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stringlengths 6
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https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | rfl | case assign.assign
Ss Ll : Stmt × State
x : String
a : State → ℕ
s : State
⊢ (Stmt.skip, State.update x (a s) s) = (Stmt.skip, State.update x (a s) s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | cases hr with
| seq_step S S₂ _ _ s₂ hS₂ =>
have hSs₁₂ :=
ih hS₂
aesop
| seq_skip => cases hS₁ | case seq_step
Ss Ll : Stmt × State
S S₁ T : Stmt
s s₁ : State
hS₁ : (S, s) ⇒ (S₁, s₁)
ih : ∀ {Rr : Stmt × State}, (S, s) ⇒ Rr → (S₁, s₁) = Rr
Rr : Stmt × State
hr : (S; T, s) ⇒ Rr
⊢ (S₁; T, s₁) = Rr | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | have hSs₁₂ :=
ih hS₂ | case seq_step.seq_step
Ss Ll : Stmt × State
S S₁ T : Stmt
s s₁ : State
hS₁ : (S, s) ⇒ (S₁, s₁)
ih : ∀ {Rr : Stmt × State}, (S, s) ⇒ Rr → (S₁, s₁) = Rr
S₂ : Stmt
s₂ : State
hS₂ : (S, s) ⇒ (S₂, s₂)
⊢ (S₁; T, s₁) = (S₂; T, s₂) | case seq_step.seq_step
Ss Ll : Stmt × State
S S₁ T : Stmt
s s₁ : State
hS₁ : (S, s) ⇒ (S₁, s₁)
ih : ∀ {Rr : Stmt × State}, (S, s) ⇒ Rr → (S₁, s₁) = Rr
S₂ : Stmt
s₂ : State
hS₂ : (S, s) ⇒ (S₂, s₂)
hSs₁₂ : (S₁, s₁) = (S₂, s₂)
⊢ (S₁; T, s₁) = (S₂; T, s₂) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | aesop | case seq_step.seq_step
Ss Ll : Stmt × State
S S₁ T : Stmt
s s₁ : State
hS₁ : (S, s) ⇒ (S₁, s₁)
ih : ∀ {Rr : Stmt × State}, (S, s) ⇒ Rr → (S₁, s₁) = Rr
S₂ : Stmt
s₂ : State
hS₂ : (S, s) ⇒ (S₂, s₂)
hSs₁₂ : (S₁, s₁) = (S₂, s₂)
⊢ (S₁; T, s₁) = (S₂; T, s₂) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | cases hS₁ | case seq_step.seq_skip
Ss Ll : Stmt × State
S₁ T : Stmt
s s₁ : State
hS₁ : (Stmt.skip, s) ⇒ (S₁, s₁)
ih : ∀ {Rr : Stmt × State}, (Stmt.skip, s) ⇒ Rr → (S₁, s₁) = Rr
⊢ (S₁; T, s₁) = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | cases hr with
| seq_step _ S _ _ s' hskip => cases hskip
| seq_skip => rfl | case seq_skip
Ss Ll : Stmt × State
T : Stmt
s : State
Rr : Stmt × State
hr : (Stmt.skip; T, s) ⇒ Rr
⊢ (T, s) = Rr | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | cases hskip | case seq_skip.seq_step
Ss Ll : Stmt × State
T : Stmt
s : State
S : Stmt
s' : State
hskip : (Stmt.skip, s) ⇒ (S, s')
⊢ (T, s) = (S; T, s') | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | rfl | case seq_skip.seq_skip
Ss Ll : Stmt × State
T : Stmt
s : State
⊢ (T, s) = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | cases hr with
| if_true => rfl
| if_false => aesop | case if_true
Ss Ll : Stmt × State
B : State → Prop
S T : Stmt
s : State
hB : B s
Rr : Stmt × State
hr : (Stmt.ifThenElse B S T, s) ⇒ Rr
⊢ (S, s) = Rr | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | rfl | case if_true.if_true
Ss Ll : Stmt × State
B : State → Prop
S T : Stmt
s : State
hB hcond✝ : B s
⊢ (S, s) = (S, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | aesop | case if_true.if_false
Ss Ll : Stmt × State
B : State → Prop
S T : Stmt
s : State
hB : B s
hcond✝ : ¬B s
⊢ (S, s) = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | cases hr with
| if_true => aesop
| if_false => rfl | case if_false
Ss Ll : Stmt × State
B : State → Prop
S T : Stmt
s : State
hB : ¬B s
Rr : Stmt × State
hr : (Stmt.ifThenElse B S T, s) ⇒ Rr
⊢ (T, s) = Rr | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | aesop | case if_false.if_true
Ss Ll : Stmt × State
B : State → Prop
S T : Stmt
s : State
hB : ¬B s
hcond✝ : B s
⊢ (T, s) = (S, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | rfl | case if_false.if_false
Ss Ll : Stmt × State
B : State → Prop
S T : Stmt
s : State
hB hcond✝ : ¬B s
⊢ (T, s) = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | cases hr with
| whileDo => rfl | case whileDo
Ss Ll : Stmt × State
B : State → Prop
S : Stmt
s : State
Rr : Stmt × State
hr : (Stmt.whileDo B S, s) ⇒ Rr
⊢ (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) = Rr | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_deterministic | [562, 1] | [592, 23] | rfl | case whileDo.whileDo
Ss Ll : Stmt × State
B : State → Prop
S : Stmt
s : State
