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optPlannerProofsScript.sml
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optPlannerProofsScript.sml
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(**
Correctness proof for optimization planner
**)
open semanticPrimitivesTheory evaluateTheory
icing_rewriterTheory icing_optimisationsTheory
icing_optimisationProofsTheory fpOptTheory fpValTreeTheory
optPlannerTheory source_to_source2Theory source_to_source2ProofsTheory
floatToRealProofsTheory icing_realIdProofsTheory;
open preamble;
val _ = new_theory "optPlannerProofs";
fun rnCases_on t = rename1 t >> Cases_on t
(** The phase repeater can only repeat rewrites **)
Theorem phase_repeater_mem:
∀ n path rws r cfg e e2.
MEM (Apply (path, rws)) r ∧
phase_repeater n apply_distributivity cfg e = (e2, r) ⇒
∃ ei. MEM (Apply (path, rws)) (SND (apply_distributivity cfg ei))
Proof
Induct_on ‘n’ >> gs[phase_repeater_def] >> rpt strip_tac
>- (qexists_tac ‘e’ >> gs[])
>> Cases_on ‘apply_distributivity cfg e’ >> gs[]
>> Cases_on ‘q ≠ e’ >> gs[]
>> rnCases_on ‘phase_repeater n apply_distributivity cfg e3’ >> gs[]
>> rveq >> gs[MEM_APPEND]
>- (qexists_tac ‘e’ >> gs[])
>> first_x_assum irule >> asm_exists_tac >> gs[]
QED
(** Side-lemmas to ease proofs later **)
fun path_tac index_def =
Induct_on ‘r’ >> simp[MAP_plan_to_path_def]
>> rpt strip_tac
>- (
Cases_on ‘h’ >> gs[]
>> Cases_on ‘p’ >> gs[index_def])
>> last_x_assum mp_tac >> impl_tac >> gs[MAP_plan_to_path_def]
>> strip_tac >> asm_exists_tac >> gs[];
Theorem MEM_MAP_plan_to_path_index:
MEM (Apply (path, rws)) (MAP_plan_to_path (listIndex n) r) ⇒
∃ spath. path = ListIndex (n, spath) ∧
MEM (Apply (spath, rws)) r
Proof
path_tac listIndex_def
QED
Theorem MEM_MAP_plan_to_path_right:
MEM (Apply (path, rws)) (MAP_plan_to_path right r) ⇒
∃ spath. path = Right spath ∧
MEM (Apply (spath, rws)) r
Proof
path_tac right_def
QED
Theorem MEM_MAP_plan_to_path_center:
MEM (Apply (path, rws)) (MAP_plan_to_path center r) ⇒
∃ spath. path = Center spath ∧
MEM (Apply (spath, rws)) r
Proof
path_tac center_def
QED
Theorem MEM_MAP_plan_to_path_left:
MEM (Apply (path, rws)) (MAP_plan_to_path left r) ⇒
∃ spath. path = Left spath ∧
MEM (Apply (spath, rws)) r
Proof
path_tac left_def
QED
Theorem MAP_plan_to_path_SUC_prod:
(λ i (_, plani). MAP_plan_to_path (listIndex (i+n)) plani) o SUC =
λ i (_, plani). MAP_plan_to_path (listIndex (i + SUC n)) plani
Proof
gs[FUN_EQ_THM] >> rpt strip_tac >> gs[SUC_ONE_ADD]
QED
Theorem MAP_plan_to_path_SUC_trip:
(λ i (_, _, plani). MAP_plan_to_path (listIndex (i+n)) plani) o SUC =
λ i (_, _, plani). MAP_plan_to_path (listIndex (i + SUC n)) plani
Proof
gs[FUN_EQ_THM] >> rpt strip_tac >> gs[SUC_ONE_ADD]
QED
Theorem MAPi_plan_to_path_prod:
MAPi ((λ i (_, plani). MAP_plan_to_path (listIndex (i + n)) plani) o SUC) =
MAPi (λ i (_, plani). MAP_plan_to_path (listIndex (i+ SUC n)) plani)
Proof
gs[MAP_plan_to_path_SUC_prod]
QED
Theorem MAPi_plan_to_path_trip:
MAPi ((λ i (_, _, plani). MAP_plan_to_path (listIndex (i + n)) plani) o SUC) =
MAPi (λ i (_, _, plani). MAP_plan_to_path (listIndex (i+ SUC n)) plani)
Proof
gs[MAP_plan_to_path_SUC_trip]
QED
Theorem MEM_list_flat_sub_exp_general:
∀ l n f path rws cfg.
MEM (Apply (path, rws))
(FLAT (MAPi (λi (_, plani).
MAP_plan_to_path (listIndex (i+n)) plani)
(MAP (λ a. f a) l))) ⇒
∃ e path. exp_size e < exp6_size l ∧
MEM (Apply (path, rws)) (SND (f e)) ∧
MEM e l
Proof
Induct_on ‘l’ >> gs[]
>> rpt strip_tac >> gs[]
>- (
qexists_tac ‘h’ >> gs[astTheory.exp_size_def]
>> Cases_on ‘f h’ >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_index >> asm_exists_tac >> gs[])
>> gs[MAPi_plan_to_path_prod]
>> res_tac >> qexists_tac ‘e’ >> gs[astTheory.exp_size_def]
>> asm_exists_tac >> gs[]
QED
Theorem MEM_list_flat_sub_exp =
Q.SPECL [‘l’, ‘0’, ‘f’] MEM_list_flat_sub_exp_general
|> GEN “l:exp list”
|> GEN “f:exp -> 'a # opt_step list”
|> REWRITE_RULE [ADD_CLAUSES];
Theorem MEM_list_flat_sub_patexp_general:
∀ l n f path rws cfg.
MEM (Apply (path, rws))
(FLAT (MAPi (λi (_, _, plani).
