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2244 lines (2150 loc) · 63.4 KB
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(*
Formalization of the maximum common induced subgraph problem
*)
Theory mcis
Libs
preamble
Ancestors
pbc graph_basic pbc_normalise
val _ = numLib.temp_prefer_num();
(* Given graphs G_p , G_t
A subset of vertices of G_p is a common induced subgraph in G_t
iff there is an injection from that set into G_t
that preserves adjacency and non-adjacency *)
Definition injective_partial_map_def:
injective_partial_map f vs ((vp,ep):graph) ((vt,et):graph) ⇔
vs ⊆ count vp ∧
INJ f vs (count vt) ∧
(∀a b. a ∈ vs ∧ b ∈ vs ⇒
(is_edge ep a b ⇔ is_edge et (f a) (f b)))
End
(* A cleaner definition *)
Theorem injective_partial_map_eq:
injective_partial_map f vs (vp,ep) (vt,et) ⇔
vs ⊆ count vp ∧ INJ f vs (count vt) ∧
(∀a b. a ∈ vs ∧ b ∈ vs ∧ is_edge ep a b ⇒
is_edge et (f a) (f b)) ∧
(∀a b. a ∈ vs ∧ b ∈ vs ∧ ¬(is_edge ep a b) ⇒
¬ is_edge et (f a) (f b))
Proof
rw[injective_partial_map_def]>>
metis_tac[]
QED
(* vs is a common induced subgraph of gp and gt *)
Definition is_cis_def:
is_cis vs (vp,ep) (vt,et) ⇔
∃f. injective_partial_map f vs (vp,ep) (vt,et)
End
Definition max_cis_size_def:
max_cis_size gp gt =
MAX_SET ({CARD vs | is_cis vs gp gt})
End
(* Vertex a is connected to vertex b with respect to vertices vs and edges e iff a R* b where R* is the RTC over edges *)
Definition is_connected_def:
is_connected vs e a b ⇔
(λx y. y ∈ vs ∧ is_edge e x y)꙳ a b
End
(* The vertex subset vs is connected if vertices are pairwise connected *)
Definition connected_subgraph_def:
connected_subgraph vs e ⇔
∀a b. a ∈ vs ∧ b ∈ vs ⇒ is_connected vs e a b
End
(* vs is a common --connected-- induced subgraph of gp and gt *)
Definition is_ccis_def:
is_ccis vs (vp,ep) (vt,et) ⇔
is_cis vs (vp,ep) (vt,et) ∧
connected_subgraph vs ep
End
Definition max_ccis_size_def:
max_ccis_size gp gt =
MAX_SET ({CARD vs | is_ccis vs gp gt})
End
Definition is_walk_def:
(is_walk ep a b [] ⇔ a = b) ∧
(is_walk ep a b (p::ps) ⇔ (is_edge ep a p ∧ is_walk ep p b ps))
End
(* Construct an explicit walk *)
Theorem is_connected_is_walk:
∀a b.
is_connected vs e a b ⇒
∃walk.
set walk ⊆ vs ∧ is_walk e a b walk
Proof
simp[is_connected_def]>>
ho_match_mp_tac RTC_INDUCT>>rw[]
>-
(qexists_tac`[]`>>simp[is_walk_def])>>
qexists_tac`a'::walk`>>simp[is_walk_def]
QED
Theorem is_walk_is_connected:
∀walk a b.
set walk ⊆ vs ∧ is_walk e a b walk ⇒
is_connected vs e a b
Proof
Induct>>rw[is_walk_def]
>- simp[is_connected_def]>>
fs[]>>
first_x_assum drule>>
simp[is_connected_def]>>
strip_tac>>
simp[Once RTC_CASES1]>>
metis_tac[]
QED
(* Encoding *)
Datatype:
enc =
Walk num num num (* Walk f g k indicates a walk of length 2^k from f to g *)
| Aux num num num num (* Aux f g h k are auxiliaries used to encode Walk *)
| Unmapped num (* x_{f,bot} *)
| Mapped num num (* x_{f,g} *)
End
(* For each a in vp, either a is unassigned or a is assigned to exactly one vertex
v in vt *)
Definition has_mapping_al1_def:
has_mapping_al1 (a:num) vt =
((strlit"al1" ^ toString a)
,(GreaterEqual,
(1, Pos (Unmapped a)) ::
GENLIST (λv. (1, Pos (Mapped a v))) vt,
1): enc pbc)
End
Definition has_mapping_am1_def:
has_mapping_am1 (a:num) vt =
((strlit"am1" ^ toString a)
,(LessEqual,
(1, Pos (Unmapped a)) ::
GENLIST (λv. (1, Pos (Mapped a v))) vt,
1): enc pbc)
End
Definition all_has_mapping_def:
all_has_mapping vp vt =
GENLIST (λa. has_mapping_al1 a vt) vp ++
GENLIST (λa. has_mapping_am1 a vt) vp
End
Definition one_one_def:
one_one u vp =
((strlit"inj" ^ toString u)
,(GreaterEqual,
(GENLIST (λb. (1, Neg (Mapped b u))) vp),
(&vp-1)): enc pbc)
End
Definition all_one_one_def:
all_one_one vp vt =
GENLIST (λu. one_one u vp) vt
End
Definition edge_map_def:
edge_map (a,b) u et =
if a = b then [] else
[
(concat [strlit"adj"; toString a; strlit"_"; toString u; strlit"_"; toString b]
,(GreaterEqual,
(1,Neg (Mapped a u)) ::
(1,Pos (Unmapped b)) ::
MAP (λv. (1,Pos (Mapped b v))) (neighbours et u),
1):enc pbc)]
End
Definition not_edge_map_def:
not_edge_map (a,b) u vt et =
if a = b then []
else
[
(concat [strlit"adj"; toString a; strlit"_"; toString u; strlit"_"; toString b]
,(GreaterEqual,
(1,Neg (Mapped a u)) ::
(1,Pos (Unmapped b)) ::
MAP (λv. (1,Pos (Mapped b v))) (not_neighbours (vt,et) u),
1):enc pbc)]
End
Definition all_full_edge_map_def:
all_full_edge_map (vp,ep) (vt,et) =
FLAT (GENLIST (λu.
