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/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Dialects/LinalgSemantics.lean | mapMCommute.mapM_commute | [282, 1] | [312, 2] | congr | case e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ (do
let r2 ← g x'
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (r2 :: __do_lift)) =
do
let b ← g x'
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (b :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | case e_a.h.e_a
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ (fun r2 => do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (r2 :: __do_lift)) =
fun b => do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (b :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ (do
let r2 ← g x'
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (r2 :: __do_lift)) =
do
let b ← g x'
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (b :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
TACTIC:
|
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Dialects/LinalgSemantics.lean | mapMCommute.mapM_commute | [282, 1] | [312, 2] | funext x' | case e_a.h.e_a
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ (fun r2 => do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (r2 :: __do_lift)) =
fun b => do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (b :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h.e_a
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ (fun r2 => do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (r2 :: __do_lift)) =
fun b => do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (b :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
TACTIC:
|
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Dialects/LinalgSemantics.lean | mapMCommute.mapM_commute | [282, 1] | [312, 2] | simp [fish] at IH | case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
TACTIC:
|
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Dialects/LinalgSemantics.lean | mapMCommute.mapM_commute | [282, 1] | [312, 2] | rewrite [<- IH] | case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM f tail >>= mapM g
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM (fun a_1 => f a_1 >>= g) tail
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
TACTIC:
|
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Dialects/LinalgSemantics.lean | mapMCommute.mapM_commute | [282, 1] | [312, 2] | simp[bind_assoc] | case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM f tail >>= mapM g
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h.e_a.h
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : mapM f tail >>= mapM g = mapM (fun a_1 => f a_1 >>= g) tail
x'✝ x' : a
⊢ (do
let r1 ← mapM f tail
let __do_lift ← mapM g r1
pure (x' :: __do_lift)) =
do
let bs ← mapM f tail >>= mapM g
pure (x' :: bs)
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
TACTIC:
|
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Dialects/LinalgSemantics.lean | mapMCommute.mapM_commute | [282, 1] | [312, 2] | apply COMMUTE | case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case e_a.h.COMMUTE
m : Type → Type u_1
a : Type
M : Monad m
LM : LawfulMonad m
f g : a → m a
COMMUTE :
∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
head : a
tail : List a
IH : fish (mapM f) (mapM g) tail = mapM (fish f g) tail
x' : a
⊢ ∀ {b : Type} (x y : a) (k : a → a → m b),
(do
let r1 ← f x
let r2 ← g y
k r1 r2) =
do
let r2 ← g y
let r1 ← f x
k r1 r2
TACTIC:
|
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Examples/IfStatementOptimization.lean | SCF_SELECT.equivalent | [38, 1] | [63, 55] | simp [LHS, RHS, INPUT] | b : FinInt 1
n m : FinInt 32
⊢ run (denoteOp Δ LHS) (INPUT b n m) = run (denoteOp Δ RHS) (INPUT b n m) | b : FinInt 1
n m : FinInt 32
⊢ run
(denoteOp Δ
(Op.mk "scf.if" [(SSAVal.SSAVal "x", MLIRType.int Signedness.Signless 32)]
[(SSAVal.SSAVal "b", MLIRType.int Signedness.Signless 1)]
[Region.mk "entry" []
[Op.mk "scf.yield" [] [(SSAVal.SSAVal "n", MLIRType.int Signedness.Signless 32)] [] (AttrDict.mk [])],
Region.mk "entry" []
[Op.mk "scf.yield" [] [(SSAVal.SSAVal "m", MLIRType.int Signedness.Signless 32)] [] (AttrDict.mk [])]]
(AttrDict.mk [])))
(SSAEnv.One
[(SSAVal.SSAVal "b", { fst := MLIRType.i1, snd := b }), (SSAVal.SSAVal "n", { fst := MLIRType.i32, snd := n }),
(SSAVal.SSAVal "m", { fst := MLIRType.i32, snd := m })]) =
run
(denoteOp Δ
(Op.mk "arith.select" [(SSAVal.SSAVal "x", MLIRType.int Signedness.Signless 32)]
[(SSAVal.SSAVal "b", MLIRType.int Signedness.Signless 1),
(SSAVal.SSAVal "n", MLIRType.int Signedness.Signless 32),
(SSAVal.SSAVal "m", MLIRType.int Signedness.Signless 32)]
[] (AttrDict.mk [])))
(SSAEnv.One
[(SSAVal.SSAVal "b", { fst := MLIRType.i1, snd := b }), (SSAVal.SSAVal "n", { fst := MLIRType.i32, snd := n }),
(SSAVal.SSAVal "m", { fst := MLIRType.i32, snd := m })]) | Please generate a tactic in lean4 to solve the state.
