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stringclasses 147
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stringlengths 7
101
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stringlengths 1
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| start
stringlengths 6
10
| end
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https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe06_InductivePredicates_Demo.lean | LoVe.IsFull_mirror_struct_induct | [415, 1] | [430, 50] | rw [mirror] | case node.node
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror (BTree.node a l r)) | case node.node
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (BTree.node a (mirror r) (mirror l)) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe06_InductivePredicates_Demo.lean | LoVe.IsFull_mirror_struct_induct | [415, 1] | [430, 50] | apply IsFull.node | case node.node
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (BTree.node a (mirror r) (mirror l)) | case node.node.hl
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror r)
case node.node.hr
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror l)
case node.node.hiff
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ mirror r = BTree.empty ↔ mirror l = BTree.empty |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe06_InductivePredicates_Demo.lean | LoVe.IsFull_mirror_struct_induct | [415, 1] | [430, 50] | { exact ih_r hr } | case node.node.hl
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror r)
case node.node.hr
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror l)
case node.node.hiff
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ mirror r = BTree.empty ↔ mirror l = BTree.empty | case node.node.hr
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror l)
case node.node.hiff
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ mirror r = BTree.empty ↔ mirror l = BTree.empty |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe06_InductivePredicates_Demo.lean | LoVe.IsFull_mirror_struct_induct | [415, 1] | [430, 50] | { apply ih_l hl } | case node.node.hr
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror l)
case node.node.hiff
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ mirror r = BTree.empty ↔ mirror l = BTree.empty | case node.node.hiff
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ mirror r = BTree.empty ↔ mirror l = BTree.empty |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe06_InductivePredicates_Demo.lean | LoVe.IsFull_mirror_struct_induct | [415, 1] | [430, 50] | { simp [mirror_Eq_empty_Iff, *] } | case node.node.hiff
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ mirror r = BTree.empty ↔ mirror l = BTree.empty | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe06_InductivePredicates_Demo.lean | LoVe.IsFull_mirror_struct_induct | [415, 1] | [430, 50] | exact ih_r hr | case node.node.hl
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror r) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe06_InductivePredicates_Demo.lean | LoVe.IsFull_mirror_struct_induct | [415, 1] | [430, 50] | apply ih_l hl | case node.node.hr
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ IsFull (mirror l) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe06_InductivePredicates_Demo.lean | LoVe.IsFull_mirror_struct_induct | [415, 1] | [430, 50] | simp [mirror_Eq_empty_Iff, *] | case node.node.hiff
α : Type
a : α
l r : BTree α
ih_l : IsFull l → IsFull (mirror l)
ih_r : IsFull r → IsFull (mirror r)
hl : IsFull l
hr : IsFull r
hiff : l = BTree.empty ↔ r = BTree.empty
⊢ mirror r = BTree.empty ↔ mirror l = BTree.empty | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.neq_fin4_of_neq | [345, 1] | [347, 39] | intro _ h | a b : ℕ
ha : a < 4
hb : b < 4
