From 2900febefef583b4afb0ebca5df0764bc9a7a28c Mon Sep 17 00:00:00 2001 From: William Spencer Date: Thu, 15 Feb 2024 22:51:42 -0600 Subject: [PATCH] Proved n_yank_r/l, using a large number of helper lemmas. The code is messy right now, and will need to be cleaned up. Also, added .lia.cache to .gitignore. --- .gitignore | 3 +- .lia.cache | Bin 524302 -> 2044149 bytes ViCaR/GeneralLemmas.v | 53 ++- examples/ZXExample.v | 987 +++++++++++++++++++++++++++++++++++++++++- 4 files changed, 1003 insertions(+), 40 deletions(-) diff --git a/.gitignore b/.gitignore index 8995ff1..99333c2 100644 --- a/.gitignore +++ b/.gitignore @@ -159,4 +159,5 @@ $RECYCLE.BIN/ _build -settings.json \ No newline at end of file +settings.json +.lia.cache diff --git a/.lia.cache b/.lia.cache index 0ecc033cc88f30cd5e92fe987af29835065da4ad..5a5df657213f593c0feac78ad28caf068d08957b 100644 GIT binary patch literal 2044149 zcmeEP2bf${+1}Z`1EDAM-b?5SUFwA1dyyi9UZtael+dIon4qADK!SpFLBxgvQUn1J zL{tz31XL8og8D0>|NDOBoLgsScW0B`?CkU0yK_%J-}ipylzZ+LC!DlQSJ%d`&b8!| z4_$cG%@1^Sb&bZ=gKKtPQ>WfNVM5Pl|I@!lQk$8AcZfoluK;O8_A;lE_T#^eyLk&_v`oChak`M3&itKk+3lFaLQQCc`E zojNt%rXdvBcGRLI!|tKjeJoUx9B2OI!Gr;Tp zn7DS0oMv+Y70En%`-gWliY*9`3AJWciuKI^IGIeH3AMKQ(Wf6bYJpf?7lKm9^6Cq| 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z^ORy(nKJys5Zq}Qs8>(rd}Q*{egY!K^(&{r0&hvFQRTt`3@eb_0@5^_g8>XrK{k>v z!Q>ER!$;D$E7s^a!=k1}Pf+lUiGf~R4T<#OzrRGA_X?Uf3r!nBDQ=#vfvGw9$4nM- zO!FEFb@;vp|4tc)I@H`#q@I292+c~ydNSpCPd}odCJz6X9#>HLrZ9*x-Rjoe+(e7U zAg!=!g#iiUY73H`11C|>ttJyPt#!0gBM0}>BLW&VlAjLdL;@L8?W0+tG*xUYrnZq= z<~I`KY{nfsPq;@=k{@p)#bf|C(lyP|N{!ro-lWS#7r!CGx*eF@E@rZYjRdq3XWB{x z)-Q_utWKi*lJ{<)EyQAt%JFF3YLwy@W^}J}>S$>!C992FC>9qn7VKhy1CK~l|N^!%i7&yae2ezA5xGrf}W-!1Cp!P%e z`SQ4ngtV@UHjP7+SunV69so+yt*_Wt2O&lS#|@EaTij7u9GNb|8DW4=#%tm;B0+H* L+36)xy!8JAYyc~R delta 24 fcmezRu%@p;p`nGbg{g(Pg{6hHg{_5s3I`tmfnx|` diff --git a/ViCaR/GeneralLemmas.v b/ViCaR/GeneralLemmas.v index 7de0640..2d8fd49 100644 --- a/ViCaR/GeneralLemmas.v +++ b/ViCaR/GeneralLemmas.v @@ -4,34 +4,49 @@ Require Import Monoidal. Local Open Scope Cat. Lemma compose_simplify_general : forall {C : Type} {Cat : Category C} - {A B M : C} (f1 f3 : A ~> B) (f2 f4 : B ~> M), - f1 ≃ f3 -> f2 ≃ f4 -> (f1 ∘ f2) ≃ (f3 ∘ f4). + {A B M : C} (f1 f3 : A ~> B) (f2 f4 : B ~> M), + f1 ≃ f3 -> f2 ≃ f4 -> (f1 ∘ f2) ≃ (f3 ∘ f4). Proof. - intros. - rewrite H, H0. - reflexivity. + intros. + rewrite H, H0. + reflexivity. Qed. Lemma stack_simplify_general : forall {C : Type} - {Cat : Category C} {MonCat : MonoidalCategory C} - {A B M N : C} (f1 f3 : A ~> B) (f2 f4 : M ~> N), - f1 ≃ f3 -> f2 ≃ f4 -> (f1 ⊗ f2) ≃ (f3 ⊗ f4). + {Cat : Category C} {MonCat : MonoidalCategory C} + {A B M N : C} (f1 f3 : A ~> B) (f2 f4 : M ~> N), + f1 ≃ f3 -> f2 ≃ f4 -> (f1 ⊗ f2) ≃ (f3 ⊗ f4). Proof. - intros. - rewrite H, H0. - reflexivity. + intros. + rewrite H, H0. + reflexivity. Qed. Lemma nwire_stack_compose_topleft_general : forall {C : Type} - {Cat : Category C} {MonCat : MonoidalCategory C} - {topIn botIn topOut botOut : C} - (f0 : botIn ~> botOut) (f1 : topIn ~> topOut), - ((c_identity topIn) ⊗ f0) ∘ (f1 ⊗ (c_identity botOut)) ≃ (f1 ⊗ f0). + {Cat : Category C} {MonCat : MonoidalCategory C} + {topIn botIn topOut botOut : C} + (f0 : botIn ~> botOut) (f1 : topIn ~> topOut), + ((c_identity topIn) ⊗ f0) ∘ (f1 ⊗ (c_identity botOut)) ≃ (f1 ⊗ f0). Proof. - intros. - rewrite <- bifunctor_comp. - rewrite left_unit; rewrite right_unit. - easy. + intros. + rewrite <- bifunctor_comp. + rewrite left_unit; rewrite right_unit. + easy. Qed. +Lemma nwire_stackcompose_topright_general : forall {C : Type} + {Cat: Category C} {MonCat : MonoidalCategory C} + {topIn botIn topOut botOut : C} + (f0 : topIn ~> topOut) (f1 : botIn ~> botOut), + (f0 ⊗ (c_identity botIn)) ∘ ((c_identity topOut) ⊗ f1) ≃ (f0 ⊗ f1). +Proof. + intros. + rewrite <- bifunctor_comp. + rewrite right_unit, left_unit. + easy. +Qed. + + + + Local Close Scope Cat. diff --git a/examples/ZXExample.v b/examples/ZXExample.v index d7af483..b9e3c3d 100644 --- a/examples/ZXExample.v +++ b/examples/ZXExample.v @@ -514,35 +514,982 @@ Proof. Unshelve. all: lia. Qed. -Lemma n_yank_l : forall n {prf1 prf2}, - cast n (n + n + n) prf1 prf2 (n_wire n ↕ n_cap n) - ⟷ (n_cup n ↕ n_wire n) ∝ n_wire n. +Lemma helper_eq: forall n m (prf: n = m), (n + n = m + m)%nat. +Proof. intros. subst. reflexivity. Qed. + +Lemma