STATE:
b : FinInt 1
n m : FinInt 32
⊢ run (denoteOp Δ LHS) (INPUT b n m) = run (denoteOp Δ RHS) (INPUT b n m)
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/02Equality.lean | reverse_mkSymm | [81, 1] | [83, 16] | simp [mkSymm] | α : Type u_1
xs : List α
⊢ List.reverse (mkSymm xs) = mkSymm xs | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
⊢ List.reverse (mkSymm xs) = mkSymm xs
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.one_add_add | [35, 1] | [40, 12] | induction n with
| zero => rfl
| succ n ih =>
dsimp
congr 1 | m n : MyNat
⊢ m + 1 + n = m + n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ m + 1 + n = m + n + 1
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.one_add_add | [35, 1] | [40, 12] | rfl | case zero
m : MyNat
⊢ m + 1 + zero = m + zero + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ m + 1 + zero = m + zero + 1
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.one_add_add | [35, 1] | [40, 12] | dsimp | case succ
m n : MyNat
ih : m + 1 + n = m + n + 1
⊢ m + 1 + succ n = m + succ n + 1 | case succ
m n : MyNat
ih : m + 1 + n = m + n + 1
⊢ m + 1 + n + 1 = m + n + 1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : m + 1 + n = m + n + 1
⊢ m + 1 + succ n = m + succ n + 1
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.one_add_add | [35, 1] | [40, 12] | congr 1 | case succ
m n : MyNat
ih : m + 1 + n = m + n + 1
⊢ m + 1 + n + 1 = m + n + 1 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : m + 1 + n = m + n + 1
⊢ m + 1 + n + 1 = m + n + 1 + 1
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.zero_add | [43, 1] | [48, 12] | induction m with
| zero => dsimp
| succ m ih =>
dsimp
rw [ih] | m : MyNat
⊢ 0 + m = m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : MyNat
⊢ 0 + m = m
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.zero_add | [43, 1] | [48, 12] | dsimp | case zero
⊢ 0 + zero = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ 0 + zero = zero
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.zero_add | [43, 1] | [48, 12] | dsimp | case succ
m : MyNat
ih : 0 + m = m
⊢ 0 + succ m = succ m | case succ
m : MyNat
ih : 0 + m = m
⊢ 0 + m + 1 = m + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m : MyNat
ih : 0 + m = m
⊢ 0 + succ m = succ m
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.zero_add | [43, 1] | [48, 12] | rw [ih] | case succ
m : MyNat
ih : 0 + m = m
⊢ 0 + m + 1 = m + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m : MyNat
ih : 0 + m = m
⊢ 0 + m + 1 = m + 1
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_assoc | [50, 1] | [53, 27] | induction o with
| zero => simp
| succ o ih => simp [ih] | m n o : MyNat
⊢ m + (n + o) = m + n + o | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n o : MyNat
⊢ m + (n + o) = m + n + o
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_assoc | [50, 1] | [53, 27] | simp | case zero
m n : MyNat
⊢ m + (n + zero) = m + n + zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m n : MyNat
⊢ m + (n + zero) = m + n + zero
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_assoc | [50, 1] | [53, 27] | simp [ih] | case succ
m n o : MyNat
ih : m + (n + o) = m + n + o
⊢ m + (n + succ o) = m + n + succ o | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n o : MyNat
ih : m + (n + o) = m + n + o
⊢ m + (n + succ o) = m + n + succ o
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_comm | [55, 1] | [63, 12] | induction n with
| zero =>
dsimp
rw [zero_add]
| succ n ih =>
dsimp
rw [one_add_add]
rw [ih] | m n : MyNat
⊢ m + n = n + m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ m + n = n + m
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_comm | [55, 1] | [63, 12] | dsimp | case zero
m : MyNat
⊢ m + zero = zero + m | case zero
m : MyNat
⊢ m = 0 + m | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ m + zero = zero + m
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_comm | [55, 1] | [63, 12] | rw [zero_add] | case zero
m : MyNat
⊢ m = 0 + m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ m = 0 + m
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_comm | [55, 1] | [63, 12] | dsimp | case succ
m n : MyNat
ih : m + n = n + m
⊢ m + succ n = succ n + m | case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + 1 + m | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : m + n = n + m
