(**************************************************************************)
include "delayed_updating/syntax/path.ma".
-include "delayed_updating/notation/functions/class_c_1.ma".
+include "delayed_updating/notation/functions/class_c_2.ma".
+include "ground/arith/nat_plus_pred.ma".
include "ground/lib/subset.ma".
-
-include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/syntax/path_height.ma".
+include "ground/lib/bool_and.ma".
+include "ground/generated/insert_eq_1.ma".
+include "ground/xoa/ex_3_2.ma".
(* CLOSED CONDITION FOR PATH ************************************************)
-axiom pcc: relation2 nat path.
+inductive pcc (o): relation2 nat path ≝
+| pcc_empty:
+ pcc o (𝟎) (𝐞)
+| pcc_d_dx (p) (n) (k):
+ (Ⓣ = o → n = ↑↓n) →
+ pcc o (n+ninj k) p → pcc o n (p◖𝗱k)
+| pcc_m_dx (p) (n):
+ pcc o n p → pcc o n (p◖𝗺)
+| pcc_L_dx (p) (n):
+ pcc o n p → pcc o (↑n) (p◖𝗟)
+| pcc_A_dx (p) (n):
+ pcc o n p → pcc o n (p◖𝗔)
+| pcc_S_dx (p) (n):
+ pcc o n p → pcc o n (p◖𝗦)
+.
interpretation
"closed condition (path)"
- 'ClassC n = (pcc n).
+ 'ClassC o n = (pcc o n).
+
+(* Advanced constructions ***************************************************)
+
+lemma pcc_false_d_dx (p) (n) (k:pnat):
+ p ϵ 𝐂❨Ⓕ,n+k❩ → p◖𝗱k ϵ 𝐂❨Ⓕ,n❩.
+#p #n #k #H0
+@pcc_d_dx [| // ]
+#H0 destruct
+qed.
+
+lemma pcc_true_d_dx (p) (n:pnat) (k:pnat):
+ p ϵ 𝐂❨Ⓣ,n+k❩ → p◖𝗱k ϵ 𝐂❨Ⓣ,n❩.
+/2 width=1 by pcc_d_dx/
+qed.
+
+(* Basic inversions ********************************************************)
+
+lemma pcc_inv_empty (o) (n):
+ (𝐞) ϵ 𝐂❨o,n❩ → 𝟎 = n.
+#o #n @(insert_eq_1 … (𝐞))
+#x * -n //
+#p #n [ #k #_ ] #_ #H0 destruct
+qed-.
+
+(**) (* alias *)
+alias symbol "DownArrow" (instance 4) = "predecessor (non-negative integers)".
+alias symbol "UpArrow" (instance 3) = "successor (non-negative integers)".
+alias symbol "and" (instance 1) = "logical and".
+
+lemma pcc_inv_d_dx (o) (p) (n) (k):
+ p◖𝗱k ϵ 𝐂❨o, n❩ →
+ ∧∧ (Ⓣ = o → n = ↑↓n)
+ & p ϵ 𝐂❨o, n+k❩.
+#o #p #n #h @(insert_eq_1 … (p◖𝗱h))
+#x * -x -n
+[|*: #x #n [ #k #Ho ] #Hx ] #H0 destruct
+/3 width=1 by conj/
+qed-.
+
+lemma pcc_inv_m_dx (o) (p) (n):
+ p◖𝗺 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
+#o #p #n @(insert_eq_1 … (p◖𝗺))
+#x * -x -n
+[|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_L_dx (o) (p) (n):
+ p◖𝗟 ϵ 𝐂❨o,n❩ →
+ ∧∧ p ϵ 𝐂❨o,↓n❩ & n = ↑↓n.
+#o #p #n @(insert_eq_1 … (p◖𝗟))
+#x * -x -n
+[|*: #x #n [ #k #_ ] #Hx ] #H0 destruct
+<npred_succ /2 width=1 by conj/
+qed-.
+
+lemma pcc_inv_A_dx (o) (p) (n):
+ p◖𝗔 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
+#o #p #n @(insert_eq_1 … (p◖𝗔))
+#x * -x -n
+[|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_S_dx (o) (p) (n):
+ p◖𝗦 ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n❩.
+#o #p #n @(insert_eq_1 … (p◖𝗦))
+#x * -x -n
+[|*: #x #n [ #k #_ ] #Hx ] #H0 destruct //
+qed-.
+
+(* Advanced destructions ****************************************************)
+
+lemma pcc_des_d_dx (o) (p) (n) (k):
+ p◖𝗱k ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨o,n+k❩.
+#o #p #n #k #H0
+elim (pcc_inv_d_dx … H0) -H0 #H1 #H2 //
+qed-.
+
+lemma pcc_des_gen (o) (p) (n):
+ p ϵ 𝐂❨o,n❩ → p ϵ 𝐂❨Ⓕ,n❩.