MAP_plan_to_path (listIndex (i+n)) plani)
(MAP (λ (p,e). (p, f e)) l))) ⇒
∃ e p path. exp_size e < exp3_size l ∧
MEM (Apply (path, rws)) (SND (f e)) ∧
MEM (p, e) l
Proof
Induct_on ‘l’ >> gs[]
>> rpt strip_tac >> gs[]
>- (
Cases_on ‘h’ >> gs[]
>> qexists_tac ‘r’ >> qexists_tac ‘q’ >> gs[astTheory.exp_size_def]
>> Cases_on ‘f r’ >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_index >> asm_exists_tac >> gs[])
>> gs[MAPi_plan_to_path_trip]
>> res_tac >> qexists_tac ‘e’ >> gs[astTheory.exp_size_def]
>> asm_exists_tac >> qexists_tac ‘p’ >> gs[]
QED
Theorem MEM_list_flat_sub_patexp =
Q.SPECL [‘l’, ‘0’] MEM_list_flat_sub_patexp_general
|> GEN “l:(pat # exp) list”
|> GEN “f:exp -> 'a # opt_step list”
|> REWRITE_RULE [ADD_CLAUSES];
(** Now prove that generate_plan_exp can only produce rewrites from its
components **)
Theorem generate_plan_upper_bound_rws:
MEM (Apply (path,rws)) (generate_plan_exp cfg e) ⇒
(∃ e. MEM (Apply (path,rws)) (SND (canonicalize cfg e))) ∨
(∃ e. MEM (Apply (path,rws)) (SND (apply_distributivity cfg e))) ∨
(∃ e. MEM (Apply (path,rws)) (SND (peephole_optimise cfg e))) ∨
(∃ e. MEM (Apply (path,rws)) (SND (balance_expression_tree cfg e)))
Proof
gs[generate_plan_exp_def, compose_plan_generation_def]
>> rnCases_on ‘canonicalize cfg e’ >> gs[]
>> rnCases_on ‘phase_repeater 100 apply_distributivity cfg e2’ >> gs[]
>> rnCases_on ‘canonicalize cfg e3’ >> gs[]
>> rnCases_on ‘canonicalize cfg e4’ >> gs[]
>> rnCases_on ‘peephole_optimise cfg e5’ >> gs[]
>> rnCases_on ‘balance_expression_tree cfg e6’ >> gs[]
>> rename1 ‘balance_expression_tree cfg e6 = (e7, r7)’
>> rpt (TOP_CASE_TAC >> gs[]) >> rveq >> strip_tac
>> imp_res_tac phase_repeater_mem
(* canonicalize cases *)
>> TRY (DISJ1_TAC >> qexists_tac ‘e’ >> gs[] >> NO_TAC)
>> TRY (DISJ1_TAC >> qexists_tac ‘e2’ >> gs[] >> NO_TAC)
>> TRY (DISJ1_TAC >> qexists_tac ‘e3’ >> gs[] >> NO_TAC)
>> TRY (DISJ1_TAC >> qexists_tac ‘e4’ >> gs[] >> NO_TAC)
(* phase repeater cases *)
>> TRY (DISJ2_TAC >> DISJ1_TAC >> qexists_tac ‘ei’ >> gs[] >> NO_TAC)
(* peephole optimise cases *)
>> TRY (ntac 2 DISJ2_TAC >> DISJ1_TAC >> qexists_tac ‘e’ >> gs[] >> NO_TAC)
>> TRY (ntac 2 DISJ2_TAC >> DISJ1_TAC >> qexists_tac ‘e2’ >> gs[] >> NO_TAC)
>> TRY (ntac 2 DISJ2_TAC >> DISJ1_TAC >> qexists_tac ‘e3’ >> gs[] >> NO_TAC)
>> TRY (ntac 2 DISJ2_TAC >> DISJ1_TAC >> qexists_tac ‘e4’ >> gs[] >> NO_TAC)
>> TRY (ntac 2 DISJ2_TAC >> DISJ1_TAC >> qexists_tac ‘e5’ >> gs[] >> NO_TAC)
(* balance expression tree cases *)
>> TRY (ntac 3 DISJ2_TAC >> qexists_tac ‘e’ >> gs[] >> NO_TAC)
>> TRY (ntac 3 DISJ2_TAC >> qexists_tac ‘e2’ >> gs[] >> NO_TAC)
>> TRY (ntac 3 DISJ2_TAC >> qexists_tac ‘e3’ >> gs[] >> NO_TAC)
>> TRY (ntac 3 DISJ2_TAC >> qexists_tac ‘e4’ >> gs[] >> NO_TAC)
>> TRY (ntac 3 DISJ2_TAC >> qexists_tac ‘e5’ >> gs[] >> NO_TAC)
>> TRY (ntac 3 DISJ2_TAC >> qexists_tac ‘e6’ >> gs[] >> NO_TAC)
QED
(** Bound for canonicalize **)
Theorem canonicalize_app_upper_bound:
∀ e op eNew r path rws rw r.
canonicalize_app e = (eNew, r) ∧
MEM (Apply (path, rws)) r ∧
MEM rw rws ⇒
MEM rw [fp_neg_times_minus_one; fp_sub_add;
fp_comm_gen FP_Add; fp_comm_gen FP_Mul;
fp_assoc_gen FP_Add; fp_assoc_gen FP_Mul]
Proof
measureInduct_on ‘exp_size e’
>> simp[Once canonicalize_app_def, CaseEq"op", CaseEq"list"]
>> rpt strip_tac >> gs[CaseEq "exp", CaseEq "fp_bop", CaseEq"list", CaseEq"op"]
>> rveq >> gs[canonicalize_sub_def]
>> qpat_x_assum ‘_ = (_, _)’ mp_tac
>> COND_CASES_TAC >> gs[] >> rpt strip_tac >> rveq >> gs[]
>- (
Cases_on ‘canonicalize_app (App (FP_bop FP_Add) [v315; v153])’
>> gs[] >> rveq >> gs[MEM]
>> imp_res_tac MEM_MAP_plan_to_path_index
>> first_x_assum $ qspec_then ‘App (FP_bop FP_Add) [v315; v153]’ mp_tac
>> gs[astTheory.exp_size_def]
>> rpt $ disch_then drule >> gs[])
>> Cases_on ‘canonicalize_app (App (FP_bop FP_Mul) [v453; v153])’
>> gs[] >> rveq >> gs[MEM]
>> imp_res_tac MEM_MAP_plan_to_path_index
>> first_x_assum $ qspec_then ‘App (FP_bop FP_Mul) [v453; v153]’ mp_tac
>> gs[astTheory.exp_size_def]
>> rpt $ disch_then drule >> gs[]
QED
fun trivial_case_tac t =
first_x_assum $ qspec_then t mp_tac >> gs[]
>> disch_then $ qspecl_then [‘spath’, ‘rws’, ‘cfg’] mp_tac >> gs[];
(** Prove upper bound on rewrites for canonicalize **)
Theorem canonicalize_upper_bound:
∀ e path rws cfg.