⊢ m + succ n = succ n + m
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_comm | [55, 1] | [63, 12] | rw [one_add_add] | case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + 1 + m | case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + m + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + 1 + m
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/06Induction.lean | MyNat.add_comm | [55, 1] | [63, 12] | rw [ih] | case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + m + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : m + n = n + m
⊢ m + n + 1 = n + m + 1
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/10Aesop.lean | ne_nil_of_mem | [8, 1] | [9, 8] | aesop | α : Type u_1
a : α
l : List α
h : a ∈ l
⊢ l ≠ [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a : α
l : List α
h : a ∈ l
⊢ l ≠ []
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/10Aesop.lean | not_mem_cons_of_ne_of_not_mem | [11, 1] | [13, 8] | aesop | α : Type u_1
a y : α
l : List α
⊢ a ≠ y → ¬a ∈ l → ¬a ∈ y :: l | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
a y : α
l : List α
⊢ a ≠ y → ¬a ∈ l → ¬a ∈ y :: l
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/10Aesop.lean | mem_map | [15, 9] | [17, 24] | induction l <;> aesop | α : Type u_1
β : Type u_2
f : α → β
b : β
l : List α
⊢ b ∈ List.map f l ↔ ∃ a, a ∈ l ∧ f a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
b : β
l : List α
⊢ b ∈ List.map f l ↔ ∃ a, a ∈ l ∧ f a = b
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/10Aesop.lean | mem_map_of_mem | [20, 1] | [22, 8] | aesop | α : Type u_1
β : Type u_2
f : α → β
a : α
l : List α
h : a ∈ l
⊢ f a ∈ List.map f l | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
a : α
l : List α
h : a ∈ l
⊢ f a ∈ List.map f l
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/10Aesop.lean | eq_nil_of_subset_nil | [25, 1] | [26, 28] | aesop (add 1% cases List) | α : Type u_1
l : List α
⊢ l ⊆ [] → l = [] | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
l : List α
⊢ l ⊆ [] → l = []
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.foo | [57, 1] | [62, 26] | intro legal | p : Piece
n✝ : ℕ
start stop : Pos n✝
⊢ Piece.LegalMove start stop p → start ≠ stop | p : Piece
n✝ : ℕ
start stop : Pos n✝
legal : Piece.LegalMove start stop p
⊢ start ≠ stop | Please generate a tactic in lean4 to solve the state.
STATE:
p : Piece
n✝ : ℕ
start stop : Pos n✝
⊢ Piece.LegalMove start stop p → start ≠ stop
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.foo | [57, 1] | [62, 26] | cases legal with
| rookHorizontal h _ => exact h
| rookVertical h _ => exact h
| king h _ _ => exact h | p : Piece
n✝ : ℕ
start stop : Pos n✝
legal : Piece.LegalMove start stop p
⊢ start ≠ stop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : Piece
n✝ : ℕ
start stop : Pos n✝
legal : Piece.LegalMove start stop p
⊢ start ≠ stop
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.foo | [57, 1] | [62, 26] | exact h | case rookHorizontal
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝ : start.y = stop.y
⊢ start ≠ stop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case rookHorizontal
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝ : start.y = stop.y
⊢ start ≠ stop
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.foo | [57, 1] | [62, 26] | exact h | case rookVertical
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝ : start.x = stop.x
⊢ start ≠ stop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case rookVertical
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝ : start.x = stop.x
⊢ start ≠ stop
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.foo | [57, 1] | [62, 26] | exact h | case king
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝¹ : Nat.dist ↑start.x ↑stop.x ≤ 1
a✝ : Nat.dist ↑start.y ↑stop.y ≤ 1
⊢ start ≠ stop | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case king
n✝ : ℕ
start stop : Pos n✝
h : start ≠ stop
a✝¹ : Nat.dist ↑start.x ↑stop.x ≤ 1
a✝ : Nat.dist ↑start.y ↑stop.y ≤ 1
⊢ start ≠ stop
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.Board.set_same | [105, 1] | [106, 13] | simp [set] | n : ℕ
pos : Pos n
p? : Option PPiece
b : Board n
⊢ set b pos p? pos = p? | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
pos : Pos n
p? : Option PPiece
b : Board n
⊢ set b pos p? pos = p?
