include "basics/relations.ma".
-inductive nat : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="nat"\ 6inductive nat : Type[0] ≝
| O : nat
| S : nat → nat.
alias num (instance 0) = "natural number".
-definition pred ≝
+\ 5img class="anchor" src="icons/tick.png" id="pred"\ 6definition pred ≝
λn. match n with [ O ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 | S p ⇒ p].
-theorem pred_Sn : ∀n.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n).
+\ 5img class="anchor" src="icons/tick.png" id="pred_Sn"\ 6theorem pred_Sn : ∀n.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n).
// qed.
-theorem injective_S : \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="injective_S"\ 6theorem injective_S : \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6.
// qed.
(*
theorem inj_S : \forall n,m:nat.(S n)=(S m) \to n=m.
//. qed. *)
-theorem not_eq_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="not_eq_S"\ 6theorem not_eq_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-definition not_zero: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="not_zero"\ 6definition not_zero: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝
λn: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. match n with [ O ⇒ \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6 | (S p) ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6 ].
-theorem not_eq_O_S : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="not_eq_O_S"\ 6theorem not_eq_O_S : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
#n @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #eqOS (change with (\ 5a href="cic:/matita/arithmetics/nat/not_zero.def(1)"\ 6not_zero\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)) >eqOS // qed.
-theorem not_eq_n_Sn: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="not_eq_n_Sn"\ 6theorem not_eq_n_Sn: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
#n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"\ 6not_eq_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem nat_case:
+\ 5img class="anchor" src="icons/tick.png" id="nat_case"\ 6theorem nat_case:
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.
(n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → P \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6) → (∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → P (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) → P n.
#n #P (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
-theorem nat_elim2 :
+\ 5img class="anchor" src="icons/tick.png" id="nat_elim2"\ 6theorem nat_elim2 :
∀R:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.
(∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 n)
→ (∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)
→ ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. R n m.
#R #ROn #RSO #RSS #n (elim n) // #n0 #Rn0m #m (cases m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
-theorem decidable_eq_nat : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m).
+\ 5img class="anchor" src="icons/tick.png" id="decidable_eq_nat"\ 6theorem decidable_eq_nat : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #n [ (cases n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | #m #Hind (cases Hind) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_eq_S.def(4)"\ 6not_eq_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
qed.
(*************************** plus ******************************)
-let rec plus n m ≝
+\ 5img class="anchor" src="icons/tick.png" id="plus"\ 6let rec plus n m ≝
match n with [ O ⇒ m | S p ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (plus p m) ].
interpretation "natural plus" 'plus x y = (plus x y).
-theorem plus_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n.
+\ 5img class="anchor" src="icons/tick.png" id="plus_O_n"\ 6theorem plus_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n.
// qed.
(*
// qed.
*)
-theorem plus_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="plus_n_O"\ 6theorem plus_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6.
#n (elim n) normalize // qed.
-theorem plus_n_Sm : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="plus_n_Sm"\ 6theorem plus_n_Sm : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
#n (elim n) normalize // qed.
(*
// qed.
*)
-theorem commutative_plus: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="commutative_plus"\ 6theorem commutative_plus: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
#n (elim n) normalize // qed.
-theorem associative_plus : \ 5a href="cic:/matita/basics/relations/associative.def(1)"\ 6associative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="associative_plus"\ 6theorem associative_plus : \ 5a href="cic:/matita/basics/relations/associative.def(1)"\ 6associative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
#n (elim n) normalize // qed.
-theorem assoc_plus1: ∀a,b,c. c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (b \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a.
+\ 5img class="anchor" src="icons/tick.png" id="assoc_plus1"\ 6theorem assoc_plus1: ∀a,b,c. c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (b \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a.
// qed.
-theorem injective_plus_r: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 (λm.n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m).
+\ 5img class="anchor" src="icons/tick.png" id="injective_plus_r"\ 6theorem injective_plus_r: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 (λm.n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m).
#n (elim n) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/injective_S.def(4)"\ 6injective_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* theorem inj_plus_r: \forall p,n,m:nat. p+n = p+m \to n=m
(*************************** times *****************************)
-let rec times n m ≝
+\ 5img class="anchor" src="icons/tick.png" id="times"\ 6let rec times n m ≝
match n with [ O ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 | S p ⇒ m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(times p m) ].
interpretation "natural times" 'times x y = (times x y).
-theorem times_Sn_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m.
+\ 5img class="anchor" src="icons/tick.png" id="times_Sn_m"\ 6theorem times_Sn_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m.
// qed.
-theorem times_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n.
+\ 5img class="anchor" src="icons/tick.png" id="times_O_n"\ 6theorem times_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n.
// qed.
-theorem times_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="times_n_O"\ 6theorem times_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
#n (elim n) // qed.
-theorem times_n_Sm : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m).
+\ 5img class="anchor" src="icons/tick.png" id="times_n_Sm"\ 6theorem times_n_Sm : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m).
#n (elim n) normalize // qed.
-theorem commutative_times : \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="commutative_times"\ 6theorem commutative_times : \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6.
#n (elim n) normalize // qed.
(* variant sym_times : \forall n,m:nat. n*m = m*n \def
symmetric_times. *)
-theorem distributive_times_plus : \ 5a href="cic:/matita/basics/relations/distributive.def(1)"\ 6distributive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="distributive_times_plus"\ 6theorem distributive_times_plus : \ 5a href="cic:/matita/basics/relations/distributive.def(1)"\ 6distributive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/plus.fix(0,0,1)"\ 6plus\ 5/a\ 6.
#n (elim n) normalize // qed.
-theorem distributive_times_plus_r :
+\ 5img class="anchor" src="icons/tick.png" id="distributive_times_plus_r"\ 6theorem distributive_times_plus_r :
∀a,b,c:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. (b\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6c)\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6a.
// qed.
-theorem associative_times: \ 5a href="cic:/matita/basics/relations/associative.def(1)"\ 6associative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="associative_times"\ 6theorem associative_times: \ 5a href="cic:/matita/basics/relations/associative.def(1)"\ 6associative\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6.
#n (elim n) normalize // qed.
-lemma times_times: ∀x,y,z. x\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(y\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6z) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(x\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6z).
+\ 5img class="anchor" src="icons/tick.png" id="times_times"\ 6lemma times_times: ∀x,y,z. x\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(y\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6z) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6(x\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6z).
// qed.
-theorem times_n_1 : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 61\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="times_n_1"\ 6theorem times_n_1 : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 61\ 5/a\ 6.
#n // qed.
(* ci servono questi risultati?
(******************** ordering relations ************************)
-inductive le (n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6) : \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="le"\ 6inductive le (n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6) : \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝
| le_n : le n n
| le_S : ∀ m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. le n m → le n (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m).
interpretation "natural 'neither less nor equal to'" 'nleq x y = (Not (le x y)).
-definition lt: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="lt"\ 6definition lt: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
interpretation "natural 'less than'" 'lt x y = (lt x y).
interpretation "natural 'not less than'" 'nless x y = (Not (lt x y)).
(* lemma eq_lt: ∀n,m. (n < m) = (S n ≤ m).
// qed. *)
-definition ge: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m.m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="ge"\ 6definition ge: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m.m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
interpretation "natural 'greater or equal to'" 'geq x y = (ge x y).
-definition gt: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m.m\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6n.
+\ 5img class="anchor" src="icons/tick.png" id="gt"\ 6definition gt: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop ≝ λn,m.m\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6n.
interpretation "natural 'greater than'" 'gt x y = (gt x y).
interpretation "natural 'not greater than'" 'ngtr x y = (Not (gt x y)).
-theorem transitive_le : \ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="transitive_le"\ 6theorem transitive_le : \ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6.
#a #b #c #leab #lebc (elim lebc) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
theorem trans_le: \forall n,m,p:nat. n \leq m \to m \leq p \to n \leq p
\def transitive_le. *)
-theorem transitive_lt: \ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="transitive_lt"\ 6theorem transitive_lt: \ 5a href="cic:/matita/basics/relations/transitive.def(2)"\ 6transitive\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6.
#a #b #c #ltab #ltbc (elim ltbc) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/qed.
(*
theorem trans_lt: \forall n,m,p:nat. lt n m \to lt m p \to lt n p
\def transitive_lt. *)
-theorem le_S_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_S_S"\ 6theorem le_S_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
#n #m #lenm (elim lenm) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_O_n : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="le_O_n"\ 6theorem le_O_n : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
#n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_n_Sn : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="le_n_Sn"\ 6theorem le_n_Sn : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,2,1)"\ 6le_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_pred_n : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="le_pred_n"\ 6theorem le_pred_n : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
#n (elim n) // qed.
-theorem monotonic_pred: \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_pred"\ 6theorem monotonic_pred: \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6.
#n #m #lenm (elim lenm) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_S_S_to_le: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_S_S_to_le"\ 6theorem le_S_S_to_le: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
(* demo *)
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
theorem lt_to_lt_S_S: ∀n,m. n < m → S n < S m.
/2/ qed. *)
-theorem lt_to_not_zero : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/not_zero.def(1)"\ 6not_zero\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="lt_to_not_zero"\ 6theorem lt_to_not_zero : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/not_zero.def(1)"\ 6not_zero\ 5/a\ 6 m.
#n #m #Hlt (elim Hlt) // qed.
(* lt vs. le *)
-theorem not_le_Sn_O: ∀ n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="not_le_Sn_O"\ 6theorem not_le_Sn_O: ∀ n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
#n @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #Hlen0 @(\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_zero.def(2)"\ 6lt_to_not_zero\ 5/a\ 6 ?? Hlen0) qed.
-theorem not_le_to_not_le_S_S: ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="not_le_to_not_le_S_S"\ 6theorem not_le_to_not_le_S_S: ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem not_le_S_S_to_not_le: ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="not_le_S_S_to_not_le"\ 6theorem not_le_S_S_to_not_le: ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem decidable_le: ∀n,m. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m).
+\ 5img class="anchor" src="icons/tick.png" id="decidable_le"\ 6theorem decidable_le: ∀n,m. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #n /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ #m * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_not_le_S_S.def(5)"\ 6not_le_to_not_le_S_S\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem decidable_lt: ∀n,m. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m).
+\ 5img class="anchor" src="icons/tick.png" id="decidable_lt"\ 6theorem decidable_lt: ∀n,m. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m).
#n #m @\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6 qed.
-theorem not_le_Sn_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="not_le_Sn_n"\ 6theorem not_le_Sn_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 n.
#n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_not_le_S_S.def(5)"\ 6not_le_to_not_le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* this is le_S_S_to_le
/2/ qed.
*)
-lemma le_gen: ∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.∀n.(∀i. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → P i) → P n.
+\ 5img class="anchor" src="icons/tick.png" id="le_gen"\ 6lemma le_gen: ∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.∀n.(∀i. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → P i) → P n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem not_le_to_lt: ∀n,m. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="not_le_to_lt"\ 6theorem not_le_to_lt: ∀n,m. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n.
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #n
[#abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
]
qed.
-theorem lt_to_not_le: ∀n,m. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → m \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="lt_to_not_le"\ 6theorem lt_to_not_le: ∀n,m. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → m \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 n.
#n #m #Hltnm (elim Hltnm) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem not_lt_to_le: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'not less than'" href="cic:/fakeuri.def(1)"\ 6≮\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="not_lt_to_le"\ 6theorem not_lt_to_le: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'not less than'" href="cic:/fakeuri.def(1)"\ 6≮\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_to_not_lt: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'not less than'" href="cic:/fakeuri.def(1)"\ 6≮\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="le_to_not_lt"\ 6theorem le_to_not_lt: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'not less than'" href="cic:/fakeuri.def(1)"\ 6≮\ 5/a\ 6 n.
#n #m #H @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ (* /3/ *) qed.
