# Theory Hoare

theory Hoare
imports Big_Step
```(* Author: Tobias Nipkow *)

section "Hoare Logic"

theory Hoare imports Big_Step begin

subsection "Hoare Logic for Partial Correctness"

type_synonym assn = "state ⇒ bool"

definition
hoare_valid :: "assn ⇒ com ⇒ assn ⇒ bool" ("⊨ {(1_)}/ (_)/ {(1_)}" 50) where
"⊨ {P}c{Q} = (∀s t. P s ∧ (c,s) ⇒ t  ⟶  Q t)"

abbreviation state_subst :: "state ⇒ aexp ⇒ vname ⇒ state"
("_[_'/_]" [1000,0,0] 999)
where "s[a/x] == s(x := aval a s)"

inductive
hoare :: "assn ⇒ com ⇒ assn ⇒ bool" ("⊢ ({(1_)}/ (_)/ {(1_)})" 50)
where
Skip: "⊢ {P} SKIP {P}"  |

Assign:  "⊢ {λs. P(s[a/x])} x::=a {P}"  |

Seq: "⟦ ⊢ {P} c⇩1 {Q};  ⊢ {Q} c⇩2 {R} ⟧
⟹ ⊢ {P} c⇩1;;c⇩2 {R}"  |

If: "⟦ ⊢ {λs. P s ∧ bval b s} c⇩1 {Q};  ⊢ {λs. P s ∧ ¬ bval b s} c⇩2 {Q} ⟧
⟹ ⊢ {P} IF b THEN c⇩1 ELSE c⇩2 {Q}"  |

While: "⊢ {λs. P s ∧ bval b s} c {P} ⟹
⊢ {P} WHILE b DO c {λs. P s ∧ ¬ bval b s}"  |

conseq: "⟦ ∀s. P' s ⟶ P s;  ⊢ {P} c {Q};  ∀s. Q s ⟶ Q' s ⟧
⟹ ⊢ {P'} c {Q'}"

lemmas [simp] = hoare.Skip hoare.Assign hoare.Seq If

lemmas [intro!] = hoare.Skip hoare.Assign hoare.Seq hoare.If

lemma strengthen_pre:
"⟦ ∀s. P' s ⟶ P s;  ⊢ {P} c {Q} ⟧ ⟹ ⊢ {P'} c {Q}"
by (blast intro: conseq)

lemma weaken_post:
"⟦ ⊢ {P} c {Q};  ∀s. Q s ⟶ Q' s ⟧ ⟹  ⊢ {P} c {Q'}"
by (blast intro: conseq)

text‹The assignment and While rule are awkward to use in actual proofs
because their pre and postcondition are of a very special form and the actual
goal would have to match this form exactly. Therefore we derive two variants
with arbitrary pre and postconditions.›

lemma Assign': "∀s. P s ⟶ Q(s[a/x]) ⟹ ⊢ {P} x ::= a {Q}"
by (simp add: strengthen_pre[OF _ Assign])

lemma While':
assumes "⊢ {λs. P s ∧ bval b s} c {P}" and "∀s. P s ∧ ¬ bval b s ⟶ Q s"
shows "⊢ {P} WHILE b DO c {Q}"
by(rule weaken_post[OF While[OF assms(1)] assms(2)])

end
```