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.Board.set_different | [108, 1] | [112, 11] | intro h | n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
⊢ pos' ≠ pos → set b pos p? pos' = b pos' | n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ set b pos p? pos' = b pos' | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
⊢ pos' ≠ pos → set b pos p? pos' = b pos'
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.Board.set_different | [108, 1] | [112, 11] | simp [set] | n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ set b pos p? pos' = b pos' | n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ pos = pos' → p? = b pos' | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ set b pos p? pos' = b pos'
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.Board.set_different | [108, 1] | [112, 11] | intros | n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ pos = pos' → p? = b pos' | n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
a✝ : pos = pos'
⊢ p? = b pos' | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
⊢ pos = pos' → p? = b pos'
TACTIC:
|
https://github.com/JLimperg/regensburg-itp-school-2023.git | 7307618fa6d85c090896d29021541cd67920d750 | Talk/11MiniChess.lean | MiniChess.Board.set_different | [108, 1] | [112, 11] | simp_all | n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
a✝ : pos = pos'
⊢ p? = b pos' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
pos' pos : Pos n
p? : Option PPiece
b : Board n
h : pos' ≠ pos
a✝ : pos = pos'
⊢ p? = b pos'
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.no_nbhrs | [50, 1] | [56, 40] | contrapose! hA | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
hA : ¬v ∈ M.A
⊢ ¬Adj (mp M) v w | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
hA :
∃ i,
i ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
i ≠ j ∧ (v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i)
⊢ v ∈ M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
hA : ¬v ∈ M.A
⊢ ¬Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.no_nbhrs | [50, 1] | [56, 40] | obtain ⟨i, hi, j, hj, _, hv⟩ := hA | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
hA :
∃ i,
i ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
i ≠ j ∧ (v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i)
⊢ v ∈ M.A | case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
⊢ v ∈ M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
hA :
∃ i,
i ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
i ≠ j ∧ (v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i)
⊢ v ∈ M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.no_nbhrs | [50, 1] | [56, 40] | cases hv with
| inl hv => exact (sub_part hi) hv.1
| inr hv => exact (sub_part hj) hv.1 | case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
⊢ v ∈ M.A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
⊢ v ∈ M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.no_nbhrs | [50, 1] | [56, 40] | exact (sub_part hi) hv.1 | case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j
⊢ v ∈ M.A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j
⊢ v ∈ M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.no_nbhrs | [50, 1] | [56, 40] | exact (sub_part hj) hv.1 | case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
⊢ v ∈ M.A | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
i : ℕ
hi : i ∈ range (M.t + 1)
j : ℕ
hj : j ∈ range (M.t + 1)
left✝ : i ≠ j
hv : v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
⊢ v ∈ M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhrs_imp | [59, 1] | [62, 22] | intro h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
⊢ Adj (mp M) v w → v ∈ M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
⊢ v ∈ M.A | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
⊢ Adj (mp M) v w → v ∈ M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhrs_imp | [59, 1] | [62, 22] | by_contra h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
⊢ v ∈ M.A | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
h : ¬v ∈ M.A
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
⊢ v ∈ M.A
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhrs_imp | [59, 1] | [62, 22] | exact no_nbhrs h h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
h : ¬v ∈ M.A
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
v w : α
M : MultiPart α
h1 : Adj (mp M) v w
h : ¬v ∈ M.A
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | intro h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
⊢ Adj (mp M) v w → i ≠ j | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : Adj (mp M) v w
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
⊢ Adj (mp M) v w → i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | cases' h with a ha | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : Adj (mp M) v w
⊢ i ≠ j | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
ha :
a ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
a ≠ j ∧ (v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M a)
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : Adj (mp M) v w
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | obtain ⟨har, b, hbr, abne, ab⟩ := ha | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
ha :
a ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
