188 lines
6.9 KiB
Plaintext
188 lines
6.9 KiB
Plaintext
(*
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* Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
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*
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* SPDX-License-Identifier: BSD-2-Clause
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*)
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theory Corres_Adjust_Preconds
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imports
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"Corres_UL"
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begin
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text \<open>
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Gadget for adjusting preconditions in a corres rule or similar.
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Probably only useful for predicates with two or more related
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preconditions, such as corres.
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Used to do some_corres_rule[adj_corres some_intro_rule],
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given e.g. some_intro_rule: @{prop "(s, t) \<in> sr \<Longrightarrow> P s \<Longrightarrow> Q t"}
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Will apply this rule to solve @{prop "Q t"} components in either
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precondition or any sub-conjunct, and will then try to put the
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assumptions @{prop "P s"}, @{prop "(s, t) \<in> sr"} into the right
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places. The premises of the rule can be in any given order.
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Concrete example at the bottom.
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\<close>
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named_theorems corres_adjust_precond_structures
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locale corres_adjust_preconds begin
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text \<open>Worker predicates. Broadly speaking, a goal
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of the form "preconds ?A ?B ?C ?D ==> P" expects to
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establish P by instantiating ?A, or failing that ?B,
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etc.
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A goal of the form finalise_preconds A exists to
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make sure that schematic conjuncts of A are resolved
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to True.\<close>
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definition
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preconds :: "bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> bool"
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where
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"preconds A B C D = (A \<and> B \<and> C \<and> D)"
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definition
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finalise_preconds :: "bool \<Rightarrow> bool"
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where
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"finalise_preconds A = True"
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text \<open>Use a precond directly to establish goal.\<close>
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lemma consume_preconds:
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"preconds A True True True \<Longrightarrow> A"
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"preconds True B True True \<Longrightarrow> B"
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"preconds True True C True \<Longrightarrow> C"
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"preconds True True True D \<Longrightarrow> D"
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by (simp_all add: preconds_def)
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lemmas consume_preconds_True = consume_preconds(1)[where A=True]
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text \<open>For use as a drule, split a set of schematic preconds
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to give two sets that can be instantiated separately.\<close>
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lemma split_preconds_left:
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"preconds (A \<and> A') (B \<and> B') (C \<and> C') (D \<and> D') \<Longrightarrow> preconds A B C D"
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"preconds (A \<and> A') (B \<and> B') (C \<and> C') True \<Longrightarrow> preconds A B C True"
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"preconds (A \<and> A') (B \<and> B') True True \<Longrightarrow> preconds A B True True"
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"preconds (A \<and> A') True True True \<Longrightarrow> preconds A True True True"
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by (simp_all add: preconds_def)
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lemma split_preconds_right:
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"preconds (A \<and> A') (B \<and> B') (C \<and> C') (D \<and> D') \<Longrightarrow> preconds A' B' C' D'"
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"preconds (A \<and> A') (B \<and> B') (C \<and> C') True \<Longrightarrow> preconds A' B' C' True"
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"preconds (A \<and> A') (B \<and> B') True True \<Longrightarrow> preconds A' B' True True"
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"preconds (A \<and> A') True True True \<Longrightarrow> preconds A' True True True"
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by (simp_all add: preconds_def)
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text \<open>For use as an erule. Initiate the precond process,
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creating a finalise goal to be solved later.\<close>
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lemma preconds_goal_initiate:
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"preconds A B C D \<Longrightarrow> (preconds A B C D \<Longrightarrow> Q)
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\<Longrightarrow> finalise_preconds (A \<and> B \<and> C \<and> D) \<Longrightarrow> Q"
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by simp
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text \<open>Finalise preconds, trying to replace conjuncts with
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True if they are not yet instantiated.\<close>
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lemma finalise_preconds:
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"finalise_preconds True"
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"finalise_preconds A \<Longrightarrow> finalise_preconds B \<Longrightarrow> finalise_preconds (A \<and> B)"
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"finalise_preconds X"
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by (simp_all add: finalise_preconds_def)
