From e03c464898a47de95b6db7235b75b6f7a2c316d0 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Wed, 10 Aug 2011 11:23:43 +0000 Subject: [PATCH] lambda-delta must be a contrib --- matita/matita/lib/lambda-delta/Makefile | 12 - matita/matita/lib/lambda-delta/ground.ma | 182 --------- matita/matita/lib/lambda-delta/names.txt | 14 - matita/matita/lib/lambda-delta/notation.ma | 78 ---- .../matita/lib/lambda-delta/reduction/cpr.ma | 52 --- .../matita/lib/lambda-delta/reduction/lpr.ma | 41 -- .../matita/lib/lambda-delta/reduction/tpr.ma | 240 ------------ .../lib/lambda-delta/reduction/tpr_lift.ma | 102 ----- .../lib/lambda-delta/reduction/tpr_tpr.ma | 351 ------------------ .../lib/lambda-delta/reduction/tpr_tps.ma | 26 -- .../lib/lambda-delta/substitution/drop.ma | 193 ---------- .../lambda-delta/substitution/drop_drop.ma | 125 ------- .../lib/lambda-delta/substitution/leq.ma | 64 ---- .../lib/lambda-delta/substitution/lift.ma | 230 ------------ .../lambda-delta/substitution/lift_lift.ma | 154 -------- .../lambda-delta/substitution/lift_weight.ma | 22 -- .../lib/lambda-delta/substitution/tps.ma | 153 -------- .../lib/lambda-delta/substitution/tps_lift.ma | 173 --------- .../lambda-delta/substitution/tps_split.ma | 58 --- .../lib/lambda-delta/substitution/tps_tps.ma | 98 ----- matita/matita/lib/lambda-delta/syntax/item.ma | 43 --- .../matita/lib/lambda-delta/syntax/length.ma | 22 -- matita/matita/lib/lambda-delta/syntax/lenv.ma | 24 -- matita/matita/lib/lambda-delta/syntax/sh.ma | 20 - matita/matita/lib/lambda-delta/syntax/term.ma | 31 -- .../matita/lib/lambda-delta/syntax/weight.ma | 44 --- matita/matita/lib/lambda-delta/xoa.conf.xml | 34 -- matita/matita/lib/lambda-delta/xoa.ma | 127 ------- .../matita/lib/lambda-delta/xoa_notation.ma | 138 ------- matita/matita/lib/lambda-delta/xoa_props.ma | 20 - 30 files changed, 2871 deletions(-) delete mode 100644 matita/matita/lib/lambda-delta/Makefile delete mode 100644 matita/matita/lib/lambda-delta/ground.ma delete mode 100644 matita/matita/lib/lambda-delta/names.txt delete mode 100644 matita/matita/lib/lambda-delta/notation.ma delete mode 100644 matita/matita/lib/lambda-delta/reduction/cpr.ma delete mode 100644 matita/matita/lib/lambda-delta/reduction/lpr.ma delete mode 100644 matita/matita/lib/lambda-delta/reduction/tpr.ma delete mode 100644 matita/matita/lib/lambda-delta/reduction/tpr_lift.ma delete mode 100644 matita/matita/lib/lambda-delta/reduction/tpr_tpr.ma delete mode 100644 matita/matita/lib/lambda-delta/reduction/tpr_tps.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/drop.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/drop_drop.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/leq.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/lift.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/lift_lift.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/lift_weight.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/tps.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/tps_lift.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/tps_split.ma delete mode 100644 matita/matita/lib/lambda-delta/substitution/tps_tps.ma delete mode 100644 matita/matita/lib/lambda-delta/syntax/item.ma delete mode 100644 matita/matita/lib/lambda-delta/syntax/length.ma delete mode 100644 matita/matita/lib/lambda-delta/syntax/lenv.ma delete mode 100644 matita/matita/lib/lambda-delta/syntax/sh.ma delete mode 100644 matita/matita/lib/lambda-delta/syntax/term.ma delete mode 100644 matita/matita/lib/lambda-delta/syntax/weight.ma delete mode 100644 matita/matita/lib/lambda-delta/xoa.conf.xml delete mode 100644 matita/matita/lib/lambda-delta/xoa.ma delete mode 100644 matita/matita/lib/lambda-delta/xoa_notation.ma delete mode 100644 matita/matita/lib/lambda-delta/xoa_props.ma diff --git a/matita/matita/lib/lambda-delta/Makefile b/matita/matita/lib/lambda-delta/Makefile deleted file mode 100644 index 8feb040db..000000000 --- a/matita/matita/lib/lambda-delta/Makefile +++ /dev/null @@ -1,12 +0,0 @@ -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) diff --git a/matita/matita/lib/lambda-delta/ground.ma b/matita/matita/lib/lambda-delta/ground.ma deleted file mode 100644 index ed5876107..000000000 --- a/matita/matita/lib/lambda-delta/ground.ma +++ /dev/null @@ -1,182 +0,0 @@ -(* - ||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 (commutative_plus p) (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 - 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_plus >minus_plus @monotonic_le_plus_r @monotonic_le_minus_l // -qed. diff --git a/matita/matita/lib/lambda-delta/names.txt b/matita/matita/lib/lambda-delta/names.txt deleted file mode 100644 index 2a030fb4f..000000000 --- a/matita/matita/lib/lambda-delta/names.txt +++ /dev/null @@ -1,14 +0,0 @@ -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 diff --git a/matita/matita/lib/lambda-delta/notation.ma b/matita/matita/lib/lambda-delta/notation.ma deleted file mode 100644 index 375d06abc..000000000 --- a/matita/matita/lib/lambda-delta/notation.ma +++ /dev/null @@ -1,78 +0,0 @@ -(* - ||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 }. diff --git a/matita/matita/lib/lambda-delta/reduction/cpr.ma b/matita/matita/lib/lambda-delta/reduction/cpr.ma deleted file mode 100644 index cfd51af8a..000000000 --- a/matita/matita/lib/lambda-delta/reduction/cpr.ma +++ /dev/null @@ -1,52 +0,0 @@ -(* - ||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 ***************************************************) diff --git a/matita/matita/lib/lambda-delta/reduction/lpr.ma b/matita/matita/lib/lambda-delta/reduction/lpr.ma deleted file mode 100644 index 92f77215c..000000000 --- a/matita/matita/lib/lambda-delta/reduction/lpr.ma +++ /dev/null @@ -1,41 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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. diff --git a/matita/matita/lib/lambda-delta/reduction/tpr.ma b/matita/matita/lib/lambda-delta/reduction/tpr.ma deleted file mode 100644 index fbf4d47c4..000000000 --- a/matita/matita/lib/lambda-delta/reduction/tpr.ma +++ /dev/null @@ -1,240 +0,0 @@ -(* - ||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. diff --git a/matita/matita/lib/lambda-delta/reduction/tpr_lift.ma b/matita/matita/lib/lambda-delta/reduction/tpr_lift.ma deleted file mode 100644 index e6fd96454..000000000 --- a/matita/matita/lib/lambda-delta/reduction/tpr_lift.ma +++ /dev/null @@ -1,102 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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. diff --git a/matita/matita/lib/lambda-delta/reduction/tpr_tpr.ma b/matita/matita/lib/lambda-delta/reduction/tpr_tpr.ma deleted file mode 100644 index ee22bfc21..000000000 --- a/matita/matita/lib/lambda-delta/reduction/tpr_tpr.ma +++ /dev/null @@ -1,351 +0,0 @@ -(* - ||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. diff --git a/matita/matita/lib/lambda-delta/reduction/tpr_tps.ma b/matita/matita/lib/lambda-delta/reduction/tpr_tps.ma deleted file mode 100644 index 96bc9b76e..000000000 --- a/matita/matita/lib/lambda-delta/reduction/tpr_tps.ma +++ /dev/null @@ -1,26 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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. diff --git a/matita/matita/lib/lambda-delta/substitution/drop.ma b/matita/matita/lib/lambda-delta/substitution/drop.ma deleted file mode 100644 index 29b57405f..000000000 --- a/matita/matita/lib/lambda-delta/substitution/drop.ma +++ /dev/null @@ -1,193 +0,0 @@ -(* - ||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; - 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. diff --git a/matita/matita/lib/lambda-delta/substitution/drop_drop.ma b/matita/matita/lib/lambda-delta/substitution/drop_drop.ma deleted file mode 100644 index 812fb7e06..000000000 --- a/matita/matita/lib/lambda-delta/substitution/drop_drop.ma +++ /dev/null @@ -1,125 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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 // (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_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 [ (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. diff --git a/matita/matita/lib/lambda-delta/substitution/leq.ma b/matita/matita/lib/lambda-delta/substitution/leq.ma deleted file mode 100644 index b0e28d48e..000000000 --- a/matita/matita/lib/lambda-delta/substitution/leq.ma +++ /dev/null @@ -1,64 +0,0 @@ -(* - ||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. diff --git a/matita/matita/lib/lambda-delta/substitution/lift.ma b/matita/matita/lib/lambda-delta/substitution/lift.ma deleted file mode 100644 index a5b15e110..000000000 --- a/matita/matita/lib/lambda-delta/substitution/lift.ma +++ /dev/null @@ -1,230 +0,0 @@ -(* - ||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 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_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/ ] - 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)). -*) diff --git a/matita/matita/lib/lambda-delta/substitution/tps_split.ma b/matita/matita/lib/lambda-delta/substitution/tps_split.ma deleted file mode 100644 index 23cdb39df..000000000 --- a/matita/matita/lib/lambda-delta/substitution/tps_split.ma +++ /dev/null @@ -1,58 +0,0 @@ -(* - ||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 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 // 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)). - -*) diff --git a/matita/matita/lib/lambda-delta/syntax/item.ma b/matita/matita/lib/lambda-delta/syntax/item.ma deleted file mode 100644 index ea7a45362..000000000 --- a/matita/matita/lib/lambda-delta/syntax/item.ma +++ /dev/null @@ -1,43 +0,0 @@ -(* - ||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. diff --git a/matita/matita/lib/lambda-delta/syntax/length.ma b/matita/matita/lib/lambda-delta/syntax/length.ma deleted file mode 100644 index 91e1bd78c..000000000 --- a/matita/matita/lib/lambda-delta/syntax/length.ma +++ /dev/null @@ -1,22 +0,0 @@ -(* - ||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). diff --git a/matita/matita/lib/lambda-delta/syntax/lenv.ma b/matita/matita/lib/lambda-delta/syntax/lenv.ma deleted file mode 100644 index c3aab910b..000000000 --- a/matita/matita/lib/lambda-delta/syntax/lenv.ma +++ /dev/null @@ -1,24 +0,0 @@ -(* - ||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). diff --git a/matita/matita/lib/lambda-delta/syntax/sh.ma b/matita/matita/lib/lambda-delta/syntax/sh.ma deleted file mode 100644 index 32840edff..000000000 --- a/matita/matita/lib/lambda-delta/syntax/sh.ma +++ /dev/null @@ -1,20 +0,0 @@ -(* - ||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 *) -}. diff --git a/matita/matita/lib/lambda-delta/syntax/term.ma b/matita/matita/lib/lambda-delta/syntax/term.ma deleted file mode 100644 index fe84e54b4..000000000 --- a/matita/matita/lib/lambda-delta/syntax/term.ma +++ /dev/null @@ -1,31 +0,0 @@ -(* - ||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). diff --git a/matita/matita/lib/lambda-delta/syntax/weight.ma b/matita/matita/lib/lambda-delta/syntax/weight.ma deleted file mode 100644 index d076dea9d..000000000 --- a/matita/matita/lib/lambda-delta/syntax/weight.ma +++ /dev/null @@ -1,44 +0,0 @@ -(* - ||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. diff --git a/matita/matita/lib/lambda-delta/xoa.conf.xml b/matita/matita/lib/lambda-delta/xoa.conf.xml deleted file mode 100644 index d25c5dcca..000000000 --- a/matita/matita/lib/lambda-delta/xoa.conf.xml +++ /dev/null @@ -1,34 +0,0 @@ - - -
- $(MATITA_RT_BASE_DIR) - -
-
- lib/lambda-delta - xoa - xoa_notation - basics/pts.ma - 2 1 - 2 2 - 3 1 - 3 2 - 3 3 - 4 3 - 4 4 - 5 3 - 5 4 - 6 6 - 7 6 - 3 - 4 - -
- diff --git a/matita/matita/lib/lambda-delta/xoa.ma b/matita/matita/lib/lambda-delta/xoa.ma deleted file mode 100644 index 68a0776dc..000000000 --- a/matita/matita/lib/lambda-delta/xoa.ma +++ /dev/null @@ -1,127 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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). - diff --git a/matita/matita/lib/lambda-delta/xoa_notation.ma b/matita/matita/lib/lambda-delta/xoa_notation.ma deleted file mode 100644 index b8270b660..000000000 --- a/matita/matita/lib/lambda-delta/xoa_notation.ma +++ /dev/null @@ -1,138 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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 }. - diff --git a/matita/matita/lib/lambda-delta/xoa_props.ma b/matita/matita/lib/lambda-delta/xoa_props.ma deleted file mode 100644 index f07bc89be..000000000 --- a/matita/matita/lib/lambda-delta/xoa_props.ma +++ /dev/null @@ -1,20 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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. -- 2.39.2