cast_n_cup_unswapped : forall n m prf1 prf2, + cast (n + n) 0 (helper_eq _ _ prf1) prf2 (n_cup_unswapped m) ∝ n_cup_unswapped n. +Proof. + intros. + subst. + rewrite cast_id_eq. + easy. +Qed. + +Lemma wire_stack_distr_compose_l : forall n m o (zx0 : ZX n m) (zx1 : ZX m o), + — ↕ (zx0 ⟷ zx1) ∝ (— ↕ zx0) ⟷ (— ↕ zx1). +Proof. + intros. + rewrite <- stack_compose_distr. + cleanup_zx. + easy. +Qed. + +Lemma wire_stack_distr_compose_r : forall n m o (zx0 : ZX n m) (zx1 : ZX m o), + (zx0 ⟷ zx1) ↕ — ∝ (zx0 ↕ —) ⟷ (zx1 ↕ —). Proof. + intros. + rewrite <- stack_compose_distr. + cleanup_zx. + easy. +Qed. + +Lemma n_cup_unswapped_grow_r : forall n prf1 prf2, + n_cup_unswapped (S n) ∝ + cast _ _ prf1 prf2 (— ↕ n_cup_unswapped n ↕ —) ⟷ ⊃. +Proof. + intros. induction n. - - intros. - rewrite n_cap_0_empty. - rewrite n_cup_0_empty. + - simpl. cleanup_zx. + apply compose_simplify; [|easy]. + prop_perm_eq. + - rewrite n_cup_unswapped_grow_l. + rewrite IHn at 1. + rewrite n_cup_unswapped_grow_l. + bundle_wires. + rewrite <- compose_assoc. + apply compose_simplify; [|easy]. + rewrite wire_stack_distr_compose_l, wire_stack_distr_compose_r. + rewrite (prop_iff_double_cast _ (1 + 0 + 1) _ _ eq_refl). + simpl_casts. + rewrite (cast_compose_mid_contract _ (1 + (n + n) + 1)). + rewrite 2!cast_contract, cast_id. + apply compose_simplify; [|easy]. + simpl_casts. + simpl (n_wire S n). + rewrite 4!stack_assoc. + rewrite (stack_assoc — (n_wire n) _). + simpl_casts. + repeat (rewrite cast_stack_distribute; apply stack_simplify; try prop_perm_eq). + rewrite cast_id; easy. + Unshelve. + all: lia. +Qed. + +Lemma nwire_stack_distr_compose_l : forall k n m o (zx0 : ZX n m) (zx1 : ZX m o), + n_wire k ↕ (zx0 ⟷ zx1) ∝ (n_wire k ↕ zx0) ⟷ (n_wire k ↕ zx1). +Proof. + intros. + rewrite <- stack_compose_distr. + cleanup_zx. + easy. +Qed. + +Lemma nwire_stack_distr_compose_r : forall k n m o (zx0 : ZX n m) (zx1 : ZX m o), + (zx0 ⟷ zx1) ↕ n_wire k ∝ (zx0 ↕ n_wire k) ⟷ (zx1 ↕ n_wire k). +Proof. + intros. + rewrite <- stack_compose_distr. + cleanup_zx. + easy. +Qed. + +Lemma n_cup_unswapped_comm_1 : forall k prf1 prf2 prf3 prf4, + cast (S k + (S k)) _ prf1 prf2 (n_wire k ↕ ⊃ ↕ n_wire k) ⟷ (n_cup_unswapped k) + ∝ cast _ _ prf3 prf4 (— ↕ n_cup_unswapped k ↕ —) ⟷ ⊃. +Proof. + intros. + rewrite <- n_cup_unswapped_grow_l. + rewrite <- n_cup_unswapped_grow_r. + easy. +Qed. + +Lemma n_wire_add_stack : forall n k, + n_wire (n + k) ∝ n_wire n ↕ n_wire k. +Proof. prop_perm_eq. Qed. + +Lemma n_wire_add_stack_rev : forall n k prf1 prf2, + n_wire (n + k) ∝ cast _ _ prf1 prf2 (n_wire k ↕ n_wire n). +Proof. prop_perm_eq. Qed. + +Lemma stack_assoc' : forall {n0 n1 n2 m0 m1 m2} (zx0 : ZX n0 m0) + (zx1 : ZX n1 m1) (zx2 : ZX n2 m2) prfn prfm, + zx0 ↕ (zx1 ↕ zx2) ∝ cast _ _ prfn prfm ((zx0 ↕ zx1) ↕ zx2). +Proof. + intros. + rewrite stack_assoc. + rewrite cast_cast_eq, cast_id. + easy. + Unshelve. + all: lia. +Qed. + +Lemma n_cup_unswapped_comm : forall n k prf1 prf2 prf3 prf4, + cast (S n + k + (S n + k)) _ prf1 prf2 (n_wire (n + k) ↕ ⊃ ↕ n_wire (n + k)) ⟷ (n_wire n ↕ n_cup_unswapped k ↕ n_wire n) + ∝ cast _ _ prf3 prf4 (n_wire (S n) ↕ n_cup_unswapped k ↕ n_wire (S n)) ⟷ (n_wire n ↕ ⊃ ↕ n_wire n). +Proof. + intros. + rewrite n_wire_add_stack at 1. + rewrite n_wire_add_stack_rev at 1. + rewrite (n_wire_add_stack_rev 1 n) at 1. + rewrite (n_wire_add_stack 1 n) at 1. + rewrite 5!stack_assoc. + repeat rewrite cast_cast_eq. + simpl_casts. + rewrite stack_assoc. + rewrite (prop_iff_double_cast (n + (k + 2 + k + n)) (n + (0 + n))). + rewrite (cast_compose_mid_contract _ (n + (k + 0 + k + n))). + repeat rewrite cast_cast_eq. + rewrite (cast_compose_mid_contract _ (n + (2 + n))). + rewrite 2!cast_cast_eq. + erewrite 3!(cast_stack_distribute _ _ _ _ _ _ (n_wire n)). + rewrite 4!cast_id_eq. + rewrite <- 2!nwire_stack_distr_compose_l. + apply stack_simplify; [easy|]. + rewrite <- wire_to_n_wire. + rewrite 3!stack_assoc'. + rewrite (stack_assoc' (_ ↕ _) (—) (n_wire n)). + repeat rewrite cast_cast_eq. + rewrite 3!