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stringclasses
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stringlengths
6
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stringlengths
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2.09M
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
linarith
case h α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l : List α h✝ : Sorted r l i✝ j✝ : ℕ a✝ : α h : i✝ < j✝ hi : i✝ < 1 hj : j✝ < 1 ⊢ False
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
intro h' hi hj
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l : List α h : Sorted r l i✝ j✝ : ℕ a₁✝ a₂✝ : α as✝ : List α h₁ : r a₁✝ a₂✝ h₂ : Sorted r (a₂✝ :: as✝) ⊢ i✝ < j✝ → ∀ (x : i✝ < List.length (a₁✝ :: a₂✝ :: as✝)) (x_1 : j✝ < List.length (a₁✝ :: a₂✝ :: as✝)), r (a₁✝ :: a₂✝ :: as✝)[i✝] (a₁✝ :: a₂✝ :: as✝)[j✝]
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l : List α h : Sorted r l i✝ j✝ : ℕ a₁✝ a₂✝ : α as✝ : List α h₁ : r a₁✝ a₂✝ h₂ : Sorted r (a₂✝ :: as✝) h' : i✝ < j✝ hi : i✝ < List.length (a₁✝ :: a₂✝ :: as✝) hj : j✝ < List.length (a₁✝ :: a₂✝ :: as✝) ⊢ r (a₁✝ :: a₂✝ :: as✝)[i✝] (a₁✝ :: a₂✝ :: as✝)[j✝]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
rename_i i j a₁ a₂ as
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l : List α h : Sorted r l i✝ j✝ : ℕ a₁✝ a₂✝ : α as✝ : List α h₁ : r a₁✝ a₂✝ h₂ : Sorted r (a₂✝ :: as✝) h' : i✝ < j✝ hi : i✝ < List.length (a₁✝ :: a₂✝ :: as✝) hj : j✝ < List.length (a₁✝ :: a₂✝ :: as✝) ⊢ r (a₁✝ :: a₂✝ :: as✝)[i✝] (a₁✝ :: a₂✝ :: as✝)[j✝]
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l : List α h : Sorted r l i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) h' : i < j hi : i < List.length (a₁ :: a₂ :: as) hj : j < List.length (a₁ :: a₂ :: as) ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[j]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
have ⟨l,hl⟩ := Nat.succ_of_lt h'
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l : List α h : Sorted r l i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) h' : i < j hi : i < List.length (a₁ :: a₂ :: as) hj : j < List.length (a₁ :: a₂ :: as) ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[j]
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) h' : i < j hi : i < List.length (a₁ :: a₂ :: as) hj : j < List.length (a₁ :: a₂ :: as) l : ℕ hl : l + 1 = j ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[j]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
rw [←hl] at hj h'
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) h' : i < j hi : i < List.length (a₁ :: a₂ :: as) hj : j < List.length (a₁ :: a₂ :: as) l : ℕ hl : l + 1 = j ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[j]
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[j]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
simp only [←hl]
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[j]
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
have : l < (a₂ :: as).length := Nat.pred_lt_pred (by simp) hj
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
have hl' : (a₁::a₂::as)[l+1] = (a₂::as)[l] := rfl
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
by_cases i = 0
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
case pos α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : i = 0 ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1] case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
simp
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j ⊢ l + 1 ≠ 0
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
simp [h,hl'] at *
case pos α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : i = 0 ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
case pos α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h : True ⊢ r a₁ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) })
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
by_cases l = 0
case pos α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h : True ⊢ r a₁ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) })
case pos α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : l = 0 ⊢ r a₁ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) }) case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r a₁ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) })
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
simp [h,h₁]
case pos α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : l = 0 ⊢ r a₁ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) })
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