⊢ a ≠ b → { val := a, isLt := ha } ≠ { val := b, isLt := hb } | a b : ℕ
ha : a < 4
hb : b < 4
a✝ : a ≠ b
h : { val := a, isLt := ha } = { val := b, isLt := hb }
⊢ False |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.neq_fin4_of_neq | [345, 1] | [347, 39] | cases h | a b : ℕ
ha : a < 4
hb : b < 4
a✝ : a ≠ b
h : { val := a, isLt := ha } = { val := b, isLt := hb }
⊢ False | case refl
a : ℕ
ha hb : a < 4
a✝ : a ≠ a
⊢ False |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.neq_fin4_of_neq | [345, 1] | [347, 39] | contradiction | case refl
a : ℕ
ha hb : a < 4
a✝ : a ≠ a
⊢ False | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 0 < 4
isLt✝ : 1 < 4
x✝ : { val := 0, isLt := isLt✝¹ } ≠ { val := 1, isLt := isLt✝ }
⊢ { val := 0, isLt := isLt✝¹ } ≠ 2 ∧
2 ≠ { val := 1, isLt := isLt✝ } ∧ { val := 0, isLt := isLt✝¹ } ≠ 3 ∧ 3 ≠ { val := 1, isLt := isLt✝ } ∧ 2 ≠ 3 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 0 < 4
isLt✝ : 2 < 4
x✝ : { val := 0, isLt := isLt✝¹ } ≠ { val := 2, isLt := isLt✝ }
⊢ { val := 0, isLt := isLt✝¹ } ≠ 1 ∧
1 ≠ { val := 2, isLt := isLt✝ } ∧ { val := 0, isLt := isLt✝¹ } ≠ 3 ∧ 3 ≠ { val := 2, isLt := isLt✝ } ∧ 1 ≠ 3 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 0 < 4
isLt✝ : 3 < 4
x✝ : { val := 0, isLt := isLt✝¹ } ≠ { val := 3, isLt := isLt✝ }
⊢ { val := 0, isLt := isLt✝¹ } ≠ 1 ∧
1 ≠ { val := 3, isLt := isLt✝ } ∧ { val := 0, isLt := isLt✝¹ } ≠ 2 ∧ 2 ≠ { val := 3, isLt := isLt✝ } ∧ 1 ≠ 2 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 1 < 4
isLt✝ : 0 < 4
x✝ : { val := 1, isLt := isLt✝¹ } ≠ { val := 0, isLt := isLt✝ }
⊢ { val := 1, isLt := isLt✝¹ } ≠ 2 ∧
2 ≠ { val := 0, isLt := isLt✝ } ∧ { val := 1, isLt := isLt✝¹ } ≠ 3 ∧ 3 ≠ { val := 0, isLt := isLt✝ } ∧ 2 ≠ 3 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 1 < 4
isLt✝ : 2 < 4
x✝ : { val := 1, isLt := isLt✝¹ } ≠ { val := 2, isLt := isLt✝ }
⊢ { val := 1, isLt := isLt✝¹ } ≠ 0 ∧
0 ≠ { val := 2, isLt := isLt✝ } ∧ { val := 1, isLt := isLt✝¹ } ≠ 3 ∧ 3 ≠ { val := 2, isLt := isLt✝ } ∧ 0 ≠ 3 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 1 < 4
isLt✝ : 3 < 4
x✝ : { val := 1, isLt := isLt✝¹ } ≠ { val := 3, isLt := isLt✝ }
⊢ { val := 1, isLt := isLt✝¹ } ≠ 0 ∧
0 ≠ { val := 3, isLt := isLt✝ } ∧ { val := 1, isLt := isLt✝¹ } ≠ 2 ∧ 2 ≠ { val := 3, isLt := isLt✝ } ∧ 0 ≠ 2 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 2 < 4
isLt✝ : 0 < 4
x✝ : { val := 2, isLt := isLt✝¹ } ≠ { val := 0, isLt := isLt✝ }
⊢ { val := 2, isLt := isLt✝¹ } ≠ 1 ∧
1 ≠ { val := 0, isLt := isLt✝ } ∧ { val := 2, isLt := isLt✝¹ } ≠ 3 ∧ 3 ≠ { val := 0, isLt := isLt✝ } ∧ 1 ≠ 3 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 2 < 4
isLt✝ : 1 < 4
x✝ : { val := 2, isLt := isLt✝¹ } ≠ { val := 1, isLt := isLt✝ }
⊢ { val := 2, isLt := isLt✝¹ } ≠ 0 ∧
0 ≠ { val := 1, isLt := isLt✝ } ∧ { val := 2, isLt := isLt✝¹ } ≠ 3 ∧ 3 ≠ { val := 1, isLt := isLt✝ } ∧ 0 ≠ 3 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 2 < 4
isLt✝ : 3 < 4
x✝ : { val := 2, isLt := isLt✝¹ } ≠ { val := 3, isLt := isLt✝ }
⊢ { val := 2, isLt := isLt✝¹ } ≠ 0 ∧
0 ≠ { val := 3, isLt := isLt✝ } ∧ { val := 2, isLt := isLt✝¹ } ≠ 1 ∧ 1 ≠ { val := 3, isLt := isLt✝ } ∧ 0 ≠ 1 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 3 < 4
isLt✝ : 0 < 4
x✝ : { val := 3, isLt := isLt✝¹ } ≠ { val := 0, isLt := isLt✝ }
⊢ { val := 3, isLt := isLt✝¹ } ≠ 1 ∧
1 ≠ { val := 0, isLt := isLt✝ } ∧ { val := 3, isLt := isLt✝¹ } ≠ 2 ∧ 2 ≠ { val := 0, isLt := isLt✝ } ∧ 1 ≠ 2 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 3 < 4
isLt✝ : 1 < 4
x✝ : { val := 3, isLt := isLt✝¹ } ≠ { val := 1, isLt := isLt✝ }
⊢ { val := 3, isLt := isLt✝¹ } ≠ 0 ∧
0 ≠ { val := 1, isLt := isLt✝ } ∧ { val := 3, isLt := isLt✝¹ } ≠ 2 ∧ 2 ≠ { val := 1, isLt := isLt✝ } ∧ 0 ≠ 2 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Homework/Homework7.lean | LoVe.exist_distinct_fin4_of_neq | [352, 1] | [369, 38] | fin_neq | isLt✝¹ : 3 < 4