(cast_stack_distribute (o':=n)). + rewrite (cast_id (n:=n)). + rewrite <- 2!nwire_stack_distr_compose_r. + apply stack_simplify; [|easy]. + rewrite (prop_iff_double_cast ((S k) + (S k)) (0)). + simpl_permlike_zx. + symmetry. + rewrite (cast_compose_mid_contract _ 2). + symmetry. + rewrite cast_id_eq. + rewrite cast_cast_eq. + rewrite <- n_cup_unswapped_comm_1. + rewrite (prop_iff_double_cast ((S k) + (S k)) (0)). + rewrite (cast_compose_mid_contract _ (k + k)). + simpl_casts. + easy. + Unshelve. + all: lia. +Qed. + + + +Lemma n_cup_unswapped_grow_k_l : forall n k prf1 prf2, + n_cup_unswapped (n + k) ∝ + cast _ _ prf1 prf2 (n_wire n ↕ (n_cup_unswapped k) ↕ n_wire n) ⟷ n_cup_unswapped n. +Proof. + intros. + induction n. + - cleanup_zx. simpl_casts. - cleanup_zx. simpl. - simpl_casts. cleanup_zx. easy. - - intros. -Admitted. + (* for k: + cleanup_zx. + simpl_casts. + bundle_wires. + rewrite <- (cast_n_cup_unswapped (n + 0) n (Nat.add_0_r _) eq_refl). + rewrite <- (nwire_removal_l (n_cup_unswapped _)) at 1. + rewrite cast_compose_distribute, cast_id_eq. + apply compose_simplify; [|easy]. + prop_perm_eq. *) + - rewrite (prop_iff_double_cast (S (n + k) + S (n + k)) 0). + rewrite cast_n_cup_unswapped. + rewrite n_cup_unswapped_grow_l, IHn. + rewrite n_cup_unswapped_grow_l. + simpl_permlike_zx. + (* simpl_casts. *) + repeat rewrite <- compose_assoc. + apply compose_simplify; [|easy]. + symmetry. + rewrite (cast_compose_mid (n + 2 + n)). + erewrite (prop_iff_double_cast ((S n + k) + (S n + k)) (n + 0 + n) _ _ eq_refl). + rewrite 2!cast_contract. + rewrite cast_compose_mid_contract, 2!cast_contract, cast_id. + rewrite cast_compose_mid_contract, 2!cast_contract, cast_id. + rewrite n_cup_unswapped_comm. + easy. + Unshelve. + all: lia. +Qed. + +Lemma n_cup_unswapped_add_comm : forall n k prf1 prf2, + n_cup_unswapped (n + k) ∝ cast _ _ prf1 prf2 (n_cup_unswapped (k + n)). +Proof. + intros. + assert (H: (k + n = n + k)%nat) by lia. + generalize dependent (k + n)%nat. + generalize dependent (n + k)%nat. + intros; subst. + rewrite cast_id_eq. + easy. +Qed. + +Lemma n_cup_unswapped_grow_k_r : forall n k prf1 prf2, + n_cup_unswapped (n + k) ∝ + cast _ _ prf1 prf2 (n_wire k ↕ (n_cup_unswapped n) ↕ n_wire k) ⟷ n_cup_unswapped k. +Proof. + intros. + rewrite n_cup_unswapped_add_comm. + rewrite n_cup_unswapped_grow_k_l. + rewrite (cast_compose_mid_contract _ (k + k)%nat). + simpl_casts. + apply compose_simplify; [|easy]. + erewrite cast_proof_independence. + reflexivity. + Unshelve. all:lia. +Qed. + + + +Lemma stack_ncup_unswapped_split : forall {n0t n0b n1t n1b} n m (zx0top : ZX n0t m) (zx0bot : ZX n0b m) + (zx1top : ZX n1t n) (zx1bot : ZX n1b n) prf1 prf2 prf3 prf4 prf5 prf6, + (zx1top ↕ zx0top) ↕ (zx0bot ↕ zx1bot) + ⟷ cast ((n + m) + (m + n)) 0 prf1 prf2 (n_cup_unswapped (n + m)) + ∝ cast _ _ prf5 prf6 (zx1top ↕ ((zx0top ↕ zx0bot) ⟷ n_cup_unswapped m) ↕ zx1bot + ⟷ cast (n + 0 + n) 0 prf3 prf4 (n_cup_unswapped n)). +Proof. + intros. + rewrite cast_compose_r. + simpl_permlike_zx. + rewrite n_cup_unswapped_grow_k_l, <- compose_assoc. + rewrite (cast_compose_mid_contract _ (n + n)). + simpl_permlike_zx. + apply compose_simplify; [|easy]. + rewrite stack_assoc, (stack_assoc' zx0top). + simpl_casts. + rewrite stack_assoc'. + rewrite cast_cast_eq. + rewrite (prop_iff_double_cast (n1t + (n0t + n0b) + n1b) (n + 0 + n)). + rewrite (cast_compose_mid_contract _ (n + (m + m) + n)). + simpl_permlike_zx. + rewrite <- 2!stack_compose_distr, 2!nwire_removal_r. + easy. + Unshelve. + all: lia. +Qed. + +(* Local Open Scope matrix_scope. *) + +Ltac is_C0 x := + assert (x = C0) by lca. + +Ltac is_C1 x := + assert (x = C1) by lca. + +Ltac print_C x := + tryif is_C0 x then idtac "0" else + tryif is_C1 x then idtac "1" else idtac "X". + +Ltac print_LHS_matU := + intros; + (let i := fresh "i" in + let j := fresh "j" in + let Hi := fresh "Hi" in + let Hj := fresh "Hj" in + intros i j Hi Hj; try solve_end; + repeat + (destruct i as [| i]; [ | apply <- Nat.succ_lt_mono in Hi ]; + try solve_end); clear Hi; + repeat + (destruct j as [| j]; [ | apply <- Nat.succ_lt_mono in Hj ]; + try solve_end); clear Hj); + match goal with |- ?x = ?y ?i ?j => autounfold with U_db; simpl; + match goal with + | |- ?x = _ => idtac i; idtac j; print_C x; idtac "" + end + end. + +Lemma sem_n_cup_unswapped_2 : + ⟦ n_cup_unswapped 2 ⟧ = + fun x y => if (x=?0) && ((y=?0) || (y=?6) || (y=?9) || (y=?15)) then C1 else C0. +Proof. + unfold n_cup_unswapped. + repeat (simpl; + rewrite cast_semantics; simpl). + rewrite 2!id_kron. + replace (list2D_to_matrix [[C1; C0; C0; C1]]) with + (fun x y => if (x =? 