\ / GNU General Public License Version 2
V_____________________________________________________________*)
+include "basics/core_notation/fintersects_2.ma".
+include "basics/finset.ma".
include "basics/vectors.ma".
+include "basics/finset.ma".
(* include "basics/relations.ma". *)
(******************************** tape ****************************************)
| cons r0 rs0 ⇒ leftof ? r0 rs0 ]
| cons l0 ls0 ⇒ rightof ? l0 ls0 ] ].
+lemma right_mk_tape :
+ ∀sig,ls,c,rs.(c = None ? → ls = [ ] ∨ rs = [ ]) → right ? (mk_tape sig ls c rs) = rs.
+#sig #ls #c #rs cases c // cases ls
+[ cases rs //
+| #l0 #ls0 #H normalize cases (H (refl ??)) #H1 [ destruct (H1) | >H1 % ] ]
+qed-.
+
+lemma left_mk_tape : ∀sig,ls,c,rs.left ? (mk_tape sig ls c rs) = ls.
+#sig #ls #c #rs cases c // cases ls // cases rs //
+qed.
+
+lemma current_mk_tape : ∀sig,ls,c,rs.current ? (mk_tape sig ls c rs) = c.
+#sig #ls #c #rs cases c // cases ls // cases rs //
+qed.
+
+lemma current_to_midtape: ∀sig,t,c. current sig t = Some ? c →
+ ∃ls,rs. t = midtape ? ls c rs.
+#sig *
+ [#c whd in ⊢ ((??%?)→?); #Hfalse destruct
+ |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
+ |#a #l #c whd in ⊢ ((??%?)→?); #Hfalse destruct
+ |#ls #a #rs #c whd in ⊢ ((??%?)→?); #H destruct
+ @(ex_intro … ls) @(ex_intro … rs) //
+ ]
+qed.
+
+(*********************************** moves ************************************)
+
inductive move : Type[0] ≝
| L : move | R : move | N : move.
+
+(*************************** turning moves into a DeqSet **********************)
+
+definition move_eq ≝ λm1,m2:move.
+ match m1 with
+ [R ⇒ match m2 with [R ⇒ true | _ ⇒ false]
+ |L ⇒ match m2 with [L ⇒ true | _ ⇒ false]
+ |N ⇒ match m2 with [N ⇒ true | _ ⇒ false]].
+
+lemma move_eq_true:∀m1,m2.
+ move_eq m1 m2 = true ↔ m1 = m2.
+*
+ [* normalize [% #_ % |2,3: % #H destruct ]
+ |* normalize [1,3: % #H destruct |% #_ % ]
+ |* normalize [1,2: % #H destruct |% #_ % ]
+qed.
+
+definition DeqMove ≝ mk_DeqSet move move_eq move_eq_true.
+
+unification hint 0 ≔ ;
+ X ≟ DeqMove
+(* ---------------------------------------- *) ⊢
+ move ≡ carr X.
+
+unification hint 0 ≔ m1,m2;
+ X ≟ DeqMove
+(* ---------------------------------------- *) ⊢
+ move_eq m1 m2 ≡ eqb X m1 m2.
+
+(************************ turning DeqMove into a FinSet ***********************)
+
+definition move_enum ≝ [L;R;N].
+
+lemma move_enum_unique: uniqueb ? [L;R;N] = true.
+// qed.
+
+lemma move_enum_complete: ∀x:move. memb ? x [L;R;N] = true.
+* // qed.
+
+definition FinMove ≝
+ mk_FinSet DeqMove [L;R;N] move_enum_unique move_enum_complete.
+
+unification hint 0 ≔ ;
+ X ≟ FinMove
+(* ---------------------------------------- *) ⊢
+ move ≡ FinSetcarr X.
(********************************** machine ***********************************)
record TM (sig:FinSet): Type[1] ≝
{ states : FinSet;
- trans : states × (option sig) → states × (option (sig × move));
+ trans : states × (option sig) → states × (option sig) × move;
start: states;
halt : states → bool
}.
-definition tape_move_left ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
- match lt with
- [ nil ⇒ leftof sig c rt
- | cons c0 lt0 ⇒ midtape sig lt0 c0 (c::rt) ].
+definition tape_move_left ≝ λsig:FinSet.λt:tape sig.
+ match t with
+ [ niltape ⇒ niltape sig
+ | leftof _ _ ⇒ t
+ | rightof a ls ⇒ midtape sig ls a [ ]
+ | midtape ls a rs ⇒
+ match ls with
+ [ nil ⇒ leftof sig a rs
+ | cons a0 ls0 ⇒ midtape sig ls0 a0 (a::rs)
+ ]
+ ].
