
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
hr : r = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [← add_eq_zero_iff_eq.mp h, mul_add_one r, hr, zero_add] at hs | case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0 | case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : r + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [add_eq_zero_iff_eq.mp hs, mul_one] at hr | case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : r + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0 | case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : 1 = 0
hs : r + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : r + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rwa [hr, add_zero] at hs | case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : 1 = 0
hs : r + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : 1 = 0
hs : r + 1 = 0
h✝ : (r + s) * (r + s) = 0
h : r + s = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [self_eq_add_right] at h0 | case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
h0 : r = r + s
⊢ r = 0 | case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
h0 : s = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
h0 : r = r + s
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [h0, zero_mul, zero_add] at hs | case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
h0 : s = 0
⊢ r = 0 | case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : 1 = 0
h : (r + s) * (r + s) = 0
h0 : s = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s) = 0
h0 : s = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [← mul_one r, hs, mul_zero] | case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : 1 = 0
h : (r + s) * (r + s) = 0
h0 : s = 0
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : 1 = 0
h : (r + s) * (r + s) = 0
h0 : s = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [add_assoc, mul_add, add_mul, hr, zero_add,
add_mul, ← add_assoc, add_assoc, hs, add_zero] at h | case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s + 1) + 1 = 0
⊢ r = 0 | case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : (r + s) * (r + s + 1) + 1 = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rw [mul_zero, zero_mul, mul_add, ← mul_assoc,
← mul_assoc, hr, zero_mul, add_zero, mul_assoc] at h0 | case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0 : r * (s * r + r * (s + 1)) * s = r * 0 * s
⊢ r = 0 | case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0 : r * s * (r * s) = 0
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0 : r * (s * r + r * (s + 1)) * s = r * 0 * s
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rcases R_elts_claim1 hS hf hr h0 with hr | h0 | h0 | case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0 : r * s * (r * s) = 0
⊢ r = 0 | case inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0 : r * s * (r * s) = 0
hr : r = 0
⊢ r = 0
case inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0✝ : r * s * (r * s) = 0
h0 : r * s = 0
⊢ r = 0
case inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0✝ : r * s * (r * s) = 0
h0 : r = r * s
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0 : r * s * (r * s) = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | exact hr | case inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0 : r * s * (r * s) = 0
hr : r = 0
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr✝ : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0 : r * s * (r * s) = 0
hr : r = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rwa [zero_mul, mul_assoc, add_eq_zero_iff_eq.mp hs, mul_one] at h0 | case inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0✝ : r * s * (r * s) = 0
h0 : r * s * (s + 1) = 0 * (s + 1)
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
hs : s * (s + 1) + 1 = 0
h : s * r + r * (s + 1) = 0
h0✝ : r * s * (r * s) = 0
h0 : r * s * (s + 1) = 0 * (s + 1)
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.R_elts_claim2 | [349, 1] | [384, 57] | rwa [mul_zero, mul_add_one r, ← mul_assoc, ← h0,
mul_add_one r, ← h0, add_add_cancel_right] at hs | case inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
h : s * r + r * (s + 1) = 0
h0✝ : r * s * (r * s) = 0
h0 : r = r * s
hs : r * (s * (s + 1) + 1) = r * 0
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
R : Type u_1
S : Type ?u.172967
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r s : R
hr : r * r = 0
h : s * r + r * (s + 1) = 0
h0✝ : r * s * (r * s) = 0
h0 : r = r * s
hs : r * (s * (s + 1) + 1) = r * 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂_solution | [386, 1] | [399, 78] | apply congrArg f at h | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : 1 = 0
⊢ 2 = 0 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : f 1 = f 0
⊢ 2 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : 1 = 0