FLAT (GENLIST (λa.
(* Check that a,u have same self-loop *)
if is_edge ep a a ⇎ is_edge et u u
then [
(concat [strlit"adj"; toString a; strlit"_"; toString u; strlit"_SELF"],
(GreaterEqual, [(1,Neg (Mapped a u))], 1):enc pbc)]
else
FLAT (MAP (λb. edge_map (a,b) u et) (neighbours ep a)) ++
FLAT (MAP (λb. not_edge_map (a,b) u vt et) (not_neighbours (vp,ep) a) )) vp)) vt)
End
(* Objective is to minimize the number of unmapped vertices *)
Definition unmapped_obj_def:
unmapped_obj vp =
SOME((GENLIST (λb. (1, Pos (Unmapped b))) vp), 0)
: ((enc lin_term # int) option)
End
Definition encode_base_def:
encode_base (vp,ep) (vt,et) =
all_has_mapping vp vt ++
all_one_one vp vt ++
all_full_edge_map (vp,ep) (vt,et)
End
Theorem b2i_geq_1[simp]:
b2i b ≥ 1 ⇔ b
Proof
Cases_on`b`>>fs[]
QED
Theorem b2i_eq_1[simp]:
b2i b = 1 ⇔ b
Proof
Cases_on`b`>>fs[]
QED
Theorem b2i_eq_0[simp]:
b2i b = 0 ⇔ ¬b
Proof
Cases_on`b`>>fs[]
QED
Theorem neg_b2i_eq_1[simp]:
1 - b2i b = 1 ⇔ ¬b
Proof
Cases_on`b`>>fs[]
QED
Theorem b2i_geq_zero[simp]:
b2i b ≥ 0
Proof
Cases_on`b`>>
simp[]
QED
Theorem b2i_add_one_geq_one[simp]:
1+ b2i b ≥ 1
Proof
Cases_on`b`>>
simp[]
QED
Theorem iSUM_zero:
(∀x. MEM x ls ⇒ x ≥ 0) ⇒
iSUM ls ≥ 0
Proof
Induct_on`ls`>> rw[iSUM_def]>>
fs[]>>
first_x_assum(qspec_then`h` assume_tac)>>
fs[]>>
intLib.ARITH_TAC
QED
Theorem iSUM_geq:
∀ls.
(∀x. MEM x ls ⇒ x ≥ 0) ∧
(∃x. MEM x ls ∧ x ≥ n)
⇒
iSUM ls ≥ n
Proof
Induct>>rw[iSUM_def]
>- (
`iSUM ls ≥ 0` by
(irule iSUM_zero>>
metis_tac[])>>
intLib.ARITH_TAC)>>
gs[]>>
last_x_assum mp_tac>>
impl_tac >- metis_tac[]>>
first_x_assum(qspec_then`h` assume_tac)>>
fs[]>>
intLib.ARITH_TAC
QED
Theorem b2i_iSUM_eq_0:
(∀x. MEM x ls ⇒ ∃y. x = b2i y) ⇒
(iSUM ls = 0 ⇔
∀j. j < LENGTH ls ⇒ EL j ls = 0)
Proof
Induct_on`ls`>>rw[iSUM_def]>>fs[]>>
first_assum(qspec_then`h` assume_tac)>>fs[]>>
Cases_on`y`>>fs[]
>- (
`iSUM ls ≥ 0` by (
match_mp_tac iSUM_zero>>
rw[]>>res_tac>>
rw[])>>
rw[EQ_IMP_THM]
>- (
last_x_assum kall_tac>>
intLib.COOPER_TAC)>>
pop_assum(qspec_then`0` mp_tac)>>simp[])
>>
rw[EQ_IMP_THM]
>-
(Cases_on`j`>>fs[])>>
first_x_assum (qspec_then`SUC j` mp_tac)>>fs[]
QED
Theorem iSUM_geq_1:
iSUM ls ≥ 1 /\
(∀x. MEM x ls ⇒ ∃y. x = b2i y) ⇒
∃i. i < LENGTH ls ∧ EL i ls = 1
Proof
Induct_on`ls`>>rw[iSUM_def]>>fs[]>>
first_assum(qspec_then`h` assume_tac)>>fs[]>>
Cases_on`y`>>fs[]>>
simp[]
>- (qexists_tac`0`>>rw[])>>
qexists_tac`SUC i`>>rw[]
QED
Theorem iSUM_eq_1:
(∀x. MEM x ls ⇒ ∃y. x = b2i y) ⇒
(iSUM ls = 1 ⇔
∃i. i < LENGTH ls ∧ EL i ls = 1 ∧
∀j. i ≠ j ∧ j < LENGTH ls ⇒ EL j ls = 0)
Proof
Induct_on`ls`>>rw[iSUM_def]>>fs[]>>
first_assum(qspec_then`h` assume_tac)>>fs[]>>
Cases_on`y`>>fs[]>>
simp[]
>-(
DEP_REWRITE_TAC[b2i_iSUM_eq_0]>>
CONJ_TAC >- metis_tac[]>>
rw[EQ_IMP_THM]
>- (
qexists_tac`0`>>rw[]>>
Cases_on`j`>>fs[])>>
`i = 0` by
(CCONTR_TAC>>
first_x_assum drule>>fs[])>>
first_x_assum(qspec_then`SUC j` assume_tac)>>rfs[]>>
Cases_on`i`>>fs[])
>>
rw[EQ_IMP_THM]
>- (
qexists_tac`SUC i`>>rw[]>>
Cases_on`j`>>fs[])>>
Cases_on`i`>>fs[]>>
qexists_tac`n`>>rw[]>>
first_x_assum(qspec_then`SUC j` assume_tac)>>fs[]
QED
Theorem iSUM_sub_b2i_geq_0':
(∀x. MEM x ls ⇒ ∃y. x = 1 - b2i y) ⇒
iSUM ls ≤ &(LENGTH ls)
Proof
Induct_on`ls`>>fs[iSUM_def]>>rw[]>>
first_assum(qspec_then`h` assume_tac)>>fs[]>>
Cases_on`y`>>fs[ADD1]>>
intLib.ARITH_TAC
QED
Theorem iSUM_sub_b2i_geq_0:
(∀x. MEM x ls ⇒ ∃y. x = 1 - b2i y) ⇒
(iSUM ls ≥ &(LENGTH ls) ⇔
∀i. i < LENGTH ls ⇒ EL i ls = 1)
Proof
Induct_on`ls`>>
fs[iSUM_def]>>rw[]>>
first_assum(qspec_then`h` assume_tac)>>fs[]>>
Cases_on`y`>>fs[]
>- (
`iSUM ls ≤ &(LENGTH ls)` by
metis_tac[iSUM_sub_b2i_geq_0']>>
rw[EQ_IMP_THM]
>-
(last_x_assum kall_tac>>intLib.ARITH_TAC)>>
first_x_assum(qspec_then`0` assume_tac)>>fs[])>>
simp[ADD1,GSYM integerTheory.INT_ADD,intLib.COOPER_PROVE``!n:int. 1 + n ≥ m + 1 ⇔ n ≥ m``]>>
rw[EQ_IMP_THM]
>-
(Cases_on`i`>>fs[])>>
first_x_assum(qspec_then`SUC i` assume_tac)>>fs[]
QED
Theorem iSUM_sub_b2i_geq:
(∀x. MEM x ls ⇒ ∃y. x = 1 - b2i y) ⇒
(iSUM ls ≥ &(LENGTH ls) − 1 ⇔
∀i. i < LENGTH ls ∧ EL i ls = 0 ⇒
∀j. i ≠ j ∧ j < LENGTH ls ⇒ EL j ls = 1)
Proof
Induct_on`ls`>>fs[iSUM_def]>>rw[]>>
simp[ADD1]>>
first_assum(qspec_then`h` assume_tac)>>fs[]>>
Cases_on`y`>>fs[]>>
simp[GSYM integerTheory.INT_ADD,intLib.COOPER_PROVE``!n:int. n +1 -1 = n``]
>- (
DEP_REWRITE_TAC[iSUM_sub_b2i_geq_0] >>
CONJ_TAC >- metis_tac[]>>
rw[EQ_IMP_THM]
>- (
Cases_on`j`>>fs[]>>
Cases_on`i`>>fs[ADD1]>>
first_x_assum drule>>fs[])>>
first_x_assum(qspec_then`0` assume_tac)>>gs[]>>
first_x_assum(qspec_then`SUC i` assume_tac)>>gs[])>>
simp[intLib.COOPER_PROVE``!n:int. 1 + n ≥ m ⇔ n ≥ m - 1``]>>
rw[EQ_IMP_THM]
>- (
Cases_on`i`>>fs[ADD1]>>
first_x_assum drule>>simp[]>>
Cases_on`j`>>fs[])>>
first_x_assum(qspec_then`SUC i` assume_tac)>>gs[]>>
first_x_assum(qspec_then`SUC j` assume_tac)>>gs[]
QED
Theorem iSUM_GENLIST_const:
∀vt c.