MEM (Apply (path, rws)) (SND (canonicalize cfg e)) ⇒
∀ rw. MEM rw rws ⇒
MEM rw [fp_neg_times_minus_one; fp_sub_add;
fp_comm_gen FP_Add; fp_comm_gen FP_Mul;
fp_assoc_gen FP_Add; fp_assoc_gen FP_Mul]
Proof
measureInduct_on ‘exp_size e’
>> Cases_on ‘e’ >> gs[canonicalize_def] >> rpt strip_tac
>> gs[astTheory.exp_size_def]
>- (
imp_res_tac MEM_list_flat_sub_exp
>> first_x_assum $ qspec_then ‘e’ mp_tac >> gs[]
>> disch_then (fn ith => first_x_assum (fn th => mp_then Any drule ith th))
>> gs[])
>- (
Cases_on ‘cfg.canOpt’ >> gs[]
>- (
Cases_on ‘canonicalize_app (App o' (MAP FST (MAP (λ a. canonicalize cfg a) l)))’ >> gs[]
>- (
imp_res_tac MEM_list_flat_sub_exp
>> first_x_assum $ qspec_then ‘e’ mp_tac >> gs[]
>> disch_then (fn ith => first_x_assum (fn th => mp_then Any drule ith th))
>> gs[])
>> imp_res_tac canonicalize_app_upper_bound >> gs[])
>> imp_res_tac MEM_list_flat_sub_exp
>> first_x_assum $ qspec_then ‘e’ mp_tac >> gs[]
>> disch_then (fn ith => first_x_assum (fn th => mp_then Any drule ith th))
>> gs[])
>- (
Cases_on ‘canonicalize cfg e0’ >> Cases_on ‘canonicalize cfg e'’
>> gs[MEM_APPEND]
>> imp_res_tac MEM_MAP_plan_to_path_left
>> imp_res_tac MEM_MAP_plan_to_path_right
>- trivial_case_tac ‘e'’
>> trivial_case_tac ‘e0’)
>- (
Cases_on ‘canonicalize cfg e1’
>> Cases_on ‘canonicalize cfg e0’
>> Cases_on ‘canonicalize cfg e'’
>> gs[MEM_APPEND]
>> imp_res_tac MEM_MAP_plan_to_path_left
>> imp_res_tac MEM_MAP_plan_to_path_center
>> imp_res_tac MEM_MAP_plan_to_path_right
>- trivial_case_tac ‘e'’
>- trivial_case_tac ‘e0’
>> trivial_case_tac ‘e1’)
>- (
Cases_on ‘canonicalize cfg e'’ >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_left
>- trivial_case_tac ‘e'’
>> imp_res_tac MEM_list_flat_sub_patexp
>> first_x_assum $ qspec_then ‘e’ mp_tac >> gs[]
>> disch_then (fn ith => first_x_assum (fn th => mp_then Any drule ith th))
>> gs[])
>- (
Cases_on ‘canonicalize cfg e0’ >> Cases_on ‘canonicalize cfg e'’
>> gs[MEM_APPEND]
>> imp_res_tac MEM_MAP_plan_to_path_left
>> imp_res_tac MEM_MAP_plan_to_path_right
>- trivial_case_tac ‘e'’
>> trivial_case_tac ‘e0’)
>- (
Cases_on ‘canonicalize cfg e'’ >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_center
>> trivial_case_tac ‘e'’)
>- (
Cases_on ‘canonicalize cfg e'’ >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_center
>> trivial_case_tac ‘e'’)
>- (
Cases_on ‘canonicalize cfg e'’ >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_center
>> trivial_case_tac ‘e'’)
>> Cases_on ‘canonicalize (cfg with canOpt := (f = Opt)) e'’ >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_center
>> first_x_assum $ qspec_then ‘e'’ mp_tac >> gs[]
>> disch_then $ qspecl_then [‘spath’, ‘rws’, ‘cfg with canOpt := (f = Opt)’] mp_tac
>> gs[]
QED
(** Intermediate theorem for postorder-traversal of expressions **)
Theorem postorder_upper_bound:
∀ f cfg e P.
(∀ e cfg eOpt plan.
f cfg e = SOME (eOpt, plan) ⇒
∀ path rws.
MEM (Apply (path, rws)) plan ⇒
P rws) ⇒
∀ eOpt plan path rws.