isLt✝ : 2 < 4
x✝ : { val := 3, isLt := isLt✝¹ } ≠ { val := 2, isLt := isLt✝ }
⊢ { val := 3, isLt := isLt✝¹ } ≠ 0 ∧
0 ≠ { val := 2, isLt := isLt✝ } ∧ { val := 3, isLt := isLt✝¹ } ≠ 1 ∧ 1 ≠ { val := 2, isLt := isLt✝ } ∧ 0 ≠ 1 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | rw [sillyLoop] | ⊢ (sillyLoop, State.update "x" 1 fun x => 0) ⟹ fun x => 0 | ⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1),
State.update "x" 1 fun x => 0) ⟹
fun x => 0 |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | apply BigStep.while_true | ⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1),
State.update "x" 1 fun x => 0) ⟹
fun x => 0 | case hcond
⊢ State.update "x" 1 (fun x => 0) "x" > State.update "x" 1 (fun x => 0) "y"
case hbody
⊢ (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1, State.update "x" 1 fun x => 0) ⟹ ?t
case hrest
⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1), ?t) ⟹ fun x => 0
case t
⊢ State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | { simp } | case hcond
⊢ State.update "x" 1 (fun x => 0) "x" > State.update "x" 1 (fun x => 0) "y"
case hbody
⊢ (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1, State.update "x" 1 fun x => 0) ⟹ ?t
case hrest
⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1), ?t) ⟹ fun x => 0
case t
⊢ State | case hbody
⊢ (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1, State.update "x" 1 fun x => 0) ⟹ ?t
case hrest
⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1), ?t) ⟹ fun x => 0
case t
⊢ State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | { apply BigStep.seq
{ apply BigStep.skip }
{ apply BigStep.assign } } | case hbody
⊢ (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1, State.update "x" 1 fun x => 0) ⟹ ?t
case hrest
⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1), ?t) ⟹ fun x => 0
case t
⊢ State | case hrest
⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1),
State.update "x" (State.update "x" 1 (fun x => 0) "x" - 1) (State.update "x" 1 fun x => 0)) ⟹
fun x => 0 |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | { simp
apply BigStep.while_false
simp } | case hrest
⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1),
State.update "x" (State.update "x" 1 (fun x => 0) "x" - 1) (State.update "x" 1 fun x => 0)) ⟹
fun x => 0 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | simp | case hcond
⊢ State.update "x" 1 (fun x => 0) "x" > State.update "x" 1 (fun x => 0) "y" | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | apply BigStep.seq | case hbody
⊢ (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1, State.update "x" 1 fun x => 0) ⟹ ?t | case hbody.hS
⊢ (Stmt.skip, State.update "x" 1 fun x => 0) ⟹ ?hbody.t
case hbody.hT
⊢ (Stmt.assign "x" fun s => s "x" - 1, ?hbody.t) ⟹ ?t
case hbody.t
⊢ State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | { apply BigStep.skip } | case hbody.hS
⊢ (Stmt.skip, State.update "x" 1 fun x => 0) ⟹ ?hbody.t
case hbody.hT
⊢ (Stmt.assign "x" fun s => s "x" - 1, ?hbody.t) ⟹ ?t
case hbody.t
⊢ State | case hbody.hT
⊢ (Stmt.assign "x" fun s => s "x" - 1, State.update "x" 1 fun x => 0) ⟹ ?t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | { apply BigStep.assign } | case hbody.hT
⊢ (Stmt.assign "x" fun s => s "x" - 1, State.update "x" 1 fun x => 0) ⟹ ?t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | apply BigStep.skip | case hbody.hS
⊢ (Stmt.skip, State.update "x" 1 fun x => 0) ⟹ ?hbody.t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | apply BigStep.assign | case hbody.hT
⊢ (Stmt.assign "x" fun s => s "x" - 1, State.update "x" 1 fun x => 0) ⟹ ?t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | simp | case hrest
