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milExampleCopyEqualScript.sml
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open HolKernel boolLib Parse bossLib metisTools wordsLib wordsTheory finite_mapTheory listTheory pred_setTheory sortingTheory milUtilityTheory milTheory milSemanticsUtilityTheory milMetaTheory milMetaIOTheory milTracesTheory milInitializationTheory milCompositionalTheory milExampleUtilityTheory milStoreTheory milExecutableExamplesTheory milExecutableUtilityTheory milExecutableTransitionTheory milExecutableInitializationTheory milExecutableIOTheory milExecutableIOTraceTheory milExecutableCompositionalTheory;
(* ===================== *)
(* Copy-on-equal example *)
(* ===================== *)
val _ = new_theory "milExampleCopyEqual";
(* ------------------- *)
(* Program definitions *)
(* ------------------- *)
Definition example_ceq:
example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
{
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero));
i_assign tc01 (e_val val_true) (o_internal (e_val r1));
i_assign tc02 (e_val val_true) (o_internal (e_val r2));
i_assign tc03 (e_val val_true) (o_internal (e_val z));
i_assign tc04 (e_val val_true) (o_internal (e_val b1));
i_assign tc05 (e_val val_true) (o_internal (e_val b2));
i_assign tc11 (e_val val_true) (o_load res_MEM tc04);
i_assign tc12 (e_val val_true) (o_store res_REG tc01 tc11);
i_assign tc21 (e_val val_true) (o_load res_REG tc03);
i_assign tc22 (e_eq (e_name tc21) (e_val val_one)) (o_load res_REG tc01);
i_assign tc23 (e_eq (e_name tc21) (e_val val_one)) (o_store res_REG tc02 tc22);
i_assign tc31 (e_val val_true) (o_load res_REG tc02);
i_assign tc32 (e_val val_true) (o_store res_MEM tc05 tc31);
i_assign tc41 (e_val val_true) (o_load res_PC tc00);
i_assign tc42 (e_val val_true) (o_internal (e_add (e_name tc41) (e_val val_four)));
i_assign tc43 (e_val val_true) (o_store res_PC tc00 tc42)
}
End
Definition example_ceq_t:
example_ceq_t t r1 r2 z b1 b2 =
example_ceq (t+1) (t+2) (t+3) (t+4) (t+5) (t+6) (t+7) (t+8) (t+9) (t+10) (t+11) (t+12)
(t+13) (t+14) (t+15) (t+16) r1 r2 z b1 b2
End
Definition example_ceq_list_1:
example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
[
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero));
i_assign tc01 (e_val val_true) (o_internal (e_val r1));
i_assign tc02 (e_val val_true) (o_internal (e_val r2));
i_assign tc03 (e_val val_true) (o_internal (e_val z));
i_assign tc04 (e_val val_true) (o_internal (e_val b1));
i_assign tc05 (e_val val_true) (o_internal (e_val b2));
i_assign tc11 (e_val val_true) (o_load res_MEM tc04);
i_assign tc12 (e_val val_true) (o_store res_REG tc01 tc11);
i_assign tc21 (e_val val_true) (o_load res_REG tc03);
i_assign tc22 (e_eq (e_name tc21) (e_val val_one)) (o_load res_REG tc01);
i_assign tc23 (e_eq (e_name tc21) (e_val val_one)) (o_store res_REG tc02 tc22);
i_assign tc31 (e_val val_true) (o_load res_REG tc02);
i_assign tc32 (e_val val_true) (o_store res_MEM tc05 tc31);
i_assign tc41 (e_val val_true) (o_load res_PC tc00);
i_assign tc42 (e_val val_true) (o_internal (e_add (e_name tc41) (e_val val_four)));
i_assign tc43 (e_val val_true) (o_store res_PC tc00 tc42)
]
End
Definition example_ceq_list_1_t:
example_ceq_list_1_t t r1 r2 z b1 b2 =
example_ceq_list_1 (t+1) (t+2) (t+3) (t+4) (t+5) (t+6) (t+7) (t+8) (t+9) (t+10) (t+11) (t+12)
(t+13) (t+14) (t+15) (t+16) r1 r2 z b1 b2
End
Definition example_ceq_list_2:
example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
[
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero));
i_assign tc01 (e_val val_true) (o_internal (e_val r1));
i_assign tc02 (e_val val_true) (o_internal (e_val r2));
