# Theory Sequence

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section ‹(Infinite) Sequences› theory Sequence imports Main begin text ‹ Lamport's Temporal Logic of Actions (TLA) is a linear-time temporal logic, and its semantics is defined over infinite sequence of states, which we simply represent by the type ‹'a seq›, defined as an abbreviation for the type ‹nat ⇒ 'a›, where ‹'a› is the type of sequence elements. This theory defines some useful notions about such sequences, and in particular concepts related to stuttering (finite repetitions of states), which are important for the semantics of TLA. We identify a finite sequence with an infinite sequence that ends in infinite stuttering. In this way, we avoid the complications of having to handle both finite and infinite sequences of states: see e.g. Devillers et al \cite{Devillers97} who discuss several variants of representing possibly infinite sequences in HOL, Isabelle and PVS. › type_synonym 'a seq = "nat ⇒ 'a" subsection "Some operators on sequences" text ‹Some general functions on sequences are provided› definition first :: "'a seq ⇒ 'a" where "first s ≡ s 0" definition second :: "('a seq) ⇒ 'a" where "second s ≡ s 1" definition suffix :: "'a seq ⇒ nat ⇒ 'a seq" (infixl "|⇩_{s}" 60) where "s |⇩_{s}i ≡ λ n. s (n+i)" definition tail :: "'a seq ⇒ 'a seq" where "tail s ≡ s |⇩_{s}1" definition app :: "'a ⇒ ('a seq) ⇒ ('a seq)" (infixl "##" 60) where "s ## σ ≡ λ n. if n=0 then s else σ (n - 1)" text ‹ ‹s |⇩_{s}i› returns the suffix of sequence @{term s} from index @{term i}. @{term first} returns the first element of a sequence while @{term second} returns the second element. @{term tail} returns the sequence starting at the second element. @{term "s ## σ"} prefixes the sequence @{term σ} by element @{term s}. › subsubsection "Properties of @{term first} and @{term second}" lemma first_tail_second: "first(tail s) = second s" by (simp add: first_def second_def tail_def suffix_def) subsubsection "Properties of @{term suffix}" lemma suffix_first: "first (s |⇩_{s}n) = s n" by (auto simp add: suffix_def first_def) lemma suffix_second: "second (s |⇩_{s}n) = s (Suc n)" by (auto simp add: suffix_def second_def) lemma suffix_plus: "s |⇩_{s}n |⇩_{s}m = s |⇩_{s}(m + n)" by (simp add: suffix_def add.assoc) lemma suffix_commute: "((s |⇩_{s}n) |⇩_{s}m) = ((s |⇩_{s}m) |⇩_{s}n)" by (simp add: suffix_plus add.commute) lemma suffix_plus_com: "s |⇩_{s}m |⇩_{s}n = s |⇩_{s}(m + n)" proof - have "s |⇩_{s}n |⇩_{s}m = s |⇩_{s}(m + n)" by (rule suffix_plus) thus "s |⇩_{s}m |⇩_{s}n = s |⇩_{s}(m + n)" by (simp add: suffix_commute) qed lemma suffix_zero[simp]: "s |⇩_{s}0 = s" by (simp add: suffix_def) lemma suffix_tail: "s |⇩_{s}1 = tail s" by (simp add: tail_def) lemma tail_suffix_suc: "s |⇩_{s}(Suc n) = tail (s |⇩_{s}n)" by (simp add: suffix_def tail_def) subsubsection "Properties of @{term app}" lemma seq_app_second: "(s ## σ) 1 = σ 0" by (simp add: app_def) lemma seq_app_first: "(s ## σ) 0 = s" by (simp add: app_def) lemma seq_app_first_tail: "(first s) ## (tail s) = s" proof (rule ext) fix x show "(first s ## tail s) x = s x" by (simp add: first_def app_def suffix_def tail_def) qed lemma seq_app_tail: "tail (x ## s) = s" by (simp add: app_def tail_def suffix_def) lemma seq_app_greater_than_zero: "n > 0 ⟹ (s ## σ) n = σ (n - 1)" by (simp add: app_def) subsection "Finite and Empty Sequences" text‹ We identify finite and empty sequences and prove lemmas about them. › definition fin :: "('a seq) ⇒ bool" where "fin s ≡ ∃ i. ∀ j ≥ i. s j = s i" abbreviation inf :: "('a seq) ⇒ bool" where "inf s ≡ ¬(fin s)" definition last :: "('a seq) ⇒ nat" where "last s ≡ LEAST i. (∀ j ≥ i. s j = s i)" definition laststate :: "('a seq) ⇒ 'a" where "laststate s ≡ s (last s)" definition emptyseq :: "('a seq) ⇒ bool" where "emptyseq ≡ λ s. ∀ i. s i = s 0" abbreviation notemptyseq :: "('a seq) ⇒ bool" where "notemptyseq s ≡ ¬(emptyseq s)" text ‹ Predicate @{term fin} holds if there is an element in the sequence such that all subsequent elements are identical, i.e. the sequence is finite. @{term "last s"} returns the smallest index from which on all elements of a finite sequence @{term s} are identical. Note that if ‹s› is not finite then an arbitrary number is returned. @{term laststate} returns the last element of a finite sequence. We assume that the sequence is finite when using @{term last} and @{term laststate}. Predicate @{term emptyseq} identifies empty sequences -- i.e. all states in the sequence are identical to the initial one, while @{term notemptyseq} holds if the given sequence is not empty. › subsubsection "Properties of @{term emptyseq}" lemma empty_is_finite: assumes "emptyseq s" shows "fin s" using assms by (auto simp: fin_def emptyseq_def) lemma empty_suffix_is_empty: assumes H: "emptyseq s" shows "emptyseq (s |⇩_{s}n)" proof (clarsimp simp: emptyseq_def) fix i from H have "(s |⇩_{s}n) i = s 0" by (simp add: emptyseq_def suffix_def) moreover from H have "(s |⇩_{s}n) 0 = s 0" by (simp add: emptyseq_def suffix_def) ultimately show "(s |⇩_{s}n) i = (s |⇩_{s}n) 0" by simp qed lemma suc_empty: assumes H1: "emptyseq (s |⇩_{s}m)" shows "emptyseq (s |⇩_{s}(Suc m))" proof - from H1 have "emptyseq ((s |⇩_{s}m) |⇩_{s}1)" by (rule empty_suffix_is_empty) thus ?thesis by (simp add: suffix_plus) qed lemma empty_suffix_exteq: assumes H:"emptyseq s" shows "(s |⇩_{s}n) m = s m" proof (unfold suffix_def) from H have "s (m+n) = s 0" by (simp add: emptyseq_def) moreover from H have "s m = s 0" by (simp add: emptyseq_def) ultimately show "s (m + n) = s m" by simp qed lemma empty_suffix_eq: assumes H: "emptyseq s" shows "(s |⇩_{s}n) = s" proof (rule ext) fix m from H show "(s |⇩_{s}n) m = s m" by (rule empty_suffix_exteq) qed lemma seq_empty_all: assumes H: "emptyseq s" shows "s i = s j" proof - from H have "s i = s 0" by (simp add: emptyseq_def) moreover from H have "s j = s 0" by (simp add: emptyseq_def) ultimately show ?thesis by simp qed subsubsection "Properties of @{term last} and @{term laststate}" lemma fin_stut_after_last: assumes H: "fin s" shows "∀j ≥ last s. s j = s (last s)" proof (clarify) fix j assume j: "j ≥ last s" from H obtain i where "∀j ≥ i. s j = s i" (is "?P i") by (auto simp: fin_def) hence "?P (last s)" unfolding last_def by (rule LeastI) with j show "s j = s (last s)" by blast qed subsection "Stuttering Invariance" text ‹ This subsection provides functions for removing stuttering steps of sequences, i.e. we formalise Lamports ‹♮› operator. Our formal definition is close to that of Wahab in the PVS prover. The key novelty with the @{term "Sequence"} theory, is the treatment of stuttering invariance, which enables verification of stuttering invariance of the operators derived using it. Such proofs require comparing sequences up to stuttering. Here, Lamport's \cite{Lamport94} method is used to mechanise the equality of sequences up to stuttering: he defines the ‹♮› operator, which collapses a sequence by removing all stuttering steps, except possibly infinite stuttering at the end of the sequence. These are left unchanged. › definition nonstutseq :: "('a seq) ⇒ bool" where "nonstutseq s ≡ ∀ i. s i = s (Suc i) ⟶ (∀ j > i. s i = s j)" definition stutstep :: "('a seq) ⇒ nat ⇒ bool" where "stutstep s n ≡ (s n = s (Suc n))" definition nextnat :: "('a seq) ⇒ nat" where "nextnat s ≡ if emptyseq s then 0 else LEAST i. s i ≠ s 0" definition nextsuffix :: "('a seq) ⇒ ('a seq)" where "nextsuffix s ≡ s |⇩_{s}(nextnat s)" fun "next" :: "nat ⇒ ('a seq) ⇒ ('a seq)" where "next 0 = id" | "next (Suc n) = nextsuffix o (next n)" definition collapse :: "('a seq) ⇒ ('a seq)" ("♮") where "♮ s ≡ λ n. (next n s) 0" text ‹ Predicate @{term nonstutseq} identifies sequences without any stuttering steps -- except possibly for infinite stuttering at the end. Further, @{term "stutstep s n"} is a predicate which holds if the element after @{term "s n"} is equal to @{term "s n"}, i.e. @{term "Suc n"} is a stuttering step. @{term "collapse s"} formalises Lamports @{term "♮"} operator. It returns the first state of the result of @{term "next n s"}. @{term "next n s"} finds suffix of the $n^{th}$ change. Hence the first element, which @{term "♮ s"} returns, is the state after the $n^{th}$ change. @{term "next n s"} is defined by primitive recursion on @{term "n"} using function composition of function @{term nextsuffix}. E.g. @{term "next 3 s"} equals @{term "nextsuffix (nextsuffix (nextsuffix s))"}. @{term "nextsuffix s"} returns the suffix of the sequence starting at the next changing state. It uses @{term "nextnat"} to obtain this. All the real computation is done in this function. Firstly, an empty sequence will obviously not contain any changes, and ‹0› is therefore returned. In this case @{term "nextsuffix"} behaves like the identify function. If the sequence is not empty then the smallest number @{term "i"} such that @{term "s i"} is different from the initial state is returned. This is achieved by @{term "Least"}. › subsubsection "Properties of @{term nonstutseq}" lemma seq_empty_is_nonstut: assumes H: "emptyseq s" shows "nonstutseq s" using H by (auto simp: nonstutseq_def seq_empty_all) lemma notempty_exist_nonstut: assumes H: "¬ emptyseq (s |⇩_{s}m)" shows "∃ i. s i ≠ s m ∧ i > m" using H proof (auto simp: emptyseq_def suffix_def) fix i assume i: "s (i + m) ≠ s m" hence "i ≠ 0" by (intro notI, simp) with i show ?thesis by auto qed subsubsection "Properties of @{term nextnat}" lemma nextnat_le_unch: assumes H: "n < nextnat s" shows "s n = s 0" proof (cases "emptyseq s") assume "emptyseq s" hence "nextnat s = 0" by (simp add: nextnat_def) with H show ?thesis by auto next assume "¬ emptyseq s" hence a1: "nextnat s = (LEAST i. s i ≠ s 0)" by (simp add: nextnat_def) show ?thesis proof (rule ccontr) assume a2: "s n ≠ s 0" (is "?P n") hence "(LEAST i. s i ≠ s 0) ≤ n" by (rule Least_le) hence "¬(n < (LEAST i. s i ≠ s 0))" by auto also from H a1 have "n < (LEAST i. s i ≠ s 0)" by simp ultimately show False by auto qed qed lemma stutnempty: assumes H: "¬ stutstep s n" shows "¬ emptyseq (s |⇩_{s}n)" proof (unfold emptyseq_def suffix_def) from H have "s (Suc n) ≠ s n" by (auto simp add: stutstep_def) hence "s (1+n) ≠ s (0+n)" by simp thus "¬(∀ i. s (i+n) = s (0+n))" by blast qed lemma notstutstep_nexnat1: assumes H: "¬ stutstep s n" shows "nextnat (s |⇩_{s}n) = 1" proof - from H have h': "nextnat (s |⇩_{s}n) = (LEAST i. (s |⇩_{s}n) i ≠ (s |⇩_{s}n) 0)" by (auto simp add: nextnat_def stutnempty) from H have "s (Suc n) ≠ s n" by (auto simp add: stutstep_def) hence "(s |⇩_{s}n) 1 ≠ (s |⇩_{s}n) 0" (is "?P 1") by (auto simp add: suffix_def) hence "Least ?P ≤ 1" by (rule Least_le) hence g1: "Least ?P = 0 ∨ Least ?P = 1" by auto with h' have g1': "nextnat (s |⇩_{s}n) = 0 ∨ nextnat (s |⇩_{s}n) = 1" by auto also have "nextnat (s |⇩_{s}n) ≠ 0" proof - from H have "¬ emptyseq (s |⇩_{s}n)" by (rule stutnempty) then obtain i where "(s |⇩_{s}n) i ≠ (s |⇩_{s}n) 0" by (auto simp add: emptyseq_def) hence "(s |⇩_{s}n) (LEAST i. (s |⇩_{s}n) i ≠ (s |⇩_{s}n) 0) ≠ (s |⇩_{s}n) 0" by (rule LeastI) with h' have g2: "(s |⇩_{s}n) (nextnat (s |⇩_{s}n)) ≠ (s |⇩_{s}n) 0" by auto show "(nextnat (s |⇩_{s}n)) ≠ 0" proof assume "(nextnat (s |⇩_{s}n)) = 0" with g2 show "False" by simp qed qed ultimately show "nextnat (s |⇩_{s}n) = 1" by auto qed lemma stutstep_notempty_notempty: assumes h1: "emptyseq (s |⇩_{s}Suc n)" (is "emptyseq ?sn") and h2: "stutstep s n" shows "emptyseq (s |⇩_{s}n)" (is "emptyseq ?s") proof (auto simp: emptyseq_def) fix k show "?s k = ?s 0" proof (cases k) assume "k = 0" thus ?thesis by simp next fix m assume k: "k = Suc m" hence "?s k = ?sn m" by (simp add: suffix_def) also from h1 have "... = ?sn 0" by (simp add: emptyseq_def) also from h2 have "... = s n" by (simp add: suffix_def stutstep_def) finally show ?thesis by (simp add: suffix_def) qed qed lemma stutstep_empty_suc: assumes "stutstep s n" shows "emptyseq (s |⇩_{s}Suc n) = emptyseq (s |⇩_{s}n)" using assms by (auto elim: stutstep_notempty_notempty suc_empty) lemma stutstep_notempty_sucnextnat: assumes h1: "¬ emptyseq (s |⇩_{s}n)" and h2: "stutstep s n" shows "(nextnat (s |⇩_{s}n)) = Suc (nextnat (s |⇩_{s}(Suc n)))" proof - from h2 have g1: "¬(s (0+n) ≠ s (Suc n))" (is "¬ ?P 0") by (auto simp add: stutstep_def) from h1 obtain i where "s (i+n) ≠ s n" by (auto simp: emptyseq_def suffix_def) with h2 have g2: "s (i+n) ≠ s (Suc n)" (is "?P i") by (simp add: stutstep_def) from g2 g1 have "(LEAST n. ?P n) = Suc (LEAST n. ?P (Suc n))" by (rule Least_Suc) from g2 g1 have "(LEAST i. s (i+n) ≠ s (Suc n)) = Suc (LEAST i. s ((Suc i)+n) ≠ s (Suc n))" by (rule Least_Suc) hence G1: "(LEAST i. s (i+n) ≠ s (Suc n)) = Suc (LEAST i. s (i+Suc n) ≠ s (Suc n))" by auto from h1 h2 have "¬ emptyseq (s |⇩_{s}Suc n)" by (simp add: stutstep_empty_suc) hence "nextnat (s |⇩_{s}Suc n) = (LEAST i. (s |⇩_{s}Suc n) i ≠ (s |⇩_{s}Suc n) 0)" by (auto simp add: nextnat_def) hence g1: "nextnat (s |⇩_{s}Suc n) = (LEAST i. s (i+(Suc n)) ≠ s (Suc n))" by (simp add: suffix_def) from h1 have "nextnat (s |⇩_{s}n) = (LEAST i. (s |⇩_{s}n) i ≠ (s |⇩_{s}n) 0)" by (auto simp add: nextnat_def) hence g2: "nextnat (s |⇩_{s}n) = (LEAST i. s (i+n) ≠ s n)" by (simp add: suffix_def) with h2 have g2': "nextnat (s |⇩_{s}n) = (LEAST i. s (i+n) ≠ s (Suc n))" by (auto simp add: stutstep_def) from G1 g1 g2' show ?thesis by auto qed lemma nextnat_empty_neq: assumes H: "¬ emptyseq s" shows "s (nextnat s) ≠ s 0" proof - from H have a1: "nextnat s = (LEAST i. s i ≠ s 0)" by (simp add: nextnat_def) from H obtain i where "s i ≠ s 0" by (auto simp: emptyseq_def) hence "s (LEAST i. s i ≠ s 0) ≠ s 0" by (rule LeastI) with a1 show ?thesis by auto qed lemma nextnat_empty_gzero: assumes H: "¬ emptyseq s" shows "nextnat s > 0" proof - from H have a1: "s (nextnat s) ≠ s 0" by (rule nextnat_empty_neq) have "nextnat s ≠ 0" proof assume "nextnat s = 0" with a1 show "False" by simp qed thus "nextnat s > 0" by simp qed subsubsection "Properties of @{term nextsuffix}" lemma empty_nextsuffix: assumes H: "emptyseq s" shows "nextsuffix s = s" using H by (simp add: nextsuffix_def nextnat_def) lemma empty_nextsuffix_id: assumes H: "emptyseq s" shows "nextsuffix s = id s" using H by (simp add: empty_nextsuffix) lemma notstutstep_nextsuffix1: assumes H: "¬ stutstep s n" shows "nextsuffix (s |⇩_{s}n) = s |⇩_{s}(Suc n)" proof (unfold nextsuffix_def) show "(s |⇩_{s}n |⇩_{s}(nextnat (s |⇩_{s}n))) = s |⇩_{s}(Suc n)" proof - from H have "nextnat (s |⇩_{s}n) = 1" by (rule notstutstep_nexnat1) hence "(s |⇩_{s}n |⇩_{s}(nextnat (s |⇩_{s}n))) = s |⇩_{s}n |⇩_{s}1" by auto thus ?thesis by (simp add: suffix_def) qed qed subsubsection "Properties of @{term next}" lemma next_suc_suffix: "next (Suc n) s = nextsuffix (next n s)" by simp lemma next_suffix_com: "nextsuffix (next n s) = (next n (nextsuffix s))" by (induct n, auto) lemma next_plus: "next (m+n) s = next m (next n s)" by (induct m, auto) lemma next_empty: assumes H: "emptyseq s" shows "next n s = s" proof (induct n) from H show "next 0 s = s" by auto next fix n assume a1: "next n s = s" have "next (Suc n) s = nextsuffix (next n s)" by auto with a1 have "next (Suc n) s = nextsuffix s" by simp with H show "next (Suc n) s = s" by (simp add: nextsuffix_def nextnat_def) qed lemma notempty_nextnotzero: assumes H: "¬emptyseq s" shows "(next (Suc 0) s) 0 ≠ s 0" proof - from H have g1: "s (nextnat s) ≠ s 0" by (rule nextnat_empty_neq) have "next (Suc 0) s = nextsuffix s" by auto hence "(next (Suc 0) s) 0 = s (nextnat s)" by (simp add: nextsuffix_def suffix_def) with g1 show ?thesis by simp qed lemma next_ex_id: "∃ i. s i = (next m s) 0" proof - have "∃ i. (s |⇩_{s}i) = (next m s)" proof (induct m) have "s |⇩_{s}0 = next 0 s" by simp thus "∃ i. (s |⇩_{s}i) = (next 0 s)" .. next fix m assume a1: "∃ i. (s |⇩_{s}i) = (next m s)" then obtain i where a1': "(s |⇩_{s}i) = (next m s)" .. have "next (Suc m) s = nextsuffix (next m s)" by auto hence "next (Suc m) s = (next m s) |⇩_{s}(nextnat (next m s))" by (simp add: nextsuffix_def) hence "∃ i. next (Suc m) s = (next m s) |⇩_{s}i" .. then obtain j where "next (Suc m) s = (next m s) |⇩_{s}j" .. with a1' have "next (Suc m) s = (s |⇩_{s}i) |⇩_{s}j" by simp hence "next (Suc m) s = (s |⇩_{s}(j+i))" by (simp add: suffix_plus) hence "(s |⇩_{s}(j+i)) = next (Suc m) s" by simp thus "∃ i. (s |⇩_{s}i) = (next (Suc m) s)" .. qed then obtain i where "(s |⇩_{s}i) = (next m s)" .. hence "(s |⇩_{s}i) 0 = (next m s) 0" by auto hence "s i = (next m s) 0" by (auto simp add: suffix_def) thus ?thesis .. qed subsubsection "Properties of @{term collapse}" lemma emptyseq_collapse_eq: assumes A1: "emptyseq s" shows "♮ s = s" proof (unfold collapse_def, rule ext) fix n from A1 have "next n s = s" by (rule next_empty) moreover from A1 have "s n = s 0" by (simp add: emptyseq_def) ultimately show "(next n s) 0 = s n" by simp qed lemma empty_collapse_empty: assumes H: "emptyseq s" shows "emptyseq (♮ s)" using H by (simp add: emptyseq_collapse_eq) lemma collapse_empty_empty: assumes H: "emptyseq (♮ s)" shows "emptyseq s" proof (rule ccontr) assume a1: "¬emptyseq s" from H have "∀ i. (next i s) 0 = s 0" by (simp add: collapse_def emptyseq_def) moreover from a1 have "(next (Suc 0) s) 0 ≠ s 0" by (rule notempty_nextnotzero) ultimately show "False" by blast qed lemma collapse_empty_iff_empty [simp]: "emptyseq (♮ s) = emptyseq s" by (auto elim: empty_collapse_empty collapse_empty_empty) subsection "Similarity of Sequences" text‹ Since adding or removing stuttering steps does not change the validity of a stuttering-invarant formula, equality is often too strong, and the weaker equality \emph{up to stuttering} is sufficient. This is often called \emph{similarity} ($\approx$) of sequences in the literature, and is required to show that logical operators are stuttering invariant. This is mechanised as: › definition seqsimilar :: "('a seq) ⇒ ('a seq) ⇒ bool" (infixl "≈" 50) where "σ ≈ τ ≡ (♮ σ) = (♮ τ)" subsubsection "Properties of @{term seqsimilar}" lemma seqsim_refl [iff]: "s ≈ s" by (simp add: seqsimilar_def) lemma seqsim_sym: assumes H: "s ≈ t" shows "t ≈ s" using H by (simp add: seqsimilar_def) lemma seqeq_imp_sim: assumes H: "s = t" shows "s ≈ t" using H by simp lemma seqsim_trans [trans]: assumes h1: "s ≈ t" and h2: "t ≈ z" shows "s ≈ z" using assms by (simp add: seqsimilar_def) theorem sim_first: assumes H: "s ≈ t" shows "first s = first t" proof - from H have "(♮ s) 0 = (♮ t) 0" by (simp add: seqsimilar_def) thus ?thesis by (simp add: collapse_def first_def) qed lemmas sim_first2 = sim_first[unfolded first_def] lemma tail_sim_second: assumes H: "tail s ≈ tail t" shows "second s = second t" proof - from H have "first (tail s) = first (tail t)" by (simp add: sim_first) thus "second s = second t" by (simp add: first_tail_second) qed lemma seqsimilarI: assumes 1: "first s = first t" and 2: "nextsuffix s ≈ nextsuffix t" shows "s ≈ t" unfolding seqsimilar_def collapse_def proof fix n show "next n s 0 = next n t 0" proof (cases n) assume "n = 0" with 1 show ?thesis by (simp add: first_def) next fix m assume m: "n = Suc m" from 2 have "next m (nextsuffix s) 0 = next m (nextsuffix t) 0" unfolding seqsimilar_def collapse_def by (rule fun_cong) with m show ?thesis by (simp add: next_suffix_com) qed qed lemma seqsim_empty_empty: assumes H1: "s ≈ t" and H2: "emptyseq s" shows "emptyseq t" proof - from H2 have "emptyseq (♮ s)" by simp with H1 have "emptyseq (♮ t)" by (simp add: seqsimilar_def) thus ?thesis by simp qed lemma seqsim_empty_iff_empty: assumes H: "s ≈ t" shows "emptyseq s = emptyseq t" proof assume "emptyseq s" with H show "emptyseq t" by (rule seqsim_empty_empty) next assume t: "emptyseq t" from H have "t ≈ s" by (rule seqsim_sym) from this t show "emptyseq s" by (rule seqsim_empty_empty) qed lemma seq_empty_eq: assumes H1: "s 0 = t 0" and H2: "emptyseq s" and H3: "emptyseq t" shows "s = t" proof (rule ext) fix n from assms have "t n = s n" by (auto simp: emptyseq_def) thus "s n = t n" by simp qed lemma seqsim_notstutstep: assumes H: "¬ (stutstep s n)" shows "(s |⇩_{s}(Suc n)) ≈ nextsuffix (s |⇩_{s}n)" using H by (simp add: notstutstep_nextsuffix1) lemma stut_nextsuf_suc: assumes H: "stutstep s n" shows "nextsuffix (s |⇩_{s}n) = nextsuffix (s |⇩_{s}(Suc n))" proof (cases "emptyseq (s |⇩_{s}n)") case True hence g1: "nextsuffix (s |⇩_{s}n) = (s |⇩_{s}n)" by (simp add: nextsuffix_def nextnat_def) from True have g2: "nextsuffix (s |⇩_{s}Suc n) = (s |⇩_{s}Suc n)" by (simp add: suc_empty nextsuffix_def nextnat_def) have "(s |⇩_{s}n) = (s |⇩_{s}Suc n)" proof fix x from True have "s (x + n) = s (0 + n)" "s (Suc x + n) = s (0 + n)" unfolding emptyseq_def suffix_def by (blast+) thus "(s |⇩_{s}n) x = (s |⇩_{s}Suc n) x" by (simp add: suffix_def) qed with g1 g2 show ?thesis by auto next case False with H have "(nextnat (s |⇩_{s}n)) = Suc (nextnat (s |⇩_{s}Suc n))" by (simp add: stutstep_notempty_sucnextnat) thus ?thesis by (simp add: nextsuffix_def suffix_plus) qed lemma seqsim_suffix_seqsim: assumes H: "s ≈ t" shows "nextsuffix s ≈ nextsuffix t" unfolding seqsimilar_def collapse_def proof fix n from H have "(next (Suc n) s) 0 = (next (Suc n) t) 0" unfolding seqsimilar_def collapse_def by (rule fun_cong) thus "next n (nextsuffix s) 0 = next n (nextsuffix t) 0" by (simp add: next_suffix_com) qed lemma seqsim_stutstep: assumes H: "stutstep s n" shows "(s |⇩_{s}(Suc n)) ≈ (s |⇩_{s}n)" (is "?sn ≈ ?s") unfolding seqsimilar_def collapse_def proof fix m show "next m (s |⇩_{s}Suc n) 0 = next m (s |⇩_{s}n) 0" proof (cases m) assume "m=0" with H show ?thesis by (simp add: suffix_def stutstep_def) next fix k assume m: "m = Suc k" with H have "next m (s |⇩_{s}Suc n) = next k (nextsuffix (s |⇩_{s}n))" by (simp add: stut_nextsuf_suc next_suffix_com) moreover from m have "next m (s |⇩_{s}n) = next k (nextsuffix (s |⇩_{s}n))" by (simp add: next_suffix_com) ultimately show "next m (s |⇩_{s}Suc n) 0 = next m (s |⇩_{s}n) 0" by simp qed qed lemma addfeqstut: "stutstep ((first t) ## t) 0" by (simp add: first_def stutstep_def app_def suffix_def) lemma addfeqsim: "((first t) ## t) ≈ t" proof - have "stutstep ((first t) ## t) 0" by (rule addfeqstut) hence "(((first t) ## t) |⇩_{s}(Suc 0)) ≈ (((first t) ## t) |⇩_{s}0)" by (rule seqsim_stutstep) hence "tail ((first t) ## t) ≈ ((first t) ## t)" by (simp add: suffix_def tail_def) hence "t ≈ ((first t) ## t)" by (simp add: tail_def app_def suffix_def) thus ?thesis by (rule seqsim_sym) qed lemma addfirststut: assumes H: "first s = second s" shows "s ≈ tail s" proof - have g1: "(first s) ## (tail s) = s" by (rule seq_app_first_tail) from H have "(first s) = first (tail s)" by (simp add: first_def second_def tail_def suffix_def) hence "(first s) ## (tail s) ≈ (tail s)" by (simp add: addfeqsim) with g1 show ?thesis by simp qed lemma app_seqsimilar: assumes h1: "s ≈ t" shows "(x ## s) ≈ (x ## t)" proof (cases "stutstep (x ## s) 0") case True from h1 have "first s = first t" by (rule sim_first) with True have a2: "stutstep (x ## t) 0" by (simp add: stutstep_def first_def app_def) from True have "((x ## s) |⇩_{s}(Suc 0)) ≈ ((x ## s) |⇩_{s}0)" by (rule seqsim_stutstep) hence "tail (x ## s) ≈ (x ## s)" by (simp add: tail_def suffix_def) hence g1: "s ≈ (x ## s)" by (simp add: app_def tail_def suffix_def) from a2 have "((x ## t) |⇩_{s}(Suc 0)) ≈ ((x ## t) |⇩_{s}0)" by (rule seqsim_stutstep) hence "tail (x ## t) ≈ (x ## t)" by (simp add: tail_def suffix_def) hence g2: "t ≈ (x ## t)" by (simp add: app_def tail_def suffix_def) from h1 g2 have "s ≈ (x ## t)" by (rule seqsim_trans) from this[THEN seqsim_sym] g1 show "(x ## s) ≈ (x ## t)" by (rule seqsim_sym[OF seqsim_trans]) next case False from h1 have "first s = first t" by (rule sim_first) with False have a2: "¬ stutstep (x ## t) 0" by (simp add: stutstep_def first_def app_def) from False have "((x ## s) |⇩_{s}(Suc 0)) ≈ nextsuffix ((x ## s) |⇩_{s}0)" by (rule seqsim_notstutstep) hence "(tail (x ## s)) ≈ nextsuffix (x ## s)" by (simp add: tail_def) hence g1: "s ≈ nextsuffix (x ## s)" by (simp add: seq_app_tail) from a2 have "((x ## t) |⇩_{s}(Suc 0)) ≈ nextsuffix ((x ## t) |⇩_{s}0)" by (rule seqsim_notstutstep) hence "(tail (x ## t)) ≈ nextsuffix (x ## t)" by (simp add: tail_def) hence g2: "t ≈ nextsuffix (x ## t)" by (simp add: seq_app_tail) with h1 have "s ≈ nextsuffix (x ## t)" by (rule seqsim_trans) from this[THEN seqsim_sym] g1 have g3: "nextsuffix (x ## s) ≈ nextsuffix (x ## t)" by (rule seqsim_sym[OF seqsim_trans]) have "first (x ## s) = first (x ## t)" by (simp add: first_def app_def) from this g3 show ?thesis by (rule seqsimilarI) qed text ‹ If two sequences are similar then for any suffix of one of them there exists a similar suffix of the other one. We will prove a stronger result below. › lemma simstep_disj1: assumes H: "s ≈ t" shows "∃ m. ((s |⇩_{s}n) ≈ (t |⇩_{s}m))" proof (induct n) from H have "((s |⇩_{s}0) ≈ (t |⇩_{s}0))" by auto thus "∃ m. ((s |⇩_{s}0) ≈ (t |⇩_{s}m))" .. next fix n assume "∃ m. ((s |⇩_{s}n) ≈ (t |⇩_{s}m))" then obtain m where a1': "(s |⇩_{s}n) ≈ (t |⇩_{s}m)" .. show "∃ m. ((s |⇩_{s}(Suc n)) ≈ (t |⇩_{s}m))" proof (cases "stutstep s n") case True hence "(s |⇩_{s}(Suc n)) ≈ (s |⇩_{s}n)" by (rule seqsim_stutstep) from this a1' have "((s |⇩_{s}(Suc n)) ≈ (t |⇩_{s}m))" by (rule seqsim_trans) thus ?thesis .. next case False hence "(s |⇩_{s}(Suc n)) ≈ nextsuffix (s |⇩_{s}n)" by (rule seqsim_notstutstep) moreover from a1' have "nextsuffix (s |⇩_{s}n) ≈ nextsuffix (t |⇩_{s}m)" by (simp add: seqsim_suffix_seqsim) ultimately have "(s |⇩_{s}(Suc n)) ≈ nextsuffix (t |⇩_{s}m)" by (rule seqsim_trans) hence "(s |⇩_{s}(Suc n)) ≈ t |⇩_{s}(m + (nextnat (t |⇩_{s}m)))" by (simp add: nextsuffix_def suffix_plus_com) thus "∃ m. (s |⇩_{s}(Suc n)) ≈ t |⇩_{s}m" .. qed qed lemma nextnat_le_seqsim: assumes n: "n < nextnat s" shows "s ≈ (s |⇩_{s}n)" proof (cases "emptyseq s") case True ― ‹case impossible› with n show ?thesis by (simp add: nextnat_def) next case False from n show ?thesis proof (induct n) show "s ≈ (s |⇩_{s}0)" by simp next fix n assume a2: "n < nextnat s ⟹ s ≈ (s |⇩_{s}n)" and a3: "Suc n < nextnat s" from a3 have g1: "s (Suc n) = s 0" by (rule nextnat_le_unch) from a3 have a3': "n < nextnat s" by simp hence "s n = s 0" by (rule nextnat_le_unch) with g1 have "stutstep s n" by (simp add: stutstep_def) hence g2: "(s |⇩_{s}n) ≈ (s |⇩_{s}(Suc n))" by (rule seqsim_stutstep[THEN seqsim_sym]) with a3' a2 show "s ≈ (s |⇩_{s}(Suc n))" by (auto elim: seqsim_trans) qed qed lemma seqsim_prev_nextnat: "s ≈ s |⇩_{s}((nextnat s) - 1)" proof (cases "emptyseq s") case True hence "s |⇩_{s}((nextnat s)-(1::nat)) = s |⇩_{s}0" by (simp add: nextnat_def) thus ?thesis by simp next case False hence "nextnat s > 0" by (rule nextnat_empty_gzero) thus ?thesis by (simp add: nextnat_le_seqsim) qed text ‹ Given a suffix ‹s |⇩_{s}n› of some sequence ‹s› that is similar to some suffix ‹t |⇩_{s}m› of sequence ‹t›, there exists some suffix ‹t |⇩_{s}m'› of ‹t› such that ‹s |⇩_{s}n› and ‹t |⇩_{s}m'› are similar and also ‹s |⇩_{s}(n+1)› is similar to either ‹t |⇩_{s}m'› or to ‹t |⇩_{s}(m'+1)›. › lemma seqsim_suffix_suc: assumes H: "s |⇩_{s}n ≈ t |⇩_{s}m" shows "∃m'. s |⇩_{s}n ≈ t |⇩_{s}m' ∧ ((s |⇩_{s}Suc n ≈ t |⇩_{s}Suc m') ∨ (s |⇩_{s}Suc n ≈ t |⇩_{s}m'))" proof (cases "stutstep s n") case True hence "s |⇩_{s}Suc n ≈ s |⇩_{s}n" by (rule seqsim_stutstep) from this H have "s |⇩_{s}Suc n ≈ t |⇩_{s}m" by (rule seqsim_trans) with H show ?thesis by blast next case False hence "¬ emptyseq (s |⇩_{s}n)" by (rule stutnempty) with H have a2: "¬ emptyseq (t |⇩_{s}m)" by (simp add: seqsim_empty_iff_empty) hence g4: "nextsuffix (t |⇩_{s}m) = (t |⇩_{s}m) |⇩_{s}Suc (nextnat (t |⇩_{s}m) - 1)" by (simp add: nextnat_empty_gzero nextsuffix_def) have g3: "(t |⇩_{s}m) ≈ (t |⇩_{s}m) |⇩_{s}(nextnat (t |⇩_{s}m) - 1)" by (rule seqsim_prev_nextnat) with H have G1: "s |⇩_{s}n ≈ (t |⇩_{s}m) |⇩_{s}(nextnat (t |⇩_{s}m) - 1)" by (rule seqsim_trans) from False have G1': "(s |⇩_{s}Suc n) = nextsuffix (s |⇩_{s}n)" by (rule notstutstep_nextsuffix1[THEN sym]) from H have "nextsuffix (s |⇩_{s}n) ≈ nextsuffix (t |⇩_{s}m)" by (rule seqsim_suffix_seqsim) with G1 G1' g4 have "s |⇩_{s}n ≈ t |⇩_{s}(m + (nextnat (t |⇩_{s}m) - 1)) ∧ s |⇩_{s}(Suc n) ≈ t |⇩_{s}Suc (m + (nextnat (t |⇩_{s}m) - 1))" by (simp add: suffix_plus_com) thus ?thesis by blast qed text ‹ The following main result about similar sequences shows that if ‹s ≈ t› holds then for any suffix ‹s |⇩_{s}n› of ‹s› there exists a suffix ‹t |⇩_{s}m› such that \begin{itemize} \item ‹s |⇩_{s}n› and ‹t |⇩_{s}m› are similar, and \item ‹s |⇩_{s}(n+1)› is similar to either ‹t |⇩_{s}(m+1)› or ‹t |⇩_{s}m›. \end{itemize} The idea is to pick the largest ‹m› such that ‹s |⇩_{s}n ≈ t |⇩_{s}m› (or some such ‹m› if ‹s |⇩_{s}n› is empty). › theorem sim_step: assumes H: "s ≈ t" shows "∃ m. s |⇩_{s}n ≈ t |⇩_{s}m ∧ ((s |⇩_{s}Suc n ≈ t |⇩_{s}Suc m) ∨ (s |⇩_{s}Suc n ≈ t |⇩_{s}m))" (is "∃m. ?Sim n m") proof (induct n) from H have "s |⇩_{s}0 ≈ t |⇩_{s}0" by simp thus "∃ m. ?Sim 0 m" by (rule seqsim_suffix_suc) next fix n assume "∃ m. ?Sim n m" hence "∃k. s |⇩_{s}Suc n ≈ t |⇩_{s}k" by blast thus "∃ m. ?Sim (Suc n) m" by (blast dest: seqsim_suffix_suc) qed end