(post_order_dfs_for_plan f cfg e) = (eOpt, plan) ⇒
(MEM (Apply (path, rws)) plan ⇒
P rws)
Proof
ho_match_mp_tac post_order_dfs_for_plan_ind >> rw[]
>> gs[post_order_dfs_for_plan_def]
>- (
Cases_on ‘post_order_dfs_for_plan f (cfg with canOpt := (sc = Opt)) e’
>> gs[] >> rveq
>> imp_res_tac MEM_MAP_plan_to_path_center
>> first_x_assum drule >> disch_then drule >> gs[])
>- (
qpat_x_assum `_ = (_,_)` mp_tac
>> COND_CASES_TAC >> gs[]
>- (
TOP_CASE_TAC >> gs[]
>- (
rpt strip_tac >> rveq
>> imp_res_tac MEM_list_flat_sub_exp
>> res_tac
>> Cases_on ‘post_order_dfs_for_plan f cfg e’ >> gs[]
>> first_x_assum drule >> gs[])
>> TOP_CASE_TAC >> gs[]
>> rpt strip_tac >> rveq >> gs[]
>> imp_res_tac MEM_list_flat_sub_exp
>> res_tac
>> Cases_on ‘post_order_dfs_for_plan f cfg e’ >> gs[]
>> first_x_assum drule >> gs[])
>> rpt strip_tac >> rveq
>> imp_res_tac MEM_list_flat_sub_exp
>> res_tac
>> Cases_on ‘post_order_dfs_for_plan f cfg e’ >> gs[]
>> first_x_assum drule >> gs[])
>- (
Cases_on ‘post_order_dfs_for_plan f cfg e'’
>> Cases_on ‘post_order_dfs_for_plan f cfg e’
>> gs[] >> rveq >> gs[MEM_APPEND]
>> imp_res_tac MEM_MAP_plan_to_path_left
>> imp_res_tac MEM_MAP_plan_to_path_right
>> res_tac)
>- (
rveq
>> imp_res_tac MEM_list_flat_sub_exp
>> res_tac
>> Cases_on ‘post_order_dfs_for_plan f cfg e’ >> gs[]
>> first_x_assum drule >> gs[])
>- (
Cases_on ‘post_order_dfs_for_plan f cfg e'’
>> Cases_on ‘post_order_dfs_for_plan f cfg e’
>> gs[] >> rveq >> gs[MEM_APPEND]
>> imp_res_tac MEM_MAP_plan_to_path_left
>> imp_res_tac MEM_MAP_plan_to_path_right
>> res_tac)
>- (
Cases_on ‘post_order_dfs_for_plan f cfg e''’
>> Cases_on ‘post_order_dfs_for_plan f cfg e'’
>> Cases_on ‘post_order_dfs_for_plan f cfg e’
>> ntac 5 $ pop_assum mp_tac
>> disch_then (fn th => mp_tac (SIMP_RULE std_ss [] th))
>> rpt strip_tac >> rveq
>> qpat_x_assum `MEM _ _` mp_tac
>> rewrite_tac[MEM_APPEND] >> strip_tac
>> imp_res_tac MEM_MAP_plan_to_path_left
>> imp_res_tac MEM_MAP_plan_to_path_center
>> imp_res_tac MEM_MAP_plan_to_path_right
>- (
res_tac >> first_x_assum drule >> rveq
>> metis_tac[])
>- (
res_tac >> first_x_assum drule >> rveq
>> metis_tac[])
>> last_x_assum $ qspecl_then [‘q''’, ‘r''’, ‘q'’, ‘r'’] mp_tac
>> impl_tac >- rewrite_tac[]
>> disch_then $ qspec_then ‘P’ mp_tac
>> impl_tac >- first_x_assum MATCH_ACCEPT_TAC
>> disch_then $ qspecl_then [‘q’, ‘r’, ‘spath’, ‘rws’] mp_tac
>> impl_tac >- asm_rewrite_tac[]
>> metis_tac[])
>- (
Cases_on ‘post_order_dfs_for_plan f cfg e’
>> gs[] >> rveq
>> imp_res_tac MEM_MAP_plan_to_path_center
>> first_x_assum drule >> disch_then drule >> gs[])
>- (
Cases_on ‘post_order_dfs_for_plan f cfg e’
>> gs[] >> rveq
>> imp_res_tac MEM_MAP_plan_to_path_center
>> first_x_assum drule >> disch_then drule >> gs[])
>- (
Cases_on ‘post_order_dfs_for_plan f cfg e’
>> gs[] >> rveq
>> imp_res_tac MEM_MAP_plan_to_path_center
>> first_x_assum drule >> disch_then drule >> gs[])
>- (
Cases_on ‘post_order_dfs_for_plan f cfg e’
>> gs[] >> rveq >> gs[MEM_APPEND]
>- (
imp_res_tac MEM_MAP_plan_to_path_left
>> first_x_assum drule >> disch_then drule >> gs[])
>> imp_res_tac MEM_list_flat_sub_patexp
>> res_tac
>> Cases_on ‘post_order_dfs_for_plan f cfg e'’ >> gs[]
>> first_x_assum drule >> gs[])
QED
Theorem plan_app_rewrite_with_each_alt:
∀ xs e eOpt plan path rws rw.
(λ (rewritten, plan).
if plan = [] then NONE
else SOME (rewritten, MAP (λ x. Apply x) plan))
(try_rewrite_with_each xs e) = SOME (eOpt, plan) ∧
MEM (Apply (path, rws)) plan ∧
MEM rw rws ⇒
MEM rw xs
Proof
Induct_on ‘xs’ >> simp[Once try_rewrite_with_each_def]
>> rpt strip_tac >> qpat_x_assum `_ = SOME _` mp_tac
>> PairCases_on ‘h’ >> gs[try_rewrite_with_each_def]
>> TOP_CASE_TAC >> gs[]
>- (rpt strip_tac >> res_tac >> metis_tac[])
>> rpt strip_tac >> rveq >> gs[]
>> ‘∃subplan. plan = (Apply (Here, [(h0,h1)]))::subplan’
by (
Cases_on ‘try_rewrite_with_each xs (rewriteFPexp [(h0, h1)] e)’
>> gs[] >> rveq >> gs[])
>> rveq >> gs[]
>> DISJ2_TAC >> first_x_assum irule
>> qexists_tac ‘rewriteFPexp [(h0,h1)] e’ >> qexists_tac ‘eOpt’
>> qexists_tac ‘path’ >> qexists_tac ‘subplan’ >> qexists_tac ‘rws’
>> gs[]
>> Cases_on ‘try_rewrite_with_each xs (rewriteFPexp [(h0, h1)] e)’
>> gs[] >> rveq >> Cases_on ‘r’ >> gs[]
QED
Theorem peephole_optimise_upper_bound:
∀ e path rws cfg.