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂_solution | [386, 1] | [399, 78] | rw [hf.map_one, hf.map_zero, zero_eq_neg] at h | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : f 1 = f 0
⊢ 2 = 0 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : 1 = 0
⊢ 2 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : f 1 = f 0
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂_solution | [386, 1] | [399, 78] | rw [← mul_one 2, h, mul_zero] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : 1 = 0
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : 1 = 0
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂_solution | [386, 1] | [399, 78] | change f 0 = ((-1 : ℤ) : S) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : Function.Bijective 𝔽₂.cast
⊢ f 𝔽₂.O.cast = ↑(𝔽₂Map 𝔽₂.O) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : Function.Bijective 𝔽₂.cast
⊢ f 0 = ↑(-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : Function.Bijective 𝔽₂.cast
⊢ f 𝔽₂.O.cast = ↑(𝔽₂Map 𝔽₂.O)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂_solution | [386, 1] | [399, 78] | rw [hf.map_zero, Int.cast_neg, Int.cast_one] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : Function.Bijective 𝔽₂.cast
⊢ f 0 = ↑(-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
hR : ∀ (r : R), r = 0 ∨ r = 1
h : Function.Bijective 𝔽₂.cast
⊢ f 0 = ↑(-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | refine ⟨𝔽₂ε.castRingHom_injective hr0 hr, λ x ↦ ?_⟩ | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
⊢ Function.Bijective (𝔽₂ε.cast r) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
⊢ ∃ a, 𝔽₂ε.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
⊢ Function.Bijective (𝔽₂ε.cast r)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | rcases R_elts_cases hS hf x with (h0 | h0) | h0 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
⊢ ∃ a, 𝔽₂ε.cast r a = x | case inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : (x + 1) * (x + 1) = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x
case inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : x * x = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x
case inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : x * (x + 1) + 1 = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
⊢ ∃ a, 𝔽₂ε.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | exact ((R_elts_claim1 hS hf hr0 h0).resolve_left hr).elim
(λ h1 ↦ ⟨1, (add_eq_zero_iff_eq.mp h1).symm⟩)
(λ h1 ↦ ⟨𝔽₂ε.Y, add_eq_iff_eq_add.mpr h1⟩) | case inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : (x + 1) * (x + 1) = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : (x + 1) * (x + 1) = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | exact ((R_elts_claim1 hS hf hr0 h0).resolve_left hr).elim
(λ h1 ↦ ⟨0, h1.symm⟩) (λ h1 ↦ ⟨𝔽₂ε.X, h1⟩) | case inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : x * x = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : x * x = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | exact absurd (R_elts_claim2 hS hf hr0 h0) hr | case inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : x * (x + 1) + 1 = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
x : R
h0 : x * (x + 1) + 1 = 0
⊢ ∃ a, 𝔽₂ε.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | change f 0 = ((-1 : ℤ) : S) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f (𝔽₂ε.cast r 𝔽₂ε.O) = ↑(𝔽₂εMap 𝔽₂ε.O) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f 0 = ↑(-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f (𝔽₂ε.cast r 𝔽₂ε.O) = ↑(𝔽₂εMap 𝔽₂ε.O)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | rw [hf.map_zero, Int.cast_neg, Int.cast_one] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f 0 = ↑(-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f 0 = ↑(-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | change f r = ((1 : ℤ) : S) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f (𝔽₂ε.cast r 𝔽₂ε.X) = ↑(𝔽₂εMap 𝔽₂ε.X) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f r = ↑1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f (𝔽₂ε.cast r 𝔽₂ε.X) = ↑(𝔽₂εMap 𝔽₂ε.X)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | have h0 := Eq2 hf.toNontrivialGood r | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f r = ↑1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f (r * r + 1) = f r ^ 2 - 1
⊢ f r = ↑1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
⊢ f r = ↑1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | rw [hr0, zero_add, hf.map_one, eq_comm, sub_eq_zero, _root_.sq_eq_one_iff] at h0 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f (r * r + 1) = f r ^ 2 - 1
⊢ f r = ↑1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f r = 1 ∨ f r = -1
⊢ f r = ↑1 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f (r * r + 1) = f r ^ 2 - 1