(* lt and le trans *)
-theorem lt_to_le_to_lt: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
+\ 5img class="anchor" src="icons/tick.png" id="lt_to_le_to_lt"\ 6theorem lt_to_le_to_lt: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
#n #m #p #H #H1 (elim H1) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_to_lt_to_lt: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
+\ 5img class="anchor" src="icons/tick.png" id="le_to_lt_to_lt"\ 6theorem le_to_lt_to_lt: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
#n #m #p #H (elim H) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem lt_S_to_lt: ∀n,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="lt_S_to_lt"\ 6theorem lt_S_to_lt: ∀n,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem ltn_to_ltO: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="ltn_to_ltO"\ 6theorem ltn_to_ltO: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(*
]
qed. *)
-theorem lt_O_n_elim: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n →
+\ 5img class="anchor" src="icons/tick.png" id="lt_O_n_elim"\ 6theorem lt_O_n_elim: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n →
∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.(∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.P (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) → P n.
#n (elim n) // #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem S_pred: ∀n. \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="S_pred"\ 6theorem S_pred: ∀n. \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n.
#n #posn (cases posn) //
qed.
*)
(* le to lt or eq *)
-theorem le_to_or_lt_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_to_or_lt_eq"\ 6theorem le_to_or_lt_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
#n #m #lenm (elim lenm) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* not eq *)
-theorem lt_to_not_eq : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="lt_to_not_eq"\ 6theorem lt_to_not_eq : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m.
#n #m #H @\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(*not lt
apply (not_le_Sn_n ? H3).
qed. *)
-theorem not_eq_to_le_to_lt: ∀n,m. n\ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6m → n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m → n\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m.
+\ 5img class="anchor" src="icons/tick.png" id="not_eq_to_le_to_lt"\ 6theorem not_eq_to_le_to_lt: ∀n,m. n\ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6m → n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6m → n\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m.
#n #m #Hneq #Hle cases (\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 ?? Hle) //
#Heq /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(*
nelim (Hneq Heq) qed. *)
(* le elimination *)
-theorem le_n_O_to_eq : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n.
+\ 5img class="anchor" src="icons/tick.png" id="le_n_O_to_eq"\ 6theorem le_n_O_to_eq : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n.
#n (cases n) // #a #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_n_O_elim: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → ∀P: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →Prop. P \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → P n.
+\ 5img class="anchor" src="icons/tick.png" id="le_n_O_elim"\ 6theorem le_n_O_elim: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → ∀P: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →Prop. P \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 → P n.
#n (cases n) // #a #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_n_Sm_elim : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m →
+\ 5img class="anchor" src="icons/tick.png" id="le_n_Sm_elim"\ 6theorem le_n_Sm_elim : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m →
∀P:Prop. (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → P) → (n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m → P) → P.
#n #m #Hle #P (elim Hle) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* le and eq *)
-theorem le_to_le_to_eq: ∀n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_to_le_to_eq"\ 6theorem le_to_le_to_eq: ∀n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 /\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le_n_O_to_eq.def(4)"\ 6le_n_O_to_eq\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem lt_O_S : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="lt_O_S"\ 6theorem lt_O_S : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(*
(* well founded induction principles *)
-theorem nat_elim1 : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="nat_elim1"\ 6theorem nat_elim1 : ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀P:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → Prop.
(∀m.(∀p. p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → P p) → P m) → P n.
#n #P #H
cut (∀q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. q \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → P q) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
(* some properties of functions *)
-definition increasing ≝ λf:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. f n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 f (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n).
+\ 5img class="anchor" src="icons/tick.png" id="increasing"\ 6definition increasing ≝ λf:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. f n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 f (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n).
-theorem increasing_to_monotonic: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="increasing_to_monotonic"\ 6theorem increasing_to_monotonic: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
\ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 f.
#f #incr #n #m #ltnm (elim ltnm) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem le_n_fn: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="le_n_fn"\ 6theorem le_n_fn: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
\ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 f n.
#f #incr #n (elim n) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem increasing_to_le: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="increasing_to_le"\ 6theorem increasing_to_le: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
\ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → ∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6i.m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 f i.
#f #incr #m (elim m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/#n * #a #lenfa
@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ?? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 a)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem increasing_to_le2: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f →
+\ 5img class="anchor" src="icons/tick.png" id="increasing_to_le2"\ 6theorem increasing_to_le2: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f →
∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. f \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6i. f i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 f (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 i).
#f #incr #m #lem (elim lem)
[@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 ? ? \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
]
qed.
-theorem increasing_to_injective: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="increasing_to_injective"\ 6theorem increasing_to_injective: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
\ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 f.
#f #incr #n #m cases(\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6 n m)
[#lenm cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 … lenm) //
qed.
(*********************** monotonicity ***************************)
-theorem monotonic_le_plus_r:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_le_plus_r"\ 6theorem monotonic_le_plus_r:
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 (λm.n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m).
#n #a #b (elim n) normalize //
#m #H #leab @\ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
theorem le_plus_r: ∀p,n,m:nat. n ≤ m → p + n ≤ p + m
≝ monotonic_le_plus_r. *)
-theorem monotonic_le_plus_l:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_le_plus_l"\ 6theorem monotonic_le_plus_l:
∀m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 (λn.n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m).
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
theorem le_plus_l: \forall p,n,m:nat. n \le m \to n + p \le m + p
\def monotonic_le_plus_l. *)
-theorem le_plus: ∀n1,n2,m1,m2:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2 → m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m2
+\ 5img class="anchor" src="icons/tick.png" id="le_plus"\ 6theorem le_plus: ∀n1,n2,m1,m2:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2 → m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m2
→ n1 \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2 \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m2.
#n1 #n2 #m1 #m2 #len #lem @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (n1\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m2))
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_plus_n :∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_n"\ 6theorem le_plus_n :∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma le_plus_a: ∀a,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_a"\ 6lemma le_plus_a: ∀a,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus.def(7)"\ 6le_plus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma le_plus_b: ∀b,n,m. n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_b"\ 6lemma le_plus_b: ∀b,n,m. n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_plus_n_r :∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_n_r"\ 6theorem le_plus_n_r :∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
-theorem eq_plus_to_le: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="eq_plus_to_le"\ 6theorem eq_plus_to_le: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
// qed.
-theorem le_plus_to_le: ∀a,n,m. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_to_le"\ 6theorem le_plus_to_le: ∀a,n,m. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#a (elim a) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_pred.def(4)"\ 6monotonic_pred\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_plus_to_le_r: ∀a,n,m. n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6a → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_to_le_r"\ 6theorem le_plus_to_le_r: ∀a,n,m. n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6a → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le.def(5)"\ 6le_plus_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* plus & lt *)
-theorem monotonic_lt_plus_r:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_lt_plus_r"\ 6theorem monotonic_lt_plus_r:
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 (λm.n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m).
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
variant lt_plus_r: \forall n,p,q:nat. p < q \to n + p < n + q \def
monotonic_lt_plus_r. *)
-theorem monotonic_lt_plus_l:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_lt_plus_l"\ 6theorem monotonic_lt_plus_l:
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 (λm.m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n).
(* /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ *) #n @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 // qed.
variant lt_plus_l: \forall n,p,q:nat. p < q \to p + n < q + n \def
monotonic_lt_plus_l. *)
-theorem lt_plus: ∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q → n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 q.
+\ 5img class="anchor" src="icons/tick.png" id="lt_plus"\ 6theorem lt_plus: ∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q → n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 q.
#n #m #p #q #ltnm #ltpq
-@(\ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6 ? (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q))/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_plus_l.def(9)"\ 6monotonic_lt_plus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+@(\ 5a href="cic:/matita/arithmetics/nat/transitive_lt.def(3)"\ 6transitive_lt\ 5/a\ 6 ? (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_plus_l.def(5)"\ 6monotonic_lt_plus_l\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem lt_plus_to_lt_l :∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q.
+\ 5img class="anchor" src="icons/tick.png" id="lt_plus_to_lt_l"\ 6theorem lt_plus_to_lt_l :∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6n → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le.def(5)"\ 6le_plus_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem lt_plus_to_lt_r :∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q.
+\ 5img class="anchor" src="icons/tick.png" id="lt_plus_to_lt_r"\ 6theorem lt_plus_to_lt_r :∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6q → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_plus_to_lt_l.def(6)"\ 6lt_plus_to_lt_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(*
*)
(* times *)
-theorem monotonic_le_times_r:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_le_times_r"\ 6theorem monotonic_le_times_r:
∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/le.ind(1,0,1)"\ 6le\ 5/a\ 6 (λm. n \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 m).
#n #x #y #lexy (elim n) normalize//(* lento /2/*)
#a #lea @\ 5a href="cic:/matita/arithmetics/nat/le_plus.def(7)"\ 6le_plus\ 5/a\ 6 //
theorem le_times_l: \forall p,n,m:nat. n \le m \to n*p \le m*p
\def monotonic_le_times_l. *)
-theorem le_times: ∀n1,n2,m1,m2:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="le_times"\ 6theorem le_times: ∀n1,n2,m1,m2:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
n1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2 → m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m2 → n1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m1 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n2\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m2.
#n1 #n2 #m1 #m2 #len #lem @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (n1\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m2)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
(* interessante *)
-theorem lt_times_n: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m.
+\ 5img class="anchor" src="icons/tick.png" id="lt_times_n"\ 6theorem lt_times_n: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6m.
#n #m #H /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_times_to_le:
+\ 5img class="anchor" src="icons/tick.png" id="le_times_to_le"\ 6theorem le_times_to_le:
∀a,n,m. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 a → a \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 a \ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 m → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#a @\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 normalize
[//
apply lt_plus.assumption.assumption.
qed. *)
-theorem monotonic_lt_times_r:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_lt_times_r"\ 6theorem monotonic_lt_times_r:
∀c:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 (λt.(c\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6t)).
#c #posc #n #m #ltnm
(elim ltnm) normalize
]
qed.
-theorem monotonic_lt_times_l:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_lt_times_l"\ 6theorem monotonic_lt_times_l:
∀c:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → \ 5a href="cic:/matita/basics/relations/monotonic.def(1)"\ 6monotonic\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/lt.def(1)"\ 6lt\ 5/a\ 6 (λt.(t\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6c)).
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_lt_times_r.def(9)"\ 6monotonic_lt_times_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem lt_to_le_to_lt_times:
+\ 5img class="anchor" src="icons/tick.png" id="lt_to_le_to_lt_times"\ 6theorem lt_to_le_to_lt_times:
∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m → p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 q → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q → n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q.
#n #m #p #q #ltnm #lepq #posq
@(\ 5a href="cic:/matita/arithmetics/nat/le_to_lt_to_lt.def(4)"\ 6le_to_lt_to_lt\ 5/a\ 6 ? (n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q))
]
qed.
-theorem lt_times:∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q → n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q.
-#n #m #p #q #ltnm #ltpq @\ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt_times.def(12)"\ 6lt_to_le_to_lt_times\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/ltn_to_ltO.def(5)"\ 6ltn_to_ltO\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+\ 5img class="anchor" src="icons/tick.png" id="lt_times"\ 6theorem lt_times:∀n,m,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6m → p\ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6q → n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q.
+#n #m #p #q #ltnm #ltpq @\ 5a href="cic:/matita/arithmetics/nat/lt_to_le_to_lt_times.def(11)"\ 6lt_to_le_to_lt_times\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/ltn_to_ltO.def(5)"\ 6ltn_to_ltO\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem lt_times_n_to_lt_l:
+\ 5img class="anchor" src="icons/tick.png" id="lt_times_n_to_lt_l"\ 6theorem lt_times_n_to_lt_l:
∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6n → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q.