MEM (Apply (path, rws)) (SND (peephole_optimise cfg e)) ⇒
∀ rw. MEM rw rws ⇒
MEM rw [fp_neg_push_mul_r; fp_times_minus_one_neg; fp_add_sub;
fp_times_two_to_add; fp_times_three_to_add; fp_times_zero;
fp_plus_zero; fp_times_one; fp_same_sub; fp_fma_intro]
Proof
simp[Once peephole_optimise_def]
>> rpt gen_tac
>> qmatch_goalsub_abbrev_tac ‘MEM _ (SND (post_order_dfs_for_plan peephole_app cfg e))’
>> rpt strip_tac
>> qspecl_then [‘peephole_app’, ‘cfg’, ‘e’,
‘λ rws. ∀ rw. MEM rw rws ⇒
rw = fp_neg_push_mul_r ∨ rw = fp_times_minus_one_neg ∨
rw = fp_add_sub ∨ rw = fp_times_two_to_add ∨
rw = fp_times_three_to_add ∨ rw = fp_times_zero ∨
rw = fp_plus_zero ∨ rw = fp_times_one ∨ rw = fp_same_sub ∨
rw = fp_fma_intro’]
mp_tac postorder_upper_bound
>> gs[]
>> impl_tac
>- (
unabbrev_all_tac
>> rpt $ pop_assum kall_tac
>> gs[] >> rpt strip_tac
>> drule plan_app_rewrite_with_each_alt
>> rpt $ disch_then drule >> gs[] >> metis_tac[])
>> Cases_on ‘post_order_dfs_for_plan peephole_app cfg e’ >> gs[]
>> rpt $ disch_then drule >> gs[]
QED
Theorem move_multiplicants_to_right_upper_bound:
∀ cfg intersect e eOpt plan path rws rw.
move_multiplicants_to_right cfg intersect e = (eOpt, plan) ∧
MEM (Apply (path, rws)) plan ∧
MEM rw rws ⇒
MEM rw [fp_comm_gen FP_Mul; fp_assoc_gen FP_Mul; fp_assoc2_gen FP_Mul]
Proof
ho_match_mp_tac move_multiplicants_to_right_ind
>> rw[]
>- gs[move_multiplicants_to_right_def]
>- (
qpat_x_assum `move_multiplicants_to_right _ _ _ = _` mp_tac
>> simp[Once move_multiplicants_to_right_def,
CaseEq "exp", CaseEq"list", CaseEq"op", CaseEq"fp_bop"]
>> rpt strip_tac >> gs[] >> rveq
>> Cases_on ‘move_multiplicants_to_right cfg [m] e2’ >> gs[]
>> qpat_x_assum `(if _ then _ else _) = _` mp_tac
>> rpt (COND_CASES_TAC >> gs[])
>> rpt strip_tac >> rveq >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_index
>> first_x_assum drule >> rpt $ disch_then drule >> gs[])
>> qpat_x_assum `move_multiplicants_to_right _ _ _ = _` mp_tac
>> simp[Once move_multiplicants_to_right_def]
>> Cases_on ‘move_multiplicants_to_right cfg [m1] e’ >> gs[]
>> simp[CaseEq "exp", CaseEq"list", CaseEq"op", CaseEq"fp_bop"]
>> rpt strip_tac >> gs[] >> rveq
>> TRY (
first_x_assum $ drule
>> rpt $ disch_then drule >> gs[] >> NO_TAC)
>> Cases_on ‘move_multiplicants_to_right cfg (m2::rest) e1’ >> gs[]
>> qpat_x_assum `(if _ then _ else _) = _` mp_tac
>> rpt (COND_CASES_TAC >> gs[])
>> rpt strip_tac >> rveq >> gs[]
>> imp_res_tac MEM_MAP_plan_to_path_index
>> first_x_assum $ drule
>> rpt $ disch_then drule >> gs[]
QED
Theorem canonicalize_for_distributivity_upper_bound:
MEM (Apply (path, rws)) (SND (canonicalize_for_distributivity cfg e)) ∧
MEM rw rws ⇒
MEM rw [fp_comm_gen FP_Mul; fp_assoc_gen FP_Mul; fp_assoc2_gen FP_Mul]
Proof
rpt strip_tac
>> qmatch_goalsub_abbrev_tac ‘MEM rw canon_rws’
>> Cases_on `canonicalize_for_distributivity cfg e` >> gs[]
>> gs[canonicalize_for_distributivity_def]
>> qpat_x_assum ‘_ = (_, _)’ mp_tac
>> qmatch_goalsub_abbrev_tac ‘post_order_dfs_for_plan canonicalize_app_fun _ _’
>> rpt strip_tac
>> qspecl_then [‘canonicalize_app_fun’, ‘cfg’, ‘e’, ‘λ rws. ∀ rw. MEM rw rws ⇒ MEM rw canon_rws’]
mp_tac postorder_upper_bound
>> gs[]
>> reverse impl_tac
>- (rpt $ disch_then drule >> gs[])
>> unabbrev_all_tac
>> rpt $ pop_assum kall_tac
>> rpt strip_tac >> gs[CaseEq"exp", CaseEq"op", CaseEq"list"]
>> rveq
>> qpat_x_assum ‘_ = SOME _’ mp_tac
>> qmatch_goalsub_abbrev_tac ‘move_multiplicants_to_right _ intersect_e1_e2 _’
>> Cases_on ‘move_multiplicants_to_right cfg' intersect_e1_e2 e1’ >> gs[]
>> Cases_on ‘move_multiplicants_to_right cfg' intersect_e1_e2 e2’ >> gs[]
>> rpt strip_tac >> rveq >> gs[MEM_APPEND]
>> imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac move_multiplicants_to_right_upper_bound >> gs[]
QED
Theorem canonicalize_for_distributivity_upper_bound_alt:
canonicalize_for_distributivity cfg e = (eOpt, plan) ∧
MEM (Apply (path, rws)) plan ∧
MEM rw rws ⇒
MEM rw [fp_comm_gen FP_Mul; fp_assoc_gen FP_Mul; fp_assoc2_gen FP_Mul]
Proof
rpt strip_tac >> irule canonicalize_for_distributivity_upper_bound
>> qexists_tac ‘cfg’ >> qexists_tac ‘e’ >> gs[]
>> asm_exists_tac >> gs[]
QED
Theorem apply_distributivity_local_plan:
apply_distributivity_local cfg e = SOME ((eOpt, res), plan) ∧
MEM (Apply (path, rws)) plan ∧
MEM rw rws ⇒
MEM (Apply (path, rws)) (SND (canonicalize_for_distributivity cfg e))∨
MEM rw [fp_comm_gen FP_Mul;
fp_distribute_gen FP_Mul FP_Add; fp_distribute_gen FP_Mul FP_Sub;
fp_distribute_gen FP_Div FP_Add; fp_distribute_gen FP_Div FP_Sub]
Proof
gs[apply_distributivity_local_def]
>> Cases_on ‘canonicalize_for_distributivity cfg e’ >> gs[CaseEq"exp", CaseEq"list", CaseEq"op"]
>> rpt strip_tac >> rveq >> gs[MEM_APPEND]
>> qpat_x_assum `_ = SOME _` mp_tac
>> COND_CASES_TAC >> gs[]
>> rpt strip_tac >> rveq >> gs[MEM_APPEND]
QED
val simple_case_tac =
last_x_assum $ mp_then Any mp_tac apply_distributivity_local_plan
>> rpt $ disch_then drule >> rpt strip_tac >> gs[]
>> first_x_assum $ mp_then Any mp_tac canonicalize_for_distributivity_upper_bound
>> disch_then drule >> gs[];
Theorem apply_distributivity_upper_bound:
∀ e path rws cfg.