+#o #p #n #H0 elim H0 -p -n //
+#p #n [ #k #Ho ] #_ #IH
+/2 width=1 by pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx, pcc_false_d_dx/
+qed-.
+
+(* Advanced inversions ******************************************************)
+
+lemma pcc_inv_empty_succ (o) (n):
+ (𝐞) ϵ 𝐂❨o,↑n❩ → ⊥.
+#o #n #H0
+lapply (pcc_inv_empty … H0) -H0 #H0
+/2 width=7 by eq_inv_zero_nsucc/
+qed-.
+
+lemma pcc_true_inv_d_dx_zero (p) (k):
+ p◖𝗱k ϵ 𝐂❨Ⓣ,𝟎❩ → ⊥.
+#p #k #H0
+elim (pcc_inv_d_dx … H0) -H0 #H0 #_
+elim (eq_inv_zero_nsucc … (H0 ?)) -H0 //
+qed-.
+
+lemma pcc_inv_L_dx_zero (o) (p):
+ p◖𝗟 ϵ 𝐂❨o,𝟎❩ → ⊥.
+#o #p #H0
+elim (pcc_inv_L_dx … H0) -H0 #_ #H0
+/2 width=7 by eq_inv_zero_nsucc/
+qed-.
+
+lemma pcc_inv_L_dx_succ (o) (p) (n):
+ p◖𝗟 ϵ 𝐂❨o,↑n❩ → p ϵ 𝐂❨o,n❩.
+#o #p #n #H0
+elim (pcc_inv_L_dx … H0) -H0 //
+qed-.
+
+(* Constructions with land **************************************************)
+
+lemma pcc_land_dx (o1) (o2) (p) (n):
+ p ϵ 𝐂❨o1,n❩ → p ϵ 𝐂❨o1∧o2,n❩.
+#o1 * /2 width=2 by pcc_des_gen/
+qed.
+
+lemma pcc_land_sn (o1) (o2) (p) (n):
+ p ϵ 𝐂❨o2,n❩ → p ϵ 𝐂❨o1∧o2,n❩.
+* /2 width=2 by pcc_des_gen/
+qed.
+
+(* Main constructions with path_append **************************************)
+
+theorem pcc_append_bi (o1) (o2) (p) (q) (m) (n):
+ p ϵ 𝐂❨o1,m❩ → q ϵ 𝐂❨o2,n❩ → p●q ϵ 𝐂❨o1∧o2,m+n❩.
+#o1 #o2 #p #q #m #n #Hm #Hn elim Hn -q -n
+/2 width=1 by pcc_m_dx, pcc_A_dx, pcc_S_dx, pcc_land_dx/
+#q #n [ #k #Ho2 ] #_ #IH
+[ @pcc_d_dx // #H0
+ elim (andb_inv_true_sn … H0) -H0 #_ #H0 >Ho2 //
+ <nplus_succ_dx <npred_succ //
+| <nplus_succ_dx /2 width=1 by pcc_L_dx/
+]
+qed.
+
+(* Inversions with path_append **********************************************)
+
+lemma pcc_false_inv_append_bi (x) (m) (n):
+ x ϵ 𝐂❨Ⓕ,m+n❩ →
+ ∃∃p,q. p ϵ 𝐂❨Ⓕ,m❩ & q ϵ 𝐂❨Ⓕ,n❩ & p●q = x.
+#x #m #n #Hx
+@(insert_eq_1 … (m+n) … Hx) -Hx #y #Hy
+generalize in match n; -n
+generalize in match m; -m
+elim Hy -x -y [|*: #x #y [ #k #_ ] #Hx #IH ] #m #n #Hy destruct
+[ elim (eq_inv_nplus_zero … Hy) -Hy #H1 #H2 destruct
+ /2 width=5 by pcc_empty, ex3_2_intro/
+| elim (IH m (n+k)) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_false_d_dx, ex3_2_intro/
+| elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_m_dx, ex3_2_intro/
+| elim (eq_inv_succ_nplus_dx … (sym_eq … Hy)) -Hy * #H1 #H2 (**) (* sym_eq *)
+ [ destruct -IH
+ /3 width=5 by pcc_empty, pcc_L_dx, ex3_2_intro/
+ | elim (IH m (↓n)) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_L_dx, ex3_2_intro/
+ ]
+| elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_A_dx, ex3_2_intro/
+| elim (IH m n) -IH // #p #q #Hp #Hq #H0 destruct -Hx
+ /3 width=5 by pcc_S_dx, ex3_2_intro/
+]
+qed-.
+
+
+(* Constructions with path_lcons ********************************************)
+
+lemma pcc_m_sn (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → (𝗺◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗺) … Hq) -Hq
+/2 width=3 by pcc_m_dx/
+qed.