a ≠ j ∧ (v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M a)
⊢ i ≠ j | case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b ∨ v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
ha :
a ∈ range (M.t + 1) ∧
∃ j,
j ∈ range (M.t + 1) ∧
a ≠ j ∧ (v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M a)
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | cases ab with
| inl ab =>
have ai := uniq_part hi har hvi ab.1; have bj := uniq_part hj hbr hwj ab.2
rwa [← ai, ← bj] at abne
| inr ab =>
have aj := uniq_part hj har hwj ab.2; have bi := uniq_part hi hbr hvi ab.1
rw [← aj, ← bi] at abne
exact abne.symm | case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b ∨ v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
⊢ i ≠ j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b ∨ v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | have ai := uniq_part hi har hvi ab.1 | case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
⊢ i ≠ j | case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | have bj := uniq_part hj hbr hwj ab.2 | case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
⊢ i ≠ j | case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
bj : j = b
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | rwa [← ai, ← bj] at abne | case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
bj : j = b
⊢ i ≠ j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inl
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M a ∧ w ∈ MultiPart.P M b
ai : i = a
bj : j = b
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | have aj := uniq_part hj har hwj ab.2 | case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
⊢ i ≠ j | case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | have bi := uniq_part hi hbr hvi ab.1 | case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
⊢ i ≠ j | case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | rw [← aj, ← bi] at abne | case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j | case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : j ≠ i
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : a ≠ b
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_adj_imp | [65, 1] | [77, 20] | exact abne.symm | case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : j ≠ i
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.intro.intro.inr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
a : ℕ
har : a ∈ range (M.t + 1)
b : ℕ
hbr : b ∈ range (M.t + 1)
abne : j ≠ i
ab : v ∈ MultiPart.P M b ∧ w ∈ MultiPart.P M a
aj : j = a
bi : i = b
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_imp_adj | [80, 1] | [82, 64] | intro h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
⊢ i ≠ j → Adj (mp M) v w | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ Adj (mp M) v w | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
⊢ i ≠ j → Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_imp_adj | [80, 1] | [82, 64] | refine' ⟨i, hi, j, hj, h, _⟩ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ Adj (mp M) v w | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_imp_adj | [80, 1] | [82, 64] | left | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j ∨ v ∈ MultiPart.P M j ∧ w ∈ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_imp_adj | [80, 1] | [82, 64] | exact ⟨hvi, hwj⟩ | case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i j : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hj : j ∈ range (M.t + 1)
hvi : v ∈ MultiPart.P M i
hwj : w ∈ MultiPart.P M j
h : i ≠ j
⊢ v ∈ MultiPart.P M i ∧ w ∈ MultiPart.P M j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.not_nbhr_same_part | [85, 1] | [89, 38] | intro h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
⊢ Adj (mp M) v w → ¬w ∈ MultiPart.P M i | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ ¬w ∈ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
⊢ Adj (mp M) v w → ¬w ∈ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.not_nbhr_same_part | [85, 1] | [89, 38] | by_contra h | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ ¬w ∈ MultiPart.P M i | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ ¬w ∈ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.not_nbhr_same_part | [85, 1] | [89, 38] | apply mp_adj_imp hi hi hv h h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ False | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ i = i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.not_nbhr_same_part | [85, 1] | [89, 38] | rfl | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ i = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
h : w ∈ MultiPart.P M i
⊢ i = i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.not_nbhr_same_part' | [92, 1] | [96, 36] | intro h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ MultiPart.P M i
⊢ ¬Adj (mp M) v w | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ MultiPart.P M i
⊢ ¬Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.not_nbhr_same_part' | [92, 1] | [96, 36] | contrapose hw | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ False | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
hw : ¬False
⊢ ¬w ∈ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ MultiPart.P M i
h1 : Adj (mp M) v w