iSUM (GENLIST (λv. c) vt) = c * &vt
Proof
Induct>>simp[iSUM_def,GENLIST_CONS,o_DEF]>>
intLib.ARITH_TAC
QED
Theorem iSUM_MAP_const:
∀ls c. iSUM (MAP (λv. c) ls) = c * &(LENGTH ls)
Proof
Induct>>simp[iSUM_def]>>
intLib.ARITH_TAC
QED
Theorem iSUM_SNOC:
∀ls.
iSUM (SNOC x ls) = x + iSUM ls
Proof
Induct>>rw[iSUM_def]>>
intLib.ARITH_TAC
QED
Theorem iSUM_GENLIST_eq_k:
∀vp vs k.
vs ⊆ count vp ⇒
iSUM (GENLIST (λb. b2i (b ∉ vs)) vp) = &vp - &CARD vs
Proof
Induct>>rw[iSUM_def]>>
reverse (Cases_on`vp ∈ vs`>>fs[GENLIST,iSUM_SNOC])
>- (
first_x_assum(qspec_then`vs` mp_tac)>>
impl_tac>- (
fs[SUBSET_DEF]>>
rw[]>>
first_x_assum drule>>fs[prim_recTheory.LESS_THM]>>
metis_tac[])>>
rw[]>>
intLib.ARITH_TAC)>>
first_x_assum(qspecl_then[`vs DIFF{vp}`] mp_tac)>>
impl_tac>- (
fs[SUBSET_DEF]>>rw[]>>
first_x_assum drule>>fs[prim_recTheory.LESS_THM])>>
rw[]>>
`(GENLIST (λb. b2i (b ∉ vs ∨ b = vp)) vp) =
(GENLIST (λb. b2i (b ∉ vs)) vp)` by
(match_mp_tac GENLIST_CONG>>fs[])>>
gvs[] >>
`FINITE vs` by (
match_mp_tac SUBSET_FINITE_I>>
qexists_tac`count (SUC vp)`>>
fs[SUBSET_DEF])>>
`CARD (vs DIFF {vp}) = CARD vs - 1` by
(DEP_REWRITE_TAC[CARD_DIFF]>>
`vs ∩ {vp} = {vp}` by
(simp[EXTENSION]>>metis_tac[])>>
simp[])>>
rw[]>>
`CARD vs > 0` by
(Cases_on`vs`>>rw[]>>gvs[EXTENSION])>>
Cases_on`CARD vs`>>fs[]>>
intLib.ARITH_TAC
QED
Theorem neg_b2i:
1 - b2i p = b2i (~ p)
Proof
Cases_on`p`>>simp[]
QED
Theorem MEM_if:
MEM x (if P then A else B) ⇔
if P then MEM x A else MEM x B
Proof
rw[]
QED
Theorem encode_base_correct:
good_graph (vp,ep) ∧
good_graph (vt,et) ∧
encode_base (vp,ep) (vt,et) = constraints ⇒
(
(∃f vs.
injective_partial_map f vs (vp,ep) (vt,et) ∧
CARD vs = k) ⇔
(∃w.
satisfies w (set (MAP SND constraints)) ∧
eval_obj (unmapped_obj vp) w = &vp - &k)
)
Proof
rw[EQ_IMP_THM]
>- (
fs[injective_partial_map_eq]>>
simp[satisfiable_def]>>
qexists_tac`λenc.
case enc of
Unmapped a => a ∉ vs
| Mapped a u => a ∈ vs ∧ f a = u
| _ => ARB` >>
rw[encode_base_def]
>- (
rename1`all_has_mapping`>>
simp[all_has_mapping_def,satisfies_def,MEM_GENLIST,MEM_MAP]>>
`∀a. a < vp ∧ a ∈ vs ⇒
iSUM (GENLIST (λv. b2i (f a = v)) vt) = 1` by (
rw[]>>
DEP_REWRITE_TAC[iSUM_eq_1,eval_lin_term_def]>>
CONJ_TAC>-
(simp[MEM_GENLIST]>>metis_tac[])>>
qexists_tac`f a`>>
CONJ_ASM1_TAC>>fs[EL_GENLIST,INJ_DEF])>>
rw[]>>
simp[satisfies_pbc_def,MAP_GENLIST,o_DEF,eval_lin_term_def,
has_mapping_al1_def,has_mapping_am1_def]>>
Cases_on`a ∈ vs`>>simp[iSUM_def,iSUM_GENLIST_const])
>- (
rename1`all_one_one`>>
simp[all_one_one_def,satisfies_def,MEM_GENLIST,one_one_def,MEM_MAP,PULL_EXISTS]>>
rw[]>>
simp[satisfies_pbc_def,MAP_GENLIST,o_DEF,eval_lin_term_def]>>
fs[INJ_DEF]>>
qmatch_goalsub_abbrev_tac`iSUM ls`>>
