+++ /dev/null
-H = @
-XOA_DIR = ../../../components/binaries/xoa
-XOA = xoa.native
-
-CONF = xoa.conf.xml
-TARGETS = xoa_natation.ma xoa.ma
-
-all: $(TARGETS)
-
-$(TARGETS): $(CONF)
- @echo " EXEC $(XOA) $(CONF)"
- $(H)MATITA_RT_BASE_DIR=../.. $(XOA_DIR)/$(XOA) $(CONF)
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "arithmetics/nat.ma".
-include "lambda-delta/xoa_props.ma".
-
-(* ARITHMETICAL PROPERTIES **************************************************)
-
-lemma plus_S_eq_O_false: ∀n,m. n + S m = 0 → False.
-#n #m <plus_n_Sm #H destruct
-qed.
-
-lemma plus_S_le_to_pos: ∀n,m,p. n + S m ≤ p → 0 < p.
-#n #m #p <plus_n_Sm #H @(lt_to_le_to_lt … H) //
-qed.
-
-lemma minus_le: ∀m,n. m - n ≤ m.
-/2/ qed.
-
-lemma le_O_to_eq_O: ∀n. n ≤ 0 → n = 0.
-/2/ qed.
-
-lemma lt_to_le: ∀a,b. a < b → a ≤ b.
-/2/ qed.
-
-lemma lt_refl_false: ∀n. n < n → False.
-#n #H elim (lt_to_not_eq … H) -H /2/
-qed.
-
-lemma lt_zero_false: ∀n. n < 0 → False.
-#n #H elim (lt_to_not_le … H) -H /2/
-qed.
-
-lemma lt_or_ge: ∀m,n. m < n ∨ n ≤ m.
-#m #n elim (decidable_lt m n) /3/
-qed.
-
-lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m.
-#m elim m -m
-[ * /2/
-| #m #IHm * [ /2/ ]
- #n elim (IHm n) -IHm #H
- [ @or3_intro0 | @or3_intro1 destruct | @or3_intro2 ] /2/ (**) (* /3/ is slow *)
- qed.
-
-lemma le_to_lt_or_eq: ∀m,n. m ≤ n → m < n ∨ m = n.
-#m #n * -n /3/
-qed.
-
-lemma plus_le_weak: ∀m,n,p. m + n ≤ p → n ≤ p.
-/2/ qed.
-
-lemma plus_lt_false: ∀m,n. m + n < m → False.
-#m #n #H1 lapply (le_plus_n_r n m) #H2
-lapply (le_to_lt_to_lt … H2 H1) -H2 H1 #H
-elim (lt_refl_false … H)
-qed.
-
-lemma monotonic_lt_minus_l: ∀p,q,n. n ≤ q → q < p → q - n < p - n.
-#p #q #n #H1 #H2
-@lt_plus_to_minus_r <plus_minus_m_m //.
-qed.
-
-lemma plus_le_minus: ∀a,b,c. a + b ≤ c → a ≤ c - b.
-/2/ qed.
-
-lemma lt_plus_minus: ∀i,u,d. u ≤ i → i < d + u → i - u < d.
-/2/ qed.
-
-lemma plus_plus_comm_23: ∀m,n,p. m + n + p = m + p + n.
-// qed.
-
-lemma le_plus_minus_comm: ∀n,m,p. p ≤ m → m + n - p = m - p + n.
-#n #m #p #lepm @plus_to_minus <associative_plus
->(commutative_plus p) <plus_minus_m_m //
-qed.
-
-lemma minus_le_minus_minus_comm: ∀b,c,a. c ≤ b → a - (b - c) = a + c - b.
-#b elim b -b
-[ #c #a #H >(le_O_to_eq_O … H) -H //
-| #b #IHb #c elim c -c //
- #c #_ #a #Hcb
- lapply (le_S_S_to_le … Hcb) -Hcb #Hcb
- <plus_n_Sm normalize /2/
-]
-qed.
-
-lemma minus_plus_comm: ∀a,b,c. a - b - c = a - (c + b).
-// qed.
-
-lemma minus_minus_comm: ∀a,b,c. a - b - c = a - c - b.
-/3/ qed.
-
-lemma le_plus_minus: ∀a,b,c. c ≤ b → a + b - c = a + (b - c).
-/2/ qed.
-
-lemma plus_minus_m_m_comm: ∀n,m. m ≤ n → n = m + (n - m).
-/2/ qed.
-
-theorem minus_plus_m_m_comm: ∀n,m. n = (m + n) - m.
-/2/ qed.
-
-lemma arith_a2: ∀a,c1,c2. c1 + c2 ≤ a → a - c1 - c2 + (c1 + c2) = a.
-/2/ qed.
-
-lemma arith_b1: ∀a,b,c1. c1 ≤ b → a - c1 - (b - c1) = a - b.
-#a #b #c1 #H >minus_plus @eq_f2 /2/
-qed.
-
-lemma arith_b2: ∀a,b,c1,c2. c1 + c2 ≤ b → a - c1 - c2 - (b - c1 - c2) = a - b.
-#a #b #c1 #c2 #H >minus_plus >minus_plus >minus_plus /2/
-qed.
-
-lemma arith_c1: ∀a,b,c1. a + c1 - (b + c1) = a - b.
-// qed.
-
-lemma arith_c1x: ∀x,a,b,c1. x + c1 + a - (b + c1) = x + a - b.
-#x #a #b #c1 >plus_plus_comm_23 //
-qed.
-
-lemma arith_d1: ∀a,b,c1. c1 ≤ b → a + c1 + (b - c1) = a + b.
-/2/ qed.
-
-lemma arith_e2: ∀a,c1,c2. a ≤ c1 → c1 + c2 - (c1 - a + c2) = a.
-/3/ qed.
-
-lemma arith_f1: ∀a,b,c1. a + b ≤ c1 → c1 - (c1 - a - b) = a + b.
-#a #b #c1 #H >minus_plus <minus_minus //
-qed.
-
-lemma arith_g1: ∀a,b,c1. c1 ≤ b → a - (b - c1) - c1 = a - b.
-/2/ qed.
-
-lemma arith_h1: ∀a1,a2,b,c1. c1 ≤ a1 → c1 ≤ b →
- a1 - c1 + a2 - (b - c1) = a1 + a2 - b.
-#a1 #a2 #b #c1 #H1 #H2 <le_plus_minus_comm /2/
-qed.
-
-lemma arith_i2: ∀a,c1,c2. c1 + c2 ≤ a → c1 + c2 + (a - c1 - c2) = a.
-/2/ qed.
-
-lemma arith_z1: ∀a,b,c1. a + c1 - b - c1 = a - b.
-// qed.
-
-(* unstable *****************************************************************)
-
-lemma arith1: ∀n,h,m,p. n + h + m ≤ p + h → n + m ≤ p.
-/2/ qed.
-
-lemma arith2: ∀j,i,e,d. d + e ≤ i → d ≤ i - e + j.
-#j #i #e #d #H lapply (plus_le_minus … H) -H /2/
-qed.
-
-lemma arith3: ∀a1,a2,b,c1. a1 + a2 ≤ b → a1 + c1 + a2 ≤ b + c1.
-/2/ qed.
-
-lemma arith4: ∀h,d,e1,e2. d ≤ e1 + e2 → d + h ≤ e1 + h + e2.
-/2/ qed.
-
-lemma arith5: ∀a,b1,b2,c1. c1 ≤ b1 → c1 ≤ a → a < b1 + b2 → a - c1 < b1 - c1 + b2.
-#a #b1 #b2 #c1 #H1 #H2 #H3
-<le_plus_minus_comm // @monotonic_lt_minus_l //
-qed.
-
-lemma arith8: ∀a,b. a < a + b + 1.
-// qed.
-
-lemma arith9: ∀a,b,c. c < a + (b + c + 1) + 1.
-// qed.
-
-lemma arith10: ∀a,b,c,d,e. a ≤ b → c + (a - d - e) ≤ c + (b - d - e).
-#a #b #c #d #e #H
->minus_plus >minus_plus @monotonic_le_plus_r @monotonic_le_minus_l //
-qed.
+++ /dev/null
-NAMING CONVENTIONS FOR METAVARIABLES
-
-H : reserved: transient premise
-IH : reserved: inductive premise
-I,J : item
-K,L : local environment
-T,U,V,W: term
-X,Y,Z : reserved: transient objet denoted by a capital letter
-
-d : relocation depth
-e : relocation height
-h : sort hierarchy parameter
-i,j : local reference position index (de Bruijn's)
-k : sort index
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-(* Syntax *******************************************************************)
-
-notation "hvbox( ⋆ )"
- non associative with precedence 90
- for @{ 'Star }.
-
-notation "hvbox( ⋆ term 90 k )"
- non associative with precedence 90
- for @{ 'Star $k }.
-
-notation "hvbox( 𝕚 { I } break (term 90 T1) . break (term 90 T) )"
- non associative with precedence 90
- for @{ 'SItem $I $T1 $T }.
-
-notation "hvbox( 𝕓 { I } break (term 90 T1) . break (term 90 T) )"
- non associative with precedence 90
- for @{ 'SBind $I $T1 $T }.
-
-notation "hvbox( 𝕗 { I } break (term 90 T1) . break (term 90 T) )"
- non associative with precedence 90
- for @{ 'SFlat $I $T1 $T }.
-
-notation "hvbox( T . break 𝕓 { I } break (term 90 T1) )"
- non associative with precedence 89
- for @{ 'DBind $T $I $T1 }.
-(*
-notation "hvbox( | L | )"
- non associative with precedence 70
- for @{ 'Length $L }.
-*)
-notation "hvbox( # term 90 x )"
- non associative with precedence 90
- for @{ 'Weight $x }.
-
-notation "hvbox( # [ x , break y ] )"
- non associative with precedence 90
- for @{ 'Weight $x $y }.
-
-(* Substitution *************************************************************)
-
-notation "hvbox( T1 break [ d , break e ] ≈ break T2 )"
- non associative with precedence 45
- for @{ 'Eq $T1 $d $e $T2 }.
-
-notation "hvbox( ↑ [ d , break e ] break T1 ≡ break T2 )"
- non associative with precedence 45
- for @{ 'RLift $T1 $d $e $T2 }.
-
-notation "hvbox( ↓ [ d , break e ] break L1 ≡ break L2 )"
- non associative with precedence 45
- for @{ 'RDrop $L1 $d $e $L2 }.
-
-notation "hvbox( L ⊢ break term 90 T1 break [ d , break e ] ≫ break T2 )"
- non associative with precedence 45
- for @{ 'PSubst $L $T1 $d $e $T2 }.
-
-(* Reduction ****************************************************************)
-
-notation "hvbox( T1 ⇒ break T2 )"
- non associative with precedence 45
- for @{ 'PRed $T1 $T2 }.
-
-notation "hvbox( L ⊢ break (term 90 T1) ⇒ break T2 )"
- non associative with precedence 45
- for @{ 'PRed $L $T1 $T2 }.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/reduction/tpr.ma".
-
-(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-
-definition cpr: lenv → term → term → Prop ≝
- λL,T1,T2. ∃∃T. T1 ⇒ T & L ⊢ T [0, |L|] ≫ T2.
-
-interpretation
- "context-sensitive parallel reduction (term)"
- 'PRed L T1 T2 = (cpr L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpr_pr: ∀T1,T2. T1 ⇒ T2 → ∀L. L ⊢ T1 ⇒ T2.
-/2/ qed.
-
-lemma cpr_tps: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 ⇒ T2.
-/3 width=5/ qed.
-
-lemma cpr_refl: ∀L,T. L ⊢ T ⇒ T.
-/2/ qed.
-
-(* NOTE: new property *)
-lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
- L ⊢ V1 ⇒ V2 → L ⊢ T1 ⇒ T2 → L ⊢ 𝕗{I} V1. T1 ⇒ 𝕗{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
-qed.