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.not_nbhr_same_part' | [92, 1] | [96, 36] | exact not_nbhr_same_part hi hv h1 | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
hw : ¬False
⊢ ¬w ∈ MultiPart.P M i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
h1 : Adj (mp M) v w
hw : ¬False
⊢ ¬w ∈ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | rw [mem_sdiff] at hw | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ M.A \ MultiPart.P M i
⊢ Adj (mp M) v w | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ M.A ∧ ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ M.A \ MultiPart.P M i
⊢ Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | cases' hw with hA hni | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ M.A ∧ ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ M.A
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hw : w ∈ M.A ∧ ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | rw [M.uni] at hA | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ M.A
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ M.A
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | rw [mem_biUnion] at hA | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : ∃ a, a ∈ range (M.t + 1) ∧ w ∈ MultiPart.P M a
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : w ∈ Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | obtain ⟨j, hj1, hj2⟩ := hA | case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : ∃ a, a ∈ range (M.t + 1) ∧ w ∈ MultiPart.P M a
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ Adj (mp M) v w | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hA : ∃ a, a ∈ range (M.t + 1) ∧ w ∈ MultiPart.P M a
hni : ¬w ∈ MultiPart.P M i
⊢ Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | refine' mp_imp_adj hi hj1 hv hj2 _ | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ Adj (mp M) v w | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ i ≠ j | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ Adj (mp M) v w
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | intro h | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ i ≠ j | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
⊢ i ≠ j
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | rw [h] at hni | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
j : ℕ
hni : ¬w ∈ MultiPart.P M j
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
hni : ¬w ∈ MultiPart.P M i
j : ℕ
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.nbhr_diff_parts | [99, 1] | [106, 77] | exact hni hj2 | case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
j : ℕ
hni : ¬w ∈ MultiPart.P M j
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v w : α
M : MultiPart α
hi : i ∈ range (M.t + 1)
hv : v ∈ MultiPart.P M i
j : ℕ
hni : ¬w ∈ MultiPart.P M j
hj1 : j ∈ range (M.t + 1)
hj2 : w ∈ MultiPart.P M j
h : i = j
⊢ False
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | ext | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ neighborFinset (mp M) v = M.A \ MultiPart.P M i | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v ↔ a✝ ∈ M.A \ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ neighborFinset (mp M) v = M.A \ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | constructor | case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v ↔ a✝ ∈ M.A \ MultiPart.P M i | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v → a✝ ∈ M.A \ MultiPart.P M i
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → a✝ ∈ neighborFinset (mp M) v | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v ↔ a✝ ∈ M.A \ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | rw [mem_neighborFinset] | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v → a✝ ∈ M.A \ MultiPart.P M i | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ Adj (mp M) v a✝ → a✝ ∈ M.A \ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ neighborFinset (mp M) v → a✝ ∈ M.A \ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | intro h | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ Adj (mp M) v a✝ → a✝ ∈ M.A \ MultiPart.P M i | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) v a✝
⊢ a✝ ∈ M.A \ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ Adj (mp M) v a✝ → a✝ ∈ M.A \ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | rw [adj_comm] at h | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) v a✝
⊢ a✝ ∈ M.A \ MultiPart.P M i | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A \ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) v a✝
⊢ a✝ ∈ M.A \ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | rw [mem_sdiff] | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A \ MultiPart.P M i | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A ∧ ¬a✝ ∈ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A \ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | refine' ⟨nbhrs_imp h, _⟩ | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A ∧ ¬a✝ ∈ MultiPart.P M i | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ ¬a✝ ∈ MultiPart.P M i | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ a✝ ∈ M.A ∧ ¬a✝ ∈ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | exact not_nbhr_same_part hv.1 hv.2 h.symm | case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ ¬a✝ ∈ MultiPart.P M i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mp