MEM (Apply (path, rws)) (SND (apply_distributivity cfg e)) ⇒
∀ rw. MEM rw rws ⇒
MEM rw [fp_comm_gen FP_Mul;
fp_distribute_gen FP_Mul FP_Add; fp_distribute_gen FP_Mul FP_Sub;
fp_distribute_gen FP_Div FP_Add; fp_distribute_gen FP_Div FP_Sub;
fp_assoc2_gen FP_Add; fp_comm_gen FP_Mul; fp_assoc_gen FP_Mul;
fp_assoc2_gen FP_Mul]
Proof
qmatch_goalsub_abbrev_tac ‘MEM _ distrib_rws’
>> simp[Once apply_distributivity_def]
>> rpt gen_tac
>> qmatch_goalsub_abbrev_tac ‘MEM _ (SND (post_order_dfs_for_plan distributivity_app cfg e))’
>> rpt strip_tac
>> qspecl_then [‘distributivity_app’, ‘cfg’, ‘e’,
‘λ rws. ∀ rw. MEM rw rws ⇒ MEM rw distrib_rws’]
mp_tac postorder_upper_bound
>> gs[]
>> reverse impl_tac
>- (
Cases_on ‘post_order_dfs_for_plan distributivity_app cfg e’ >> gs[]
>> rpt $ disch_then drule >> gs[])
>> unabbrev_all_tac
>> rpt $ pop_assum kall_tac
>> rpt strip_tac
>> gs[CaseEq"exp", CaseEq"option", CaseEq"prod"] >> rveq
>~ [‘App op es’]
>- (
gs[CaseEq"op", CaseEq"option", CaseEq"prod", CaseEq"fp_bop", CaseEq"list",
CaseEq"exp"]
>> rveq >> gs[]
>> TRY (simple_case_tac >> NO_TAC)
>> Cases_on ‘canonicalize_for_distributivity cfg' e2_2’
>> Cases_on ‘canonicalize_for_distributivity cfg' (App (FP_bop FP_Add) [e1; e2_1])’
>> gs[CaseEq"option", CaseEq"prod"] >> rveq >> gs[]
>- simple_case_tac
>- (
Cases_on ‘canonicalize_for_distributivity cfg' (App (FP_bop FP_Add) [e1_can_dist; q])’
>> gs[CaseEq"option", CaseEq"prod"] >> rveq >> gs[]
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac canonicalize_for_distributivity_upper_bound_alt
>> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac canonicalize_for_distributivity_upper_bound_alt
>> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> simple_case_tac)
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac canonicalize_for_distributivity_upper_bound_alt
>> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac canonicalize_for_distributivity_upper_bound_alt
>> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> first_x_assum $ mp_then Any mp_tac apply_distributivity_local_plan
>> rpt $ disch_then drule >> rpt strip_tac >> gs[]
>> first_x_assum $ mp_then Any mp_tac canonicalize_for_distributivity_upper_bound
>> disch_then drule >> gs[])
>- (imp_res_tac canonicalize_for_distributivity_upper_bound_alt >> gs[])
>> qpat_x_assum ‘apply_distributivity_local _ (App _ _) = SOME _’
$ mp_then Any mp_tac apply_distributivity_local_plan
>> rpt $ disch_then drule >> rpt strip_tac >> gs[]
>> first_x_assum $ mp_then Any mp_tac canonicalize_for_distributivity_upper_bound_alt
>> rpt $ disch_then drule >> gs[])
>> Cases_on ‘canonicalize_for_distributivity cfg' (App (FP_bop FP_Add) [e1_can_dist; e2_can_dist])’
>> gs[CaseEq"option", CaseEq"prod"] >> rveq >> gs[]
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac canonicalize_for_distributivity_upper_bound_alt
>> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac canonicalize_for_distributivity_upper_bound_alt
>> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> first_x_assum $ mp_then Any mp_tac apply_distributivity_local_plan
>> rpt $ disch_then drule >> rpt strip_tac >> gs[]
>> first_x_assum $ mp_then Any mp_tac canonicalize_for_distributivity_upper_bound
>> disch_then drule >> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> first_x_assum $ mp_then Any mp_tac apply_distributivity_local_plan
>> rpt $ disch_then drule >> rpt strip_tac >> gs[]
>> first_x_assum $ mp_then Any mp_tac canonicalize_for_distributivity_upper_bound
>> disch_then drule >> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac canonicalize_for_distributivity_upper_bound_alt
>> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> qpat_x_assum `canonicalize_for_distributivity _ _ = (_, r)` $
mp_then Any mp_tac canonicalize_for_distributivity_upper_bound_alt
>> rpt $ disch_then drule >> rpt strip_tac >> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> first_x_assum $ mp_then Any mp_tac apply_distributivity_local_plan
>> rpt $ disch_then drule >> rpt strip_tac >> gs[]
>> first_x_assum $ mp_then Any mp_tac canonicalize_for_distributivity_upper_bound
>> disch_then drule >> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> last_x_assum $ mp_then Any mp_tac apply_distributivity_local_plan
>> disch_then kall_tac
>> last_x_assum $ mp_then Any mp_tac apply_distributivity_local_plan
>> rpt $ disch_then drule >> rpt strip_tac >> gs[]
>> first_x_assum $ mp_then Any mp_tac canonicalize_for_distributivity_upper_bound
>> disch_then drule >> gs[])
>- (
imp_res_tac MEM_MAP_plan_to_path_index
>> imp_res_tac canonicalize_for_distributivity_upper_bound_alt
>> gs[])
>> qpat_x_assum ‘apply_distributivity_local _ (App _ _) = SOME _’
$ mp_then Any mp_tac apply_distributivity_local_plan
>> rpt $ disch_then drule >> rpt strip_tac >> gs[]
>> first_x_assum $ mp_then Any mp_tac canonicalize_for_distributivity_upper_bound_alt
>> disch_then drule >> gs[])
>> simple_case_tac
QED
Theorem balance_expression_tree_upper_bound:
MEM (Apply (path, rws)) (SND(balance_expression_tree cfg e)) ⇒
∀ rw.
