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(*
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* Copyright (c) Facebook, Inc. and its affiliates.
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*
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* This source code is licensed under the MIT license found in the
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* LICENSE file in the root directory of this source tree.
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*)
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(* Proofs about llvm to llair translation *)
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open HolKernel boolLib bossLib Parse lcsymtacs;
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open listTheory arithmeticTheory pred_setTheory finite_mapTheory wordsTheory integer_wordTheory;
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open rich_listTheory pathTheory;
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open settingsTheory miscTheory memory_modelTheory;
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open llvmTheory llvm_propTheory llvm_ssaTheory llairTheory llair_propTheory llvm_to_llairTheory;
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new_theory "llvm_to_llair_prop";
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set_grammar_ancestry ["llvm", "llair", "llair_prop", "llvm_to_llair", "llvm_ssa"];
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numLib.prefer_num ();
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Inductive v_rel:
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(∀w. v_rel (FlatV (PtrV w)) (FlatV (IntV (w2i w) llair$pointer_size))) ∧
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(∀w. v_rel (FlatV (W1V w)) (FlatV (IntV (w2i w) 1))) ∧
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(∀w. v_rel (FlatV (W8V w)) (FlatV (IntV (w2i w) 8))) ∧
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(∀w. v_rel (FlatV (W32V w)) (FlatV (IntV (w2i w) 32))) ∧
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(∀w. v_rel (FlatV (W64V w)) (FlatV (IntV (w2i w) 64))) ∧
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(∀vs1 vs2.
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list_rel v_rel vs1 vs2
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⇒
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v_rel (AggV vs1) (AggV vs2))
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End
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Definition take_to_call_def:
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(take_to_call [] = []) ∧
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(take_to_call (i::is) =
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if terminator i ∨ is_call i then [i] else i :: take_to_call is)
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End
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Definition dest_llair_lab_def:
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dest_llair_lab (Lab_name f b) = (f, b)
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End
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Inductive pc_rel:
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(* LLVM side points to a normal instruction *)
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(∀prog emap ip bp d b idx b' prev_i fname.
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(* Both are valid pointers to blocks n the same function *)
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dest_fn ip.f = fst (dest_llair_lab bp) ∧
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alookup prog ip.f = Some d ∧
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alookup d.blocks ip.b = Some b ∧
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ip.i = Offset idx ∧
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idx < length b.body ∧
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get_block (translate_prog prog) bp b' ∧
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(* The LLVM side is at the start of a block, or immediately following a
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* call, which will also start a new block in llair *)
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(ip.i ≠ Offset 0 ⇒ get_instr prog (ip with i := Offset (idx - 1)) (Inl prev_i) ∧ is_call prev_i) ∧
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ip.f = Fn fname ∧
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(∃regs_to_keep.
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b' = fst (translate_instrs fname emap regs_to_keep (take_to_call (drop idx b.body))))
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⇒
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pc_rel prog emap ip bp) ∧
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(* If the LLVM side points to phi instructions, the llair side
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* should point to a block generated from them *)
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(∀prog emap ip bp b from_l phis.
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get_instr prog ip (Inr (from_l, phis)) ∧
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(* TODO: constrain b to be generated from the phis *)
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get_block (translate_prog prog) bp b
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⇒
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pc_rel prog emap ip bp)
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End
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Definition untranslate_reg_def:
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untranslate_reg (Var_name x t) = Reg x
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End
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Inductive complete_trace_rel:
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(∀i. complete_trace_rel (Some i) (Complete i)) ∧
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(complete_trace_rel None Partial) ∧
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(complete_trace_rel None Stuck)
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End
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(* Define when an LLVM state is related to a llair one.
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* Parameterised on a map for locals relating LLVM registers to llair
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* expressions that compute the value in that register. This corresponds to part
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* of the translation's state.
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*)
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Definition mem_state_rel_def:
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mem_state_rel prog emap (s:llvm$state) (s':llair$state) ⇔
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(* Live LLVM registers are mapped and have a related value in the emap
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* (after evaluating) *)
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(∀r. r ∈ live prog s.ip ⇒
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(∃v v' e.
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v_rel v.value v' ∧
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flookup s.locals r = Some v ∧
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flookup emap r = Some e ∧ eval_exp s' e v' ∧
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(* Each register used in e is dominated by an assignment to that
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* register for the entire live range of r. *)
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(∀ip1 r'. ip1.f = s.ip.f ∧ r ∈ live prog ip1 ∧ r' ∈ exp_uses e ⇒
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∃ip2. untranslate_reg r' ∈ assigns prog ip2 ∧ dominates prog ip2 ip1))) ∧
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reachable prog s.ip ∧
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erase_tags s.heap = s'.heap ∧
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complete_trace_rel s.exited s'.status
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End
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(* Define when an LLVM state is related to a llair one
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* Parameterised on a map for locals relating LLVM registers to llair
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* expressions that compute the value in that register. This corresponds to part
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* of the translation's state.
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*)
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Definition state_rel_def:
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state_rel prog emap (s:llvm$state) (s':llair$state) ⇔
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pc_rel prog emap s.ip s'.bp ∧
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mem_state_rel prog emap s s' ∧
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s'.status = get_observation prog s
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End
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Theorem mem_state_ignore_bp:
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∀prog emap s s' b. mem_state_rel prog emap s (s' with bp := b) ⇔ mem_state_rel prog emap s s'
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Proof
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rw [mem_state_rel_def] >> eq_tac >> rw [] >>
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first_x_assum drule >> rw [] >>
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`eval_exp (s' with bp := b) e v' ⇔ eval_exp s' e v'`
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by (irule eval_exp_ignores >> rw []) >>
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metis_tac []
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QED
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Theorem mem_state_rel_no_update:
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∀prog emap s1 s1' v res_v r i i'.
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assigns prog s1.ip = {} ∧
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mem_state_rel prog emap s1 s1' ∧
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v_rel v.value res_v ∧
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i ∈ next_ips prog s1.ip
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⇒
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mem_state_rel prog emap (s1 with ip := i) s1'
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Proof
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rw [mem_state_rel_def]
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>- (
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first_x_assum (qspec_then `r` mp_tac) >> simp [Once live_gen_kill, PULL_EXISTS] >>
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metis_tac [next_ips_same_func])
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>- metis_tac [next_ips_reachable]
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QED
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Theorem mem_state_rel_update:
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∀prog emap s1 s1' v res_v r e i.
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is_ssa prog ∧
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assigns prog s1.ip = {r} ∧
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mem_state_rel prog emap s1 s1' ∧
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eval_exp s1' e res_v ∧
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v_rel v.value res_v ∧
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i ∈ next_ips prog s1.ip ∧
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(∀r_use. r_use ∈ exp_uses e ⇒
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∃r_tmp. r_use ∈ exp_uses (translate_arg emap (Variable r_tmp)) ∧ r_tmp ∈ live prog s1.ip)
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⇒
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mem_state_rel prog (emap |+ (r, e))
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(s1 with <|ip := i; locals := s1.locals |+ (r, v) |>)
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s1'
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Proof
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rw [mem_state_rel_def]
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>- (
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rw [FLOOKUP_UPDATE]
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>- (
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HINT_EXISTS_TAC >> rw [] >>
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first_x_assum drule >> rw [] >>
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first_x_assum drule >> rw [] >>
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fs [exp_uses_def, translate_arg_def] >>
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pop_assum (qspec_then `s1.ip` mp_tac) >> simp [] >>
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disch_then drule >> rw [] >>
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`dominates prog s1.ip ip1`
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by (
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irule ssa_dominates_live_range_lem >> rw [] >>
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metis_tac [next_ips_same_func]) >>
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metis_tac [dominates_trans]) >>
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`i.f = s1.ip.f` by metis_tac [next_ips_same_func] >> simp [] >>
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first_x_assum irule >>
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simp [Once live_gen_kill, PULL_EXISTS, METIS_PROVE [] ``x ∨ y ⇔ (~y ⇒ x)``] >>
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metis_tac [])
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>- metis_tac [next_ips_reachable]
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QED
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Theorem mem_state_rel_update_keep:
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∀prog emap s1 s1' v res_v r i ty.