⊢ (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) = (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_skip | [597, 1] | [602, 12] | intro h | S : Stmt
s t : State
⊢ ¬(Stmt.skip, s) ⇒ (S, t) | S : Stmt
s t : State
h : (Stmt.skip, s) ⇒ (S, t)
⊢ False |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_skip | [597, 1] | [602, 12] | cases h | S : Stmt
s t : State
h : (Stmt.skip, s) ⇒ (S, t)
⊢ False | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | apply Iff.intro | S T : Stmt
s : State
Ut : Stmt × State
⊢ (S; T, s) ⇒ Ut ↔ (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) | case mp
S T : Stmt
s : State
Ut : Stmt × State
⊢ (S; T, s) ⇒ Ut → (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s)
case mpr
S T : Stmt
s : State
Ut : Stmt × State
⊢ (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) → (S; T, s) ⇒ Ut |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | { intro hST
cases hST with
| seq_step _ S' _ _ s' hS =>
apply Or.intro_left
apply Exists.intro S'
apply Exists.intro s'
aesop
| seq_skip =>
apply Or.intro_right
aesop } | case mp
S T : Stmt
s : State
Ut : Stmt × State
⊢ (S; T, s) ⇒ Ut → (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s)
case mpr
S T : Stmt
s : State
Ut : Stmt × State
⊢ (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) → (S; T, s) ⇒ Ut | case mpr
S T : Stmt
s : State
Ut : Stmt × State
⊢ (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) → (S; T, s) ⇒ Ut |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | {
intro hor
cases hor with
| inl hex =>
cases hex with
| intro S' hex' =>
cases hex' with
| intro s' hand =>
cases hand with
| intro hS hUt =>
rw [hUt]
apply SmallStep.seq_step
assumption
| inr hand =>
cases hand with
| intro hS hUt =>
rw [hS, hUt]
apply SmallStep.seq_skip } | case mpr
S T : Stmt
s : State
Ut : Stmt × State
⊢ (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) → (S; T, s) ⇒ Ut | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | intro hST | case mp
S T : Stmt
s : State
Ut : Stmt × State
⊢ (S; T, s) ⇒ Ut → (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) | case mp
S T : Stmt
s : State
Ut : Stmt × State
hST : (S; T, s) ⇒ Ut
⊢ (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | cases hST with
| seq_step _ S' _ _ s' hS =>
apply Or.intro_left
apply Exists.intro S'
apply Exists.intro s'
aesop
| seq_skip =>
apply Or.intro_right
aesop | case mp
S T : Stmt
s : State
Ut : Stmt × State
hST : (S; T, s) ⇒ Ut
⊢ (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | apply Or.intro_left | case mp.seq_step
S T : Stmt
s : State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
⊢ (∃ S'_1 t, (S, s) ⇒ (S'_1, t) ∧ (S'; T, s') = (S'_1; T, t)) ∨ S = Stmt.skip ∧ (S'; T, s') = (T, s) | case mp.seq_step.h
S T : Stmt
s : State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
⊢ ∃ S'_1 t, (S, s) ⇒ (S'_1, t) ∧ (S'; T, s') = (S'_1; T, t) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | apply Exists.intro S' | case mp.seq_step.h
S T : Stmt
s : State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
⊢ ∃ S'_1 t, (S, s) ⇒ (S'_1, t) ∧ (S'; T, s') = (S'_1; T, t) | case mp.seq_step.h
S T : Stmt
s : State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
⊢ ∃ t, (S, s) ⇒ (S', t) ∧ (S'; T, s') = (S'; T, t) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | apply Exists.intro s' | case mp.seq_step.h
S T : Stmt
s : State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
⊢ ∃ t, (S, s) ⇒ (S', t) ∧ (S'; T, s') = (S'; T, t) | case mp.seq_step.h
S T : Stmt
s : State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
⊢ (S, s) ⇒ (S', s') ∧ (S'; T, s') = (S'; T, s') |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | aesop | case mp.seq_step.h
S T : Stmt
s : State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
⊢ (S, s) ⇒ (S', s') ∧ (S'; T, s') = (S'; T, s') | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | apply Or.intro_right | case mp.seq_skip
T : Stmt
s : State
⊢ (∃ S' t, (Stmt.skip, s) ⇒ (S', t) ∧ (T, s) = (S'; T, t)) ∨ Stmt.skip = Stmt.skip ∧ (T, s) = (T, s) | case mp.seq_skip.h