`vp = LENGTH ls` by
simp[Abbr`ls`]>>
simp[]>>
DEP_REWRITE_TAC[iSUM_sub_b2i_geq]>>
simp[Abbr`ls`]>>
CONJ_TAC>- (
simp[MEM_GENLIST]>>
metis_tac[])>>
rw[]>>
gs[EL_GENLIST]>>
metis_tac[])
>- (
rename1`all_full_edge_map`>>
simp[all_full_edge_map_def,satisfies_def,MEM_GENLIST,MEM_FLAT]>>
rw[]>>
gvs[MEM_FLAT,MEM_GENLIST,MEM_MAP]>>
pop_assum mp_tac>>
qmatch_goalsub_abbrev_tac`if P then _ else _`>>
IF_CASES_TAC
>- (
rw[Abbr`P`]>>
simp[satisfies_pbc_def,iSUM_def,eval_lin_term_def]>>
Cases_on`a ∈ vs`>>simp[iSUM_def]>>
`f a ≠ u` by metis_tac[MEM_neighbours]>>
simp[])>>
simp[MEM_FLAT,MEM_MAP,PULL_EXISTS,MEM_if]>>
rw[]
>- (
(* edge_map constraint *)
gvs[edge_map_def,MEM_if,MEM_neighbours]>>
simp[satisfies_pbc_def,MAP_MAP_o,o_DEF,eval_lin_term_def]>>
`b < vp` by
(fs[good_graph_eq,is_edge_thm]>>
metis_tac[])>>
simp[]>>
reverse (Cases_on`b ∈ vs`)>>fs[]
>- (
simp[iSUM_def,iSUM_MAP_const]>>
Cases_on`a ∈ vs ∧ f a = u`>>simp[])>>
reverse (Cases_on`f a = u`>>rw[]>>simp[iSUM_def])
>- (
simp[intLib.COOPER_PROVE``!n:int. 1 + n ≥ 1 ⇔ n ≥ 0``]>>
match_mp_tac iSUM_zero>>
simp[MEM_MAP,MEM_neighbours]>>
rw[]>>
simp[])>>
Cases_on`a ∈ vs`>>fs[]
>- (
match_mp_tac iSUM_geq>>
rw[]
>-
(fs[MEM_MAP]>>pairarg_tac>>simp[])>>
simp[MEM_MAP,MEM_FILTER,LAMBDA_PROD,PULL_EXISTS,EXISTS_PROD,MEM_neighbours]>>
qexists_tac`f b`>>simp[]>>
fs[INJ_DEF])>>
simp[intLib.COOPER_PROVE``!n:int. 1 + n ≥ 1 ⇔ n ≥ 0``]>>
match_mp_tac iSUM_zero>>
simp[MEM_MAP,MEM_neighbours]>>
rw[]>>
simp[])
>- (
(* not_edge_map constraint *)
gvs[not_edge_map_def,MEM_if,MEM_not_neighbours]>>
simp[satisfies_pbc_def,MAP_MAP_o,o_DEF,eval_lin_term_def]>>
reverse (Cases_on`b ∈ vs`)>>fs[]
>- (
simp[iSUM_def,iSUM_MAP_const]>>
Cases_on`a ∈ vs ∧ f a = u`>>simp[])>>
reverse (Cases_on`f a = u`>>rw[]>>simp[iSUM_def])
>- (
simp[intLib.COOPER_PROVE``!n:int. 1 + n ≥ 1 ⇔ n ≥ 0``]>>
match_mp_tac iSUM_zero>>
simp[MEM_MAP,MEM_not_neighbours]>>
rw[]>>
simp[])>>
Cases_on`a ∈ vs`>>fs[]
>- (
match_mp_tac iSUM_geq>>
rw[]
>-
(fs[MEM_MAP]>>pairarg_tac>>simp[])>>
simp[MEM_MAP,MEM_FILTER,LAMBDA_PROD,PULL_EXISTS,EXISTS_PROD,MEM_not_neighbours]>>
fs[INJ_DEF])>>
simp[intLib.COOPER_PROVE``!n:int. 1 + n ≥ 1 ⇔ n ≥ 0``]>>
match_mp_tac iSUM_zero>>
simp[MEM_MAP,MEM_not_neighbours]>>
rw[]>>
simp[]))
>- (
simp[eval_obj_def,unmapped_obj_def,MAP_GENLIST, o_DEF,eval_lin_term_def]>>
DEP_REWRITE_TAC[iSUM_GENLIST_eq_k]>>
fs[])
)>>
fs[satisfiable_def,injective_partial_map_eq]>>
qexists_tac`λn. @m. m < vt ∧ w (Mapped n m)`>>
qabbrev_tac`dom = {n | n < vp ∧ ¬ w (Unmapped n)}`>>
qexists_tac `dom`>>
simp[]>>
reverse CONJ_TAC >-
(
fs[eval_obj_def,unmapped_obj_def,MAP_GENLIST,o_DEF,neg_b2i,eval_lin_term_def]>>
qpat_x_assum`_ = _` mp_tac>>