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is_ssa prog ∧
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assigns prog s1.ip = {r} ∧
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mem_state_rel prog emap s1 s1' ∧
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v_rel v.value res_v ∧
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reachable prog s1.ip ∧
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i ∈ next_ips prog s1.ip
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⇒
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mem_state_rel prog (emap |+ (r, Var (translate_reg r ty)))
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(s1 with <|ip := i; locals := s1.locals |+ (r, v)|>)
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(s1' with locals := s1'.locals |+ (translate_reg r ty, res_v))
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Proof
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rw [mem_state_rel_def]
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>- (
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rw [FLOOKUP_UPDATE]
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>- (
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simp [Once eval_exp_cases] >>
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qexists_tac `res_v` >> rw [exp_uses_def] >>
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rw [FLOOKUP_UPDATE] >>
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Cases_on `r` >> simp [translate_reg_def, untranslate_reg_def] >>
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`∃ip. ip.f = ip1.f ∧ Reg s ∈ uses prog ip`
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by (
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qabbrev_tac `x = (ip1.f = i.f)` >>
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fs [live_def] >> qexists_tac `last (ip1::path')` >> rw [] >>
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irule good_path_same_func >>
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qexists_tac `ip1::path'` >> rw [MEM_LAST] >>
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metis_tac []) >>
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metis_tac [ssa_dominates_live_range]) >>
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first_x_assum (qspec_then `r'` mp_tac) >>
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simp [Once live_gen_kill, PULL_EXISTS] >>
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impl_tac >> rw []
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>- metis_tac [] >>
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ntac 3 HINT_EXISTS_TAC >> rw []
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>- (
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`DRESTRICT (s1' with locals := s1'.locals |+ (translate_reg r ty,res_v)).locals (exp_uses e) =
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DRESTRICT s1'.locals (exp_uses e)`
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suffices_by metis_tac [eval_exp_ignores_unused] >>
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rw [] >>
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first_x_assum (qspecl_then [`s1.ip`, `translate_reg r ty`] mp_tac) >> simp [Once live_gen_kill] >>
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impl_tac >- metis_tac [] >> rw [] >>
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`ip2 = s1.ip`
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by (
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fs [is_ssa_def, EXTENSION, IN_DEF] >>
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Cases_on `r` >> fs [translate_reg_def, untranslate_reg_def] >>
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metis_tac [reachable_dominates_same_func]) >>
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metis_tac [dominates_irrefl])
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>- (
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first_x_assum irule >> rw [] >>
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metis_tac [next_ips_same_func]))
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>- metis_tac [next_ips_reachable]
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QED
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Theorem v_rel_bytes:
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∀v v'. v_rel v v' ⇒ llvm_value_to_bytes v = llair_value_to_bytes v'
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Proof
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ho_match_mp_tac v_rel_ind >>
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rw [v_rel_cases, llvm_value_to_bytes_def, llair_value_to_bytes_def] >>
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rw [value_to_bytes_def, llvmTheory.unconvert_value_def, w2n_i2n,
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llairTheory.unconvert_value_def, llairTheory.pointer_size_def,
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llvmTheory.pointer_size_def] >>
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pop_assum mp_tac >>
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qid_spec_tac `vs1` >>
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Induct_on `vs2` >> rw [] >> rw []
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QED
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Theorem translate_constant_correct_lem:
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(∀c s prog emap s' (g : glob_var |-> β # word64).
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mem_state_rel prog emap s s'
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⇒
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∃v'. eval_exp s' (translate_const c) v' ∧ v_rel (eval_const g c) v') ∧
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(∀(cs : (ty # const) list) s prog emap s' (g : glob_var |-> β # word64).
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mem_state_rel prog emap s s'
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⇒
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∃v'. list_rel (eval_exp s') (map (translate_const o snd) cs) v' ∧ list_rel v_rel (map (eval_const g o snd) cs) v') ∧
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(∀(tc : ty # const) s prog emap s' (g : glob_var |-> β # word64).
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mem_state_rel prog emap s s'
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⇒
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∃v'. eval_exp s' (translate_const (snd tc)) v' ∧ v_rel (eval_const g (snd tc)) v')
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Proof
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ho_match_mp_tac const_induction >> rw [translate_const_def] >>
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simp [Once eval_exp_cases, eval_const_def]
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>- (
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Cases_on `s` >> simp [eval_const_def, translate_size_def, v_rel_cases] >>
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metis_tac [truncate_2comp_i2w_w2i, dimindex_1, dimindex_8, dimindex_32, dimindex_64])
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>- (
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simp [v_rel_cases, PULL_EXISTS, MAP_MAP_o] >>
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fs [combinTheory.o_DEF, pairTheory.LAMBDA_PROD] >>
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metis_tac [])
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>- (
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simp [v_rel_cases, PULL_EXISTS, MAP_MAP_o] >>
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fs [combinTheory.o_DEF, pairTheory.LAMBDA_PROD] >>
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metis_tac [])
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(* TODO: unimplemented stuff *)
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>- cheat
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>- cheat
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>- cheat
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>- cheat
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QED
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Theorem translate_constant_correct:
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∀c s prog emap s' g.
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mem_state_rel prog emap s s'
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⇒
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∃v'. eval_exp s' (translate_const c) v' ∧ v_rel (eval_const g c) v'
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Proof
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metis_tac [translate_constant_correct_lem]
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QED
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Theorem translate_const_no_reg[simp]:
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∀c. r ∉ exp_uses (translate_const c)
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Proof
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ho_match_mp_tac translate_const_ind >>
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rw [translate_const_def, exp_uses_def, MEM_MAP, METIS_PROVE [] ``x ∨ y ⇔ (~x ⇒ y)``] >>
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TRY pairarg_tac >> fs []
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>- metis_tac []
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>- metis_tac [] >>
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(* TODO: unimplemented stuff *)
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cheat
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QED
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Theorem translate_arg_correct:
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∀s a v prog emap s'.
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mem_state_rel prog emap s s' ∧
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eval s a = Some v ∧
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arg_to_regs a ⊆ live prog s.ip
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⇒
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∃v'. eval_exp s' (translate_arg emap a) v' ∧ v_rel v.value v'
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Proof
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Cases_on `a` >> rw [eval_def, translate_arg_def] >> rw []
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>- metis_tac [translate_constant_correct] >>
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CASE_TAC >> fs [PULL_EXISTS, mem_state_rel_def, arg_to_regs_def] >>
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res_tac >> rfs [] >> metis_tac []
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QED
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Theorem is_allocated_mem_state_rel:
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∀prog emap s1 s1'.