-definition tape_move_right ≝ λsig:FinSet.λlt:list sig.λc:sig.λrt:list sig.
- match rt with
- [ nil ⇒ rightof sig c lt
- | cons c0 rt0 ⇒ midtape sig (c::lt) c0 rt0 ].
+definition tape_move_right ≝ λsig:FinSet.λt:tape sig.
+ match t with
+ [ niltape ⇒ niltape sig
+ | rightof _ _ ⇒ t
+ | leftof a rs ⇒ midtape sig [ ] a rs
+ | midtape ls a rs ⇒
+ match rs with
+ [ nil ⇒ rightof sig a ls
+ | cons a0 rs0 ⇒ midtape sig (a::ls) a0 rs0
+ ]
+ ].
+
+definition tape_write ≝ λsig.λt: tape sig.λs:option sig.
+ match s with
+ [ None ⇒ t
+ | Some s0 ⇒ midtape ? (left ? t) s0 (right ? t)
+ ].
-definition tape_move ≝ λsig.λt: tape sig.λm:option (sig × move).
+definition tape_move ≝ λsig.λt: tape sig.λm:move.
match m with
- [ None ⇒ t
- | Some m' ⇒
- let 〈s,m1〉 ≝ m' in
- match m1 with
- [ R ⇒ tape_move_right ? (left ? t) s (right ? t)
- | L ⇒ tape_move_left ? (left ? t) s (right ? t)
- | N ⇒ midtape ? (left ? t) s (right ? t)
- ] ].
+ [ R ⇒ tape_move_right ? t
+ | L ⇒ tape_move_left ? t
+ | N ⇒ t
+ ].
+
+definition tape_move_mono ≝
+ λsig,t,mv.
+ tape_move sig (tape_write sig t (\fst mv)) (\snd mv).
record config (sig,states:FinSet): Type[0] ≝
{ cstate : states;
definition step ≝ λsig.λM:TM sig.λc:config sig (states sig M).
let current_char ≝ current ? (ctape ?? c) in
- let 〈news,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
- mk_config ?? news (tape_move sig (ctape ?? c) mv).
+ let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
+ mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
+(*
+lemma step_eq : ∀sig,M,c.
+ let current_char ≝ current ? (ctape ?? c) in
+ let 〈news,a,mv〉 ≝ trans sig M 〈cstate ?? c,current_char〉 in
+ step sig M c =
+ mk_config ?? news (tape_move sig (tape_write ? (ctape ?? c) a) mv).
+#sig #M #c
+ whd in match (tape_move_mono sig ??);
+*)
+
(******************************** loop ****************************************)
let rec loop (A:Type[0]) n (f:A→A) p a on n ≝
match n with
notation "M ⊨ [q: R1,R2]" non associative with precedence 45 for @{ 'cmodels $M $q $R1 $R2}.
interpretation "conditional realizability" 'cmodels M q R1 R2 = (accRealize ? M q R1 R2).
+(*************************** guarded realizablity *****************************)
+definition GRealize ≝ λsig.λM:TM sig.λPre:tape sig →Prop.λR:relation (tape sig).
+∀t.Pre t → ∃i.∃outc.
+ loopM sig M i (initc sig M t) = Some ? outc ∧ R t (ctape ?? outc).
+
+definition accGRealize ≝ λsig.λM:TM sig.λacc:states sig M.
+λPre: tape sig → Prop.λRtrue,Rfalse.
+∀t.Pre t → ∃i.∃outc.
+ loopM sig M i (initc sig M t) = Some ? outc ∧
+ (cstate ?? outc = acc → Rtrue t (ctape ?? outc)) ∧
+ (cstate ?? outc ≠ acc → Rfalse t (ctape ?? outc)).
+
+lemma WRealize_to_GRealize : ∀sig.∀M: TM sig.∀Pre,R.
+ (∀t.Pre t → M ↓ t) → M ⊫ R → GRealize sig M Pre R.
+#sig #M #Pre #R #HT #HW #t #HPre cases (HT … t HPre) #i * #outc #Hloop
+@(ex_intro … i) @(ex_intro … outc) % // @(HW … i) //
+qed.
+
+lemma Realize_to_GRealize : ∀sig,M.∀P,R.
+ M ⊨ R → GRealize sig M P R.
+#alpha #M #Pre #R #HR #t #HPre
+cases (HR t) -HR #k * #outc * #Hloop #HR
+@(ex_intro ?? k) @(ex_intro ?? outc) %
+ [ @Hloop | @HR ]
+qed.