# Theory Intensional

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section ‹Representing Intensional Logic› theory Intensional imports Main begin text‹ In higher-order logic, every proof rule has a corresponding tautology, i.e. the \emph{deduction theorem} holds. Isabelle/HOL implements this since object-level implication ($\longrightarrow$) and meta-level entailment ($\Longrightarrow$) commute, viz. the proof rule ‹impI:› @{thm impI}. However, the deduction theorem does not hold for most modal and temporal logics \cite[page 95]{Lamport02}\cite{Merz98}. For example $A \vdash \Box A$ holds, meaning that if $A$ holds in any world, then it always holds. However, $\vdash A \longrightarrow \Box A$, stating that $A$ always holds if it initially holds, is not valid. Merz \cite{Merz98} overcame this problem by creating an @{term Intensional} logic. It exploits Isabelle's axiomatic type class feature \cite{Wenzel00b} by creating a type class @{term world}, which provides Skolem constants to associate formulas with the world they hold in. The class is trivial, not requiring any axioms. › class world text ‹ @{term world} is a type class of possible worlds. It is a subclass of all HOL types @{term type}. No axioms are provided, since its only purpose is to avoid silly use of the @{term Intensional} syntax. › subsection‹Abstract Syntax and Definitions› type_synonym ('w,'a) expr = "'w ⇒ 'a" type_synonym 'w form = "('w, bool) expr" text ‹The intention is that @{typ 'a} will be used for unlifted types (class @{term type}), while @{typ 'w} is lifted (class @{term world}). › definition Valid :: "('w::world) form ⇒ bool" where "Valid A ≡ ∀w. A w" definition const :: "'a ⇒ ('w::world, 'a) expr" where unl_con: "const c w ≡ c" definition lift :: "['a ⇒ 'b, ('w::world, 'a) expr] ⇒ ('w,'b) expr" where unl_lift: "lift f x w ≡ f (x w)" definition lift2 :: "['a ⇒ 'b ⇒ 'c, ('w::world,'a) expr, ('w,'b) expr] ⇒ ('w,'c) expr" where unl_lift2: "lift2 f x y w ≡ f (x w) (y w)" definition lift3 :: "['a ⇒ 'b => 'c ⇒ 'd, ('w::world,'a) expr, ('w,'b) expr, ('w,'c) expr] ⇒ ('w,'d) expr" where unl_lift3: "lift3 f x y z w ≡ f (x w) (y w) (z w)" definition lift4 :: "['a ⇒ 'b => 'c ⇒ 'd ⇒ 'e, ('w::world,'a) expr, ('w,'b) expr, ('w,'c) expr,('w,'d) expr] ⇒ ('w,'e) expr" where unl_lift4: "lift4 f x y z zz w ≡ f (x w) (y w) (z w) (zz w)" text ‹ @{term "Valid F"} asserts that the lifted formula @{term F} holds everywhere. @{term const} allows lifting of a constant, while @{term lift} through @{term lift4} allow functions with arity 1--4 to be lifted. (Note that there is no way to define a generic lifting operator for functions of arbitrary arity.) › definition RAll :: "('a ⇒ ('w::world) form) ⇒ 'w form" (binder "Rall " 10) where unl_Rall: "(Rall x. A x) w ≡ ∀x. A x w" definition REx :: "('a ⇒ ('w::world) form) ⇒ 'w form" (binder "Rex " 10) where unl_Rex: "(Rex x. A x) w ≡ ∃x. A x w" definition REx1 :: "('a ⇒ ('w::world) form) ⇒ 'w form" (binder "Rex! " 10) where unl_Rex1: "(Rex! x. A x) w ≡ ∃!x. A x w" text ‹ @{term RAll}, @{term REx} and @{term REx1} introduces ``rigid'' quantification over values (of non-world types) within ``intensional'' formulas. @{term RAll} is universal quantification, @{term REx} is existential quantifcation. @{term REx1} requires unique existence. › text ‹ We declare the ``unlifting rules'' as rewrite rules that will be applied automatically. › lemmas intensional_rews[simp] = unl_con unl_lift unl_lift2 unl_lift3 unl_lift4 unl_Rall unl_Rex unl_Rex1 subsection‹Concrete Syntax› nonterminal lift and liftargs text‹ The non-terminal @{term lift} represents lifted expressions. The idea is to use Isabelle's macro mechanism to convert between the concrete and abstract syntax. › syntax "" :: "id ⇒ lift" ("_") "" :: "longid ⇒ lift" ("_") "" :: "var ⇒ lift" ("_") "_applC" :: "[lift, cargs] ⇒ lift" ("(1_/ _)" [1000, 1000] 999) "" :: "lift ⇒ lift" ("'(_')") "_lambda" :: "[idts, 'a] ⇒ lift" ("(3%_./ _)" [0, 3] 3) "_constrain" :: "[lift, type] ⇒ lift" ("(_::_)" [4, 0] 3) "" :: "lift ⇒ liftargs" ("_") "_liftargs" :: "[lift, liftargs] ⇒ liftargs" ("_,/ _") "_Valid" :: "lift ⇒ bool" ("(⊢ _)" 5) "_holdsAt" :: "['a, lift] ⇒ bool" ("(_ ⊨ _)" [100,10] 10) (* Syntax for lifted expressions outside the scope of ⊢ or ⊨.*) "LIFT" :: "lift ⇒ 'a" ("LIFT _") (* generic syntax for lifted constants and functions *) "_const" :: "'a ⇒ lift" ("(#_)" [1000] 999) "_lift" :: "['a, lift] ⇒ lift" ("(_<_>)" [1000] 999) "_lift2" :: "['a, lift, lift] ⇒ lift" ("(_<_,/ _>)" [1000] 999) "_lift3" :: "['a, lift, lift, lift] ⇒ lift" ("(_<_,/ _,/ _>)" [1000] 999) "_lift4" :: "['a, lift, lift, lift,lift] ⇒ lift" ("(_<_,/ _,/ _,/ _>)" [1000] 999) (* concrete syntax for common infix functions: reuse same symbol *) "_liftEqu" :: "[lift, lift] ⇒ lift" ("(_ =/ _)" [50,51] 50) "_liftNeq" :: "[lift, lift] ⇒ lift" (infixl "≠" 50) "_liftNot" :: "lift ⇒ lift" ("¬ _" [90] 90) "_liftAnd" :: "[lift, lift] ⇒ lift" (infixr "∧" 35) "_liftOr" :: "[lift, lift] ⇒ lift" (infixr "∨" 30) "_liftImp" :: "[lift, lift] ⇒ lift" (infixr "⟶" 25) "_liftIf" :: "[lift, lift, lift] ⇒ lift" ("(if (_)/ then (_)/ else (_))" 10) "_liftPlus" :: "[lift, lift] ⇒ lift" ("(_ +/ _)" [66,65] 65) "_liftMinus" :: "[lift, lift] ⇒ lift" ("(_ -/ _)" [66,65] 65) "_liftTimes" :: "[lift, lift] ⇒ lift" ("(_ */ _)" [71,70] 70) "_liftDiv" :: "[lift, lift] ⇒ lift" ("(_ div _)" [71,70] 70) "_liftMod" :: "[lift, lift] ⇒ lift" ("(_ mod _)" [71,70] 70) "_liftLess" :: "[lift, lift] ⇒ lift" ("(_/ < _)" [50, 51] 50) "_liftLeq" :: "[lift, lift] ⇒ lift" ("(_/ ≤ _)" [50, 51] 50) "_liftMem" :: "[lift, lift] ⇒ lift" ("(_/ ∈ _)" [50, 51] 50) "_liftNotMem" :: "[lift, lift] ⇒ lift" ("(_/ ∉ _)" [50, 51] 50) "_liftFinset" :: "liftargs => lift" ("{(_)}") (** TODO: syntax for lifted collection / comprehension **) "_liftPair" :: "[lift,liftargs] ⇒ lift" ("(1'(_,/ _'))") (* infix syntax for list operations *) "_liftCons" :: "[lift, lift] ⇒ lift" ("(_ #/ _)" [65,66] 65) "_liftApp" :: "[lift, lift] ⇒ lift" ("(_ @/ _)" [65,66] 65) "_liftList" :: "liftargs ⇒ lift" ("[(_)]") (* Rigid quantification (syntax level) *) "_ARAll" :: "[idts, lift] ⇒ lift" ("(3! _./ _)" [0, 10] 10) "_AREx" :: "[idts, lift] ⇒ lift" ("(3? _./ _)" [0, 10] 10) "_AREx1" :: "[idts, lift] ⇒ lift" ("(3?! _./ _)" [0, 10] 10) "_RAll" :: "[idts, lift] ⇒ lift" ("(3∀_./ _)" [0, 10] 10) "_REx" :: "[idts, lift] ⇒ lift" ("(3∃_./ _)" [0, 10] 10) "_REx1" :: "[idts, lift] ⇒ lift" ("(3∃!_./ _)" [0, 10] 10) translations "_const" ⇌ "CONST const" translations "_lift" ⇌ "CONST lift" "_lift2" ⇌ "CONST lift2" "_lift3" ⇌ "CONST lift3" "_lift4" ⇌ "CONST lift4" "_Valid" ⇌ "CONST Valid" translations "_RAll x A" ⇌ "Rall x. A" "_REx x A" ⇌ "Rex x. A" "_REx1 x A" ⇌ "Rex! x. A" translations "_ARAll" ⇀ "_RAll" "_AREx" ⇀ "_REx" "_AREx1" ⇀ "_REx1" "w ⊨ A" ⇀ "A w" "LIFT A" ⇀ "A::_⇒_" translations "_liftEqu" ⇌ "_lift2 (=)" "_liftNeq u v" ⇌ "_liftNot (_liftEqu u v)" "_liftNot" ⇌ "_lift (CONST Not)" "_liftAnd" ⇌ "_lift2 (&)" "_liftOr" ⇌ "_lift2 ((|) )" "_liftImp" ⇌ "_lift2 (-->)" "_liftIf" ⇌ "_lift3 (CONST If)" "_liftPlus" ⇌ "_lift2 (+)" "_liftMinus" ⇌ "_lift2 (-)" "_liftTimes" ⇌ "_lift2 (*)" "_liftDiv" ⇌ "_lift2 (div)" "_liftMod" ⇌ "_lift2 (mod)" "_liftLess" ⇌ "_lift2 (<)" "_liftLeq" ⇌ "_lift2 (<=)" "_liftMem" ⇌ "_lift2 (:)" "_liftNotMem x xs" ⇌ "_liftNot (_liftMem x xs)" translations "_liftFinset (_liftargs x xs)" ⇌ "_lift2 (CONST insert) x (_liftFinset xs)" "_liftFinset x" ⇌ "_lift2 (CONST insert) x (_const (CONST Set.empty))" "_liftPair x (_liftargs y z)" ⇌ "_liftPair x (_liftPair y z)" "_liftPair" ⇌ "_lift2 (CONST Pair)" "_liftCons" ⇌ "_lift2 (CONST Cons)" "_liftApp" ⇌ "_lift2 (@)" "_liftList (_liftargs x xs)" ⇌ "_liftCons x (_liftList xs)" "_liftList x" ⇌ "_liftCons x (_const [])" "w ⊨ ¬ A" ↽ "_liftNot A w" "w ⊨ A ∧ B" ↽ "_liftAnd A B w" "w ⊨ A ∨ B" ↽ "_liftOr A B w" "w ⊨ A ⟶ B" ↽ "_liftImp A B w" "w ⊨ u = v" ↽ "_liftEqu u v w" "w ⊨ ∀x. A" ↽ "_RAll x A w" "w ⊨ ∃x. A" ↽ "_REx x A w" "w ⊨ ∃!x. A" ↽ "_REx1 x A w" syntax (ASCII) "_Valid" :: "lift ⇒ bool" ("(|- _)" 5) "_holdsAt" :: "['a, lift] ⇒ bool" ("(_ |= _)" [100,10] 10) "_liftNeq" :: "[lift, lift] ⇒ lift" ("(_ ~=/ _)" [50,51] 50) "_liftNot" :: "lift ⇒ lift" ("(~ _)" [90] 90) "_liftAnd" :: "[lift, lift] ⇒ lift" ("(_ &/ _)" [36,35] 35) "_liftOr" :: "[lift, lift] ⇒ lift" ("(_ |/ _)" [31,30] 30) "_liftImp" :: "[lift, lift] ⇒ lift" ("(_ -->/ _)" [26,25] 25) "_liftLeq" :: "[lift, lift] ⇒ lift" ("(_/ <= _)" [50, 51] 50) "_liftMem" :: "[lift, lift] ⇒ lift" ("(_/ : _)" [50, 51] 50) "_liftNotMem" :: "[lift, lift] ⇒ lift" ("(_/ ~: _)" [50, 51] 50) "_RAll" :: "[idts, lift] ⇒ lift" ("(3ALL _./ _)" [0, 10] 10) "_REx" :: "[idts, lift] ⇒ lift" ("(3EX _./ _)" [0, 10] 10) "_REx1" :: "[idts, lift] ⇒ lift" ("(3EX! _./ _)" [0, 10] 10) subsection ‹Lemmas and Tactics› lemma intD[dest]: "⊢ A ⟹ w ⊨ A" proof - assume a:"⊢ A" from a have "∀w. w ⊨ A" by (auto simp add: Valid_def) thus ?thesis .. qed lemma intI [intro!]: assumes P1:"(⋀ w. w ⊨ A)" shows "⊢ A" using assms by (auto simp: Valid_def) text‹ Basic unlifting introduces a parameter @{term w} and applies basic rewrites, e.g @{term "⊢ F = G"} becomes @{term "F w = G w"} and @{term "⊢ F ⟶ G"} becomes @{term "F w ⟶ G w"}. › method_setup int_unlift = ‹ Scan.succeed (fn ctxt => SIMPLE_METHOD' (resolve_tac ctxt @{thms intI} THEN' rewrite_goal_tac ctxt @{thms intensional_rews})) › "method to unlift and followed by intensional rewrites" lemma inteq_reflection: assumes P1: "⊢ x=y" shows "(x ≡ y)" proof - from P1 have P2: "∀w. x w = y w" by (unfold Valid_def unl_lift2) hence P3:"x=y" by blast thus "x ≡ y" by (rule "eq_reflection") qed lemma int_simps: "⊢ (x=x) = #True" "⊢ (¬ #True) = #False" "⊢ (¬ #False) = #True" "⊢ (¬¬ P) = P" "⊢ ((¬ P) = P) = #False" "⊢ (P = (¬P)) = #False" "⊢ (P ≠ Q) = (P = (¬ Q))" "⊢ (#True=P) = P" "⊢ (P=#True) = P" "⊢ (#True ⟶ P) = P" "⊢ (#False ⟶ P) = #True" "⊢ (P ⟶ #True) = #True" "⊢ (P ⟶ P) = #True" "⊢ (P ⟶ #False) = (¬P)" "⊢ (P ⟶ ~P) = (¬P)" "⊢ (P ∧ #True) = P" "⊢ (#True ∧ P) = P" "⊢ (P ∧ #False) = #False" "⊢ (#False ∧ P) = #False" "⊢ (P ∧ P) = P" "⊢ (P ∧ ~P) = #False" "⊢ (¬P ∧ P) = #False" "⊢ (P ∨ #True) = #True" "⊢ (#True ∨ P) = #True" "⊢ (P ∨ #False) = P" "⊢ (#False ∨ P) = P" "⊢ (P ∨ P) = P" "⊢ (P ∨ ¬P) = #True" "⊢ (¬P ∨ P) = #True" "⊢ (∀ x. P) = P" "⊢ (∃ x. P) = P" by auto lemmas intensional_simps[simp] = int_simps[THEN inteq_reflection] method_setup int_rewrite = ‹ Scan.succeed (fn ctxt => SIMPLE_METHOD' (rewrite_goal_tac ctxt @{thms intensional_simps})) › "rewrite method at intensional level" lemma Not_Rall: "⊢ (¬(∀ x. F x)) = (∃ x. ¬F x)" by auto lemma Not_Rex: "⊢ (¬(∃ x. F x)) = (∀ x. ¬F x)" by auto lemma TrueW [simp]: "⊢ #True" by auto lemma int_eq: "⊢ X = Y ⟹ X = Y" by (auto simp: inteq_reflection) lemma int_iffI: assumes "⊢ F ⟶ G" and "⊢ G ⟶ F" shows "⊢ F = G" using assms by force lemma int_iffD1: assumes h: "⊢ F = G" shows "⊢ F ⟶ G" using h by auto lemma int_iffD2: assumes h: "⊢ F = G" shows "⊢ G ⟶ F" using h by auto lemma lift_imp_trans: assumes "⊢ A ⟶ B" and "⊢ B ⟶ C" shows "⊢ A ⟶ C" using assms by force lemma lift_imp_neg: assumes "⊢ A ⟶ B" shows "⊢ ¬B ⟶ ¬A" using assms by auto lemma lift_and_com: "⊢ (A ∧ B) = (B ∧ A)" by auto end

# Theory Semantics

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section ‹Semantics› theory Semantics imports Sequence Intensional begin text ‹ This theory mechanises a \emph{shallow} embedding of \tlastar{} using the ‹Sequence› and ‹Intensional› theories. A shallow embedding represents \tlastar{} using Isabelle/HOL predicates, while a \emph{deep} embedding would represent \tlastar{} formulas and pre-formulas as mutually inductive datatypes\footnote{See e.g. \cite{Wildmoser04} for a discussion about deep vs. shallow embeddings in Isabelle/HOL.}. The choice of a shallow over a deep embedding is motivated by the following factors: a shallow embedding is usually less involved, and existing Isabelle theories and tools can be applied more directly to enhance automation; due to the lifting in the ‹Intensional› theory, a shallow embedding can reuse standard logical operators, whilst a deep embedding requires a different set of operators for both formulas and pre-formulas. Finally, since our target is system verification rather than proving meta-properties of \tlastar{}, which requires a deep embedding, a shallow embedding is more fit for purpose. › subsection "Types of Formulas" text ‹ To mechanise the \tlastar{} semantics, the following type abbreviations are used: › type_synonym ('a,'b) formfun = "'a seq ⇒ 'b" type_synonym 'a formula = "('a,bool) formfun" type_synonym ('a,'b) stfun = "'a ⇒ 'b" type_synonym 'a stpred = "('a,bool) stfun" instance "fun" :: (type,type) world .. instance "prod" :: (type,type) world .. text ‹ Pair and function are instantiated to be of type class world. This allows use of the lifted intensional logic for formulas, and standard logical connectives can therefore be used. › subsection "Semantics of TLA*" text ‹The semantics of \tlastar{} is defined.› definition always :: "('a::world) formula ⇒ 'a formula" where "always F ≡ λ s. ∀ n. (s |⇩_{s}n) ⊨ F" definition nexts :: "('a::world) formula ⇒ 'a formula" where "nexts F ≡ λ s. (tail s) ⊨ F" definition before :: "('a::world,'b) stfun ⇒ ('a,'b) formfun" where "before f ≡ λ s. (first s) ⊨ f" definition after :: "('a::world,'b) stfun ⇒ ('a,'b) formfun" where "after f ≡ λ s. (second s) ⊨ f" definition unch :: "('a::world,'b) stfun ⇒ 'a formula" where "unch v ≡ λ s. s ⊨ (after v) = (before v)" definition action :: "('a::world) formula ⇒ ('a,'b) stfun ⇒ 'a formula" where "action P v ≡ λ s. ∀ i. ((s |⇩_{s}i) ⊨ P) ∨ ((s |⇩_{s}i) ⊨ unch v)" subsubsection "Concrete Syntax" text‹This is the concrete syntax for the (abstract) operators above.