-
-lemma cpr_delta: ∀L,K,V1,V2,V,i.
- ↓[0, i] L ≡ K. 𝕓{Abbr} V1 → K ⊢ V1 [0, |L| - i - 1] ≫ V2 →
- ↑[0, i + 1] V2 ≡ V → L ⊢ #i ⇒ V.
-#L #K #V1 #V2 #V #i #HLK #HV12 #HV2
-@ex2_1_intro [2: // | skip ] /3 width=8/ (**) (* /4/ is too slow *)
-qed.
-
-lemma cpr_cast: ∀L,V,T1,T2.
- L ⊢ T1 ⇒ T2 → L ⊢ 𝕗{Cast} V. T1 ⇒ T2.
-#L #V #T1 #T2 * /3/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "lambda-delta/reduction/tpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-
-inductive lpr: lenv → lenv → Prop ≝
-| lpr_sort: lpr (⋆) (⋆)
-| lpr_item: ∀K1,K2,I,V1,V2.
- lpr K1 K2 → V1 ⇒ V2 → lpr (K1. 𝕓{I} V1) (K2. 𝕓{I} V2) (*𝕓*)
-.
-
-interpretation
- "context-free parallel reduction (environment)"
- 'PRed L1 L2 = (lpr L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpr_inv_item1_aux: ∀L1,L2. L1 ⇒ L2 → ∀K1,I,V1. L1 = K1. 𝕓{I} V1 →
- ∃∃K2,V2. K1 ⇒ K2 & V1 ⇒ V2 & L2 = K2. 𝕓{I} V2.
-#L1 #L2 * -L1 L2
-[ #K1 #I #V1 #H destruct
-| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct - K1 I V1 /2 width=5/
-]
-qed.
-
-lemma lpr_inv_item1: ∀K1,I,V1,L2. K1. 𝕓{I} V1 ⇒ L2 →
- ∃∃K2,V2. K1 ⇒ K2 & V1 ⇒ V2 & L2 = K2. 𝕓{I} V2.
-/2/ qed.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/tps.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-inductive tpr: term → term → Prop ≝
-| tpr_sort : ∀k. tpr (⋆k) (⋆k)
-| tpr_lref : ∀i. tpr (#i) (#i)
-| tpr_bind : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
- tpr (𝕓{I} V1. T1) (𝕓{I} V2. T2)
-| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
- tpr (𝕗{I} V1. T1) (𝕗{I} V2. T2)
-| tpr_beta : ∀V1,V2,W,T1,T2.
- tpr V1 V2 → tpr T1 T2 →
- tpr (𝕚{Appl} V1. 𝕚{Abst} W. T1) (𝕚{Abbr} V2. T2)
-| tpr_delta: ∀V1,V2,T1,T2,T.
- tpr V1 V2 → tpr T1 T2 → ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T →
- tpr (𝕚{Abbr} V1. T1) (𝕚{Abbr} V2. T)
-| tpr_theta: ∀V,V1,V2,W1,W2,T1,T2.
- tpr V1 V2 → ↑[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 →
- tpr (𝕚{Appl} V1. 𝕚{Abbr} W1. T1) (𝕚{Abbr} W2. 𝕚{Appl} V. T2)
-| tpr_zeta : ∀V,T,T1,T2. ↑[0,1] T1 ≡ T → tpr T1 T2 →
- tpr (𝕚{Abbr} V. T) T2
-| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (𝕚{Cast} V. T1) T2
-.
-
-interpretation
- "context-free parallel reduction (term)"
- 'PRed T1 T2 = (tpr T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tpr_refl: ∀T. T ⇒ T.
-#T elim T -T //
-#I elim I -I /2/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma tpr_inv_sort1_aux: ∀U1,U2. U1 ⇒ U2 → ∀k. U1 = ⋆k → U2 = ⋆k.
-#U1 #U2 * -U1 U2
-[ #k0 #k #H destruct -k0 //
-| #i #k #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
-| #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct
-| #V1 #V2 #T1 #T2 #T #_ #_ #_ #k #H destruct
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct
-| #V #T #T1 #T2 #_ #_ #k #H destruct
-| #V #T1 #T2 #_ #k #H destruct
-]
-qed.
-
-lemma tpr_inv_sort1: ∀k,U2. ⋆k ⇒ U2 → U2 = ⋆k.
-/2/ qed.
-
-lemma tpr_inv_lref1_aux: ∀U1,U2. U1 ⇒ U2 → ∀i. U1 = #i → U2 = #i.
-#U1 #U2 * -U1 U2
-[ #k #i #H destruct
-| #j #i #H destruct -j //
-| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
-| #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
-| #V1 #V2 #T1 #T2 #T #_ #_ #_ #i #H destruct
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
-| #V #T #T1 #T2 #_ #_ #i #H destruct
-| #V #T1 #T2 #_ #i #H destruct
-]
-qed.
-
-lemma tpr_inv_lref1: ∀i,U2. #i ⇒ U2 → U2 = #i.
-/2/ qed.
-
-lemma tpr_inv_abbr1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,T1. U1 = 𝕚{Abbr} V1. T1 →
- ∨∨ ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Abbr} V2. T2
- | ∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 &
- ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T &
- U2 = 𝕚{Abbr} V2. T
- | ∃∃T. ↑[0,1] T ≡ T1 & T ⇒ U2.
-#U1 #U2 * -U1 U2
-[ #k #V #T #H destruct
-| #i #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #V #T #H destruct -I V1 T1 /3 width=5/
-| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct
-| #V1 #V2 #W #T1 #T2 #_ #_ #V #T #H destruct
-| #V1 #V2 #T1 #T2 #T #HV12 #HT12 #HT2 #V0 #T0 #H destruct -V1 T1 /3 width=7/
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #V0 #T0 #H destruct
-| #V #T #T1 #T2 #HT1 #HT12 #V0 #T0 #H destruct -V T /3/
-| #V #T1 #T2 #_ #V0 #T0 #H destruct
-]
-qed.
-
-lemma tpr_inv_abbr1: ∀V1,T1,U2. 𝕚{Abbr} V1. T1 ⇒ U2 →
- ∨∨ ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Abbr} V2. T2
- | ∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 &
- ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T &
- U2 = 𝕚{Abbr} V2. T
- | ∃∃T. ↑[0,1] T ≡ T1 & tpr T U2.
-/2/ qed.
-
-lemma tpr_inv_abst1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,T1. U1 = 𝕚{Abst} V1. T1 →
- ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Abst} V2. T2.
-#U1 #U2 * -U1 U2
-[ #k #V #T #H destruct
-| #i #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #V #T #H destruct -I V1 T1 /2 width=5/
-| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct
-| #V1 #V2 #W #T1 #T2 #_ #_ #V #T #H destruct
-| #V1 #V2 #T1 #T2 #T #_ #_ #_ #V0 #T0 #H destruct
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #V0 #T0 #H destruct
-| #V #T #T1 #T2 #_ #_ #V0 #T0 #H destruct
-| #V #T1 #T2 #_ #V0 #T0 #H destruct
-]
-qed.
-
-lemma tpr_inv_abst1: ∀V1,T1,U2. 𝕚{Abst} V1. T1 ⇒ U2 →
- ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Abst} V2. T2.
-/2/ qed.
-
-lemma tpr_inv_bind1: ∀V1,T1,U2,I. 𝕓{I} V1. T1 ⇒ U2 →
- ∨∨ ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕓{I} V2. T2
- | ∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 &
- ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T &
- U2 = 𝕚{Abbr} V2. T & I = Abbr
- | ∃∃T. ↑[0,1] T ≡ T1 & tpr T U2 & I = Abbr.
-#V1 #T1 #U2 * #H
-[ elim (tpr_inv_abbr1 … H) -H * /3 width=7/
-| /3/
-]
-qed.
-
-lemma tpr_inv_appl1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,U0. U1 = 𝕚{Appl} V1. U0 →
- ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 &
- U2 = 𝕚{Appl} V2. T2
- | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 &
- U0 = 𝕚{Abst} W. T1 &
- U2 = 𝕓{Abbr} V2. T2
- | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 &
- ↑[0,1] V2 ≡ V &
- U0 = 𝕚{Abbr} W1. T1 &
- U2 = 𝕚{Abbr} W2. 𝕚{Appl} V. T2.
-#U1 #U2 * -U1 U2
-[ #k #V #T #H destruct
-| #i #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #V #T #H destruct -I V1 T1 /3 width=5/
-| #V1 #V2 #W #T1 #T2 #HV12 #HT12 #V #T #H destruct -V1 T /3 width=8/
-| #V1 #V2 #T1 #T2 #T #_ #_ #_ #V0 #T0 #H destruct
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #V0 #T0 #H
- destruct -V1 T0 /3 width=12/
-| #V #T #T1 #T2 #_ #_ #V0 #T0 #H destruct
-| #V #T1 #T2 #_ #V0 #T0 #H destruct
-]
-qed.
-
-lemma tpr_inv_appl1: ∀V1,U0,U2. 𝕚{Appl} V1. U0 ⇒ U2 →
- ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 &
- U2 = 𝕚{Appl} V2. T2
- | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 &
- U0 = 𝕚{Abst} W. T1 &
- U2 = 𝕓{Abbr} V2. T2
- | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 &
- ↑[0,1] V2 ≡ V &
- U0 = 𝕚{Abbr} W1. T1 &
- U2 = 𝕚{Abbr} W2. 𝕚{Appl} V. T2.
-/2/ qed.
-
-lemma tpr_inv_cast1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,T1. U1 = 𝕚{Cast} V1. T1 →
- (∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Cast} V2. T2)
- ∨ T1 ⇒ U2.
-#U1 #U2 * -U1 U2
-[ #k #V #T #H destruct
-| #i #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #V #T #H destruct -I V1 T1 /3 width=5/
-| #V1 #V2 #W #T1 #T2 #_ #_ #V #T #H destruct
-| #V1 #V2 #T1 #T2 #T #_ #_ #_ #V0 #T0 #H destruct
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #V0 #T0 #H destruct
-| #V #T #T1 #T2 #_ #_ #V0 #T0 #H destruct
-| #V #T1 #T2 #HT12 #V0 #T0 #H destruct -V T1 /2/
-]
-qed.
-
-lemma tpr_inv_cast1: ∀V1,T1,U2. 𝕚{Cast} V1. T1 ⇒ U2 →
- (∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Cast} V2. T2)
- ∨ T1 ⇒ U2.
-/2/ qed.
-
-lemma tpr_inv_flat1: ∀V1,U0,U2,I. 𝕗{I} V1. U0 ⇒ U2 →
- ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 &
- U2 = 𝕗{I} V2. T2
- | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 &
- U0 = 𝕚{Abst} W. T1 &
- U2 = 𝕓{Abbr} V2. T2 & I = Appl
- | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 &
- ↑[0,1] V2 ≡ V &
- U0 = 𝕚{Abbr} W1. T1 &
- U2 = 𝕚{Abbr} W2. 𝕚{Appl} V. T2 &
- I = Appl
- | (U0 ⇒ U2 ∧ I = Cast).
-#V1 #U0 #U2 * #H
-[ elim (tpr_inv_appl1 … H) -H * /3 width=12/
-| elim (tpr_inv_cast1 … H) -H [1: *] /3 width=5/
-]
-qed.
-
-lemma tpr_inv_lref2_aux: ∀T1,T2. T1 ⇒ T2 → ∀i. T2 = #i →
- ∨∨ T1 = #i
- | ∃∃V,T,T0. ↑[O,1] T0 ≡ T & T0 ⇒ #i &
- T1 = 𝕚{Abbr} V. T
- | ∃∃V,T. T ⇒ #i & T1 = 𝕚{Cast} V. T.