T : Stmt
s : State
⊢ Stmt.skip = Stmt.skip ∧ (T, s) = (T, s) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | aesop | case mp.seq_skip.h
T : Stmt
s : State
⊢ Stmt.skip = Stmt.skip ∧ (T, s) = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | intro hor | case mpr
S T : Stmt
s : State
Ut : Stmt × State
⊢ (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s) → (S; T, s) ⇒ Ut | case mpr
S T : Stmt
s : State
Ut : Stmt × State
hor : (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s)
⊢ (S; T, s) ⇒ Ut |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | cases hor with
| inl hex =>
cases hex with
| intro S' hex' =>
cases hex' with
| intro s' hand =>
cases hand with
| intro hS hUt =>
rw [hUt]
apply SmallStep.seq_step
assumption
| inr hand =>
cases hand with
| intro hS hUt =>
rw [hS, hUt]
apply SmallStep.seq_skip | case mpr
S T : Stmt
s : State
Ut : Stmt × State
hor : (∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)) ∨ S = Stmt.skip ∧ Ut = (T, s)
⊢ (S; T, s) ⇒ Ut | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | cases hex with
| intro S' hex' =>
cases hex' with
| intro s' hand =>
cases hand with
| intro hS hUt =>
rw [hUt]
apply SmallStep.seq_step
assumption | case mpr.inl
S T : Stmt
s : State
Ut : Stmt × State
hex : ∃ S' t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)
⊢ (S; T, s) ⇒ Ut | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | cases hex' with
| intro s' hand =>
cases hand with
| intro hS hUt =>
rw [hUt]
apply SmallStep.seq_step
assumption | case mpr.inl.intro
S T : Stmt
s : State
Ut : Stmt × State
S' : Stmt
hex' : ∃ t, (S, s) ⇒ (S', t) ∧ Ut = (S'; T, t)
⊢ (S; T, s) ⇒ Ut | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | cases hand with
| intro hS hUt =>
rw [hUt]
apply SmallStep.seq_step
assumption | case mpr.inl.intro.intro
S T : Stmt
s : State
Ut : Stmt × State
S' : Stmt
s' : State
hand : (S, s) ⇒ (S', s') ∧ Ut = (S'; T, s')
⊢ (S; T, s) ⇒ Ut | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | rw [hUt] | case mpr.inl.intro.intro.intro
S T : Stmt
s : State
Ut : Stmt × State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
hUt : Ut = (S'; T, s')
⊢ (S; T, s) ⇒ Ut | case mpr.inl.intro.intro.intro
S T : Stmt
s : State
Ut : Stmt × State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
hUt : Ut = (S'; T, s')
⊢ (S; T, s) ⇒ (S'; T, s') |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | apply SmallStep.seq_step | case mpr.inl.intro.intro.intro
S T : Stmt
s : State
Ut : Stmt × State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
hUt : Ut = (S'; T, s')
⊢ (S; T, s) ⇒ (S'; T, s') | case mpr.inl.intro.intro.intro.hS
S T : Stmt
s : State
Ut : Stmt × State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
hUt : Ut = (S'; T, s')
⊢ (S, s) ⇒ (S', s') |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | assumption | case mpr.inl.intro.intro.intro.hS
S T : Stmt
s : State
Ut : Stmt × State
S' : Stmt
s' : State
hS : (S, s) ⇒ (S', s')
hUt : Ut = (S'; T, s')
⊢ (S, s) ⇒ (S', s') | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | cases hand with
| intro hS hUt =>
rw [hS, hUt]
apply SmallStep.seq_skip | case mpr.inr
S T : Stmt
s : State
Ut : Stmt × State
hand : S = Stmt.skip ∧ Ut = (T, s)
⊢ (S; T, s) ⇒ Ut | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | rw [hS, hUt] | case mpr.inr.intro
S T : Stmt
s : State
Ut : Stmt × State
hS : S = Stmt.skip
hUt : Ut = (T, s)
⊢ (S; T, s) ⇒ Ut | case mpr.inr.intro
S T : Stmt
s : State
Ut : Stmt × State
hS : S = Stmt.skip
hUt : Ut = (T, s)
⊢ (Stmt.skip; T, s) ⇒ (T, s) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_seq_Iff | [604, 9] | [638, 37] | apply SmallStep.seq_skip | case mpr.inr.intro
S T : Stmt
s : State
Ut : Stmt × State
hS : S = Stmt.skip
hUt : Ut = (T, s)
⊢ (Stmt.skip; T, s) ⇒ (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | apply Iff.intro | B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us ↔ B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) | case mp
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us → B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s)