› syntax "_always" :: "lift ⇒ lift" ("(□_)" [90] 90) "_nexts" :: "lift ⇒ lift" ("(○_)" [90] 90) "_action" :: "[lift,lift] ⇒ lift" ("(□[_]'_(_))" [20,1000] 90) "_before" :: "lift ⇒ lift" ("($_)" [100] 99) "_after" :: "lift ⇒ lift" ("(_$)" [100] 99) "_prime" :: "lift ⇒ lift" ("(_`)" [100] 99) "_unch" :: "lift ⇒ lift" ("(Unchanged _)" [100] 99) "TEMP" :: "lift ⇒ 'b" ("(TEMP _)") syntax (ASCII) "_always" :: "lift ⇒ lift" ("([]_)" [90] 90) "_nexts" :: "lift ⇒ lift" ("(Next _)" [90] 90) "_action" :: "[lift,lift] ⇒ lift" ("([][_]'_(_))" [20,1000] 90) translations "_always" ⇌ "CONST always" "_nexts" ⇌ "CONST nexts" "_action" ⇌ "CONST action" "_before" ⇌ "CONST before" "_after" ⇌ "CONST after" "_prime" ⇀ "CONST after" "_unch" ⇌ "CONST unch" "TEMP F" ⇀ "(F:: (nat ⇒ _) ⇒ _)" subsection "Abbreviations" text ‹Some standard temporal abbreviations, with their concrete syntax.› definition actrans :: "('a::world) formula ⇒ ('a,'b) stfun ⇒ 'a formula" where "actrans P v ≡ TEMP(P ∨ unch v)" definition eventually :: "('a::world) formula ⇒ 'a formula" where "eventually F ≡ LIFT(¬□(¬F))" definition angle_action :: "('a::world) formula ⇒ ('a,'b) stfun ⇒ 'a formula" where "angle_action P v ≡ LIFT(¬□[¬P]_v)" definition angle_actrans :: "('a::world) formula ⇒ ('a,'b) stfun ⇒ 'a formula" where "angle_actrans P v ≡ TEMP (¬ actrans (LIFT(¬P)) v)" definition leadsto :: "('a::world) formula ⇒ 'a formula ⇒ 'a formula" where "leadsto P Q ≡ LIFT □(P ⟶ eventually Q)" subsubsection "Concrete Syntax" syntax (ASCII) "_actrans" :: "[lift,lift] ⇒ lift" ("([_]'_(_))" [20,1000] 90) "_eventually" :: "lift ⇒ lift" ("(<>_)" [90] 90) "_angle_action" :: "[lift,lift] ⇒ lift" ("(<><_>'_(_))" [20,1000] 90) "_angle_actrans" :: "[lift,lift] ⇒ lift" ("(<_>'_(_))" [20,1000] 90) "_leadsto" :: "[lift,lift] ⇒ lift" ("(_ ~> _)" [26,25] 25) syntax "_eventually" :: "lift ⇒ lift" ("(◇_)" [90] 90) "_angle_action" :: "[lift,lift] ⇒ lift" ("(◇⟨_⟩'_(_))" [20,1000] 90) "_angle_actrans" :: "[lift,lift] ⇒ lift" ("(⟨_⟩'_(_))" [20,1000] 90) "_leadsto" :: "[lift,lift] ⇒ lift" ("(_ ↝ _)" [26,25] 25) translations "_actrans" ⇌ "CONST actrans" "_eventually" ⇌ "CONST eventually" "_angle_action" ⇌ "CONST angle_action" "_angle_actrans" ⇌ "CONST angle_actrans" "_leadsto" ⇌ "CONST leadsto" subsection "Properties of Operators" text ‹The following lemmas show that these operators have the expected semantics.› lemma eventually_defs: "(w ⊨ ◇ F) = (∃ n. (w |⇩_{s}n) ⊨ F)" by (simp add: eventually_def always_def) lemma angle_action_defs: "(w ⊨ ◇⟨P⟩_v) = (∃ i. ((w |⇩_{s}i) ⊨ P) ∧ ((w |⇩_{s}i) ⊨ v$ ≠ $v))" by (simp add: angle_action_def action_def unch_def) lemma unch_defs: "(w ⊨ Unchanged v) = (((second w) ⊨ v) = ((first w) ⊨ v))" by (simp add: unch_def before_def nexts_def after_def tail_def suffix_def first_def second_def) lemma linalw: assumes h1: "a ≤ b" and h2: "(w |⇩_{s}a) ⊨ □A" shows "(w |⇩_{s}b) ⊨ □A" proof (clarsimp simp: always_def) fix n from h1 obtain k where g1: "b = a + k" by (auto simp: le_iff_add) with h2 show "(w |⇩_{s}b |⇩_{s}n) ⊨ A" by (auto simp: always_def suffix_plus ac_simps) qed subsection "Invariance Under Stuttering" text ‹ A key feature of \tlastar{} is that specification at different abstraction levels can be compared. The soundness of this relies on the stuttering invariance of formulas. Since the embedding is shallow, it cannot be shown that a generic \tlastar{} formula is stuttering invariant. However, this section will show that each operator is stuttering invariant or preserves stuttering invariance in an appropriate sense, which can be used to show stuttering invariance for given specifications. Formula ‹F› is stuttering invariant if for any two similar behaviours (i.e., sequences of states), ‹F› holds in one iff it holds in the other. The definition is generalised to arbitrary expressions, and not just predicates. › definition stutinv :: "('a,'b) formfun ⇒ bool" where "stutinv F ≡ ∀ σ τ. σ ≈ τ ⟶ (σ ⊨ F) = (τ ⊨ F)" text‹ The requirement for stuttering invariance is too strong for pre-formulas. For example, an action formula specifies a relation between the first two states of a behaviour, and will rarely be satisfied by a stuttering step. This is why pre-formulas are ``protected'' by (square or angle) brackets in \tlastar{}: the only place a pre-formula ‹P› can be used is inside an action: ‹□[P]_v›. To show that ‹□[P]_v› is stuttering invariant, is must be shown that a slightly weaker predicate holds for @{term P}. For example, if @{term P} contains a term of the form ‹○○Q›, then it is not a well-formed pre-formula, thus ‹□[P]_v› is not stuttering invariant. This weaker version of stuttering invariance has been named \emph{near stuttering invariance}. › definition nstutinv :: "('a,'b) formfun ⇒ bool" where "nstutinv P ≡ ∀ σ τ. (first σ = first τ) ∧ (tail σ) ≈ (tail τ) ⟶ (σ ⊨ P) = (τ ⊨ P)" syntax "_stutinv" :: "lift ⇒ bool" ("(STUTINV _)" [40] 40) "_nstutinv" :: "lift ⇒ bool" ("(NSTUTINV _)" [40] 40) translations "_stutinv" ⇌ "CONST stutinv" "_nstutinv" ⇌ "CONST nstutinv" text ‹ Predicate @{term "stutinv F"} formalises stuttering invariance for formula @{term F}. That is if two sequences are similar @{term "s ≈ t"} (equal up to stuttering) then the validity of @{term F} under both @{term s} and @{term t} are equivalent. Predicate @{term "nstutinv P"} should be read as \emph{nearly stuttering invariant} -- and is required for some stuttering invariance proofs. › lemma stutinv_strictly_stronger: assumes h: "STUTINV F" shows "NSTUTINV F" unfolding nstutinv_def proof (clarify) fix s t :: "nat ⇒ 'a" assume a1: "first s = first t" and a2: "(tail s) ≈ (tail t)" have "s ≈ t" proof - have tg1: "(first s) ## (tail s) = s" by (rule seq_app_first_tail) have tg2: "(first t) ## (tail t) = t" by (rule seq_app_first_tail) with a1 have tg2': "(first s) ## (tail t) = t" by simp from a2 have "(first s) ## (tail s) ≈ (first s) ## (tail t)" by (rule app_seqsimilar) with tg1 tg2' show ?thesis by simp qed with h show "(s ⊨ F) = (t ⊨ F)" by (simp add: stutinv_def) qed subsubsection "Properties of @{term stutinv}" text ‹ This subsection proves stuttering invariance, preservation of stuttering invariance and introduction of stuttering invariance for different formulas. First, state predicates are stuttering invariant. › theorem stut_before: "STUTINV $F" proof (clarsimp simp: stutinv_def) fix s t :: "'a seq" assume a1: "s ≈ t" hence "(first s) = (first t)" by (rule sim_first) thus "(s ⊨ $F) = (t ⊨ $F)" by (simp add: before_def) qed lemma nstut_after: "NSTUTINV F$" proof (clarsimp simp: nstutinv_def) fix s t :: "'a seq" assume a1: "tail s ≈ tail t" thus "(s ⊨ F$) = (t ⊨ F$)" by (simp add: after_def tail_sim_second) qed text‹The always operator preserves stuttering invariance.› theorem stut_always: assumes H:"STUTINV F" shows "STUTINV □F" proof (clarsimp simp: stutinv_def) fix s t :: "'a seq" assume a2: "s ≈ t" show "(s ⊨ (□ F)) = (t ⊨ (□ F))" proof assume a1: "t ⊨ □ F" show "s ⊨ □ F" proof (clarsimp simp: always_def) fix n from a2[THEN sim_step] obtain m where m: "s |⇩_{s}n ≈ t |⇩_{s}m" by blast from a1 have "(t |⇩_{s}m) ⊨ F" by (simp add: always_def) with H m show "(s |⇩_{s}n) ⊨ F" by (simp add: stutinv_def) qed next assume a1: "s ⊨ (□ F)" show "t ⊨ (□ F)" proof (clarsimp simp: always_def) fix n from a2[THEN seqsim_sym, THEN sim_step] obtain m where m: "t |⇩_{s}n ≈ s |⇩_{s}m" by blast from a1 have "(s |⇩_{s}m) ⊨ F" by (simp add: always_def) with H m show "(t |⇩_{s}n) ⊨ F" by (simp add: stutinv_def) qed qed qed text ‹ Assuming that formula @{term P} is nearly suttering invariant then ‹□[P]_v› will be stuttering invariant. › lemma stut_action_lemma: assumes H: "NSTUTINV P" and st: "s ≈ t" and P: "t ⊨ □[P]_v" shows "s ⊨ □[P]_v" proof (clarsimp simp: action_def) fix n assume "¬ ((s |⇩_{s}n) ⊨ Unchanged v)" hence v: "v (s (Suc n)) ≠ v (s n)" by (simp add: unch_defs first_def second_def suffix_def) from st[THEN sim_step] obtain m where a2': "s |⇩_{s}n ≈ t |⇩_{s}m ∧ (s |⇩_{s}Suc n ≈ t |⇩_{s}Suc m ∨ s |⇩_{s}Suc n ≈ t |⇩_{s}m)" .. hence g1: "(s |⇩_{s}n ≈ t |⇩_{s}m)" by simp hence g1'': "first (s |⇩_{s}n) = first (t |⇩_{s}m)" by (simp add: sim_first) hence g1': "s n = t m" by (simp add: suffix_def first_def) from a2' have g2: "s |⇩_{s}Suc n ≈ t |⇩_{s}Suc m ∨ s |⇩_{s}Suc n ≈ t |⇩_{s}m" by simp from P have a1': "((t |⇩_{s}m) ⊨ P) ∨ ((t |⇩_{s}m) ⊨ Unchanged v)" by (simp add: action_def) from g2 show "(s |⇩_{s}n) ⊨ P" proof assume "s |⇩_{s}Suc n ≈ t |⇩_{s}m" hence "first (s |⇩_{s}Suc n) = first (t |⇩_{s}m)" by (simp add: sim_first) hence "s (Suc n) = t m" by (simp add: suffix_def first_def) with g1' v show ?thesis by simp ― ‹by contradiction› next assume a3: "s |⇩_{s}Suc n ≈ t |⇩_{s}Suc m" hence "first (s |⇩_{s}Suc n) = first (t |⇩_{s}Suc m)" by (simp add: sim_first) hence a3': "s (Suc n) = t (Suc m)" by (simp add: suffix_def first_def) from a1' show ?thesis proof assume "(t |⇩_{s}m) ⊨ Unchanged v" hence "v (t (Suc m)) = v (t m)" by (simp add: unch_defs first_def second_def suffix_def) with g1' a3' v show ?thesis by simp ― ‹again, by contradiction› next assume a4: "(t |⇩_{s}m) ⊨ P" from a3 have "tail (s |⇩_{s}n) ≈ tail (t |⇩_{s}m)" by (simp add: tail_def suffix_plus) with H g1'' a4 show ?thesis by (auto simp: nstutinv_def) qed qed qed theorem stut_action: assumes H: "NSTUTINV P" shows "STUTINV □[P]_v" proof (clarsimp simp: stutinv_def) fix s t :: "'a seq" assume st: "s ≈ t" show "(s ⊨ □[P]_v) = (t ⊨ □[P]_v)" proof assume "t ⊨ □[P]_v" with H st show "s ⊨ □[P]_v" by (rule stut_action_lemma) next assume "s ⊨ □[P]_v" with H st[THEN seqsim_sym] show "t ⊨ □[P]_v" by (rule stut_action_lemma) qed qed text ‹ The lemmas below shows that propositional and predicate operators preserve stuttering invariance. › lemma stut_and: "⟦STUTINV F;STUTINV G⟧ ⟹ STUTINV (F ∧ G)" by (simp add: stutinv_def) lemma stut_or: "⟦STUTINV F;STUTINV G⟧ ⟹ STUTINV (F ∨ G)" by (simp add: stutinv_def) lemma stut_imp: "⟦STUTINV F;STUTINV G⟧ ⟹ STUTINV (F ⟶ G)" by (simp add: stutinv_def) lemma stut_eq: "⟦STUTINV F;STUTINV G⟧ ⟹ STUTINV (F = G)" by (simp add: stutinv_def) lemma stut_noteq: "⟦STUTINV F;STUTINV G⟧ ⟹ STUTINV (F ≠ G)" by (simp add: stutinv_def) lemma stut_not: "STUTINV F ⟹ STUTINV (¬ F)" by (simp add: stutinv_def) lemma stut_all: "(⋀x. STUTINV (F x)) ⟹ STUTINV (∀ x. F x)" by (simp add: stutinv_def) lemma stut_ex: "(⋀x. STUTINV (F x)) ⟹ STUTINV (∃ x. F x)" by (simp add: stutinv_def) lemma stut_const: "STUTINV #c" by (simp add: stutinv_def) lemma stut_fun1: "STUTINV X ⟹ STUTINV (f <X>)" by (simp add: stutinv_def) lemma stut_fun2: "⟦STUTINV X;STUTINV Y⟧ ⟹ STUTINV (f <X,Y>)" by (simp add: stutinv_def) lemma stut_fun3: "⟦STUTINV X;STUTINV Y;STUTINV Z⟧ ⟹ STUTINV (f <X,Y,Z>)" by (simp add: stutinv_def) lemma stut_fun4: "⟦STUTINV X;STUTINV Y;STUTINV Z; STUTINV W⟧ ⟹ STUTINV (f <X,Y,Z,W>)" by (simp add: stutinv_def) lemma stut_plus: "⟦STUTINV x;STUTINV y⟧ ⟹ STUTINV (x+y)" by (simp add: stutinv_def) subsubsection "Properties of @{term nstutinv}" text ‹ This subsection shows analogous properties about near stuttering invariance. If a formula @{term F} is stuttering invariant then ‹○F› is nearly stuttering invariant. › lemma nstut_nexts: assumes H: "STUTINV F" shows "NSTUTINV ○F" using H by (simp add: stutinv_def nstutinv_def nexts_def) text ‹ The lemmas below shows that propositional and predicate operators preserves near stuttering invariance. › lemma nstut_and: "⟦NSTUTINV F;NSTUTINV G⟧ ⟹ NSTUTINV (F ∧ G)" by (auto simp: nstutinv_def) lemma nstut_or: "⟦NSTUTINV F;NSTUTINV G⟧ ⟹ NSTUTINV (F ∨ G)" by (auto simp: nstutinv_def) lemma nstut_imp: "⟦NSTUTINV F;NSTUTINV G⟧ ⟹ NSTUTINV (F ⟶ G)" by (auto simp: nstutinv_def) lemma nstut_eq: "⟦NSTUTINV F; NSTUTINV G⟧ ⟹ NSTUTINV (F = G)" by (force simp: nstutinv_def) lemma nstut_not: "NSTUTINV F ⟹ NSTUTINV (¬ F)" by (auto simp: nstutinv_def) lemma nstut_noteq: "⟦NSTUTINV F; NSTUTINV G⟧ ⟹ NSTUTINV (F ≠ G)" by (simp add: nstut_eq nstut_not) lemma nstut_all: "(⋀x. NSTUTINV (F x)) ⟹ NSTUTINV (∀ x. F x)" by (auto simp: nstutinv_def) lemma nstut_ex: "(⋀x. NSTUTINV (F x)) ⟹ NSTUTINV (∃ x. F x)" by (auto simp: nstutinv_def) lemma nstut_const: "NSTUTINV #c" by (auto simp: nstutinv_def) lemma nstut_fun1: "NSTUTINV X ⟹ NSTUTINV (f <X>)" by (force simp: nstutinv_def) lemma nstut_fun2: "⟦NSTUTINV X; NSTUTINV Y⟧ ⟹ NSTUTINV (f <X,Y>)" by (force simp: nstutinv_def) lemma nstut_fun3: "⟦NSTUTINV X; NSTUTINV Y; NSTUTINV Z⟧ ⟹ NSTUTINV (f <X,Y,Z>)" by (force simp: nstutinv_def) lemma nstut_fun4: "⟦NSTUTINV X; NSTUTINV Y; NSTUTINV Z; NSTUTINV W⟧ ⟹ NSTUTINV (f <X,Y,Z,W>)" by (force simp: nstutinv_def) lemma nstut_plus: "⟦NSTUTINV x;NSTUTINV y⟧ ⟹ NSTUTINV (x+y)" by (simp add: nstut_fun2) subsubsection "Abbreviations" text ‹ We show the obvious fact that the same properties holds for abbreviated operators. › lemmas nstut_before = stut_before[THEN stutinv_strictly_stronger] lemma nstut_unch: "NSTUTINV (Unchanged v)" proof (unfold unch_def) have g1: "NSTUTINV v$" by (rule nstut_after) have "NSTUTINV $v" by (rule stut_before[THEN stutinv_strictly_stronger]) with g1 show "NSTUTINV (v$ = $v)" by (rule nstut_eq) qed text‹ Formulas ‹[P]_v› are not \tlastar{} formulas by themselves, but we need to reason about them when they appear wrapped inside ‹□[-]_v›. We only require that it preserves nearly stuttering invariance. Observe that ‹[P]_v› trivially holds for a stuttering step, so it cannot be stuttering invariant. › lemma nstut_actrans: "NSTUTINV P ⟹ NSTUTINV [P]_v" by (simp add: actrans_def nstut_unch nstut_or) lemma stut_eventually: "STUTINV F ⟹ STUTINV ◇F" by (simp add: eventually_def stut_not stut_always) lemma stut_leadsto: "⟦STUTINV F; STUTINV G⟧ ⟹ STUTINV (F ↝ G)" by (simp add: leadsto_def stut_always stut_eventually stut_imp) lemma stut_angle_action: "NSTUTINV P ⟹ STUTINV ◇⟨P⟩_v" by (simp add: angle_action_def nstut_not stut_action stut_not) lemma nstut_angle_acttrans: "NSTUTINV P ⟹ NSTUTINV ⟨P⟩_v" by (simp add: angle_actrans_def nstut_not nstut_actrans) lemmas stutinvs = stut_before stut_always stut_action stut_and stut_or stut_imp stut_eq stut_noteq stut_not stut_all stut_ex stut_eventually stut_leadsto stut_angle_action stut_const stut_fun1 stut_fun2 stut_fun3 stut_fun4 lemmas nstutinvs = nstut_after nstut_nexts nstut_actrans nstut_unch nstut_and nstut_or nstut_imp nstut_eq nstut_noteq nstut_not nstut_all nstut_ex nstut_angle_acttrans stutinv_strictly_stronger nstut_fun1 nstut_fun2 nstut_fun3 nstut_fun4 stutinvs[THEN stutinv_strictly_stronger] lemmas bothstutinvs = stutinvs nstutinvs end