-#T1 #T2 * -T1 T2
-[ #k #i #H destruct
-| #j #i /2/
-| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
-| #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
-| #V1 #V2 #T1 #T2 #T #_ #_ #_ #i #H destruct
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
-| #V #T #T1 #T2 #HT1 #HT12 #i #H destruct /3 width=6/
-| #V #T1 #T2 #HT12 #i #H destruct /3/
-]
-qed.
-
-lemma tpr_inv_lref2: ∀T1,i. T1 ⇒ #i →
- ∨∨ T1 = #i
- | ∃∃V,T,T0. ↑[O,1] T0 ≡ T & T0 ⇒ #i &
- T1 = 𝕓{Abbr} V. T
- | ∃∃V,T. T ⇒ #i & T1 = 𝕗{Cast} V. T.
-/2/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "lambda-delta/substitution/tps_lift.ma".
-include "lambda-delta/reduction/tpr.ma".
-
-(* Relocation properties ****************************************************)
-
-lemma tpr_lift: ∀T1,T2. T1 ⇒ T2 →
- ∀d,e,U1. ↑[d, e] T1 ≡ U1 → ∀U2. ↑[d, e] T2 ≡ U2 → U1 ⇒ U2.
-#T1 #T2 #H elim H -H T1 T2
-[ #k #d #e #U1 #HU1 #U2 #HU2
- lapply (lift_mono … HU1 … HU2) -HU1 #H destruct -U1;
- lapply (lift_inv_sort1 … HU2) -HU2 #H destruct -U2 //
-| #i #d #e #U1 #HU1 #U2 #HU2
- lapply (lift_mono … HU1 … HU2) -HU1 #H destruct -U1;
- lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct -U2 //
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
- elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
- elim (lift_inv_bind1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct -X2 /3/
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
- elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct -X2 /3/
-| #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1;
- elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X;
- elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct -X2 /3/
-| #V1 #V2 #T1 #T2 #T0 #HV12 #HT12 #HT2 #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
- elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
- elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct -X2;
- elim (lift_total T2 (d + 1) e) #U2 #HTU2
- @tpr_delta
- [4: @(tps_lift_le … HT2 … HTU2 HTU0 ?) /2/ |1: skip |2: /2/ |3: /2/ ] (**) (*/3. is too slow *)
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
- elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1;
- elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X;
- elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct -X2;
- elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct -X;
- elim (lift_trans_ge … HV2 … HV3 ?) -HV2 HV3 V // /3/
-| #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX #T0 #HT20
- elim (lift_inv_bind1 … HX) -HX #V3 #T3 #_ #HT3 #HX destruct -X;
- elim (lift_trans_ge … HT1 … HT3 ?) -HT1 HT3 T // /3 width=6/
-| #V #T1 #T2 #_ #IHT12 #d #e #X #HX #T #HT2
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct -X /3/
-]
-qed.
-
-lemma tpr_inv_lift: ∀T1,T2. T1 ⇒ T2 →
- ∀d,e,U1. ↑[d, e] U1 ≡ T1 →
- ∃∃U2. ↑[d, e] U2 ≡ T2 & U1 ⇒ U2.
-#T1 #T2 #H elim H -H T1 T2
-[ #k #d #e #U1 #HU1
- lapply (lift_inv_sort2 … HU1) -HU1 #H destruct -U1 /2/
-| #i #d #e #U1 #HU1
- lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct -U1 /3/
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
- elim (lift_inv_bind2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X;
- elim (IHV12 … HV01) -IHV12 HV01;
- elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X;
- elim (IHV12 … HV01) -IHV12 HV01;
- elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
-| #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y;
- elim (IHV12 … HV01) -IHV12 HV01;
- elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
-| #V1 #V2 #T1 #T2 #T0 #_ #_ #HT20 #IHV12 #IHT12 #d #e #X #HX
- elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct -X;
- elim (IHV12 … HWV1) -IHV12 HWV1 #W2 #HWV2 #HW12
- elim (IHT12 … HUT1) -IHT12 HUT1 #U2 #HUT2 #HU12
- elim (tps_inv_lift1_le … HT20 … HUT2 ?) -HT20 HUT2 // [3: /2 width=5/ |2: skip ] #U0 #HU20 #HUT0
- @ex2_1_intro [2: /2/ |1: skip |3: /2/ ] (**) (* /3 width=5/ is slow *)
-| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
- elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y;
- elim (IHV12 … HV01) -IHV12 HV01 #V3 #HV32 #HV03
- elim (IHW12 … HW01) -IHW12 HW01 #W3 #HW32 #HW03
- elim (IHT12 … HT01) -IHT12 HT01 #T3 #HT32 #HT03
- elim (lift_trans_le … HV32 … HV2 ?) -HV32 HV2 V2 // #V2 #HV32 #HV2
- @ex2_1_intro [2: /3/ |1: skip |3: /2/ ] (**) (* /4 width=5/ is slow *)
-| #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX
- elim (lift_inv_bind2 … HX) -HX #V0 #T0 #_ #HT0 #H destruct -X;
- elim (lift_div_le … HT1 … HT0 ?) -HT1 HT0 T // #T #HT0 #HT1
- elim (IHT12 … HT1) -IHT12 HT1 /3 width=5/
-| #V #T1 #T2 #_ #IHT12 #d #e #X #HX
- elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct -X;
- elim (IHT12 … HT01) -IHT12 HT01 /3/
-]
-qed.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/lift_weight.ma".
-include "lambda-delta/substitution/tps_tps.ma".
-include "lambda-delta/reduction/tpr_lift.ma".
-include "lambda-delta/reduction/tpr_tps.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Confluence lemmas ********************************************************)
-
-lemma tpr_conf_sort_sort: ∀k. ∃∃X. ⋆k ⇒ X & ⋆k ⇒ X.
-/2/ qed.
-
-lemma tpr_conf_lref_lref: ∀i. ∃∃X. #i ⇒ X & #i ⇒ X.
-/2/ qed.
-
-lemma tpr_conf_bind_bind:
- ∀I,V0,V1,T0,T1,V2,T2. (
- ∀X0:term. #X0 < #V0 + #T0 + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
- ∃∃X. 𝕓{I} V1. T1 ⇒ X & 𝕓{I} V2. T2 ⇒ X.
-#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
-elim (IH … HT01 … HT02) -HT01 HT02 IH /3 width=5/
-qed.
-
-lemma tpr_conf_bind_delta:
- ∀V0,V1,T0,T1,V2,T2,T. (
- ∀X0:term. #X0 < #V0 + #T0 + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → V0 ⇒ V2 →
- T0 ⇒ T1 → T0 ⇒ T2 → ⋆. 𝕓{Abbr} V2 ⊢ T2 [O,1] ≫ T →
- ∃∃X. 𝕓{Abbr} V1. T1 ⇒ X & 𝕓{Abbr} V2. T ⇒ X.
-#V0 #V1 #T0 #T1 #V2 #T2 #T #IH #HV01 #HV02 #HT01 #HT02 #HT2
-elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
-elim (IH … HT01 … HT02) -HT01 HT02 IH // -V0 T0 #T0 #HT10 #HT20
-elim (tpr_tps_bind … HV2 HT20 … HT2) -HT20 HT2 /3 width=5/
-qed.
-
-lemma tpr_conf_bind_zeta:
- ∀X2,V0,V1,T0,T1,T. (
- ∀X0:term. #X0 < #V0 + #T0 +1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → T0 ⇒ T1 → T ⇒ X2 → ↑[O, 1] T ≡ T0 →
- ∃∃X. 𝕓{Abbr} V1. T1 ⇒ X & X2 ⇒ X.
-#X2 #V0 #V1 #T0 #T1 #T #IH #HV01 #HT01 #HTX2 #HT0
-elim (tpr_inv_lift … HT01 … HT0) -HT01 #U #HUT1 #HTU
-lapply (tw_lift … HT0) -HT0 #HT0
-elim (IH … HTX2 … HTU) -HTX2 HTU IH /3/
-qed.
-
-lemma tpr_conf_flat_flat:
- ∀I,V0,V1,T0,T1,V2,T2. (
- ∀X0:term. #X0 < #V0 + #T0 + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
- ∃∃T0. 𝕗{I} V1. T1 ⇒ T0 & 𝕗{I} V2. T2 ⇒ T0.
-#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
-elim (IH … HT01 … HT02) -HT01 HT02 /3 width=5/
-qed.
-
-lemma tpr_conf_flat_beta:
- ∀V0,V1,T1,V2,W0,U0,T2. (
- ∀X0:term. #X0 < #V0 + (#W0 + #U0 + 1) + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → V0 ⇒ V2 →
- U0 ⇒ T2 → 𝕓{Abst} W0. U0 ⇒ T1 →
- ∃∃X. 𝕗{Appl} V1. T1 ⇒ X & 𝕓{Abbr} V2. T2 ⇒ X.
-#V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H
-elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct -T1;
-elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
-elim (IH … HT02 … HU01) -HT02 HU01 IH /3 width=5/
-qed.
-
-lemma tpr_conf_flat_theta:
- ∀V0,V1,T1,V2,V,W0,W2,U0,U2. (
- ∀X0:term. #X0 < #V0 + (#W0 + #U0 + 1) + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → V0 ⇒ V2 → ↑[O,1] V2 ≡ V →
- W0 ⇒ W2 → U0 ⇒ U2 → 𝕓{Abbr} W0. U0 ⇒ T1 →
- ∃∃X. 𝕗{Appl} V1. T1 ⇒ X & 𝕓{Abbr} W2. 𝕗{Appl} V. U2 ⇒ X.
-#V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H
-elim (IH … HV01 … HV02) -HV01 HV02 // #VV #HVV1 #HVV2
-elim (lift_total VV 0 1) #VVV #HVV
-lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
-elim (tpr_inv_abbr1 … H) -H *
-(* case 1: bind *)
-[ -HV2 HVV2 #WW #UU #HWW0 #HUU0 #H destruct -T1;
- elim (IH … HW02 … HWW0) -HW02 HWW0 // #W #HW2 #HWW
- elim (IH … HU02 … HUU0) -HU02 HUU0 IH // #U #HU2 #HUU
- @ex2_1_intro [2: @tpr_theta |1:skip |3: @tpr_bind ] /2 width=7/ (**) (* /4 width=7/ is too slow *)
-(* case 2: delta *)
-| -HV2 HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct -T1;
- elim (IH … HW02 … HWW2) -HW02 HWW2 // #W #HW02 #HWW2
- elim (IH … HU02 … HUU02) -HU02 HUU02 IH // #U #HU2 #HUUU2
- elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -HUU2 HUUU2 #UUU #HUUU2 #HUUU1
- @ex2_1_intro
- [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
- |1:skip
- |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/
- ] (**) (* /5 width=14/ is too slow *)
-(* case 3: zeta *)
-| -HW02 HVV HVVV #UU1 #HUU10 #HUUT1
- elim (tpr_inv_lift … HU02 … HUU10) -HU02 #UU #HUU2 #HUU1
- lapply (tw_lift … HUU10) -HUU10 #HUU10
- elim (IH … HUUT1 … HUU1) -HUUT1 HUU1 IH // -HUU10 #U #HU2 #HUUU2
- @ex2_1_intro
- [2: @tpr_flat
- |1: skip
- |3: @tpr_zeta [2: @lift_flat |1: skip |3: @tpr_flat ]
- ] /2 width=5/ (**) (* /5 width=5/ is too slow *)
-]
-qed.
-
-lemma tpr_conf_flat_cast:
- ∀X2,V0,V1,T0,T1. (
- ∀X0:term. #X0 < #V0 + # T0 + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → T0 ⇒ T1 → T0 ⇒ X2 →
- ∃∃X. 𝕗{Cast} V1. T1 ⇒ X & X2 ⇒ X.