⊢ f r = ↑1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | refine (h0.resolve_right λ h1 ↦ hr ?_).trans Int.cast_one.symm | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f r = 1 ∨ f r = -1
⊢ f r = ↑1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f r = 1 ∨ f r = -1
h1 : f r = -1
⊢ r = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f r = 1 ∨ f r = -1
⊢ f r = ↑1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₂ε_solution | [401, 1] | [425, 78] | exact map_eq_neg_one_reduced_imp hS hf h1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f r = 1 ∨ f r = -1
h1 : f r = -1
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r ≠ 0
hr0 : r * r = 0
h : Function.Bijective (𝔽₂ε.cast r)
h0 : f r = 1 ∨ f r = -1
h1 : f r = -1
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | have X : (1 : R) ≠ 0 := λ X ↦ hS <| by
apply congrArg f at X; rw [hf.map_one, hf.map_zero, zero_eq_neg] at X
rw [← mul_one 2, X, mul_zero] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | obtain ⟨hr0, hr1⟩ := (reduced_𝔽₄_iff hS hf).mp hr | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | have hr' : r * r + r = 1 := by rwa [← mul_add_one r, ← add_eq_zero_iff_eq] | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [← eq_sub_iff_add_eq'] at hr0 | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [hr0, ← neg_sub, mul_neg, neg_inj, mul_sub_one, sub_eq_iff_eq_add'] at hr1 | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | let ι := ℤφ.castRingHom hr1 | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | have h0 : ∀ x, f (𝔽₄.cast r x) = ι (𝔽₄Map x)
| 𝔽₄.O => by change f 0 = ι (-1)
rw [hf.map_zero, ι.map_neg, ι.map_one]
| 𝔽₄.I => (hf.map_one).trans ι.map_zero.symm
| 𝔽₄.X => (ℤφ.cast_φ _).symm
| 𝔽₄.Y => by change f (r + 1) = ((1 : ℤ) : S) + (-1 : ℤ) • f r
rw [Int.cast_one, neg_one_zsmul, hr0, sub_eq_add_neg] | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
h0 : ∀ (x : 𝔽₄), f (𝔽₄.cast r x) = ι (𝔽₄Map x)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | let ρ := RingEquiv.ofBijective (𝔽₄.castRingHom hr') h | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
h0 : ∀ (x : 𝔽₄), f (𝔽₄.cast r x) = ι (𝔽₄Map x)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
h0 : ∀ (x : 𝔽₄), f (𝔽₄.cast r x) = ι (𝔽₄Map x)
ρ : 𝔽₄ ≃+* R := RingEquiv.ofBijective (𝔽₄.castRingHom hr') h
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
h0 : ∀ (x : 𝔽₄), f (𝔽₄.cast r x) = ι (𝔽₄Map x)
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | exact ⟨ρ.symm, ι, λ x ↦ h0 _ ▸ congrArg f (Equiv.apply_symm_apply ρ.toEquiv _).symm⟩ | case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
h0 : ∀ (x : 𝔽₄), f (𝔽₄.cast r x) = ι (𝔽₄Map x)
ρ : 𝔽₄ ≃+* R := RingEquiv.ofBijective (𝔽₄.castRingHom hr') h
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
h0 : ∀ (x : 𝔽₄), f (𝔽₄.cast r x) = ι (𝔽₄Map x)
ρ : 𝔽₄ ≃+* R := RingEquiv.ofBijective (𝔽₄.castRingHom hr') h
⊢ ∃ φ ι, ∀ (x : R), f x = ι (𝔽₄Map (φ x))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | apply congrArg f at X | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 = 0
⊢ 2 = 0 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : f 1 = f 0
⊢ 2 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 = 0
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [hf.map_one, hf.map_zero, zero_eq_neg] at X | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : f 1 = f 0
⊢ 2 = 0 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 = 0
⊢ 2 = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : f 1 = f 0
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [← mul_one 2, X, mul_zero] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 = 0
⊢ 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 = 0
⊢ 2 = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rwa [← mul_add_one r, ← add_eq_zero_iff_eq] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
⊢ r * r + r = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
⊢ r * r + r = 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | refine ⟨𝔽₄.castRingHom_injective hr' X, λ x ↦ ?_⟩ | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
⊢ Function.Bijective (𝔽₄.cast r) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
⊢ Function.Bijective (𝔽₄.cast r)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | have h0 {x} (h0 : x * x = 0) : x = 0 := R_elts_claim2 hS hf h0 hr | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
⊢ ∃ a, 𝔽₄.cast r a = x | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rcases R_elts_cases hS hf x with (h1 | h1) | h1 | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : (x + 1) * (x + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
case inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = 0
⊢ ∃ a, 𝔽₄.cast r a = x
case inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rcases R_elts_cases hS hf (x + r) with (h2 | h2) | h2 | case inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r + 1) * (x + r + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
case inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r) * (x + r) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
case inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r) * (x + r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [_root_.add_comm x r, add_assoc, mul_add, add_mul, add_mul, add_assoc, add_assoc,
add_assoc, h1, add_zero, mul_add_one r, ← add_assoc (x * r), add_left_comm, hr'] at h2 | case inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r) * (x + r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r) * (x + r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rcases R_elts_cases hS hf (x * r) with (h3 | h3) | h3 | case inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : (x * r + 1) * (x * r + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r * (x * r) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | exact ⟨1, (add_eq_zero_iff_eq.mp (h0 h1)).symm⟩ | case inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : (x + 1) * (x + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : (x + 1) * (x + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | exact ⟨0, (h0 h1).symm⟩ | case inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = 0
⊢ ∃ a, 𝔽₄.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | exact ⟨𝔽₄.Y, (add_eq_zero_iff_eq.mp <| (add_assoc x r 1).symm.trans (h0 h2)).symm⟩ | case inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r + 1) * (x + r + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r + 1) * (x + r + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | exact ⟨𝔽₄.X, (add_eq_zero_iff_eq.mp (h0 h2)).symm⟩ | case inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r) * (x + r) = 0
⊢ ∃ a, 𝔽₄.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : (x + r) * (x + r) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | apply h0 at h3 | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : (x * r + 1) * (x * r + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : (x * r + 1) * (x * r + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [add_right_comm, h3, zero_add] at h2 | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : x * r + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [mul_zero, mul_add_one r, ← mul_assoc, h2, zero_mul, zero_add] at h3 | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : r * (x * r + 1) = r * 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : r * (x * r + 1) = r * 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [h3, zero_mul, zero_add] at hr | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | exact absurd hr X | case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inl
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 0
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | apply h0 at h3 | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r * (x * r) = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r * (x * r) = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [h3, zero_add, add_eq_zero_iff_eq] at h2 | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | have h4 : r = 0 := by rw [← one_mul r, ← h2, mul_assoc, h3, mul_zero] | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
h4 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [h4, zero_mul, zero_add] at hr | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
h4 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
h4 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
h4 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | exact absurd hr X | case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
h4 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inl.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
h4 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [← one_mul r, ← h2, mul_assoc, h3, mul_zero] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
⊢ r = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : r * x = 1
h3 : x * r = 0
⊢ r = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [add_right_comm, add_eq_zero_iff_eq] at h2 | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + 1 = r * x
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + r * x + 1 = 0
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [mul_add_one x, add_assoc, add_eq_zero_iff_eq] at h1 | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + 1 = r * x
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * (x + 1) + 1 = 0
h2 : x * r + 1 = r * x
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [h2, ← mul_assoc, mul_assoc x, add_eq_iff_eq_add'.mp hr', mul_add_one x, add_mul,
h1, add_assoc, add_add_cancel_right, ← add_one_mul _ x, h2, mul_assoc, h1] at h3 | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r * (x + 1) = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : x * r * (x * r + 1) + 1 = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [zero_mul, mul_assoc, add_one_mul x, h1, add_add_cancel_middle₂, mul_one] at h3 | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r * (x + 1) * x = 0 * x