MEM rw rws ⇒
MEM rw [fp_assoc2_gen FP_Add ; fp_assoc2_gen FP_Mul]
Proof
simp[Once balance_expression_tree_def]
>> rpt gen_tac
>> qmatch_goalsub_abbrev_tac ‘MEM _ (SND (post_order_dfs_for_plan balance_app cfg e))’
>> rpt strip_tac
>> qspecl_then [‘balance_app’, ‘cfg’, ‘e’,
‘λ rws. ∀ rw. MEM rw rws ⇒ MEM rw [fp_assoc2_gen FP_Add; fp_assoc2_gen FP_Mul]’]
mp_tac postorder_upper_bound
>> gs[]
>> reverse impl_tac
>- (
Cases_on ‘post_order_dfs_for_plan balance_app cfg e’ >> gs[]
>> rpt $ disch_then drule >> gs[])
>> unabbrev_all_tac >> rpt $ pop_assum kall_tac
>> gs[CaseEq"exp", CaseEq"op", CaseEq"fp_bop", CaseEq"list"]
>> rpt strip_tac >> gs[]
QED
Theorem rewriteFPexp_weakening:
(∀ rw. MEM rw rws2 ⇒
∀ (st1:'a semanticPrimitives$state) st2 env e r.
is_rewriteFPexp_correct [rw] st1 st2 env e r) ∧
(∀ rw. MEM rw rws1 ⇒ MEM rw rws2) ⇒
∀ (st1:'a semanticPrimitives$state) st2 env e r.
is_rewriteFPexp_correct rws1 st1 st2 env e r
Proof
Induct_on ‘rws1’ >> gs[empty_rw_correct]
>> rpt strip_tac
>> ‘h :: rws1 = [h] ++ rws1’ by gs[]
>> pop_assum $ rewrite_tac o single
>> irule rewriteExp_compositional >> reverse conj_tac
>- (
first_x_assum $ qspec_then ‘h’ mp_tac >> gs[] >> strip_tac
>> first_x_assum drule >> gs[])
>> last_x_assum irule
>> conj_tac >> gs[]
QED
Theorem is_real_id_exp_weakening:
(∀ rw. MEM rw rws2 ⇒
∀ (st1:'a semanticPrimitives$state) st2 env e r.
is_real_id_exp [rw] st1 st2 env e r) ∧
(∀ rw. MEM rw rws1 ⇒ MEM rw rws2) ⇒
∀ (st1:'a semanticPrimitives$state) st2 env e r.
is_real_id_exp rws1 st1 st2 env e r
Proof
Induct_on ‘rws1’ >> gs[empty_rw_real_id]
>> rpt strip_tac
>> ‘h :: rws1 = [h] ++ rws1’ by gs[]
>> pop_assum $ rewrite_tac o single
>> irule real_valued_id_compositional >> reverse conj_tac
>- (
first_x_assum $ qspec_then ‘h’ mp_tac >> gs[] >> strip_tac
>> first_x_assum drule >> gs[])
>> last_x_assum irule
>> conj_tac >> gs[]
QED
Theorem optPlanner_correct_float_single:
∀ e st1 st2 env cfg exps r.