#n #p #q #Hlt (elim (\ 5a href="cic:/matita/arithmetics/nat/decidable_lt.def(7)"\ 6decidable_lt\ 5/a\ 6 p q)) //
#nltpq @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 ? ? (\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 ? ? Hlt))
applyS \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_lt_to_le.def(6)"\ 6not_lt_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem lt_times_n_to_lt_r:
+\ 5img class="anchor" src="icons/tick.png" id="lt_times_n_to_lt_r"\ 6theorem lt_times_n_to_lt_r:
∀n,p,q:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n\ 5a title="natural times" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6q → p \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 q.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_times_n_to_lt_l.def(9)"\ 6lt_times_n_to_lt_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(************************** minus ******************************)
-let rec minus n m ≝
+\ 5img class="anchor" src="icons/tick.png" id="minus"\ 6let rec minus n m ≝
match n with
[ O ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6
| S p ⇒
interpretation "natural minus" 'minus x y = (minus x y).
-theorem minus_S_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
+\ 5img class="anchor" src="icons/tick.png" id="minus_S_S"\ 6theorem minus_S_S: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
// qed.
-theorem minus_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
+\ 5img class="anchor" src="icons/tick.png" id="minus_O_n"\ 6theorem minus_O_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
#n (cases n) // qed.
-theorem minus_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="minus_n_O"\ 6theorem minus_n_O: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
#n (cases n) // qed.
-theorem minus_n_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
+\ 5img class="anchor" src="icons/tick.png" id="minus_n_n"\ 6theorem minus_n_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
#n (elim n) // qed.
-theorem minus_Sn_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
+\ 5img class="anchor" src="icons/tick.png" id="minus_Sn_n"\ 6theorem minus_Sn_n: ∀n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
#n (elim n) normalize // qed.
-theorem minus_Sn_m: ∀m,n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m).
+\ 5img class="anchor" src="icons/tick.png" id="minus_Sn_m"\ 6theorem minus_Sn_m: ∀m,n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m).
(* qualcosa da capire qui
#n #m #lenm nelim lenm napplyS refl_eq. *)
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6
napplyS (not_eq_to_le_to_lt n (S m) H H1)
qed. *)
-theorem eq_minus_S_pred: ∀n,m. n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6(n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m).
+\ 5img class="anchor" src="icons/tick.png" id="eq_minus_S_pred"\ 6theorem eq_minus_S_pred: ∀n,m. n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6(n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 normalize //
qed.
-theorem plus_minus:
+\ 5img class="anchor" src="icons/tick.png" id="plus_minus"\ 6theorem plus_minus:
∀m,n,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6
[//
]
qed.
-theorem minus_plus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
+\ 5img class="anchor" src="icons/tick.png" id="minus_plus_m_m"\ 6theorem minus_plus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m)\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le.con(0,1,1)"\ 6le_n\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"\ 6plus_minus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem plus_minus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="plus_minus_m_m"\ 6theorem plus_minus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
#n #m #lemn @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/plus_minus.def(5)"\ 6plus_minus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_plus_minus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_minus_m_m"\ 6theorem le_plus_minus_m_m: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
#n (elim n) // #a #Hind #m (cases m) // normalize #n/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem minus_to_plus :∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="minus_to_plus"\ 6theorem minus_to_plus :∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 p → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p.
#n #m #p #lemn #eqp (applyS \ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6) //
qed.
-theorem plus_to_minus :∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 p.
+\ 5img class="anchor" src="icons/tick.png" id="plus_to_minus"\ 6theorem plus_to_minus :∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 p.
#n #m #p #eqp @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 (applyS (\ 5a href="cic:/matita/arithmetics/nat/minus_plus_m_m.def(6)"\ 6minus_plus_m_m\ 5/a\ 6 p m))
qed.
-theorem minus_pred_pred : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m →
+\ 5img class="anchor" src="icons/tick.png" id="minus_pred_pred"\ 6theorem minus_pred_pred : ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 m →
\ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/pred.def(1)"\ 6pred\ 5/a\ 6 m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 m.
#n #m #posn #posm @(\ 5a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(4)"\ 6lt_O_n_elim\ 5/a\ 6 n posn) @(\ 5a href="cic:/matita/arithmetics/nat/lt_O_n_elim.def(4)"\ 6lt_O_n_elim\ 5/a\ 6 m posm) //.
qed.
(* monotonicity and galois *)
-theorem monotonic_le_minus_l:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_le_minus_l"\ 6theorem monotonic_le_minus_l:
∀p,q,n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. q \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → q\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6n.
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 #p #q
[#lePO @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 ? lePO) //
]
qed.
-theorem le_minus_to_plus: ∀n,m,p. n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
+\ 5img class="anchor" src="icons/tick.png" id="le_minus_to_plus"\ 6theorem le_minus_to_plus: ∀n,m,p. n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n\ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m.
#n #m #p #lep @\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6
[|@\ 5a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(6)"\ 6le_plus_minus_m_m\ 5/a\ 6 | @\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_l.def(6)"\ 6monotonic_le_plus_l\ 5/a\ 6 // ]
qed.
-theorem le_minus_to_plus_r: ∀a,b,c. c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b → a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 c → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b.
-#a #b #c #Hlecb #H >(\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 … Hlecb) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_minus_to_plus.def(10)"\ 6le_minus_to_plus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+\ 5img class="anchor" src="icons/tick.png" id="le_minus_to_plus_r"\ 6theorem le_minus_to_plus_r: ∀a,b,c. c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b → a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 c → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 b.
+#a #b #c #Hlecb #H >(\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 … Hlecb) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_minus_to_plus.def(7)"\ 6le_minus_to_plus\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem le_plus_to_minus: ∀n,m,p. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p.
-#n #m #p #lep /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_to_minus"\ 6theorem le_plus_to_minus: ∀n,m,p. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p.
+#n #m #p #lep /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(5)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_plus_to_minus_r: ∀a,b,c. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c → a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6b.
+\ 5img class="anchor" src="icons/tick.png" id="le_plus_to_minus_r"\ 6theorem le_plus_to_minus_r: ∀a,b,c. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c → a \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 c \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6b.
#a #b #c #H @(\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_le_r.def(6)"\ 6le_plus_to_le_r\ 5/a\ 6 … b) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem lt_minus_to_plus: ∀a,b,c. a \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 b \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b.
+\ 5img class="anchor" src="icons/tick.png" id="lt_minus_to_plus"\ 6theorem lt_minus_to_plus: ∀a,b,c. a \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 b \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b.
#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6
@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 …H)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(7)"\ 6le_plus_to_minus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem lt_minus_to_plus_r: ∀a,b,c. a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 c → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 b.
-#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6 …))
+\ 5img class="anchor" src="icons/tick.png" id="lt_minus_to_plus_r"\ 6theorem lt_minus_to_plus_r: ∀a,b,c. a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 b \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 c → a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 c \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 b.
+#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(7)"\ 6le_plus_to_minus\ 5/a\ 6 …))
@\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6 //
qed.
-theorem lt_plus_to_minus: ∀n,m,p. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
-#n #m #p #lenm #H normalize <\ 5a href="cic:/matita/arithmetics/nat/minus_Sn_m.def(5)"\ 6minus_Sn_m\ 5/a\ 6 // @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6 //
+\ 5img class="anchor" src="icons/tick.png" id="lt_plus_to_minus"\ 6theorem lt_plus_to_minus: ∀n,m,p. m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → n \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 p.
+#n #m #p #lenm #H normalize <\ 5a href="cic:/matita/arithmetics/nat/minus_Sn_m.def(5)"\ 6minus_Sn_m\ 5/a\ 6 // @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(7)"\ 6le_plus_to_minus\ 5/a\ 6 //
qed.
-theorem lt_plus_to_minus_r: ∀a,b,c. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 b.
+\ 5img class="anchor" src="icons/tick.png" id="lt_plus_to_minus_r"\ 6theorem lt_plus_to_minus_r: ∀a,b,c. a \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 b \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c → a \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 c \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 b.
#a #b #c #H @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus_r.def(7)"\ 6le_plus_to_minus_r\ 5/a\ 6 //
qed.
-theorem monotonic_le_minus_r:
+\ 5img class="anchor" src="icons/tick.png" id="monotonic_le_minus_r"\ 6theorem monotonic_le_minus_r:
∀p,q,n:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. q \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 p → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6q.
-#p #q #n #lepq @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(10)"\ 6le_plus_to_minus\ 5/a\ 6
-@(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(9)"\ 6le_plus_minus_m_m\ 5/a\ 6 ? q)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+#p #q #n #lepq @\ 5a href="cic:/matita/arithmetics/nat/le_plus_to_minus.def(7)"\ 6le_plus_to_minus\ 5/a\ 6
+@(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_plus_minus_m_m.def(6)"\ 6le_plus_minus_m_m\ 5/a\ 6 ? q)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_plus_r.def(3)"\ 6monotonic_le_plus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem eq_minus_O: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="eq_minus_O"\ 6theorem eq_minus_O: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
-#n #m #lenm @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_r.def(11)"\ 6monotonic_le_minus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+#n #m #lenm @(\ 5a href="cic:/matita/arithmetics/nat/le_n_O_elim.def(4)"\ 6le_n_O_elim\ 5/a\ 6 (n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_r.def(8)"\ 6monotonic_le_minus_r\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem distributive_times_minus: \ 5a href="cic:/matita/basics/relations/distributive.def(1)"\ 6distributive\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/minus.fix(0,0,1)"\ 6minus\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="distributive_times_minus"\ 6theorem distributive_times_minus: \ 5a href="cic:/matita/basics/relations/distributive.def(1)"\ 6distributive\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/times.fix(0,0,2)"\ 6times\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/minus.fix(0,0,1)"\ 6minus\ 5/a\ 6.
#a #b #c
(cases (\ 5a href="cic:/matita/arithmetics/nat/decidable_lt.def(7)"\ 6decidable_lt\ 5/a\ 6 b c)) #Hbc
- [> \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 //
+ [> \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(9)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(9)"\ 6eq_minus_O\ 5/a\ 6 //
@\ 5a href="cic:/matita/arithmetics/nat/monotonic_le_times_r.def(8)"\ 6monotonic_le_times_r\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_plus_b.def(8)"\ 6le_plus_b\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 (applyS \ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6) <\ 5a href="cic:/matita/arithmetics/nat/distributive_times_plus.def(7)"\ 6distributive_times_plus\ 5/a\ 6
@\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 (applyS \ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_lt_to_le.def(6)"\ 6not_lt_to_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem minus_plus: ∀n,m,p. n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6(m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p).
+\ 5img class="anchor" src="icons/tick.png" id="minus_plus"\ 6theorem minus_plus: ∀n,m,p. n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n \ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6(m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p).
#n #m #p
cases (\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6 (m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p) n) #Hlt
[@\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6
@\ 5a href="cic:/matita/arithmetics/nat/minus_to_plus.def(8)"\ 6minus_to_plus\ 5/a\ 6 //
|cut (n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6p) [@(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 … (\ 5a href="cic:/matita/arithmetics/nat/le_n_Sn.def(1)"\ 6le_n_Sn\ 5/a\ 6 …)) @\ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6 //]
- #H >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(12)"\ 6eq_minus_O\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(9)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ #H >\ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(9)"\ 6eq_minus_O\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/eq_minus_O.def(9)"\ 6eq_minus_O\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/monotonic_le_minus_l.def(5)"\ 6monotonic_le_minus_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
]
qed.
>associative_plus <plus_minus_m_m //
qed. *)
-theorem minus_minus: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n →
+\ 5img class="anchor" src="icons/tick.png" id="minus_minus"\ 6theorem minus_minus: ∀n,m,p:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. p \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n →
p\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6(n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6m) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 n\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6(m\ 5a title="natural minus" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6p).