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mem_state_rel prog emap s1 s1'
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⇒
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(∀i. is_allocated i s1.heap ⇔ is_allocated i s1'.heap)
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Proof
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rw [mem_state_rel_def, is_allocated_def, erase_tags_def] >>
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pop_assum mp_tac >> pop_assum (mp_tac o GSYM) >> rw []
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QED
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Theorem restricted_i2w_11:
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∀i (w:'a word). INT_MIN (:'a) ≤ i ∧ i ≤ INT_MAX (:'a) ⇒ (i2w i : 'a word) = i2w (w2i w) ⇒ i = w2i w
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Proof
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rw [i2w_def]
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>- (
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Cases_on `n2w (Num (-i)) = INT_MINw` >>
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rw [w2i_neg, w2i_INT_MINw] >>
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fs [word_L_def] >>
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`?j. 0 ≤ j ∧ i = -j` by intLib.COOPER_TAC >>
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rw [] >>
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fs [] >>
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`INT_MIN (:'a) < dimword (:'a)` by metis_tac [INT_MIN_LT_DIMWORD] >>
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`Num j MOD dimword (:'a) = Num j`
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by (irule LESS_MOD >> intLib.COOPER_TAC) >>
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fs []
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>- intLib.COOPER_TAC
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>- (
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`Num j < INT_MIN (:'a)` by intLib.COOPER_TAC >>
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fs [w2i_n2w_pos, integerTheory.INT_OF_NUM]))
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>- (
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fs [GSYM INT_MAX, INT_MAX_def] >>
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`Num i < INT_MIN (:'a)` by intLib.COOPER_TAC >>
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rw [w2i_n2w_pos, integerTheory.INT_OF_NUM] >>
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|
intLib.COOPER_TAC)
|
|
|
QED
|
|
|
|
|
|
Theorem translate_sub_correct:
|
|
|
∀prog emap s1 s1' nsw nuw ty v1 v1' v2 v2' e2' e1' result.
|
|
|
mem_state_rel prog emap s1 s1' ∧
|
|
|
do_sub nuw nsw v1 v2 ty = Some result ∧
|
|
|
eval_exp s1' e1' v1' ∧
|
|
|
v_rel v1.value v1' ∧
|
|
|
eval_exp s1' e2' v2' ∧
|
|
|
v_rel v2.value v2'
|
|
|
⇒
|
|
|
∃v3'.
|
|
|
eval_exp s1' (Sub (translate_ty ty) e1' e2') v3' ∧
|
|
|
v_rel result.value v3'
|
|
|
Proof
|
|
|
rw [] >>
|
|
|
simp [Once eval_exp_cases] >>
|
|
|
fs [do_sub_def] >> rw [] >>
|
|
|
rfs [v_rel_cases] >> rw [] >> fs [] >>
|
|
|
BasicProvers.EVERY_CASE_TAC >> fs [PULL_EXISTS, translate_ty_def, translate_size_def] >>
|
|
|
pairarg_tac >> fs [] >>
|
|
|
fs [pairTheory.PAIR_MAP, wordsTheory.FST_ADD_WITH_CARRY] >>
|
|
|
rw [] >>
|
|
|
qmatch_goalsub_abbrev_tac `w2i (-1w * w1 + w2)` >>
|
|
|
qexists_tac `w2i w2` >> qexists_tac `w2i w1` >> simp [] >>
|
|
|
unabbrev_all_tac >> rw []
|
|
|
>- (
|
|
|
irule restricted_i2w_11 >> simp [word_sub_i2w] >>
|
|
|
`dimindex (:1) = 1` by rw [] >>
|
|
|
drule truncate_2comp_i2w_w2i >>
|
|
|
rw [word_sub_i2w] >>
|
|
|
metis_tac [w2i_ge, w2i_le, SIMP_CONV (srw_ss()) [] ``INT_MIN (:1)``,
|
|
|
SIMP_CONV (srw_ss()) [] ``INT_MAX (:1)``])
|
|
|
>- (
|
|
|
irule restricted_i2w_11 >> simp [word_sub_i2w] >>
|
|
|
`dimindex (:8) = 8` by rw [] >>
|
|
|
drule truncate_2comp_i2w_w2i >>
|
|
|
rw [word_sub_i2w] >>
|
|
|
metis_tac [w2i_ge, w2i_le, SIMP_CONV (srw_ss()) [] ``INT_MIN (:8)``,
|
|
|
SIMP_CONV (srw_ss()) [] ``INT_MAX (:8)``])
|
|
|
>- (
|
|
|
irule restricted_i2w_11 >> simp [word_sub_i2w] >>
|
|
|
`dimindex (:32) = 32` by rw [] >>
|
|
|
drule truncate_2comp_i2w_w2i >>
|
|
|
rw [word_sub_i2w] >>
|
|
|
metis_tac [w2i_ge, w2i_le, SIMP_CONV (srw_ss()) [] ``INT_MIN (:32)``,
|
|
|
SIMP_CONV (srw_ss()) [] ``INT_MAX (:32)``])
|
|
|
>- (
|
|
|
irule restricted_i2w_11 >> simp [word_sub_i2w] >>
|
|
|
`dimindex (:64) = 64` by rw [] >>
|
|
|
drule truncate_2comp_i2w_w2i >>
|
|
|
rw [word_sub_i2w] >>
|
|
|
metis_tac [w2i_ge, w2i_le, SIMP_CONV (srw_ss()) [] ``INT_MIN (:64)``,
|
|
|
SIMP_CONV (srw_ss()) [] ``INT_MAX (:64)``])
|
|
|
QED
|
|
|
|
|
|
Theorem translate_extract_correct:
|
|
|
∀prog emap s1 s1' a v v1' e1' cs ns result.
|
|
|
mem_state_rel prog emap s1 s1' ∧
|
|
|
map (λci. signed_v_to_num (eval_const s1.globals ci)) cs = map Some ns ∧
|
|
|
extract_value v ns = Some result ∧
|
|
|
eval_exp s1' e1' v1' ∧
|
|
|
v_rel v v1'
|
|
|
⇒
|
|
|
∃v2'.
|
|
|
eval_exp s1' (foldl (λe c. Select e (translate_const c)) e1' cs) v2' ∧
|
|
|
v_rel result v2'
|
|
|
Proof
|
|
|
Induct_on `cs` >> rw [] >> fs [extract_value_def]
|
|
|
>- metis_tac [] >>
|
|
|
first_x_assum irule >>
|
|
|
Cases_on `ns` >> fs [] >>
|
|
|
qmatch_goalsub_rename_tac `translate_const c` >>
|
|
|
`?v2'. eval_exp s1' (translate_const c) v2' ∧ v_rel (eval_const s1.globals c) v2'`
|
|
|
by metis_tac [translate_constant_correct] >>
|
|
|
Cases_on `v` >> fs [extract_value_def] >>
|
|
|
qpat_x_assum `v_rel (AggV _) _` mp_tac >>
|
|
|
simp [Once v_rel_cases] >> rw [] >>
|
|
|
simp [Once eval_exp_cases, PULL_EXISTS] >>
|
|
|
fs [LIST_REL_EL_EQN] >>
|
|
|
qmatch_assum_rename_tac `_ = map Some is` >>
|
|
|
Cases_on `eval_const s1.globals c` >> fs [signed_v_to_num_def, signed_v_to_int_def] >> rw [] >>
|
|
|
`?i. v2' = FlatV i` by fs [v_rel_cases] >> fs [] >>
|
|
|
qmatch_assum_rename_tac `option_join _ = Some x` >>
|
|
|
`?size. i = IntV (&x) size` suffices_by metis_tac [] >> rw [] >>
|
|
|
qpat_x_assum `v_rel _ _` mp_tac >>
|
|
|
simp [v_rel_cases] >> rw [] >> fs [signed_v_to_int_def] >> rw [] >>
|
|
|
intLib.COOPER_TAC
|
|
|
QED
|
|
|
|
|
|
Theorem translate_update_correct:
|
|
|
∀prog emap s1 s1' a v1 v1' v2 v2' e2 e2' e1' cs ns result.
|
|
|
mem_state_rel prog emap s1 s1' ∧
|
|
|
map (λci. signed_v_to_num (eval_const s1.globals ci)) cs = map Some ns ∧
|
|
|
insert_value v1 v2 ns = Some result ∧
|
|
|
eval_exp s1' e1' v1' ∧
|
|
|
v_rel v1 v1' ∧
|
|
|
eval_exp s1' e2' v2' ∧
|
|
|
v_rel v2 v2'
|
|
|
⇒
|
|
|
∃v3'.