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
h : Adj (mp M) a✝ v
⊢ ¬a✝ ∈ MultiPart.P M i
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | rw [mem_neighborFinset] | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → a✝ ∈ neighborFinset (mp M) v | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → Adj (mp M) v a✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → a✝ ∈ neighborFinset (mp M) v
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_nbhd | [109, 1] | [114, 61] | exact nbhr_diff_parts hv.1 hv.2 | case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → Adj (mp M) v a✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.mpr
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
a✝ : α
⊢ a✝ ∈ M.A \ MultiPart.P M i → Adj (mp M) v a✝
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg | [121, 1] | [122, 76] | rw [degree] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ degree (mp M) v = card (M.A \ MultiPart.P M i) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (neighborFinset (mp M) v) = card (M.A \ MultiPart.P M i) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ degree (mp M) v = card (M.A \ MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg | [121, 1] | [122, 76] | rw [mp_nbhd hv] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (neighborFinset (mp M) v) = card (M.A \ MultiPart.P M i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (neighborFinset (mp M) v) = card (M.A \ MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_diff | [125, 1] | [126, 101] | rw [mp_deg hv] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ degree (mp M) v = card M.A - card (MultiPart.P M i) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (M.A \ MultiPart.P M i) = card M.A - card (MultiPart.P M i) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ degree (mp M) v = card M.A - card (MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_diff | [125, 1] | [126, 101] | exact card_sdiff (sub_part hv.1) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (M.A \ MultiPart.P M i) = card M.A - card (MultiPart.P M i) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
i : ℕ
v : α
M : MultiPart α
hv : i ∈ range (M.t + 1) ∧ v ∈ MultiPart.P M i
⊢ card (M.A \ MultiPart.P M i) = card M.A - card (MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum | [129, 1] | [135, 36] | nth_rw 1 [M.uni] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in M.A, degree (mp M) v = ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i, degree (mp M) v =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in M.A, degree (mp M) v = ∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum | [129, 1] | [135, 36] | rw [sum_biUnion (pair_disjoint M)] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i, degree (mp M) v =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), ∑ i in MultiPart.P M x, degree (mp M) i =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ v in Finset.biUnion (range (M.t + 1)) fun i => MultiPart.P M i, degree (mp M) v =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum | [129, 1] | [135, 36] | apply Finset.sum_congr rfl _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), ∑ i in MultiPart.P M x, degree (mp M) i =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) →
∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∑ x in range (M.t + 1), ∑ i in MultiPart.P M x, degree (mp M) i =
∑ i in range (M.t + 1), card (MultiPart.P M i) * card (M.A \ MultiPart.P M i)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum | [129, 1] | [135, 36] | intro x hx | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) →
∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
⊢ ∀ (x : ℕ),
x ∈ range (M.t + 1) →
∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum | [129, 1] | [135, 36] | rw [Finset.card_eq_sum_ones, sum_mul, one_mul] | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = ∑ x_1 in MultiPart.P M x, card (M.A \ MultiPart.P M x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = card (MultiPart.P M x) * card (M.A \ MultiPart.P M x)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum | [129, 1] | [135, 36] | apply Finset.sum_congr rfl _ | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = ∑ x_1 in MultiPart.P M x, card (M.A \ MultiPart.P M x) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∀ (x_1 : α), x_1 ∈ MultiPart.P M x → degree (mp M) x_1 = card (M.A \ MultiPart.P M x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∑ i in MultiPart.P M x, degree (mp M) i = ∑ x_1 in MultiPart.P M x, card (M.A \ MultiPart.P M x)
TACTIC:
|
https://github.com/jt496/Turan_4.git | 329b6acff8f9b8f41609e3e5758ed80c61047eb5 | Turan4/Multipartite.lean | SimpleGraph.mp_deg_sum | [129, 1] | [135, 36] | intro v hv | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∀ (x_1 : α), x_1 ∈ MultiPart.P M x → degree (mp M) x_1 = card (M.A \ MultiPart.P M x) | α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
v : α
hv : v ∈ MultiPart.P M x
⊢ degree (mp M) v = card (M.A \ MultiPart.P M x) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
M : MultiPart α
x : ℕ
hx : x ∈ range (M.t + 1)
⊢ ∀ (x_1 : α), x_1 ∈ MultiPart.P M x → degree (mp M) x_1 = card (M.A \ MultiPart.P M x)
TACTIC:
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.