#n #m #p #lepm #lemn
@\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/plus_to_minus.def(7)"\ 6plus_to_minus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/associative_plus.def(4)"\ 6associative_plus\ 5/a\ 6 <\ 5a href="cic:/matita/arithmetics/nat/plus_minus_m_m.def(7)"\ 6plus_minus_m_m\ 5/a\ 6 //
(*********************** boolean arithmetics ********************)
include "basics/bool.ma".
-let rec eqb n m ≝
+\ 5img class="anchor" src="icons/tick.png" id="eqb"\ 6let rec eqb n m ≝
match n with
[ O ⇒ match m with [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 | S q ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6]
| S p ⇒ match m with [ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 | S q ⇒ eqb p q]
].
-theorem eqb_elim : ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqb_elim"\ 6theorem eqb_elim : ∀ n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
(n\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6m → (P \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6)) → (n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m → (P \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6)) → (P (\ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m)).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6
[#n (cases n) normalize /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
]
qed.
-theorem eqb_n_n: ∀n. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="eqb_n_n"\ 6theorem eqb_n_n: ∀n. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#n (elim n) normalize // qed.
-theorem eqb_true_to_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="eqb_true_to_eq"\ 6theorem eqb_true_to_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
#n #m @(\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6 n m) // #_ #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem eqb_false_to_not_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="eqb_false_to_not_eq"\ 6theorem eqb_false_to_not_eq: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m.
#n #m @(\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6 n m) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem eq_to_eqb_true: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="eq_to_eqb_true"\ 6theorem eq_to_eqb_true: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
// qed.
-theorem not_eq_to_eqb_false: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="not_eq_to_eqb_false"\ 6theorem not_eq_to_eqb_false: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
n \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
#n #m #noteq @\ 5a href="cic:/matita/arithmetics/nat/eqb_elim.def(5)"\ 6eqb_elim\ 5/a\ 6// #Heq @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-let rec leb n m ≝
+\ 5img class="anchor" src="icons/tick.png" id="leb"\ 6let rec leb n m ≝
match n with
[ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6
| (S p) ⇒
[ O ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6
| (S q) ⇒ leb p q]].
-theorem leb_elim: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. ∀P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="leb_elim"\ 6theorem leb_elim: ∀n,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. ∀P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
(n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → P \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) → (n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → P \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) → P (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m).
@\ 5a href="cic:/matita/arithmetics/nat/nat_elim2.def(2)"\ 6nat_elim2\ 5/a\ 6 normalize
[/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
]
qed.
-theorem leb_true_to_le:∀n,m.\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="leb_true_to_le"\ 6theorem leb_true_to_le:∀n,m.\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #_ #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem leb_false_to_not_le:∀n,m.
+\ 5img class="anchor" src="icons/tick.png" id="leb_false_to_not_le"\ 6theorem leb_false_to_not_le:∀n,m.
\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m.
#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #_ #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem le_to_leb_true: ∀n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="le_to_leb_true"\ 6theorem le_to_leb_true: ∀n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #H #H1 @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem not_le_to_leb_false: ∀n,m. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="not_le_to_leb_false"\ 6theorem not_le_to_leb_false: ∀n,m. n \ 5a title="natural 'neither less nor equal to'" href="cic:/fakeuri.def(1)"\ 6≰\ 5/a\ 6 m → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
#n #m @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 // #H #H1 @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem lt_to_leb_false: ∀n,m. m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="lt_to_leb_false"\ 6theorem lt_to_leb_false: ∀n,m. m \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n → \ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/lt_to_not_le.def(7)"\ 6lt_to_not_le\ 5/a\ 6, \ 5a href="cic:/matita/arithmetics/nat/not_le_to_leb_false.def(7)"\ 6not_le_to_leb_false\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* serve anche ltb?
qed. *)
(* min e max *)
-definition min: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
+\ 5img class="anchor" src="icons/tick.png" id="min"\ 6definition min: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
λn.λm. if (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m) then n else m.
-definition max: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
+\ 5img class="anchor" src="icons/tick.png" id="max"\ 6definition max: \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 →\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 ≝
λn.λm. if (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m) then m else n.
-lemma commutative_min: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="commutative_min"\ 6lemma commutative_min: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6.
#n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6
[@\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_le_to_eq.def(5)"\ 6le_to_le_to_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|#notle >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 …) // @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
] qed.
-lemma le_minr: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="le_minr"\ 6lemma le_minr: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m.
#i #n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma le_minl: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
+\ 5img class="anchor" src="icons/tick.png" id="le_minl"\ 6lemma le_minl: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_minr.def(7)"\ 6le_minr\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma to_min: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m.
+\ 5img class="anchor" src="icons/tick.png" id="to_min"\ 6lemma to_min: ∀i,n,m. i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 n → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 m → i \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/min.def(2)"\ 6min\ 5/a\ 6 n m.
#i #n #m #lein #leim normalize (cases (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m))
normalize // qed.
-lemma commutative_max: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="commutative_max"\ 6lemma commutative_max: \ 5a href="cic:/matita/basics/relations/commutative.def(1)"\ 6commutative\ 5/a\ 6 ? \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6.
#n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6
[@\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_to_le_to_eq.def(5)"\ 6le_to_le_to_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|#notle >(\ 5a href="cic:/matita/arithmetics/nat/le_to_leb_true.def(7)"\ 6le_to_leb_true\ 5/a\ 6 …) // @(\ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6 ? (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m)) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/not_le_to_lt.def(5)"\ 6not_le_to_lt\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
] qed.
-lemma le_maxl: ∀i,n,m. \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
+\ 5img class="anchor" src="icons/tick.png" id="le_maxl"\ 6lemma le_maxl: ∀i,n,m. \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
#i #n #m normalize @\ 5a href="cic:/matita/arithmetics/nat/leb_elim.def(6)"\ 6leb_elim\ 5/a\ 6 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/transitive_le.def(3)"\ 6transitive_le\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma le_maxr: ∀i,n,m. \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
+\ 5img class="anchor" src="icons/tick.png" id="le_maxr"\ 6lemma le_maxr: ∀i,n,m. \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/arithmetics/nat/le_maxl.def(7)"\ 6le_maxl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma to_max: ∀i,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
+\ 5img class="anchor" src="icons/tick.png" id="to_max"\ 6lemma to_max: ∀i,n,m. n \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i → \ 5a href="cic:/matita/arithmetics/nat/max.def(2)"\ 6max\ 5/a\ 6 n m \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 i.
#i #n #m #leni #lemi normalize (cases (\ 5a href="cic:/matita/arithmetics/nat/leb.fix(0,0,1)"\ 6leb\ 5/a\ 6 n m))
normalize // qed.
include "basics/relations.ma".
(********** bool **********)
-inductive bool: Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="bool"\ 6inductive bool: Type[0] ≝
| true : bool
| false : bool.
(* destruct does not work *)
-theorem not_eq_true_false : \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="not_eq_true_false"\ 6theorem not_eq_true_false : \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
@\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #Heq change with match \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 with [true ⇒ \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6|false ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6]
>Heq // qed.
-definition notb : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
+\ 5img class="anchor" src="icons/tick.png" id="notb"\ 6definition notb : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
λ b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. match b with [true ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6|false ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 ].
interpretation "boolean not" 'not x = (notb x).
-theorem notb_elim: ∀ b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="notb_elim"\ 6theorem notb_elim: ∀ b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
match b with
[ true ⇒ P \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6
| false ⇒ P \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6] → P (\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b).
#b #P elim b normalize // qed.
-theorem notb_notb: ∀b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 (\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b.
+\ 5img class="anchor" src="icons/tick.png" id="notb_notb"\ 6theorem notb_notb: ∀b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 (\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b.
#b elim b // qed.
-theorem injective_notb: \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="injective_notb"\ 6theorem injective_notb: \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6.
#b1 #b2 #H // qed.
-theorem noteq_to_eqnot: ∀b1,b2. b1 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b2 → b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b2.
+\ 5img class="anchor" src="icons/tick.png" id="noteq_to_eqnot"\ 6theorem noteq_to_eqnot: ∀b1,b2. b1 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b2 → b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b2.
* * // #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-theorem eqnot_to_noteq: ∀b1,b2. b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b2 → b1 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b2.
+\ 5img class="anchor" src="icons/tick.png" id="eqnot_to_noteq"\ 6theorem eqnot_to_noteq: ∀b1,b2. b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b2 → b1 \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b2.
* * normalize // #H @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … \ 5a href="cic:/matita/basics/bool/not_eq_true_false.def(3)"\ 6not_eq_true_false\ 5/a\ 6) //
qed.
-definition andb : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
+\ 5img class="anchor" src="icons/tick.png" id="andb"\ 6definition andb : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
λb1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. match b1 with [ true ⇒ b2 | false ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 ].
interpretation "boolean and" 'and x y = (andb x y).
-theorem andb_elim: ∀ b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. ∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="andb_elim"\ 6theorem andb_elim: ∀ b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. ∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
match b1 with [ true ⇒ P b2 | false ⇒ P \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6] → P (b1 \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2).
#b1 #b2 #P (elim b1) normalize // qed.
-theorem andb_true_l: ∀ b1,b2. (b1 \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="andb_true_l"\ 6theorem andb_true_l: ∀ b1,b2. (b1 \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#b1 (cases b1) normalize // qed.
-theorem andb_true_r: ∀b1,b2. (b1 \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="andb_true_r"\ 6theorem andb_true_r: ∀b1,b2. (b1 \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#b1 #b2 (cases b1) normalize // (cases b2) // qed.
-theorem andb_true: ∀b1,b2. (b1 \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="andb_true"\ 6theorem andb_true: ∀b1,b2. (b1 \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/andb_true_r.def(4)"\ 6andb_true_r\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/andb_true_l.def(4)"\ 6andb_true_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-definition orb : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
+\ 5img class="anchor" src="icons/tick.png" id="orb"\ 6definition orb : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
λb1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.match b1 with [ true ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 | false ⇒ b2].
interpretation "boolean or" 'or x y = (orb x y).
-theorem orb_elim: ∀ b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. ∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="orb_elim"\ 6theorem orb_elim: ∀ b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. ∀ P:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → Prop.
match b1 with [ true ⇒ P \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 | false ⇒ P b2] → P (\ 5a href="cic:/matita/basics/bool/orb.def(1)"\ 6orb\ 5/a\ 6 b1 b2).
#b1 #b2 #P (elim b1) normalize // qed.
-lemma orb_true_r1: ∀b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="orb_true_r1"\ 6lemma orb_true_r1: ∀b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → (b1 \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#b1 #b2 #H >H // qed.
-lemma orb_true_r2: ∀b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="orb_true_r2"\ 6lemma orb_true_r2: ∀b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → (b1 \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#b1 #b2 #H >H cases b1 // qed.
-lemma orb_true_l: ∀b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="orb_true_l"\ 6lemma orb_true_l: ∀b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
(b1 \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 b2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → (b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 (b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6).
* normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-definition xorb : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
+\ 5img class="anchor" src="icons/tick.png" id="xorb"\ 6definition xorb : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
λb1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
match b1 with
[ true ⇒ match b2 with [ true ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 | false ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 ]
notation < "hvbox('if' \nbsp term 46 e \nbsp break 'then' \nbsp term 46 t \nbsp break 'else' \nbsp term 49 f \nbsp)" non associative with precedence 46
for @{ match $e with [ true ⇒ $t | false ⇒ $f] }.
-definition ite ≝ λA:Type[0].λe.λt,f:A.match e with [ true ⇒ t | false ⇒ f ].
+\ 5img class="anchor" src="icons/tick.png" id="bool_to_decidable_eq"\ 6theorem bool_to_decidable_eq:
+ ∀b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. \ 5a href="cic:/matita/basics/logic/decidable.def(1)"\ 6decidable\ 5/a\ 6 (b1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b2).