|
|
|
eval_exp s1' (translate_updatevalue e1' e2' cs) v3' ∧
|
|
|
v_rel result v3'
|
|
|
Proof
|
|
|
Induct_on `cs` >> rw [] >> fs [insert_value_def, translate_updatevalue_def]
|
|
|
>- metis_tac [] >>
|
|
|
simp [Once eval_exp_cases, PULL_EXISTS] >>
|
|
|
Cases_on `ns` >> fs [] >>
|
|
|
Cases_on `v1` >> fs [insert_value_def] >>
|
|
|
rename [`insert_value (el x _) _ ns`] >>
|
|
|
Cases_on `insert_value (el x l) v2 ns` >> fs [] >> rw [] >>
|
|
|
qpat_x_assum `v_rel (AggV _) _` mp_tac >> simp [Once v_rel_cases] >> rw [] >>
|
|
|
simp [v_rel_cases] >>
|
|
|
qmatch_goalsub_rename_tac `translate_const c` >>
|
|
|
qexists_tac `vs2` >> simp [] >>
|
|
|
`?v4'. eval_exp s1' (translate_const c) v4' ∧ v_rel (eval_const s1.globals c) v4'`
|
|
|
by metis_tac [translate_constant_correct] >>
|
|
|
`?idx_size. v4' = FlatV (IntV (&x) idx_size)`
|
|
|
by (
|
|
|
pop_assum mp_tac >> simp [Once v_rel_cases] >>
|
|
|
rw [] >> fs [signed_v_to_num_def, signed_v_to_int_def] >>
|
|
|
intLib.COOPER_TAC) >>
|
|
|
first_x_assum drule >>
|
|
|
disch_then drule >>
|
|
|
disch_then drule >>
|
|
|
disch_then (qspecl_then [`el x vs2`, `v2'`, `e2'`, `Select e1' (translate_const c)`] mp_tac) >>
|
|
|
simp [Once eval_exp_cases] >>
|
|
|
metis_tac [EVERY2_LUPDATE_same, LIST_REL_LENGTH, LIST_REL_EL_EQN]
|
|
|
QED
|
|
|
|
|
|
Theorem prog_ok_nonterm:
|
|
|
∀prog i ip.
|
|
|
prog_ok prog ∧ get_instr prog ip (Inl i) ∧ ¬terminator i ⇒ inc_pc ip ∈ next_ips prog ip
|
|
|
Proof
|
|
|
rw [next_ips_cases, IN_DEF, get_instr_cases, PULL_EXISTS] >>
|
|
|
`terminator (last b.body) ∧ b.body ≠ []` by metis_tac [prog_ok_def] >>
|
|
|
Cases_on `length b.body = idx + 1`
|
|
|
>- (
|
|
|
drule LAST_EL >>
|
|
|
rw [] >> fs [DECIDE ``PRE (x + 1) = x``]) >>
|
|
|
Cases_on `el idx b.body` >>
|
|
|
fs [instr_next_ips_def, terminator_def] >>
|
|
|
rw [EXISTS_OR_THM, inc_pc_def, inc_bip_def]
|
|
|
QED
|
|
|
|
|
|
Theorem translate_instr_to_exp_correct:
|
|
|
∀emap instr r t s1 s1' s2 prog l.
|
|
|
is_ssa prog ∧ prog_ok prog ∧
|
|
|
classify_instr instr = Exp r t ∧
|
|
|
mem_state_rel prog emap s1 s1' ∧
|
|
|
get_instr prog s1.ip (Inl instr) ∧
|
|
|
step_instr prog s1 instr l s2 ⇒
|
|
|
∃pv emap' s2'.
|
|
|
l = Tau ∧
|
|
|
s2.ip = inc_pc s1.ip ∧
|
|
|
mem_state_rel prog emap' s2 s2' ∧
|
|
|
(r ∉ regs_to_keep ⇒ s1' = s2' ∧ emap' = emap |+ (r, translate_instr_to_exp emap instr)) ∧
|
|
|
(r ∈ regs_to_keep ⇒
|
|
|
emap' = emap |+ (r,Var (translate_reg r t)) ∧
|
|
|
step_inst s1' (Move [(translate_reg r t, translate_instr_to_exp emap instr)]) Tau s2')
|
|
|
Proof
|
|
|
recInduct translate_instr_to_exp_ind >>
|
|
|
simp [translate_instr_to_exp_def, classify_instr_def] >>
|
|
|
conj_tac
|
|
|
>- ( (* Sub *)
|
|
|
rw [step_instr_cases, get_instr_cases, update_result_def] >>
|
|
|
qpat_x_assum `Sub _ _ _ _ _ _ = el _ _` (assume_tac o GSYM) >>
|
|
|
`bigunion (image arg_to_regs {a1; a2}) ⊆ live prog s1.ip`
|
|
|
by (
|
|
|
simp [Once live_gen_kill, SUBSET_DEF, uses_cases, IN_DEF, get_instr_cases,
|
|
|
instr_uses_def] >>
|
|
|
metis_tac []) >>
|
|
|
fs [] >>
|
|
|
first_x_assum (mp_then.mp_then mp_then.Any mp_tac translate_arg_correct) >>
|
|
|
disch_then drule >> disch_then drule >>
|
|
|
first_x_assum (mp_then.mp_then mp_then.Any mp_tac translate_arg_correct) >>
|
|
|
disch_then drule >> disch_then drule >> rw [] >>
|
|
|
drule translate_sub_correct >> disch_then drule >>
|
|
|
disch_then (qspecl_then [`v'`, `v''`] mp_tac) >> simp [] >>
|
|
|
disch_then drule >> disch_then drule >> rw [] >>
|
|
|
rename1 `eval_exp _ (Sub _ _ _) res_v` >>
|
|
|
rename1 `r ∈ _` >>
|
|
|
Cases_on `r ∈ regs_to_keep` >> rw []
|
|
|
>- (
|
|
|
simp [step_inst_cases, PULL_EXISTS] >>
|
|
|
qexists_tac `res_v` >> rw []
|
|
|
>- simp [inc_pc_def, llvmTheory.inc_pc_def]
|
|
|
>- (
|
|
|
rw [update_results_def, GSYM FUPDATE_EQ_FUPDATE_LIST] >>
|
|
|
simp [llvmTheory.inc_pc_def] >>
|
|
|
irule mem_state_rel_update_keep >> rw []
|
|
|
>- rw [assigns_cases, EXTENSION, IN_DEF, get_instr_cases, instr_assigns_def]
|
|
|
>- (
|
|
|
drule prog_ok_nonterm >>
|
|
|
simp [get_instr_cases, PULL_EXISTS] >>
|
|
|
ntac 3 (disch_then drule) >>
|
|
|
simp [terminator_def, next_ips_cases, IN_DEF, inc_pc_def])
|
|
|
>- fs [mem_state_rel_def]))
|
|
|
>- rw [inc_pc_def, llvmTheory.inc_pc_def]
|
|
|
>- (
|
|
|
simp [llvmTheory.inc_pc_def] >>
|
|
|
irule mem_state_rel_update >> rw []
|
|
|
>- (
|
|
|
fs [exp_uses_def]
|
|
|
>| [Cases_on `a1`, Cases_on `a2`] >>
|
|
|
fs [translate_arg_def] >>
|
|
|
rename1 `flookup _ r_tmp` >>
|
|
|
qexists_tac `r_tmp` >> rw [] >>
|
|
|
simp [Once live_gen_kill] >> disj2_tac >>
|
|
|
simp [uses_cases, IN_DEF, get_instr_cases, instr_uses_def, arg_to_regs_def])
|
|
|
>- rw [assigns_cases, EXTENSION, IN_DEF, get_instr_cases, instr_assigns_def]
|
|
|
>- (
|
|
|
drule prog_ok_nonterm >>
|
|
|
simp [get_instr_cases, PULL_EXISTS] >>
|
|
|
ntac 3 (disch_then drule) >>
|
|
|