-#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 HT02 IH /3/
-qed.
-
-lemma tpr_conf_beta_beta:
- ∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
- ∀X0:term. #X0 < #V0 + (#W0 + #T0 + 1) + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
- ∃∃X. 𝕓{Abbr} V1. T1 ⇒X & 𝕓{Abbr} V2. T2 ⇒ X.
-#W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
-elim (IH … HV01 … HV02) -HV01 HV02 //
-elim (IH … HT01 … HT02) -HT01 HT02 IH /3 width=5/
-qed.
-
-lemma tpr_conf_delta_delta:
- ∀V0,V1,T0,T1,TT1,V2,T2,TT2. (
- ∀X0:term. #X0 < #V0 +#T0 + 1→
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
- ⋆. 𝕓{Abbr} V1 ⊢ T1 [O, 1] ≫ TT1 →
- ⋆. 𝕓{Abbr} V2 ⊢ T2 [O, 1] ≫ TT2 →
- ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & 𝕓{Abbr} V2. TT2 ⇒ X.
-#V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
-elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
-elim (IH … HT01 … HT02) -HT01 HT02 IH // #T #HT1 #HT2
-elim (tpr_tps_bind … HV1 HT1 … HTT1) -HT1 HTT1 #U1 #TTU1 #HTU1
-elim (tpr_tps_bind … HV2 HT2 … HTT2) -HT2 HTT2 #U2 #TTU2 #HTU2
-elim (tps_conf … HTU1 … HTU2) -HTU1 HTU2 #U #HU1 #HU2
-@ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
-qed.
-
-lemma tpr_conf_delta_zeta:
- ∀X2,V0,V1,T0,T1,TT1,T2. (
- ∀X0:term. #X0 < #V0 +#T0 + 1→
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → T0 ⇒ T1 → ⋆. 𝕓{Abbr} V1 ⊢ T1 [O,1] ≫ TT1 →
- T2 ⇒ X2 → ↑[O, 1] T2 ≡ T0 →
- ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & X2 ⇒ X.
-#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HTX2 #HTT20
-elim (tpr_inv_lift … HT01 … HTT20) -HT01 #TT2 #HTT21 #HTT2
-lapply (tps_inv_lift1_eq … HTT1 … HTT21) -HTT1 #HTT1 destruct -T1;
-lapply (tw_lift … HTT20) -HTT20 #HTT20
-elim (IH … HTX2 … HTT2) -HTX2 HTT2 IH /3/
-qed.
-
-lemma tpr_conf_theta_theta:
- ∀VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. (
- ∀X0:term. #X0 < #V0 + (#W0 + #T0 + 1) + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- V0 ⇒ V1 → V0 ⇒ V2 → W0 ⇒ W1 → W0 ⇒ W2 → T0 ⇒ T1 → T0 ⇒ T2 →
- ↑[O, 1] V1 ≡ VV1 → ↑[O, 1] V2 ≡ VV2 →
- ∃∃X. 𝕓{Abbr} W1. 𝕗{Appl} VV1. T1 ⇒ X & 𝕓{Abbr} W2. 𝕗{Appl} VV2. T2 ⇒ X.
-#VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2
-elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
-elim (IH … HW01 … HW02) -HW01 HW02 // #W #HW1 #HW2
-elim (IH … HT01 … HT02) -HT01 HT02 IH // #T #HT1 #HT2
-elim (lift_total V 0 1) #VV #HVV
-lapply (tpr_lift … HV1 … HVV1 … HVV) -HV1 HVV1 #HVV1
-lapply (tpr_lift … HV2 … HVV2 … HVV) -HV2 HVV2 HVV #HVV2
-@ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
-qed.
-
-lemma tpr_conf_zeta_zeta:
- ∀V0:term. ∀X2,TT0,T0,T1,T2. (
- ∀X0:term. #X0 < #V0 + #TT0 + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- T0 ⇒ T1 → T2 ⇒ X2 →
- ↑[O, 1] T0 ≡ TT0 → ↑[O, 1] T2 ≡ TT0 →
- ∃∃X. T1 ⇒ X & X2 ⇒ X.
-#V0 #X2 #TT0 #T0 #T1 #T2 #IH #HT01 #HTX2 #HTT0 #HTT20
-lapply (lift_inj … HTT0 … HTT20) -HTT0 #H destruct -T0;
-lapply (tw_lift … HTT20) -HTT20 #HTT20
-elim (IH … HT01 … HTX2) -HT01 HTX2 IH /2/
-qed.
-
-lemma tpr_conf_tau_tau:
- ∀V0,T0:term. ∀X2,T1. (
- ∀X0:term. #X0 < #V0 + #T0 + 1 →
- ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- T0 ⇒ T1 → T0 ⇒ X2 →
- ∃∃X. T1 ⇒ X & X2 ⇒ X.
-#V0 #T0 #X2 #T1 #IH #HT01 #HT02
-elim (IH … HT01 … HT02) -HT01 HT02 IH /2/
-qed.
-
-(* Confluence ***************************************************************)
-
-lemma tpr_conf_aux:
- ∀Y0:term. (
- ∀X0:term. #X0 < #Y0 → ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X
- ) →
- ∀X0,X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → X0 = Y0 →
- ∃∃X. X1 ⇒ X & X2 ⇒ X.
-#Y0 #IH #X0 #X1 #X2 * -X0 X1
-[ #k1 #H1 #H2 destruct -Y0;
- lapply (tpr_inv_sort1 … H1) -H1
-(* case 1: sort, sort *)
- #H1 destruct -X2 //
-| #i1 #H1 #H2 destruct -Y0;
- lapply (tpr_inv_lref1 … H1) -H1
-(* case 2: lref, lref *)
- #H1 destruct -X2 //
-| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
- elim (tpr_inv_bind1 … H1) -H1 *
-(* case 3: bind, bind *)
- [ #V2 #T2 #HV02 #HT02 #H destruct -X2
- /3 width=7 by tpr_conf_bind_bind/ (**) (* /3 width=7/ is too slow *)
-(* case 4: bind, delta *)
- | #V2 #T2 #T #HV02 #HT02 #HT2 #H1 #H2 destruct -X2 I
- /3 width=9 by tpr_conf_bind_delta/ (**) (* /3 width=9/ is too slow *)
-(* case 5: bind, zeta *)
- | #T #HT0 #HTX2 #H destruct -I
- /3 width=8 by tpr_conf_bind_zeta/ (**) (* /3 width=8/ is too slow *)
- ]
-| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
- elim (tpr_inv_flat1 … H1) -H1 *
-(* case 6: flat, flat *)
- [ #V2 #T2 #HV02 #HT02 #H destruct -X2
- /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
-(* case 7: flat, beta *)
- | #V2 #W #U0 #T2 #HV02 #HT02 #H1 #H2 #H3 destruct -T0 X2 I
- /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
-(* case 8: flat, theta *)
- | #V2 #V #W0 #W2 #U0 #U2 #HV02 #HW02 #HT02 #HV2 #H1 #H2 #H3 destruct -T0 X2 I
- /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
-(* case 9: flat, tau *)
- | #HT02 #H destruct -I
- /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
- ]
-| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
- elim (tpr_inv_appl1 … H1) -H1 *
-(* case 10: beta, flat (repeated) *)
- [ #V2 #T2 #HV02 #HT02 #H destruct -X2
- @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/
-(* case 11: beta, beta *)
- | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct -W0 T0 X2
- /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
-(* case 12, beta, theta (excluded) *)
- | #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H destruct
- ]
-| #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct -Y0;
- elim (tpr_inv_abbr1 … H1) -H1 *
-(* case 13: delta, bind (repeated) *)
- [ #V2 #T2 #HV02 #T02 #H destruct -X2
- @ex2_1_comm /3 width=9 by tpr_conf_bind_delta/
-(* case 14: delta, delta *)
- | #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
- /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
-(* case 15: delta, zata *)
- | #T2 #HT20 #HTX2
- /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
- ]
-| #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HVV1 #HW01 #HT01 #H1 #H2 destruct -Y0;
- elim (tpr_inv_appl1 … H1) -H1 *
-(* case 16: theta, flat (repeated) *)
- [ #V2 #T2 #HV02 #HT02 #H destruct -X2
- @ex2_1_comm /3 width=11 by tpr_conf_flat_theta/
-(* case 17: theta, beta (repeated) *)
- | #V2 #WW0 #TT0 #T2 #_ #_ #H destruct
-(* case 18: theta, theta *)
- | #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct -W0 T0 X2
- /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
- ]
-| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct -Y0;
- elim (tpr_inv_abbr1 … H1) -H1 *
-(* case 19: zeta, bind (repeated) *)
- [ #V2 #T2 #HV02 #T02 #H destruct -X2
- @ex2_1_comm /3 width=8 by tpr_conf_bind_zeta/
-(* case 20: zeta, delta (repeated) *)
- | #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
- @ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/
-(* case 21: zeta, zeta *)
- | #T2 #HTT20 #HTX2
- /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
- ]
-| #V0 #T0 #T1 #HT01 #H1 #H2 destruct -Y0;
- elim (tpr_inv_cast1 … H1) -H1
-(* case 22: tau, flat (repeated) *)
- [ * #V2 #T2 #HV02 #HT02 #H destruct -X2
- @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
-(* case 23: tau, tau *)
- | #HT02
- /2 by tpr_conf_tau_tau/
- ]
-]
-qed.
-
-theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ⇒ T1 → T0 ⇒ T2 →
- ∃∃T. T1 ⇒ T & T2 ⇒ T.
-#T @(tw_wf_ind … T) -T /3 width=6 by tpr_conf_aux/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "lambda-delta/reduction/lpr.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-axiom tpr_tps_lpr: ∀L1,L2. L1 ⇒ L2 → ∀T1,T2. T1 ⇒ T2 →
- ∀d,e,U1. L1 ⊢ T1 [d, e] ≫ U1 →
- ∃∃U2. U1 ⇒ U2 & L2 ⊢ T2 [d, e] ≫ U2.
-
-lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ⇒ V2 → T1 ⇒ T2 →
- ⋆. 𝕓{I} V1 ⊢ T1 [0, 1] ≫ U1 →
- ∃∃U2. U1 ⇒ U2 & ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ U2.
-/3 width=5/ qed.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/leq.ma".
-include "lambda-delta/substitution/lift.ma".
-
-(* DROPPING *****************************************************************)
-
-inductive drop: lenv → nat → nat → lenv → Prop ≝
-| drop_sort: ∀d,e. drop (⋆) d e (⋆)
-| drop_comp: ∀L1,L2,I,V. drop L1 0 0 L2 → drop (L1. 𝕓{I} V) 0 0 (L2. 𝕓{I} V)
-| drop_drop: ∀L1,L2,I,V,e. drop L1 0 e L2 → drop (L1. 𝕓{I} V) 0 (e + 1) L2
-| drop_skip: ∀L1,L2,I,V1,V2,d,e.
- drop L1 d e L2 → ↑[d,e] V2 ≡ V1 →
- drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
-.
-
-interpretation "dropping" 'RDrop L1 d e L2 = (drop L1 d e L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
-#d #e #L1 #L2 #H elim H -H d e L1 L2
-[ //
-| #L1 #L2 #I #V #_ #IHL12 #H1 #H2
- >(IHL12 H1 H2) -IHL12 H1 H2 L1 //
-| #L1 #L2 #I #V #e #_ #_ #_ #H
- elim (plus_S_eq_O_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H
- elim (plus_S_eq_O_false … H)
-]
-qed.
-
-lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5/ qed.
-
-lemma drop_inv_sort1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
- L2 = ⋆.