case mpr
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) → (Stmt.ifThenElse B S T, s) ⇒ Us |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | { intro h
cases h with
| if_true _ _ _ _ hB => aesop
| if_false _ _ _ _ hB => aesop } | case mp
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us → B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s)
case mpr
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) → (Stmt.ifThenElse B S T, s) ⇒ Us | case mpr
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) → (Stmt.ifThenElse B S T, s) ⇒ Us |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | { intro hor
cases hor with
| inl hand =>
cases hand with
| intro hB hUs =>
rw [hUs]
apply SmallStep.if_true
assumption
| inr hand =>
cases hand with
| intro hB hUs =>
rw [hUs]
apply SmallStep.if_false
assumption } | case mpr
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) → (Stmt.ifThenElse B S T, s) ⇒ Us | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | intro h | case mp
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us → B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) | case mp
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
h : (Stmt.ifThenElse B S T, s) ⇒ Us
⊢ B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | cases h with
| if_true _ _ _ _ hB => aesop
| if_false _ _ _ _ hB => aesop | case mp
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
h : (Stmt.ifThenElse B S T, s) ⇒ Us
⊢ B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | aesop | case mp.if_true
B : State → Prop
S T : Stmt
s : State
hB : B s
⊢ B s ∧ (S, s) = (S, s) ∨ ¬B s ∧ (S, s) = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | aesop | case mp.if_false
B : State → Prop
S T : Stmt
s : State
hB : ¬B s
⊢ B s ∧ (T, s) = (S, s) ∨ ¬B s ∧ (T, s) = (T, s) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | intro hor | case mpr
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
⊢ B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s) → (Stmt.ifThenElse B S T, s) ⇒ Us | case mpr
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hor : B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | cases hor with
| inl hand =>
cases hand with
| intro hB hUs =>
rw [hUs]
apply SmallStep.if_true
assumption
| inr hand =>
cases hand with
| intro hB hUs =>
rw [hUs]
apply SmallStep.if_false
assumption | case mpr
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hor : B s ∧ Us = (S, s) ∨ ¬B s ∧ Us = (T, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | cases hand with
| intro hB hUs =>
rw [hUs]
apply SmallStep.if_true
assumption | case mpr.inl
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hand : B s ∧ Us = (S, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | rw [hUs] | case mpr.inl.intro
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : B s
hUs : Us = (S, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us | case mpr.inl.intro
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : B s
hUs : Us = (S, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ (S, s) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | apply SmallStep.if_true | case mpr.inl.intro
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : B s
hUs : Us = (S, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ (S, s) | case mpr.inl.intro.hcond
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : B s
hUs : Us = (S, s)
⊢ B s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | assumption | case mpr.inl.intro.hcond
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : B s
hUs : Us = (S, s)
⊢ B s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | cases hand with
| intro hB hUs =>
rw [hUs]
apply SmallStep.if_false
assumption | case mpr.inr
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hand : ¬B s ∧ Us = (T, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | rw [hUs] | case mpr.inr.intro
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : ¬B s
hUs : Us = (T, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ Us | case mpr.inr.intro