simp [terminator_def, next_ips_cases, IN_DEF, inc_pc_def]) >>
|
|
|
metis_tac [])) >>
|
|
|
conj_tac
|
|
|
>- ( (* Extractvalue *)
|
|
|
rw [step_instr_cases] >>
|
|
|
simp [llvmTheory.inc_pc_def, update_result_def, FLOOKUP_UPDATE] >>
|
|
|
drule translate_extract_correct >> rpt (disch_then drule) >>
|
|
|
drule translate_arg_correct >> disch_then drule >>
|
|
|
`arg_to_regs a ⊆ live prog s1.ip`
|
|
|
by (
|
|
|
fs [get_instr_cases] >>
|
|
|
qpat_x_assum `Extractvalue _ _ _ = el _ _` (mp_tac o GSYM) >>
|
|
|
simp [Once live_gen_kill, SUBSET_DEF, uses_cases, IN_DEF, get_instr_cases,
|
|
|
instr_uses_def]) >>
|
|
|
simp [] >> strip_tac >>
|
|
|
disch_then drule >> simp [] >> rw [] >>
|
|
|
rename1 `eval_exp _ (foldl _ _ _) res_v` >>
|
|
|
rw [inc_bip_def, inc_pc_def] >>
|
|
|
rename1 `r ∈ _` >>
|
|
|
Cases_on `r ∈ regs_to_keep` >> rw []
|
|
|
>- (
|
|
|
simp [step_inst_cases, PULL_EXISTS] >>
|
|
|
qexists_tac `res_v` >> rw [] >>
|
|
|
rw [update_results_def] >>
|
|
|
(* TODO: unfinished *)
|
|
|
cheat)
|
|
|
>- cheat) >>
|
|
|
conj_tac
|
|
|
>- ( (* Updatevalue *)
|
|
|
rw [step_instr_cases] >>
|
|
|
simp [llvmTheory.inc_pc_def, update_result_def, FLOOKUP_UPDATE] >>
|
|
|
drule translate_update_correct >> rpt (disch_then drule) >>
|
|
|
first_x_assum (mp_then.mp_then mp_then.Any mp_tac translate_arg_correct) >>
|
|
|
disch_then drule >>
|
|
|
first_x_assum (mp_then.mp_then mp_then.Any mp_tac translate_arg_correct) >>
|
|
|
disch_then drule >>
|
|
|
`arg_to_regs a1 ⊆ live prog s1.ip ∧
|
|
|
arg_to_regs a2 ⊆ live prog s1.ip`
|
|
|
by (
|
|
|
fs [get_instr_cases] >>
|
|
|
qpat_x_assum `Insertvalue _ _ _ _ = el _ _` (mp_tac o GSYM) >>
|
|
|
ONCE_REWRITE_TAC [live_gen_kill] >>
|
|
|
simp [SUBSET_DEF, uses_cases, IN_DEF, get_instr_cases,
|
|
|
instr_uses_def]) >>
|
|
|
simp [] >> strip_tac >> strip_tac >>
|
|
|
disch_then (qspecl_then [`v'`, `v''`] mp_tac) >> simp [] >>
|
|
|
disch_then drule >> disch_then drule >>
|
|
|
rw [] >>
|
|
|
rename1 `eval_exp _ (translate_updatevalue _ _ _) res_v` >>
|
|
|
rw [inc_pc_def, inc_bip_def] >>
|
|
|
rename1 `r ∈ _` >>
|
|
|
Cases_on `r ∈ regs_to_keep` >> rw []
|
|
|
>- (
|
|
|
simp [step_inst_cases, PULL_EXISTS] >>
|
|
|
qexists_tac `res_v` >> rw [] >>
|
|
|
rw [update_results_def] >>
|
|
|
(* TODO: unfinished *)
|
|
|
cheat)
|
|
|
>- cheat) >>
|
|
|
cheat
|
|
|
QED
|
|
|
|
|
|
Triviality eval_exp_help:
|
|
|
(s1 with heap := h).locals = s1.locals
|
|
|
Proof
|
|
|
rw []
|
|
|
QED
|
|
|
|
|
|
Theorem erase_tags_set_bytes:
|
|
|
∀p v l h. erase_tags (set_bytes p v l h) = set_bytes () v l (erase_tags h)
|
|
|
Proof
|
|
|
Induct_on `v` >> rw [set_bytes_def] >>
|
|
|
irule (METIS_PROVE [] ``x = y ⇒ f a b c x = f a b c y``) >>
|
|
|
rw [erase_tags_def]
|
|
|
QED
|
|
|
|
|
|
(*
|
|
|
Theorem translate_instr_to_inst_correct:
|
|
|
∀prog emap instr s1 s1' s2.
|
|
|
classify_instr instr = Non_exp ∧
|
|
|
state_rel prog emap s1 s1' ∧
|
|
|
get_instr prog s1.ip instr ∧
|
|
|
step_instr prog s1 instr s2 ⇒
|
|
|
∃s2'.
|
|
|
step_inst s1' (translate_instr_to_inst emap instr) s2' ∧
|
|
|
state_rel prog emap s2 s2'
|
|
|
|
|
|
Proof
|
|
|
|
|
|
rw [step_instr_cases] >>
|
|
|
fs [classify_instr_def, translate_instr_to_inst_def]
|
|
|
>- ( (* Load *)
|
|
|
cheat)
|
|
|
>- ( (* Store *)
|
|
|
simp [step_inst_cases, PULL_EXISTS] >>
|
|
|
drule get_instr_live >> rw [uses_def] >>
|
|
|
drule translate_arg_correct >> disch_then drule >> disch_then drule >>
|
|
|
qpat_x_assum `eval _ _ = Some _` mp_tac >>
|
|
|
drule translate_arg_correct >> disch_then drule >> disch_then drule >>
|
|
|
rw [] >>
|
|
|
qpat_x_assum `v_rel (FlatV _) _` mp_tac >> simp [Once v_rel_cases] >> rw [] >>
|
|
|
HINT_EXISTS_TAC >> rw [] >>
|
|
|
qexists_tac `freeable` >> rw [] >>
|
|
|
HINT_EXISTS_TAC >> rw []
|
|
|
>- metis_tac [v_rel_bytes]
|
|
|
>- (
|
|
|
fs [w2n_i2n, pointer_size_def] >>
|
|
|
metis_tac [v_rel_bytes, is_allocated_state_rel, ADD_COMM]) >>
|
|
|
fs [state_rel_def] >>
|
|
|
rw []
|
|
|
>- cheat
|
|
|
>- (
|
|
|
fs [llvmTheory.inc_pc_def] >>
|
|
|
`r ∈ live prog s1.ip`
|
|
|
by (
|
|
|
drule live_gen_kill >>
|
|
|
rw [next_ips_def, assigns_def, uses_def, inc_pc_def]) >>
|
|
|
first_x_assum drule >> rw [] >>
|
|
|
metis_tac [eval_exp_ignores, eval_exp_help])
|
|
|
>- (
|
|
|
rw [llvmTheory.inc_pc_def, w2n_i2n, pointer_size_def, erase_tags_set_bytes] >>
|
|
|
metis_tac[v_rel_bytes]))
|
|
|
>- cheat
|
|
|
>- cheat
|
|
|
>- cheat
|
|
|
QED
|
|
|
|
|
|
|
|
|
simp [step_inst_cases, PULL_EXISTS] >>
|
|
|
Cases_on `r` >> simp [translate_reg_def] >>
|
|
|
drule get_instr_live >> rw [uses_def] >>
|
|
|
drule translate_arg_correct >> disch_then drule >> disch_then drule >>
|
|
|
simp [Once v_rel_cases] >> rw [] >>
|
|
|
qexists_tac `IntV (w2i w) pointer_size` >> rw [] >>
|
|
|
qexists_tac `freeable` >> rw []
|
|
|
>- (fs [w2n_i2n, pointer_size_def] >> metis_tac [is_allocated_state_rel]) >>
|