+#b1 #b2 (cases b1) (cases b2) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ %2 /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/bool/eqnot_to_noteq.def(4)"\ 6eqnot_to_noteq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="true_or_false"\ 6theorem true_or_false:
+∀b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+#b (cases b) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-interpretation "if then else" 'ite e t f = (ite ? e t f).
include "basics/logic.ma".
-inductive Sigma: Type[1] ≝
+\ 5img class="anchor" src="icons/tick.png" id="Sigma"\ 6inductive Sigma: Type[1] ≝
mk_Sigma: ∀p1: Type[0]. p1 → Sigma.
-definition p1: Sigma → Type[0].
+\ 5img class="anchor" src="icons/tick.png" id="p1"\ 6definition p1: \ 5a href="cic:/matita/basics/jmeq/Sigma.ind(1,0,0)"\ 6Sigma\ 5/a\ 6 → Type[0].
#S cases S #Y #_ @Y
qed.
-definition p2: ∀S:Sigma. p1 S.
+\ 5img class="anchor" src="icons/tick.png" id="p2"\ 6definition p2: ∀S:\ 5a href="cic:/matita/basics/jmeq/Sigma.ind(1,0,0)"\ 6Sigma\ 5/a\ 6. \ 5a href="cic:/matita/basics/jmeq/p1.def(1)"\ 6p1\ 5/a\ 6 S.
#S cases S #Y #x @x
qed.
-inductive jmeq (A:Type[0]) (x:A) : ∀B:Type[0]. B →Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="jmeq"\ 6inductive jmeq (A:Type[0]) (x:A) : ∀B:Type[0]. B →Prop ≝
jmrefl : jmeq A x A x.
-definition eqProp ≝ λA:Prop.eq A.
+\ 5img class="anchor" src="icons/tick.png" id="eqProp"\ 6definition eqProp ≝ λA:Prop.\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 A.
-lemma K : ∀A.∀x:A.∀h:x=x. eqProp ? h (refl A x).
-#A #x #h @(refl ? h: eqProp ? ? ?).
+\ 5img class="anchor" src="icons/tick.png" id="K"\ 6lemma K : ∀A.∀x:A.∀h:x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x. \ 5a href="cic:/matita/basics/jmeq/eqProp.def(1)" title="null"\ 6eqProp\ 5/a\ 6 ? h (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A x).
+#A #x #h @(\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? h: \ 5a href="cic:/matita/basics/jmeq/eqProp.def(1)"\ 6eqProp\ 5/a\ 6 ? ? ?).
qed.
-definition cast:
- ∀A,B:Type[0].∀E:A=B. A → B.
+\ 5img class="anchor" src="icons/tick.png" id="cast"\ 6definition cast:
+ ∀A,B:Type[0].∀E:A\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6B. A → B.
#A #B #E cases E #X @X
qed.
-lemma tech1:
- ∀Aa,Bb:Sigma.∀E:Aa=Bb.
- cast (p1 Aa) (p1 Bb) ? (p2 Aa) = p2 Bb.
+\ 5img class="anchor" src="icons/tick.png" id="tech1"\ 6lemma tech1:
+ ∀Aa,Bb:\ 5a href="cic:/matita/basics/jmeq/Sigma.ind(1,0,0)"\ 6Sigma\ 5/a\ 6.∀E:Aa\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6Bb.
+ \ 5a href="cic:/matita/basics/jmeq/cast.def(1)"\ 6cast\ 5/a\ 6 (\ 5a href="cic:/matita/basics/jmeq/p1.def(1)"\ 6p1\ 5/a\ 6 Aa) (\ 5a href="cic:/matita/basics/jmeq/p1.def(1)"\ 6p1\ 5/a\ 6 Bb) ? (\ 5a href="cic:/matita/basics/jmeq/p2.def(2)"\ 6p2\ 5/a\ 6 Aa) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/jmeq/p2.def(2)"\ 6p2\ 5/a\ 6 Bb.
[2: >E %
| #Aa #Bb #E >E cases Bb #B #b normalize % ]
qed.
-lemma gral: ∀A.∀x,y:A.
- mk_Sigma A x = mk_Sigma A y → x=y.
- #A #x #y #E lapply (tech1 ?? E)
- generalize in ⊢ (??(???%?)? → ?) #E1
- normalize in E1; >(K ?? E1) normalize
+\ 5img class="anchor" src="icons/tick.png" id="gral"\ 6lemma gral: ∀A.∀x,y:A.
+ \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A y → x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y.
+ #A #x #y #E lapply (\ 5a href="cic:/matita/basics/jmeq/tech1.def(3)"\ 6tech1\ 5/a\ 6 ?? E)
+ generalize in ⊢ (??(???%?)? → ?); #E1
+ normalize in E1; >(\ 5a href="cic:/matita/basics/jmeq/K.def(2)"\ 6K\ 5/a\ 6 ?? E1) normalize
#H @H
qed.
-axiom daemon: False.
+\ 5img class="anchor" src="icons/tick.png" id="daemon"\ 6axiom daemon: \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
-lemma jm_to_eq_sigma:
- ∀A,x,y. jmeq A x A y → mk_Sigma A x = mk_Sigma A y.
+\ 5img class="anchor" src="icons/tick.png" id="jm_to_eq_sigma"\ 6lemma jm_to_eq_sigma:
+ ∀A,x,y. \ 5a href="cic:/matita/basics/jmeq/jmeq.ind(1,0,2)"\ 6jmeq\ 5/a\ 6 A x A y → \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A y.
#A #x #y #E cases E in ⊢ (???%); %
qed.
-definition curry:
+\ 5img class="anchor" src="icons/tick.png" id="curry"\ 6definition curry:
∀A,x.
- (∀y. jmeq A x A y → Type[0]) →
- (∀y. mk_Sigma A x = mk_Sigma A y → Type[0]).
- #A #x #f #y #E @(f y) >(gral ??? E) %
+ (∀y. \ 5a href="cic:/matita/basics/jmeq/jmeq.ind(1,0,2)"\ 6jmeq\ 5/a\ 6 A x A y → Type[0]) →
+ (∀y. \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A y → Type[0]).
+ #A #x #f #y #E @(f y) >(\ 5a href="cic:/matita/basics/jmeq/gral.def(4)"\ 6gral\ 5/a\ 6 ??? E) %
qed.
-lemma G : ∀A.∀x:A.∀P:∀y:A.mk_Sigma A x = mk_Sigma A y →Type[0].
- P x (refl ? (mk_Sigma A x)) → ∀y.∀h:mk_Sigma A x = mk_Sigma A y.
+\ 5img class="anchor" src="icons/tick.png" id="G"\ 6lemma G : ∀A.∀x:A.∀P:∀y:A.\ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A y →Type[0].
+ P x (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A x)) → ∀y.∀h:\ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A y.
P y h.
- #A #x #P #H #y #E lapply (gral ??? E) #G generalize in match E;
+ #A #x #P #H #y #E lapply (\ 5a href="cic:/matita/basics/jmeq/gral.def(4)"\ 6gral\ 5/a\ 6 ??? E) #G generalize in match E;
@(match G
- return λy.λ_. ∀e:mk_Sigma A x = mk_Sigma A y. P y e
+ return λy.λ_. ∀e:\ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/jmeq/Sigma.con(0,1,0)"\ 6mk_Sigma\ 5/a\ 6 A y. P y e
with
[refl ⇒ ?])
- #E <(sym_eq ??? (K ?? E)) @H
+ #E <(\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 ??? (\ 5a href="cic:/matita/basics/jmeq/K.def(2)"\ 6K\ 5/a\ 6 ?? E)) @H
qed.
-definition PP: ∀A:Prop.∀P:A → Type[0]. A → Type[0].
+\ 5img class="anchor" src="icons/tick.png" id="PP"\ 6definition PP: ∀A:Prop.∀P:A → Type[0]. A → Type[0].
#A #P #a @(P a)
qed.
-lemma E : ∀A.∀x:A.∀P:∀y:A.jmeq A x A y→Type[0].
- PP ? (P x) (jmrefl A x) → ∀y.∀h:jmeq A x A y.PP ? (P y) h.
- #A #a #P #H #b #E letin F ≝ (jm_to_eq_sigma ??? E)
- lapply (G ?? (curry ?? P) ?? F)
+\ 5img class="anchor" src="icons/tick.png" id="E"\ 6lemma E : ∀A.∀x:A.∀P:∀y:A.\ 5a href="cic:/matita/basics/jmeq/jmeq.ind(1,0,2)"\ 6jmeq\ 5/a\ 6 A x A y→Type[0].
+ \ 5a href="cic:/matita/basics/jmeq/PP.def(1)"\ 6PP\ 5/a\ 6 ? (P x) (\ 5a href="cic:/matita/basics/jmeq/jmeq.con(0,1,2)"\ 6jmrefl\ 5/a\ 6 A x) → ∀y.∀h:\ 5a href="cic:/matita/basics/jmeq/jmeq.ind(1,0,2)"\ 6jmeq\ 5/a\ 6 A x A y.\ 5a href="cic:/matita/basics/jmeq/PP.def(1)"\ 6PP\ 5/a\ 6 ? (P y) h.
+ #A #a #P #H #b #E letin F ≝ (\ 5a href="cic:/matita/basics/jmeq/jm_to_eq_sigma.def(1)"\ 6jm_to_eq_sigma\ 5/a\ 6 ??? E)
+ lapply (\ 5a href="cic:/matita/basics/jmeq/G.def(5)"\ 6G\ 5/a\ 6 ?? (\ 5a href="cic:/matita/basics/jmeq/curry.def(5)"\ 6curry\ 5/a\ 6 ?? P) ?? F)
[ normalize //
- | #H whd in H; @(H : PP ? (P b) ?) ]
+ | #H whd in H; @(H : \ 5a href="cic:/matita/basics/jmeq/PP.def(1)"\ 6PP\ 5/a\ 6 ? (P b) ?) ]
qed.
-lemma jmeq_elim : ∀A.∀x:A.∀P:∀y:A.jmeq A x A y→Type[0].
- P x (jmrefl A x) → ∀y.∀h:jmeq A x A y.P y h ≝ E.
+\ 5img class="anchor" src="icons/tick.png" id="jmeq_elim"\ 6lemma jmeq_elim : ∀A.∀x:A.∀P:∀y:A.\ 5a href="cic:/matita/basics/jmeq/jmeq.ind(1,0,2)"\ 6jmeq\ 5/a\ 6 A x A y→Type[0].
+ P x (\ 5a href="cic:/matita/basics/jmeq/jmeq.con(0,1,2)"\ 6jmrefl\ 5/a\ 6 A x) → ∀y.∀h:\ 5a href="cic:/matita/basics/jmeq/jmeq.ind(1,0,2)"\ 6jmeq\ 5/a\ 6 A x A y.P y h ≝ \ 5a href="cic:/matita/basics/jmeq/E.def(6)"\ 6E\ 5/a\ 6.
\ No newline at end of file
include "arithmetics/nat.ma".
-inductive list (A:Type[0]) : Type[0] :=
+\ 5img class="anchor" src="icons/tick.png" id="list"\ 6inductive list (A:Type[0]) : Type[0] :=
| nil: list A
| cons: A -> list A -> list A.
interpretation "nil" 'nil = (nil ?).
interpretation "cons" 'cons hd tl = (cons ? hd tl).
-definition not_nil: ∀A:Type[0].\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="not_nil"\ 6definition not_nil: ∀A:Type[0].\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A → Prop ≝
λA.λl.match l with [ nil ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6 | cons hd tl ⇒ \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6 ].