`GENLIST (λb. b2i (w (Unmapped b))) vp =
GENLIST (λb. b2i (b ∉ dom)) vp` by
(match_mp_tac GENLIST_CONG>>rw[Abbr`dom`])>>
simp[]>>
DEP_REWRITE_TAC[iSUM_GENLIST_eq_k]>>
rw[]
>-
fs[Abbr`dom`,SUBSET_DEF]>>
intLib.ARITH_TAC)>>
CONJ_TAC>-
simp[Abbr`dom`,SUBSET_DEF]>>
fs[satisfies_def,encode_base_def,SF DNF_ss]>>
`∀n. n < vp ∧ ¬w (Unmapped n) ⇒
∃m. m < vt ∧ w (Mapped n m) ∧
∀m'. m' < vt ∧ w (Mapped n m') ⇔ m = m'` by (
fs[all_has_mapping_def,MEM_GENLIST,has_mapping_al1_def,
has_mapping_am1_def,PULL_EXISTS,MEM_MAP,PULL_EXISTS,SF DNF_ss]>>
rw[]>>
first_x_assum drule>>
first_x_assum drule>>
simp[satisfies_pbc_def,MAP_GENLIST,o_DEF,eval_lin_term_def]>>
simp[iSUM_def]>>
rw[]>>
`iSUM (GENLIST (λv. b2i (w (Mapped n v))) vt) = 1` by intLib.ARITH_TAC>>
pop_assum mp_tac>>
DEP_REWRITE_TAC[iSUM_eq_1]>>
CONJ_TAC>-
(simp[MEM_GENLIST]>>metis_tac[])>>
rw[]>>gs[EL_GENLIST]>>
asm_exists_tac>>fs[]>>
CCONTR_TAC>>gs[]>>
Cases_on`i=m'`>>gs[]>>
first_x_assum drule>>
fs[])>>
rw[]
>- (
fs[Abbr`dom`]>>
rw[INJ_DEF]
>- (
first_x_assum drule>>strip_tac>>
rfs[])>>
fs[all_one_one_def,MEM_GENLIST,PULL_EXISTS,one_one_def,MEM_MAP]>>
res_tac>>
gvs[]>>
last_x_assum drule>>
simp[satisfies_pbc_def,MAP_GENLIST,o_DEF,eval_lin_term_def]>>
qmatch_goalsub_abbrev_tac`iSUM ls`>>
`vp = LENGTH ls` by
simp[Abbr`ls`]>>
simp[]>>
DEP_REWRITE_TAC[iSUM_sub_b2i_geq]>>
simp[Abbr`ls`]>>
CONJ_TAC>- (
simp[MEM_GENLIST]>>
metis_tac[])>>
rw[]>>
first_x_assum drule>>
simp[EL_GENLIST]>>
disch_then(qspec_then`n` mp_tac)>>
simp[])
>- (
fs[Abbr`dom`,good_graph_eq]>>
first_assum(qspec_then`a` mp_tac)>>
first_x_assum(qspec_then`b` drule)>>
simp[]>> rw[]>>
gvs[]>>
fs[all_full_edge_map_def,satisfies_def,MEM_GENLIST,MEM_FLAT,edge_map_def,PULL_EXISTS,MEM_MAP,FORALL_PROD]>>
`is_edge ep b a` by
fs[is_edge_thm]>>
first_x_assum (drule_at (Pos (el 2)))>>
qpat_x_assum`m < _` assume_tac>>
disch_then drule>>
qmatch_goalsub_abbrev_tac`if P then _ else _`>>
IF_CASES_TAC
>- (
fs[Abbr`P`]>>
simp[satisfies_pbc_def,iSUM_def,eval_lin_term_def])>>
simp[MEM_FLAT,MEM_MAP,MEM_neighbours,SF DNF_ss,MEM_if]>>
Cases_on` b = a` >-
metis_tac[]>>
strip_tac>> pop_assum kall_tac>>
pop_assum drule>>
simp[satisfies_pbc_def,iSUM_def,MAP_MAP_o,o_DEF,LAMBDA_PROD,MEM_neighbours,eval_lin_term_def]>>
strip_tac>>
gs[]>>
drule iSUM_geq_1>>
simp[MEM_MAP,PULL_EXISTS,MEM_FILTER,FORALL_PROD]>>
impl_tac >- metis_tac[]>>
strip_tac>>
gs[EL_MAP]>>
qmatch_asmsub_abbrev_tac`Mapped _ ee`>>
`m' = ee` by (
unabbrev_all_tac>>
metis_tac[MEM_EL,MEM_neighbours,is_edge_thm])>>
rw[]>>
`MEM ee (neighbours et m)` by
metis_tac[EL_MEM,Abbr`ee`]>>
fs[MEM_neighbours]>>
metis_tac[is_edge_thm])
>- (
fs[Abbr`dom`,good_graph_eq]>>
first_assum(qspec_then`a` mp_tac)>>