|
|
fs [state_rel_def] >> rw []
|
|
|
>- cheat
|
|
|
>- (
|
|
|
fs [llvmTheory.inc_pc_def, update_results_def, update_result_def] >>
|
|
|
rw [] >> fs [FLOOKUP_UPDATE] >> rw []
|
|
|
>- (
|
|
|
cheat)
|
|
|
>- (
|
|
|
`r ∈ live prog s1.ip`
|
|
|
by (
|
|
|
drule live_gen_kill >>
|
|
|
rw [next_ips_def, assigns_def, uses_def, inc_pc_def]) >>
|
|
|
first_x_assum drule >> rw [] >>
|
|
|
qexists_tac `v` >>
|
|
|
qexists_tac `v'` >>
|
|
|
qexists_tac `e` >>
|
|
|
rw []
|
|
|
metis_tac [eval_exp_ignores, eval_exp_help])
|
|
|
|
|
|
|
|
|
>- fs [update_results_def, llvmTheory.inc_pc_def, update_result_def]
|
|
|
|
|
|
*)
|
|
|
|
|
|
Definition translate_trace_def:
|
|
|
(translate_trace types Tau = Tau ) ∧
|
|
|
(translate_trace types (W gv bytes) = W (translate_glob_var gv (types gv)) bytes)
|
|
|
End
|
|
|
|
|
|
Definition untranslate_glob_var_def:
|
|
|
untranslate_glob_var (Var_name n ty) = Glob_var n
|
|
|
End
|
|
|
|
|
|
Definition untranslate_trace_def:
|
|
|
(untranslate_trace Tau = Tau ) ∧
|
|
|
(untranslate_trace (W gv bytes) = W (untranslate_glob_var gv) bytes)
|
|
|
End
|
|
|
|
|
|
Theorem un_translate_glob_inv:
|
|
|
∀x t. untranslate_glob_var (translate_glob_var x t) = x
|
|
|
Proof
|
|
|
Cases_on `x` >> rw [untranslate_glob_var_def, translate_glob_var_def]
|
|
|
QED
|
|
|
|
|
|
Theorem un_translate_trace_inv:
|
|
|
∀x. untranslate_trace (translate_trace types x) = x
|
|
|
Proof
|
|
|
Cases >> rw [translate_trace_def, untranslate_trace_def] >>
|
|
|
metis_tac [un_translate_glob_inv]
|
|
|
QED
|
|
|
|
|
|
Theorem take_to_call_lem:
|
|
|
∀i idx body.
|
|
|
idx < length body ∧ el idx body = i ∧ ¬terminator i ∧ ¬is_call i ⇒
|
|
|
take_to_call (drop idx body) = i :: take_to_call (drop (idx + 1) body)
|
|
|
Proof
|
|
|
Induct_on `idx` >> rw []
|
|
|
>- (Cases_on `body` >> fs [take_to_call_def] >> rw []) >>
|
|
|
Cases_on `body` >> fs [] >>
|
|
|
first_x_assum drule >> simp [ADD1]
|
|
|
QED
|
|
|
|
|
|
Theorem translate_instrs_correct1:
|
|
|
∀prog s1 tr s2.
|
|
|
multi_step prog s1 tr s2 ⇒
|
|
|
!s1' b' emap regs_to_keep d b types idx.
|
|
|
prog_ok prog ∧
|
|
|
is_ssa prog ∧
|
|
|
mem_state_rel prog emap s1 s1' ∧
|
|
|
alookup prog s1.ip.f = Some d ∧
|
|
|
alookup d.blocks s1.ip.b = Some b ∧
|
|
|
s1.ip.i = Offset idx ∧
|
|
|
b' = fst (translate_instrs (dest_fn s1.ip.f) emap regs_to_keep (take_to_call (drop idx b.body)))
|
|
|
⇒
|
|
|
∃emap s2' tr'.
|
|
|
step_block (translate_prog prog) s1' b'.cmnd tr' b'.term s2' ∧
|
|
|
filter ($≠ Tau) tr' = filter ($≠ Tau) (map (translate_trace types) tr) ∧
|
|
|
state_rel prog emap s2 s2'
|
|
|
Proof
|
|
|
ho_match_mp_tac multi_step_ind >> rw_tac std_ss []
|
|
|
>- (
|
|
|
fs [last_step_def]
|
|
|
>- ( (* Phi (not handled here) *)
|
|
|
fs [get_instr_cases])
|
|
|
>- ( (* Terminator *)
|
|
|
`l = Tau`
|
|
|
by (
|
|
|
fs [llvmTheory.step_cases] >>
|
|
|
`i' = i''` by metis_tac [get_instr_func, sumTheory.INL_11] >>
|
|
|
fs [step_instr_cases] >> rfs [terminator_def]) >>
|
|
|
fs [get_instr_cases] >> rw [] >>
|
|
|
`el idx b.body = el 0 (drop idx b.body)` by rw [EL_DROP] >>
|
|
|
fs [] >>
|
|
|
Cases_on `drop idx b.body` >> fs [DROP_NIL] >> rw [] >>
|
|
|
simp [take_to_call_def, translate_instrs_def] >>
|
|
|
Cases_on `el idx b.body` >> fs [terminator_def, classify_instr_def, translate_trace_def] >> rw []
|
|
|
>- ( (* Ret *)
|
|
|
cheat)
|
|
|
>- ( (* Br *)
|
|
|
simp [translate_instr_to_term_def, Once step_block_cases] >>
|
|
|
simp [step_term_cases, PULL_EXISTS] >>
|
|
|
fs [llvmTheory.step_cases] >>
|
|
|
drule get_instr_live >> disch_tac >>
|
|
|
drule translate_arg_correct >>
|
|
|
fs [step_instr_cases] >> fs [] >>
|
|
|
TRY (fs [get_instr_cases] >> NO_TAC) >>
|
|
|
`a = a'` by fs [get_instr_cases] >>
|
|
|
disch_then drule >>
|
|
|
impl_tac
|
|
|
>- (
|
|
|
fs [SUBSET_DEF, IN_DEF] >> rfs [uses_cases, get_instr_cases, instr_uses_def] >>
|
|
|
fs [IN_DEF]) >>
|
|
|
disch_tac >> fs [] >>
|
|
|
fs [v_rel_cases, GSYM PULL_EXISTS] >>
|
|
|
qexists_tac `emap` >> qexists_tac `w2i tf` >> simp [] >> conj_tac
|
|
|
>- metis_tac [] >>
|
|
|
Cases_on `s1'.bp` >> fs [dest_llair_lab_def] >>
|
|
|
rename1 `el _ _ = Br e lab1 lab2` >>
|
|
|
qexists_tac `dest_fn s1.ip.f` >>
|
|
|
qexists_tac `if 0 = w2i tf then dest_label lab2 else dest_label lab1` >> simp [] >>
|
|
|
qpat_abbrev_tac `target = if tf = 0w then l2 else l1` >>
|
|
|
qpat_abbrev_tac `target' = if 0 = w2i tf then dest_label lab2 else dest_label lab1` >>
|
|
|
rw [] >>
|
|
|
`translate_label (dest_fn s1.ip.f) target = Lab_name (dest_fn s1.ip.f) target' `
|
|
|
by (
|
|
|
fs [get_instr_cases] >> rw [] >>
|
|
|
unabbrev_all_tac >> rw [] >> fs [word_0_w2i] >>
|
|
|
Cases_on `l2` >> Cases_on `l1` >> rw [translate_label_def, dest_label_def] >>
|
|
|
`0 = w2i (0w:word1)` by rw [word_0_w2i] >>
|
|
|
fs [w2i_11]) >>
|
|
|
rw [state_rel_def]
|
|
|
>- (Cases_on `lab2` >> rw [Abbr `target'`, translate_label_def, dest_label_def])