+
+lemma acc_Realize_to_acc_GRealize: ∀sig,M.∀q:states sig M.∀P,R1,R2.
+ M ⊨ [q:R1,R2] → accGRealize sig M q P R1 R2.
+#alpha #M #q #Pre #R1 #R2 #HR #t #HPre
+cases (HR t) -HR #k * #outc * * #Hloop #HRtrue #HRfalse
+@(ex_intro ?? k) @(ex_intro ?? outc) %
+ [ % [@Hloop] @HRtrue | @HRfalse]
+qed.
+
+(******************************** monotonicity ********************************)
+lemma Realize_to_Realize : ∀alpha,M,R1,R2.
+ R1 ⊆ R2 → Realize alpha M R1 → Realize alpha M R2.
+#alpha #M #R1 #R2 #Himpl #HR1 #intape
+cases (HR1 intape) -HR1 #k * #outc * #Hloop #HR1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma WRealize_to_WRealize: ∀sig,M,R1,R2.
+ R1 ⊆ R2 → WRealize sig M R1 → WRealize ? M R2.
+#alpha #M #R1 #R2 #Hsub #HR1 #intape #i #outc #Hloop
+@Hsub @(HR1 … i) @Hloop
+qed.
+
+lemma GRealize_to_GRealize : ∀alpha,M,P,R1,R2.
+ R1 ⊆ R2 → GRealize alpha M P R1 → GRealize alpha M P R2.
+#alpha #M #P #R1 #R2 #Himpl #HR1 #intape #HP
+cases (HR1 intape HP) -HR1 #k * #outc * #Hloop #HR1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma GRealize_to_GRealize_2 : ∀alpha,M,P1,P2,R1,R2.
+ P2 ⊆ P1 → R1 ⊆ R2 → GRealize alpha M P1 R1 → GRealize alpha M P2 R2.
+#alpha #M #P1 #P2 #R1 #R2 #Himpl1 #Himpl2 #H1 #intape #HP
+cases (H1 intape (Himpl1 … HP)) -H1 #k * #outc * #Hloop #H1
+@(ex_intro ?? k) @(ex_intro ?? outc) % /2/
+qed.
+
+lemma acc_Realize_to_acc_Realize: ∀sig,M.∀q:states sig M.∀R1,R2,R3,R4.
+ R1 ⊆ R3 → R2 ⊆ R4 → M ⊨ [q:R1,R2] → M ⊨ [q:R3,R4].
+#alpha #M #q #R1 #R2 #R3 #R4 #Hsub13 #Hsub24 #HRa #intape
+cases (HRa intape) -HRa #k * #outc * * #Hloop #HRtrue #HRfalse
+@(ex_intro ?? k) @(ex_intro ?? outc) %
+ [ % [@Hloop] #Hq @Hsub13 @HRtrue // | #Hq @Hsub24 @HRfalse //]
+qed.
+
+(**************************** A canonical relation ****************************)
+
+definition R_TM ≝ λsig.λM:TM sig.λq.λt1,t2.
+∃i,outc.
+ loopM ? M i (mk_config ?? q t1) = Some ? outc ∧
+ t2 = (ctape ?? outc).
+
+lemma R_TM_to_R: ∀sig,M,R. ∀t1,t2.
+ M ⊫ R → R_TM ? M (start sig M) t1 t2 → R t1 t2.
+#sig #M #R #t1 #t2 whd in ⊢ (%→?); #HMR * #i * #outc *
+#Hloop #Ht2 >Ht2 @(HMR … Hloop)
+qed.
+
(******************************** NOP Machine *********************************)
(* NO OPERATION
definition nop ≝
λalpha:FinSet.mk_TM alpha nop_states
- (λp.let 〈q,a〉 ≝ p in 〈q,None ?〉)
+ (λp.let 〈q,a〉 ≝ p in 〈q,None ?,N〉)
start_nop (λ_.true).
definition R_nop ≝ λalpha.λt1,t2:tape alpha.t2 = t1.
λp. let 〈s,a〉 ≝ p in
match s with
[ inl s1 ⇒
- if halt sig M1 s1 then 〈inr … (start sig M2), None ?〉
- else let 〈news1,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,m〉
- | inr s2 ⇒ let 〈news2,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,m〉
+ if halt sig M1 s1 then 〈inr … (start sig M2), None ?,N〉
+ else let 〈news1,newa,m〉 ≝ trans sig M1 〈s1,a〉 in 〈inl … news1,newa,m〉
+ | inr s2 ⇒ let 〈news2,newa,m〉 ≝ trans sig M2 〈s2,a〉 in 〈inr … news2,newa,m〉
].
definition seq ≝ λsig. λM1,M2 : TM sig.