i_assign tc03 (e_val val_true) (o_internal (e_val z));
i_assign tc04 (e_val val_true) (o_internal (e_val b1));
i_assign tc05 (e_val val_true) (o_internal (e_val b2));
i_assign tc21 (e_val val_true) (o_load res_REG tc03);
i_assign tc22 (e_eq (e_name tc21) (e_val val_one)) (o_load res_REG tc01);
i_assign tc23 (e_eq (e_name tc21) (e_val val_one)) (o_store res_REG tc02 tc22);
i_assign tc31 (e_val val_true) (o_load res_REG tc02);
i_assign tc32 (e_val val_true) (o_store res_MEM tc05 tc31);
i_assign tc11 (e_val val_true) (o_load res_MEM tc04);
i_assign tc12 (e_val val_true) (o_store res_REG tc01 tc11);
i_assign tc41 (e_val val_true) (o_load res_PC tc00);
i_assign tc42 (e_val val_true) (o_internal (e_add (e_name tc41) (e_val val_four)));
i_assign tc43 (e_val val_true) (o_store res_PC tc00 tc42)
]
End
Definition example_ceq_list_2_t:
example_ceq_list_2_t t r1 r2 z b1 b2 =
example_ceq_list_2 (t+1) (t+2) (t+3) (t+4) (t+5) (t+6) (t+7) (t+8) (t+9) (t+10) (t+11) (t+12)
(t+13) (t+14) (t+15) (t+16) r1 r2 z b1 b2
End
Definition example_ceq_list_3:
example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
[
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero));
i_assign tc01 (e_val val_true) (o_internal (e_val r1));
i_assign tc02 (e_val val_true) (o_internal (e_val r2));
i_assign tc03 (e_val val_true) (o_internal (e_val z));
i_assign tc04 (e_val val_true) (o_internal (e_val b1));
i_assign tc05 (e_val val_true) (o_internal (e_val b2));
i_assign tc41 (e_val val_true) (o_load res_PC tc00);
i_assign tc42 (e_val val_true) (o_internal (e_add (e_name tc41) (e_val val_four)));
i_assign tc43 (e_val val_true) (o_store res_PC tc00 tc42);
i_assign tc11 (e_val val_true) (o_load res_MEM tc04);
i_assign tc12 (e_val val_true) (o_store res_REG tc01 tc11);
i_assign tc21 (e_val val_true) (o_load res_REG tc03);
i_assign tc22 (e_eq (e_name tc21) (e_val val_one)) (o_load res_REG tc01);
i_assign tc23 (e_eq (e_name tc21) (e_val val_one)) (o_store res_REG tc02 tc22);
i_assign tc31 (e_val val_true) (o_load res_REG tc02);
i_assign tc32 (e_val val_true) (o_store res_MEM tc05 tc31)
]
End
Definition example_ceq_list_3_t:
example_ceq_list_3_t t r1 r2 z b1 b2 =
example_ceq_list_3 (t+1) (t+2) (t+3) (t+4) (t+5) (t+6) (t+7) (t+8) (t+9) (t+10) (t+11) (t+12) (t+13) (t+14) (t+15) (t+16) r1 r2 z b1 b2
End
Definition example_ceq_list_4:
example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
[
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero));
i_assign tc01 (e_val val_true) (o_internal (e_val r1));
i_assign tc02 (e_val val_true) (o_internal (e_val r2));
i_assign tc03 (e_val val_true) (o_internal (e_val z));
i_assign tc04 (e_val val_true) (o_internal (e_val b1));
i_assign tc05 (e_val val_true) (o_internal (e_val b2));
i_assign tc11 (e_val val_true) (o_load res_MEM tc04);
i_assign tc41 (e_val val_true) (o_load res_PC tc00);
i_assign tc42 (e_val val_true) (o_internal (e_add (e_name tc41) (e_val val_four)));
i_assign tc43 (e_val val_true) (o_store res_PC tc00 tc42);
i_assign tc12 (e_val val_true) (o_store res_REG tc01 tc11);
i_assign tc21 (e_val val_true) (o_load res_REG tc03);
i_assign tc22 (e_eq (e_name tc21) (e_val val_one)) (o_load res_REG tc01);
i_assign tc23 (e_eq (e_name tc21) (e_val val_one)) (o_store res_REG tc02 tc22);
i_assign tc31 (e_val val_true) (o_load res_REG tc02);
i_assign tc32 (e_val val_true) (o_store res_MEM tc05 tc31)
]
End
Definition example_ceq_list_4_t:
example_ceq_list_4_t t r1 r2 z b1 b2 =
example_ceq_list_4 (t+1) (t+2) (t+3) (t+4) (t+5) (t+6) (t+7) (t+8) (t+9) (t+10) (t+11) (t+12) (t+13) (t+14) (t+15) (t+16) r1 r2 z b1 b2
End
(* ------------------------ *)
(* Utility and basic lemmas *)
(* ------------------------ *)
Theorem t_and_lt[local]:
!(t:num).