-theorem nil_cons:
- ∀A:Type[0].∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a:A. a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 [\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
- #A #l #a @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #Heq (change with (\ 5a href="cic:/matita/basics/list/not_nil.def(1)"\ 6not_nil\ 5/a\ 6 ? (a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l))) >Heq //
+\ 5img class="anchor" src="icons/tick.png" id="nil_cons"\ 6theorem nil_cons:
+ ∀A:Type[0].∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a:A. a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
+ #A #l #a @\ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6 #Heq (change with (\ 5a href="cic:/matita/basics/list/not_nil.def(1)"\ 6not_nil\ 5/a\ 6 ? (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l))) >Heq //
qed.
(*
[ nil => []
| (cons hd tl) => hd :: id_list A tl ]. *)
-let rec append A (l1: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) l2 on l1 ≝
+\ 5img class="anchor" src="icons/tick.png" id="append"\ 6let rec append A (l1: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) l2 on l1 ≝
match l1 with
[ nil ⇒ l2
- | cons hd tl ⇒ hd :\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 append A tl l2 ].
+ | cons hd tl ⇒ hd \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 append A tl l2 ].
-definition hd ≝ λA.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.λd:A.
+\ 5img class="anchor" src="icons/tick.png" id="hd"\ 6definition hd ≝ λA.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.λd:A.
match l with [ nil ⇒ d | cons a _ ⇒ a].
-definition tail ≝ λA.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
- match l with [ nil ⇒ [\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 | cons hd tl ⇒ tl].
+\ 5img class="anchor" src="icons/tick.png" id="tail"\ 6definition tail ≝ λA.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 | cons hd tl ⇒ tl].
interpretation "append" 'append l1 l2 = (append ? l1 l2).
-theorem append_nil: ∀A.∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 [\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
+\ 5img class="anchor" src="icons/tick.png" id="append_nil"\ 6theorem append_nil: ∀A.∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
#A #l (elim l) normalize // qed.
-theorem associative_append:
+\ 5img class="anchor" src="icons/tick.png" id="associative_append"\ 6theorem associative_append:
∀A.\ 5a href="cic:/matita/basics/relations/associative.def(1)"\ 6associative\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (\ 5a href="cic:/matita/basics/list/append.fix(0,1,1)"\ 6append\ 5/a\ 6 A).
#A #l1 #l2 #l3 (elim l1) normalize // qed.
a :: (l1 @ l2) = (a :: l1) @ l2.
//; nqed. *)
-theorem append_cons:∀A.∀a:A.∀l,l1.l\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l1)\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6(l\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6[\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6))\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l1.\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5span class="autotactic"\ 6\ 5/span\ 6
+\ 5img class="anchor" src="icons/tick.png" id="append_cons"\ 6theorem append_cons:∀A.∀a:A.∀l,l1.l\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l1)\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6(l\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6))\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l1.\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5span class="autotactic"\ 6\ 5/span\ 6
#A #a #l1 #l2 >\ 5a href="cic:/matita/basics/list/associative_append.def(4)"\ 6associative_append\ 5/a\ 6 // qed.
-theorem nil_append_elim: ∀A.∀l1,l2: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀P:?→?→Prop.
- l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6[\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 → P (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 A) (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 A) → P l1 l2.
+\ 5img class="anchor" src="icons/tick.png" id="nil_append_elim"\ 6theorem nil_append_elim: ∀A.∀l1,l2: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀P:?→?→Prop.
+ l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 → P (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 A) (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 A) → P l1 l2.
#A #l1 #l2 #P (cases l1) normalize //
#a #l3 #heq destruct
qed.
-theorem nil_to_nil: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
- l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 [\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 → l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 [\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 [\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="nil_to_nil"\ 6theorem nil_to_nil: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 → l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
#A #l1 #l2 #isnil @(\ 5a href="cic:/matita/basics/list/nil_append_elim.def(4)"\ 6nil_append_elim\ 5/a\ 6 A l1 l2) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
(* iterators *)
-let rec map (A,B:Type[0]) (f: A → B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 B ≝
- match l with [ nil ⇒ \ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 ? | cons x tl ⇒ f x :\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 (map A B f tl)].
+\ 5img class="anchor" src="icons/tick.png" id="map"\ 6let rec map (A,B:Type[0]) (f: A → B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 B ≝
+ match l with [ nil ⇒ \ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 ? | cons x tl ⇒ f x \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 (map A B f tl)].
-lemma map_append : ∀A,B,f,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="map_append"\ 6lemma map_append : ∀A,B,f,l1,l2.
(\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f l1) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f (l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2).
#A #B #f #l1 elim l1
[ #l2 @\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6
| #h #t #IH #l2 normalize //
] qed.
-let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l :B ≝
+\ 5img class="anchor" src="icons/tick.png" id="foldr"\ 6let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l :B ≝
match l with [ nil ⇒ b | cons a l ⇒ f a (foldr A B f b l)].
-definition filter ≝
+\ 5img class="anchor" src="icons/tick.png" id="filter"\ 6definition filter ≝
λT.λp:T → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
- \ 5a href="cic:/matita/basics/list/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 T (\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 T) (λx,l0.if (p x) then (x:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l0) else l0) (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 T).
+ \ 5a href="cic:/matita/basics/list/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 T (\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 T) (λx,l0.if (p x) then (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l0) else l0) (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 T).
-definition compose ≝ λA,B,C.λf:A→B→C.λl1,l2.
- \ 5a href="cic:/matita/basics/list/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 ?? (λi,acc.(\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (f i) l2)\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6acc) [ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 l1.
+\ 5img class="anchor" src="icons/tick.png" id="compose"\ 6definition compose ≝ λA,B,C.λf:A→B→C.λl1,l2.
+ \ 5a href="cic:/matita/basics/list/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 ?? (λi,acc.(\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (f i) l2)\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6acc) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 l1.
-lemma filter_true : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
- \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a :\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
+\ 5img class="anchor" src="icons/tick.png" id="filter_true"\ 6lemma filter_true : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
#A #l #a #p #pa (elim l) normalize >pa normalize // qed.
-lemma filter_false : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 →
- \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
+\ 5img class="anchor" src="icons/tick.png" id="filter_false"\ 6lemma filter_false : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 →
+ \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
#A #l #a #p #pa (elim l) normalize >pa normalize // qed.
-theorem eq_map : ∀A,B,f,g,l. (∀x.f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g x) → \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B g l.
+\ 5img class="anchor" src="icons/tick.png" id="eq_map"\ 6theorem eq_map : ∀A,B,f,g,l. (∀x.f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g x) → \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B g l.
#A #B #f #g #l #eqfg (elim l) normalize // qed.
(*
]. *)
(**************************** reverse *****************************)
-let rec rev_append S (l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S) on l1 ≝
+\ 5img class="anchor" src="icons/tick.png" id="rev_append"\ 6let rec rev_append S (l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S) on l1 ≝
match l1 with
[ nil ⇒ l2
- | cons a tl ⇒ rev_append S tl (a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l2)
+ | cons a tl ⇒ rev_append S tl (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l2)
]
.
-definition reverse ≝λS.λl.\ 5a href="cic:/matita/basics/list/rev_append.fix(0,1,1)"\ 6rev_append\ 5/a\ 6 S l [\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="reverse"\ 6definition reverse ≝λS.λl.\ 5a href="cic:/matita/basics/list/rev_append.fix(0,1,1)"\ 6rev_append\ 5/a\ 6 S l \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
-lemma reverse_single : ∀S,a. \ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S (a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6[\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6[\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6).
+\ 5img class="anchor" src="icons/tick.png" id="reverse_single"\ 6lemma reverse_single : ∀S,a. \ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6).
// qed.
-lemma rev_append_def : ∀S,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="rev_append_def"\ 6lemma rev_append_def : ∀S,l1,l2.
\ 5a href="cic:/matita/basics/list/rev_append.fix(0,1,1)"\ 6rev_append\ 5/a\ 6 S l1 l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S l1) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l2 .
#S #l1 elim l1 normalize //
qed.
-lemma reverse_cons : ∀S,a,l. \ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S (a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S l)\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6[\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6).
+\ 5img class="anchor" src="icons/tick.png" id="reverse_cons"\ 6lemma reverse_cons : ∀S,a,l. \ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S l)\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6).
#S #a #l whd in ⊢ (??%?); //
qed.
-lemma reverse_append: ∀S,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="reverse_append"\ 6lemma reverse_append: ∀S,l1,l2.
\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S (l1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S l2)\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S l1).
#S #l1 elim l1 [normalize // | #a #tl #Hind #l2 >\ 5a href="cic:/matita/basics/list/reverse_cons.def(7)"\ 6reverse_cons\ 5/a\ 6
>\ 5a href="cic:/matita/basics/list/reverse_cons.def(7)"\ 6reverse_cons\ 5/a\ 6 // qed.
-lemma reverse_reverse : ∀S,l. \ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S (\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
+\ 5img class="anchor" src="icons/tick.png" id="reverse_reverse"\ 6lemma reverse_reverse : ∀S,l. \ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S (\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
#S #l elim l // #a #tl #Hind >\ 5a href="cic:/matita/basics/list/reverse_cons.def(7)"\ 6reverse_cons\ 5/a\ 6 >\ 5a href="cic:/matita/basics/list/reverse_append.def(8)"\ 6reverse_append\ 5/a\ 6
normalize // qed.
(* an elimination principle for lists working on the tail;
useful for strings *)
-lemma list_elim_left: ∀S.∀P:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S → Prop. P (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 S) →
-(∀a.∀tl.P tl → P (tl\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6[\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6))) → ∀l. P l.
+\ 5img class="anchor" src="icons/tick.png" id="list_elim_left"\ 6lemma list_elim_left: ∀S.∀P:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S → Prop. P (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 S) →
+(∀a.∀tl.P tl → P (tl\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6))) → ∀l. P l.
#S #P #Pnil #Pstep #l <(\ 5a href="cic:/matita/basics/list/reverse_reverse.def(9)"\ 6reverse_reverse\ 5/a\ 6 … l)
generalize in match (\ 5a href="cic:/matita/basics/list/reverse.def(2)"\ 6reverse\ 5/a\ 6 S l); #l elim l //
#a #tl #H >\ 5a href="cic:/matita/basics/list/reverse_cons.def(7)"\ 6reverse_cons\ 5/a\ 6 @Pstep //
(**************************** length *******************************)
-let rec length (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l ≝
+\ 5img class="anchor" src="icons/tick.png" id="length"\ 6let rec length (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l ≝
match l with
[ nil ⇒ \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6
| cons a tl ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (length A tl)].
notation "|M|" non associative with precedence 60 for @{'norm $M}.
interpretation "norm" 'norm l = (length ? l).
-lemma length_append: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
- |l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 |l1\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6|l2\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="length_append"\ 6lemma length_append: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l1\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l2\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.
#A #l1 elim l1 // normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
qed.
-let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (d:A) ≝
+\ 5img class="anchor" src="icons/tick.png" id="nth"\ 6let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (d:A) ≝
match n with
[O ⇒ \ 5a href="cic:/matita/basics/list/hd.def(1)"\ 6hd\ 5/a\ 6 A l d
|S m ⇒ nth m A (\ 5a href="cic:/matita/basics/list/tail.def(1)"\ 6tail\ 5/a\ 6 A l) d].
(***************************** fold *******************************)
-let rec fold (A,B:Type[0]) (op:B → B → B) (b:B) (p:A→\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (f:A→B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l :B ≝
+\ 5img class="anchor" src="icons/tick.png" id="fold"\ 6let rec fold (A,B:Type[0]) (op:B → B → B) (b:B) (p:A→\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (f:A→B) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l :B ≝
match l with
[ nil ⇒ b
| cons a l ⇒ if (p a) then (op (f a) (fold A B op b p f l))
interpretation "\fold" 'fold op nil p f l = (fold ? ? op nil p f l).