-#d #e #L1 #L2 * -d e L1 L2
-[ //
-| #L1 #L2 #I #V #_ #H destruct
-| #L1 #L2 #I #V #e #_ #H destruct
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
-]
-qed.
-
-lemma drop_inv_sort1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
-/2 width=5/ qed.
-
-lemma drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
- ∀K,I,V. L1 = K. 𝕓{I} V →
- (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
- (0 < e ∧ ↓[d, e - 1] K ≡ L2).
-#d #e #L1 #L2 * -d e L1 L2
-[ #d #e #_ #K #I #V #H destruct
-| #L1 #L2 #I #V #HL12 #H #K #J #W #HX destruct -L1 I V
- >(drop_inv_refl … HL12) -HL12 K /3/
-| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
-]
-qed.
-
-lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
- (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
- (0 < e ∧ ↓[0, e - 1] K ≡ L2).
-/2/ qed.
-
-lemma drop_inv_drop1: ∀e,K,I,V,L2.
- ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
-#e #K #I #V #L2 #H #He
-elim (drop_inv_O1 … H) -H * // #H destruct -e;
-elim (lt_refl_false … He)
-qed.
-
-lemma drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
- ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
- ↑[d - 1, e] V2 ≡ V1 &
- L2 = K2. 𝕓{I} V2.
-#d #e #L1 #L2 * -d e L1 L2
-[ #d #e #_ #I #K #V #H destruct
-| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
-| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
- /2 width=5/
-]
-qed.
-
-lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
- ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
- ↑[d - 1, e] V2 ≡ V1 &
- L2 = K2. 𝕓{I} V2.
-/2/ qed.
-
-lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
- ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
- ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
- ↑[d - 1, e] V2 ≡ V1 &
- L1 = K1. 𝕓{I} V1.
-#d #e #L1 #L2 * -d e L1 L2
-[ #d #e #_ #I #K #V #H destruct
-| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
-| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
-| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
- /2 width=5/
-]
-qed.
-
-lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
- ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
- L1 = K1. 𝕓{I} V1.
-/2/ qed.
-
-(* Basic properties *********************************************************)
-
-lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
-#L elim L -L /2/
-qed.
-
-lemma drop_drop_lt: ∀L1,L2,I,V,e.
- ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
-#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
-qed.
-
-lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 →
- ∀I,K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{I} V →
- d ≤ i → i < d + e →
- ∃∃K2. K1 [0, d + e - i - 1] ≈ K2 &
- ↓[0, i] L2 ≡ K2. 𝕓{I} V.
-#L1 #L2 #d #e #H elim H -H L1 L2 d e
-[ #d #e #I #K1 #V #i #H
- lapply (drop_inv_sort1 … H) -H #H destruct
-| #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #I #K1 #V #i #_ #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #V #e #HL12 #IHL12 #J #K1 #W #i #H #_ #Hie
- elim (drop_inv_O1 … H) -H * #Hi #HLK1
- [ -IHL12 Hie; destruct -i K1 J W;
- <minus_n_O <minus_plus_m_m /2/
- | -HL12;
- elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
- ]
-| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #I #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
- lapply (plus_S_le_to_pos … Hdi) #Hi
- lapply (drop_inv_drop1 … H ?) -H // #HLK1
- elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
-]
-qed.
-
-(* Basic forvard lemmas *****************************************************)
-
-lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
- ↓[O, e + 1] L1 ≡ K2.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
- elim (drop_inv_O1 … H) -H * #He #H
- [ -IHL1; destruct -e K2 I2 V2 /2/
- | @drop_drop >(plus_minus_m_m e 1) /2/
- ]
-]
-qed.
-
-lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e.
- ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
- elim (drop_inv_O1 … H) -H * #He #H
- [ -IHL1; destruct -e K2 I2 V2 //
- | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
- ]
-]
-qed.
-
-lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
-#L1 elim L1 -L1
-[ #L2 #e #H >(drop_inv_sort1 … H) -H //
-| #K1 #I1 #V1 #IHL1 #L2 #e #H
- elim (drop_inv_O1 … H) -H * #He #H
- [ -IHL1; destruct -e L2 //
- | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
- >minus_le_minus_minus_comm //
- ]
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "lambda-delta/substitution/lift_lift.ma".
-include "lambda-delta/substitution/drop.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Main properties **********************************************************)
-
-theorem drop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 →
- ∀L2. ↓[d, e] L ≡ L2 → L1 = L2.
-#d #e #L #L1 #H elim H -H d e L L1
-[ #d #e #L2 #H
- >(drop_inv_sort1 … H) -H L2 //
-| #K1 #K2 #I #V #HK12 #_ #L2 #HL12
- <(drop_inv_refl … HK12) -HK12 K2
- <(drop_inv_refl … HL12) -HL12 L2 //
-| #L #K #I #V #e #_ #IHLK #L2 #H
- lapply (drop_inv_drop1 … H ?) -H /2/
-| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
- elim (drop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct -X
- >(lift_inj … HVT1 … HVT2) -HVT1 HVT2
- >(IHLK1 … HLK2) -IHLK1 HLK2 //
-]
-qed.
-
-theorem drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
- ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
- ↓[0, e2 - e1] L1 ≡ L2.
-#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
-[ #d #e #e2 #L2 #H
- >(drop_inv_sort1 … H) -H L2 //
-| #K1 #K2 #I #V #HK12 #_ #e2 #L2 #H #_ <minus_n_O
- <(drop_inv_refl … HK12) -HK12 K2 //
-| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
- lapply (drop_inv_drop1 … H ?) -H /2/ #HL2
- <minus_plus_comm /3/
-| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
- lapply (transitive_le 1 … Hdee2) // #He2
- lapply (drop_inv_drop1 … H ?) -H // -He2 #HL2
- lapply (transitive_le (1 + e) … Hdee2) // #Hee2
- @drop_drop_lt >minus_minus_comm /3/ (**) (* explicit constructor *)
-]
-qed.
-
-theorem drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
- ∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 →
- e2 < d1 → let d ≝ d1 - e2 - 1 in
- ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 &
- ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2.
-#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
-[ #d #e #e2 #K2 #I #V2 #H
- lapply (drop_inv_sort1 … H) -H #H destruct
-| #L1 #L2 #I #V #_ #_ #e2 #K2 #J #V2 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H
- elim (lt_zero_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d
- elim (drop_inv_O1 … H) -H *
- [ -IHL12 He2d #H1 #H2 destruct -e2 K2 J V /2 width=5/
- | -HL12 -HV12 #He #HLK
- elim (IHL12 … HLK ?) -IHL12 HLK [ <minus_minus /3 width=5/ | /2/ ] (**) (* a bit slow *)
- ]
-]
-qed.
-
-theorem drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
- ∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 →
- ∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2.
-#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
-[ #d #e #e2 #L2 #H
- >(drop_inv_sort1 … H) -H L2 /2/
-| #K1 #K2 #I #V #HK12 #_ #e2 #L2 #HL2 #H
- >(drop_inv_refl … HK12) -HK12 K1;
- lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/
-| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
- lapply (le_O_to_eq_O … H) -H #H destruct -e2;
- elim (IHL12 … HL2 ?) -IHL12 HL2 // #L0 #H #HL0
- lapply (drop_inv_refl … H) -H #H destruct -L1 /3 width=5/
-| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
- elim (drop_inv_O1 … H) -H *
- [ -He2d IHL12 #H1 #H2 destruct -e2 L /3 width=5/
- | -HL12 HV12 #He2 #HL2
- elim (IHL12 … HL2 ?) -IHL12 HL2 L2
- [ >minus_le_minus_minus_comm // /3/ | /2/ ]
- ]
-]
-qed.
-
-theorem drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
- ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2.
-#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
-[ #d #e #e2 #L2 #H
- >(drop_inv_sort1 … H) -H L2 //
-| #K1 #K2 #I #V #HK12 #_ #e2 #L2 #H #_ normalize
- >(drop_inv_refl … HK12) -HK12 K1 //
-| /3/
-| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2
- lapply (lt_to_le_to_lt 0 … Hde2) // #He2
- lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
- lapply (drop_inv_drop1 … H ?) -H // #HL2
- @drop_drop_lt // >le_plus_minus // @IHL12 /2/ (**) (* explicit constructor *)
-]
-qed.
-
-theorem drop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L.
- ↓[d1, e1] L1 ≡ L → ↓[0, e2] L ≡ L2 → d1 ≤ e2 →
- ↓[0, e2 + e1] L1 ≡ L2.
-#e1 #e1 #e2 >commutative_plus /2 width=5/
-qed.
-
-axiom drop_div: ∀e1,L1,L. ↓[0, e1] L1 ≡ L → ∀e2,L2. ↓[0, e2] L2 ≡ L →
- ∃∃L0. ↓[0, e1] L0 ≡ L2 & ↓[e1, e2] L0 ≡ L1.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/syntax/length.ma".
-
-(* LOCAL ENVIRONMENT EQUALITY ***********************************************)
-
-inductive leq: lenv → nat → nat → lenv → Prop ≝
-| leq_sort: ∀d,e. leq (⋆) d e (⋆)
-| leq_comp: ∀L1,L2,I1,I2,V1,V2.
- leq L1 0 0 L2 → leq (L1. 𝕓{I1} V1) 0 0 (L2. 𝕓{I2} V2)
-| leq_eq: ∀L1,L2,I,V,e. leq L1 0 e L2 → leq (L1. 𝕓{I} V) 0 (e + 1) (L2.𝕓{I} V)
-| leq_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
- leq L1 d e L2 → leq (L1. 𝕓{I1} V1) (d + 1) e (L2. 𝕓{I2} V2)
-.
-
-interpretation "local environment equality" 'Eq L1 d e L2 = (leq L1 d e L2).
-
-(* Basic properties *********************************************************)
-
-lemma leq_refl: ∀d,e,L. L [d, e] ≈ L.
-#d elim d -d
-[ #e elim e -e [ #L elim L -L /2/ | #e #IHe #L elim L -L /2/ ]
-| #d #IHd #e #L elim L -L /2/
-]
-qed.
-
-lemma leq_sym: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L2 [d, e] ≈ L1.
-#L1 #L2 #d #e #H elim H -H L1 L2 d e /2/
-qed.
-
-lemma leq_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≈ L2 → 0 < d →
- ∀I1,I2,V1,V2. L1. 𝕓{I1} V1 [d, e] ≈ L2. 𝕓{I2} V2.
-
-#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2/
-qed.
-
-lemma leq_fwd_length: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → |L1| = |L2|.
-#L1 #L2 #d #e #H elim H -H L1 L2 d e; normalize //
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma leq_inv_sort1_aux: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 #d #e #H elim H -H L1 L2 d e
-[ //
-| #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #H destruct
-| #L1 #L2 #I #V #e #_ #_ #H destruct
-| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #_ #H destruct
-qed.
-
-lemma leq_inv_sort1: ∀L2,d,e. ⋆ [d, e] ≈ L2 → L2 = ⋆.
-/2 width=5/ qed.
-
-lemma leq_inv_sort2: ∀L1,d,e. L1 [d, e] ≈ ⋆ → L1 = ⋆.
-/3/ qed.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/syntax/term.ma".
-
-(* RELOCATION ***************************************************************)
-
-inductive lift: term → nat → nat → term → Prop ≝
-| lift_sort : ∀k,d,e. lift (⋆k) d e (⋆k)
-| lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i)
-| lift_lref_ge: ∀i,d,e. d ≤ i → lift (#i) d e (#(i + e))
-| lift_bind : ∀I,V1,V2,T1,T2,d,e.
- lift V1 d e V2 → lift T1 (d + 1) e T2 →
- lift (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
-| lift_flat : ∀I,V1,V2,T1,T2,d,e.
- lift V1 d e V2 → lift T1 d e T2 →
- lift (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
-.