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : ¬B s
hUs : Us = (T, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ (T, s) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | apply SmallStep.if_false | case mpr.inr.intro
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : ¬B s
hUs : Us = (T, s)
⊢ (Stmt.ifThenElse B S T, s) ⇒ (T, s) | case mpr.inr.intro.hcond
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : ¬B s
hUs : Us = (T, s)
⊢ ¬B s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.SmallStep_if_Iff | [640, 9] | [663, 23] | assumption | case mpr.inr.intro.hcond
B : State → Prop
S T : Stmt
s : State
Us : Stmt × State
hB : ¬B s
hUs : Us = (T, s)
⊢ ¬B s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_seq | [675, 1] | [686, 19] | apply RTC.lift (fun Ss ↦ (Prod.fst Ss; T, Prod.snd Ss)) _ h | S T : Stmt
s u : State
h : RTC SmallStep (S, s) (Stmt.skip, u)
⊢ RTC SmallStep (S; T, s) (Stmt.skip; T, u) | S T : Stmt
s u : State
h : RTC SmallStep (S, s) (Stmt.skip, u)
⊢ ∀ (a b : Stmt × State), a ⇒ b → (fun Ss => (Ss.fst; T, Ss.snd)) a ⇒ (fun Ss => (Ss.fst; T, Ss.snd)) b |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_seq | [675, 1] | [686, 19] | intro Ss Ss' hrtc | S T : Stmt
s u : State
h : RTC SmallStep (S, s) (Stmt.skip, u)
⊢ ∀ (a b : Stmt × State), a ⇒ b → (fun Ss => (Ss.fst; T, Ss.snd)) a ⇒ (fun Ss => (Ss.fst; T, Ss.snd)) b | S T : Stmt
s u : State
h : RTC SmallStep (S, s) (Stmt.skip, u)
Ss Ss' : Stmt × State
hrtc : Ss ⇒ Ss'
⊢ (fun Ss => (Ss.fst; T, Ss.snd)) Ss ⇒ (fun Ss => (Ss.fst; T, Ss.snd)) Ss' |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_seq | [675, 1] | [686, 19] | cases Ss with
| mk S s =>
cases Ss' with
| mk S' s' =>
apply SmallStep.seq_step
assumption | S T : Stmt
s u : State
h : RTC SmallStep (S, s) (Stmt.skip, u)
Ss Ss' : Stmt × State
hrtc : Ss ⇒ Ss'
⊢ (fun Ss => (Ss.fst; T, Ss.snd)) Ss ⇒ (fun Ss => (Ss.fst; T, Ss.snd)) Ss' | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_seq | [675, 1] | [686, 19] | cases Ss' with
| mk S' s' =>
apply SmallStep.seq_step
assumption | case mk
S✝ T : Stmt
s✝ u : State
h : RTC SmallStep (S✝, s✝) (Stmt.skip, u)
Ss' : Stmt × State
S : Stmt
s : State
hrtc : (S, s) ⇒ Ss'
⊢ (fun Ss => (Ss.fst; T, Ss.snd)) (S, s) ⇒ (fun Ss => (Ss.fst; T, Ss.snd)) Ss' | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_seq | [675, 1] | [686, 19] | apply SmallStep.seq_step | case mk.mk
S✝ T : Stmt
s✝ u : State
h : RTC SmallStep (S✝, s✝) (Stmt.skip, u)
S : Stmt
s : State
S' : Stmt
s' : State
hrtc : (S, s) ⇒ (S', s')
⊢ (fun Ss => (Ss.fst; T, Ss.snd)) (S, s) ⇒ (fun Ss => (Ss.fst; T, Ss.snd)) (S', s') | case mk.mk.hS
S✝ T : Stmt
s✝ u : State
h : RTC SmallStep (S✝, s✝) (Stmt.skip, u)
S : Stmt
s : State
S' : Stmt
s' : State
hrtc : (S, s) ⇒ (S', s')
⊢ ((S, s).fst, (S, s).snd) ⇒ ((S', s').fst, (S', s').snd) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_seq | [675, 1] | [686, 19] | assumption | case mk.mk.hS
S✝ T : Stmt
s✝ u : State
h : RTC SmallStep (S✝, s✝) (Stmt.skip, u)
S : Stmt
s : State
S' : Stmt
s' : State
hrtc : (S, s) ⇒ (S', s')
⊢ ((S, s).fst, (S, s).snd) ⇒ ((S', s').fst, (S', s').snd) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | induction hS with
| skip => exact RTC.refl
| assign =>
apply RTC.single
apply SmallStep.assign
| seq S T s t u hS hT ihS ihT =>
apply RTC.trans
{ exact RTC_SmallStep_seq ihS }
{ apply RTC.head
apply SmallStep.seq_skip
assumption }
| if_true B S T s t hB hst ih =>
apply RTC.head
{ apply SmallStep.if_true
assumption }
{ assumption }
| if_false B S T s t hB hst ih =>
apply RTC.head
{ apply SmallStep.if_false
assumption }
{ assumption }
| while_true B S s t u hB hS hw ihS ihw =>
apply RTC.head
{ apply SmallStep.whileDo }
{ apply RTC.head
{ apply SmallStep.if_true
assumption }
{ apply RTC.trans
{ exact RTC_SmallStep_seq ihS }
{ apply RTC.head
apply SmallStep.seq_skip
assumption } } }
| while_false B S s hB =>
apply RTC.tail
apply RTC.single
apply SmallStep.whileDo
apply SmallStep.if_false
assumption | Ss : Stmt × State
t : State
hS : Ss ⟹ t
⊢ RTC SmallStep Ss (Stmt.skip, t) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | exact RTC.refl | case skip