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text \<open>Shape of the whole process for application to regular corres goals.\<close>
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lemma corres_adjust_pre:
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"corres_underlying R nf nf' rs P Q f f'
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\<Longrightarrow> (\<And>s s'. (s, s') \<in> R \<Longrightarrow> preconds (P1 s) (Q1 s') True True \<Longrightarrow> P s)
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\<Longrightarrow> (\<And>s s'. (s, s') \<in> R \<Longrightarrow> preconds (Q2 s') (P2 s) True True \<Longrightarrow> Q s')
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\<Longrightarrow> corres_underlying R nf nf' rs (\<lambda>s. P1 s \<and> P2 s) (\<lambda>s'. Q1 s' \<and> Q2 s') f f'"
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apply (erule stronger_corres_guard_imp)
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apply (simp add: preconds_def)+
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done
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ML \<open>
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structure Corres_Adjust_Preconds = struct
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val def_intros = @{thms conjI pred_conjI}
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(* apply an intro rule, splitting preconds assumptions to
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provide unique assumptions for each goal. *)
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fun intro_split ctxt intros i =
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((resolve_tac ctxt intros
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THEN_ALL_NEW (TRY o assume_tac ctxt))
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THEN_ALL_NEW (fn j => (EVERY (replicate (j - i) (dresolve_tac ctxt @{thms split_preconds_left} j)))
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THEN dresolve_tac ctxt @{thms split_preconds_right} j)) i
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fun handle_preconds ctxt intros =
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TRY o (eresolve_tac ctxt [@{thm preconds_goal_initiate}]
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THEN' REPEAT_ALL_NEW (eresolve_tac ctxt @{thms consume_preconds_True}
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ORELSE' intro_split ctxt (intros @ def_intros)
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ORELSE' eresolve_tac ctxt @{thms consume_preconds})
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THEN' REPEAT_ALL_NEW (resolve_tac ctxt @{thms finalise_preconds})
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)
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fun mk_adj_preconds ctxt intros rule = let
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val xs = [rule] RL (Named_Theorems.get ctxt @{named_theorems corres_adjust_precond_structures})
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val x = case xs of
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[] => raise THM ("no unifier with corres_adjust_precond_structures", 1, [rule])
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| xs => hd xs
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in x
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|> ALLGOALS (handle_preconds ctxt intros)
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|> Seq.hd
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|> Simplifier.simplify (clear_simpset ctxt addsimps @{thms conj_assoc simp_thms(21-22)})
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end
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val setup =
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Attrib.setup @{binding "adj_corres"}
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((Attrib.thms -- Args.context)
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>> (fn (intros, ctxt) => Thm.rule_attribute [] (K (mk_adj_preconds ctxt intros))))
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"use argument theorems to adjust a corres theorem."
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end
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\<close>
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end
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declare corres_adjust_preconds.corres_adjust_pre[corres_adjust_precond_structures]
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setup Corres_Adjust_Preconds.setup
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experiment begin
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definition
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test_sr :: "(nat \<times> nat) set"
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where
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"test_sr = {(x, y). y = 2 * x}"
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lemma test_corres:
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"corres_underlying test_sr nf nf' dc (\<lambda>x. x < 40) (\<lambda>y. y < 30 \<and> y = 6)
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(modify (\<lambda>x. x + 2)) (modify (\<lambda>y. 10))"
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by (simp add: corres_underlying_def simpler_modify_def test_sr_def)
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lemma test_adj_precond:
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"(x, y) \<in> test_sr \<Longrightarrow> x = 3 \<Longrightarrow> y = 6"
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by (simp add: test_sr_def)
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ML \<open>
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Corres_Adjust_Preconds.mk_adj_preconds @{context} [@{thm test_adj_precond}] @{thm test_corres}
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\<close>
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lemma foo_adj_corres:
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"corres_underlying test_sr nf nf' dc (\<lambda>s. s < 40 \<and> s = 3) (\<lambda>s'. s' < 30) (modify (\<lambda>x. x + 2))
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(modify (\<lambda>y. 10))"
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by (rule test_corres[adj_corres test_adj_precond])
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text \<open>A more general demo of what it does.\<close>
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lemma
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assumes my_corres: "corres_underlying my_sr nf nf' rvrel P Q a c"
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assumes my_adj: "\<And>s s'. (s,s') \<in> my_sr \<Longrightarrow> P s \<Longrightarrow> Q s'"
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shows "corres_underlying my_sr nf nf' rvrel (\<lambda>s. P s \<and> P s) (\<lambda>s'. True) a c"
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apply (rule my_corres[adj_corres my_adj])
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done
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end
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end
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