t + 1 < t + 2 /\ t + 2 < t + 3 /\ t + 3 < t + 4 /\
t + 4 < t + 5 /\ t + 5 < t + 6 /\ t + 6 < t + 7 /\
t + 7 < t + 8 /\ t + 8 < t + 9 /\ t + 9 < t + 10 /\
t + 10 < t + 11 /\ t + 11 < t + 12 /\ t + 12 < t + 13 /\
t + 13 < t + 14 /\ t + 14 < t + 15 /\ t + 15 < t + 16
Proof
rw [] >> DECIDE_TAC
QED
Theorem example_ceq_list_1_set:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
set (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [example_ceq_list_1,example_ceq]
QED
Theorem example_ceq_list_1_t_set:
!t r1 r2 z b1 b2. example_ceq_t t r1 r2 z b1 b2 = set (example_ceq_list_1_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_1_t,example_ceq_t,example_ceq_list_1_set]
QED
Theorem example_ceq_list_2_set:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
set (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [example_ceq_list_2,example_ceq] >> rw [EXTENSION] >> METIS_TAC []
QED
Theorem example_ceq_list_2_t_set:
!t r1 r2 z b1 b2. example_ceq_t t r1 r2 z b1 b2 = set (example_ceq_list_2_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_2_t,example_ceq_t,example_ceq_list_2_set]
QED
Theorem example_ceq_list_3_set:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
set (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [example_ceq_list_3,example_ceq] >> rw [EXTENSION] >> METIS_TAC []
QED
Theorem example_ceq_list_3_t_set:
!t r1 r2 z b1 b2. example_ceq_t t r1 r2 z b1 b2 = set (example_ceq_list_3_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_3_t,example_ceq_t,example_ceq_list_3_set]
QED
Theorem example_ceq_list_4_set:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 =
set (example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [example_ceq_list_4,example_ceq] >> rw [EXTENSION] >> METIS_TAC []
QED
Theorem example_ceq_list_4_t_set:
!t r1 r2 z b1 b2. example_ceq_t t r1 r2 z b1 b2 = set (example_ceq_list_4_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_4_t,example_ceq_t,example_ceq_list_4_set]
QED
Theorem example_ceq_list_1_map:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
MAP bound_name_instr (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
[tc00; tc01; tc02; tc03; tc04; tc05; tc11; tc12; tc21; tc22; tc23; tc31; tc32; tc41; tc42; tc43]
Proof
rw [example_ceq_list_1,bound_name_instr] >> fs []
QED
Theorem example_ceq_list_2_map:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
MAP bound_name_instr (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
[tc00; tc01; tc02; tc03; tc04; tc05; tc21; tc22; tc23; tc31; tc32; tc11; tc12; tc41; tc42; tc43]
Proof
rw [example_ceq_list_2,bound_name_instr] >> fs []
QED
Theorem example_ceq_list_3_map:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
MAP bound_name_instr (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
[tc00; tc01; tc02; tc03; tc04; tc05; tc41; tc42; tc43; tc11; tc12; tc21; tc22; tc23; tc31; tc32]
Proof
rw [example_ceq_list_3,bound_name_instr] >> fs []
QED
Theorem example_ceq_list_4_map:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
MAP bound_name_instr (example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
[tc00; tc01; tc02; tc03; tc04; tc05; tc11; tc41; tc42; tc43; tc12; tc21; tc22; tc23; tc31; tc32]
Proof
rw [example_ceq_list_4,bound_name_instr] >> fs []
QED
Theorem example_ceq_list_1_NO_DUPLICATES:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
NO_DUPLICATES (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [NO_DUPLICATES] >>
once_rewrite_tac [example_ceq_list_1_map] >>
rw [ALL_DISTINCT] >> DECIDE_TAC
QED
Theorem example_ceq_list_1_t_NO_DUPLICATES:
!t r1 r2 z b1 b2. NO_DUPLICATES (example_ceq_list_1_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_1_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
METIS_TAC [example_ceq_list_1_NO_DUPLICATES]
QED
Theorem example_ceq_list_2_NO_DUPLICATES:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
NO_DUPLICATES (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [NO_DUPLICATES] >>
once_rewrite_tac [example_ceq_list_2_map] >>
rw [ALL_DISTINCT] >> DECIDE_TAC
QED
Theorem example_ceq_list_2_t_NO_DUPLICATES:
!t r1 r2 z b1 b2. NO_DUPLICATES (example_ceq_list_2_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_2_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
METIS_TAC [example_ceq_list_2_NO_DUPLICATES]
QED
Theorem example_ceq_list_3_NO_DUPLICATES:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
NO_DUPLICATES (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [NO_DUPLICATES] >>
once_rewrite_tac [example_ceq_list_3_map] >>
rw [ALL_DISTINCT] >> DECIDE_TAC
QED
Theorem example_ceq_list_3_t_NO_DUPLICATES:
!t r1 r2 z b1 b2. NO_DUPLICATES (example_ceq_list_3_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_3_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
METIS_TAC [example_ceq_list_3_NO_DUPLICATES]
QED
Theorem example_ceq_list_4_NO_DUPLICATES:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
NO_DUPLICATES (example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [NO_DUPLICATES] >>
once_rewrite_tac [example_ceq_list_4_map] >>
rw [ALL_DISTINCT] >> DECIDE_TAC
QED
Theorem example_ceq_list_4_t_NO_DUPLICATES:
!t r1 r2 z b1 b2. NO_DUPLICATES (example_ceq_list_4_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_4_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
METIS_TAC [example_ceq_list_4_NO_DUPLICATES]
QED
Theorem example_ceq_list_1_SORTED:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
SORTED bound_name_instr_le (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
Proof
rw [example_ceq_list_1,bound_name_instr_le,name_le,bound_name_instr]
QED
Theorem example_ceq_list_1_t_SORTED:
!t r1 r2 z b1 b2. SORTED bound_name_instr_le (example_ceq_list_1_t t r1 r2 z b1 b2)
Proof
rw [example_ceq_list_1_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
METIS_TAC [example_ceq_list_1_SORTED]
QED
Theorem example_ceq_list_1_bound_names_program_list:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
bound_names_program_list (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
[tc00; tc01; tc02; tc03; tc04; tc05; tc11; tc12; tc21; tc22; tc23; tc31; tc32; tc41; tc42; tc43]
Proof
rw [bound_names_program_list,example_ceq_list_1,bound_name_instr]
QED
Theorem example_ceq_bound_names_program:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
bound_names_program (example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
{tc00; tc01; tc02; tc03; tc04; tc05; tc11; tc12; tc21; tc22; tc23; tc31; tc32; tc41; tc42; tc43}
Proof
rw [example_ceq_list_1_set] >>
rw [GSYM bound_names_program_list_correct] >>
rw [example_ceq_list_1_bound_names_program_list]
QED
Theorem example_ceq_list_1_DROP:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 il0.
DROP (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2
Proof
rw [example_ceq_list_1] >>
Induct_on `il0` >> rw []
QED
Theorem example_ceq_list_2_DROP:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 il0.
DROP (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2
Proof
rw [example_ceq_list_2] >>
Induct_on `il0` >> rw []
QED
Theorem example_ceq_list_3_DROP:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 il0.
DROP (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2
Proof
rw [example_ceq_list_3] >>
Induct_on `il0` >> rw []
QED
Theorem example_ceq_list_4_DROP:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 il0.
DROP (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2
Proof
rw [example_ceq_list_4] >>
Induct_on `il0` >> rw []
QED
Theorem example_ceq_list_1_t_DROP:
!t r1 r2 z b1 b2 il0.
DROP (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_1_t t r1 r2 z b1 b2) =
example_ceq_list_1_t t r1 r2 z b1 b2
Proof
once_rewrite_tac [example_ceq_list_1_t] >>
rw [example_ceq_list_1_DROP]
QED
Theorem example_ceq_list_2_t_DROP:
!t r1 r2 z b1 b2 il0.
DROP (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_2_t t r1 r2 z b1 b2) =
example_ceq_list_2_t t r1 r2 z b1 b2
Proof
once_rewrite_tac [example_ceq_list_2_t] >>
rw [example_ceq_list_2_DROP]
QED
Theorem example_ceq_list_3_t_DROP:
!t r1 r2 z b1 b2 il0.
DROP (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_3_t t r1 r2 z b1 b2) =
example_ceq_list_3_t t r1 r2 z b1 b2
Proof
once_rewrite_tac [example_ceq_list_3_t] >>
rw [example_ceq_list_3_DROP]
QED
Theorem example_ceq_list_4_t_DROP:
!t r1 r2 z b1 b2 il0.