# Theory PreFormulas

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section "Reasoning about PreFormulas" theory PreFormulas imports Semantics begin text‹ Semantic separation of formulas and pre-formulas requires a deep embedding. We introduce a syntactically distinct notion of validity, written ‹|~ A›, for pre-formulas. Although it is semantically identical to ‹⊢ A›, it helps users distinguish pre-formulas from formulas in \tlastar{} proofs. › definition PreValid :: "('w::world) form ⇒ bool" where "PreValid A ≡ ∀ w. w ⊨ A" syntax "_PreValid" :: "lift ⇒ bool" ("(|~ _)" 5) translations "_PreValid" ⇌ "CONST PreValid" lemma prefD[dest]: "|~ A ⟹ w ⊨ A" by (simp add: PreValid_def) lemma prefI[intro!]: "(⋀ w. w ⊨ A) ⟹ |~ A" by (simp add: PreValid_def) method_setup pref_unlift = ‹ Scan.succeed (fn ctxt => SIMPLE_METHOD' (resolve_tac ctxt @{thms prefI} THEN' rewrite_goal_tac ctxt @{thms intensional_rews})) › "int_unlift for PreFormulas" lemma prefeq_reflection: assumes P1: "|~ x=y" shows "(x ≡ y)" using P1 by (intro eq_reflection) force lemma pref_True[simp]: "|~ #True" by auto lemma pref_eq: "|~ X = Y ⟹ X = Y" by (auto simp: prefeq_reflection) lemma pref_iffI: assumes "|~ F ⟶ G" and "|~ G ⟶ F" shows "|~ F = G" using assms by force lemma pref_iffD1: assumes "|~ F = G" shows "|~ F ⟶ G" using assms by auto lemma pref_iffD2: assumes "|~ F = G" shows "|~ G ⟶ F" using assms by auto lemma unl_pref_imp: assumes "|~ F ⟶ G" shows "⋀ w. w ⊨ F ⟹ w ⊨ G" using assms by auto lemma pref_imp_trans: assumes "|~ F ⟶ G" and "|~ G ⟶ H" shows "|~ F ⟶ H" using assms by force subsection "Lemmas about ‹Unchanged›" text ‹ Many of the \tlastar{} axioms only require a state function witness which leaves the state space unchanged. An obvious witness is the @{term id} function. The lemmas require that the given formula is invariant under stuttering. › lemma pre_id_unch: assumes h: "stutinv F" shows "|~ F ∧ Unchanged id ⟶ ○F" proof (pref_unlift, clarify) fix s assume a1: "s ⊨ F" and a2: "s ⊨ Unchanged id" from a2 have "(id (second s) = id (first s))" by (simp add: unch_defs) hence "s ≈ (tail s)" by (simp add: addfirststut) with h a1 have "(tail s) ⊨ F" by (simp add: stutinv_def) thus "s ⊨ ○F" by (unfold nexts_def) qed lemma pre_ex_unch: assumes h: "stutinv F" shows "∃(v::'a::world ⇒ 'a). ( |~ F ∧ Unchanged v ⟶ ○F)" using pre_id_unch[OF h] by blast lemma unch_pair: "|~ Unchanged (x,y) = (Unchanged x ∧ Unchanged y)" by (auto simp: unch_def before_def after_def nexts_def) lemmas unch_eq1 = unch_pair[THEN pref_eq] lemmas unch_eq2 = unch_pair[THEN prefeq_reflection] lemma angle_actrans_sem: "|~ ⟨F⟩_v = (F ∧ v$ ≠ $v)" by (auto simp: angle_actrans_def actrans_def unch_def) lemmas angle_actrans_sem_eq = angle_actrans_sem[THEN pref_eq] subsection "Lemmas about ‹after›" lemma after_const: "|~ (#c)` = #c" by (auto simp: nexts_def before_def after_def) lemma after_fun1: "|~ f<x>` = f<x`>" by (auto simp: nexts_def before_def after_def) lemma after_fun2: "|~ f<x,y>` = f <x`,y`>" by (auto simp: nexts_def before_def after_def) lemma after_fun3: "|~ f<x,y,z>` = f <x`,y`,z`>" by (auto simp: nexts_def before_def after_def) lemma after_fun4: "|~ f<x,y,z,zz>` = f <x`,y`,z`,zz`>" by (auto simp: nexts_def before_def after_def) lemma after_forall: "|~ (∀ x. P x)` = (∀ x. (P x)`)" by (auto simp: nexts_def before_def after_def) lemma after_exists: "|~ (∃ x. P x)` = (∃ x. (P x)`)" by (auto simp: nexts_def before_def after_def) lemma after_exists1: "|~ (∃! x. P x)` = (∃! x. (P x)`)" by (auto simp: nexts_def before_def after_def) lemmas all_after = after_const after_fun1 after_fun2 after_fun3 after_fun4 after_forall after_exists after_exists1 lemmas all_after_unl = all_after[THEN prefD] lemmas all_after_eq = all_after[THEN prefeq_reflection] subsection "Lemmas about ‹before›" lemma before_const: "⊢ $(#c) = #c" by (auto simp: before_def) lemma before_fun1: "⊢ $(f<x>) = f <$x>" by (auto simp: before_def) lemma before_fun2: "⊢ $(f<x,y>) = f <$x,$y>" by (auto simp: before_def) lemma before_fun3: "⊢ $(f<x,y,z>) = f <$x,$y,$z>" by (auto simp: before_def) lemma before_fun4: "⊢ $(f<x,y,z,zz>) = f <$x,$y,$z,$zz>" by (auto simp: before_def) lemma before_forall: "⊢ $(∀ x. P x) = (∀ x. $(P x))" by (auto simp: before_def) lemma before_exists: "⊢ $(∃ x. P x) = (∃ x. $(P x))" by (auto simp: before_def) lemma before_exists1: "⊢ $(∃! x. P x) = (∃! x. $(P x))" by (auto simp: before_def) lemmas all_before = before_const before_fun1 before_fun2 before_fun3 before_fun4 before_forall before_exists before_exists1 lemmas all_before_unl = all_before[THEN intD] lemmas all_before_eq = all_before[THEN inteq_reflection] subsection "Some general properties" lemma angle_actrans_conj: "|~ (⟨F ∧ G⟩_v) = (⟨F⟩_v ∧ ⟨G⟩_v)" by (auto simp: angle_actrans_def actrans_def unch_def) lemma angle_actrans_disj: "|~ (⟨F ∨ G⟩_v) = (⟨F⟩_v ∨ ⟨G⟩_v)" by (auto simp: angle_actrans_def actrans_def unch_def) lemma int_eq_true: "⊢ P ⟹ ⊢ P = #True" by auto lemma pref_eq_true: "|~ P ⟹ |~ P = #True" by auto subsection "Unlifting attributes and methods" text ‹Attribute which unlifts an intensional formula or preformula› ML ‹ fun unl_rewr ctxt thm = let val unl = (thm RS @{thm intD}) handle THM _ => (thm RS @{thm prefD}) handle THM _ => thm val rewr = rewrite_rule ctxt @{thms intensional_rews} in unl |> rewr end; › attribute_setup unlifted = ‹ Scan.succeed (Thm.rule_attribute [] (unl_rewr o Context.proof_of)) › "unlift intensional formulas" attribute_setup unlift_rule = ‹ Scan.succeed (Thm.rule_attribute [] (Context.proof_of #> (fn ctxt => Object_Logic.rulify ctxt o unl_rewr ctxt))) › "unlift and rulify intensional formulas" text ‹ Attribute which turns an intensional formula or preformula into a rewrite rule. Formulas ‹F› that are not equalities are turned into ‹F ≡ #True›. › ML ‹ fun int_rewr thm = (thm RS @{thm inteq_reflection}) handle THM _ => (thm RS @{thm prefeq_reflection}) handle THM _ => ((thm RS @{thm int_eq_true}) RS @{thm inteq_reflection}) handle THM _ => ((thm RS @{thm pref_eq_true}) RS @{thm prefeq_reflection}); › attribute_setup simp_unl = ‹ Attrib.add_del (Thm.declaration_attribute (fn th => Simplifier.map_ss (Simplifier.add_simp (int_rewr th)))) (K (NONE, NONE)) (* note only adding -- removing is ignored *) › "add thm unlifted from rewrites from intensional formulas or preformulas" attribute_setup int_rewrite = ‹Scan.succeed (Thm.rule_attribute [] (fn _ => int_rewr))› "produce rewrites from intensional formulas or preformulas" end