|
|
|
>- (Cases_on `lab1` >> rw [Abbr `target'`, translate_label_def, dest_label_def])
|
|
|
>- (
|
|
|
rw [pc_rel_cases] >> cheat)
|
|
|
>- (
|
|
|
fs [mem_state_rel_def] >> rw []
|
|
|
>- (
|
|
|
qpat_x_assum `!r. r ∈ live _ _ ⇒ P r` mp_tac >>
|
|
|
simp [Once live_gen_kill] >> disch_then (qspec_then `r` mp_tac) >>
|
|
|
impl_tac >> rw []
|
|
|
>- (
|
|
|
rw [PULL_EXISTS] >>
|
|
|
disj1_tac >>
|
|
|
qexists_tac `<|f := s1.ip.f; b := Some target; i := Phi_ip s1.ip.b|>` >>
|
|
|
rw [next_ips_cases, IN_DEF, assigns_cases]
|
|
|
>- (
|
|
|
disj1_tac >>
|
|
|
qexists_tac `Br a l1 l2` >>
|
|
|
rw [instr_next_ips_def, Abbr `target`] >>
|
|
|
cheat) >>
|
|
|
CCONTR_TAC >> fs [] >>
|
|
|
imp_res_tac get_instr_func >> fs [] >> rw [] >>
|
|
|
fs [instr_assigns_def])
|
|
|
>- (
|
|
|
rpt HINT_EXISTS_TAC >> rw [] >>
|
|
|
qmatch_goalsub_abbrev_tac `eval_exp s3 _` >>
|
|
|
`s1'.locals = s3.locals` by fs [Abbr `s3`] >>
|
|
|
metis_tac [eval_exp_ignores]))
|
|
|
>- cheat
|
|
|
>- cheat)
|
|
|
>- (
|
|
|
cheat))
|
|
|
>- ( (* Invoke *)
|
|
|
cheat)
|
|
|
>- ( (* Unreachable *)
|
|
|
cheat)
|
|
|
>- ( (* Exit *)
|
|
|
cheat)
|
|
|
>- ( (* Throw *)
|
|
|
cheat))
|
|
|
>- ( (* Call *)
|
|
|
cheat)
|
|
|
>- ( (* Stuck *)
|
|
|
cheat))
|
|
|
>- ( (* Middle of the block *)
|
|
|
fs [llvmTheory.step_cases] >> TRY (fs [get_instr_cases] >> NO_TAC) >>
|
|
|
`i' = i` by metis_tac [get_instr_func, sumTheory.INL_11] >> fs [] >>
|
|
|
rename [`step_instr _ _ _ _ s2`, `state_rel _ _ s3 _`,
|
|
|
`mem_state_rel _ _ s1 s1'`] >>
|
|
|
Cases_on `∃r t. classify_instr i = Exp r t` >> fs []
|
|
|
>- ( (* instructions that compile to expressions *)
|
|
|
drule translate_instr_to_exp_correct >>
|
|
|
ntac 5 (disch_then drule) >>
|
|
|
disch_then (qspec_then `regs_to_keep` mp_tac) >>
|
|
|
rw [] >> fs [translate_trace_def] >>
|
|
|
`reachable prog (inc_pc s1.ip)`
|
|
|
by metis_tac [prog_ok_nonterm, next_ips_reachable, mem_state_rel_def] >>
|
|
|
first_x_assum drule >>
|
|
|
simp [inc_pc_def, inc_bip_def] >>
|
|
|
disch_then (qspecl_then [`regs_to_keep`, `types`] mp_tac) >> rw [] >>
|
|
|
rename1 `state_rel prog emap3 s3 s3'` >>
|
|
|
qexists_tac `emap3` >> qexists_tac `s3'` >> rw [] >>
|
|
|
`take_to_call (drop idx b.body) = i :: take_to_call (drop (idx + 1) b.body)`
|
|
|
by (
|
|
|
irule take_to_call_lem >> simp [] >>
|
|
|
fs [get_instr_cases]) >>
|
|
|
simp [translate_instrs_def] >>
|
|
|
Cases_on `r ∉ regs_to_keep` >> fs [] >> rw []
|
|
|
>- metis_tac [] >>
|
|
|
qexists_tac `Tau::tr'` >> rw [] >>
|
|
|
simp [Once step_block_cases] >> disj2_tac >>
|
|
|
pairarg_tac >> rw [] >> fs [] >>
|
|
|
metis_tac [])
|
|
|
>- ( (* Non-expression instructions *)
|
|
|
cheat))
|
|
|
QED
|
|
|
|
|
|
Theorem multi_step_to_step_block:
|
|
|
∀prog s1 tr s2 s1'.
|
|
|
prog_ok prog ∧ is_ssa prog ∧
|
|
|
multi_step prog s1 tr s2 ∧
|
|
|
state_rel prog emap s1 s1'
|
|
|
⇒
|
|
|
∃s2' emap2 b tr'.
|
|
|
get_block (translate_prog prog) s1'.bp b ∧
|
|
|
step_block (translate_prog prog) s1' b.cmnd tr' b.term s2' ∧
|
|
|
filter ($≠ Tau) tr' = filter ($≠ Tau) (map (translate_trace types) tr) ∧
|
|
|
state_rel prog emap2 s2 s2'
|
|
|
Proof
|
|
|
rw [] >> pop_assum mp_tac >> simp [Once state_rel_def] >> rw [pc_rel_cases]
|
|
|
>- (
|
|
|
(* Non-phi instruction *)
|
|
|
drule translate_instrs_correct1 >> simp [] >>
|
|
|
disch_then drule >>
|
|
|
disch_then (qspecl_then [`regs_to_keep`, `types`] mp_tac) >> simp [] >>
|
|
|
rw [] >>
|
|
|
qexists_tac `s2'` >> simp [] >>
|
|
|
ntac 3 HINT_EXISTS_TAC >>
|
|
|
rw [] >> fs [dest_fn_def]) >>
|
|
|
(* Phi instruction *)
|
|
|
reverse (fs [Once multi_step_cases])
|
|
|
>- metis_tac [get_instr_func, sumTheory.sum_distinct] >>
|
|
|
qpat_x_assum `last_step _ _ _ _` mp_tac >>
|
|
|
simp [last_step_def] >> simp [Once llvmTheory.step_cases] >>
|
|
|
rw [] >> imp_res_tac get_instr_func >> fs [] >> rw [] >>
|
|
|
fs [translate_trace_def] >>
|
|
|
(* TODO: unfinished *)
|
|
|
cheat
|
|
|
QED
|
|
|
|
|
|
Theorem step_block_to_multi_step:
|
|
|
∀prog s1 s1' tr s2' b.
|
|
|
state_rel prog emap s1 s1' ∧
|
|
|
get_block (translate_prog prog) s1'.bp b ∧
|
|
|
step_block (translate_prog prog) s1' b.cmnd tr b.term s2'
|
|
|
⇒
|
|
|
∃s2.
|
|
|
multi_step prog s1 (map untranslate_trace tr) s2 ∧
|
|
|
state_rel prog emap s2 s2'
|
|
|
Proof
|
|
|
cheat
|
|
|
QED
|
|
|
|
|
|
Theorem trans_trace_not_tau:
|
|
|
∀types. ($≠ Tau) ∘ translate_trace types = ($≠ Tau)
|
|
|
Proof
|
|
|
rw [FUN_EQ_THM] >> eq_tac >> rw [translate_trace_def] >>
|
|
|
TRY (Cases_on `y`) >> fs [translate_trace_def]
|
|
|
QED
|
|
|
|
|
|
Theorem untrans_trace_not_tau:
|
|
|
∀types. ($≠ Tau) ∘ untranslate_trace = ($≠ Tau)
|
|
|
Proof
|
|
|
rw [FUN_EQ_THM] >> eq_tac >> rw [untranslate_trace_def] >>
|
|
|
TRY (Cases_on `y`) >> fs [untranslate_trace_def]
|
|
|
QED
|
|
|
|
|
|
Theorem translate_prog_correct_lem1:
|
|
|
∀path.