0)&&((y =? 0) || (y =? 3)) then C1 else C0) by solve_matrix. + solve_matrix. +Qed. + + + + + +Lemma swap_2cup_transport : + ⟦ n_cup_unswapped 2 ⟧ × (swap ⊗ (Matrix.I (2^2))) + = ⟦ n_cup_unswapped 2 ⟧ × ((Matrix.I (2^2)) ⊗ swap). +Proof. + apply mat_equiv_eq; auto with wf_db. + rewrite sem_n_cup_unswapped_2. + by_cell; try lca. +Qed. + +Lemma swap_2cup_flip : + ⨉ ↕ n_wire 2 ⟷ n_cup_unswapped 2 ∝ n_wire 2 ↕ ⨉ ⟷ n_cup_unswapped 2. +Proof. + prop_exists_nonzero 1. + rewrite (compose_semantics (⨉ ↕ n_wire 2) (n_cup_unswapped 2)). (* For some reason, it needs *) + rewrite (compose_semantics (n_wire 2 ↕ ⨉) (n_cup_unswapped 2)). (* its arguments explicitly *) + rewrite Mscale_1_l. + rewrite 2!stack_semantics, n_wire_semantics. + simpl (⟦ ⨉ ⟧). + apply swap_2cup_transport. +Qed. + +Tactic Notation "preplace" constr(zx0) "with" constr(zx1) := + (let H := fresh "H" in + enough (H: zx0 ∝ zx1); + [rewrite H; clear H| ]). + +Tactic Notation "preplace" constr(zx0) "with" constr(zx1) "in" hyp(target) := + (let H := fresh "H" in + enough (H: zx0 ∝ zx1); + [rewrite H in target; clear H| ]). + +Tactic Notation "preplace" constr(zx0) "with" constr(zx1) "in" "*" := + (let H := fresh "H" in + enough (H: zx0 ∝ zx1); + [rewrite H in *; clear H| ]). + +Tactic Notation "preplace" constr(zx0) "with" constr(zx1) "by" tactic(slv) := + (let H := fresh "H" in + assert (H: zx0 ∝ zx1) by slv; + rewrite H; clear H). + +Tactic Notation "preplace" constr(zx0) "with" constr(zx1) "in" hyp(target) "by" tactic(slv) := + (let H := fresh "H" in + assert (H: zx0 ∝ zx1) by slv; + rewrite H in target; clear H). + +Tactic Notation "preplace" constr(zx0) "with" constr(zx1) "in" "*" "by" tactic(slv) := + (let H := fresh "H" in + assert (H: zx0 ∝ zx1) by slv; + rewrite H in *; clear H). + + + +Lemma n_cup_unswapped_split_stack : forall {n00 m0 n01 m1 n10 n11 nbot} + (zx00 : ZX n00 m0) (zx01 : ZX n01 m1) (zx10 : ZX n10 m1) (zx11 : ZX n11 m0) + prf1 prf2 prf3 prf4 prf5 prf6, + (zx00 ↕ zx01) ↕ (cast nbot (m0 + m1) prf1 prf2 (zx10 ↕ zx11)) ⟷ n_cup_unswapped (m0 + m1) + ∝ + cast _ 0 prf5 prf6 (n_wire n00 + ↕ (zx01 ↕ zx10 ⟷ n_cup_unswapped m1) ↕ n_wire n11 ⟷ + cast _ 0 prf3 prf4 (zx00 ↕ zx11 ⟷ n_cup_unswapped m0)). +Proof. + intros. + rewrite cast_stack_r. + rewrite n_cup_unswapped_grow_k_l. + rewrite <- compose_assoc. + rewrite stack_assoc, (stack_assoc' zx01). + simpl_casts. + rewrite stack_assoc', cast_cast_eq. + rewrite <- cast_compose_mid_contract. + rewrite <- 2!stack_compose_distr, 2!nwire_removal_r. + rewrite pull_out_bot, <- (nwire_removal_l zx11), stack_compose_distr. + rewrite cast_compose_distribute. + rewrite (cast_compose_mid_contract _ (n00 + 0 + n11)). + rewrite compose_assoc. + apply compose_simplify; [easy | ]. + simpl_casts. + rewrite cast_compose_mid_contract, cast_id_eq. + apply compose_simplify; [|easy]. + rewrite nwire_removal_l. + easy. + Unshelve. + all: lia. +Qed. + +Lemma n_cup_unswapped_split_stack' : forall {n00 m0 n01 m1 n10 n11 ntop} + (zx00 : ZX n00 m0) (zx01 : ZX n01 m1) (zx10 : ZX n10 m1) (zx11 : ZX n11 m0) + prf1 prf2 prf3 prf4 prf5 prf6, + (cast ntop (m1 + m0) prf1 prf2 (zx00 ↕ zx01)) ↕ (zx10 ↕ zx11) ⟷ n_cup_unswapped (m1 + m0) + ∝ + cast _ 0 prf5 prf6 (n_wire n00 + ↕ (zx01 ↕ zx10 ⟷ n_cup_unswapped m1) ↕ n_wire n11 ⟷ + cast _ 0 prf3 prf4 (zx00 ↕ zx11 ⟷ n_cup_unswapped m0)). +Proof. + intros. + rewrite (prop_iff_double_cast (ntop + (n10 + n11)) 0). + rewrite (cast_compose_mid_contract _ (m0 + m1 + (m0 + m1))). + rewrite cast_n_cup_unswapped by lia. + subst ntop. + rewrite cast_stack_distribute, cast_cast_eq, cast_id_eq. + rewrite n_cup_unswapped_split_stack. + simpl_casts. + easy. + Unshelve. + all: lia. +Qed. + +Lemma n_cup_unswapped_split_stack_cast : forall {n00 m0 n01 m1 n10 n11 ntop nbot} + (zx00 : ZX n00 m0) (zx01 : ZX n01 m1) (zx10 : ZX n10 m1) (zx11 : ZX n11 m0) + prf1 prf2 prf3 prf4 prf5 prf6 prf7 prf8, + (cast ntop (m0 + m1) prf1 prf2 (zx00 ↕ zx01)) + ↕ (cast nbot (m0 + m1) prf3 prf4 (zx10 ↕ zx11)) + ⟷ n_cup_unswapped (m0 + m1) ∝ + cast _ 0 prf5 prf6 ( + n_wire n00 ↕ (zx01 ↕ zx10 ⟷ n_cup_unswapped m1) ↕ n_wire n11 + ⟷ cast _ 0 prf7 prf8 (zx00 ↕ zx11 ⟷ n_cup_unswapped m0)). +Proof. + intros. + subst ntop. + rewrite cast_id_eq. + apply n_cup_unswapped_split_stack. +Qed. + + +Lemma n_cup_unswapped_split_stack_n_wire_bot : forall {n0 n1} + (zx0 : ZX n0 n0) (zx1 : ZX n1 n1) prf1 prf2 prf3 prf4, + zx0 ↕ zx1 ↕ n_wire (n0 + n1) ⟷ n_cup_unswapped (n0 + n1) + ∝ cast _ 0 prf1 prf2 ( + n_wire n0 ↕ (zx1 ↕ n_wire n1 ⟷ n_cup_unswapped n1) ↕ n_wire n0 ⟷ + cast _ 0 prf3 prf4 (zx0 ↕ n_wire n0 ⟷ n_cup_unswapped n0)). +Proof. + intros. + rewrite n_wire_add_stack_rev. + rewrite n_cup_unswapped_split_stack. + easy. + Unshelve. + all: lia. +Qed. + + + +Lemma n_cup_unswapped_split_stack_n_wire_top : forall {n0 n1} + (zx0 : ZX n0 n0) (zx1 : ZX n1 n1) prf1 prf2 prf3 prf4, + n_wire (n0 + n1) ↕ (zx0 ↕ zx1) ⟷ n_cup_unswapped (n0 + n1) + ∝ cast _ 0 prf1 prf2 ( + n_wire n1 ↕ (n_wire n0 ↕ zx0 ⟷ n_cup_unswapped n0) ↕ n_wire n1 ⟷ + cast _ 0 prf3 prf4 (n_wire n1 ↕ zx1 ⟷ n_cup_unswapped n1)). +Proof. + intros. + rewrite n_wire_add_stack_rev. + rewrite n_cup_unswapped_split_stack'. + easy. + Unshelve. + all: lia. +Qed. + +Lemma n_cup_unswapped_split_stack_n_wire_bot' : forall {n0 n1 ntop} + (zx0 : ZX n0 n0) (zx1 : ZX n1 n1) prf1 prf2 prf3 prf4 prf5 prf6, + cast ntop _ prf1 prf2 (zx0 ↕ zx1) ↕ n_wire (n1 + n0) ⟷ n_cup_unswapped (n1 + n0) + ∝ cast _ 0 prf3 prf4 ( + n_wire n0 ↕ (zx1 ↕ n_wire n1 ⟷ n_cup_unswapped n1) ↕ n_wire n0 ⟷ + cast _ 0 prf5 prf6 (zx0 ↕ n_wire n0 ⟷ n_cup_unswapped n0)). +Proof. + intros. + rewrite n_wire_add_stack. + rewrite n_cup_unswapped_split_stack'. + easy. + Unshelve. + all: lia. +Qed. + +Lemma n_cup_unswapped_split_stack_n_wire_top' : forall {n0 n1 ntop} + (zx0 : ZX n0 n0) (zx1 : ZX n1 n1) prf1 prf2 prf3 prf4 prf5 prf6, + n_wire (n1 + n0) ↕ cast ntop _ prf1 prf2 (zx0 ↕ zx1) ⟷ n_cup_unswapped (n1 + n0) + ∝ cast _ 0 prf3 prf4 ( + n_wire n1 ↕ (n_wire n0 ↕ zx0 ⟷ n_cup_unswapped n0) ↕ n_wire n1 ⟷ + cast _ 0 prf5 prf6 (n_wire n1 ↕ zx1 ⟷ n_cup_unswapped n1)). +Proof. + intros. + rewrite n_wire_add_stack. + rewrite n_cup_unswapped_split_stack. + easy. + Unshelve. + all: lia. +Qed. + + +Lemma n_cup_unswapped_grow_r_back : forall n prf1 prf2, + (— ↕ n_cup_unswapped (n) ↕ — ⟷ ⊃) + ∝ cast _ _ prf1 prf2 (n_cup_unswapped (S n)). +Proof. + intros. + rewrite (n_cup_unswapped_grow_r n). + rewrite cast_compose_l. + rewrite cast_cast_eq, 2!cast_id_eq. + easy. + Unshelve. + all: lia. +Qed. + +Lemma n_cup_unswapped_grow_k_r_back : forall n k prf1 prf2 prf3 prf4, + (n_wire k ↕ n_cup_unswapped n ↕ n_wire k) + ⟷ cast (k + 0 + k) 0 prf1 prf2 (n_cup_unswapped k) + ∝ cast _ 0 prf3 prf4 (n_cup_unswapped (n + k)). +Proof. + intros. + rewrite n_cup_unswapped_grow_k_r. + rewrite (cast_compose_mid_contract _ (k + 0 + k)). + rewrite cast_cast_eq, cast_id_eq. + easy. + Unshelve. + all: lia. +Qed. + +Lemma compose_n_wire_comm : forall {n m} (zx : ZX n m), + n_wire n ⟷ zx ∝ zx ⟷ n_wire m. +Proof. + intros. + cleanup_zx; easy. +Qed. + +Lemma compose_stack_n_wire_comm : forall {n0 m0 n1 m1} (zx0 : ZX n0 m0) (zx1 : ZX n1 m1), + zx0 ↕ n_wire n1 ⟷ (n_wire m0 ↕ zx1) ∝ n_wire n0 ↕ zx1 ⟷ (zx0 ↕ n_wire m1). +Proof. + intros. + cleanup_zx. + easy. +Qed. + +Lemma top_to_bottom_cup_flip : forall k, + top_to_bottom k ↕ n_wire k ⟷ n_cup_unswapped k + ∝ n_wire k ↕ top_to_bottom k ⟷ n_cup_unswapped k. +Proof. + destruct k; [prop_perm_eq|]. + induction k; + [ apply compose_simplify; [prop_perm_eq | easy] | ]. + rewrite top_to_bottom_grow_r at 1. + rewrite nwire_stack_distr_compose_r. + rewrite compose_assoc. + rewrite (n_wire_add_stack 2 k) at 2. + rewrite (n_cup_unswapped_split_stack' (n_wire k) ⨉ (n_wire 2) (n_wire k)). + rewrite swap_2cup_flip. + bundle_wires; cleanup_zx. + rewrite nwire_stack_distr_compose_l, nwire_stack_distr_compose_r. + rewrite compose_assoc. + rewrite n_cup_unswapped_grow_k_r_back. + rewrite (stack_assoc' (n_wire k)). + rewrite 2!cast_stack_l. + rewrite (stack_assoc _ ⨉), cast_cast_eq. + rewrite (cast_compose_mid_contract _ (S (S k) + S (S k))). + simpl_casts. + rewrite 2!cast_stack_distribute. + simpl_permlike_zx. + bundle_wires; rewrite cast_n_wire. + rewrite <- compose_assoc, compose_stack_n_wire_comm. + rewrite (n_wire_add_stack 1 (S k)) at 2. + rewrite compose_assoc. + rewrite (n_cup_unswapped_split_stack' (top_to_bottom (S k)) — (n_wire 1) (n_wire S k)). + rewrite IHk. + rewrite <- n_cup_unswapped_split_stack'. + rewrite <- compose_assoc. + rewrite <- stack_compose_distr. + rewrite <- wire_to_n_wire. + rewrite <- (top_to_bottom_grow_l k). + apply compose_simplify; [|easy]. + apply