(λs.match s with
[ inl _ ⇒ false | inr s2 ⇒ halt sig M2 s2]).
-notation "a · b" non associative with precedence 65 for @{ 'middot $a $b}.
+notation "a · b" right associative with precedence 65 for @{ 'middot $a $b}.
interpretation "sequential composition" 'middot a b = (seq ? a b).
definition lift_confL ≝
#sig #S1 #S2 #halt #c cases c #s #t %
qed.
-lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftL : ∀sig,M1,M2,s,a,news,newa,move.
halt ? M1 s = false →
- trans sig M1 〈s,a〉 = 〈news,move〉 →
- trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,move〉.
-#sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #move
+ trans sig M1 〈s,a〉 = 〈news,newa,move〉 →
+ trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inl … news,newa,move〉.
+#sig (*#M1*) * #Q1 #T1 #init1 #halt1 #M2 #s #a #news #newa #move
#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
qed.
-lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,move.
+lemma trans_seq_liftR : ∀sig,M1,M2,s,a,news,newa,move.
halt ? M2 s = false →
- trans sig M2 〈s,a〉 = 〈news,move〉 →
- trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,move〉.
-#sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #move
+ trans sig M2 〈s,a〉 = 〈news,newa,move〉 →
+ trans sig (seq sig M1 M2) 〈inr … s,a〉 = 〈inr … news,newa,move〉.
+#sig #M1 * #Q2 #T2 #init2 #halt2 #s #a #news #newa #move
#Hhalt #Htrans whd in ⊢ (??%?); >Hhalt >Htrans %
qed.
#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
lapply (refl ? (trans ?? 〈s,current sig t〉))
cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
- #s0 #m0 cases t
+ * #s0 #a0 #m0 cases t
[ #Heq #Hhalt
| 2,3: #s1 #l1 #Heq #Hhalt
|#ls #s1 #rs #Heq #Hhalt ]
#sig #M1 (* * #Q1 #T1 #init1 #halt1 *) #M2 * #s #t
lapply (refl ? (trans ?? 〈s,current sig t〉))
cases (trans ?? 〈s,current sig t〉) in ⊢ (???% → %);
- #s0 #m0 cases t
+ * #s0 #a0 #m0 cases t
[ #Heq #Hhalt
| 2,3: #s1 #l1 #Heq #Hhalt
|#ls #s1 #rs #Heq #Hhalt ]
lemma trans_liftL_true : ∀sig,M1,M2,s,a.
halt ? M1 s = true →
- trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?〉.
+ trans sig (seq sig M1 M2) 〈inl … s,a〉 = 〈inr … (start ? M2),None ?,N〉.
#sig #M1 #M2 #s #a #Hhalt whd in ⊢ (??%?); >Hhalt %
qed.
#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
% [@Hloop |@Hsub @Houtc]
qed.
+
+(* composition with guards *)
+theorem sem_seq_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2.
+ GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 →
+ (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) →
+ GRealize sig (M1 · M2) Pre1 (R1 ∘ R2).
+#sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #HGR1 #HGR2 #Hinv #t1 #HPre1
+cases (HGR1 t1 HPre1) #k1 * #outc1 * #Hloop1 #HM1
+cases (HGR2 (ctape sig (states ? M1) outc1) ?)
+ [2: @(Hinv … HPre1 HM1)]
+#k2 * #outc2 * #Hloop2 #HM2
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+%
+[@(loop_merge ???????????
+ (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
+ (step sig M1) (step sig (seq sig M1 M2))
+ (λc.halt sig M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
+ [ * *
+ [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+ | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+ || #c0 #Hhalt <step_seq_liftL //
+ | #x <p_halt_liftL %
+ |6:cases outc1 #s1 #t1 %
+ |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt <step_seq_liftR // ]
+ |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
+ generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
+ >(trans_liftL_true sig M1 M2 ??)
+ [ whd in ⊢ (??%?); whd in ⊢ (???%);
+ @config_eq whd in ⊢ (???%); //
+ | @(loop_Some ?????? Hloop10) ]
+ ]
+| @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+ % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R //
+]
+qed.
+
+theorem sem_seq_app_guarded: ∀sig.∀M1,M2:TM sig.∀Pre1,Pre2,R1,R2,R3.