DROP (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_4_t t r1 r2 z b1 b2) =
example_ceq_list_4_t t r1 r2 z b1 b2
Proof
once_rewrite_tac [example_ceq_list_4_t] >>
rw [example_ceq_list_4_DROP]
QED
Theorem example_ceq_list_1_HD:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
HD (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero))
Proof
rw [example_ceq_list_1]
QED
Theorem example_ceq_list_2_HD:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
HD (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero))
Proof
rw [example_ceq_list_2]
QED
Theorem example_ceq_list_3_HD:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
HD (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero))
Proof
rw [example_ceq_list_3]
QED
Theorem example_ceq_list_4_HD:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
HD (example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
i_assign tc00 (e_val val_true) (o_internal (e_val val_zero))
Proof
rw [example_ceq_list_4]
QED
Theorem example_ceq_list_1_t_HD:
!t r1 r2 z b1 b2.
HD (example_ceq_list_1_t t r1 r2 z b1 b2) = i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero))
Proof
rw [example_ceq_list_1_t,example_ceq_list_1_HD]
QED
Theorem example_ceq_list_2_t_HD:
!t r1 r2 z b1 b2.
HD (example_ceq_list_2_t t r1 r2 z b1 b2) = i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero))
Proof
rw [example_ceq_list_2_t,example_ceq_list_2_HD]
QED
Theorem example_ceq_list_3_t_HD:
!t r1 r2 z b1 b2.
HD (example_ceq_list_3_t t r1 r2 z b1 b2) = i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero))
Proof
rw [example_ceq_list_3_t,example_ceq_list_3_HD]
QED
Theorem example_ceq_list_4_t_HD:
!t r1 r2 z b1 b2.
HD (example_ceq_list_4_t t r1 r2 z b1 b2) = i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero))
Proof
rw [example_ceq_list_4_t,example_ceq_list_4_HD]
QED
Theorem example_ceq_list_1_NTH:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 il0.
NTH (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
SOME (i_assign tc00 (e_val val_true) (o_internal (e_val val_zero)))
Proof
rw [] >>
`LENGTH il0 < LENGTH (il0 ++ example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [LENGTH_APPEND,example_ceq_list_1] >>
rw [NTH_EL_LENGTH] >>
rw [GSYM HD_DROP] >>
`LENGTH il0 = PRE (SUC (LENGTH il0))` by rw [] >>
METIS_TAC [example_ceq_list_1_HD,example_ceq_list_1_DROP]
QED
Theorem example_ceq_list_2_NTH:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 il0.
NTH (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
SOME (i_assign tc00 (e_val val_true) (o_internal (e_val val_zero)))
Proof
rw [] >>
`LENGTH il0 < LENGTH (il0 ++ example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [LENGTH_APPEND,example_ceq_list_2] >>
rw [NTH_EL_LENGTH] >>
rw [GSYM HD_DROP] >>
`LENGTH il0 = PRE (SUC (LENGTH il0))` by rw [] >>
METIS_TAC [example_ceq_list_2_HD,example_ceq_list_2_DROP]
QED
Theorem example_ceq_list_3_NTH:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 il0.
NTH (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
SOME (i_assign tc00 (e_val val_true) (o_internal (e_val val_zero)))
Proof
rw [] >>
`LENGTH il0 < LENGTH (il0 ++ example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [LENGTH_APPEND,example_ceq_list_3] >>
rw [NTH_EL_LENGTH] >>
rw [GSYM HD_DROP] >>
`LENGTH il0 = PRE (SUC (LENGTH il0))` by rw [] >>
METIS_TAC [example_ceq_list_3_HD,example_ceq_list_3_DROP]
QED
Theorem example_ceq_list_4_NTH:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 il0.
NTH (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) =
SOME (i_assign tc00 (e_val val_true) (o_internal (e_val val_zero)))
Proof
rw [] >>
`LENGTH il0 < LENGTH (il0 ++ example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [LENGTH_APPEND,example_ceq_list_4] >>
rw [NTH_EL_LENGTH] >>
rw [GSYM HD_DROP] >>
`LENGTH il0 = PRE (SUC (LENGTH il0))` by rw [] >>
METIS_TAC [example_ceq_list_4_HD,example_ceq_list_4_DROP]
QED
Theorem example_ceq_list_1_t_NTH:
!t r1 r2 z b1 b2 il0.
NTH (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_1_t t r1 r2 z b1 b2) =
SOME (i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero)))
Proof
once_rewrite_tac [example_ceq_list_1_t] >> rw [example_ceq_list_1_NTH]
QED
Theorem example_ceq_list_2_t_NTH:
!t r1 r2 z b1 b2 il0.
NTH (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_2_t t r1 r2 z b1 b2) =
SOME (i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero)))
Proof
once_rewrite_tac [example_ceq_list_2_t] >> rw [example_ceq_list_2_NTH]
QED
Theorem example_ceq_list_3_t_NTH:
!t r1 r2 z b1 b2 il0.