stack_simplify; [|easy]. + prop_perm_eq. + Unshelve. + all: lia. +Qed. + +Lemma bottom_to_top_cup_flip : forall k, + bottom_to_top k ↕ n_wire k ⟷ n_cup_unswapped k + ∝ n_wire k ↕ bottom_to_top k ⟷ n_cup_unswapped k. +Proof. + intros k. + destruct k; [prop_perm_eq | ]. + induction k; + [ apply compose_simplify; [prop_perm_eq | easy] | ]. + rewrite bottom_to_top_grow_r at 1. + rewrite nwire_stack_distr_compose_r. + rewrite compose_assoc. + rewrite (n_wire_add_stack_rev 2 k) at 2. + rewrite (n_cup_unswapped_split_stack ⨉ (n_wire k) (n_wire k) (n_wire 2)). + rewrite swap_2cup_flip. + rewrite <- n_cup_unswapped_split_stack. + rewrite <- compose_assoc, <- n_wire_add_stack, compose_stack_n_wire_comm. + preplace (n_wire (2 + k)) with (n_wire (1 + (S k))) by easy. + rewrite (n_wire_add_stack_rev 1 (S k)). + rewrite compose_assoc. + rewrite (n_cup_unswapped_split_stack — (bottom_to_top (S k)) (n_wire S k) (n_wire 1)). + rewrite IHk. + rewrite <- n_cup_unswapped_split_stack. + rewrite <- compose_assoc. + apply compose_simplify; [|easy]. + rewrite <- stack_compose_distr. + apply stack_simplify; [prop_perm_eq | ]. + rewrite <- wire_to_n_wire. + rewrite cast_compose_r, cast_cast_eq. + rewrite (cast_compose_partial_contract_l _ (S k + 1)). + rewrite <- (bottom_to_top_grow_l k). + easy. + Unshelve. + all: lia. +Qed. + +Lemma a_swap_cup_flip : forall k, + a_swap k ↕ n_wire k ⟷ n_cup_unswapped k + ∝ n_wire k ↕ a_swap k ⟷ n_cup_unswapped k. +Proof. + intros k. + destruct k; [prop_perm_eq|]. + simpl a_swap. + rewrite nwire_stack_distr_compose_r, compose_assoc. + rewrite (n_cup_unswapped_split_stack_n_wire_bot —). + rewrite top_to_bottom_cup_flip. + rewrite <- (cast_cast_eq _ _ (k + 1 + (k + 1)) 0 (1 + k + (1 + k)) 0). + preplace (— ↕ n_wire 1) with (n_wire 1 ↕ —) by prop_perm_eq. + rewrite <- (n_cup_unswapped_split_stack_n_wire_top (top_to_bottom k) —). + rewrite (cast_compose_mid_contract _ (1 + k + (1 + k))). + rewrite cast_n_cup_unswapped, cast_stack_distribute, cast_n_wire by lia. + rewrite <- compose_assoc, compose_stack_n_wire_comm, compose_assoc. + rewrite bottom_to_top_cup_flip. + rewrite <- compose_assoc. + apply compose_simplify; [|easy]. + rewrite <- stack_compose_distr. + rewrite nwire_removal_r. + apply stack_simplify; prop_perm_eq. + solve_modular_permutation_equalities. + Unshelve. + all: lia. +Qed. + +Lemma n_swap_zxperm : forall n, + ZXperm n (n_swap n). +Proof. + induction n; simpl; auto with zxperm_db. +Qed. + +#[export] Hint Resolve n_swap_zxperm : zxperm_db. + +Lemma perm_of_n_swap : forall n, + perm_of_zx (n_swap n) = fun k => if n <=? k then k else (n - S k)%nat. +Proof. + (* destruct n; [simpl; solve_modular_permutation_equalities|]. *) + induction n; simpl perm_of_zx; cleanup_perm_of_zx; + [|rewrite IHn]; solve_modular_permutation_equalities. +Qed. + +#[export] Hint Rewrite perm_of_n_swap : perm_of_zx_cleanup_db. + +Lemma n_swap_cup_flip : forall k, + n_swap k ↕ n_wire k ⟷ n_cup_unswapped k + ∝ n_wire k ↕ n_swap k ⟷ n_cup_unswapped k. +Proof. + intros k. + (* destruct k; [prop_perm_eq|]. *) + induction k; + [prop_perm_eq | ]. + (* [apply compose_simplify; [prop_perm_eq | easy] | ]. *) + simpl (n_swap (S k)). + rewrite nwire_stack_distr_compose_r, compose_assoc. + rewrite (n_cup_unswapped_split_stack_n_wire_bot —), IHk. + preplace (— ↕ n_wire 1) with (n_wire 1 ↕ —) by prop_perm_eq. + rewrite <- (n_cup_unswapped_split_stack_n_wire_top' (n_swap k) —). + rewrite <- compose_assoc, compose_stack_n_wire_comm, compose_assoc. + rewrite bottom_to_top_cup_flip, <- compose_assoc. + apply compose_simplify; [|easy]. + rewrite <- stack_compose_distr, nwire_removal_l. + apply stack_simplify; [easy|]. + prop_perm_eq. + solve_modular_permutation_equalities. + Unshelve. + all: lia. +Qed. + + + + + + +Local Open Scope ZX_scope. + + + +Lemma n_yank_1_l_helper_helper : forall n, + (⊃) ⊤ ↕ n_wire n ⟷ (— ↕ n_wire (1 + n)) ∝ n_wire n ⟷ (⊂ ↕ n_wire n). +Proof. + intros n. + simpl. + rewrite nwire_removal_l, nwire_removal_r. + easy. +Qed. + +Lemma n_yank_1_l_helper : forall n prf1 prf2 prf3 prf4 prf5 prf6 prf7 prf8, + cast (S n + (n + n)) (1 + n + S n + S n) prf1 prf2 (n_wire (1 + n) ↕ (n_wire n ↕ ⊃ ↕ n_wire n) ⊤) + ⟷ (((cast (S n + S n) 2 prf3 prf4 (— ↕ n_cup_unswapped n ↕ —)) ⟷ ⊃) ↕ (— ↕ n_wire n)) + ∝ cast _ _ prf7 prf8 ( + (— ↕ (n_wire n ↕ n_wire n) ↕ n_wire n) ⟷ (cast (1 + (n + n) + n) (1 + 0 + n) prf5 prf6 (— ↕ (n_cup_unswapped n) ↕ n_wire n) ) + ⟷ (— ↕ ⊂ ↕ n_wire n) ⟷ (⊃ ↕ — ↕ n_wire n) + ). +Proof. + intros. + rewrite nwire_stack_distr_compose_r. + rewrite <- compose_assoc. + rewrite (cast_compose_mid_contract _ (2 + 1 + n)). + apply compose_simplify; [| + rewrite stack_assoc, cast_cast_eq, cast_id; easy]. + rewrite 2!stack_transpose, n_wire_transpose. + rewrite 2!stack_assoc', (stack_assoc — (n_wire n) (n_wire n)). + simpl_permlike_zx. + rewrite cast_stack_l, cast_cast_eq. + rewrite (stack_assoc _ ((⊃) ⊤) (n_wire n)). + rewrite (prop_iff_double_cast (S (n + n) + (0 + n)) (1 + (1 + (1 + n)))). + rewrite (cast_compose_mid_contract _ ((n + n) + 1 + (1 + (1 + n)))). + + rewrite 2!cast_cast_eq. + rewrite cast_stack_distribute. + rewrite (cast_stack_l (mTop':=2)). + rewrite (stack_assoc _ —). + rewrite 2!cast_cast_eq. + rewrite (cast_stack_distribute (o':= 1 + (1 + n))). + rewrite <- stack_compose_distr. + + simpl_permlike_zx. + rewrite n_yank_1_l_helper_helper. + rewrite <- (nwire_removal_r (— ↕ n_cup_unswapped n)). + rewrite stack_compose_distr. + rewrite compose_assoc. + apply compose_simplify; [ + bundle_wires; prop_perm_eq|]. + cleanup_zx. + enough (Hrw : — ↕ n_cup_unswapped n ↕ n_wire n ⟷ (— ↕ ⊂ ↕ n_wire n) + ∝ @cast (1 + (n + n) + n) (1 + 2 + n) (1 + (n + n) + 0 + n) (1 + 0 + 2 + n) + (ltac:(lia)) (ltac:(lia)) ( + — ↕ n_cup_unswapped n ↕ ⦰ ↕ n_wire n ⟷ (— ↕ ⦰ ↕ ⊂ ↕ n_wire n))). + - rewrite Hrw. + repeat rewrite <- stack_compose_distr. + rewrite 2!nwire_removal_l. + rewrite (stack_assoc' _ ⊂ (n_wire n)). + simpl_casts. + do 2 (apply stack_simplify; [|easy]). + apply stack_simplify; cleanup_zx; easy. + + - cleanup_zx. simpl_permlike_zx. + rewrite (cast_compose_mid_contract _ (1 + 0 + n)), cast_id_eq. + apply compose_simplify; [|easy]. + rewrite cast_stack_l, cast_cast_eq, cast_id_eq. + easy. + + Unshelve. all: lia. +Qed. + + + +Lemma n_yank_1_l : forall n, + (— ↕ n_wire n) ↕ ((n_cup_unswapped (S n)) ⊤) ⟷ zx_inv_associator ⟷ ((n_cup_unswapped (S n)) ↕ (— ↕ n_wire n)) + ∝ — ↕ (n_wire n ↕ ((n_cup_unswapped n) ⊤) ⟷ zx_inv_associator ⟷ ((n_cup_unswapped n) ↕ n_wire n)). +Proof. + intros n. + rewrite n_cup_unswapped_grow_l at 1. + rewrite compose_transpose. + rewrite n_cup_unswapped_grow_r. + unfold zx_inv_associator. + simpl_permlike_zx. + (* bundle_wires. *) + rewrite nwire_stack_distr_compose_l. + rewrite (cast_compose_mid_contract _ ((S n + (n + n)))). + rewrite (cast_stack_distribute). + simpl_permlike_zx. + rewrite compose_assoc. + simpl_casts. + + rewrite n_yank_1_l_helper, cast_id_eq. + rewrite compose_assoc. + rewrite <- 3!stack_compose_distr. + rewrite yank_l. + simpl (n_wire (1 + n)). + rewrite (stack_assoc —). + simpl_casts. + rewrite wire_to_n_wire, nwire_removal_r, <- wire_to_n_wire. + rewrite (stack_assoc —). + simpl_casts. + bundle_wires. + rewrite 2!nwire_removal_l. + (* rewrite <- (wire_stack_distr_compose_l _ _ _ (n_cup_unswapped n ↕ n_wire n) (n_wire n)). *) + replace (S n + (n + n))%nat with (1 + (n + (n + n)))%nat by easy. + rewrite (wire_stack_distr_compose_l (n + 0)). + simpl_casts. + bundle_wires; rewrite nwire_removal_r. + rewrite (prop_iff_double_cast (1 + (n + 0)) (0 + (1 + n))). + rewrite 2!(cast_compose_mid_contract _ (1 + (n + n + n))). + rewrite 2!cast_cast_eq, 2!cast_id_eq. + easy. + Unshelve. all: lia. +Qed. + + +Lemma n_yank_l_unswapped : forall n {prf1 prf2}, + (n_wire n) ↕ ((n_cup_unswapped n) ⊤) ⟷ zx_inv_associator ⟷ ((n_cup_unswapped n) ↕ (n_wire n)) + ∝ cast _ _ prf1 prf2 (n_wire n). +Proof. + intros. + induction n. + - unfold zx_inv_associator, n_cup_unswapped. + prop_perm_eq. + - simpl (n_wire S n). + rewrite n_yank_1_l, IHn. + prop_perm_eq. + Unshelve. + all: lia. +Qed. + +Lemma compose_zx_inv_associator_r : forall {n0 n m o} (zx : ZX n0 (n + (m + o))) prf1 prf2, + zx ⟷ zx_inv_associator ∝ cast n0 (n + m + o) prf1 prf2 zx. +Proof. + intros. + rewrite (prop_iff_double_cast n0 (n + (m + o))). + rewrite (cast_compose_mid_contract _ (n + (m + o))). + rewrite cast_cast_eq, 2!cast_id_eq. + rewrite <- (nwire_removal_r zx) at 2. + apply compose_simplify; [easy | prop_perm_eq]. + Unshelve. + all: lia. +Qed. + +Lemma n_yank_l_unswapped': forall n {prf1 prf2}, + cast n (n+n+n) prf1 prf2 (n_wire n ↕ (n_cup_unswapped n) ⊤) ⟷ ((n_cup_unswapped n) ↕ (n_wire n)) + ∝ n_wire n. +Proof. + intros. + rewrite (prop_iff_double_cast (n + 0) (0 + n)). + rewrite <- n_yank_l_unswapped. + rewrite compose_zx_inv_associator_r. + rewrite (cast_compose_mid_contract _ (n + n + n)), cast_id_eq, cast_cast_eq. + easy. + Unshelve. + all: lia. +Qed. + + + +(* Lemma n_swap_zxperm : forall n, + ZXperm n (n_swap n). +Proof. + induction n; simpl; auto with zxperm_db. +Qed. + +#[export] Hint Resolve n_swap_zxperm : zxperm_db. + +Lemma perm_of_n_swap : forall n, + perm_of_zx (n_swap n) = fun k => if n <=? k then k else (n - 1 - k)%nat. +Proof. + intros n. + induction n. + - easy. + - simpl (n_swap (S n)). + cleanup_perm_of_zx. + simpl perm_of_zx. + rewrite IHn. + solve_modular_permutation_equalities. +Qed. + +#[export] Hint Rewrite perm_of_n_swap : perm_of_zx_cleanup_db. *) + +Lemma n_swap_grow_r' : forall n prf1 prf2, + n_swap (S n) ∝ top_to_bottom (S n) ⟷ cast (S n) (S n) prf1 prf2 (n_swap n ↕ —). +Proof. + intros. + prop_perm_eq. + solve_modular_permutation_equalities. +Qed. + +Lemma n_swap_n_swap : forall n, + n_swap n ⟷ n_swap n ∝ n_wire n. +Proof. + prop_perm_eq. + solve_modular_permutation_equalities. +Qed. + +#[export] Hint Rewrite n_swap_n_swap : perm_inv_db. + +(* Lemma n_cup_unswapped_n_swap_n_swap_helper : forall n {prf1 prf2 prf3 prf4 prf5 prf6}, + (— ↕ n_swap n ↕ (cast (S n) (S n) prf5 prf6 (n_swap n ↕ — )) + ⟷ (cast (S n + S n) 2 prf1 prf2 (— ↕ n_cup_unswapped n ↕ —) ⟷ ⊃)) + ∝ cast _ _ prf3 prf4 (— ↕ (n_swap n ↕ n_swap n ⟷ n_cup_unswapped n) ↕ — + ⟷ ⊃). +Proof. + intros. + simpl_casts. + rewrite stack_assoc', (stack_assoc —). + simpl_casts. + rewrite (cast_compose_l _ _ _ ⊃). + rewrite <- cast_compose_mid_contract. + simpl_casts. + rewrite <- compose_assoc. + apply compose_simplify; [|easy]. + rewrite wire_stack_distr_compose_l, wire_stack_distr_compose_r. + easy. + Unshelve. all: lia. +Qed. *) + +Lemma n_cup_unswapped_n_swap_n_swap : forall n, + n_swap n ↕ n_swap n ⟷ n_cup_unswapped n ∝ n_cup_unswapped n. +Proof. + intros n. + rewrite <- (nwire_removal_l (n_swap n)) at 1. + rewrite <- (nwire_removal_r (n_swap n)) at 2. + rewrite stack_compose_distr. + rewrite compose_assoc, n_swap_cup_flip, <- compose_assoc. + rewrite <- stack_compose_distr, n_swap_n_swap. + rewrite nwire_removal_l, <- n_wire_add_stack, nwire_removal_l. + easy. +Qed. + +Lemma n_cup_inv_n_swap_n_wire' : forall n, n_cup n ∝ n_wire n ↕ n_swap n ⟷ n_cup_unswapped n. +Proof. + intros n. + unfold n_cup. + apply n_swap_cup_flip. +Qed. + +Lemma n_swap_transpose : forall n, + (n_swap n) ⊤ ∝ n_swap n. +Proof. + intros n. + prop_perm_eq. + perm_eq_by_WF_inv_inj (perm_of_zx (n_swap n)) n; + [apply zxperm_permutation | apply perm_of_zx_WF + | rewrite perm_of_transpose_is_linv | ]; + auto with zxperm_db. + cleanup_perm_of_zx. + solve_modular_permutation_equalities. +Qed. + +Lemma n_yank_l : forall n {prf1 prf2}, + cast n (n + n + n) prf1 prf2 (n_wire n ↕ n_cap n) + ⟷ (n_cup n ↕ n_wire n) ∝ n_wire n. +Proof. + intros. + unfold n_cap. + rewrite n_cup_inv_n_swap_n_wire' at 2. + unfold n_cup. + (* rewrite n_cup_inv_n_swap_n_wire. *) + simpl. + rewrite n_wire_transpose, n_swap_transpose. + rewrite nwire_stack_distr_compose_l. + rewrite stack_assoc'. + rewrite (cast_compose_mid_contract n (n + n + n) (n + n + n)). + rewrite cast_cast_eq, cast_id_eq. + rewrite nwire_stack_distr_compose_r. + rewrite <- compose_assoc, (compose_assoc _ _ (_ ↕ n_swap n ↕ _)). + rewrite <- 2!stack_compose_distr, n_swap_n_swap, nwire_removal_l. + rewrite <- 2!n_wire_add_stack, nwire_removal_r. + apply n_yank_l_unswapped'. + Unshelve. + all: lia. +Qed. Lemma n_yank_r : forall n {prf1 prf2 prf3 prf4}, cast n n prf3 prf4 (cast n (n + (n + n)) prf1 prf2 (n_cap n ↕ n_wire n) ⟷ (n_wire n ↕ n_cup n)) ∝ n_wire n. Proof. - induction n. - - intros. - rewrite n_cap_0_empty. - rewrite n_cup_0_empty. - simpl_casts. - cleanup_zx. simpl. - simpl_casts. cleanup_zx. - easy. - - intros. -Admitted. + intros. + apply transpose_diagrams. + rewrite cast_transpose, compose_transpose. + rewrite (cast_compose_mid_contract _ (n+n+n)). + rewrite cast_transpose, cast_cast_eq, cast_id_eq. + rewrite 2!stack_transpose. + rewrite n_wire_transpose. + unfold n_cap. + rewrite Proportional.transpose_involutive. + apply n_yank_l. + Unshelve. + all: lia. +Qed. Lemma zx_triangle_1 : forall n, zx_inv_right_unitor ⟷ (n_wire n ↕ n_cap n) ⟷ zx_inv_associator