is_optimise_with_plan_correct (generate_plan_exp cfg e) st1 st2 env cfg exps r
Proof
rpt strip_tac
>> irule is_optimise_with_plan_correct_lift
>> rpt gen_tac >> strip_tac
>> irule is_rewriteFPexp_correct_lift_perform_rewrites
>> irule lift_rewriteFPexp_correct_list
>> first_x_assum $ mp_then Any assume_tac generate_plan_upper_bound_rws
>> gs[]
>> imp_res_tac balance_expression_tree_upper_bound
>> imp_res_tac canonicalize_upper_bound
>> imp_res_tac apply_distributivity_upper_bound
>> imp_res_tac peephole_optimise_upper_bound
>> irule rewriteFPexp_weakening >> asm_exists_tac >> gs[]
>> rpt strip_tac >> rveq
>> gs[fp_times_two_to_add_correct_unfold, fp_times_two_to_add_correct,
fp_times_three_to_add_correct_unfold, fp_times_three_to_add_correct,
fp_times_one_correct_unfold, fp_times_one_correct,
fp_times_minus_one_neg_correct_unfold,
fp_times_minus_one_neg_correct, fp_sub_add_correct_unfold,
fp_sub_add_correct, fp_plus_zero_correct_unfold,
fp_plus_zero_correct, fp_neg_times_minus_one_correct_unfold,
fp_neg_times_minus_one_correct, fp_neg_push_mul_r_correct_unfold,
fp_neg_push_mul_r_correct, fp_fma_intro_correct_unfold,
fp_fma_intro_correct, fp_distribute_gen_correct_unfold_add,
fp_distribute_gen_correct_unfold, fp_distribute_gen_correct,
fp_comm_gen_correct_unfold_mul, fp_comm_gen_correct_unfold_add,
fp_comm_gen_correct_unfold, fp_comm_gen_correct,
fp_assoc_gen_correct_unfold_mul, fp_assoc_gen_correct_unfold_add,
fp_assoc_gen_correct_unfold, fp_assoc_gen_correct,
fp_assoc2_gen_correct_unfold_mul, fp_assoc2_gen_correct_unfold_add,
fp_assoc2_gen_correct_unfold, fp_assoc2_gen_correct,
fp_add_sub_correct_unfold, fp_add_sub_correct,
reverse_tuple_def, fp_undistribute_gen_def, fp_times_zero_def,
fp_times_two_to_add_def, fp_times_three_to_add_def,
fp_times_one_def,
fp_times_minus_one_neg_def, fp_times_into_div_def, fp_sub_add_def,
fp_same_sub_def, fp_plus_zero_def,
fp_neg_times_minus_one_def, fp_neg_push_mul_r_def,
fp_mul_sub_undistribute_def, fp_mul_sub_distribute_def,
fp_mul_comm_def, fp_mul_assoc_def, fp_mul_assoc2_def,
fp_mul_add_undistribute_def, fp_mul_add_distribute_def,
fp_fma_intro_def, fp_div_sub_undistribute_def,
fp_div_sub_distribute_def, fp_div_add_undistribute_def,
fp_div_add_distribute_def, fp_distribute_gen_def, fp_comm_gen_def,
fp_assoc_gen_def, fp_assoc2_gen_def, fp_add_sub_def,
fp_add_comm_def, fp_add_assoc_def, fp_add_assoc2_def,
fp_times_zero_correct, fp_same_sub_correct]
QED
Theorem optPlanner_correct_real_single:
∀ e st1 st2 env cfg exps r.
is_real_id_optimise_with_plan (generate_plan_exp cfg e) st1 st2 env cfg exps r
Proof
rpt gen_tac
>> irule is_real_id_perform_rewrites_optimise_with_plan_lift
>> rpt gen_tac >> strip_tac
>> irule is_real_id_list_perform_rewrites_lift
>> irule lift_real_id_exp_list_strong
>> first_x_assum $ mp_then Any assume_tac generate_plan_upper_bound_rws
>> gs[]
>> imp_res_tac balance_expression_tree_upper_bound
>> imp_res_tac canonicalize_upper_bound
>> imp_res_tac apply_distributivity_upper_bound
>> imp_res_tac peephole_optimise_upper_bound
>> irule is_real_id_exp_weakening >> asm_exists_tac >> gs[]
>> rpt strip_tac >> rveq
>> gs[fp_times_two_to_add_real_id_unfold, fp_times_two_to_add_real_id,
fp_times_three_to_add_real_id_unfold, fp_times_three_to_add_real_id,
fp_times_one_real_id_unfold, fp_times_one_real_id,
fp_times_minus_one_neg_real_id_unfold,
fp_times_minus_one_neg_real_id, fp_sub_add_real_id_unfold,
fp_sub_add_real_id, fp_plus_zero_real_id_unfold,
fp_plus_zero_real_id, fp_neg_times_minus_one_real_id_unfold,
fp_neg_times_minus_one_real_id, fp_neg_push_mul_r_real_id_unfold,
fp_neg_push_mul_r_real_id, fma_intro_real_id,
fma_intro_real_id, fp_distribute_gen_real_id_unfold,
Q.SPECL [‘FP_Sub’, ‘FP_Mul’] fp_distribute_gen_real_id,
Q.SPECL [‘FP_Add’, ‘FP_Div’] fp_distribute_gen_real_id,
Q.SPECL [‘FP_Sub’, ‘FP_Div’] fp_distribute_gen_real_id,
fp_comm_gen_real_id_unfold_mul, fp_comm_gen_real_id_unfold_add,
Q.SPEC ‘FP_Sub’ fp_comm_gen_real_id, fp_assoc_gen_real_id_unfold_mul,
fp_assoc_gen_real_id_unfold_add, fp_assoc_gen_real_id,
fp_assoc2_gen_real_id_unfold_mul, fp_assoc2_gen_real_id_unfold_add,
fp_assoc2_gen_real_id, fp_add_sub_real_id_unfold, fp_add_sub_real_id,
reverse_tuple_def, fp_undistribute_gen_def, fp_times_zero_def,
fp_times_two_to_add_def, fp_times_three_to_add_def,
fp_times_one_def,
fp_times_minus_one_neg_def, fp_times_into_div_def, fp_sub_add_def,
fp_same_sub_def, fp_plus_zero_def,
fp_neg_times_minus_one_def, fp_neg_push_mul_r_def,
fp_mul_sub_undistribute_def, fp_mul_sub_distribute_def,
fp_mul_comm_def, fp_mul_assoc_def, fp_mul_assoc2_def,
fp_mul_add_undistribute_def, fp_mul_add_distribute_def,
fp_fma_intro_def, fp_div_sub_undistribute_def,
fp_div_sub_distribute_def, fp_div_add_undistribute_def,
fp_div_add_distribute_def, fp_distribute_gen_def, fp_comm_gen_def,
fp_assoc_gen_def, fp_assoc2_gen_def, fp_add_sub_def,
fp_add_comm_def, fp_add_assoc_def, fp_add_assoc2_def,
fp_times_zero_real_id, fp_same_sub_real_id]
QED
val _ = export_theory();