NTH (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_3_t t r1 r2 z b1 b2) =
SOME (i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero)))
Proof
once_rewrite_tac [example_ceq_list_3_t] >> rw [example_ceq_list_3_NTH]
QED
Theorem example_ceq_list_4_t_NTH:
!t r1 r2 z b1 b2 il0.
NTH (PRE (SUC (LENGTH il0))) (il0 ++ example_ceq_list_4_t t r1 r2 z b1 b2) =
SOME (i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero)))
Proof
once_rewrite_tac [example_ceq_list_4_t] >> rw [example_ceq_list_4_NTH]
QED
Theorem example_ceq_compositional_program:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
!t. t < tc00 ==>
compositional_program (example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) t
Proof
rw [compositional_program,example_ceq] >>
fs [bound_name_instr,free_names_instr,names_e,names_o,val_true,val_false,sem_expr_correct] >>
rw [] >> METIS_TAC [bound_name_instr]
QED
Theorem example_ceq_t_compositional_program:
!t r1 r2 z b1 b2 t'. t' <= t ==> compositional_program (example_ceq_t t r1 r2 z b1 b2) t'
Proof
rw [example_ceq_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`t' < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_compositional_program]
QED
Theorem example_ceq_instr_lt:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 State i i'.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
well_formed_state State ==>
max_name_in_State State < tc00 ==>
instr_in_State i State ==>
i' IN (example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) ==>
bound_name_instr i < bound_name_instr i'
Proof
rw [] >>
Cases_on `State` >>
rename1 `State_st I0 s0 C0 F0` >>
fs [max_name_in_State,instr_in_State] >>
`FINITE I0` by METIS_TAC [wfs_FINITE] >>
`bound_name_instr i IN bound_names_program I0`
by (rw [bound_names_program] >> METIS_TAC []) >>
METIS_TAC [example_ceq_compositional_program,compositional_program_state_lt_bound_name_instr]
QED
Theorem example_ceq_t_instr_lt:
!t r1 r2 z b1 b2 State i i'.
well_formed_state State ==>
max_name_in_State State <= t ==>
instr_in_State i State ==>
i' IN (example_ceq_t t r1 r2 z b1 b2) ==>
bound_name_instr i < bound_name_instr i'
Proof
rw [example_ceq_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_State State < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_instr_lt]
QED
Theorem example_ceq_not_Completed:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 State i.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
well_formed_state State ==>
max_name_in_State State < tc00 ==>
i IN (example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) ==>
~(Completed (union_program_state State (example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)) i)
Proof
rw [] >>
Cases_on `i` >>
rename1 `i_assign t c mop` >>
`names_e c <> {} \/ c = e_val val_true` by fs [example_ceq,names_e] >>
`compositional_program
(example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)
(max_name_in_State State)`
by METIS_TAC [example_ceq_compositional_program] >-
METIS_TAC [compositional_program_guard_variables_not_completed] >>
METIS_TAC [compositional_program_true_guard_not_completed]
QED
Theorem example_ceq_list_1_not_Completed_list:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl i.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
MEM i (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) ==>
~(Completed_list sem_expr (append_program_state_list stl (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)) i)
Proof
rw [] >>
Cases_on `stl` >>
rename1 `State_st_list il0 s0 fl0 cl0` >>
fs [State_st_list_well_formed_ok] >>
rw [Completed_list_correct,append_program_state_list_correct,GSYM example_ceq_list_1_set] >>
`i IN example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2`
by rw [example_ceq_list_1_set] >>
`max_name_in_State (state_list_to_state (State_st_list il0 s0 fl0 cl0)) < tc00`
by rw [max_name_in_state_list_correct] >>
METIS_TAC [example_ceq_not_Completed]
QED
Theorem example_ceq_list_2_not_Completed_list:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl i.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
MEM i (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) ==>
~(Completed_list sem_expr (append_program_state_list stl (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)) i)
Proof
rw [] >>
Cases_on `stl` >>
rename1 `State_st_list il0 s0 fl0 cl0` >>
fs [State_st_list_well_formed_ok] >>
rw [Completed_list_correct,append_program_state_list_correct,GSYM example_ceq_list_2_set] >>
`i IN example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2`
by rw [example_ceq_list_2_set] >>
`max_name_in_State (state_list_to_state (State_st_list il0 s0 fl0 cl0)) < tc00`
by rw [max_name_in_state_list_correct] >>
METIS_TAC [example_ceq_not_Completed]
QED
Theorem example_ceq_list_3_not_Completed_list:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl i.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
MEM i (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) ==>