# Theory Rules

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section "A Proof System for TLA* " theory Rules imports PreFormulas begin text‹ We prove soundness of the proof system of \tlastar{}, from which the system verification rules from Lamport's original TLA paper will be derived. This theory is still state-independent, thus state-dependent enableness proofs, required for proofs based on fairness assumptions, and flexible quantification, are not discussed here. The \tlastar{} paper \cite{Merz99} suggest both a \emph{hetereogeneous} and a \emph{homogenous} proof system for \tlastar{}. The homogeneous version eliminates the auxiliary definitions from the ‹Preformula› theory, creating a single provability relation. This axiomatisation is based on the fact that a pre-formula can only be used via the ‹sq› rule. In a nutshell, ‹sq› is applied to ‹pax1› to ‹pax5›, and ‹nex›, ‹pre› and ‹pmp› are changed to accommodate this. It is argued that while the hetereogenous version is easier to understand, the homogenous system avoids the introduction of an auxiliary provability relation. However, the price to pay is that reasoning about pre-formulas (in particular, actions) has to be performed in the scope of temporal operators such as ‹□[P]_v›, which is notationally quite heavy, We prefer here the heterogeneous approach, which exposes the pre-formulas and lets us use standard HOL rules more directly. › subsection "The Basic Axioms" theorem fmp: assumes "⊢ F" and "⊢ F ⟶ G" shows "⊢ G" using assms[unlifted] by auto theorem pmp: assumes "|~ F" and "|~ F ⟶ G" shows "|~ G" using assms[unlifted] by auto theorem sq: assumes "|~ P" shows "⊢ □[P]_v" using assms[unlifted] by (auto simp: action_def) theorem pre: assumes "⊢ F" shows "|~ F" using assms by auto theorem nex: assumes h1: "⊢ F" shows "|~ ○F" using assms by (auto simp: nexts_def) theorem ax0: "⊢ # True" by auto theorem ax1: "⊢ □F ⟶ F" proof (clarsimp simp: always_def) fix w assume "∀n. (w |⇩_{s}n) ⊨ F" hence "(w |⇩_{s}0) ⊨ F" .. thus "w ⊨ F" by simp qed theorem ax2: "⊢ □F ⟶ □[□F]_v" by (auto simp: always_def action_def suffix_plus) theorem ax3: assumes H: "|~ F ∧ Unchanged v ⟶ ○F" shows "⊢ □[F ⟶ ○F]_v ⟶ (F ⟶ □F)" proof (clarsimp simp: always_def) fix w n assume a1: "w ⊨ □[F ⟶ ○F]_v" and a2: "w ⊨ F" show "(w |⇩_{s}n) ⊨ F" proof (induct n) from a2 show "(w |⇩_{s}0) ⊨ F" by simp next fix m assume a3: "(w |⇩_{s}m) ⊨ F" with a1 H[unlifted] show "(w |⇩_{s}(Suc m)) ⊨ F" by (auto simp: nexts_def action_def tail_suffix_suc) qed qed theorem ax4: "⊢ □[P ⟶ Q]_v ⟶ (□[P]_v ⟶ □[Q]_v)" by (force simp: action_def) theorem ax5: "⊢ □[v` ≠ $v]_v" by (auto simp: action_def unch_def) theorem pax0: "|~ # True" by auto theorem pax1 [simp_unl]: "|~ (○¬F) = (¬○F)" by (auto simp: nexts_def) theorem pax2: "|~ ○(F ⟶ G) ⟶ (○F ⟶ ○G)" by (auto simp: nexts_def) theorem pax3: "|~ □F ⟶ ○□F" by (auto simp: always_def nexts_def tail_def suffix_plus) theorem pax4: "|~ □[P]_v = ([P]_v ∧ ○□[P]_v)" proof (auto) fix w assume "w ⊨ □[P]_v" from this[unfolded action_def] have "((w |⇩_{s}0) ⊨ P) ∨ ((w |⇩_{s}0) ⊨ Unchanged v)" .. thus "w ⊨ [P]_v" by (simp add: actrans_def) next fix w assume "w ⊨ □[P]_v" thus "w ⊨ ○□[P]_v" by (auto simp: nexts_def action_def tail_def suffix_plus) next fix w assume 1: "w ⊨ [P]_v" and 2: "w ⊨ ○□[P]_v" show "w ⊨ □[P]_v" proof (auto simp: action_def) fix i assume 3: "¬ ((w |⇩_{s}i) ⊨ Unchanged v)" show "(w |⇩_{s}i) ⊨ P" proof (cases i) assume "i = 0" with 1 3 show ?thesis by (simp add: actrans_def) next fix j assume "i = Suc j" with 2 3 show ?thesis by (auto simp: nexts_def action_def tail_def suffix_plus) qed qed qed theorem pax5: "|~ ○□F ⟶ □[○F]_v" by (auto simp: nexts_def always_def action_def tail_def suffix_plus) text ‹ Theorem to show that universal quantification distributes over the always operator. Since the \tlastar{} paper only addresses the propositional fragment, this theorem does not appear there. › theorem allT: "⊢ (∀x. □(F x)) = (□(∀x. F x))" by (auto simp: always_def) theorem allActT: "⊢ (∀x. □[F x]_v) = (□[(∀x. F x)]_v)" by (force simp: action_def) subsection "Derived Theorems" text‹ This section includes some derived theorems based on the axioms, taken from the \tlastar{} paper~\cite{Merz99}. We mimic the proofs given there and avoid semantic reasoning whenever possible. The ‹alw› theorem of~\cite{Merz99} states that if F holds in all worlds then it always holds, i.e. $F \vDash \Box F$. However, the derivation of this theorem (using the proof rules above) relies on access of the set of free variables (FV), which is not available in a shallow encoding. However, we can prove a similar rule ‹alw2› using an additional hypothesis @{term "|~ F ∧ Unchanged v ⟶ ○F"}. › theorem alw2: assumes h1: "⊢ F" and h2: "|~ F ∧ Unchanged v ⟶ ○F" shows "⊢ □F" proof - from h1 have g2: "|~ ○F" by (rule nex) hence g3: "|~ F ⟶ ○F" by auto hence g4:"⊢ □[(F ⟶ ○F)]_v" by (rule sq) from h2 have "⊢ □[(F ⟶ ○F)]_v ⟶ F ⟶ □F" by (rule ax3) with g4[unlifted] have g5: "⊢ F ⟶ □F" by auto with h1[unlifted] show ?thesis by auto qed text‹ Similar theorem, assuming that @{term "F"} is stuttering invariant. › theorem alw3: assumes h1: "⊢ F" and h2: "stutinv F" shows "⊢ □F" proof - from h2 have "|~ F ∧ Unchanged id ⟶ ○F" by (rule pre_id_unch) with h1 show ?thesis by (rule alw2) qed text‹ In a deep embedding, we could prove that all (proper) \tlastar{} formulas are stuttering invariant and then get rid of the second hypothesis of rule ‹alw3›. In fact, the rule is even true for pre-formulas, as shown by the following rule, whose proof relies on semantical reasoning. › theorem alw: assumes H1: "⊢ F" shows "⊢ □F" using H1 by (auto simp: always_def) theorem alw_valid_iff_valid: "(⊢ □F) = (⊢ F)" proof assume "⊢ □F" from this ax1 show "⊢ F" by (rule fmp) qed (rule alw) text ‹ \cite{Merz99} proves the following theorem using the deduction theorem of \tlastar{}: ‹(⊢ F ⟹ ⊢ G) ⟹ ⊢ []F ⟶ G›, which can only be proved by induction on the formula structure, in a deep embedding. › theorem T1[simp_unl]: "⊢ □□F = []F" proof (auto simp: always_def suffix_plus) fix w n assume "∀m k. (w |⇩_{s}(k+m)) ⊨ F" hence "(w |⇩_{s}(n+0)) ⊨ F" by blast thus "(w |⇩_{s}n) ⊨ F" by simp qed theorem T2[simp_unl]: "⊢ □□[P]_v = □[P]_v" proof - have 1: "|~ □[P]_v ⟶ ○□[P]_v" using pax4 by force hence "⊢ □[□[P]_v ⟶ ○□[P]_v]_v" by (rule sq) moreover have "⊢ □[ □[P]_v ⟶ ○□[P]_v ]_v ⟶ □[P]_v ⟶ □□[P]_v" by (rule ax3) (auto elim: 1[unlift_rule]) moreover have "⊢ □□[P]_v ⟶ □[P]_v" by (rule ax1) ultimately show ?thesis by force qed theorem T3[simp_unl]: "⊢ □[[P]_v]_v = □[P]_v" proof - have "|~ P ⟶ [P]_v" by (auto simp: actrans_def) hence "⊢ □[(P ⟶ [P]_v)]_v" by (rule sq) with ax4 have "⊢ □[P]_v ⟶ □[[P]_v]_v" by force moreover have "|~ [P]_v ⟶ v`≠ $v ⟶ P" by (auto simp: unch_def actrans_def) hence "⊢ □[[P]_v ⟶ v`≠ $v ⟶ P]_v" by (rule sq) with ax5 have "⊢ □[[P]_v]_v ⟶ □[P]_v" by (force intro: ax4[unlift_rule]) ultimately show ?thesis by force qed theorem M2: assumes h: "|~ F ⟶ G" shows "⊢ □[F]_v ⟶ □[G]_v" using sq[OF h] ax4 by force theorem N1: assumes h: "⊢ F ⟶ G" shows "|~ ○F ⟶ ○G" by (rule pmp[OF nex[OF h] pax2]) theorem T4: "⊢ □[P]_v ⟶ □[[P]_v]_w" proof - have "⊢ □□[P]_v ⟶ □[□□[P]_v]_w" by (rule ax2) moreover from pax4 have "|~ □□[P]_v ⟶ [P]_v" unfolding T2[int_rewrite] by force hence "⊢ □[□□[P]_v]_w ⟶ □[[P]_v]_w" by (rule M2) ultimately show ?thesis unfolding T2[int_rewrite] by (rule lift_imp_trans) qed theorem T5: "⊢ □[[P]_w]_v ⟶ □[[P]_v]_w" proof - have "|~ [[P]_w]_v ⟶ [[P]_v]_w" by (auto simp: actrans_def) hence "⊢ □[[[P]_w]_v]_w ⟶ □[[[P]_v]_w]_w" by (rule M2) with T4 show ?thesis unfolding T3[int_rewrite] by (rule lift_imp_trans) qed theorem T6: "⊢ □F ⟶ □[○F]_v" proof - from ax1 have "|~ ○(□F ⟶ F)" by (rule nex) with pax2 have "|~ ○□F ⟶ ○F" by force with pax3 have "|~ □F ⟶ ○F" by (rule pref_imp_trans) hence "⊢ □[□F]_v ⟶ □[○F]_v" by (rule M2) with ax2 show ?thesis by (rule lift_imp_trans) qed theorem T7: assumes h: "|~ F ∧ Unchanged v ⟶ ○F" shows "|~ (F ∧ ○□F) = □F" proof - have "⊢ □[○F ⟶ F ⟶ ○F]_v" by (rule sq) auto with ax4 have "⊢ □[○F]_v ⟶ □[(F ⟶ ○F)]_v" by force with ax3[OF h, unlifted] have "⊢ □[○F]_v ⟶ (F ⟶ □F)" by force with pax5 have "|~ F ∧ ○□F ⟶ □F" by force with ax1[of "TEMP F",unlifted] pax3[of "TEMP F",unlifted] show ?thesis by force qed theorem T8: "|~ ○(F ∧ G) = (○F ∧ ○G)" proof - have "|~ ○(F ∧ G) ⟶ ○F" by (rule N1) auto moreover have "|~ ○(F ∧ G) ⟶ ○G" by (rule N1) auto moreover have "⊢ F ⟶ G ⟶ F ∧ G" by auto from nex[OF this] have "|~ ○F ⟶ ○G ⟶ ○(F ∧ G)" by (force intro: pax2[unlift_rule]) ultimately show ?thesis by force qed lemma T9: "|~ □[A]_v ⟶ [A]_v" using pax4 by force theorem H1: assumes h1: "⊢ □[P]_v" and h2: "⊢ □[P ⟶ Q]_v" shows "⊢ □[Q]_v" using assms ax4[unlifted] by force theorem H2: assumes h1: "⊢ F" shows "⊢ □[F]_v" using h1 by (blast dest: pre sq) theorem H3: assumes h1: "⊢ □[P ⟶ Q]_v" and h2: "⊢ □[Q ⟶ R]_v" shows "⊢ □[P ⟶ R]_v" proof - have "|~ (P ⟶ Q) ⟶ (Q ⟶ R) ⟶ (P ⟶ R)" by auto hence "⊢ □[(P ⟶ Q) ⟶ (Q ⟶ R) ⟶ (P ⟶ R)]_v" by (rule sq) with h1 have "⊢ □[(Q ⟶ R) ⟶ (P ⟶ R)]_v" by (rule H1) with h2 show ?thesis by (rule H1) qed theorem H4: "⊢ □[[P]_v ⟶ P]_v" proof - have "|~ v` ≠ $v ⟶ ([P]_v ⟶ P)" by (auto simp: unch_def actrans_def) hence "⊢ □[v` ≠ $v ⟶ ([P]_v ⟶ P)]_v" by (rule sq) with ax5 show ?thesis by (rule H1) qed theorem H5: "⊢ □[□F ⟶ ○□F]_v" by (rule pax3[THEN sq]) subsection "Some other useful derived theorems" theorem P1: "|~ □F ⟶ ○F" proof - have "|~ ○□F ⟶ ○F" by (rule N1[OF ax1]) with pax3 show ?thesis by (rule pref_imp_trans) qed theorem P2: "|~ □F ⟶ F ∧ ○F" using ax1[of F] P1[of F] by force theorem P4: "⊢ □F ⟶ □[F]_v" proof - have "⊢ □[□F]_v ⟶ □[F]_v" by (rule M2[OF pre[OF ax1]]) with ax2 show ?thesis by (rule lift_imp_trans) qed theorem P5: "⊢ □[P]_v ⟶ □[□[P]_v]_w" proof - have "⊢ □□[P]_v ⟶ □[□[P]_v]_w" by (rule P4) thus ?thesis by (unfold T2[int_rewrite]) qed theorem M0: "⊢ □F ⟶ □[F ⟶ ○F]_v" proof - from P1 have "|~ □F ⟶ F ⟶ ○F" by force hence "⊢ □[□F]_v ⟶ □[F ⟶ ○F]_v" by (rule M2) with ax2 show ?thesis by (rule lift_imp_trans) qed theorem M1: "⊢ □F ⟶ □[F ∧ ○F]_v" proof - have "|~ □F ⟶ F ∧ ○F" by (rule P2) hence "⊢ □[□F]_v ⟶ □[F ∧ ○F]_v" by (rule M2) with ax2 show ?thesis by (rule lift_imp_trans) qed theorem M3: assumes h: "⊢ F" shows "⊢ □[○F]_v" using alw[OF h] T6 by (rule fmp) lemma M4: "⊢ □[○(F ∧ G) = (○F ∧ ○G)]_v" by (rule sq[OF T8]) theorem M5: "⊢ □[ □[P]_v ⟶ ○□[P]_v ]_w" proof (rule sq) show "|~ □[P]_v ⟶ ○□[P]_v" by (auto simp: pax4[unlifted]) qed theorem M6: "⊢ □[F ∧ G]_v ⟶ □[F]_v ∧ □[G]_v" proof - have "⊢ □[F ∧ G]_v ⟶ □[F]_v" by (rule M2) auto moreover have "⊢ □[F ∧ G]_v ⟶ □[G]_v" by (rule M2) auto ultimately show ?thesis by force qed theorem M7: "⊢ □[F]_v ∧ □[G]_v ⟶ □[F ∧ G]_v" proof - have "|~ F ⟶ G ⟶ F ∧ G" by auto hence "⊢ □[F]_v ⟶ □[G ⟶ F ∧ G]_v" by (rule M2) with ax4 show ?thesis by force qed theorem M8: "⊢ □[F ∧ G]_v = (□[F]_v ∧ □[G]_v)" by (rule int_iffI[OF M6 M7]) theorem M9: "|~ □F ⟶ F ∧ ○□F" using pre[OF ax1[of "F"]] pax3[of "F"] by force theorem M10: assumes h: "|~ F ∧ Unchanged v ⟶ ○F" shows "|~ F ∧ ○□F ⟶ □F" using T7[OF h] by auto theorem M11: assumes h: "|~ [A]_f ⟶ [B]_g" shows "⊢ □[A]_f ⟶ □[B]_g" proof - from h have "⊢ □[[A]_f]_g ⟶ □[[B]_g]_g" by (rule M2) with T4 show ?thesis by force qed theorem M12: "⊢ (□[A]_f ∧ □[B]_g) = □[[A]_f ∧ [B]_g]_(f,g)" proof - have "⊢ □[A]_f ∧ □[B]_g ⟶ □[[A]_f ∧ [B]_g]_(f,g)" by (auto simp: M8[int_rewrite] elim: T4[unlift_rule]) moreover have "|~ [[A]_f ∧ [B]_g]_(f,g) ⟶ [A]_f" by (auto simp: actrans_def unch_def all_before_eq all_after_eq) hence "⊢ □[[A]_f ∧ [B]_g]_(f,g) ⟶ □[A]_f" by (rule M11) moreover have "|~ [[A]_f ∧ [B]_g]_(f,g) ⟶ [B]_g" by (auto simp: actrans_def unch_def all_before_eq all_after_eq) hence "⊢ □[[A]_f ∧ [B]_g]_(f,g) ⟶ □[B]_g" by (rule M11) ultimately show ?thesis by force qed text ‹ We now derive Lamport's 6 simple temporal logic rules (STL1)-(STL6) \cite{Lamport94}. Firstly, STL1 is the same as @{thm alw} derived above. › lemmas STL1 = alw text ‹ STL2 and STL3 have also already been derived. › lemmas STL2 = ax1 lemmas STL3 = T1 text ‹ As with the derivation of @{thm alw}, a purely syntactic derivation of (STL4) relies on an additional argument -- either using ‹Unchanged› or ‹STUTINV›. › theorem STL4_2: assumes h1: "⊢ F ⟶ G" and h2: "|~ G ∧ Unchanged v ⟶ ○G" shows "⊢ □F ⟶ □G" proof - from ax1[of F] h1 have "⊢ □F ⟶ G" by (rule lift_imp_trans) moreover from h1 have "|~ ○F ⟶ ○G" by (rule N1) hence "|~ ○F ⟶ G ⟶ ○G" by auto hence "⊢ □[○F]_v ⟶ □[G ⟶ ○G]_v" by (rule M2) with T6 have "⊢ □F ⟶ □[G ⟶ ○G]_v" by (rule lift_imp_trans) moreover from h2 have "⊢ □[G ⟶ ○G]_v ⟶ G ⟶ □G" by (rule ax3) ultimately show ?thesis by force qed lemma STL4_3: assumes h1: "⊢ F ⟶ G" and h2: "STUTINV G" shows "⊢ □F ⟶ □G" using h1 h2[THEN pre_id_unch] by (rule STL4_2) text ‹Of course, the original rule can be derived semantically› lemma STL4: assumes h: "⊢ F ⟶ G" shows "⊢ □F ⟶ □G" using h by (force simp: always_def) text ‹Dual rule for ‹◇›› lemma STL4_eve: assumes h: "⊢ F ⟶ G" shows "⊢ ◇F ⟶ ◇G" using h by (force simp: eventually_defs) text‹ Similarly, a purely syntactic derivation of (STL5) requires extra hypotheses. › theorem STL5_2: assumes h1: "|~ F ∧ Unchanged f ⟶ ○F" and h2: "|~ G ∧ Unchanged g ⟶ ○G" shows "⊢ □(F ∧ G) = (□F ∧ □G)" proof (rule int_iffI) have "⊢ F ∧ G ⟶ F" by auto from this h1 have "⊢ □(F ∧ G) ⟶ □F" by (rule STL4_2) moreover have "⊢ F ∧ G ⟶ G" by auto from this h2 have "⊢ □(F ∧ G) ⟶ □G" by (rule STL4_2) ultimately show "⊢ □(F ∧ G) ⟶ □F ∧ □G" by force next have "|~ Unchanged (f,g) ⟶ Unchanged f ∧ Unchanged g" by (auto simp: unch_defs) with h1[unlifted] h2[unlifted] T8[of F G, unlifted] have h3: "|~ (F ∧ G) ∧ Unchanged (f,g) ⟶ ○(F ∧ G)" by force from ax1[of F] ax1[of G] have "⊢ □F ∧ □G ⟶ F ∧ G" by force moreover from ax2[of F] ax2[of G] have "⊢ □F ∧ □G ⟶ □[□F]_(f,g) ∧ □[□G]_(f,g)" by force with M8 have "⊢ □F ∧ □G ⟶ □[□F ∧ □G]_(f,g)" by force moreover from P1[of F] P1[of G] have "|~ □F ∧ □G ⟶ F ∧ G ⟶ ○(F ∧ G)" unfolding T8[int_rewrite] by force hence "⊢ □[ □F ∧ □G ]_(f,g) ⟶ □[F ∧ G ⟶ ○(F ∧ G)]_(f,g)" by (rule M2) from this ax3[OF h3] have "⊢ □[ □F ∧ □G ]_(f,g) ⟶ F ∧ G ⟶ □(F ∧ G)" by (rule lift_imp_trans) ultimately show "⊢ □F ∧ □G ⟶ □(F ∧ G)" by force qed theorem STL5_21: assumes h1: "stutinv F" and h2: "stutinv G" shows "⊢ □(F ∧ G) = (□F ∧ □G)" using h1[THEN pre_id_unch] h2[THEN pre_id_unch] by (rule STL5_2) text ‹We also derive STL5 semantically.› lemma STL5: "⊢ □(F ∧ G) = (□F ∧ □G)" by (auto simp: always_def) text ‹Elimination rule corresponding to ‹STL5› in unlifted form.› lemma box_conjE: assumes "s ⊨ □F" and "s ⊨ □G" and "s ⊨ □(F ∧ G) ⟹ P" shows "P" using assms by (auto simp: STL5[unlifted]) lemma box_thin: assumes h1: "s ⊨ □F" and h2: "PROP W" shows "PROP W" using h2 . text ‹Finally, we derive STL6 (only semantically)› lemma STL6: "⊢ ◇□(F ∧ G) = (◇□F ∧ ◇□G)" proof auto fix w assume a1: "w ⊨ ◇□F" and a2: "w ⊨ ◇□G" from a1 obtain nf where nf: "(w |⇩_{s}nf) ⊨ □F" by (auto simp: eventually_defs) from a2 obtain ng where ng: "(w |⇩_{s}ng) ⊨ □G" by (auto simp: eventually_defs) let ?n = "max nf ng" have "nf ≤ ?n" by simp from this nf have "(w |⇩_{s}?n) ⊨ □F" by (rule linalw) moreover have "ng ≤ ?n" by simp from this ng have "(w |⇩_{s}?n) ⊨ □G" by (rule linalw) ultimately have "(w |⇩_{s}?n) ⊨ □(F ∧ G)" by (rule box_conjE) thus "w ⊨ ◇□(F ∧ G)" by (auto simp: eventually_defs) next fix w assume h: "w ⊨ ◇□(F ∧ G)" have "⊢ F ∧ G ⟶ F" by auto hence "⊢ ◇□(F ∧ G) ⟶ ◇□F" by (rule STL4_eve[OF STL4]) with h show "w ⊨ ◇□F" by auto next fix w assume h: "w ⊨ ◇□(F ∧ G)" have "⊢ F ∧ G ⟶ G" by auto hence "⊢ ◇□(F ∧ G) ⟶ ◇□G" by (rule STL4_eve[OF STL4]) with h show "w ⊨ ◇□G" by auto qed lemma MM0: "⊢ □(F ⟶ G) ⟶ □F ⟶ □G" proof - have "⊢ □(F ∧ (F ⟶ G)) ⟶ □G" by (rule STL4) auto thus ?thesis by (auto simp: STL5[int_rewrite]) qed lemma MM1: assumes h: "⊢ F = G" shows "⊢ □F = □G" by (auto simp: h[int_rewrite]) theorem MM2: "⊢ □A ∧ □(B ⟶ C) ⟶ □(A ∧ B ⟶ C)" proof - have "⊢ □(A ∧ (B ⟶ C)) ⟶ □(A ∧ B ⟶ C)" by (rule STL4) auto thus ?thesis by (auto simp: STL5[int_rewrite]) qed theorem MM3: "⊢ □¬A ⟶ □(A ∧ B ⟶ C)" by (rule STL4) auto theorem MM4[simp_unl]: "⊢ □#F = #F" proof (cases "F") assume "F" hence 1: "⊢ #F" by auto hence "⊢ □#F" by (rule alw) with 1 show ?thesis by force next assume "¬F" hence 1: "⊢ ¬ #F" by auto from ax1 have "⊢ ¬ #F ⟶ ¬ □#F" by (rule lift_imp_neg) with 1 show ?thesis by force qed theorem MM4b[simp_unl]: "⊢ □¬#F = ¬#F" proof - have "⊢ ¬#F = #(¬F)" by auto hence "⊢ □¬#F = □#(¬F)" by (rule MM1) thus ?thesis by auto qed theorem MM5: "⊢ □F ∨ □G ⟶ □(F ∨ G)" proof - have "⊢ □F ⟶ □(F ∨ G)" by (rule STL4) auto moreover have "⊢ □G ⟶ □(F ∨ G)" by (rule STL4) auto ultimately show ?thesis by force qed theorem MM6: "⊢ □F ∨ □G ⟶ □(□F ∨ □G)" proof - have "⊢ □□F ∨ □□G ⟶ □(□F ∨ □G)" by (rule MM5) thus ?thesis by simp qed lemma MM10: assumes h: "|~ F = G" shows "⊢ □[F]_v = □[G]_v" by (auto simp: h[int_rewrite]) lemma MM9: assumes h: "⊢ F = G" shows "⊢ □[F]_v = □[G]_v" by (rule MM10[OF pre[OF h]]) theorem MM11: "⊢ □[¬(P ∧ Q)]_v ⟶ □[P]_v ⟶ □[P ∧ ¬Q]_v" proof - have "⊢ □[¬(P ∧ Q)]_v ⟶ □[P ⟶ P ∧ ¬Q]_v" by (rule M2) auto from this ax4 show ?thesis by (rule lift_imp_trans) qed theorem MM12[simp_unl]: "⊢ □[□[P]_v]_v = □[P]_v" proof (rule int_iffI) have "|~ □[P]_v ⟶ [P]_v" by (auto simp: pax4[unlifted]) hence "⊢ □[□[P]_v]_v ⟶ □[[P]_v]_v" by (rule M2) thus "⊢ □[□[P]_v]_v ⟶ □[P]_v" by (unfold T3[int_rewrite]) next have "⊢ □□[P]_v ⟶ □[□□[P]_v]_v" by (rule ax2) thus "⊢ □[P]_v ⟶ □[□[P]_v]_v" by auto qed subsection "Theorems about the eventually operator" ― ‹rules to push negation inside modal operators, sometimes useful for rewriting› theorem dualization: "⊢ ¬□F = ◇¬F" "⊢ ¬◇F = □¬F" "⊢ ¬□[A]_v = ◇⟨¬A⟩_v" "⊢ ¬◇⟨A⟩_v = □[¬A]_v" unfolding eventually_def angle_action_def by simp_all lemmas dualization_rew = dualization[int_rewrite] lemmas dualization_unl = dualization[unlifted] theorem E1: "⊢ ◇(F ∨ G) = (◇F ∨ ◇G)" proof - have "⊢ □¬(F ∨ G) = □(¬F ∧ ¬G)" by (rule MM1) auto thus ?thesis unfolding eventually_def STL5[int_rewrite] by force qed theorem E3: "⊢ F ⟶ ◇F" unfolding eventually_def by (force dest: ax1[unlift_rule]) theorem E4: "⊢ □F ⟶ ◇F" by (rule lift_imp_trans[OF ax1 E3]) theorem E5: "⊢ □F ⟶ □◇F" proof - have "⊢ □□F ⟶ □◇F" by (rule STL4[OF E4]) thus ?thesis by simp qed theorem E6: "⊢ □F ⟶ ◇□F" using E4[of "TEMP □F"] by simp theorem E7: assumes h: "|~ ¬F ∧ Unchanged v ⟶ ○¬F" shows "|~ ◇F ⟶ F ∨ ○◇F" proof - from h have "|~ ¬F ∧ ○□¬F ⟶ □¬F" by (rule M10) thus ?thesis by (auto simp: eventually_def) qed theorem E8: "⊢ ◇(F ⟶ G) ⟶ □F ⟶ ◇G" proof - have "⊢ □(F ∧ ¬G) ⟶ □¬(F ⟶ G)" by (rule STL4) auto thus ?thesis unfolding eventually_def STL5[int_rewrite] by auto qed theorem E9: "⊢ □(F ⟶ G) ⟶ ◇F ⟶ ◇G" proof - have "⊢ □(F ⟶ G) ⟶ □(¬G ⟶ ¬F)" by (rule STL4) auto with MM0[of "TEMP ¬G" "TEMP ¬F"] show ?thesis unfolding eventually_def by force qed theorem E10[simp_unl]: "⊢ ◇◇F = ◇F" by (simp add: eventually_def) theorem E22: assumes h: "⊢ F = G" shows "⊢ ◇F = ◇G" by (auto simp: h[int_rewrite]) theorem E15[simp_unl]: "⊢ ◇#F = #F" by (simp add: eventually_def) theorem E15b[simp_unl]: "⊢ ◇¬#F = ¬#F" by (simp add: eventually_def) theorem E16: "⊢ ◇□F ⟶ ◇F" by (rule STL4_eve[OF ax1]) text ‹An action version of STL6› lemma STL6_act: "⊢ ◇(□[F]_v ∧ □[G]_w) = (◇□[F]_v ∧ ◇□[G]_w)" proof - have "⊢ (◇□(□[F]_v ∧ □[G]_w)) = ◇(□□[F]_v ∧ □□[G]_w)" by (rule E22[OF STL5]) thus ?thesis by (auto simp: STL6[int_rewrite]) qed lemma SE1: "⊢ □F ∧ ◇G ⟶ ◇(□F ∧ G)" proof - have "⊢ □¬(□F ∧ G) ⟶ □(□F ⟶ ¬G)" by (rule STL4) auto with MM0 show ?thesis by (force simp: eventually_def) qed lemma SE2: "⊢ □F ∧ ◇G ⟶ ◇(F ∧ G)" proof - have "⊢ □F ∧ G ⟶ F ∧ G" by (auto elim: ax1[unlift_rule]) hence "⊢ ◇(□F ∧ G) ⟶ ◇(F ∧ G)" by (rule STL4_eve) with SE1 show ?thesis by (rule lift_imp_trans) qed lemma SE3: "⊢ □F ∧ ◇G ⟶ ◇(G ∧ F)" proof - have "⊢ ◇(F ∧ G) ⟶ ◇(G ∧ F)" by (rule STL4_eve) auto with SE2 show ?thesis by (rule lift_imp_trans) qed lemma SE4: assumes h1: "s ⊨ □F" and h2: "s ⊨ ◇G" and h3: "⊢ □F ∧ G ⟶ H" shows "s ⊨ ◇H" using h1 h2 h3[THEN STL4_eve] SE1 by force theorem E17: "⊢ □◇□F ⟶ □◇F" by (rule STL4[OF STL4_eve[OF ax1]]) theorem E18: "⊢ □◇□F ⟶ ◇□F" by (rule ax1) theorem E19: "⊢ ◇□F ⟶ □◇□F" proof - have "⊢ (□F ∧ ¬□F) = #False" by auto hence "⊢ ◇□(□F ∧ ¬□F) = ◇□#False" by (rule E22[OF MM1]) thus ?thesis unfolding STL6[int_rewrite] by (auto simp: eventually_def) qed theorem E20: "⊢ ◇□F ⟶ □◇F" by (rule lift_imp_trans[OF E19 E17]) theorem E21[simp_unl]: "⊢ □◇□F = ◇□F" by (rule int_iffI[OF E18 E19]) theorem E27[simp_unl]: "⊢ ◇□◇F = □◇F" using E21 unfolding eventually_def by force lemma E28: "⊢ ◇□F ∧ □◇G ⟶ □◇(F ∧ G)" proof - have "⊢ ◇□(□F ∧ ◇G) ⟶ ◇□◇(F ∧ G)" by (rule STL4_eve[OF STL4[OF SE2]]) thus ?thesis by (simp add: STL6[int_rewrite]) qed lemma E23: "|~ ○F ⟶ ◇F" using P1 by (force simp: eventually_def) lemma E24: "⊢ ◇□Q ⟶ □[◇Q]_v" by (rule lift_imp_trans[OF E20 P4]) lemma E25: "⊢ ◇⟨A⟩_v ⟶ ◇A" using P4 by (force simp: eventually_def angle_action_def) lemma E26: "⊢ □◇⟨A⟩_v ⟶ □◇A" by (rule STL4[OF E25]) lemma allBox: "(s ⊨ □(∀x. F x)) = (∀x. s ⊨ □(F x))" unfolding allT[unlifted] .. lemma E29: "|~ ○◇F ⟶ ◇F" unfolding eventually_def using pax3 by force lemma E30: assumes h1: "⊢ F ⟶ □F" and h2: "⊢ ◇F" shows "⊢ ◇□F" using h2 h1[THEN STL4_eve] by (rule fmp) lemma E31: "⊢ □(F ⟶ □F) ∧ ◇F ⟶ ◇□F" proof - have "⊢ □(F ⟶ □F) ∧ ◇F ⟶ ◇(□(F ⟶ □F) ∧ F)" by (rule SE1) moreover have "⊢ □(F ⟶ □F) ∧ F ⟶ □F" using ax1[of "TEMP F ⟶ □F"] by auto hence "⊢ ◇(□(F ⟶ □F) ∧ F) ⟶ ◇□F" by (rule STL4_eve) ultimately show ?thesis by (rule lift_imp_trans) qed lemma allActBox: "(s ⊨ □[(∀x. F x)]_v) = (∀x. s ⊨ □[(F x)]_v)" unfolding allActT[unlifted] .. theorem exEE: "⊢ (∃x. ◇(F x)) = ◇(∃x. F x)" proof - have "⊢ ¬(∃ x. ◇(F x)) = ¬◇(∃ x. F x)" by (auto simp: eventually_def Not_Rex[int_rewrite] allBox) thus ?thesis by force qed theorem exActE: "⊢ (∃x. ◇⟨F x⟩_v) = ◇⟨(∃x. F x)⟩_v" proof - have "⊢ ¬(∃x. ◇⟨F x⟩_v) = ¬◇⟨(∃x. F x)⟩_v" by (auto simp: angle_action_def Not_Rex[int_rewrite] allActBox) thus ?thesis by force qed subsection "Theorems about the leadsto operator" theorem LT1: "⊢ F ↝ F" unfolding leadsto_def by (rule alw[OF E3]) theorem LT2: assumes h: "⊢ F ⟶ G" shows "⊢ F ⟶ ◇G" by (rule lift_imp_trans[OF h E3]) theorem LT3: assumes h: "⊢ F ⟶ G" shows "⊢ F ↝ G" unfolding leadsto_def by (rule alw[OF LT2[OF h]]) theorem LT4: "⊢ F ⟶ (F ↝ G) ⟶ ◇G" unfolding leadsto_def using ax1[of "TEMP F ⟶ ◇G"] by auto theorem LT5: "⊢ □(F ⟶ ◇G) ⟶ ◇F ⟶ ◇G" using E9[of "F" "TEMP ◇G"] by simp theorem LT6: "⊢ ◇F ⟶ (F ↝ G) ⟶ ◇G" unfolding leadsto_def using LT5[of "F" "G"] by auto theorem LT9[simp_unl]: "⊢ □(F ↝ G) = (F ↝ G)" by (auto simp: leadsto_def) theorem LT7: "⊢ □◇F ⟶ (F ↝ G) ⟶ □◇G" proof - have "⊢ □◇F ⟶ □((F ↝ G) ⟶ ◇G)" by (rule STL4[OF LT6]) from lift_imp_trans[OF this MM0] show ?thesis by simp qed theorem LT8: "⊢ □◇G ⟶ (F ↝ G)" unfolding leadsto_def by (rule STL4) auto theorem LT13: "⊢ (F ↝ G) ⟶ (G ↝ H) ⟶ (F ↝ H)" proof - have "⊢ ◇G ⟶ (G ↝ H) ⟶ ◇H" by (rule LT6) hence "⊢ □(F ⟶ ◇G) ⟶ □((G ↝ H) ⟶ (F ⟶ ◇H))" by (intro STL4) auto from lift_imp_trans[OF this MM0] show ?thesis by (simp add: leadsto_def) qed theorem LT11: "⊢ (F ↝ G) ⟶ (F ↝ (G ∨ H))" proof - have "⊢ G ↝ (G ∨ H)" by (rule LT3) auto with LT13[of "F" "G" "TEMP (G ∨ H)"] show ?thesis by force qed theorem LT12: "⊢ (F ↝ H) ⟶ (F ↝ (G ∨ H))" proof - have "⊢ H ↝ (G ∨ H)" by (rule LT3) auto with LT13[of "F" "H" "TEMP (G ∨ H)"] show ?thesis by force qed theorem LT14: "⊢ ((F ∨ G) ↝ H) ⟶ (F ↝ H)" unfolding leadsto_def by (rule STL4) auto theorem LT15: "⊢ ((F ∨ G) ↝ H) ⟶ (G ↝ H)" unfolding leadsto_def by (rule STL4) auto theorem LT16: "⊢ (F ↝ H) ⟶ (G ↝ H) ⟶ ((F ∨ G) ↝ H)" proof - have "⊢ □(F ⟶ ◇H) ⟶ □((G ⟶ ◇H) ⟶ (F ∨ G ⟶ ◇H))" by (rule STL4) auto from lift_imp_trans[OF this MM0] show ?thesis by (unfold leadsto_def) qed theorem LT17: "⊢ ((F ∨ G) ↝ H) = ((F ↝ H) ∧ (G ↝ H))" by (auto elim: LT14[unlift_rule] LT15[unlift_rule] LT16[unlift_rule]) theorem LT10: assumes h: "⊢ (F ∧ ¬G) ↝ G" shows "⊢ F ↝ G" proof - from h have "⊢ ((F ∧ ¬G) ∨ G) ↝ G" by (auto simp: LT17[int_rewrite] LT1[int_rewrite]) moreover have "⊢ F ↝ ((F ∧ ¬G) ∨ G)" by (rule LT3, auto) ultimately show ?thesis by (force elim: LT13[unlift_rule]) qed theorem LT18: "⊢ (A ↝ (B ∨ C)) ⟶ (B ↝ D) ⟶ (C ↝ D) ⟶ (A ↝ D)" proof - have "⊢ (B ↝ D) ⟶ (C ↝ D) ⟶ ((B ∨ C) ↝ D)" by (rule LT16) thus ?thesis by (force elim: LT13[unlift_rule]) qed theorem LT19: "⊢ (A ↝ (D ∨ B)) ⟶ (B ↝ D) ⟶ (A ↝ D)" using LT18[of "A" "D" "B" "D"] LT1[of "D"] by force theorem LT20: "⊢ (A ↝ (B ∨ D)) ⟶ (B ↝ D) ⟶ (A ↝ D)" using LT18[of "A" "B" "D" "D"] LT1[of "D"] by force theorem LT21: "⊢ ((∃x. F x) ↝ G) = (∀x. (F x ↝ G))" proof - have "⊢ □((∃x. F x) ⟶ ◇G) = □(∀x. (F x ⟶ ◇G))" by (rule MM1) auto thus ?thesis by (unfold leadsto_def allT[int_rewrite]) qed theorem LT22: "⊢ (F ↝ (G ∨ H)) ⟶ □¬G ⟶ (F ↝ H)" proof - have "⊢ □¬G ⟶ (G ↝ H)" unfolding leadsto_def by (rule STL4) auto thus ?thesis by (force elim: LT20[unlift_rule]) qed lemma LT23: "|~ (P ⟶ ○Q) ⟶ (P ⟶ ◇Q)" by (auto dest: E23[unlift_rule]) theorem LT24: "⊢ □I ⟶ ((P ∧ I) ↝ Q) ⟶ P ↝ Q" proof - have "⊢ □I ⟶ □((P ∧ I ⟶ ◇Q) ⟶ (P ⟶ ◇Q))" by (rule STL4) auto from lift_imp_trans[OF this MM0] show ?thesis by (unfold leadsto_def) qed theorem LT25[simp_unl]: "⊢ (F ↝ #False) = □¬F" unfolding leadsto_def proof (rule MM1) show "⊢ (F ⟶ ◇#False) = ¬F" by simp qed lemma LT28: assumes h: "|~ P ⟶ ○P ∨ ○Q" shows "|~ (P ⟶ ○P) ∨ ◇Q" using h E23[of Q] by force lemma LT29: assumes h1: "|~ P ⟶ ○P ∨ ○Q" and h2: "|~ P ∧ Unchanged v ⟶ ○P" shows "⊢ P ⟶ □P ∨ ◇Q" proof - from h1[THEN LT28] have "|~ □¬Q ⟶ (P ⟶ ○P)" unfolding eventually_def by auto hence "⊢ □[□¬Q]_v ⟶ □[P ⟶ ○P]_v" by (rule M2) moreover have "⊢ ¬◇Q ⟶ □[□¬Q]_v" unfolding dualization_rew by (rule ax2) moreover note ax3[OF h2] ultimately show ?thesis by force qed lemma LT30: assumes h: "|~ P ∧ N ⟶ ○P ∨ ○Q" shows "|~ N ⟶ (P ⟶ ○P) ∨ ◇Q" using h E23 by force lemma LT31: assumes h1: "|~ P ∧ N ⟶ ○P ∨ ○Q" and h2: "|~ P ∧ Unchanged v ⟶ ○P" shows"⊢ □N ⟶ P ⟶ □P ∨ ◇Q" proof - from h1[THEN LT30] have "|~ N ⟶ □¬Q ⟶ P ⟶ ○P" unfolding eventually_def by auto hence "⊢ □[N ⟶ □¬Q ⟶ P ⟶ ○P]_v" by (rule sq) hence "⊢ □[N]_v ⟶ □[□¬Q]_v ⟶ □[P ⟶ ○P]_v" by (force intro: ax4[unlift_rule]) with P4 have "⊢ □N ⟶ □[□¬Q]_v ⟶ □[P ⟶ ○P]_v" by (rule lift_imp_trans) moreover have "⊢ ¬◇Q ⟶ □[□¬Q]_v" unfolding dualization_rew by (rule ax2) moreover note ax3[OF h2] ultimately show ?thesis by force qed lemma LT33: "⊢ ((#P ∧ F) ↝ G) = (#P ⟶ (F ↝ G))" by (cases "P", auto simp: leadsto_def) lemma AA1: "⊢ □[#False]_v ⟶ ¬◇⟨Q⟩_v" unfolding dualization_rew by (rule M2) auto lemma AA2: "⊢ □[P]_v ∧ ◇⟨Q⟩_v ⟶ ◇⟨P ∧ Q⟩_v" proof - have "⊢ □[P ⟶ ~(P ∧ Q) ⟶ ¬Q]_v" by (rule sq) (auto simp: actrans_def) hence "⊢ □[P]_v ⟶ □[~(P ∧ Q)]_v ⟶ □[¬Q]_v" by (force intro: ax4[unlift_rule]) thus ?thesis by (auto simp: angle_action_def) qed lemma AA3: "⊢ □P ∧ □[P ⟶ Q]_v ∧ ◇⟨A⟩_v ⟶ ◇Q" proof - have "⊢ □P ∧ □[P ⟶ Q]_v ⟶ □[P ∧ (P ⟶ Q)]_v" by (auto dest: P4[unlift_rule] simp: M8[int_rewrite]) moreover have "⊢ □[P ∧ (P ⟶ Q)]_v ⟶ □[Q]_v" by (rule M2) auto ultimately have "⊢ □P ∧ □[P ⟶ Q]_v ⟶ □[Q]_v" by (rule lift_imp_trans) moreover have "⊢ ◇(Q ∧ A) ⟶ ◇Q" by (rule STL4_eve) auto hence "⊢ ◇⟨Q ∧ A⟩_v ⟶ ◇Q" by (force dest: E25[unlift_rule]) with AA2 have "⊢ □[Q]_v ∧ ◇⟨A⟩_v ⟶ ◇Q" by (rule lift_imp_trans) ultimately show ?thesis by force qed lemma AA4: "⊢ ◇⟨⟨A⟩_v⟩_w ⟶ ◇⟨⟨A⟩_w⟩_v" unfolding angle_action_def angle_actrans_def using T5 by force lemma AA7: assumes h: "|~ F ⟶ G" shows "⊢ ◇⟨F⟩_v ⟶ ◇⟨G⟩_v" proof - from h have "⊢ □[¬G]_v ⟶ □[¬F]_v" by (intro M2) auto thus ?thesis unfolding angle_action_def by force qed lemma AA6: "⊢ □[P ⟶ Q]_v ∧ ◇⟨P⟩_v ⟶ ◇⟨Q⟩_v" proof - have "⊢ ◇⟨(P ⟶ Q) ∧ P⟩_v ⟶ ◇⟨Q⟩_v" by (rule AA7) auto with AA2 show ?thesis by (rule lift_imp_trans) qed lemma AA8: "⊢ □[P]_v ∧ ◇⟨A⟩_v ⟶ ◇⟨□[P]_v ∧ A⟩_v" proof - have "⊢ □[□[P]_v]_v ∧ ◇⟨A⟩_v ⟶ ◇⟨□[P]_v ∧ A⟩_v" by (rule AA2) with P5 show ?thesis by force qed lemma AA9: "⊢ □[P]_v ∧ ◇⟨A⟩_v ⟶ ◇⟨[P]_v ∧ A⟩_v" proof - have "⊢ □[[P]_v]_v ∧ ◇⟨A⟩_v ⟶ ◇⟨[P]_v ∧ A⟩_v" by (rule AA2) thus ?thesis by simp qed lemma AA10: "⊢ ¬(□[P]_v ∧ ◇⟨¬P⟩_v)" unfolding angle_action_def by auto lemma AA11: "⊢ ¬◇⟨v$ = $v⟩_v" unfolding dualization_rew by (rule ax5) lemma AA15: "⊢ ◇⟨P ∧ Q⟩_v ⟶ ◇⟨P⟩_v" by (rule AA7) auto lemma AA16: "⊢ ◇⟨P ∧ Q⟩_v ⟶ ◇⟨Q⟩_v" by (rule AA7) auto lemma AA13: "⊢ ◇⟨P⟩_v ⟶ ◇⟨v$ ≠ $v⟩_v" proof - have "⊢ □[v$ ≠ $v]_v ∧ ◇⟨P⟩_v ⟶ ◇⟨v$ ≠ $v ∧ P⟩_v" by (rule AA2) hence "⊢ ◇⟨P⟩_v ⟶ ◇⟨v$ ≠ $v ∧ P⟩_v" by (simp add: ax5[int_rewrite]) from this AA15 show ?thesis by (rule lift_imp_trans) qed lemma AA14: "⊢ ◇⟨P ∨ Q⟩_v = (◇⟨P⟩_v ∨ ◇⟨Q⟩_v)" proof - have "⊢ □[¬(P ∨ Q)]_v = □[¬P ∧ ¬Q]_v" by (rule MM10) auto hence "⊢ □[¬(P ∨ Q)]_v = (□[¬P]_v ∧ □[¬Q]_v)" by (unfold M8[int_rewrite]) thus ?thesis unfolding angle_action_def by auto qed lemma AA17: "⊢ ◇⟨[P]_v ∧ A⟩_v ⟶ ◇⟨P ∧ A⟩_v" proof - have "⊢ □[v$ ≠ $v ∧ ¬(P ∧ A)]_v ⟶ □[¬([P]_v ∧ A)]_v" by (rule M2) (auto simp: actrans_def unch_def) with ax5[of "v"] show ?thesis unfolding angle_action_def M8[int_rewrite] by force qed lemma AA19: "⊢ □P ∧ ◇⟨A⟩_v ⟶ ◇⟨P ∧ A⟩_v" using P4 by (force intro: AA2[unlift_rule]) lemma AA20: assumes h1: "|~ P ⟶ ○P ∨ ○Q" and h2: "|~ P ∧ A ⟶ ○Q" and h3: "|~ P ∧ Unchanged w ⟶ ○P" shows "⊢ □(□P ⟶ ◇⟨A⟩_v) ⟶ (P ↝ Q)" proof - from h2 E23 have "|~ P ∧ A ⟶ ◇Q" by force hence "⊢ ◇⟨P ∧ A⟩_v ⟶ ◇⟨◇Q⟩_v" by (rule AA7) with E25[of "TEMP ◇Q" "v"] have "⊢ ◇⟨P ∧ A⟩_v ⟶ ◇Q" by force with AA19 have "⊢ □P ∧ ◇⟨A⟩_v ⟶ ◇Q" by (rule lift_imp_trans) with LT29[OF h1 h3] have "⊢ (□P ⟶ ◇⟨A⟩_v) ⟶ (P ⟶ ◇Q)" by force thus ?thesis unfolding leadsto_def by (rule STL4) qed lemma AA21: "|~ ◇⟨○F⟩_v ⟶ ○◇F" using pax5[of "TEMP ¬F" "v"] unfolding angle_action_def eventually_def by auto theorem AA24[simp_unl]: "⊢ ◇⟨⟨P⟩_f⟩_f = ◇⟨P⟩_f" unfolding angle_action_def angle_actrans_def by simp lemma AA22: assumes h1: "|~ P ∧ N ⟶ ○P ∨ ○Q" and h2: "|~ P ∧ N ∧ ⟨A⟩_v ⟶ ○Q" and h3: "|~ P ∧ Unchanged w ⟶ ○P" shows "⊢ □N ∧ □(□P ⟶ ◇⟨A⟩_v) ⟶ (P ↝ Q)" proof - from h2 have "|~ ⟨(N ∧ P) ∧ A⟩_v ⟶ ○Q" by (auto simp: angle_actrans_sem[int_rewrite]) from pref_imp_trans[OF this E23] have "⊢ ◇⟨⟨(N ∧ P) ∧ A⟩_v⟩_v ⟶ ◇⟨◇Q⟩_v" by (rule AA7) hence "⊢ ◇⟨(N ∧ P) ∧ A⟩_v ⟶ ◇Q" by (force dest: E25[unlift_rule]) with AA19 have "⊢ □(N ∧ P) ∧ ◇⟨A⟩_v ⟶ ◇Q" by (rule lift_imp_trans) hence "⊢ □N ∧ □P ∧ ◇⟨A⟩_v ⟶ ◇Q" by (auto simp: STL5[int_rewrite]) with LT31[OF h1 h3] have "⊢ □N ∧ (□P ⟶ ◇⟨A⟩_v) ⟶ (P ⟶ ◇Q)" by force hence "⊢ □(□N ∧ (□P ⟶ ◇⟨A⟩_v)) ⟶ □(P ⟶ ◇Q)" by (rule STL4) thus ?thesis by (simp add: leadsto_def STL5[int_rewrite]) qed lemma AA23: assumes "|~ P ∧ N ⟶ ○P ∨ ○Q" and "|~ P ∧ N ∧ ⟨A⟩_v ⟶ ○Q" and "|~ P ∧ Unchanged w ⟶ ○P" shows "⊢ □N ∧ □◇⟨A⟩_v ⟶ (P ↝ Q)" proof - have "⊢ □◇⟨A⟩_v ⟶ □(□P ⟶ ◇⟨A⟩_v)" by (rule STL4) auto with AA22[OF assms] show ?thesis by force qed lemma AA25: assumes h: "|~ ⟨P⟩_v ⟶ ⟨Q⟩_w" shows "⊢ ◇⟨P⟩_v ⟶ ◇⟨Q⟩_w" proof - from h have "⊢ ◇⟨⟨P⟩_v⟩_v ⟶ ◇⟨⟨P⟩_w⟩_v" by (intro AA7) (auto simp: angle_actrans_def actrans_def) with AA4 have "⊢ ◇⟨P⟩_v ⟶ ◇⟨⟨P⟩_v⟩_w" by force from this AA7[OF h] have "⊢ ◇⟨P⟩_v ⟶ ◇⟨⟨Q⟩_w⟩_w" by (rule lift_imp_trans) thus ?thesis by simp qed lemma AA26: assumes h: "|~ ⟨A⟩_v = ⟨B⟩_w" shows "⊢ ◇⟨A⟩_v = ◇⟨B⟩_w" proof - from h have "|~ ⟨A⟩_v ⟶ ⟨B⟩_w" by auto hence "⊢ ◇⟨A⟩_v ⟶ ◇⟨B⟩_w" by (rule AA25) moreover from h have "|~ ⟨B⟩_w ⟶ ⟨A⟩_v" by auto hence "⊢ ◇⟨B⟩_w ⟶ ◇⟨A⟩_v" by (rule AA25) ultimately show ?thesis by force qed theorem AA28[simp_unl]: "⊢ ◇◇⟨A⟩_v = ◇⟨A⟩_v" unfolding eventually_def angle_action_def by simp theorem AA29: "⊢ □[N]_v ∧ □◇⟨A⟩_v ⟶ □◇⟨N ∧ A⟩_v" proof - have "⊢ □(□[N]_v ∧ ◇⟨A⟩_v) ⟶ □◇⟨N ∧ A⟩_v" by (rule STL4[OF AA2]) thus ?thesis by (simp add: STL5[int_rewrite]) qed theorem AA30[simp_unl]: "⊢ ◇⟨◇⟨P⟩_f⟩_f = ◇⟨P⟩_f" unfolding angle_action_def by simp theorem AA31: "⊢ ◇⟨○F⟩_v ⟶ ◇F" using pref_imp_trans[OF AA21 E29] by auto lemma AA32[simp_unl]: "⊢ □◇□[A]_v = ◇□[A]_v" using E21[of "TEMP □[A]_v"] by simp lemma AA33[simp_unl]: "⊢ ◇□◇⟨A⟩_v = □◇⟨A⟩_v" using E27[of "TEMP ◇⟨A⟩_v"] by simp subsection "Lemmas about the next operator" lemma N2: assumes h: "⊢ F = G" shows "|~ ○F = ○G" by (simp add: h[int_rewrite]) lemmas next_and = T8 lemma next_or: "|~ ○(F ∨ G) = (○F ∨ ○G)" proof (rule pref_iffI) have "|~ ○((F ∨ G) ∧ ¬F) ⟶ ○G" by (rule N1) auto thus "|~ ○(F ∨ G) ⟶ ○F ∨ ○G" by (auto simp: T8[int_rewrite]) next have "|~ ○F ⟶ ○(F ∨ G)" by (rule N1) auto moreover have "|~ ○G ⟶ ○(F ∨ G)" by (rule N1) auto ultimately show "|~ ○F ∨ ○G ⟶ ○(F ∨ G)" by force qed lemma next_imp: "|~ ○(F ⟶ G) = (○F ⟶ ○G)" proof (rule pref_iffI) have "|~ ○G ⟶ ○(F ⟶ G)" by (rule N1) auto moreover have "|~ ○¬F ⟶ ○(F ⟶ G)" by (rule N1) auto ultimately show "|~ (○F ⟶ ○G) ⟶ ○(F ⟶ G)" by force qed (rule pax2) lemmas next_not = pax1 lemma next_eq: "|~ ○(F = G) = (○F = ○G)" proof - have "|~ ○(F = G) = ○((F ⟶ G) ∧ (G ⟶ F))" by (rule N2) auto from this[int_rewrite] show ?thesis by (auto simp: next_and[int_rewrite] next_imp[int_rewrite]) qed lemma next_noteq: "|~ ○(F ≠ G) = (○F ≠ ○G)" by (simp add: next_eq[int_rewrite]) lemma next_const[simp_unl]: "|~ ○#P = #P" proof (cases "P") assume "P" hence 1: "⊢ #P" by auto hence "|~ ○#P" by (rule nex) with 1 show ?thesis by force next assume "¬P" hence 1: "⊢ ¬#P" by auto hence "|~ ○¬#P" by (rule nex) with 1 show ?thesis by force qed text ‹ The following are proved semantically because they are essentially first-order theorems. › lemma next_fun1: "|~ ○f<x> = f<○x>" by (auto simp: nexts_def) lemma next_fun2: "|~ ○f<x,y> = f<○x,○y>" by (auto simp: nexts_def) lemma next_fun3: "|~ ○f<x,y,z> = f<○x,○y,○z>" by (auto simp: nexts_def) lemma next_fun4: "|~ ○f<x,y,z,zz> = f<○x,○y,○z,○zz>" by (auto simp: nexts_def) lemma next_forall: "|~ ○(∀ x. P x) = (∀ x. ○ P x)" by (auto simp: nexts_def) lemma next_exists: "|~ ○(∃ x. P x) = (∃ x. ○ P x)" by (auto simp: nexts_def) lemma next_exists1: "|~ ○(∃! x. P x) = (∃! x. ○ P x)" by (auto simp: nexts_def) text ‹ Rewrite rules to push the ``next'' operator inward over connectives. (Note that axiom ‹pax1› and theorem ‹next_const› are anyway active as rewrite rules.) › lemmas next_commutes[int_rewrite] = next_and next_or next_imp next_eq next_fun1 next_fun2 next_fun3 next_fun4 next_forall next_exists next_exists1 lemmas ifs_eq[int_rewrite] = after_fun3 next_fun3 before_fun3 lemmas next_always = pax3 lemma t1: "|~ ○$x = x$" by (simp add: before_def after_def nexts_def first_tail_second) text ‹ Theorem ‹next_eventually› should not be used "blindly". › lemma next_eventually: assumes h: "stutinv F" shows "|~ ◇F ⟶ ¬F ⟶ ○◇F" proof - from h have 1: "stutinv (TEMP ¬F)" by (rule stut_not) have "|~ □¬F = (¬F ∧ ○□¬F)" unfolding T7[OF pre_id_unch[OF 1], int_rewrite] by simp thus ?thesis by (auto simp: eventually_def) qed lemma next_action: "|~ □[P]_v ⟶ ○□[P]_v" using pax4[of P v] by auto subsection "Higher Level Derived Rules" text ‹ In most verification tasks the low-level rules discussed above are not used directly. Here, we derive some higher-level rules more suitable for verification. In particular, variants of Lamport's rules ‹TLA1›, ‹TLA2›, ‹INV1› and ‹INV2› are derived, where ‹|~› is used where appropriate. › theorem TLA1: assumes H: "|~ P ∧ Unchanged f ⟶ ○P" shows "⊢ □P = (P ∧ □[P ⟶ ○P]_f)" proof (rule int_iffI) from ax1[of P] M0[of P f] show "⊢ □P ⟶ P ∧ □[P ⟶ ○P]_f" by force next from ax3[OF H] show "⊢ P ∧ □[P ⟶ ○P]_f ⟶ □P" by auto qed theorem TLA2: assumes h1: "⊢ P ⟶ Q" and h2: "|~ P ∧ ○P ∧ [A]_f ⟶ [B]_g" shows "⊢ □P ∧ □[A]_f ⟶ □Q ∧ □[B]_g" proof - from h2 have "⊢ □[P ∧ ○P ∧ [A]_f]_g ⟶ □[[B]_g]_g" by (rule M2) hence "⊢ □[P ∧ ○P]_g ∧ □[[A]_f]_g ⟶ □[B]_g" by (auto simp add: M8[int_rewrite]) with M1[of P g] T4[of A f g] have "⊢ □P ∧ □[A]_f ⟶ □[B]_g" by force with h1[THEN STL4] show ?thesis by force qed theorem INV1: assumes H: "|~ I ∧ [N]_f ⟶ ○I" shows "⊢ I ∧ □[N]_f ⟶ □I" proof - from H have "|~ [N]_f ⟶ I ⟶ ○I" by auto hence "⊢ □[[N]_f]_f ⟶ □[I ⟶ ○I]_f" by (rule M2) moreover from H have "|~ I ∧ Unchanged f ⟶ ○I" by (auto simp: actrans_def) hence "⊢ □[I ⟶ ○I]_f ⟶ I ⟶ □I" by (rule ax3) ultimately show ?thesis by force qed theorem INV2: "⊢ □I ⟶ □[N]_f = □[N ∧ I ∧ ○I]_f" proof - from M1[of I f] have "⊢ □I ⟶ (□[N]_f = □[N]_f ∧ □[I ∧ ○I]_f)" by auto thus ?thesis by (auto simp: M8[int_rewrite]) qed lemma R1: assumes H: "|~ Unchanged w ⟶ Unchanged v" shows "⊢ □[F]_w ⟶ □[F]_v" proof - from H have "|~ [F]_w ⟶ [F]_v" by (auto simp: actrans_def) thus ?thesis by (rule M11) qed theorem invmono: assumes h1: "⊢ I ⟶ P" and h2: "|~ P ∧ [N]_f ⟶ ○P" shows "⊢ I ∧ □[N]_f ⟶ □P" using h1 INV1[OF h2] by force theorem preimpsplit: assumes "|~ I ∧ N ⟶ Q" and "|~ I ∧ Unchanged v ⟶ Q" shows "|~ I ∧ [N]_v ⟶ Q" using assms[unlift_rule] by (auto simp: actrans_def) theorem refinement1: assumes h1: "⊢ P ⟶ Q" and h2: "|~ I ∧ ○I ∧ [A]_f ⟶ [B]_g" shows "⊢ P ∧ □I ∧ □[A]_f ⟶ Q ∧ □[B]_g" proof - have "⊢ I ⟶ #True" by simp from this h2 have "⊢ □I ∧ □[A]_f ⟶ □#True ∧ □[B]_g" by (rule TLA2) with h1 show ?thesis by force qed theorem inv_join: assumes "⊢ P ⟶ □Q" and "⊢ P ⟶ □R" shows "⊢ P ⟶ □(Q ∧ R)" using assms[unlift_rule] unfolding STL5[int_rewrite] by force lemma inv_cases: "⊢ □(A ⟶ B) ∧ □(¬A ⟶ B) ⟶ □B" proof - have "⊢ □((A ⟶ B) ∧ (¬A ⟶ B)) ⟶ □B" by (rule STL4) auto thus ?thesis by (simp add: STL5[int_rewrite]) qed end