Ss : Stmt × State
t s✝ : State
⊢ RTC SmallStep (Stmt.skip, s✝) (Stmt.skip, s✝) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.single | case assign
Ss : Stmt × State
t : State
x✝ : String
a✝ : State → ℕ
s✝ : State
⊢ RTC SmallStep (Stmt.assign x✝ a✝, s✝) (Stmt.skip, State.update x✝ (a✝ s✝) s✝) | case assign.hab
Ss : Stmt × State
t : State
x✝ : String
a✝ : State → ℕ
s✝ : State
⊢ (Stmt.assign x✝ a✝, s✝) ⇒ (Stmt.skip, State.update x✝ (a✝ s✝) s✝) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply SmallStep.assign | case assign.hab
Ss : Stmt × State
t : State
x✝ : String
a✝ : State → ℕ
s✝ : State
⊢ (Stmt.assign x✝ a✝, s✝) ⇒ (Stmt.skip, State.update x✝ (a✝ s✝) s✝) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.trans | case seq
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; T, s) (Stmt.skip, u) | case seq.hab
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; T, s) ?seq.b
case seq.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep ?seq.b (Stmt.skip, u)
case seq.b
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ Stmt × State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { exact RTC_SmallStep_seq ihS } | case seq.hab
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; T, s) ?seq.b
case seq.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep ?seq.b (Stmt.skip, u)
case seq.b
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ Stmt × State | case seq.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.skip; T, t) (Stmt.skip, u) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { apply RTC.head
apply SmallStep.seq_skip
assumption } | case seq.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.skip; T, t) (Stmt.skip, u) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | exact RTC_SmallStep_seq ihS | case seq.hab
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; T, s) ?seq.b | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.head | case seq.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.skip; T, t) (Stmt.skip, u) | case seq.hbc.hab
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ (Stmt.skip; T, t) ⇒ ?seq.hbc.b
case seq.hbc.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep ?seq.hbc.b (Stmt.skip, u)
case seq.hbc.b
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ Stmt × State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply SmallStep.seq_skip | case seq.hbc.hab
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ (Stmt.skip; T, t) ⇒ ?seq.hbc.b
case seq.hbc.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep ?seq.hbc.b (Stmt.skip, u)
case seq.hbc.b
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ Stmt × State | case seq.hbc.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (T, t) (Stmt.skip, u) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | assumption | case seq.hbc.hbc
Ss : Stmt × State
t✝ : State
S T : Stmt
s t u : State
hS : (S, s) ⟹ t
hT : (T, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihT : RTC SmallStep (T, t) (Stmt.skip, u)
⊢ RTC SmallStep (T, t) (Stmt.skip, u) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.head | case if_true
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ RTC SmallStep (Stmt.ifThenElse B S T, s) (Stmt.skip, t) | case if_true.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ (Stmt.ifThenElse B S T, s) ⇒ ?if_true.b
case if_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ RTC SmallStep ?if_true.b (Stmt.skip, t)
case if_true.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ Stmt × State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { apply SmallStep.if_true
assumption } | case if_true.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ (Stmt.ifThenElse B S T, s) ⇒ ?if_true.b
case if_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ RTC SmallStep ?if_true.b (Stmt.skip, t)