apply trans'
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r a₁ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) })
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r a₁ ?neg.a₂✝ case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r ?neg.a₂✝ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) }) case neg.a₂ α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ α
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
. exact h₁
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r a₁ ?neg.a₂✝ case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r ?neg.a₂✝ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) }) case neg.a₂ α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ α
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r a₂ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) })
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
exact h₁
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r a₁ ?neg.a₂✝
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
change r (a₂::as)[0] (a₂::as)[l]
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r a₂ (List.get (a₂ :: as) { val := l, isLt := (_ : l < List.length (a₂ :: as)) })
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r (a₂ :: as)[0] (a₂ :: as)[l]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
apply totally_ordered_of_sorted h₂
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ r (a₂ :: as)[0] (a₂ :: as)[l]
case neg.a.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ 0 < l
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
apply Nat.one_le_iff_ne_zero.mpr h
case neg.a.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝¹ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) h' hl' h✝ : True h : ¬l = 0 ⊢ 0 < l
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
have ⟨k,hk⟩ := Nat.exists_eq_succ_of_ne_zero h
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ hk : i = Nat.succ k ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
rw [hk] at hi h'
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ h' : i < l + 1 hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ hk : i = Nat.succ k ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
simp only [hk]
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k ⊢ r (a₁ :: a₂ :: as)[i] (a₁ :: a₂ :: as)[l + 1]
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k ⊢ r (a₁ :: a₂ :: as)[Nat.succ k] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
have : k < (a₂::as).length := Nat.pred_lt_pred (by simp) hi
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k ⊢ r (a₁ :: a₂ :: as)[Nat.succ k] (a₁ :: a₂ :: as)[l + 1]
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) ⊢ r (a₁ :: a₂ :: as)[Nat.succ k] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
have hk' : (a₁::a₂::as)[k+1] = (a₂::as)[k] := rfl
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) ⊢ r (a₁ :: a₂ :: as)[Nat.succ k] (a₁ :: a₂ :: as)[l + 1]
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) hk' : (a₁ :: a₂ :: as)[k + 1] = (a₂ :: as)[k] ⊢ r (a₁ :: a₂ :: as)[Nat.succ k] (a₁ :: a₂ :: as)[l + 1]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
rw [hk',hl']
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) hk' : (a₁ :: a₂ :: as)[k + 1] = (a₂ :: as)[k] ⊢ r (a₁ :: a₂ :: as)[Nat.succ k] (a₁ :: a₂ :: as)[l + 1]
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) hk' : (a₁ :: a₂ :: as)[k + 1] = (a₂ :: as)[k] ⊢ r (a₂ :: as)[k] (a₂ :: as)[l]
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
apply totally_ordered_of_sorted h₂
case neg α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) hk' : (a₁ :: a₂ :: as)[k + 1] = (a₂ :: as)[k] ⊢ r (a₂ :: as)[k] (a₂ :: as)[l]
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) hk' : (a₁ :: a₂ :: as)[k + 1] = (a₂ :: as)[k] ⊢ k < l
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
apply Nat.pred_lt_pred (by simp) h'
case neg.a α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) hk' : (a₁ :: a₂ :: as)[k + 1] = (a₂ :: as)[k] ⊢ k < l
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
simp
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k ⊢ Nat.succ k ≠ 0
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred2.lean
Notes.totally_ordered_of_sorted
[180, 1]
[216, 42]
simp