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r * (x + 1) * x = 0 * x
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [h3, zero_mul, zero_add] at hr | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | exact absurd hr X | case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr.inr
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : 1 = 0
X : 1 ≠ 0
hr0 : f r + f (r + 1) = 1
hr1 : f r * f (r + 1) = -1
hr' : r * r + r = 1
x : R
h0 : ∀ {x : R}, x * x = 0 → x = 0
h1 : x * x = x + 1
h2 : x * r + 1 = r * x
h3 : r = 0
⊢ ∃ a, 𝔽₄.cast r a = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | change f 0 = ι (-1) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f (𝔽₄.cast r 𝔽₄.O) = ι (𝔽₄Map 𝔽₄.O) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f 0 = ι (-1) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f (𝔽₄.cast r 𝔽₄.O) = ι (𝔽₄Map 𝔽₄.O)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [hf.map_zero, ι.map_neg, ι.map_one] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f 0 = ι (-1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f 0 = ι (-1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | change f (r + 1) = ((1 : ℤ) : S) + (-1 : ℤ) • f r | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f (𝔽₄.cast r 𝔽₄.Y) = ι (𝔽₄Map 𝔽₄.Y) | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f (r + 1) = ↑1 + -1 • f r | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f (𝔽₄.cast r 𝔽₄.Y) = ι (𝔽₄Map 𝔽₄.Y)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A5/A5Cases/Case3.lean | IMOSL.IMO2012A5.Case3.SCharNeTwo.𝔽₄_solution | [427, 1] | [481, 87] | rw [Int.cast_one, neg_one_zsmul, hr0, sub_eq_add_neg] | R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f (r + 1) = ↑1 + -1 • f r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
S : Type u_2
inst✝³ : Ring R
inst✝² : CharTwo R
inst✝¹ : Ring S
inst✝ : NoZeroDivisors S
hS : 2 ≠ 0
f : R → S
hf : ReducedGood f
r : R
hr : r * (r + 1) + 1 = 0
X : 1 ≠ 0
hr0 : f (r + 1) = 1 - f r
hr1 : f r * f r = f r + 1
hr' : r * r + r = 1
h : Function.Bijective (𝔽₄.cast r)
ι : ℤφ →+* S := ℤφ.castRingHom hr1
⊢ f (r + 1) = ↑1 + -1 • f r
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | refine ⟨λ h ↦ funext λ n ↦ le_antisymm ?_ ?_,
λ h m n ↦ h.symm ▸ dvd_refl (m * m + n)⟩ | f : ℕ+ → ℕ+
⊢ (∀ (m n : ℕ+), m * m + f n ∣ m * f m + n) ↔ f = id | case refine_1
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ f n ≤ id n
case refine_2
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ id n ≤ f n | Please generate a tactic in lean4 to solve the state.
STATE:
f : ℕ+ → ℕ+
⊢ (∀ (m n : ℕ+), m * m + f n ∣ m * f m + n) ↔ f = id
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | specialize h (f n) n | case refine_1
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ f n ≤ id n | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * f n + f n ∣ f n * f (f n) + n
⊢ f n ≤ id n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rw [← mul_add_one (α := ℕ+)] at h | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * f n + f n ∣ f n * f (f n) + n
⊢ f n ≤ id n | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * (f n + 1) ∣ f n * f (f n) + n
⊢ f n ≤ id n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * f n + f n ∣ f n * f (f n) + n
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | replace h := (dvd_mul_right _ _).trans h | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * (f n + 1) ∣ f n * f (f n) + n
⊢ f n ≤ id n | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ f n * f (f n) + n
⊢ f n ≤ id n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n * (f n + 1) ∣ f n * f (f n) + n
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rw [PNat.dvd_iff, PNat.add_coe, PNat.mul_coe,
Nat.dvd_add_right ⟨_, rfl⟩, ← PNat.dvd_iff] at h | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ f n * f (f n) + n
⊢ f n ≤ id n | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ n
⊢ f n ≤ id n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ f n * f (f n) + n
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | exact PNat.le_of_dvd h | case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ n
⊢ f n ≤ id n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : ℕ+ → ℕ+
n : ℕ+
h : f n ∣ n
⊢ f n ≤ id n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rcases eq_or_ne n 1 with rfl | h0 | case refine_2
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ id n ≤ f n | case refine_2.inl
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
⊢ id 1 ≤ f 1
case refine_2.inr
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
h0 : n ≠ 1
⊢ id n ≤ f n | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