case if_true.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ Stmt × State | case if_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ RTC SmallStep (S, s) (Stmt.skip, t) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { assumption } | case if_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ RTC SmallStep (S, s) (Stmt.skip, t) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply SmallStep.if_true | case if_true.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ (Stmt.ifThenElse B S T, s) ⇒ ?if_true.b | case if_true.hab.hcond
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ B s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | assumption | case if_true.hab.hcond
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ B s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | assumption | case if_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : B s
hst : (S, s) ⟹ t
ih : RTC SmallStep (S, s) (Stmt.skip, t)
⊢ RTC SmallStep (S, s) (Stmt.skip, t) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.head | case if_false
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ RTC SmallStep (Stmt.ifThenElse B S T, s) (Stmt.skip, t) | case if_false.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ (Stmt.ifThenElse B S T, s) ⇒ ?if_false.b
case if_false.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ RTC SmallStep ?if_false.b (Stmt.skip, t)
case if_false.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ Stmt × State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { apply SmallStep.if_false
assumption } | case if_false.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ (Stmt.ifThenElse B S T, s) ⇒ ?if_false.b
case if_false.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ RTC SmallStep ?if_false.b (Stmt.skip, t)
case if_false.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ Stmt × State | case if_false.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ RTC SmallStep (T, s) (Stmt.skip, t) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { assumption } | case if_false.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ RTC SmallStep (T, s) (Stmt.skip, t) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply SmallStep.if_false | case if_false.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ (Stmt.ifThenElse B S T, s) ⇒ ?if_false.b | case if_false.hab.hcond
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ ¬B s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | assumption | case if_false.hab.hcond
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ ¬B s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | assumption | case if_false.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S T : Stmt
s t : State
hB : ¬B s
hst : (T, s) ⟹ t
ih : RTC SmallStep (T, s) (Stmt.skip, t)
⊢ RTC SmallStep (T, s) (Stmt.skip, t) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.head | case while_true
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.whileDo B S, s) (Stmt.skip, u) | case while_true.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ (Stmt.whileDo B S, s) ⇒ ?while_true.b
case while_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep ?while_true.b (Stmt.skip, u)
case while_true.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ Stmt × State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { apply SmallStep.whileDo } | case while_true.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ (Stmt.whileDo B S, s) ⇒ ?while_true.b
case while_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep ?while_true.b (Stmt.skip, u)
case while_true.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ Stmt × State | case while_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) (Stmt.skip, u) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { apply RTC.head
{ apply SmallStep.if_true
assumption }
{ apply RTC.trans
{ exact RTC_SmallStep_seq ihS }
{ apply RTC.head
apply SmallStep.seq_skip
assumption } } } | case while_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) (Stmt.skip, u) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply SmallStep.whileDo | case while_true.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ (Stmt.whileDo B S, s) ⇒ ?while_true.b | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.head | case while_true.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) (Stmt.skip, u) | case while_true.hbc.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) ⇒ ?while_true.hbc.b