-(* Basic destructions *******************************************************)
+lemma pcc_L_sn (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → (𝗟◗q) ϵ 𝐂❨o,↑n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗟) … Hq) -Hq
+/2 width=3 by pcc_L_dx/
+qed.
-axiom pcc_empty:
- (𝐞) ϵ 𝐂❨𝟎❩.
+lemma pcc_A_sn (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → (𝗔◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗔) … Hq) -Hq
+/2 width=3 by pcc_A_dx/
+qed.
-axiom pcc_d (p) (d) (n:pnat):
- p ϵ 𝐂❨d+n❩ → p◖𝗱n ϵ 𝐂❨d❩.
+lemma pcc_S_sn (o) (q) (n):
+ q ϵ 𝐂❨o,n❩ → (𝗦◗q) ϵ 𝐂❨o,n❩.
+#o #q #n #Hq
+lapply (pcc_append_bi (Ⓣ) … (𝐞◖𝗦) … Hq) -Hq
+/2 width=3 by pcc_S_dx/
+qed.
-axiom pcc_L (p) (d):
- p ϵ 𝐂❨d❩ → p◖𝗟 ϵ 𝐂❨↑d❩.
+(* Main inversions **********************************************************)
-axiom pcc_A (p) (d):
- p ϵ 𝐂❨d❩ → p◖𝗔 ϵ 𝐂❨d❩.
+theorem pcc_mono (o1) (o2) (q) (n1):
+ q ϵ 𝐂❨o1,n1❩ → ∀n2. q ϵ 𝐂❨o2,n2❩ → n1 = n2.
+#o1 #o2 #q1 #n1 #Hn1 elim Hn1 -q1 -n1
+[|*: #q1 #n1 [ #k1 #_ ] #_ #IH ] #n2 #Hn2
+[ <(pcc_inv_empty … Hn2) -n2 //
+| lapply (pcc_des_d_dx … Hn2) -Hn2 #Hn2
+ lapply (IH … Hn2) -q1 #H0
+ /2 width=2 by eq_inv_nplus_bi_dx/
+| lapply (pcc_inv_m_dx … Hn2) -Hn2 #Hn2
+ <(IH … Hn2) -q1 -n2 //
+| elim (pcc_inv_L_dx … Hn2) -Hn2 #Hn2 #H0
+ >(IH … Hn2) -q1 //
+| lapply (pcc_inv_A_dx … Hn2) -Hn2 #Hn2
+ <(IH … Hn2) -q1 -n2 //
+| lapply (pcc_inv_S_dx … Hn2) -Hn2 #Hn2
+ <(IH … Hn2) -q1 -n2 //
+]
+qed-.
-axiom pcc_S (p) (d):
- p ϵ 𝐂❨d❩ → p◖𝗦 ϵ 𝐂❨d❩.
+theorem pcc_inj_L_sn (o1) (o2) (p1) (p2) (q1) (n):
+ q1 ϵ 𝐂❨o1,n❩ → ∀q2. q2 ϵ 𝐂❨o2,n❩ →
+ p1●𝗟◗q1 = p2●𝗟◗q2 → q1 = q2.
+#o1 #o2 #p1 #p2 #q1 #n #Hq1 elim Hq1 -q1 -n
+[|*: #q1 #n1 [ #k1 #_ ] #_ #IH ] * //
+[1,3,5,7,9,11: #l2 #q2 ] #Hq2
+<list_append_lcons_sn <list_append_lcons_sn #H0
+elim (eq_inv_list_lcons_bi ????? H0) -H0 #H0 #H1 destruct
+[ elim (pcc_inv_L_dx_zero … Hq2)
+| lapply (pcc_des_d_dx … Hq2) -Hq2 #Hq2
+ <(IH … Hq2) //
+| lapply (pcc_inv_m_dx … Hq2) -Hq2 #Hq2
+ <(IH … Hq2) //
+| lapply (pcc_inv_L_dx_succ … Hq2) -Hq2 #Hq2
+ <(IH … Hq2) //
+| lapply (pcc_inv_A_dx … Hq2) -Hq2 #Hq2
+ <(IH … Hq2) //
+| lapply (pcc_inv_S_dx … Hq2) -Hq2 #Hq2
+ <(IH … Hq2) //
+| elim (pcc_inv_empty_succ … Hq2)
+]
+qed-.
-axiom pcc_des_gen (p) (d):
- p ϵ 𝐂❨d❩ → d + ♯p = ❘p❘.
+theorem pcc_inv_L_sn (o) (q) (n) (m):
+ (𝗟◗q) ϵ 𝐂❨o,n❩ → q ϵ 𝐂❨o,m❩ →
+ ∧∧ ↓n = m & n = ↑↓n.
+#o #q #n #m #H1q #H2q
+lapply (pcc_L_sn … H2q) -H2q #H2q
+<(pcc_mono … H2q … H1q) -q -n
+/2 width=1 by conj/
+qed-.
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