-
-interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e T2).
-
-(* Basic properties *********************************************************)
-
-lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i.
-#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
-qed.
-
-lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T.
-#T elim T -T
-[ //
-| #i #d elim (lt_or_ge i d) /2/
-| #I elim I -I /2/
-]
-qed.
-
-lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
-#T1 elim T1 -T1
-[ /2/
-| #i #d #e elim (lt_or_ge i d) /3/
-| * #I #V1 #T1 #IHV1 #IHT1 #d #e
- elim (IHV1 d e) -IHV1 #V2 #HV12
- [ elim (IHT1 (d+1) e) -IHT1 /3/
- | elim (IHT1 d e) -IHT1 /3/
- ]
-]
-qed.
-
-lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
- d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
- ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2.
-#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
-[ /3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
-| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
- lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
- <(arith_d1 i e2 e1) // /3/
-| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
- elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
- elim (IHT (d2+1) … ? ? He12) /3 width = 5/
-| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
- elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
- elim (IHT d2 … ? ? He12) /3 width = 5/
-]
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
-#d #e #T1 #T2 #H elim H -H d e T1 T2 /3/
-qed.
-
-lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
-/2/ qed.
-
-lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
-#d #e #T1 #T2 * -d e T1 T2 //
-[ #i #d #e #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-]
-qed.
-
-lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5/ qed.
-
-lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
- (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-#d #e #T1 #T2 * -d e T1 T2
-[ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3/
-| #j #d #e #Hj #i #Hi destruct /3/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-]
-qed.
-
-lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
- (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-/2/ qed.
-
-lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i.
-#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
-elim (lt_refl_false … Hdd)
-qed.
-
-lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
-#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
-elim (lt_refl_false … Hdd)
-qed.
-
-lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
- ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
- ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
- T2 = 𝕓{I} V2. U2.
-#d #e #T1 #T2 * -d e T1 T2
-[ #k #d #e #I #V1 #U1 #H destruct
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
-]
-qed.
-
-lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
- T2 = 𝕓{I} V2. U2.
-/2/ qed.
-
-lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
- ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
- ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
- T2 = 𝕗{I} V2. U2.
-#d #e #T1 #T2 * -d e T1 T2
-[ #k #d #e #I #V1 #U1 #H destruct
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
-]
-qed.
-
-lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
- T2 = 𝕗{I} V2. U2.
-/2/ qed.
-
-lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
-#d #e #T1 #T2 * -d e T1 T2 //
-[ #i #d #e #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-]
-qed.
-
-lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
-/2 width=5/ qed.
-
-lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
- (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-#d #e #T1 #T2 * -d e T1 T2
-[ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3/
-| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-]
-qed.
-
-lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
- (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-/2/ qed.
-
-lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i.
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
-elim (plus_lt_false … Hdd)
-qed.
-
-lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
-elim (plus_lt_false … Hdd)
-qed.
-
-lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
- ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
- ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
- T1 = 𝕓{I} V1. U1.
-#d #e #T1 #T2 * -d e T1 T2
-[ #k #d #e #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
-]
-qed.
-
-lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 →
- ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
- T1 = 𝕓{I} V1. U1.
-/2/ qed.
-
-lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
- ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
- ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
- T1 = 𝕗{I} V1. U1.
-#d #e #T1 #T2 * -d e T1 T2
-[ #k #d #e #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width = 5/
-]
-qed.
-
-lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 →
- ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
- T1 = 𝕗{I} V1. U1.
-/2/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "lambda-delta/substitution/lift.ma".
-
-(* RELOCATION ***************************************************************)
-
-(* Main properies ***********************************************************)
-
-theorem lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
-#d #e #T1 #U #H elim H -H d e T1 U
-[ #k #d #e #X #HX
- lapply (lift_inv_sort2 … HX) -HX //
-| #i #d #e #Hid #X #HX
- lapply (lift_inv_lref2_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #HX
- lapply (lift_inv_lref2_ge … HX ?) -HX /2/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
-]
-qed.
-
-theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
- d1 ≤ d2 →
- ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
-#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
-[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
- lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
- [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
- | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
- @(ex2_1_intro … #(i - e2))
- [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
- | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
- ]
- ]
-| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
- lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
- elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
- >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
-| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
- lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
- elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
- elim (IHU … HU2 ?) /3 width = 5/
-]
-qed.
-
-theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
-#d #e #T #U1 #H elim H -H d e T U1
-[ #k #d #e #X #HX
- lapply (lift_inv_sort1 … HX) -HX //
-| #i #d #e #Hid #X #HX
- lapply (lift_inv_lref1_lt … HX ?) -HX //
-| #i #d #e #Hdi #X #HX
- lapply (lift_inv_lref1_ge … HX ?) -HX //
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
-]
-qed.
-
-theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
- d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
-[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
- >(lift_inv_sort1 … HT2) -HT2 //
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref1_lt … HT2 Hid2) /2/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
- lapply (lift_inv_lref1_ge … HT2 ?) -HT2
- [ @(transitive_le … Hd21 ?) -Hd21 /2/
- | -Hd21 /2/
- ]
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2/
-]
-qed.
-
-theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
- ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
- elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
- lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
- lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
- >plus_plus_comm_23 /4/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
-]
-qed.
-
-theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
- ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
-#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
- lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
- lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
- lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
- lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
- [2: >plus_plus_comm_23] /4/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
- <plus_minus /3 width=5/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
- elim (IHV12 … HV20 ?) -IHV12 HV20 //
- elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "lambda-delta/syntax/weight.ma".
-include "lambda-delta/substitution/lift.ma".
-
-(* RELOCATION ***************************************************************)
-
-lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #T1 = #T2.
-#d #e #T1 #T2 #H elim H -d e T1 T2; normalize //
-qed.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/drop.ma".
-
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-inductive tps: lenv → term → nat → nat → term → Prop ≝
-| tps_sort : ∀L,k,d,e. tps L (⋆k) d e (⋆k)
-| tps_lref : ∀L,i,d,e. tps L (#i) d e (#i)
-| tps_subst: ∀L,K,V,U1,U2,i,d,e.
- d ≤ i → i < d + e →
- ↓[0, i] L ≡ K. 𝕓{Abbr} V → tps K V 0 (d + e - i - 1) U1 →
- ↑[0, i + 1] U1 ≡ U2 → tps L (#i) d e U2
-| tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
- tps L V1 d e V2 → tps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
- tps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
-| tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
- tps L V1 d e V2 → tps L T1 d e T2 →
- tps L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
-.
-
-interpretation "partial telescopic substritution"
- 'PSubst L T1 d e T2 = (tps L T1 d e T2).
-
-(* Basic properties *********************************************************)
-
-lemma tps_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
- ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
-#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
-[ //
-| //
-| #L1 #K1 #V #V1 #V2 #i #d #e #Hdi #Hide #HLK1 #_ #HV12 #IHV12 #L2 #HL12
- elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // #K2 #HK12 #HLK2
- @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ] (**) (* /3 width=6/ is too slow *)
-| /4/
-| /3/
-]
-qed.
-
-lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
-#T elim T -T //
-#I elim I -I /2/
-qed.
-
-lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
- ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
- L ⊢ T1 [d2, e2] ≫ T2.
-#L #T1 #T #d1 #e1 #H elim H -L T1 T d1 e1
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d1 #e1 #Hid1 #Hide1 #HLK #_ #HV12 #IHV12 #d2 #e2 #Hd12 #Hde12
- lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
- lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 #Hide2
- @tps_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2/ ] (**) (* /4 width=6/ is too slow *)
-| /4/
-| /4/
-]
-qed.
-
-lemma tps_weak_top: ∀L,T1,T2,d,e.
- L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
-#L #T1 #T #d #e #H elim H -L T1 T d e
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d #e #Hdi #_ #HLK #_ #HV12 #IHV12
- lapply (drop_fwd_drop2_length … HLK) #Hi
- lapply (le_to_lt_to_lt … Hdi Hi) #Hd
- lapply (plus_minus_m_m_comm (|L|) d ?) [ /2/ ] -Hd #Hd
- lapply (drop_fwd_O1_length … HLK) normalize #HKL
- lapply (tps_weak … IHV12 0 (|L| - i - 1) ? ?) -IHV12 // -HKL /2 width=6/
-| normalize /2/
-| /2/
-]
-qed.
-
-lemma tps_weak_all: ∀L,T1,T2,d,e.
- L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2.
-#L #T1 #T #d #e #HT12
-lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
-lapply (tps_weak_top … HT12) //
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma tps_inv_lref1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. T1 = #i →
- T2 = #i ∨
- ∃∃K,V1,V2,i. d ≤ i & i < d + e &
- ↓[O, i] L ≡ K. 𝕓{Abbr} V1 &
- K ⊢ V1 [O, d + e - i - 1] ≫ V2 &
- ↑[O, i + 1] V2 ≡ T2.
-#L #T1 #T2 #d #e * -L T1 T2 d e
-[ #L #k #d #e #i #H destruct
-| /2/
-| #L #K #V1 #V2 #T2 #i #d #e #Hdi #Hide #HLK #HV12 #HVT2 #j #H destruct -i /3 width=9/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-]
-qed.
-
-lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 →
- T2 = #i ∨
- ∃∃K,V1,V2,i. d ≤ i & i < d + e &
- ↓[O, i] L ≡ K. 𝕓{Abbr} V1 &
- K ⊢ V1 [O, d + e - i - 1] ≫ V2 &
- ↑[O, i + 1] V2 ≡ T2.
-/2/ qed.
-
-lemma tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
- ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
- L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
- U2 = 𝕓{I} V2. T2.
-#d #e #L #U1 #U2 * -d e L U1 U2
-[ #L #k #d #e #I #V1 #T1 #H destruct
-| #L #i #d #e #I #V1 #T1 #H destruct
-| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
-]
-qed.
-
-lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
- L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
- U2 = 𝕓{I} V2. T2.
-/2/ qed.
-
-lemma tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
- ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
- U2 = 𝕗{I} V2. T2.
-#d #e #L #U1 #U2 * -d e L U1 U2
-[ #L #k #d #e #I #V1 #T1 #H destruct
-| #L #i #d #e #I #V1 #T1 #H destruct
-| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
-| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
-]
-qed.
-
-lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
- ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
- U2 = 𝕗{I} V2. T2.
-/2/ qed.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/drop_drop.ma".
-include "lambda-delta/substitution/tps.ma".
-
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-(* Relocation properties ****************************************************)
-
-lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
- ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
- ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
- dt + et ≤ d →
- L ⊢ U1 [dt, et] ≫ U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
-[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #_ #HV12 #IHV12 #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
- lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct -U1;
- elim (lift_trans_ge … HV12 … HVU2 ?) -HV12 HVU2 V2 // <minus_plus #V2 #HV12 #HVU2
- elim (drop_trans_le … HLK … HKV ?) -HLK HKV K /2/ #X #HLK #H
- elim (drop_inv_skip2 … H ?) -H /2/ -Hid #K #W #HKV #HVW #H destruct -X
- @tps_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2 width=6/ ] (**) (* explicit constructor *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
- @tps_bind [ /2 width=6/ | @IHT12 [3,4,5: /2/ |1,2: skip | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
- /3 width=6/
-]
-qed.
-
-lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
- ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
- ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
- d ≤ dt →
- L ⊢ U1 [dt + e, et] ≫ U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
-[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
- lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
-| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #HV1 #HV12 #_ #L #U1 #U2 #d #e #HLK #H #HVU2 #Hddt
- <(arith_c1x ? ? ? e) in HV1 #HV1 (**) (* explicit athmetical rewrite and ?'s *)
- lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
- lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct -U1;
- lapply (lift_trans_be … HV12 … HVU2 ? ?) -HV12 HVU2 V2 /2/ >plus_plus_comm_23 #HV1U2
- lapply (drop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid #HLKV
- @tps_subst [4,5: /2/ |6,7,8: // |1,2,3: skip ] (**) (* /3 width=8/ is too slow *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
- @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
-| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
- /3 width=5/
-]
-qed.