α : Type r : α → α → Prop inst✝² : DecidableRel r inst✝¹ : Trans r r r inst✝ : Transitive r l✝ : List α h✝ : Sorted r l✝ i j : ℕ a₁ a₂ : α as : List α h₁ : r a₁ a₂ h₂ : Sorted r (a₂ :: as) hi✝ : i < List.length (a₁ :: a₂ :: as) hj✝ : j < List.length (a₁ :: a₂ :: as) l : ℕ hj : l + 1 < List.length (a₁ :: a₂ :: as) hl : l + 1 = j this✝ : l < List.length (a₂ :: as) hl' : (a₁ :: a₂ :: as)[l + 1] = (a₂ :: as)[l] h : ¬i = 0 k : ℕ h' : Nat.succ k < l + 1 hi : Nat.succ k < List.length (a₁ :: a₂ :: as) hk : i = Nat.succ k this : k < List.length (a₂ :: as) hk' : (a₁ :: a₂ :: as)[k + 1] = (a₂ :: as)[k] ⊢ Nat.succ k ≠ 0
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
leq_succ_of_leq
[96, 1]
[106, 45]
match n with | 0 => have meq0 : m = 0 := Nat.eq_zero_of_le_zero h rw [meq0] simp | n'+1 => simp [plus] calc m ≤ n'+1 := by exact h _ ≤ (n'+1)+1 := by exact Nat.le_succ _
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n : ℕ h : m ≤ n ⊢ m ≤ plus n 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
leq_succ_of_leq
[96, 1]
[106, 45]
have meq0 : m = 0 := Nat.eq_zero_of_le_zero h
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n : ℕ h : m ≤ 0 ⊢ m ≤ plus 0 1
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n : ℕ h : m ≤ 0 meq0 : m = 0 ⊢ m ≤ plus 0 1
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
leq_succ_of_leq
[96, 1]
[106, 45]
rw [meq0]
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n : ℕ h : m ≤ 0 meq0 : m = 0 ⊢ m ≤ plus 0 1
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n : ℕ h : m ≤ 0 meq0 : m = 0 ⊢ 0 ≤ plus 0 1
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
leq_succ_of_leq
[96, 1]
[106, 45]
simp
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n : ℕ h : m ≤ 0 meq0 : m = 0 ⊢ 0 ≤ plus 0 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
leq_succ_of_leq
[96, 1]
[106, 45]
simp [plus]
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n n' : ℕ h : m ≤ n' + 1 ⊢ m ≤ plus (n' + 1) 1
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n n' : ℕ h : m ≤ n' + 1 ⊢ m ≤ Nat.succ (n' + 1)
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
leq_succ_of_leq
[96, 1]
[106, 45]
calc m ≤ n'+1 := by exact h _ ≤ (n'+1)+1 := by exact Nat.le_succ _
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n n' : ℕ h : m ≤ n' + 1 ⊢ m ≤ Nat.succ (n' + 1)
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
leq_succ_of_leq
[96, 1]
[106, 45]
exact h
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n n' : ℕ h : m ≤ n' + 1 ⊢ m ≤ n' + 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
leq_succ_of_leq
[96, 1]
[106, 45]
exact Nat.le_succ _
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n n' : ℕ h : m ≤ n' + 1 ⊢ n' + 1 ≤ n' + 1 + 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
succ_leq_succ_of_leq
[108, 1]
[114, 39]
match a with | 0 => simp [plus]; assumption | b+1 => simp [plus] apply Nat.succ_le_succ exact succ_leq_succ_of_leq m n b h
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n ⊢ plus m a ≤ plus n a
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
succ_leq_succ_of_leq
[108, 1]
[114, 39]
simp [plus]
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n ⊢ plus m 0 ≤ plus n 0
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n ⊢ m ≤ n
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
succ_leq_succ_of_leq
[108, 1]
[114, 39]
assumption
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n ⊢ m ≤ n
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
succ_leq_succ_of_leq
[108, 1]
[114, 39]
simp [plus]
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n b : ℕ ⊢ plus m (b + 1) ≤ plus n (b + 1)
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n b : ℕ ⊢ Nat.succ (plus m b) ≤ Nat.succ (plus n b)
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
succ_leq_succ_of_leq
[108, 1]
[114, 39]
apply Nat.succ_le_succ
α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n b : ℕ ⊢ Nat.succ (plus m b) ≤ Nat.succ (plus n b)
case a α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n b : ℕ ⊢ plus m b ≤ plus n b
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/MoreTacs.lean
succ_leq_succ_of_leq
[108, 1]
[114, 39]
exact succ_leq_succ_of_leq m n b h
case a α β γ δ : Type f : α → β g : β → γ h✝ : γ → δ m n a : ℕ h : m ≤ n b : ℕ ⊢ plus m b ≤ plus n b
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/Theorems.lean
exp_eq_exp'
[98, 1]
[104, 25]
match e with | 0 => simp [exp,exp'] | n+1 => simp only [exp,exp'] simp only [Nat.add] rw [exp_eq_exp' b n]
b e : ℕ ⊢ exp b e = exp' b e
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/Theorems.lean
exp_eq_exp'
[98, 1]
[104, 25]
simp [exp,exp']
b e : ℕ ⊢ exp b 0 = exp' b 0
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/Theorems.lean
exp_eq_exp'
[98, 1]
[104, 25]
simp only [exp,exp']