⊢ (Stmt.whileDo (fun s => s "x" > s "y") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1),
State.update "x" (State.update "x" 1 (fun x => 0) "x" - 1) (State.update "x" 1 fun x => 0)) ⟹
fun x => 0 | case hrest
⊢ (Stmt.whileDo (fun s => s "y" < s "x") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1), fun x => 0) ⟹ fun x => 0 |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | apply BigStep.while_false | case hrest
⊢ (Stmt.whileDo (fun s => s "y" < s "x") (Stmt.skip; Stmt.assign "x" fun s => s "x" - 1), fun x => 0) ⟹ fun x => 0 | case hrest.hcond
⊢ ¬0 < 0 |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.sillyLoop_from_1_BigStep | [201, 1] | [212, 13] | simp | case hrest.hcond
⊢ ¬0 < 0 | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | induction hl generalizing r with
| skip s =>
cases hr with
| skip => rfl
| assign x a s =>
cases hr with
| assign => rfl
| seq S T s l₀ l hS hT ihS ihT =>
cases hr with
| seq _ _ _ r₀ _ hS' hT' =>
cases ihS hS' with
| refl =>
cases ihT hT' with
| refl => rfl
| if_true B S T s l hB hS ih =>
cases hr with
| if_true _ _ _ _ _ hB' hS' => apply ih hS'
| if_false _ _ _ _ _ hB' hS' => aesop
| if_false B S T s l hB hT ih =>
cases hr with
| if_true _ _ _ _ _ hB' hS' => aesop
| if_false _ _ _ _ _ hB' hS' => apply ih hS'
| while_true B S s l₀ l hB hS hw ihS ihw =>
cases hr with
| while_true _ _ _ r₀ hB' hB' hS' hw' =>
cases ihS hS' with
| refl =>
cases ihw hw' with
| refl => rfl
| while_false _ _ _ hB' => aesop
| while_false B S s hB =>
cases hr with
| while_true _ _ _ s' _ hB' hS hw => aesop
| while_false _ _ _ hB' => rfl | Ss : Stmt × State
l r : State
hl : Ss ⟹ l
hr : Ss ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases hr with
| skip => rfl | case skip
Ss : Stmt × State
l s r : State
hr : (Stmt.skip, s) ⟹ r
⊢ s = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | rfl | case skip.skip
Ss : Stmt × State
l s : State
⊢ s = s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases hr with
| assign => rfl | case assign
Ss : Stmt × State
l : State
x : String
a : State → ℕ
s r : State
hr : (Stmt.assign x a, s) ⟹ r
⊢ State.update x (a s) s = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | rfl | case assign.assign
Ss : Stmt × State
l : State
x : String
a : State → ℕ
s : State
⊢ State.update x (a s) s = State.update x (a s) s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases hr with
| seq _ _ _ r₀ _ hS' hT' =>
cases ihS hS' with
| refl =>
cases ihT hT' with
| refl => rfl | case seq
Ss : Stmt × State
l✝ : State
S T : Stmt
s l₀ l : State
hS : (S, s) ⟹ l₀
hT : (T, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihT : ∀ {r : State}, (T, l₀) ⟹ r → l = r
r : State
hr : (S; T, s) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases ihS hS' with
| refl =>
cases ihT hT' with
| refl => rfl | case seq.seq
Ss : Stmt × State
l✝ : State
S T : Stmt
s l₀ l : State
hS : (S, s) ⟹ l₀
hT : (T, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihT : ∀ {r : State}, (T, l₀) ⟹ r → l = r
r r₀ : State
hS' : (S, s) ⟹ r₀
hT' : (T, r₀) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases ihT hT' with
| refl => rfl | case seq.seq.refl
Ss : Stmt × State
l✝ : State
S T : Stmt
s l₀ l : State
hS : (S, s) ⟹ l₀
hT : (T, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihT : ∀ {r : State}, (T, l₀) ⟹ r → l = r
r : State
hS' : (S, s) ⟹ l₀
hT' : (T, l₀) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | rfl | case seq.seq.refl.refl
Ss : Stmt × State
l✝ : State
S T : Stmt
s l₀ l : State
hS : (S, s) ⟹ l₀
hT : (T, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihT : ∀ {r : State}, (T, l₀) ⟹ r → l = r
hS' : (S, s) ⟹ l₀
hT' : (T, l₀) ⟹ l
⊢ l = l | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases hr with
| if_true _ _ _ _ _ hB' hS' => apply ih hS'
| if_false _ _ _ _ _ hB' hS' => aesop | case if_true