+ GRealize sig M1 Pre1 R1 → GRealize sig M2 Pre2 R2 →
+ (∀t1,t2.Pre1 t1 → R1 t1 t2 → Pre2 t2) → R1 ∘ R2 ⊆ R3 →
+ GRealize sig (M1 · M2) Pre1 R3.
+#sig #M1 #M2 #Pre1 #Pre2 #R1 #R2 #R3 #HR1 #HR2 #Hinv #Hsub
+#t #HPre1 cases (sem_seq_guarded … HR1 HR2 Hinv t HPre1)
+#k * #outc * #Hloop #Houtc @(ex_intro … k) @(ex_intro … outc)
+% [@Hloop |@Hsub @Houtc]
+qed.
+
+theorem acc_sem_seq : ∀sig.∀M1,M2:TM sig.∀R1,Rtrue,Rfalse,acc.
+ M1 ⊨ R1 → M2 ⊨ [ acc: Rtrue, Rfalse ] →
+ M1 · M2 ⊨ [ inr … acc: R1 ∘ Rtrue, R1 ∘ Rfalse ].
+#sig #M1 #M2 #R1 #Rtrue #Rfalse #acc #HR1 #HR2 #t
+cases (HR1 t) #k1 * #outc1 * #Hloop1 #HM1
+cases (HR2 (ctape sig (states ? M1) outc1)) #k2 * #outc2 * * #Hloop2
+#HMtrue #HMfalse
+@(ex_intro … (k1+k2)) @(ex_intro … (lift_confR … outc2))
+% [ %
+[@(loop_merge …
+ (loop_lift ??? (lift_confL sig (states sig M1) (states sig M2))
+ (step sig M1) (step sig (seq sig M1 M2))
+ (λc.halt sig M1 (cstate … c))
+ (λc.halt_liftL ?? (halt sig M1) (cstate … c)) … Hloop1))
+ [ * *
+ [ #sl #tl whd in ⊢ (??%? → ?); #Hl %
+ | #sr #tr whd in ⊢ (??%? → ?); #Hr destruct (Hr) ]
+ || #c0 #Hhalt <step_seq_liftL //
+ | #x <p_halt_liftL %
+ |6:cases outc1 #s1 #t1 %
+ |7:@(loop_lift … (initc ?? (ctape … outc1)) … Hloop2)
+ [ * #s2 #t2 %
+ | #c0 #Hhalt <step_seq_liftR // ]
+ |whd in ⊢ (??(???%)?);whd in ⊢ (??%?);
+ generalize in match Hloop1; cases outc1 #sc1 #tc1 #Hloop10
+ >(trans_liftL_true sig M1 M2 ??)
+ [ whd in ⊢ (??%?); whd in ⊢ (???%); //
+ | @(loop_Some ?????? Hloop10) ]
+ ]
+| >(config_expand … outc2) in ⊢ (%→?); whd in ⊢ (??%?→?);
+ #Hqtrue destruct (Hqtrue)
+ @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+ % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R /2/ ]
+| >(config_expand … outc2) in ⊢ (%→?); whd in ⊢ (?(??%?)→?); #Hqfalse
+ @(ex_intro … (ctape ? (FinSum (states ? M1) (states ? M2)) (lift_confL … outc1)))
+ % // >eq_ctape_lift_conf_L >eq_ctape_lift_conf_R @HMfalse
+ @(not_to_not … Hqfalse) //
+]
+qed.
+
+lemma acc_sem_seq_app : ∀sig.∀M1,M2:TM sig.∀R1,Rtrue,Rfalse,R2,R3,acc.
+ M1 ⊨ R1 → M2 ⊨ [acc: Rtrue, Rfalse] →
+ (∀t1,t2,t3. R1 t1 t3 → Rtrue t3 t2 → R2 t1 t2) →
+ (∀t1,t2,t3. R1 t1 t3 → Rfalse t3 t2 → R3 t1 t2) →
+ M1 · M2 ⊨ [inr … acc : R2, R3].
+#sig #M1 #M2 #R1 #Rtrue #Rfalse #R2 #R3 #acc
+#HR1 #HRacc #Hsub1 #Hsub2
+#t cases (acc_sem_seq … HR1 HRacc t)
+#k * #outc * * #Hloop #Houtc1 #Houtc2 @(ex_intro … k) @(ex_intro … outc)
+% [% [@Hloop
+ |#H cases (Houtc1 H) #t3 * #Hleft #Hright @Hsub1 // ]
+ |#H cases (Houtc2 H) #t3 * #Hleft #Hright @Hsub2 // ]
+qed.
projects / helm.git / blobdiff