b e n : ℕ ⊢ exp b (n + 1) = exp' b (n + 1)
b e n : ℕ ⊢ exp b (Nat.add n 0) * b = exp' b (Nat.add n 0) * b
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/Theorems.lean
exp_eq_exp'
[98, 1]
[104, 25]
simp only [Nat.add]
b e n : ℕ ⊢ exp b (Nat.add n 0) * b = exp' b (Nat.add n 0) * b
b e n : ℕ ⊢ exp b n * b = exp' b n * b
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/Theorems.lean
exp_eq_exp'
[98, 1]
[104, 25]
rw [exp_eq_exp' b n]
b e n : ℕ ⊢ exp b n * b = exp' b n * b
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.sorted_tail_of_sorted
[49, 1]
[53, 30]
match h with | single => apply nil | longer _ h'' => exact h''
α : Type f : α → α → Bool r : α → α → Prop a : α as : List α h : Sorted f (a :: as) ⊢ Sorted f as
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.sorted_tail_of_sorted
[49, 1]
[53, 30]
apply nil
α : Type f : α → α → Bool r : α → α → Prop a : α as : List α h : Sorted f (a :: as) ⊢ Sorted f []
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.sorted_tail_of_sorted
[49, 1]
[53, 30]
exact h''
α : Type f : α → α → Bool r : α → α → Prop a : α as : List α h : Sorted f (a :: as) a₂✝ : α as✝ : List α h✝ : f a a₂✝ = true h'' : Sorted f (a₂✝ :: as✝) ⊢ Sorted f (a₂✝ :: as✝)
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.len_insert_eq_succ_len
[80, 1]
[87, 62]
match l with | [] => simp [insert] | a'::as => match h : f a a' with | true => simp [insert, h] | false => simp [insert, h]; apply len_insert_eq_succ_len
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α ⊢ List.length (insert f a l) = List.length l + 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.len_insert_eq_succ_len
[80, 1]
[87, 62]
simp [insert]
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α ⊢ List.length (insert f a []) = List.length [] + 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.len_insert_eq_succ_len
[80, 1]
[87, 62]
match h : f a a' with | true => simp [insert, h] | false => simp [insert, h]; apply len_insert_eq_succ_len
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α a' : α as : List α ⊢ List.length (insert f a (a' :: as)) = List.length (a' :: as) + 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.len_insert_eq_succ_len
[80, 1]
[87, 62]
simp [insert, h]
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α a' : α as : List α h : f a a' = true ⊢ List.length (insert f a (a' :: as)) = List.length (a' :: as) + 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.len_insert_eq_succ_len
[80, 1]
[87, 62]
simp [insert, h]
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α a' : α as : List α h : f a a' = false ⊢ List.length (insert f a (a' :: as)) = List.length (a' :: as) + 1
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α a' : α as : List α h : f a a' = false ⊢ List.length (insert f a as) = List.length as + 1
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.len_insert_eq_succ_len
[80, 1]
[87, 62]
apply len_insert_eq_succ_len
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α a' : α as : List α h : f a a' = false ⊢ List.length (insert f a as) = List.length as + 1
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.ordered_cons_insert_of_unordered
[136, 1]
[143, 66]
simpa [insert]
α : Type f : α → α → Bool r : α → α → Prop a a' : α as : List α h' : f a' a = true h : Sorted f [a'] ⊢ f a' (insert f a [])[0] = true
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.ordered_cons_insert_of_unordered
[136, 1]
[143, 66]
simpa [insert, h'']
α : Type f : α → α → Bool r : α → α → Prop a a' : α as : List α h' : f a' a = true a'' : α as' : List α h : Sorted f (a' :: a'' :: as') h'' : f a a'' = true ⊢ f a' (insert f a (a'' :: as'))[0] = true
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.ordered_cons_insert_of_unordered
[136, 1]
[143, 66]
simp [insert, h'']
α : Type f : α → α → Bool r : α → α → Prop a a' : α as : List α h' : f a' a = true a'' : α as' : List α h : Sorted f (a' :: a'' :: as') h'' : f a a'' = false ⊢ f a' (insert f a (a'' :: as'))[0] = true
α : Type f : α → α → Bool r : α → α → Prop a a' : α as : List α h' : f a' a = true a'' : α as' : List α h : Sorted f (a' :: a'' :: as') h'' : f a a'' = false ⊢ f a' a'' = true
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.ordered_cons_insert_of_unordered
[136, 1]
[143, 66]
apply ordered_of_sorted f h
α : Type f : α → α → Bool r : α → α → Prop a a' : α as : List α h' : f a' a = true a'' : α as' : List α h : Sorted f (a' :: a'' :: as') h'' : f a a'' = false ⊢ f a' a'' = true
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