first_x_assum(qspec_then`b` drule)>>
simp[]>> rw[]>>
gvs[]>>
fs[all_full_edge_map_def,satisfies_def,MEM_FLAT,MEM_GENLIST,PULL_EXISTS,MEM_MAP,MEM_not_neighbours,not_edge_map_def]>>
first_x_assum (drule_at (Pos (el 1)))>>
qpat_x_assum`a < vp` assume_tac>>
disch_then drule>>simp[]>>
qmatch_goalsub_abbrev_tac`if P then _ else _`>>
IF_CASES_TAC
>- (
fs[Abbr`P`]>>
simp[satisfies_pbc_def,iSUM_def,eval_lin_term_def])>>
simp[MEM_FLAT,MEM_MAP,MEM_not_neighbours,SF DNF_ss,MEM_if]>>
Cases_on` b = a` >-
metis_tac[]>>
strip_tac>>
pop_assum (drule_at (Pos (el 2)))>>
simp[satisfies_pbc_def,iSUM_def,MAP_MAP_o,o_DEF,LAMBDA_PROD,eval_lin_term_def]>>
strip_tac>>
drule iSUM_geq_1>>
simp[MEM_MAP,PULL_EXISTS,MEM_FILTER,FORALL_PROD]>>
impl_tac >- metis_tac[]>>
strip_tac>>
gs[EL_MAP]>>
qmatch_asmsub_abbrev_tac`Mapped b ee`>>
`m = ee` by (
unabbrev_all_tac>>
metis_tac[MEM_EL,MEM_not_neighbours])>>
rw[]>>
`MEM ee (not_neighbours (vt,et) m')` by
metis_tac[EL_MEM,Abbr`ee`]>>
fs[MEM_not_neighbours])
QED
Theorem satisfies_pbc_geq_1:
satisfies_pbc w (GreaterEqual,xs, 1) ∧
EVERY ($= 1) (MAP FST xs) ⇒
∃x. MEM x (MAP SND xs) ∧ eval_lit w x = 1
Proof
rw[satisfies_pbc_def,eval_lin_term_def]>>
drule iSUM_geq_1>>
impl_tac>- (
fs[EVERY_MEM,MEM_MAP,PULL_EXISTS,EXISTS_PROD]>>
strip_tac>>Cases>>simp[]>>
strip_tac>>first_x_assum drule>>simp[]
>- metis_tac[]>>
rw[]>>
qexists_tac`(¬ (w a))`>>
Cases_on`w a`>>simp[])>>
rw[]>>
rfs[EL_MAP]>>
qexists_tac `SND (EL i xs)`>>fs[MEM_MAP]>>
Cases_on`EL i xs`>>rfs[EVERY_MEM,MEM_MAP]
>- metis_tac[EL_MEM,SND]>>
fs[PULL_EXISTS]>>
first_x_assum(qspec_then`EL i xs` mp_tac)>>fs[]>>
metis_tac[MEM_EL]
QED
(* Encode variable x <-> y_1 ∧ y_2 ...., where y_i are literals *)
Definition iff_and_def:
iff_and x ys =
(GreaterEqual,(1, Pos x)::MAP (λy.(1, negate y)) ys,1):'a pbc ::
MAP (λy.
(GreaterEqual, [(1, Neg x); (1,y)], 1)) ys
End
Theorem eval_lit_negate:
eval_lit w (negate x) = 1 - eval_lit w x
Proof
Cases_on`x`>>simp[]
QED
Theorem iff_and:
satisfies w (set (iff_and x ys)) ⇒
(w x ⇔ EVERY (λy. eval_lit w y = 1) ys)
Proof
rw[iff_and_def]>>
fs[satisfies_pbc_def,satisfies_def,MEM_MAP,PULL_EXISTS,eval_lin_term_def]>>
rw[EQ_IMP_THM]
>- (
rw[EVERY_MEM]>>first_x_assum drule>>
Cases_on`y`>>Cases_on`w a`>>simp[iSUM_def])>>
drule iSUM_geq_1>>
impl_tac>- (
rw[MEM_MAP]>- metis_tac[]>>
Cases_on`y'`>>simp[]
>- (
qexists_tac`(¬ (w a))`>>
Cases_on`w a`>>simp[])>>
metis_tac[])>>
rw[]>>Cases_on`i`>>
gs[MAP_MAP_o,EL_MAP]>>
fs[EVERY_EL,eval_lit_negate]>>
first_x_assum drule>>
rw[]>>gvs[]
QED
(* Encode variable x <-> y_1 ∨ y_2 ...., where y_i are literals *)
Definition iff_or_def:
iff_or x ys =
(GreaterEqual, (1, Neg x)::MAP (λy.(1, y)) ys, 1):'a pbc ::
MAP (λy.