~(Completed_list sem_expr (append_program_state_list stl (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)) i)
Proof
rw [] >>
Cases_on `stl` >>
rename1 `State_st_list il0 s0 fl0 cl0` >>
fs [State_st_list_well_formed_ok] >>
rw [Completed_list_correct,append_program_state_list_correct,GSYM example_ceq_list_3_set] >>
`i IN example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2`
by rw [example_ceq_list_3_set] >>
`max_name_in_State (state_list_to_state (State_st_list il0 s0 fl0 cl0)) < tc00`
by rw [max_name_in_state_list_correct] >>
METIS_TAC [example_ceq_not_Completed]
QED
Theorem example_ceq_list_4_not_Completed_list:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl i.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
MEM i (example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2) ==>
~(Completed_list sem_expr (append_program_state_list stl (example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)) i)
Proof
rw [] >>
Cases_on `stl` >>
rename1 `State_st_list il0 s0 fl0 cl0` >>
fs [State_st_list_well_formed_ok] >>
rw [Completed_list_correct,append_program_state_list_correct,GSYM example_ceq_list_4_set] >>
`i IN example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2`
by rw [example_ceq_list_4_set] >>
`max_name_in_State (state_list_to_state (State_st_list il0 s0 fl0 cl0)) < tc00`
by rw [max_name_in_state_list_correct] >>
METIS_TAC [example_ceq_not_Completed]
QED
Theorem example_ceq_list_1_not_Completed_list_HD:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
~(Completed_list sem_expr (append_program_state_list stl (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2))
(i_assign tc00 (e_val val_true) (o_internal (e_val val_zero))))
Proof
rw [] >>
`MEM (i_assign tc00 (e_val val_true) (o_internal (e_val val_zero)))
(example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [example_ceq_list_1] >>
METIS_TAC [example_ceq_list_1_not_Completed_list]
QED
Theorem example_ceq_list_2_not_Completed_list_HD:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
~(Completed_list sem_expr (append_program_state_list stl (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2))
(i_assign tc00 (e_val val_true) (o_internal (e_val val_zero))))
Proof
rw [] >>
`MEM (i_assign tc00 (e_val val_true) (o_internal (e_val val_zero)))
(example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [example_ceq_list_2] >>
METIS_TAC [example_ceq_list_2_not_Completed_list]
QED
Theorem example_ceq_list_3_not_Completed_list_HD:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
~(Completed_list sem_expr (append_program_state_list stl (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2))
(i_assign tc00 (e_val val_true) (o_internal (e_val val_zero))))
Proof
rw [] >>
`MEM (i_assign tc00 (e_val val_true) (o_internal (e_val val_zero)))
(example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [example_ceq_list_3] >>
METIS_TAC [example_ceq_list_3_not_Completed_list]
QED
Theorem example_ceq_list_4_not_Completed_list_HD:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
~(Completed_list sem_expr (append_program_state_list stl (example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2))
(i_assign tc00 (e_val val_true) (o_internal (e_val val_zero))))
Proof
rw [] >>
`MEM (i_assign tc00 (e_val val_true) (o_internal (e_val val_zero)))
(example_ceq_list_4 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [example_ceq_list_4] >>
METIS_TAC [example_ceq_list_4_not_Completed_list]
QED
Theorem example_ceq_list_1_t_not_Completed_list_HD:
!t r1 r2 z b1 b2 stl.
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl <= t ==>
~(Completed_list sem_expr
(append_program_state_list stl (example_ceq_list_1_t t r1 r2 z b1 b2))
(i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero))))
Proof
rw [example_ceq_list_1_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_state_list stl < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_list_1_not_Completed_list_HD]
QED
Theorem example_ceq_list_2_t_not_Completed_list_HD:
!t r1 r2 z b1 b2 stl.
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl <= t ==>
~(Completed_list sem_expr
(append_program_state_list stl (example_ceq_list_2_t t r1 r2 z b1 b2))
(i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero))))
Proof
rw [example_ceq_list_2_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_state_list stl < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_list_2_not_Completed_list_HD]
QED
Theorem example_ceq_list_3_t_not_Completed_list_HD:
!t r1 r2 z b1 b2 stl.
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl <= t ==>
~(Completed_list sem_expr
(append_program_state_list stl (example_ceq_list_3_t t r1 r2 z b1 b2))
(i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero))))
Proof
rw [example_ceq_list_3_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_state_list stl < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_list_3_not_Completed_list_HD]
QED
Theorem example_ceq_list_4_t_not_Completed_list_HD:
!t r1 r2 z b1 b2 stl.