-
-lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
- ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
- dt + et ≤ d →
- ∃∃T2. K ⊢ T1 [dt, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
-[ #L #k #dt #et #K #d #e #_ #T1 #H #_
- lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-| #L #i #dt #et #K #d #e #_ #T1 #H #_
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
-| #L #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #_ #HV12 #IHV12 #K #d #e #HLK #T1 #H #Hdetd
- lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
- lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1;
- elim (drop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #W #HKL #HKVL #HWV
- elim (IHV12 … HKVL … HWV ?) -HKVL HWV /2/ -Hdetd #W1 #HW1 #HWV1
- elim (lift_trans_le … HWV1 … HV12 ?) -HWV1 HV12 V1 // >arith_a2 /3 width=6/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
- elim (IHV12 … HLK … HWV1 ?) -IHV12 //
- elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @drop_skip // |2: skip ] -HLK HWV1 Hdetd /3 width=5/ (**) (* just /3 width=5/ is too slow *)
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
- elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
- elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
-]
-qed.
-
-lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
- ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
- d + e ≤ dt →
- ∃∃T2. K ⊢ T1 [dt - e, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
-[ #L #k #dt #et #K #d #e #_ #T1 #H #_
- lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-| #L #i #dt #et #K #d #e #_ #T1 #H #_
- elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
-| #L #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #HV1 #HV12 #_ #K #d #e #HLK #T1 #H #Hdedt
- lapply (transitive_le … Hdedt … Hdti) #Hdei
- lapply (plus_le_weak … Hdedt) -Hdedt #Hedt
- lapply (plus_le_weak … Hdei) #Hei
- <(arith_h1 ? ? ? e ? ?) in HV1 // #HV1
- lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1;
- lapply (drop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
- elim (lift_split … HV12 d (i - e + 1) ? ? ?) -HV12; [2,3,4: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
- @ex2_1_intro
- [2: @tps_subst [4: /2/ |6,7,8: // |1,2,3: skip |5: @arith5 // ]
- |1: skip
- | //
- ] (**) (* explicitc constructors *)
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
- lapply (plus_le_weak … Hdetd) #Hedt
- elim (IHV12 … HLK … HWV1 ?) -IHV12 // #W2 #HW12 #HWV2
- elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @drop_skip // |2: skip |3: /2/ ]
- <plus_minus // /3 width=5/
-| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
- elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
- elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
-]
-qed.
-
-lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
- L ⊢ U1 [d, e] ≫ U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H elim H -H L U1 U2 d e
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #_ #_ #_ #_ #T1 #H
- elim (lift_inv_lref2 … H) -H * #H
- [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi H #H
- elim (lt_refl_false … H)
- | lapply (lt_to_le_to_lt … Hide … H) -Hide H #H
- elim (lt_refl_false … H)
- ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
- >IHV12 // >IHT12 //
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
- >IHV12 // >IHT12 //
-]
-qed.
-(*
- Theorem subst0_gen_lift_ge : (u,t1,x:?; i,h,d:?) (subst0 i u (lift h d t1) x) ->
- (le (plus d h) i) ->
- (EX t2 | x = (lift h d t2) & (subst0 (minus i h) u t1 t2)).
-
- Theorem subst0_gen_lift_rev_ge: (t1,v,u2:?; i,h,d:?)
- (subst0 i v t1 (lift h d u2)) ->
- (le (plus d h) i) ->
- (EX u1 | (subst0 (minus i h) v u1 u2) &
- t1 = (lift h d u1)
- ).
-
-
- Theorem subst0_gen_lift_rev_lelt: (t1,v,u2:?; i,h,d:?)
- (subst0 i v t1 (lift h d u2)) ->
- (le d i) -> (lt i (plus d h)) ->
- (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
-*)
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/tps_lift.ma".
-
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-(* Split properties *********************************************************)
-
-lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
- ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
-#L #T1 #T2 #d #e #H elim H -L T1 T2 d e
-[ /2/
-| /2/
-| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #HV1 #HV12 #IHV12 #j #Hdj #Hjde
- elim (lt_or_ge i j) #Hij
- [ -HV1 Hide;
- lapply (drop_fwd_drop2 … HLK) #HLK'
- elim (IHV12 (j - i - 1) ? ?) -IHV12; normalize /2/ -Hjde <minus_n_O >arith_b2 // #W1 #HVW1 #HWV1
- generalize in match HVW1 generalize in match Hij -HVW1 (**) (* rewriting in the premises, rewrites in the goal too *)
- >(plus_minus_m_m_comm … Hdj) in ⊢ (% → % → ?) -Hdj #Hij' #HVW1
- elim (lift_total W1 0 (i + 1)) #W2 #HW12
- lapply (tps_lift_ge … HWV1 … HLK' HW12 HV12 ?) -HWV1 HLK' HV12 // >arith_a2 /3 width=6/
- | -IHV12 Hdi Hdj;
- generalize in match HV1 generalize in match Hide -HV1 Hide (**) (* rewriting in the premises, rewrites in the goal too *)
- >(plus_minus_m_m_comm … Hjde) in ⊢ (% → % → ?) -Hjde #Hide #HV1
- @ex2_1_intro [2: @tps_lref |1: skip | /2 width=6/ ] (**) (* /3 width=6 is too slow *)
- ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
- elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
- -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
- @ex2_1_intro [2,3: @tps_bind | skip ] (**) (* explicit constructors *)
- [3: @HV1 |4: @HT1 |5: // |1,2: skip | /3 width=5/ ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
- elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
- -Hdi Hide /3 width=5/
-]
-qed.
-
-lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
- ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
- d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
- ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tps_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
-lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
-elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
-qed.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/substitution/tps_split.ma".
-
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
-
-(* Main properties **********************************************************)
-(*
-theorem tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
- L ⊢ T1 [d, e] ≫ T2.
-#L #T1 #T #d #e #H elim H -L T1 T d e
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
- lapply (drop_fwd_drop2 … HLK) #H
- elim (tps_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
- @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (tps_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
- @tps_bind /2/ @IHT12 @(tps_leq_repl … HT2) /2/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
- elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
-]
-qed.
-*)
-
-axiom tps_conf_subst_subst_lt: ∀L,K1,V1,W1,T1,i1,d,e,T2,K2,V2,W2,i2.
- ↓[O, i1] L ≡ K1. 𝕓{Abbr} V1 → ↓[O, i2] L≡ K2. 𝕓{Abbr} V2 →
- K1 ⊢ V1 [O, d + e - i1 - 1] ≫ W1 → K2 ⊢ V2 [O, d + e - i2 - 1] ≫ W2 →
- ↑[O, i1 + 1] W1 ≡ T1 → ↑[O, i2 + 1] W2 ≡ T2 →
- d ≤ i1 → i1 < d + e → d ≤ i2 → i2 < d + e → i1 < i2 →
- ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
-(*
-#L #K1 #V1 #W1 #T1 #i1 #d #e #T2 #K2 #V2 #W2 #i2
-#HLK1 #HLK2 #HVW1 #HVW2 #HWT1 #HWT2 #Hdi1 #Hi1de #Hdi2 #Hi2de #Hi12
-*)
-
-theorem tps_conf: ∀L,T0,T1,d,e. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
- ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
-#L #T0 #T1 #d #e #H elim H -H L T0 T1 d e
-[ /2/
-| /2/
-| #L #K1 #V1 #W1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVW1 #HWT1 #IHVW1 #T2 #H
- elim (tps_inv_lref1 … H) -H
- [ -IHVW1 #HX destruct -T2
- @ex2_1_intro [2: // | skip ] /2 width=6/ (**) (* /3 width=9/ is slow *)
- | * #K2 #V2 #W2 #i2 #Hdi2 #Hi2de #HLK2 #HVW2 #HWT2
- elim (lt_or_eq_or_gt i1 i2) #Hi12
- [ @tps_conf_subst_subst_lt /width=19/
- | -HVW1; destruct -i2;
- lapply (transitive_le ? ? (i1 + 1) Hdi2 ?) -Hdi2 // #Hdi2
- lapply (drop_mono … HLK1 … HLK2) -HLK1 Hdi1 Hi1de #H destruct -V1 K1;
- elim (IHVW1 … HVW2) -IHVW1 HVW2 #W #HW1 #HW2
- lapply (drop_fwd_drop2 … HLK2) -HLK2 #HLK2
- elim (lift_total W 0 (i1 + 1)) #T #HWT
- lapply (tps_lift_ge … HW1 … HLK2 HWT1 HWT ?) -HW1 HWT1 //
- lapply (tps_lift_ge … HW2 … HLK2 HWT2 HWT ?) -HW2 HWT2 HLK2 HWT // normalize #HT2 #HT1
- lapply (tps_weak … HT1 d e ? ?) -HT1 [ >arith_i2 // | // ]
- lapply (tps_weak … HT2 d e ? ?) -HT2 [ >arith_i2 // | // ]
- /2/
- | @ex2_1_comm @tps_conf_subst_subst_lt /width=19/
- ]
- ]
-| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
- elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
- elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
- @ex2_1_intro
- [2: @tps_bind [4: @(tps_leq_repl … HT1) /2/ |2: skip ]
- |1: skip
- |3: @tps_bind [2: @(tps_leq_repl … HT2) /2/ ]
- ] // (**) (* /5/ is too slow *)
-| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
- elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
- elim (IHV01 … HV02) -IHV01 HV02;
- elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/
-]
-qed.
-
-(*
- Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
- (u1,u:?; i:?) (subst0 i u u1 u2) ->
- (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
-
- Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
- (u1,u:?; i:?) (subst0 i u u2 u1) ->
- (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
-
-*)
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-(* THE FORMAL SYSTEM λδ - MATITA SOURCE SCRIPTS
- * Specification started: 2011 April 17
- * - Patience on me so that I gain peace and perfection! -
- * [ suggested invocation to start formal specifications with ]
- *)
-
-include "lambda-delta/ground.ma".
-include "lambda-delta/notation.ma".
-
-(* BINARY ITEMS *************************************************************)
-
-(* binary binding items *)
-inductive bind2: Type[0] ≝
-| Abbr: bind2 (* abbreviation *)
-| Abst: bind2 (* abstraction *)
-.
-
-(* binary non-binding items *)
-inductive flat2: Type[0] ≝
-| Appl: flat2 (* application *)
-| Cast: flat2 (* explicit type annotation *)
-.
-
-(* binary items *)
-inductive item2: Type[0] ≝
-| Bind: bind2 → item2 (* binding item *)
-| Flat: flat2 → item2 (* non-binding item *)
-.
-
-coercion item2_of_bind2: ∀I:bind2.item2 ≝ Bind on _I:bind2 to item2.
-
-coercion item2_of_flat2: ∀I:flat2.item2 ≝ Flat on _I:flat2 to item2.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/syntax/lenv.ma".
-
-(* LENGTH *******************************************************************)
-
-(* the length of a local environment *)
-let rec length L ≝ match L with
-[ LSort ⇒ 0
-| LPair L _ _ ⇒ length L + 1
-].
-
-interpretation "length (local environment)" 'card L = (length L).
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/syntax/term.ma".
-
-(* LOCAL ENVIRONMENTS *******************************************************)
-
-(* local environments *)
-inductive lenv: Type[0] ≝
-| LSort: lenv (* empty *)
-| LPair: lenv → bind2 → term → lenv (* binary binding construction *)
-.
-
-interpretation "sort (local environment)" 'Star = LSort.
-
-interpretation "environment binding construction (binary)" 'DBind L I T = (LPair L I T).