(GreaterEqual, [(1, Pos x); (1, negate y)], 1)) ys
End
Theorem iff_or:
satisfies w (set (iff_or x ys)) ⇒
(w x ⇔ EXISTS (λy. eval_lit w y = 1) ys)
Proof
rw[iff_or_def]>>
fs[satisfies_pbc_def,satisfies_def,MEM_MAP,PULL_EXISTS,eval_lin_term_def]>>
rw[EQ_IMP_THM]
>- (
drule iSUM_geq_1>>
impl_tac>- (
rw[] >- (qexists_tac`F`>>simp[])>>
gvs[MEM_MAP]>>
Cases_on`y'`>>simp[]
>- metis_tac[]>>
qexists_tac`¬ (w a)`>>
Cases_on`w a`>>simp[])>>
rw[]>>Cases_on`i`>>gs[]>>
gs[MAP_MAP_o,EL_MAP]>>
simp[EXISTS_MEM,MEM_EL]>>
metis_tac[EL_MEM])>>
fs[EXISTS_MEM]>>
first_x_assum drule>>simp[iSUM_def,eval_lit_negate]>>
Cases_on`w x`>>simp[]
QED
Theorem iff_or_satisfies:
(w x ⇔ EXISTS (λy. eval_lit w y = 1) ys) ⇒
satisfies w (set (iff_or x ys))
Proof
rw[iff_or_def]>>
fs[satisfies_pbc_def,satisfies_def,MEM_MAP,PULL_EXISTS,eval_lin_term_def]>>
rw[]
>- (
Cases_on`w x`>>gs[iSUM_def]
>- (
fs[EXISTS_MEM]>>
match_mp_tac iSUM_geq>>simp[MEM_MAP,PULL_EXISTS]>>
first_x_assum (irule_at Any)>>rw[]>>
Cases_on`y'`>>Cases_on`w a`>>simp[])>>
qmatch_goalsub_abbrev_tac`b2i A`>>
`~A` by simp[Abbr`A`,NOT_EXISTS]>>
simp[intLib.COOPER_PROVE``!n:int. 1 + n ≥ 1 ⇔ n ≥ 0``]>>
match_mp_tac iSUM_zero>>
simp[MEM_MAP]>>
rw[]>>
simp[]>>
Cases_on`y'`>>Cases_on`w a`>>simp[])>>
Cases_on`w x`>>gs[iSUM_def]
>-
(Cases_on`y`>>Cases_on`w a`>>simp[])>>
fs[EVERY_MEM]>>
first_x_assum drule>>
Cases_on`y`>>Cases_on`w a`>>simp[]>>
qmatch_goalsub_abbrev_tac`b2i A`>>Cases_on`A`>>simp[]
QED
(* encoding for the base case f-g *)
Definition walk_base_def:
walk_base ep f g =
if is_edge ep f g then
(* x_{f,g}^1 <-> !x_f,bot ∧ !x_g,bot *)
iff_and (Walk f g 0) [Neg (Unmapped f); Neg (Unmapped g)]
else
[(GreaterEqual, [(1,Neg (Walk f g 0))], 1): enc pbc]
End
Definition walk_aux_def:
walk_aux f g h k =
if f = h ∨ g = h then [] (* Ignore trivial cases *)
else
(* x_{f,h,g}^k <-> x_{h,g}^{k-1} *)
if g < h then
(* f < g < h *)
iff_and (Aux f h g k) [Pos (Walk f h (k-1)) ; Pos (Walk g h (k-1))]
else if h < f then
(* h < f < g *)
iff_and (Aux f h g k) [Pos (Walk h f (k-1)) ; Pos (Walk h g (k-1))]
else
(* f < h < g *)
iff_and (Aux f h g k) [Pos (Walk f h (k-1)) ; Pos (Walk h g (k-1))]
End
Definition walk_ind_def:
walk_ind f g k vp =
iff_or (Walk f g k)
(Pos (Walk f g (k-1)) ::
FLAT (GENLIST (λh.
if f = h ∨ g = h then [] else [Pos (Aux f h g k)]) vp))
End
(* x_{f,g}^k <-> x_{f,g}^{k-1} ∨ x_{f,h,g}^k *)
Definition walk_k_def:
(walk_k (vp,ep) 0 =
FLAT (GENLIST (λf.
FLAT (GENLIST (λg.
if f < g then
walk_base ep f g
else []) vp)) vp)) ∧
(walk_k (vp,ep) k =
FLAT (GENLIST (λf.
FLAT (GENLIST (λg.
if f < g then
FLAT (GENLIST (λh.
walk_aux f g h k) vp) ++
walk_ind f g k vp
else [])
vp)) vp) ++
walk_k (vp,ep) (k-1)
)
End
Theorem is_walk_append:
∀walk a b c.
is_walk ep a b walk ∧
is_walk ep b c walk' ⇒
is_walk ep a c (walk++walk')
Proof
Induct>>
rw[is_walk_def]>>
metis_tac[]
QED
Theorem is_walk_SNOC:
∀walk a b c.
is_walk e a b walk ∧
is_edge e b c ⇒
is_walk e a c (SNOC c walk)
Proof
Induct>>rw[is_walk_def]>>
first_x_assum match_mp_tac>>
metis_tac[]
QED
Theorem good_graph_is_walk_REVERSE:
∀walk a b.
good_graph (v,e) ∧
is_walk e a b walk ⇒
is_walk e b a (TL (REVERSE (a::walk)))
Proof
Induct>>rw[is_walk_def]>>
fs[]>>
first_x_assum drule>>
strip_tac>>
drule is_walk_SNOC>>
`is_edge e h a` by
(fs[good_graph_eq,is_edge_def]>>
Cases_on`lookup a e`>>fs[])>>
disch_then drule>>
simp[SNOC_APPEND]>>
Cases_on`REVERSE walk`>>simp[]
QED
Theorem is_walk_TAKE:
∀walk i a b.
i < LENGTH walk ∧
is_walk ep a b walk ⇒
is_walk ep a (EL i walk) (TAKE (i+1) walk)
Proof