-theorem fold_true:
+\ 5img class="anchor" src="icons/tick.png" id="fold_true"\ 6theorem fold_true:
∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A→B. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
- \fold[op,nil]_{i ∈ a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- op (f a) \fold[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ op (f a) \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
#A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
-theorem fold_false:
+\ 5img class="anchor" src="icons/tick.png" id="fold_false"\ 6theorem fold_false:
∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f.
-p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → \fold[op,nil]_{i ∈ a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- \fold[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
+p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
#A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
-theorem fold_filter:
+\ 5img class="anchor" src="icons/tick.png" id="fold_filter"\ 6theorem fold_filter:
∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A →B.
- \fold[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- \fold[op,nil]_{i ∈ (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l)\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 A p l)\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
#A #B #a #l #p #op #nil #f elim l //
#a #tl #Hind cases(\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p a)) #pa
[ >\ 5a href="cic:/matita/basics/list/filter_true.def(3)"\ 6filter_true\ 5/a\ 6 // > \ 5a href="cic:/matita/basics/list/fold_true.def(3)"\ 6fold_true\ 5/a\ 6 // >\ 5a href="cic:/matita/basics/list/fold_true.def(3)"\ 6fold_true\ 5/a\ 6 //
| >\ 5a href="cic:/matita/basics/list/filter_false.def(3)"\ 6filter_false\ 5/a\ 6 // >\ 5a href="cic:/matita/basics/list/fold_false.def(3)"\ 6fold_false\ 5/a\ 6 // ]
qed.
-record Aop (A:Type[0]) (nil:A) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="Aop"\ 6record Aop (A:Type[0]) (nil:A) : Type[0] ≝
{op :2> A → A → A;
nill:∀a. op nil a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a;
nilr:∀a. op a nil \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a;
assoc: ∀a,b,c.op a (op b c) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 op (op a b) c
}.
-theorem fold_sum: ∀A,B. ∀I,J:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀nil.∀op:\ 5a href="cic:/matita/basics/list/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f.
- op (\fold[op,nil]_{i∈I\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i)) (\fold[op,nil]_{i∈J\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- \fold[op,nil]_{i∈(I\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6J)\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
+\ 5img class="anchor" src="icons/tick.png" id="fold_sum"\ 6theorem fold_sum: ∀A,B. ∀I,J:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀nil.∀op:\ 5a href="cic:/matita/basics/list/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f.
+ op (\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i∈I\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i)) (\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i∈J\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i∈(I\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6J)\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
#A #B #I #J #nil #op #f (elim I) normalize
[>\ 5a href="cic:/matita/basics/list/nill.fix(0,2,2)"\ 6nill\ 5/a\ 6 //|#a #tl #Hind <\ 5a href="cic:/matita/basics/list/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 //]
qed.
\ No newline at end of file
include "basics/list.ma".
include "arithmetics/nat.ma".
-let rec length (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l ≝
+\ 5img class="anchor" src="icons/tick.png" id="length"\ 6let rec length (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l ≝
match l with
[ nil ⇒ \ 5a title="natural number" href="cic:/fakeuri.def(1)"\ 60\ 5/a\ 6
| cons a tl ⇒ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (length A tl)].
notation "|M|" non associative with precedence 60 for @{'norm $M}.
interpretation "norm" 'norm l = (length ? l).
-let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (d:A) ≝
+\ 5img class="anchor" src="icons/tick.png" id="nth"\ 6let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (d:A) ≝
match n with
[O ⇒ \ 5a href="cic:/matita/basics/list/hd.def(1)"\ 6hd\ 5/a\ 6 A l d
- |S m ⇒ nth m A (\ 5a href="cic:/matita/basics/list/tail.def(1)"\ 6tail\ 5/a\ 6 A l) d].
+ |S m ⇒ nth m A (\ 5a href="cic:/matita/basics/list/tail.def(1)"\ 6tail\ 5/a\ 6 A l) d].
\ No newline at end of file
(* propositional equality *)
-inductive eq (A:Type[2]) (x:A) : A → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="eq"\ 6inductive eq (A:Type[2]) (x:A) : A → Prop ≝
refl: eq A x x.
interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
interpretation "leibniz reflexivity" 'refl = refl.
-lemma eq_rect_r:
+\ 5img class="anchor" src="icons/tick.png" id="eq_rect_r"\ 6lemma eq_rect_r:
∀A.∀a,x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.∀P: ∀x:A. \ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a → Type[3]. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → P x p.
#A #a #x #p (cases p) // qed.
-lemma eq_ind_r :
+\ 5img class="anchor" src="icons/tick.png" id="eq_ind_r"\ 6lemma eq_ind_r :
∀A.∀a.∀P: ∀x:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a → Prop. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → ∀x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.P x p.
#A #a #P #p #x0 #p0; @(\ 5a href="cic:/matita/basics/logic/eq_rect_r.def(1)"\ 6eq_rect_r\ 5/a\ 6 ? ? ? p0) //; qed.
-lemma eq_rect_Type0_r:
+\ 5img class="anchor" src="icons/tick.png" id="eq_rect_Type0_r"\ 6lemma eq_rect_Type0_r:
∀A.∀a.∀P: ∀x:A. \ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a → Type[0]. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → ∀x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.P x p.
#A #a #P #H #x #p lapply H lapply P
cases p; //; qed.
-lemma eq_rect_Type1_r:
+\ 5img class="anchor" src="icons/tick.png" id="eq_rect_Type1_r"\ 6lemma eq_rect_Type1_r:
∀A.∀a.∀P: ∀x:A. \ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a → Type[1]. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → ∀x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.P x p.
#A #a #P #H #x #p lapply H lapply P
cases p; //; qed.
-lemma eq_rect_Type2_r:
+\ 5img class="anchor" src="icons/tick.png" id="eq_rect_Type2_r"\ 6lemma eq_rect_Type2_r:
∀A.∀a.∀P: ∀x:A. \ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a → Type[2]. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → ∀x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.P x p.
#A #a #P #H #x #p lapply H lapply P
cases p; //; qed.
-lemma eq_rect_Type3_r:
+\ 5img class="anchor" src="icons/tick.png" id="eq_rect_Type3_r"\ 6lemma eq_rect_Type3_r:
∀A.∀a.∀P: ∀x:A. \ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a → Type[3]. P a (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A a) → ∀x.∀p:\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 ? x a.P x p.
#A #a #P #H #x #p lapply H lapply P
cases p; //; qed.
-theorem rewrite_l: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → P y.
+\ 5img class="anchor" src="icons/tick.png" id="rewrite_l"\ 6theorem rewrite_l: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → P y.
#A #x #P #Hx #y #Heq (cases Heq); //; qed.
-theorem sym_eq: ∀A.∀x,y:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x.
+\ 5img class="anchor" src="icons/tick.png" id="sym_eq"\ 6theorem sym_eq: ∀A.∀x,y:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x.
#A #x #y #Heq @(\ 5a href="cic:/matita/basics/logic/rewrite_l.def(1)"\ 6rewrite_l\ 5/a\ 6 A x (λz.z\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x)) // qed.
-theorem rewrite_r: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x → P y.
+\ 5img class="anchor" src="icons/tick.png" id="rewrite_r"\ 6theorem rewrite_r: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x → P y.
#A #x #P #Hx #y #Heq (cases (\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 ? ? ? Heq)); //; qed.
-theorem eq_coerc: ∀A,B:Type[0].A→(A\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6B)→B.
+\ 5img class="anchor" src="icons/tick.png" id="eq_coerc"\ 6theorem eq_coerc: ∀A,B:Type[0].A→(A\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6B)→B.
#A #B #Ha #Heq (elim Heq); //; qed.
-theorem trans_eq : ∀A.∀x,y,z:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 z → x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 z.
+\ 5img class="anchor" src="icons/tick.png" id="trans_eq"\ 6theorem trans_eq : ∀A.∀x,y,z:A. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 z → x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 z.
#A #x #y #z #H1 #H2 >H1; //; qed.
-theorem eq_f: ∀A,B.∀f:A→B.∀x,y:A. x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y → f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y.
+\ 5img class="anchor" src="icons/tick.png" id="eq_f"\ 6theorem eq_f: ∀A,B.∀f:A→B.∀x,y:A. x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y → f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y.
#A #B #f #x #y #H >H; //; qed.
(* deleterio per auto? *)
-theorem eq_f2: ∀A,B,C.∀f:A→B→C.
+\ 5img class="anchor" src="icons/tick.png" id="eq_f2"\ 6theorem eq_f2: ∀A,B,C.∀f:A→B→C.
∀x1,x2:A.∀y1,y2:B. x1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x2 → y1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y2 → f x1 y1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f x2 y2.
#A #B #C #f #x1 #x2 #y1 #y2 #E1 #E2 >E1; >E2; //; qed.
-lemma eq_f3: ∀A,B,C,D.∀f:A→B→C->D.
+\ 5img class="anchor" src="icons/tick.png" id="eq_f3"\ 6lemma eq_f3: ∀A,B,C,D.∀f:A→B→C->D.
∀x1,x2:A.∀y1,y2:B. ∀z1,z2:C. x1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x2 → y1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y2 → z1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6z2 → f x1 y1 z1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f x2 y2 z2.
#A #B #C #D #f #x1 #x2 #y1 #y2 #z1 #z2 #E1 #E2 #E3 >E1; >E2; >E3 //; qed.
(********** connectives ********)
-inductive True: Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="True"\ 6inductive True: Prop ≝
I : True.
-inductive False: Prop ≝ .
+\ 5img class="anchor" src="icons/tick.png" id="False"\ 6inductive False: Prop ≝ .
(* ndefinition Not: Prop → Prop ≝
λA. A → False. *)
-inductive Not (A:Prop): Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="Not"\ 6inductive Not (A:Prop): Prop ≝
nmk: (A → \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6) → Not A.
interpretation "logical not" 'not x = (Not x).
-theorem absurd : ∀A:Prop. A → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6A → \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="absurd"\ 6theorem absurd : ∀A:Prop. A → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6A → \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
#A #H #Hn (elim Hn); /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/; qed.
(*
#A; #C; #H; #Hn; nelim (Hn H).
nqed. *)
-theorem not_to_not : ∀A,B:Prop. (A → B) → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6B →\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6A.
+\ 5img class="anchor" src="icons/tick.png" id="not_to_not"\ 6theorem not_to_not : ∀A,B:Prop. (A → B) → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6B →\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6A.
/\ 5span class="autotactic"\ 64\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/; qed.
(* inequality *)
interpretation "leibnitz's non-equality" 'neq t x y = (Not (eq t x y)).
-theorem sym_not_eq: ∀A.∀x,y:A. x \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 y → y \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 x.
+\ 5img class="anchor" src="icons/tick.png" id="sym_not_eq"\ 6theorem sym_not_eq: ∀A.∀x,y:A. x \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 y → y \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 x.
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/; qed.
(* and *)
-inductive And (A,B:Prop) : Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="And"\ 6inductive And (A,B:Prop) : Prop ≝
conj : A → B → And A B.
interpretation "logical and" 'and x y = (And x y).
-theorem proj1: ∀A,B:Prop. A \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 B → A.
+\ 5img class="anchor" src="icons/tick.png" id="proj1"\ 6theorem proj1: ∀A,B:Prop. A \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 B → A.
#A #B #AB (elim AB) //; qed.
-theorem proj2: ∀ A,B:Prop. A \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 B → B.
+\ 5img class="anchor" src="icons/tick.png" id="proj2"\ 6theorem proj2: ∀ A,B:Prop. A \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 B → B.
#A #B #AB (elim AB) //; qed.
(* or *)
-inductive Or (A,B:Prop) : Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="Or"\ 6inductive Or (A,B:Prop) : Prop ≝
or_introl : A → (Or A B)
| or_intror : B → (Or A B).
interpretation "logical or" 'or x y = (Or x y).