|
|
|
okpath (multi_step prog) path ∧ finite path
|
|
|
⇒
|
|
|
∀emap s1'.
|
|
|
prog_ok prog ∧
|
|
|
is_ssa prog ∧
|
|
|
state_rel prog emap (first path) s1'
|
|
|
⇒
|
|
|
∃path' emap.
|
|
|
finite path' ∧
|
|
|
okpath (step (translate_prog prog)) path' ∧
|
|
|
first path' = s1' ∧
|
|
|
LMAP (filter ($≠ Tau)) (labels path') =
|
|
|
LMAP (map (translate_trace types) o filter ($≠ Tau)) (labels path) ∧
|
|
|
state_rel prog emap (last path) (last path')
|
|
|
Proof
|
|
|
ho_match_mp_tac finite_okpath_ind >> rw []
|
|
|
>- (qexists_tac `stopped_at s1'` >> rw [] >> metis_tac []) >>
|
|
|
fs [] >>
|
|
|
drule multi_step_to_step_block >> ntac 3 (disch_then drule) >>
|
|
|
disch_then (qspec_then `types` mp_tac) >> rw [] >>
|
|
|
first_x_assum drule >> rw [] >>
|
|
|
qexists_tac `pcons s1' tr' path'` >> rw [] >>
|
|
|
rw [FILTER_MAP, combinTheory.o_DEF, trans_trace_not_tau] >>
|
|
|
HINT_EXISTS_TAC >> simp [] >>
|
|
|
simp [step_cases] >> qexists_tac `b` >> simp [] >>
|
|
|
fs [state_rel_def, mem_state_rel_def] >> simp [get_observation_def] >>
|
|
|
fs [Once multi_step_cases, last_step_def] >> rw [] >>
|
|
|
metis_tac [get_instr_func, exit_no_step]
|
|
|
QED
|
|
|
|
|
|
Theorem translate_prog_correct_lem2:
|
|
|
∀path'.
|
|
|
okpath (step (translate_prog prog)) path' ∧ finite path'
|
|
|
⇒
|
|
|
∀s1.
|
|
|
prog_ok prog ∧
|
|
|
state_rel prog emap s1 (first path')
|
|
|
⇒
|
|
|
∃path.
|
|
|
finite path ∧
|
|
|
okpath (multi_step prog) path ∧
|
|
|
first path = s1 ∧
|
|
|
labels path = LMAP (map untranslate_trace) (labels path') ∧
|
|
|
state_rel prog emap (last path) (last path')
|
|
|
Proof
|
|
|
ho_match_mp_tac finite_okpath_ind >> rw []
|
|
|
>- (qexists_tac `stopped_at s1` >> rw []) >>
|
|
|
fs [step_cases] >>
|
|
|
drule step_block_to_multi_step >> ntac 2 (disch_then drule) >> rw [] >>
|
|
|
first_x_assum drule >> rw [] >>
|
|
|
qexists_tac `pcons s1 (map untranslate_trace r) path` >> rw []
|
|
|
QED
|
|
|
|
|
|
Theorem translate_global_var_11:
|
|
|
∀path.
|
|
|
okpath (step (translate_prog prog)) path ∧ finite path
|
|
|
⇒
|
|
|
∀x t1 bytes t2 l.
|
|
|
labels path = fromList l ∧
|
|
|
MEM (W (Var_name x t1) bytes) (flat l) ∧
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MEM (W (Var_name x t2) bytes) (flat l)
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⇒
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t1 = t2
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Proof
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cheat
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QED
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Theorem translate_prog_correct:
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∀prog s1 s1'.
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prog_ok prog ∧ is_ssa prog ∧
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state_rel prog emap s1 s1'
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⇒
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multi_step_sem prog s1 = image (I ## map untranslate_trace) (sem (translate_prog prog) s1')
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Proof
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rw [sem_def, multi_step_sem_def, EXTENSION] >> eq_tac >> rw []
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>- (
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drule translate_prog_correct_lem1 >> ntac 4 (disch_then drule) >>
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disch_then (qspec_then `types` mp_tac) >> rw [pairTheory.EXISTS_PROD] >>
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PairCases_on `x` >> rw [] >>
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qexists_tac `map (translate_trace types) x1` >> rw []
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>- rw [MAP_MAP_o, combinTheory.o_DEF, un_translate_trace_inv] >>
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qexists_tac `path'` >> rw [] >>
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fs [IN_DEF, observation_prefixes_cases, toList_some] >> rw [] >>
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`?labs. labels path' = fromList labs`
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by (
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fs [GSYM finite_labels] >>
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imp_res_tac llistTheory.LFINITE_toList >>
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fs [toList_some]) >>
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fs [] >>
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rfs [lmap_fromList, combinTheory.o_DEF, MAP_MAP_o] >>
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simp [FILTER_FLAT, MAP_FLAT, MAP_MAP_o, combinTheory.o_DEF, FILTER_MAP]
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>- fs [state_rel_def, mem_state_rel_def]
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>- fs [state_rel_def, mem_state_rel_def] >>
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rename [`labels path' = fromList l'`, `labels path = fromList l`,
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|
`state_rel _ _ (last path) (last path')`, `lsub ≼ flat l`] >>
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|
cheat)
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|
(*
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`INJ (translate_trace types) (set l2' ∪ set (flat l2)) UNIV`
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by (
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simp [INJ_DEF] >> rpt gen_tac >>
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Cases_on `x` >> Cases_on `y` >> simp [translate_trace_def] >>
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Cases_on `a` >> Cases_on `a'` >> simp [translate_glob_var_def]) >>
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fs [INJ_MAP_EQ_IFF, inj_map_prefix_iff] >> rw [] >>
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|
fs [state_rel_def, mem_state_rel_def])
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*)
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|
>- (
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|
|
fs [toList_some] >>
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|
|
drule translate_prog_correct_lem2 >> simp [] >>
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|
|
disch_then drule >> rw [] >>
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|
qexists_tac `path'` >> rw [] >>
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|
|
fs [IN_DEF, observation_prefixes_cases, toList_some] >> rw [] >>
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|
|
rfs [lmap_fromList] >>
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|
|
simp [GSYM MAP_FLAT, FILTER_MAP, untrans_trace_not_tau]
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|
|
>- fs [state_rel_def, mem_state_rel_def]
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|
|
>- fs [state_rel_def, mem_state_rel_def] >>
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|
|
qexists_tac `map untranslate_trace l2'` >>
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|
|
simp [GSYM MAP_FLAT, FILTER_MAP, untrans_trace_not_tau] >>
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|
|
`INJ untranslate_trace (set l2' ∪ set (flat l2)) UNIV`
|
|
|
by (
|
|
|
drule is_prefix_subset >> rw [SUBSET_DEF] >>
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|
|
`set l2' ∪ set (flat l2) = set (flat l2)` by (rw [EXTENSION] >> metis_tac []) >>
|
|
|
simp [] >>
|
|
|
simp [INJ_DEF] >> rpt gen_tac >>
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|
|
Cases_on `x` >> Cases_on `y` >> simp [untranslate_trace_def] >>
|
|
|
Cases_on `a` >> Cases_on `a'` >> simp [untranslate_glob_var_def] >>
|
|
|
metis_tac [translate_global_var_11]) >>
|
|
|
fs [INJ_MAP_EQ_IFF, inj_map_prefix_iff] >> rw [] >>
|
|
|
fs [state_rel_def, mem_state_rel_def])
|
|
|
QED
|
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|
export_theory ();
|