Ss : Stmt × State
l✝ : State
B : State → Prop
S T : Stmt
s l : State
hB : B s
hS : (S, s) ⟹ l
ih : ∀ {r : State}, (S, s) ⟹ r → l = r
r : State
hr : (Stmt.ifThenElse B S T, s) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | apply ih hS' | case if_true.if_true
Ss : Stmt × State
l✝ : State
B : State → Prop
S T : Stmt
s l : State
hB : B s
hS : (S, s) ⟹ l
ih : ∀ {r : State}, (S, s) ⟹ r → l = r
r : State
hB' : B s
hS' : (S, s) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | aesop | case if_true.if_false
Ss : Stmt × State
l✝ : State
B : State → Prop
S T : Stmt
s l : State
hB : B s
hS : (S, s) ⟹ l
ih : ∀ {r : State}, (S, s) ⟹ r → l = r
r : State
hB' : ¬B s
hS' : (T, s) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases hr with
| if_true _ _ _ _ _ hB' hS' => aesop
| if_false _ _ _ _ _ hB' hS' => apply ih hS' | case if_false
Ss : Stmt × State
l✝ : State
B : State → Prop
S T : Stmt
s l : State
hB : ¬B s
hT : (T, s) ⟹ l
ih : ∀ {r : State}, (T, s) ⟹ r → l = r
r : State
hr : (Stmt.ifThenElse B S T, s) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | aesop | case if_false.if_true
Ss : Stmt × State
l✝ : State
B : State → Prop
S T : Stmt
s l : State
hB : ¬B s
hT : (T, s) ⟹ l
ih : ∀ {r : State}, (T, s) ⟹ r → l = r
r : State
hB' : B s
hS' : (S, s) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | apply ih hS' | case if_false.if_false
Ss : Stmt × State
l✝ : State
B : State → Prop
S T : Stmt
s l : State
hB : ¬B s
hT : (T, s) ⟹ l
ih : ∀ {r : State}, (T, s) ⟹ r → l = r
r : State
hB' : ¬B s
hS' : (T, s) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases hr with
| while_true _ _ _ r₀ hB' hB' hS' hw' =>
cases ihS hS' with
| refl =>
cases ihw hw' with
| refl => rfl
| while_false _ _ _ hB' => aesop | case while_true
Ss : Stmt × State
l✝ : State
B : State → Prop
S : Stmt
s l₀ l : State
hB : B s
hS : (S, s) ⟹ l₀
hw : (Stmt.whileDo B S, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihw : ∀ {r : State}, (Stmt.whileDo B S, l₀) ⟹ r → l = r
r : State
hr : (Stmt.whileDo B S, s) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases ihS hS' with
| refl =>
cases ihw hw' with
| refl => rfl | case while_true.while_true
Ss : Stmt × State
l✝ : State
B : State → Prop
S : Stmt
s l₀ l : State
hB : B s
hS : (S, s) ⟹ l₀
hw : (Stmt.whileDo B S, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihw : ∀ {r : State}, (Stmt.whileDo B S, l₀) ⟹ r → l = r
r r₀ : State
hB' : B s
hS' : (S, s) ⟹ r₀
hw' : (Stmt.whileDo B S, r₀) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases ihw hw' with
| refl => rfl | case while_true.while_true.refl
Ss : Stmt × State
l✝ : State
B : State → Prop
S : Stmt
s l₀ l : State
hB : B s
hS : (S, s) ⟹ l₀
hw : (Stmt.whileDo B S, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihw : ∀ {r : State}, (Stmt.whileDo B S, l₀) ⟹ r → l = r
r : State
hB' : B s
hS' : (S, s) ⟹ l₀
hw' : (Stmt.whileDo B S, l₀) ⟹ r
⊢ l = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | rfl | case while_true.while_true.refl.refl
Ss : Stmt × State
l✝ : State
B : State → Prop
S : Stmt
s l₀ l : State
hB : B s
hS : (S, s) ⟹ l₀
hw : (Stmt.whileDo B S, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihw : ∀ {r : State}, (Stmt.whileDo B S, l₀) ⟹ r → l = r
hB' : B s
hS' : (S, s) ⟹ l₀
hw' : (Stmt.whileDo B S, l₀) ⟹ l
⊢ l = l | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | aesop | case while_true.while_false
Ss : Stmt × State
l✝ : State
B : State → Prop
S : Stmt
s l₀ l : State
hB : B s
hS : (S, s) ⟹ l₀
hw : (Stmt.whileDo B S, l₀) ⟹ l
ihS : ∀ {r : State}, (S, s) ⟹ r → l₀ = r
ihw : ∀ {r : State}, (Stmt.whileDo B S, l₀) ⟹ r → l = r
hB' : ¬B s
⊢ l = s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | cases hr with
| while_true _ _ _ s' _ hB' hS hw => aesop
| while_false _ _ _ hB' => rfl | case while_false
Ss : Stmt × State