# Theory Liveness

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section "Liveness" theory Liveness imports Rules begin text‹This theory derives proof rules for liveness properties.› definition enabled :: "'a formula ⇒ 'a formula" where "enabled F ≡ λ s. ∃ t. ((first s) ## t) ⊨ F" syntax "_Enabled" :: "lift ⇒ lift" ("(Enabled _)" [90] 90) translations "_Enabled" ⇌ "CONST enabled" definition WeakF :: "('a::world) formula ⇒ ('a,'b) stfun ⇒ 'a formula" where "WeakF F v ≡ TEMP ◇□Enabled ⟨F⟩_v ⟶ □◇⟨F⟩_v" definition StrongF :: "('a::world) formula ⇒ ('a,'b) stfun ⇒ 'a formula" where "StrongF F v ≡ TEMP □◇Enabled ⟨F⟩_v ⟶ □◇⟨F⟩_v" text‹ Lamport's TLA defines the above notions for actions. In \tlastar{}, (pre-)formulas generalise TLA's actions and the above definition is the natural generalisation of enabledness to pre-formulas. In particular, we have chosen to define ‹enabled› such that it yields itself a temporal formula, although its value really just depends on the first state of the sequence it is evaluated over. Then, the definitions of weak and strong fairness are exactly as in TLA. › syntax "_WF" :: "[lift,lift] ⇒ lift" ("(WF'(_')'_(_))" [20,1000] 90) "_SF" :: "[lift,lift] ⇒ lift" ("(SF'(_')'_(_))" [20,1000] 90) "_WFsp" :: "[lift,lift] ⇒ lift" ("(WF '(_')'_(_))" [20,1000] 90) "_SFsp" :: "[lift,lift] ⇒ lift" ("(SF '(_')'_(_))" [20,1000] 90) translations "_WF" ⇌ "CONST WeakF" "_SF" ⇌ "CONST StrongF" "_WFsp" ⇀ "CONST WeakF" "_SFsp" ⇀ "CONST StrongF" subsection "Properties of @{term enabled}" theorem enabledI: "⊢ F ⟶ Enabled F" proof (clarsimp) fix w assume "w ⊨ F" with seq_app_first_tail[of w] have "((first w) ## tail w) ⊨ F" by simp thus "w ⊨ Enabled F" by (auto simp: enabled_def) qed theorem enabledE: assumes "s ⊨ Enabled F" and "⋀u. (first s ## u) ⊨ F ⟹ Q" shows "Q" using assms unfolding enabled_def by blast lemma enabled_mono: assumes "w ⊨ Enabled F" and "⊢ F ⟶ G" shows "w ⊨ Enabled G" using assms[unlifted] unfolding enabled_def by blast lemma Enabled_disj1: "⊢ Enabled F ⟶ Enabled (F ∨ G)" by (auto simp: enabled_def) lemma Enabled_disj2: "⊢ Enabled F ⟶ Enabled (G ∨ F)" by (auto simp: enabled_def) lemma Enabled_conj1: "⊢ Enabled (F ∧ G) ⟶ Enabled F" by (auto simp: enabled_def) lemma Enabled_conj2: "⊢ Enabled (G ∧ F) ⟶ Enabled F" by (auto simp: enabled_def) lemma Enabled_disjD: "⊢ Enabled (F ∨ G) ⟶ Enabled F ∨ Enabled G" by (auto simp: enabled_def) lemma Enabled_disj: "⊢ Enabled (F ∨ G) = (Enabled F ∨ Enabled G)" by (auto simp: enabled_def) lemmas enabled_disj_rew = Enabled_disj[int_rewrite] lemma Enabled_ex: "⊢ Enabled (∃ x. F x) = (∃ x. Enabled (F x))" by (force simp: enabled_def) subsection "Fairness Properties" lemma WF_alt: "⊢ WF(A)_v = (□◇¬Enabled ⟨A⟩_v ∨ □◇⟨A⟩_v)" proof - have "⊢ WF(A)_v = (¬◇□ Enabled ⟨A⟩_v ∨ □◇⟨A⟩_v)" by (auto simp: WeakF_def) thus ?thesis by (simp add: dualization_rew) qed lemma SF_alt: "⊢ SF(A)_v = (◇□¬Enabled ⟨A⟩_v ∨ □◇⟨A⟩_v)" proof - have "⊢ SF(A)_v = (¬□◇ Enabled ⟨A⟩_v ∨ □◇⟨A⟩_v)" by (auto simp: StrongF_def) thus ?thesis by (simp add: dualization_rew) qed lemma alwaysWFI: "⊢ WF(A)_v ⟶ □WF(A)_v" unfolding WF_alt[int_rewrite] by (rule MM6) theorem WF_always[simp_unl]: "⊢ □WF(A)_v = WF(A)_v" by (rule int_iffI[OF ax1 alwaysWFI]) theorem WF_eventually[simp_unl]: "⊢ ◇WF(A)_v = WF(A)_v" proof - have 1: "⊢ ¬WF(A)_v = (◇□Enabled ⟨A⟩_v ∧ ¬ □◇⟨A⟩_v)" by (auto simp: WeakF_def) have "⊢ □¬WF(A)_v = ¬WF(A)_v" by (simp add: 1[int_rewrite] STL5[int_rewrite] dualization_rew) thus ?thesis by (auto simp: eventually_def) qed lemma alwaysSFI: "⊢ SF(A)_v ⟶ □SF(A)_v" proof - have "⊢ □◇□¬Enabled ⟨A⟩_v ∨ □◇⟨A⟩_v ⟶ □(□◇□¬Enabled ⟨A⟩_v ∨ □◇⟨A⟩_v)" by (rule MM6) thus ?thesis unfolding SF_alt[int_rewrite] by simp qed theorem SF_always[simp_unl]: "⊢ □SF(A)_v = SF(A)_v" by (rule int_iffI[OF ax1 alwaysSFI]) theorem SF_eventually[simp_unl]: "⊢ ◇SF(A)_v = SF(A)_v" proof - have 1: "⊢ ¬SF(A)_v = (□◇Enabled ⟨A⟩_v ∧ ¬ □◇⟨A⟩_v)" by (auto simp: StrongF_def) have "⊢ □¬SF(A)_v = ¬SF(A)_v" by (simp add: 1[int_rewrite] STL5[int_rewrite] dualization_rew) thus ?thesis by (auto simp: eventually_def) qed theorem SF_imp_WF: "⊢ SF (A)_v ⟶ WF (A)_v" unfolding WeakF_def StrongF_def by (auto dest: E20[unlift_rule]) lemma enabled_WFSF: "⊢ □Enabled ⟨F⟩_v ⟶ (WF(F)_v = SF(F)_v)" proof - have "⊢ □Enabled ⟨F⟩_v ⟶ ◇□Enabled ⟨F⟩_v" by (rule E3) hence "⊢ □Enabled ⟨F⟩_v ⟶ WF(F)_v ⟶ SF(F)_v" by (auto simp: WeakF_def StrongF_def) moreover have "⊢ □Enabled ⟨F⟩_v ⟶ □◇Enabled ⟨F⟩_v" by (rule STL4[OF E3]) hence "⊢ □Enabled ⟨F⟩_v ⟶ SF(F)_v ⟶ WF(F)_v" by (auto simp: WeakF_def StrongF_def) ultimately show ?thesis by force qed theorem WF1_general: assumes h1: "|~ P ∧ N ⟶ ○P ∨ ○Q" and h2: "|~ P ∧ N ∧ ⟨A⟩_v ⟶ ○Q" and h3: "⊢ P ∧ N ⟶ Enabled ⟨A⟩_v" and h4: "|~ P ∧ Unchanged w ⟶ ○P" shows "⊢ □N ∧ WF(A)_v ⟶ (P ↝ Q)" proof - have "⊢ □(□N ∧ WF(A)_v) ⟶ □(□P ⟶ ◇⟨A⟩_v)" proof (rule STL4) have "⊢ □(P ∧ N) ⟶ ◇□Enabled ⟨A⟩_v" by (rule lift_imp_trans[OF h3[THEN STL4] E3]) hence "⊢ □P ∧ □N ∧ WF(A)_v ⟶ □◇⟨A⟩_v" by (auto simp: WeakF_def STL5[int_rewrite]) with ax1[of "TEMP ◇⟨A⟩_v"] show "⊢ □N ∧ WF(A)_v ⟶ □P ⟶ ◇⟨A⟩_v" by force qed hence "⊢ □N ∧ WF(A)_v ⟶ □(□P ⟶ ◇⟨A⟩_v)" by (simp add: STL5[int_rewrite]) with AA22[OF h1 h2 h4] show ?thesis by force qed text ‹Lamport's version of the rule is derived as a special case.› theorem WF1: assumes h1: "|~ P ∧ [N]_v ⟶ ○P ∨ ○Q" and h2: "|~ P ∧ ⟨N ∧ A⟩_v ⟶ ○Q" and h3: "⊢ P ⟶ Enabled ⟨A⟩_v" and h4: "|~ P ∧ Unchanged v ⟶ ○P" shows "⊢ □[N]_v ∧ WF(A)_v ⟶ (P ↝ Q)" proof - have "⊢ □□[N]_v ∧ WF(A)_v ⟶ (P ↝ Q)" proof (rule WF1_general) from h1 T9[of N v] show "|~ P ∧ □[N]_v ⟶ ○P ∨ ○Q" by force next from T9[of N v] have "|~ P ∧ □[N]_v ∧ ⟨A⟩_v ⟶ P ∧ ⟨N ∧ A⟩_v" by (auto simp: actrans_def angle_actrans_def) from this h2 show "|~ P ∧ □[N]_v ∧ ⟨A⟩_v ⟶ ○Q" by (rule pref_imp_trans) next from h3 T9[of N v] show "⊢ P ∧ □[N]_v ⟶ Enabled ⟨A⟩_v" by force qed (rule h4) thus ?thesis by simp qed text ‹ The corresponding rule for strong fairness has an additional hypothesis ‹□F›, which is typically a conjunction of other fairness properties used to prove that the helpful action eventually becomes enabled. › theorem SF1_general: assumes h1: "|~ P ∧ N ⟶ ○P ∨ ○Q" and h2: "|~ P ∧ N ∧ ⟨A⟩_v ⟶ ○Q" and h3: "⊢ □P ∧ □N ∧ □F ⟶ ◇Enabled ⟨A⟩_v" and h4: "|~ P ∧ Unchanged w ⟶ ○P" shows "⊢ □N ∧ SF(A)_v ∧ □F ⟶ (P ↝ Q)" proof - have "⊢ □(□N ∧ SF(A)_v ∧ □F) ⟶ □(□P ⟶ ◇⟨A⟩_v)" proof (rule STL4) have "⊢ □(□P ∧ □N ∧ □F) ⟶ □◇Enabled ⟨A⟩_v" by (rule STL4[OF h3]) hence "⊢ □P ∧ □N ∧ □F ∧ SF(A)_v ⟶ □◇⟨A⟩_v" by (auto simp: StrongF_def STL5[int_rewrite]) with ax1[of "TEMP ◇⟨A⟩_v"] show "⊢ □N ∧ SF(A)_v ∧ □F ⟶ □P ⟶ ◇⟨A⟩_v" by force qed hence "⊢ □N ∧ SF(A)_v ∧ □F ⟶ □(□P ⟶ ◇⟨A⟩_v)" by (simp add: STL5[int_rewrite]) with AA22[OF h1 h2 h4] show ?thesis by force qed theorem SF1: assumes h1: "|~ P ∧ [N]_v ⟶ ○P ∨ ○Q" and h2: "|~ P ∧ ⟨N ∧ A⟩_v ⟶ ○Q" and h3: "⊢ □P ∧ □[N]_v ∧ □F ⟶ ◇Enabled ⟨A⟩_v" and h4: "|~ P ∧ Unchanged v ⟶ ○P" shows "⊢ □[N]_v ∧ SF(A)_v ∧ □F ⟶ (P ↝ Q)" proof - have "⊢ □□[N]_v ∧ SF(A)_v ∧ □F ⟶ (P ↝ Q)" proof (rule SF1_general) from h1 T9[of N v] show "|~ P ∧ □[N]_v ⟶ ○P ∨ ○Q" by force next from T9[of N v] have "|~ P ∧ □[N]_v ∧ ⟨A⟩_v ⟶ P ∧ ⟨N ∧ A⟩_v" by (auto simp: actrans_def angle_actrans_def) from this h2 show "|~ P ∧ □[N]_v ∧ ⟨A⟩_v ⟶ ○Q" by (rule pref_imp_trans) next from h3 show "⊢ □P ∧ □□[N]_v ∧ □F ⟶ ◇Enabled ⟨A⟩_v" by simp qed (rule h4) thus ?thesis by simp qed text ‹ Lamport proposes the following rule as an introduction rule for ‹WF› formulas. › theorem WF2: assumes h1: "|~ ⟨N ∧ B⟩_f ⟶ ⟨M⟩_g" and h2: "|~ P ∧ ○P ∧ ⟨N ∧ A⟩_f ⟶ B" and h3: "⊢ P ∧ Enabled ⟨M⟩_g ⟶ Enabled ⟨A⟩_f" and h4: "⊢ □[N ∧ ¬B]_f ∧ WF(A)_f ∧ □F ∧ ◇□Enabled ⟨M⟩_g ⟶ ◇□P" shows "⊢ □[N]_f ∧ WF(A)_f ∧ □F ⟶ WF(M)_g" proof - have "⊢ □[N]_f ∧ WF(A)_f ∧ □F ∧ ◇□Enabled ⟨M⟩_g ∧ ¬□◇⟨M⟩_g ⟶ □◇⟨M⟩_g" proof - have 1: "⊢ □[N]_f ∧ WF(A)_f ∧ □F ∧ ◇□Enabled ⟨M⟩_g ∧ ¬□◇⟨M⟩_g ⟶ ◇□P" proof - have A: "⊢ □[N]_f ∧ WF(A)_f ∧ □F ∧ ◇□Enabled ⟨M⟩_g ∧ ¬□◇⟨M⟩_g ⟶ □(□[N]_f ∧ WF(A)_f ∧ □F) ∧ ◇□(◇□Enabled ⟨M⟩_g ∧ □[¬M]_g)" unfolding STL6[int_rewrite] (* important to do this before STL5 is applied *) by (auto simp: STL5[int_rewrite] dualization_rew) have B: "⊢ □(□[N]_f ∧ WF(A)_f ∧ □F) ∧ ◇□(◇□Enabled ⟨M⟩_g ∧ □[¬M]_g) ⟶ ◇((□[N]_f ∧ WF(A)_f ∧ □F) ∧ □(◇□Enabled ⟨M⟩_g ∧ □[¬M]_g))" by (rule SE2) from lift_imp_trans[OF A B] have "⊢ □[N]_f ∧ WF(A)_f ∧ □F ∧ ◇□Enabled ⟨M⟩_g ∧ ¬□◇⟨M⟩_g ⟶ ◇((□[N]_f ∧ WF(A)_f ∧ □F) ∧ (◇□Enabled ⟨M⟩_g ∧ □[¬M]_g))" by (simp add: STL5[int_rewrite]) moreover from h1 have "|~ [N]_f ⟶ [¬M]_g ⟶ [N ∧ ¬B]_f" by (auto simp: actrans_def angle_actrans_def) hence "⊢ □[[N]_f]_f ⟶ □[[¬M]_g ⟶ [N ∧ ¬B]_f]_f" by (rule M2) from lift_imp_trans[OF this ax4] have "⊢ □[N]_f ∧ □[¬M]_g ⟶ □[N ∧ ¬B]_f" by (force intro: T4[unlift_rule]) with h4 have "⊢ (□[N]_f ∧ WF(A)_f ∧ □F) ∧ (◇□Enabled ⟨M⟩_g ∧ □[¬M]_g) ⟶ ◇□P" by force from STL4_eve[OF this] have "⊢ ◇((□[N]_f ∧ WF(A)_f ∧ □F) ∧ (◇□Enabled ⟨M⟩_g ∧ □[¬M]_g)) ⟶ ◇□P" by simp ultimately show ?thesis by (rule lift_imp_trans) qed have 2: "⊢ □[N]_f ∧ WF(A)_f ∧ ◇□Enabled ⟨M⟩_g ∧ ◇□P ⟶ □◇⟨M⟩_g" proof - have A: "⊢ ◇□P ∧ ◇□Enabled ⟨M⟩_g ∧ WF(A)_f ⟶ □◇⟨A⟩_f" using h3[THEN STL4, THEN STL4_eve] by (auto simp: STL6[int_rewrite] WeakF_def) have B: "⊢ □[N]_f ∧ ◇□P ∧ □◇⟨A⟩_f ⟶ □◇⟨M⟩_g" proof - from M1[of P f] have "⊢ □P ∧ □◇⟨N ∧ A⟩_f ⟶ □◇⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f" by (force intro: AA29[unlift_rule]) hence "⊢ ◇□(□P ∧ □◇⟨N ∧ A⟩_f) ⟶ ◇□□◇⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f" by (rule STL4_eve[OF STL4]) hence "⊢ ◇□P ∧ □◇⟨N ∧ A⟩_f ⟶ □◇⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f" by (simp add: STL6[int_rewrite]) with AA29[of N f A] have B1: "⊢ □[N]_f ∧ ◇□P ∧ □◇⟨A⟩_f ⟶ □◇⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f" by force from h2 have "|~ ⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f ⟶ ⟨N ∧ B⟩_f" by (auto simp: angle_actrans_sem[unlifted]) from B1 this[THEN AA25, THEN STL4] have "⊢ □[N]_f ∧ ◇□P ∧ □◇⟨A⟩_f ⟶ □◇⟨N ∧ B⟩_f" by (rule lift_imp_trans) moreover have "⊢ □◇⟨N ∧ B⟩_f ⟶ □◇⟨M⟩_g" by (rule h1[THEN AA25, THEN STL4]) ultimately show ?thesis by (rule lift_imp_trans) qed from A B show ?thesis by force qed from 1 2 show ?thesis by force qed thus ?thesis by (auto simp: WeakF_def) qed text ‹ Lamport proposes an analogous theorem for introducing strong fairness, and its proof is very similar, in fact, it was obtained by copy and paste, with minimal modifications. › theorem SF2: assumes h1: "|~ ⟨N ∧ B⟩_f ⟶ ⟨M⟩_g" and h2: "|~ P ∧ ○P ∧ ⟨N ∧ A⟩_f ⟶ B" and h3: "⊢ P ∧ Enabled ⟨M⟩_g ⟶ Enabled ⟨A⟩_f" and h4: "⊢ □[N ∧ ¬B]_f ∧ SF(A)_f ∧ □F ∧ □◇Enabled ⟨M⟩_g ⟶ ◇□P" shows "⊢ □[N]_f ∧ SF(A)_f ∧ □F ⟶ SF(M)_g" proof - have "⊢ □[N]_f ∧ SF(A)_f ∧ □F ∧ □◇Enabled ⟨M⟩_g ∧ ¬□◇⟨M⟩_g ⟶ □◇⟨M⟩_g" proof - have 1: "⊢ □[N]_f ∧ SF(A)_f ∧ □F ∧ □◇Enabled ⟨M⟩_g ∧ ¬□◇⟨M⟩_g ⟶ ◇□P" proof - have A: "⊢ □[N]_f ∧ SF(A)_f ∧ □F ∧ □◇Enabled ⟨M⟩_g ∧ ¬□◇⟨M⟩_g ⟶ □(□[N]_f ∧ SF(A)_f ∧ □F) ∧ ◇□(□◇Enabled ⟨M⟩_g ∧ □[¬M]_g)" unfolding STL6[int_rewrite] (* important to do this before STL5 is applied *) by (auto simp: STL5[int_rewrite] dualization_rew) have B: "⊢ □(□[N]_f ∧ SF(A)_f ∧ □F) ∧ ◇□(□◇Enabled ⟨M⟩_g ∧ □[¬M]_g) ⟶ ◇((□[N]_f ∧ SF(A)_f ∧ □F) ∧ □(□◇Enabled ⟨M⟩_g ∧ □[¬M]_g))" by (rule SE2) from lift_imp_trans[OF A B] have "⊢ □[N]_f ∧ SF(A)_f ∧ □F ∧ □◇Enabled ⟨M⟩_g ∧ ¬□◇⟨M⟩_g ⟶ ◇((□[N]_f ∧ SF(A)_f ∧ □F) ∧ (□◇Enabled ⟨M⟩_g ∧ □[¬M]_g))" by (simp add: STL5[int_rewrite]) moreover from h1 have "|~ [N]_f ⟶ [¬M]_g ⟶ [N ∧ ¬B]_f" by (auto simp: actrans_def angle_actrans_def) hence "⊢ □[[N]_f]_f ⟶ □[[¬M]_g ⟶ [N ∧ ¬B]_f]_f" by (rule M2) from lift_imp_trans[OF this ax4] have "⊢ □[N]_f ∧ □[¬M]_g ⟶ □[N ∧ ¬B]_f" by (force intro: T4[unlift_rule]) with h4 have "⊢ (□[N]_f ∧ SF(A)_f ∧ □F) ∧ (□◇Enabled ⟨M⟩_g ∧ □[¬M]_g) ⟶ ◇□P" by force from STL4_eve[OF this] have "⊢ ◇((□[N]_f ∧ SF(A)_f ∧ □F) ∧ (□◇Enabled ⟨M⟩_g ∧ □[¬M]_g)) ⟶ ◇□P" by simp ultimately show ?thesis by (rule lift_imp_trans) qed have 2: "⊢ □[N]_f ∧ SF(A)_f ∧ □◇Enabled ⟨M⟩_g ∧ ◇□P ⟶ □◇⟨M⟩_g" proof - have "⊢ □◇(P ∧ Enabled ⟨M⟩_g) ∧ SF(A)_f ⟶ □◇⟨A⟩_f" using h3[THEN STL4_eve, THEN STL4] by (auto simp: StrongF_def) with E28 have A: "⊢ ◇□P ∧ □◇Enabled ⟨M⟩_g ∧ SF(A)_f ⟶ □◇⟨A⟩_f" by force have B: "⊢ □[N]_f ∧ ◇□P ∧ □◇⟨A⟩_f ⟶ □◇⟨M⟩_g" proof - from M1[of P f] have "⊢ □P ∧ □◇⟨N ∧ A⟩_f ⟶ □◇⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f" by (force intro: AA29[unlift_rule]) hence "⊢ ◇□(□P ∧ □◇⟨N ∧ A⟩_f) ⟶ ◇□□◇⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f" by (rule STL4_eve[OF STL4]) hence "⊢ ◇□P ∧ □◇⟨N ∧ A⟩_f ⟶ □◇⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f" by (simp add: STL6[int_rewrite]) with AA29[of N f A] have B1: "⊢ □[N]_f ∧ ◇□P ∧ □◇⟨A⟩_f ⟶ □◇⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f" by force from h2 have "|~ ⟨(P ∧ ○P) ∧ (N ∧ A)⟩_f ⟶ ⟨N ∧ B⟩_f" by (auto simp: angle_actrans_sem[unlifted]) from B1 this[THEN AA25, THEN STL4] have "⊢ □[N]_f ∧ ◇□P ∧ □◇⟨A⟩_f ⟶ □◇⟨N ∧ B⟩_f" by (rule lift_imp_trans) moreover have "⊢ □◇⟨N ∧ B⟩_f ⟶ □◇⟨M⟩_g" by (rule h1[THEN AA25, THEN STL4]) ultimately show ?thesis by (rule lift_imp_trans) qed from A B show ?thesis by force qed from 1 2 show ?thesis by force qed thus ?thesis by (auto simp: StrongF_def) qed text ‹This is the lattice rule from TLA› theorem wf_leadsto: assumes h1: "wf r" and h2: "⋀x. ⊢ F x ↝ (G ∨ (∃y. #((y,x) ∈ r) ∧ F y))" shows "⊢ F x ↝ G" using h1 proof (rule wf_induct) fix x assume ih: "∀y. (y, x) ∈ r ⟶ (⊢ F y ↝ G)" show "⊢ F x ↝ G" proof - from ih have "⊢ (∃y. #((y,x) ∈ r) ∧ F y) ↝ G" by (force simp: LT21[int_rewrite] LT33[int_rewrite]) with h2 show ?thesis by (force intro: LT19[unlift_rule]) qed qed subsection "Stuttering Invariance" theorem stut_Enabled: "STUTINV Enabled ⟨F⟩_v" by (auto simp: enabled_def stutinv_def dest!: sim_first) theorem stut_WF: "NSTUTINV F ⟹ STUTINV WF(F)_v" by (auto simp: WeakF_def stut_Enabled bothstutinvs) theorem stut_SF: "NSTUTINV F ⟹ STUTINV SF(F)_v" by (auto simp: StrongF_def stut_Enabled bothstutinvs) lemmas livestutinv = stut_WF stut_SF stut_Enabled end