⊢ id n ≤ f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | exact (f 1).one_le | case refine_2.inl
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
⊢ id 1 ≤ f 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inl
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
⊢ id 1 ≤ f 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rcases PNat.exists_eq_succ_of_ne_one h0 with ⟨k, rfl⟩ | case refine_2.inr
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
h0 : n ≠ 1
⊢ id n ≤ f n | case refine_2.inr.intro
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
k : ℕ+
h0 : k + 1 ≠ 1
⊢ id (k + 1) ≤ f (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
n : ℕ+
h0 : n ≠ 1
⊢ id n ≤ f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | replace h := PNat.le_of_dvd (h (k + 1) (k + 1)) | case refine_2.inr.intro
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
k : ℕ+
h0 : k + 1 ≠ 1
⊢ id (k + 1) ≤ f (k + 1) | case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : (k + 1) * (k + 1) + f (k + 1) ≤ (k + 1) * f (k + 1) + (k + 1)
⊢ id (k + 1) ≤ f (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
f : ℕ+ → ℕ+
h : ∀ (m n : ℕ+), m * m + f n ∣ m * f m + n
k : ℕ+
h0 : k + 1 ≠ 1
⊢ id (k + 1) ≤ f (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | rw [add_one_mul (α := ℕ+), add_right_comm,
add_assoc, add_one_mul (α := ℕ+), add_assoc] at h | case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : (k + 1) * (k + 1) + f (k + 1) ≤ (k + 1) * f (k + 1) + (k + 1)
⊢ id (k + 1) ≤ f (k + 1) | case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : k * (k + 1) + (f (k + 1) + (k + 1)) ≤ k * f (k + 1) + (f (k + 1) + (k + 1))
⊢ id (k + 1) ≤ f (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : (k + 1) * (k + 1) + f (k + 1) ≤ (k + 1) * f (k + 1) + (k + 1)
⊢ id (k + 1) ≤ f (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/N1/N1.lean | IMOSL.IMO2013N1.final_solution | [20, 1] | [39, 73] | exact le_of_mul_le_mul_left' <| le_of_add_le_add_right (α := ℕ+) h | case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : k * (k + 1) + (f (k + 1) + (k + 1)) ≤ k * f (k + 1) + (f (k + 1) + (k + 1))
⊢ id (k + 1) ≤ f (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.inr.intro
f : ℕ+ → ℕ+
k : ℕ+
h0 : k + 1 ≠ 1
h : k * (k + 1) + (f (k + 1) + (k + 1)) ≤ k * f (k + 1) + (f (k + 1) + (k + 1))
⊢ id (k + 1) ≤ f (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.le_max_one_sq | [48, 1] | [51, 76] | apply (le_abs_self a).trans | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ a ≤ max 1 (a ^ 2) | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ max 1 (a ^ 2) | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ a ≤ max 1 (a ^ 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.le_max_one_sq | [48, 1] | [51, 76] | rw [← sq_abs, le_max_iff] | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ max 1 (a ^ 2) | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ 1 ∨ |a| ≤ |a| ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ max 1 (a ^ 2)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.le_max_one_sq | [48, 1] | [51, 76] | exact (le_total |a| 1).imp_right λ h ↦ le_self_pow h (Nat.succ_ne_zero 1) | R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ 1 ∨ |a| ≤ |a| ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
a : R
⊢ |a| ≤ 1 ∨ |a| ≤ |a| ^ 2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rcases le_total a 0 with h | h | R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : a ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
case inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rcases le_total b 0 with h0 | h0 | case inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inr.inl
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : b ≤ 0
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | rw [posPart_eq_self.mpr h, posPart_eq_self.mpr h0, eq_comm] | case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ | case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ = a * b | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a⁺ * b⁺ = (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2012/A7/A7PiRing.lean | IMOSL.IMO2012A7.pos_part_mul_pos_part_main_formula | [53, 1] | [70, 50] | refine (congr_arg _ (min_eq_left <| le_min ?_ ?_)).trans
(posPart_eq_self.mpr (mul_nonneg h h0)) | case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ = a * b | case inr.inr.refine_1
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a * b ≤ max a (a * b ^ 2)
case inr.inr.refine_2
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ a * b ≤ max b (a ^ 2 * b) | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.inr
R : Type u_1
inst✝ : LinearOrderedRing R
a b : R
h : 0 ≤ a
h0 : 0 ≤ b
⊢ (min (a * b) (min (max a (a * b ^ 2)) (max b (a ^ 2 * b))))⁺ = a * b
TACTIC:
|
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