l : State
B : State → Prop
S : Stmt
s : State
hB : ¬B s
r : State
hr : (Stmt.whileDo B S, s) ⟹ r
⊢ s = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | aesop | case while_false.while_true
Ss : Stmt × State
l : State
B : State → Prop
S : Stmt
s : State
hB : ¬B s
r s' : State
hB' : B s
hS : (S, s) ⟹ s'
hw : (Stmt.whileDo B S, s') ⟹ r
⊢ s = r | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_deterministic | [225, 1] | [263, 47] | rfl | case while_false.while_false
Ss : Stmt × State
l : State
B : State → Prop
S : Stmt
s : State
hB hB' : ¬B s
⊢ s = s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | apply Iff.intro | s t : State
⊢ (Stmt.skip, s) ⟹ t ↔ t = s | case mp
s t : State
⊢ (Stmt.skip, s) ⟹ t → t = s
case mpr
s t : State
⊢ t = s → (Stmt.skip, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | { intro h
cases h with
| skip => rfl } | case mp
s t : State
⊢ (Stmt.skip, s) ⟹ t → t = s
case mpr
s t : State
⊢ t = s → (Stmt.skip, s) ⟹ t | case mpr
s t : State
⊢ t = s → (Stmt.skip, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | { intro h
rw [h]
apply BigStep.skip } | case mpr
s t : State
⊢ t = s → (Stmt.skip, s) ⟹ t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | intro h | case mp
s t : State
⊢ (Stmt.skip, s) ⟹ t → t = s | case mp
s t : State
h : (Stmt.skip, s) ⟹ t
⊢ t = s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | cases h with
| skip => rfl | case mp
s t : State
h : (Stmt.skip, s) ⟹ t
⊢ t = s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | rfl | case mp.skip
s : State
⊢ s = s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | intro h | case mpr
s t : State
⊢ t = s → (Stmt.skip, s) ⟹ t | case mpr
s t : State
h : t = s
⊢ (Stmt.skip, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | rw [h] | case mpr
s t : State
h : t = s
⊢ (Stmt.skip, s) ⟹ t | case mpr
s t : State
h : t = s
⊢ (Stmt.skip, s) ⟹ s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_skip_Iff | [273, 9] | [283, 27] | apply BigStep.skip | case mpr
s t : State
h : t = s
⊢ (Stmt.skip, s) ⟹ s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | apply Iff.intro | x : String
a : State → ℕ
s t : State
⊢ (Stmt.assign x a, s) ⟹ t ↔ t = State.update x (a s) s | case mp
x : String
a : State → ℕ
s t : State
⊢ (Stmt.assign x a, s) ⟹ t → t = State.update x (a s) s
case mpr
x : String
a : State → ℕ
s t : State
⊢ t = State.update x (a s) s → (Stmt.assign x a, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | { intro h
cases h with
| assign => rfl } | case mp
x : String
a : State → ℕ
s t : State
⊢ (Stmt.assign x a, s) ⟹ t → t = State.update x (a s) s
case mpr
x : String
a : State → ℕ
s t : State
⊢ t = State.update x (a s) s → (Stmt.assign x a, s) ⟹ t | case mpr
x : String
a : State → ℕ
s t : State
⊢ t = State.update x (a s) s → (Stmt.assign x a, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | { intro h
rw [h]
apply BigStep.assign } | case mpr
x : String
a : State → ℕ
s t : State
⊢ t = State.update x (a s) s → (Stmt.assign x a, s) ⟹ t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | intro h | case mp
x : String
a : State → ℕ
s t : State
⊢ (Stmt.assign x a, s) ⟹ t → t = State.update x (a s) s | case mp
x : String
a : State → ℕ
s t : State
h : (Stmt.assign x a, s) ⟹ t
⊢ t = State.update x (a s) s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | cases h with
| assign => rfl | case mp
x : String
a : State → ℕ
s t : State
h : (Stmt.assign x a, s) ⟹ t
⊢ t = State.update x (a s) s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | rfl | case mp.assign
x : String
a : State → ℕ
s : State
⊢ State.update x (a s) s = State.update x (a s) s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | intro h | case mpr
x : String
a : State → ℕ
s t : State
⊢ t = State.update x (a s) s → (Stmt.assign x a, s) ⟹ t | case mpr