# Theory State

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section ‹Representing state in TLA*› theory State imports Liveness begin text‹ We adopt the hidden state appraoch, as used in the existing Isabelle/HOL TLA embedding \cite{Merz98}. This approach is also used in \cite{Ehmety01}. Here, a state space is defined by its projections, and everything else is unknown. Thus, a variable is a projection of the state space, and has the same type as a state function. Moreover, strong typing is achieved, since the projection function may have any result type. To achieve this, the state space is represented by an undefined type, which is an instance of the ‹world› class to enable use with the ‹Intensional› theory. › typedecl state instance state :: world .. type_synonym 'a statefun = "(state,'a) stfun" type_synonym statepred = "bool statefun" type_synonym 'a tempfun = "(state,'a) formfun" type_synonym temporal = "state formula" text ‹ Formalizing type state would require formulas to be tagged with their underlying state space and would result in a system that is much harder to use. (Unlike Hoare logic or Unity, TLA has quantification over state variables, and therefore one usually works with different state spaces within a single specification.) Instead, state is just an anonymous type whose only purpose is to provide Skolem constants. Moreover, we do not define a type of state variables separate from that of arbitrary state functions, again in order to simplify the definition of flexible quantification later on. Nevertheless, we need to distinguish state variables, mainly to define the enabledness of actions. The user identifies (tuples of) ``base'' state variables in a specification via the ``meta predicate'' ‹basevars›, which is defined here. › definition stvars :: "'a statefun ⇒ bool" where basevars_def: "stvars ≡ surj" syntax "PRED" :: "lift ⇒ 'a" ("PRED _") "_stvars" :: "lift ⇒ bool" ("basevars _") translations "PRED P" ⇀ "(P::state => _)" "_stvars" ⇌ "CONST stvars" text ‹ Base variables may be assigned arbitrary (type-correct) values. In the following lemma, note that ‹vs› may be a tuple of variables. The correct identification of base variables is up to the user who must take care not to introduce an inconsistency. For example, @{term "basevars (x,x)"} would definitely be inconsistent. › lemma basevars: "basevars vs ⟹ ∃u. vs u = c" proof (unfold basevars_def surj_def) assume "∀y. ∃x. y = vs x" then obtain x where "c = vs x" by blast thus "∃u. vs u = c" by blast qed lemma baseE: assumes H1: "basevars v" and H2:"⋀x. v x = c ⟹ Q" shows "Q" using H1[THEN basevars] H2 by auto text ‹A variant written for sequences rather than single states.› lemma first_baseE: assumes H1: "basevars v" and H2: "⋀x. v (first x) = c ⟹ Q" shows "Q" using H1[THEN basevars] H2 by (force simp: first_def) lemma base_pair1: assumes h: "basevars (x,y)" shows "basevars x" proof (auto simp: basevars_def) fix c from h[THEN basevars] obtain s where "(LIFT (x,y)) s = (c, arbitrary)" by auto thus "c ∈ range x" by auto qed lemma base_pair2: assumes h: "basevars (x,y)" shows "basevars y" proof (auto simp: basevars_def) fix d from h[THEN basevars] obtain s where "(LIFT (x,y)) s = (arbitrary, d)" by auto thus "d ∈ range y" by auto qed lemma base_pair: "basevars (x,y) ⟹ basevars x ∧ basevars y" by (auto elim: base_pair1 base_pair2) text ‹ Since the @{typ unit} type has just one value, any state function of unit type satisfies the predicate ‹basevars›. The following theorem can sometimes be useful because it gives a trivial solution for ‹basevars› premises. › lemma unit_base: "basevars (v::state ⇒ unit)" by (auto simp: basevars_def) text ‹ A pair of the form ‹(x,x)› will generally not satisfy the predicate ‹basevars› -- except for pathological cases such as ‹x::unit›. › lemma fixes x :: "state ⇒ bool" assumes h1: "basevars (x,x)" shows "False" proof - from h1 have "∃u. (LIFT (x,x)) u = (False,True)" by (rule basevars) thus False by auto qed lemma fixes x :: "state ⇒ nat" assumes h1: "basevars (x,x)" shows "False" proof - from h1 have "∃u. (LIFT (x,x)) u = (0,1)" by (rule basevars) thus False by auto qed text ‹ The following theorem reduces the reasoning about the existence of a state sequence satisfiyng an enabledness predicate to finding a suitable value ‹c› at the successor state for the base variables of the specification. This rule is intended for reasoning about standard TLA specifications, where ‹Enabled› is applied to actions, not arbitrary pre-formulas. › lemma base_enabled: assumes h1: "basevars vs" and h2: "⋀u. vs (first u) = c ⟹ ((first s) ## u) ⊨ F" shows "s ⊨ Enabled F" using h1 proof (rule first_baseE) fix t assume "vs (first t) = c" hence "((first s) ## t) ⊨ F" by (rule h2) thus "s ⊨ Enabled F" unfolding enabled_def by blast qed subsection "Temporal Quantifiers" text‹ In \cite{Lamport94}, Lamport gives a stuttering invariant definition of quantification over (flexible) variables. It relies on similarity of two sequences (as supported in our @{theory TLA.Sequence} theory), and equivalence of two sequences up to a variable (the bound variable). However, sequence equaivalence up to a variable, requires state equaivalence up to a variable. Our state representation above does not support this, hence we cannot encode Lamport's definition in our theory. Thus, we need to axiomatise quantification over (flexible) variables. Note that with a state representation supporting this, our theory should allow such an encoding. › consts EEx :: "('a statefun ⇒ temporal) ⇒ temporal" (binder "Eex " 10) AAll :: "('a statefun ⇒ temporal) ⇒ temporal" (binder "Aall " 10) syntax "_EEx" :: "[idts, lift] => lift" ("(3∃∃ _./ _)" [0,10] 10) "_AAll" :: "[idts, lift] => lift" ("(3∀∀ _./ _)" [0,10] 10) translations "_EEx v A" == "Eex v. A" "_AAll v A" == "Aall v. A" axiomatization where eexI: "⊢ F x ⟶ (∃∃ x. F x)" and eexE: "⟦s ⊨ (∃∃ x. F x) ; basevars vs; (!! x. ⟦ basevars (x,vs); s ⊨ F x ⟧ ⟹ s ⊨ G)⟧ ⟹ (s ⊨ G)" and all_def: "⊢ (∀∀ x. F x) = (¬(∃∃ x. ¬(F x)))" and eexSTUT: "STUTINV F x ⟹ STUTINV (∃∃ x. F x)" and history: "⊢ (I ∧ □[A]_v) = (∃∃ h. ($h = ha) ∧ I ∧ □[A ∧ h$=hb]_(h,v))" lemmas eexI_unl = eexI[unlift_rule] ― ‹@{text "w ⊨ F x ⟹ w ⊨ (∃∃ x. F x)"}› text ‹ ‹tla_defs› can be used to unfold TLA definitions into lowest predicate level. This is particularly useful for reasoning about enabledness of formulas. › lemmas tla_defs = unch_def before_def after_def first_def second_def suffix_def tail_def nexts_def app_def angle_actrans_def actrans_def end

# Theory Even

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section ‹A simple illustrative example› theory Even imports State begin text‹ A trivial example illustrating invariant proofs in the logic, and how Isabelle/HOL can help with specification. It proves that ‹x› is always even in a program where ‹x› is initialized as 0 and always incremented by 2. › inductive_set Even :: "nat set" where even_zero: "0 ∈ Even" | even_step: "n ∈ Even ⟹ Suc (Suc n) ∈ Even" locale Program = fixes x :: "state ⇒ nat" and init :: "temporal" and act :: "temporal" and phi :: "temporal" defines "init ≡ TEMP $x = # 0" and "act ≡ TEMP x` = Suc<Suc<$x>>" and "phi ≡ TEMP init ∧ □[act]_x" lemma (in Program) stutinvprog: "STUTINV phi" by (auto simp: phi_def init_def act_def stutinvs nstutinvs) lemma (in Program) inveven: "⊢ phi ⟶ □($x ∈ # Even)" unfolding phi_def proof (rule invmono) show "⊢ init ⟶ $x ∈ #Even" by (auto simp: init_def even_zero) next show "|~ $x ∈ #Even ∧ [act]_x ⟶ ○($x ∈ #Even)" by (auto simp: act_def even_step tla_defs) qed end

# Theory Inc

(* Title: A Definitional Encoding of TLA in Isabelle/HOL Authors: Gudmund Grov <ggrov at inf.ed.ac.uk> Stephan Merz <Stephan.Merz at loria.fr> Year: 2011 Maintainer: Gudmund Grov <ggrov at inf.ed.ac.uk> *) section ‹Lamport's Inc example› theory Inc imports State begin text‹ This example illustrates use of the embedding by mechanising the running example of Lamports original TLA paper \cite{Lamport94}. › datatype pcount = a | b | g locale Firstprogram = fixes x :: "state ⇒ nat" and y :: "state ⇒ nat" and init :: "temporal" and m1 :: "temporal" and m2 :: "temporal" and phi :: "temporal" and Live :: "temporal" defines "init ≡ TEMP $x = # 0 ∧ $y = # 0" and "m1 ≡ TEMP x` = Suc<$x> ∧ y` = $y" and "m2 ≡ TEMP y` = Suc<$y> ∧ x` = $x" and "Live ≡ TEMP WF(m1)_(x,y) ∧ WF(m2)_(x,y)" and "phi ≡ TEMP (init ∧ □[m1 ∨ m2]_(x,y) ∧ Live)" assumes bvar: "basevars (x,y)" lemma (in Firstprogram) "STUTINV phi" by (auto simp: phi_def init_def m1_def m2_def Live_def stutinvs nstutinvs livestutinv) lemma (in Firstprogram) enabled_m1: "⊢ Enabled ⟨m1⟩_(x,y)" proof (clarify) fix s show "s ⊨ Enabled ⟨m1⟩_(x,y)" by (rule base_enabled[OF bvar]) (auto simp: m1_def tla_defs) qed lemma (in Firstprogram) enabled_m2: "⊢ Enabled ⟨m2⟩_(x,y)" proof (clarify) fix s show "s ⊨ Enabled ⟨m2⟩_(x,y)" by (rule base_enabled[OF bvar]) (auto simp: m2_def tla_defs) qed locale Secondprogram = Firstprogram + fixes sem :: "state ⇒ nat" and pc1 :: "state ⇒ pcount" and pc2 :: "state ⇒ pcount" and vars and initPsi :: "temporal" and alpha1 :: "temporal" and alpha2 :: "temporal" and beta1 :: "temporal" and beta2 :: "temporal" and gamma1 :: "temporal" and gamma2 :: "temporal" and n1 :: "temporal" and n2 :: "temporal" and Live2 :: "temporal" and psi :: "temporal" and I :: "temporal" defines "vars ≡ LIFT (x,y,sem,pc1,pc2)" and "initPsi ≡ TEMP $pc1 = # a ∧ $pc2 = # a ∧ $x = # 0 ∧ $y = # 0 ∧ $sem = # 1" and "alpha1 ≡ TEMP $pc1 =#a ∧ # 0 < $sem ∧ pc1$ = #b ∧ sem$ = $sem - # 1 ∧ Unchanged (x,y,pc2)" and "alpha2 ≡ TEMP $pc2 =#a ∧ # 0 < $sem ∧ pc2` = #b ∧ sem$ = $sem - # 1 ∧ Unchanged (x,y,pc1)" and "beta1 ≡ TEMP $pc1 =#b ∧ pc1` = #g ∧ x` = Suc<$x> ∧ Unchanged (