simp [insert, h']
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = true ⊢ Sorted f (insert f a (a' :: as))
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = true ⊢ Sorted f (a :: a' :: as)
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
apply longer h' h
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = true ⊢ Sorted f (a :: a' :: as)
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
simp [insert, h']
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ Sorted f (insert f a (a' :: as))
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ Sorted f (a' :: insert f a as)
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
apply cons f
α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ Sorted f (a' :: insert f a as)
case h₁ α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ Sorted f (insert f a as) case h₃ α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ f a' (insert f a as)[0] = true case h₂ α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ List.length (insert f a as) > 0
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
apply insert_sorted_of_sorted <| sorted_tail_of_sorted f a' as h
case h₁ α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ Sorted f (insert f a as)
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
apply ordered_cons_insert_of_unordered f h
case h₃ α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ f a' (insert f a as)[0] = true
case h₃ α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ f a' a = true
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
apply asym
case h₃ α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ f a' a = true
case h₃.a α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ (!f a a') = true
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
simp
case h₃.a α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ (!f a a') = true
case h₃.a α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ f a a' = false
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.insert_sorted_of_sorted
[147, 1]
[159, 39]
assumption
case h₃.a α : Type f : α → α → Bool r : α → α → Prop a : α l : List α inst✝ : Asymmetric f a' : α as : List α h : Sorted f (a' :: as) h' : f a a' = false ⊢ f a a' = false
no goals
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.sorted_of_insertSort
[163, 1]
[169, 63]
dsimp [insertSort]
α : Type f : α → α → Bool r : α → α → Prop l : List α inst✝ : Asymmetric f a : α as : List α ⊢ Sorted f (insertSort f (a :: as))
α : Type f : α → α → Bool r : α → α → Prop l : List α inst✝ : Asymmetric f a : α as : List α ⊢ Sorted f (insert f a (insertSort f as))
https://github.com/UofSC-Spring-2023-SCHC-411-H01/notes.git
72bc9ee3a0c93523fec747000cb89a6983f00afb
Notes4/IndPred.lean
Notes.sorted_of_insertSort
[163, 1]
[169, 63]
apply insert_sorted_of_sorted f <| sorted_of_insertSort as
α : Type f : α → α → Bool r : α → α → Prop l : List α inst✝ : Asymmetric f a : α as : List α ⊢ Sorted f (insert f a (insertSort f as))
no goals
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
have : Fintype V := Fintype.ofFinite V
V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G G : Digraph V ⊢ motive G
V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G G : Digraph V this : Fintype V ⊢ motive G
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
generalize hs : G.edgeSet.toFinset = s
V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G G : Digraph V this : Fintype V ⊢ motive G
V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G G : Digraph V this : Fintype V s : Finset (V × V) hs : Set.toFinset (edgeSet G) = s ⊢ motive G
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
rw [← Set.toFinset_empty, Set.toFinset_inj, ← edgeSet_bot, edgeSet_inj] at hs
case empty V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this : Fintype V G : Digraph V hs : Set.toFinset (edgeSet G) = ∅ ⊢ motive G
case empty V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this : Fintype V G : Digraph V hs✝ : Set.toFinset (edgeSet G) = ∅ hs : G = ⊥ ⊢ motive G
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
rwa [hs]
case empty V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this : Fintype V G : Digraph V hs✝ : Set.toFinset (edgeSet G) = ∅ hs : G = ⊥ ⊢ motive G
no goals
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
have : e ∈ G.edgeSet
case insert V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s ⊢ motive G
case this V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s ⊢ e ∈ edgeSet G case insert V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G ⊢ motive G