case while_true.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep ?while_true.hbc.b (Stmt.skip, u)
case while_true.hbc.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ Stmt × State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { apply SmallStep.if_true
assumption } | case while_true.hbc.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) ⇒ ?while_true.hbc.b
case while_true.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep ?while_true.hbc.b (Stmt.skip, u)
case while_true.hbc.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ Stmt × State | case while_true.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; Stmt.whileDo B S, s) (Stmt.skip, u) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { apply RTC.trans
{ exact RTC_SmallStep_seq ihS }
{ apply RTC.head
apply SmallStep.seq_skip
assumption } } | case while_true.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; Stmt.whileDo B S, s) (Stmt.skip, u) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply SmallStep.if_true | case while_true.hbc.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ (Stmt.ifThenElse B (S; Stmt.whileDo B S) Stmt.skip, s) ⇒ ?while_true.hbc.b | case while_true.hbc.hab.hcond
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ B s |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | assumption | case while_true.hbc.hab.hcond
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ B s | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.trans | case while_true.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; Stmt.whileDo B S, s) (Stmt.skip, u) | case while_true.hbc.hbc.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; Stmt.whileDo B S, s) ?while_true.hbc.hbc.b
case while_true.hbc.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep ?while_true.hbc.hbc.b (Stmt.skip, u)
case while_true.hbc.hbc.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ Stmt × State |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { exact RTC_SmallStep_seq ihS } | case while_true.hbc.hbc.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; Stmt.whileDo B S, s) ?while_true.hbc.hbc.b
case while_true.hbc.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep ?while_true.hbc.hbc.b (Stmt.skip, u)
case while_true.hbc.hbc.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ Stmt × State | case while_true.hbc.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.skip; Stmt.whileDo B S, t) (Stmt.skip, u) |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | { apply RTC.head
apply SmallStep.seq_skip
assumption } | case while_true.hbc.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.skip; Stmt.whileDo B S, t) (Stmt.skip, u) | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | exact RTC_SmallStep_seq ihS | case while_true.hbc.hbc.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (S; Stmt.whileDo B S, s) ?while_true.hbc.hbc.b | no goals |
https://github.com/BrownCS1951x/fpv2023.git | 9aaf6b5c454aa9a70fc4e6807adf3123b001ea66 | LoVe/Lectures/LoVe09_OperationalSemantics_Demo.lean | LoVe.RTC_SmallStep_of_BigStep | [688, 1] | [728, 17] | apply RTC.head | case while_true.hbc.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep (Stmt.skip; Stmt.whileDo B S, t) (Stmt.skip, u) | case while_true.hbc.hbc.hbc.hab
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ (Stmt.skip; Stmt.whileDo B S, t) ⇒ ?while_true.hbc.hbc.hbc.b
case while_true.hbc.hbc.hbc.hbc
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ RTC SmallStep ?while_true.hbc.hbc.hbc.b (Stmt.skip, u)
case while_true.hbc.hbc.hbc.b
Ss : Stmt × State
t✝ : State
B : State → Prop
S : Stmt
s t u : State
hB : B s
hS : (S, s) ⟹ t
hw : (Stmt.whileDo B S, t) ⟹ u
ihS : RTC SmallStep (S, s) (Stmt.skip, t)
ihw : RTC SmallStep (Stmt.whileDo B S, t) (Stmt.skip, u)
⊢ Stmt × State |
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