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl <= t ==>
~(Completed_list sem_expr
(append_program_state_list stl (example_ceq_list_4_t t r1 r2 z b1 b2))
(i_assign (t+1) (e_val val_true) (o_internal (e_val val_zero))))
Proof
rw [example_ceq_list_4_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_state_list stl < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_list_4_not_Completed_list_HD]
QED
(* ---------------------- *)
(* Well-formedness lemmas *)
(* ---------------------- *)
Theorem example_ceq_compositional_well_formed_state:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 State.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
well_formed_state State ==>
max_name_in_State State < tc00 ==>
well_formed_state (union_program_state State (example_ceq tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2))
Proof
rw [] >>
METIS_TAC [
compositional_program_union_program_state_well_formed,
example_ceq_compositional_program
]
QED
Theorem example_ceq_t_compositional_well_formed_state:
!t r1 r2 z b1 b2 State.
well_formed_state State ==>
max_name_in_State State <= t ==>
well_formed_state (union_program_state State (example_ceq_t t r1 r2 z b1 b2))
Proof
rw [example_ceq_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_State State < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_compositional_well_formed_state]
QED
Theorem example_ceq_list_1_well_formed_ok:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
State_st_list_well_formed_ok
(append_program_state_list stl (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2))
Proof
rw [] >>
`NO_DUPLICATES (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [example_ceq_list_1_NO_DUPLICATES] >>
`compositional_program (set (example_ceq_list_1 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)) (max_name_in_state_list stl)`
by METIS_TAC [example_ceq_compositional_program,example_ceq_list_1_set] >>
METIS_TAC [compositional_program_append_program_state_list_well_formed_ok]
QED
Theorem example_ceq_list_1_t_well_formed_ok:
!t r1 r2 z b1 b2 stl.
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl <= t ==>
State_st_list_well_formed_ok (append_program_state_list stl (example_ceq_list_1_t t r1 r2 z b1 b2))
Proof
rw [example_ceq_list_1_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_state_list stl < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_list_1_well_formed_ok]
QED
Theorem example_ceq_list_2_well_formed_ok:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
State_st_list_well_formed_ok
(append_program_state_list stl (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2))
Proof
rw [] >>
`NO_DUPLICATES (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [example_ceq_list_2_NO_DUPLICATES] >>
`compositional_program (set (example_ceq_list_2 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)) (max_name_in_state_list stl)`
by METIS_TAC [example_ceq_compositional_program,example_ceq_list_2_set] >>
METIS_TAC [compositional_program_append_program_state_list_well_formed_ok]
QED
Theorem example_ceq_list_2_t_well_formed_ok:
!t r1 r2 z b1 b2 stl.
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl <= t ==>
State_st_list_well_formed_ok (append_program_state_list stl (example_ceq_list_2_t t r1 r2 z b1 b2))
Proof
rw [example_ceq_list_2_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_state_list stl < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_list_2_well_formed_ok]
QED
Theorem example_ceq_list_3_well_formed_ok:
!tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2 stl.
tc00 < tc01 /\ tc01 < tc02 /\ tc02 < tc03 /\ tc03 < tc04 /\ tc04 < tc05 /\ tc05 < tc11 /\
tc11 < tc12 /\ tc12 < tc21 /\ tc21 < tc22 /\ tc22 < tc23 /\ tc23 < tc31 /\ tc31 < tc32 /\
tc32 < tc41 /\ tc41 < tc42 /\ tc42 < tc43 ==>
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl < tc00 ==>
State_st_list_well_formed_ok
(append_program_state_list stl (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2))
Proof
rw [] >>
`NO_DUPLICATES (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)`
by rw [example_ceq_list_3_NO_DUPLICATES] >>
`compositional_program (set (example_ceq_list_3 tc00 tc01 tc02 tc03 tc04 tc05 tc11 tc12 tc21 tc22 tc23 tc31 tc32 tc41 tc42 tc43 r1 r2 z b1 b2)) (max_name_in_state_list stl)`
by METIS_TAC [example_ceq_compositional_program,example_ceq_list_3_set] >>
METIS_TAC [compositional_program_append_program_state_list_well_formed_ok]
QED
Theorem example_ceq_list_3_t_well_formed_ok:
!t r1 r2 z b1 b2 stl.
State_st_list_well_formed_ok stl ==>
max_name_in_state_list stl <= t ==>
State_st_list_well_formed_ok (append_program_state_list stl (example_ceq_list_3_t t r1 r2 z b1 b2))
Proof
rw [example_ceq_list_3_t] >>
MP_TAC (Q.SPEC `t` t_and_lt) >>
`max_name_in_state_list stl < t + 1` by DECIDE_TAC >>
METIS_TAC [example_ceq_list_3_well_formed_ok]