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
apply hdelete G e.1 e.2 this
case insert V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G ⊢ motive G
case insert V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G ⊢ motive (deleteEdges G {(e.fst, e.snd)})
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
apply ih
case insert V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G ⊢ motive (deleteEdges G {(e.fst, e.snd)})
case insert.hs V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G ⊢ Set.toFinset (edgeSet (deleteEdges G {(e.fst, e.snd)})) = s
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
ext ⟨v, w⟩
case insert.hs V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G ⊢ Set.toFinset (edgeSet (deleteEdges G {(e.fst, e.snd)})) = s
case insert.hs.a.mk V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G v w : V ⊢ (v, w) ∈ Set.toFinset (edgeSet (deleteEdges G {(e.fst, e.snd)})) ↔ (v, w) ∈ s
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
simp only [Prod.mk.eta, Set.mem_toFinset, mem_edgeSet, deleteEdges_adj, Set.mem_singleton_iff]
case insert.hs.a.mk V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G v w : V ⊢ (v, w) ∈ Set.toFinset (edgeSet (deleteEdges G {(e.fst, e.snd)})) ↔ (v, w) ∈ s
case insert.hs.a.mk V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s this : e ∈ edgeSet G v w : V ⊢ G.Adj v w ∧ ¬(v, w) = e ↔ (v, w) ∈ s
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
simp only [Set.mem_toFinset, mem_edgeSet, Finset.mem_insert, eq_iff_iff] at hs
case insert.hs.a.mk V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : ((v, w) ∈ Set.toFinset (edgeSet G)) = ((v, w) ∈ insert e s) ⊢ G.Adj v w ∧ ¬(v, w) = e ↔ (v, w) ∈ s
case insert.hs.a.mk V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s ⊢ G.Adj v w ∧ ¬(v, w) = e ↔ (v, w) ∈ s
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
rw [hs]
case insert.hs.a.mk V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s ⊢ G.Adj v w ∧ ¬(v, w) = e ↔ (v, w) ∈ s
case insert.hs.a.mk V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s ⊢ ((v, w) = e ∨ (v, w) ∈ s) ∧ ¬(v, w) = e ↔ (v, w) ∈ s
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
constructor
case insert.hs.a.mk V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s ⊢ ((v, w) = e ∨ (v, w) ∈ s) ∧ ¬(v, w) = e ↔ (v, w) ∈ s
case insert.hs.a.mk.mp V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s ⊢ ((v, w) = e ∨ (v, w) ∈ s) ∧ ¬(v, w) = e → (v, w) ∈ s case insert.hs.a.mk.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s ⊢ (v, w) ∈ s → ((v, w) = e ∨ (v, w) ∈ s) ∧ ¬(v, w) = e
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
rw [← Set.mem_toFinset, hs]
case this V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s ⊢ e ∈ edgeSet G
case this V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s ⊢ e ∈ insert e s
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
exact Finset.mem_insert_self e s
case this V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V hs : Set.toFinset (edgeSet G) = insert e s ⊢ e ∈ insert e s
no goals
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
rintro ⟨(rfl | ha), hne⟩
case insert.hs.a.mk.mp V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s ⊢ ((v, w) = e ∨ (v, w) ∈ s) ∧ ¬(v, w) = e → (v, w) ∈ s
case insert.hs.a.mk.mp.intro.inl V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V s : Finset (V × V) ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V v w : V he : ¬(v, w) ∈ s this : (v, w) ∈ edgeSet G hs : G.Adj v w ↔ (v, w) = (v, w) ∨ (v, w) ∈ s hne : ¬(v, w) = (v, w) ⊢ (v, w) ∈ s case insert.hs.a.mk.mp.intro.inr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s hne : ¬(v, w) = e ha : (v, w) ∈ s ⊢ (v, w) ∈ s
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
simp at hne
case insert.hs.a.mk.mp.intro.inl V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V s : Finset (V × V) ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V v w : V he : ¬(v, w) ∈ s this : (v, w) ∈ edgeSet G hs : G.Adj v w ↔ (v, w) = (v, w) ∨ (v, w) ∈ s hne : ¬(v, w) = (v, w) ⊢ (v, w) ∈ s
no goals
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
assumption
case insert.hs.a.mk.mp.intro.inr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s hne : ¬(v, w) = e ha : (v, w) ∈ s ⊢ (v, w) ∈ s
no goals
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
intro h
case insert.hs.a.mk.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s ⊢ (v, w) ∈ s → ((v, w) = e ∨ (v, w) ∈ s) ∧ ¬(v, w) = e