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/ground.ma".
-
-(* SORT HIERARCHY ***********************************************************)
-
-(* sort hierarchy specifications *)
-record sh: Type[0] ≝ {
- next: nat → nat; (* next sort in the hierarchy *)
- next_lt: ∀k. k < next k (* strict monotonicity condition *)
-}.
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/syntax/item.ma".
-
-(* TERMS ********************************************************************)
-
-(* terms *)
-inductive term: Type[0] ≝
-| TSort: nat → term (* sort: starting at 0 *)
-| TLRef: nat → term (* reference by index: starting at 0 *)
-| TPair: item2 → term → term → term (* binary item construction *)
-.
-
-interpretation "sort (term)" 'Star k = (TSort k).
-
-interpretation "local reference (term)" 'Weight i = (TLRef i).
-
-interpretation "term construction (binary)" 'SItem I T1 T2 = (TPair I T1 T2).
-
-interpretation "term binding construction (binary)" 'SBind I T1 T2 = (TPair (Bind I) T1 T2).
-
-interpretation "term flat construction (binary)" 'SFlat I T1 T2 = (TPair (Flat I) T1 T2).
+++ /dev/null
-(*
- ||M|| This file is part of HELM, an Hypertextual, Electronic
- ||A|| Library of Mathematics, developed at the Computer Science
- ||T|| Department of the University of Bologna, Italy.
- ||I||
- ||T||
- ||A|| This file is distributed under the terms of the
- \ / GNU General Public License Version 2
- \ /
- V_______________________________________________________________ *)
-
-include "lambda-delta/syntax/lenv.ma".
-
-(* WEIGHTS ******************************************************************)
-
-(* the weight of a term *)
-let rec tw T ≝ match T with
-[ TSort _ ⇒ 1
-| TLRef _ ⇒ 1
-| TPair _ V T ⇒ tw V + tw T + 1
-].
-
-interpretation "weight (term)" 'Weight T = (tw T).
-
-(* the weight of a local environment *)
-let rec lw L ≝ match L with
-[ LSort ⇒ 0
-| LPair L _ V ⇒ lw L + #V
-].
-
-interpretation "weight (local environment)" 'Weight L = (lw L).
-
-(* the weight of a closure *)
-definition cw: lenv → term → ? ≝ λL,T. #L + #T.
-
-interpretation "weight (closure)" 'Weight L T = (cw L T).
-
-axiom tw_wf_ind: ∀P:term→Prop.
- (∀T2. (∀T1. # T1 < # T2 → P T1) → P T2) →
- ∀T. P T.
-
-axiom cw_wf_ind: ∀P:lenv→term→Prop.
- (∀L2,T2. (∀L1,T1. #[L1,T1] < #[L2,T2] → P L1 T1) → P L2 T2) →
- ∀L,T. P L T.
+++ /dev/null
-<?xml version="1.0" encoding="utf-8"?>
-<helm_registry>
- <section name="matita">
- <key name="rt_base_dir">$(MATITA_RT_BASE_DIR)</key>
-<!--
- <key name="system">false</key>
- <key name="map_unicode_to_tex">false</key>
- <key name="do_heavy_checks">true</key>
- <key name="include_path">lib</key>
--->
- </section>
- <section name="xoa">
- <key name="output_dir">lib/lambda-delta</key>
- <key name="objects">xoa</key>
- <key name="notations">xoa_notation</key>
- <key name="include">basics/pts.ma</key>
- <key name="ex">2 1</key>
- <key name="ex">2 2</key>
- <key name="ex">3 1</key>
- <key name="ex">3 2</key>
- <key name="ex">3 3</key>
- <key name="ex">4 3</key>
- <key name="ex">4 4</key>
- <key name="ex">5 3</key>
- <key name="ex">5 4</key>
- <key name="ex">6 6</key>
- <key name="ex">7 6</key>
- <key name="or">3</key>
- <key name="or">4</key>
-<!--
- <key name="and">3</key>
--->
- </section>
-</helm_registry>
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was generated by xoa.native: do not edit *********************)
-
-include "basics/pts.ma".
-
-(* multiple existental quantifier (2, 1) *)
-
-inductive ex2_1 (A0:Type[0]) (P0,P1:A0→Prop) : Prop ≝
- | ex2_1_intro: ∀x0. P0 x0 → P1 x0 → ex2_1 ? ? ?
-.
-
-interpretation "multiple existental quantifier (2, 1)" 'Ex P0 P1 = (ex2_1 ? P0 P1).
-
-(* multiple existental quantifier (2, 2) *)
-
-inductive ex2_2 (A0,A1:Type[0]) (P0,P1:A0→A1→Prop) : Prop ≝
- | ex2_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → ex2_2 ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (2, 2)" 'Ex P0 P1 = (ex2_2 ? ? P0 P1).
-
-(* multiple existental quantifier (3, 1) *)
-
-inductive ex3_1 (A0:Type[0]) (P0,P1,P2:A0→Prop) : Prop ≝
- | ex3_1_intro: ∀x0. P0 x0 → P1 x0 → P2 x0 → ex3_1 ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (3, 1)" 'Ex P0 P1 P2 = (ex3_1 ? P0 P1 P2).
-
-(* multiple existental quantifier (3, 2) *)
-
-inductive ex3_2 (A0,A1:Type[0]) (P0,P1,P2:A0→A1→Prop) : Prop ≝
- | ex3_2_intro: ∀x0,x1. P0 x0 x1 → P1 x0 x1 → P2 x0 x1 → ex3_2 ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (3, 2)" 'Ex P0 P1 P2 = (ex3_2 ? ? P0 P1 P2).
-
-(* multiple existental quantifier (3, 3) *)
-
-inductive ex3_3 (A0,A1,A2:Type[0]) (P0,P1,P2:A0→A1→A2→Prop) : Prop ≝
- | ex3_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → ex3_3 ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (3, 3)" 'Ex P0 P1 P2 = (ex3_3 ? ? ? P0 P1 P2).
-
-(* multiple existental quantifier (4, 3) *)
-
-inductive ex4_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3:A0→A1→A2→Prop) : Prop ≝
- | ex4_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → ex4_3 ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (4, 3)" 'Ex P0 P1 P2 P3 = (ex4_3 ? ? ? P0 P1 P2 P3).
-
-(* multiple existental quantifier (4, 4) *)
-
-inductive ex4_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3:A0→A1→A2→A3→Prop) : Prop ≝
- | ex4_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → ex4_4 ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (4, 4)" 'Ex P0 P1 P2 P3 = (ex4_4 ? ? ? ? P0 P1 P2 P3).
-
-(* multiple existental quantifier (5, 3) *)
-
-inductive ex5_3 (A0,A1,A2:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→Prop) : Prop ≝
- | ex5_3_intro: ∀x0,x1,x2. P0 x0 x1 x2 → P1 x0 x1 x2 → P2 x0 x1 x2 → P3 x0 x1 x2 → P4 x0 x1 x2 → ex5_3 ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (5, 3)" 'Ex P0 P1 P2 P3 P4 = (ex5_3 ? ? ? P0 P1 P2 P3 P4).
-
-(* multiple existental quantifier (5, 4) *)
-
-inductive ex5_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→Prop) : Prop ≝
- | ex5_4_intro: ∀x0,x1,x2,x3. P0 x0 x1 x2 x3 → P1 x0 x1 x2 x3 → P2 x0 x1 x2 x3 → P3 x0 x1 x2 x3 → P4 x0 x1 x2 x3 → ex5_4 ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4).
-
-(* multiple existental quantifier (6, 6) *)
-
-inductive ex6_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
- | ex6_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → ex6_6 ? ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (6, 6)" 'Ex P0 P1 P2 P3 P4 P5 = (ex6_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5).
-
-(* multiple existental quantifier (7, 6) *)
-
-inductive ex7_6 (A0,A1,A2,A3,A4,A5:Type[0]) (P0,P1,P2,P3,P4,P5,P6:A0→A1→A2→A3→A4→A5→Prop) : Prop ≝
- | ex7_6_intro: ∀x0,x1,x2,x3,x4,x5. P0 x0 x1 x2 x3 x4 x5 → P1 x0 x1 x2 x3 x4 x5 → P2 x0 x1 x2 x3 x4 x5 → P3 x0 x1 x2 x3 x4 x5 → P4 x0 x1 x2 x3 x4 x5 → P5 x0 x1 x2 x3 x4 x5 → P6 x0 x1 x2 x3 x4 x5 → ex7_6 ? ? ? ? ? ? ? ? ? ? ? ? ?
-.
-
-interpretation "multiple existental quantifier (7, 6)" 'Ex P0 P1 P2 P3 P4 P5 P6 = (ex7_6 ? ? ? ? ? ? P0 P1 P2 P3 P4 P5 P6).
-
-(* multiple disjunction connective (3) *)
-
-inductive or3 (P0,P1,P2:Prop) : Prop ≝
- | or3_intro0: P0 → or3 ? ? ?
- | or3_intro1: P1 → or3 ? ? ?
- | or3_intro2: P2 → or3 ? ? ?
-.
-
-interpretation "multiple disjunction connective (3)" 'Or P0 P1 P2 = (or3 P0 P1 P2).
-
-(* multiple disjunction connective (4) *)
-
-inductive or4 (P0,P1,P2,P3:Prop) : Prop ≝
- | or4_intro0: P0 → or4 ? ? ? ?
- | or4_intro1: P1 → or4 ? ? ? ?
- | or4_intro2: P2 → or4 ? ? ? ?
- | or4_intro3: P3 → or4 ? ? ? ?
-.
-
-interpretation "multiple disjunction connective (4)" 'Or P0 P1 P2 P3 = (or4 P0 P1 P2 P3).
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* This file was generated by xoa.native: do not edit *********************)
-
-(* multiple existental quantifier (2, 1) *)
-
-notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) }.
-
-notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) }.
-
-(* multiple existental quantifier (2, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) }.
-
-(* multiple existental quantifier (3, 1) *)
-
-notation > "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.$P0) (λ${ident x0}.$P1) (λ${ident x0}.$P2) }.
-
-notation < "hvbox(∃∃ ident x0 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.$P0) (λ${ident x0}:$T0.$P1) (λ${ident x0}:$T0.$P2) }.
-
-(* multiple existental quantifier (3, 2) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.$P0) (λ${ident x0}.λ${ident x1}.$P1) (λ${ident x0}.λ${ident x1}.$P2) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.$P2) }.
-
-(* multiple existental quantifier (3, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) }.
-
-(* multiple existental quantifier (4, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) }.
-
-(* multiple existental quantifier (4, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) }.
-
-(* multiple existental quantifier (5, 3) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.$P4) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.$P4) }.
-
-(* multiple existental quantifier (5, 4) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.$P4) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }.
-
-(* multiple existental quantifier (6, 6) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) }.
-
-(* multiple existental quantifier (7, 6) *)
-
-notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P4) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P5) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.λ${ident x5}.$P6) }.
-
-notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 , ident x5 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5 break & term 19 P6)"
- non associative with precedence 20
- for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P4) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P5) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.λ${ident x5}:$T5.$P6) }.
-
-(* multiple disjunction connective (3) *)
-
-notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2)"
- non associative with precedence 30
- for @{ 'Or $P0 $P1 $P2 }.
-
-(* multiple disjunction connective (4) *)
-
-notation "hvbox(∨∨ term 29 P0 break | term 29 P1 break | term 29 P2 break | term 29 P3)"
- non associative with precedence 30
- for @{ 'Or $P0 $P1 $P2 $P3 }.
-
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "lambda-delta/xoa_notation.ma".
-include "lambda-delta/xoa.ma".
-
-lemma ex2_1_comm: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0.
-#A0 #P0 #P1 * /2/
-qed.
projects / helm.git / commitdiff