x : String
a : State → ℕ
s t : State
h : t = State.update x (a s) s
⊢ (Stmt.assign x a, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | rw [h] | case mpr
x : String
a : State → ℕ
s t : State
h : t = State.update x (a s) s
⊢ (Stmt.assign x a, s) ⟹ t | case mpr
x : String
a : State → ℕ
s t : State
h : t = State.update x (a s) s
⊢ (Stmt.assign x a, s) ⟹ State.update x (a s) s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_assign_Iff | [285, 9] | [295, 29] | apply BigStep.assign | case mpr
x : String
a : State → ℕ
s t : State
h : t = State.update x (a s) s
⊢ (Stmt.assign x a, s) ⟹ State.update x (a s) s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | apply Iff.intro | S T : Stmt
s u : State
⊢ (S; T, s) ⟹ u ↔ ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u | case mp
S T : Stmt
s u : State
⊢ (S; T, s) ⟹ u → ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u
case mpr
S T : Stmt
s u : State
⊢ (∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u) → (S; T, s) ⟹ u |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | { intro h
cases h with
| seq =>
apply Exists.intro
apply And.intro <;>
assumption } | case mp
S T : Stmt
s u : State
⊢ (S; T, s) ⟹ u → ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u
case mpr
S T : Stmt
s u : State
⊢ (∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u) → (S; T, s) ⟹ u | case mpr
S T : Stmt
s u : State
⊢ (∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u) → (S; T, s) ⟹ u |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | { intro h
cases h with
| intro s' h' =>
cases h' with
| intro hS hT =>
apply BigStep.seq <;>
assumption } | case mpr
S T : Stmt
s u : State
⊢ (∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u) → (S; T, s) ⟹ u | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | intro h | case mp
S T : Stmt
s u : State
⊢ (S; T, s) ⟹ u → ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u | case mp
S T : Stmt
s u : State
h : (S; T, s) ⟹ u
⊢ ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | cases h with
| seq =>
apply Exists.intro
apply And.intro <;>
assumption | case mp
S T : Stmt
s u : State
h : (S; T, s) ⟹ u
⊢ ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | apply Exists.intro | case mp.seq
S T : Stmt
s u t✝ : State
hS✝ : (S, s) ⟹ t✝
hT✝ : (T, t✝) ⟹ u
⊢ ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u | case mp.seq.h
S T : Stmt
s u t✝ : State
hS✝ : (S, s) ⟹ t✝
hT✝ : (T, t✝) ⟹ u
⊢ (S, s) ⟹ ?mp.seq.w ∧ (T, ?mp.seq.w) ⟹ u
case mp.seq.w
S T : Stmt
s u t✝ : State
hS✝ : (S, s) ⟹ t✝
hT✝ : (T, t✝) ⟹ u
⊢ State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | apply And.intro <;>
assumption | case mp.seq.h
S T : Stmt
s u t✝ : State
hS✝ : (S, s) ⟹ t✝
hT✝ : (T, t✝) ⟹ u
⊢ (S, s) ⟹ ?mp.seq.w ∧ (T, ?mp.seq.w) ⟹ u
case mp.seq.w
S T : Stmt
s u t✝ : State
hS✝ : (S, s) ⟹ t✝
hT✝ : (T, t✝) ⟹ u
⊢ State | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | intro h | case mpr
S T : Stmt
s u : State
⊢ (∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u) → (S; T, s) ⟹ u | case mpr
S T : Stmt
s u : State
h : ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u
⊢ (S; T, s) ⟹ u |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | cases h with
| intro s' h' =>
cases h' with
| intro hS hT =>
apply BigStep.seq <;>
assumption | case mpr
S T : Stmt
s u : State
h : ∃ t, (S, s) ⟹ t ∧ (T, t) ⟹ u
⊢ (S; T, s) ⟹ u | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | cases h' with
| intro hS hT =>
apply BigStep.seq <;>
assumption | case mpr.intro
S T : Stmt
s u s' : State
h' : (S, s) ⟹ s' ∧ (T, s') ⟹ u
⊢ (S; T, s) ⟹ u | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_seq_Iff | [297, 9] | [314, 25] | apply BigStep.seq <;>
assumption | case mpr.intro.intro
S T : Stmt
s u s' : State
hS : (S, s) ⟹ s'
hT : (T, s') ⟹ u