case insert.hs.a.mk.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s h : (v, w) ∈ s ⊢ ((v, w) = e ∨ (v, w) ∈ s) ∧ ¬(v, w) = e
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
simp only [h, or_true, true_and]
case insert.hs.a.mk.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s h : (v, w) ∈ s ⊢ ((v, w) = e ∨ (v, w) ∈ s) ∧ ¬(v, w) = e
case insert.hs.a.mk.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s h : (v, w) ∈ s ⊢ ¬(v, w) = e
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
rintro rfl
case insert.hs.a.mk.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V e : V × V s : Finset (V × V) he : ¬e ∈ s ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V this : e ∈ edgeSet G v w : V hs : G.Adj v w ↔ (v, w) = e ∨ (v, w) ∈ s h : (v, w) ∈ s ⊢ ¬(v, w) = e
case insert.hs.a.mk.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V s : Finset (V × V) ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V v w : V h : (v, w) ∈ s he : ¬(v, w) ∈ s this : (v, w) ∈ edgeSet G hs : G.Adj v w ↔ (v, w) = (v, w) ∨ (v, w) ∈ s ⊢ False
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction
[6, 1]
[35, 24]
exact absurd h he
case insert.hs.a.mk.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), G.Adj v w → motive (deleteEdges G {(v, w)}) → motive G this✝ : Fintype V s : Finset (V × V) ih : ∀ (G : Digraph V), Set.toFinset (edgeSet G) = s → motive G G : Digraph V v w : V h : (v, w) ∈ s he : ¬(v, w) ∈ s this : (v, w) ∈ edgeSet G hs : G.Adj v w ↔ (v, w) = (v, w) ∨ (v, w) ∈ s ⊢ False
no goals
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction'
[39, 1]
[55, 19]
exact hbot
case hbot V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) ⊢ motive ⊥
no goals
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction'
[39, 1]
[55, 19]
convert hdelete (G.deleteEdges {(v, w)}) v w (by simp) ih
case hdelete V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) ⊢ motive G
case h.e'_1 V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) ⊢ G = deleteEdges G {(v, w)} ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction'
[39, 1]
[55, 19]
ext a b
case h.e'_1 V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) ⊢ G = deleteEdges G {(v, w)} ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}
case h.e'_1.Adj.h.h.a V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ↔ (deleteEdges G {(v, w)} ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}).Adj a b
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction'
[39, 1]
[55, 19]
simp only [Set.le_eq_subset, ge_iff_le, sup_adj, deleteEdges_adj, Set.mem_singleton_iff, Prod.mk.injEq, not_and]
case h.e'_1.Adj.h.h.a V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ↔ (deleteEdges G {(v, w)} ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}).Adj a b
case h.e'_1.Adj.h.h.a V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ↔ G.Adj a b ∧ (a = v → ¬b = w) ∨ (↑(OrderIso.symm (edgeSetIso V)) {(v, w)}).Adj a b
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction'
[39, 1]
[55, 19]
rw [edgeSetIso_symm_adj]
case h.e'_1.Adj.h.h.a V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ↔ G.Adj a b ∧ (a = v → ¬b = w) ∨ (↑(OrderIso.symm (edgeSetIso V)) {(v, w)}).Adj a b
case h.e'_1.Adj.h.h.a V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ↔ G.Adj a b ∧ (a = v → ¬b = w) ∨ (a, b) ∈ {(v, w)}
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction'
[39, 1]
[55, 19]
simp only [Set.mem_singleton_iff, Prod.mk.injEq]
case h.e'_1.Adj.h.h.a V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ↔ G.Adj a b ∧ (a = v → ¬b = w) ∨ (a, b) ∈ {(v, w)}
case h.e'_1.Adj.h.h.a V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ↔ G.Adj a b ∧ (a = v → ¬b = w) ∨ a = v ∧ b = w
https://github.com/kmill/msri2023_graphs.git
c87c12d835cabc843f7f9c83d4c947c11d8ff043
GraphProjects/DigraphInduction.lean
Digraph.deleteEdges_induction'
[39, 1]
[55, 19]
constructor
case h.e'_1.Adj.h.h.a V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ↔ G.Adj a b ∧ (a = v → ¬b = w) ∨ a = v ∧ b = w
case h.e'_1.Adj.h.h.a.mp V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b → G.Adj a b ∧ (a = v → ¬b = w) ∨ a = v ∧ b = w case h.e'_1.Adj.h.h.a.mpr V : Type u_1 inst✝ : Finite V motive : Digraph V → Prop hbot : motive ⊥ hdelete : ∀ (G : Digraph V) (v w : V), ¬G.Adj v w → motive G → motive (G ⊔ ↑(OrderIso.symm (edgeSetIso V)) {(v, w)}) G : Digraph V v w : V hvw : G.Adj v w ih : motive (deleteEdges G {(v, w)}) a b : V ⊢ G.Adj a b ∧ (a = v → ¬b = w) ∨ a = v ∧ b = w → G.Adj a b