⊢ (S; T, s) ⟹ u | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | apply Iff.intro | B : State → Prop
S T : Stmt
s t : State
⊢ (Stmt.ifThenElse B S T, s) ⟹ t ↔ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t | case mp
B : State → Prop
S T : Stmt
s t : State
⊢ (Stmt.ifThenElse B S T, s) ⟹ t → B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t
case mpr
B : State → Prop
S T : Stmt
s t : State
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t → (Stmt.ifThenElse B S T, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | { intro h
cases h with
| if_true _ _ _ _ _ hB hS =>
apply Or.intro_left
aesop
| if_false _ _ _ _ _ hB hT =>
apply Or.intro_right
aesop } | case mp
B : State → Prop
S T : Stmt
s t : State
⊢ (Stmt.ifThenElse B S T, s) ⟹ t → B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t
case mpr
B : State → Prop
S T : Stmt
s t : State
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t → (Stmt.ifThenElse B S T, s) ⟹ t | case mpr
B : State → Prop
S T : Stmt
s t : State
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t → (Stmt.ifThenElse B S T, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | { intro h
cases h with
| inl h =>
cases h with
| intro hB hS =>
apply BigStep.if_true <;>
assumption
| inr h =>
cases h with
| intro hB hT =>
apply BigStep.if_false <;>
assumption } | case mpr
B : State → Prop
S T : Stmt
s t : State
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t → (Stmt.ifThenElse B S T, s) ⟹ t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | intro h | case mp
B : State → Prop
S T : Stmt
s t : State
⊢ (Stmt.ifThenElse B S T, s) ⟹ t → B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t | case mp
B : State → Prop
S T : Stmt
s t : State
h : (Stmt.ifThenElse B S T, s) ⟹ t
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | cases h with
| if_true _ _ _ _ _ hB hS =>
apply Or.intro_left
aesop
| if_false _ _ _ _ _ hB hT =>
apply Or.intro_right
aesop | case mp
B : State → Prop
S T : Stmt
s t : State
h : (Stmt.ifThenElse B S T, s) ⟹ t
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | apply Or.intro_left | case mp.if_true
B : State → Prop
S T : Stmt
s t : State
hB : B s
hS : (S, s) ⟹ t
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t | case mp.if_true.h
B : State → Prop
S T : Stmt
s t : State
hB : B s
hS : (S, s) ⟹ t
⊢ B s ∧ (S, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | aesop | case mp.if_true.h
B : State → Prop
S T : Stmt
s t : State
hB : B s
hS : (S, s) ⟹ t
⊢ B s ∧ (S, s) ⟹ t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | apply Or.intro_right | case mp.if_false
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hT : (T, s) ⟹ t
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t | case mp.if_false.h
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hT : (T, s) ⟹ t
⊢ ¬B s ∧ (T, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | aesop | case mp.if_false.h
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hT : (T, s) ⟹ t
⊢ ¬B s ∧ (T, s) ⟹ t | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | intro h | case mpr
B : State → Prop
S T : Stmt
s t : State
⊢ B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t → (Stmt.ifThenElse B S T, s) ⟹ t | case mpr
B : State → Prop
S T : Stmt
s t : State
h : B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t
⊢ (Stmt.ifThenElse B S T, s) ⟹ t |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.BigStep_if_Iff | [316, 9] | [341, 25] | cases h with
| inl h =>
cases h with
| intro hB hS =>
apply BigStep.if_true <;>
assumption
| inr h =>
cases h with
| intro hB hT =>
apply BigStep.if_false <;>
assumption | case mpr
B : State → Prop
S T : Stmt
s t : State
h : B s ∧ (S, s) ⟹ t ∨ ¬B s ∧ (T, s) ⟹ t
⊢ (Stmt.ifThenElse B S T, s) ⟹ t | no goals |
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