-definition decidable : Prop → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="decidable"\ 6definition decidable : Prop → Prop ≝
λ A:Prop. A \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 A.
(* exists *)
-inductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="ex"\ 6inductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
ex_intro: ∀ x:A. P x → ex A P.
interpretation "exists" 'exists x = (ex ? x).
-inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="ex2"\ 6inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
ex_intro2: ∀ x:A. P x → Q x → ex2 A P Q.
(* iff *)
-definition iff :=
+\ 5img class="anchor" src="icons/tick.png" id="iff"\ 6definition iff :=
λ A,B. (A → B) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (B → A).
interpretation "iff" 'iff a b = (iff a b).
-lemma iff_sym: ∀A,B. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → B \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 A.
+\ 5img class="anchor" src="icons/tick.png" id="iff_sym"\ 6lemma iff_sym: ∀A,B. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → B \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 A.
#A #B * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma iff_trans:∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → B \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 C → A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 C.
+\ 5img class="anchor" src="icons/tick.png" id="iff_trans"\ 6lemma iff_trans:∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → B \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 C → A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 C.
#A #B #C * #H1 #H2 * #H3 #H4 % /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ qed.
-lemma iff_not: ∀A,B. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6B.
+\ 5img class="anchor" src="icons/tick.png" id="iff_not"\ 6lemma iff_not: ∀A,B. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6B.
#A #B * #H1 #H2 % /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma iff_and_l: ∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → C \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 C \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 B.
+\ 5img class="anchor" src="icons/tick.png" id="iff_and_l"\ 6lemma iff_and_l: ∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → C \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 C \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 B.
#A #B #C * #H1 #H2 % * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma iff_and_r: ∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → A \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 C \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 C.
+\ 5img class="anchor" src="icons/tick.png" id="iff_and_r"\ 6lemma iff_and_r: ∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → A \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 C \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 C.
#A #B #C * #H1 #H2 % * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma iff_or_l: ∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → C \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 C \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 B.
+\ 5img class="anchor" src="icons/tick.png" id="iff_or_l"\ 6lemma iff_or_l: ∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → C \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 C \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 B.
#A #B #C * #H1 #H2 % * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma iff_or_r: ∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → A \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 C \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 C.
+\ 5img class="anchor" src="icons/tick.png" id="iff_or_r"\ 6lemma iff_or_r: ∀A,B,C. A \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B → A \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 C \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 B \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 C.
#A #B #C * #H1 #H2 % * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(* cose per destruct: da rivedere *)
-definition R0 ≝ λT:Type[0].λt:T.t.
+\ 5img class="anchor" src="icons/tick.png" id="R0"\ 6definition R0 ≝ λT:Type[0].λt:T.t.
-definition R1 ≝ \ 5a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="R1"\ 6definition R1 ≝ \ 5a href="cic:/matita/basics/logic/eq_rect_Type0.fix(0,5,1)"\ 6eq_rect_Type0\ 5/a\ 6.
(* used for lambda-delta *)
-definition R2 :
+\ 5img class="anchor" src="icons/tick.png" id="R2"\ 6definition R2 :
∀T0:Type[0].
∀a0:T0.
∀T1:∀x0:T0. a0\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x0 → Type[0].
@a2
qed.
-definition R3 :
+\ 5img class="anchor" src="icons/tick.png" id="R3"\ 6definition R3 :
∀T0:Type[0].
∀a0:T0.
∀T1:∀x0:T0. a0\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x0 → Type[0].
@a3
qed.
-definition R4 :
+\ 5img class="anchor" src="icons/tick.png" id="R4"\ 6definition R4 :
∀T0:Type[0].
∀a0:T0.
∀T1:∀x0:T0. \ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 T0 a0 x0 → Type[0].
@a4
qed.
-definition eqProp ≝ λA:Prop.\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 A.
+\ 5img class="anchor" src="icons/tick.png" id="eqProp"\ 6definition eqProp ≝ λA:Prop.\ 5a href="cic:/matita/basics/logic/eq.ind(1,0,2)"\ 6eq\ 5/a\ 6 A.
(* Example to avoid indexing and the consequential creation of ill typed
terms during paramodulation *)
-example lemmaK : ∀A.∀x:A.∀h:x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x. \ 5a href="cic:/matita/basics/logic/eqProp.def(1)"\ 6eqProp\ 5/a\ 6 ? h (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A x).
+\ 5img class="anchor" src="icons/tick.png" id="lemmaK"\ 6example lemmaK : ∀A.∀x:A.∀h:x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x. \ 5a href="cic:/matita/basics/logic/eqProp.def(1)"\ 6eqProp\ 5/a\ 6 ? h (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 A x).
#A #x #h @(\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? h: \ 5a href="cic:/matita/basics/logic/eqProp.def(1)"\ 6eqProp\ 5/a\ 6 ? ? ?).
qed.
-theorem streicherK : ∀T:Type[2].∀t:T.∀P:t \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 t → Type[3].P (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? t) → ∀p.P p.
+\ 5img class="anchor" src="icons/tick.png" id="streicherK"\ 6theorem streicherK : ∀T:Type[2].∀t:T.∀P:t \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 t → Type[3].P (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? t) → ∀p.P p.
#T #t #P #H #p >(\ 5a href="cic:/matita/basics/logic/lemmaK.def(2)"\ 6lemmaK\ 5/a\ 6 T t p) @H
qed.
include "basics/logic.ma".
(********** relations **********)
-definition relation : Type[0] → Type[0]
+\ 5img class="anchor" src="icons/tick.png" id="relation"\ 6definition relation : Type[0] → Type[0]
≝ λA.A→A→Prop.
-definition reflexive: ∀A.∀R :\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="reflexive"\ 6definition reflexive: ∀A.∀R :\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
≝ λA.λR.∀x:A.R x x.
-definition symmetric: ∀A.∀R: \ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="symmetric"\ 6definition symmetric: ∀A.∀R: \ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
≝ λA.λR.∀x,y:A.R x y → R y x.
-definition transitive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="transitive"\ 6definition transitive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
≝ λA.λR.∀x,y,z:A.R x y → R y z → R x z.
-definition irreflexive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="irreflexive"\ 6definition irreflexive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
≝ λA.λR.∀x:A.\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(R x x).
-definition cotransitive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="cotransitive"\ 6definition cotransitive: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
≝ λA.λR.∀x,y:A.R x y → ∀z:A. R x z \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 R z y.
-definition tight_apart: ∀A.∀eq,ap:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="tight_apart"\ 6definition tight_apart: ∀A.∀eq,ap:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
≝ λA.λeq,ap.∀x,y:A. (\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(ap x y) → eq x y) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6
(eq x y → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(ap x y)).
-definition antisymmetric: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="antisymmetric"\ 6definition antisymmetric: ∀A.∀R:\ 5a href="cic:/matita/basics/relations/relation.def(1)"\ 6relation\ 5/a\ 6 A.Prop
≝ λA.λR.∀x,y:A. R x y → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(R y x).
(********** functions **********)
-definition compose ≝
+\ 5img class="anchor" src="icons/tick.png" id="compose"\ 6definition compose ≝
λA,B,C:Type[0].λf:B→C.λg:A→B.λx:A.f (g x).
interpretation "function composition" 'compose f g = (compose ? ? ? f g).
-definition injective: ∀A,B:Type[0].∀ f:A→B.Prop
+\ 5img class="anchor" src="icons/tick.png" id="injective"\ 6definition injective: ∀A,B:Type[0].∀ f:A→B.Prop
≝ λA,B.λf.∀x,y:A.f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y → x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y.
-definition surjective: ∀A,B:Type[0].∀f:A→B.Prop
-≝λA,B.λf.∀z:B.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6x:A.z \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f x.
+\ 5img class="anchor" src="icons/tick.png" id="surjective"\ 6definition surjective: ∀A,B:Type[0].∀f:A→B.Prop
+≝λA,B.λf.∀z:B.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6x:A\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6z \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f x.
-definition commutative: ∀A:Type[0].∀f:A→A→A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="commutative"\ 6definition commutative: ∀A:Type[0].∀f:A→A→A.Prop
≝ λA.λf.∀x,y.f x y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y x.
-definition commutative2: ∀A,B:Type[0].∀f:A→A→B.Prop
+\ 5img class="anchor" src="icons/tick.png" id="commutative2"\ 6definition commutative2: ∀A,B:Type[0].∀f:A→A→B.Prop
≝ λA,B.λf.∀x,y.f x y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f y x.
-definition associative: ∀A:Type[0].∀f:A→A→A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="associative"\ 6definition associative: ∀A:Type[0].∀f:A→A→A.Prop
≝ λA.λf.∀x,y,z.f (f x y) z \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 f x (f y z).
(* functions and relations *)
-definition monotonic : ∀A:Type[0].∀R:A→A→Prop.
+\ 5img class="anchor" src="icons/tick.png" id="monotonic"\ 6definition monotonic : ∀A:Type[0].∀R:A→A→Prop.
∀f:A→A.Prop ≝
λA.λR.λf.∀x,y:A.R x y → R (f x) (f y).
(* functions and functions *)
-definition distributive: ∀A:Type[0].∀f,g:A→A→A.Prop
+\ 5img class="anchor" src="icons/tick.png" id="distributive"\ 6definition distributive: ∀A:Type[0].∀f,g:A→A→A.Prop
≝ λA.λf,g.∀x,y,z:A. f x (g y z) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g (f x y) (f x z).
-definition distributive2: ∀A,B:Type[0].∀f:A→B→B.∀g:B→B→B.Prop
+\ 5img class="anchor" src="icons/tick.png" id="distributive2"\ 6definition distributive2: ∀A,B:Type[0].∀f:A→B→B.∀g:B→B→B.Prop
≝ λA,B.λf,g.∀x:A.∀y,z:B. f x (g y z) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g (f x y) (f x z).
-lemma injective_compose : ∀A,B,C,f,g.
+\ 5img class="anchor" src="icons/tick.png" id="injective_compose"\ 6lemma injective_compose : ∀A,B,C,f,g.
\ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 A B f → \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 B C g → \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 A C (λx.g (f x)).
-/3/; qed.
+/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/; qed.
(* extensional equality *)
-definition exteqP: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="exteqP"\ 6definition exteqP: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝
λA.λP,Q.∀a.\ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (P a) (Q a).
-definition exteqR: ∀A,B:Type[0].∀R,S:A→B→Prop.Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="exteqR"\ 6definition exteqR: ∀A,B:Type[0].∀R,S:A→B→Prop.Prop ≝
λA,B.λR,S.∀a.∀b.\ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (R a b) (S a b).
-definition exteqF: ∀A,B:Type[0].∀f,g:A→B.Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="exteqF"\ 6definition exteqF: ∀A,B:Type[0].∀f,g:A→B.Prop ≝
λA,B.λf,g.∀a.f a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g a.
notation " x \eqP y " non associative with precedence 45
for @{'eqF ? ? f g}.
interpretation "functional extentional equality"
-'eqF A B f g = (exteqF A B f g).
+'eqF A B f g = (exteqF A B f g).
\ No newline at end of file
include "basics/logic.ma".
(* void *)
-inductive void : Type[0] ≝.
+\ 5img class="anchor" src="icons/tick.png" id="void"\ 6inductive void : Type[0] ≝.
(* unit *)
-inductive unit : Type[0] ≝ it: unit.
+\ 5img class="anchor" src="icons/tick.png" id="unit"\ 6inductive unit : Type[0] ≝ it: unit.
(* sum *)
-inductive Sum (A,B:Type[0]) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="Sum"\ 6inductive Sum (A,B:Type[0]) : Type[0] ≝
inl : A → Sum A B
| inr : B → Sum A B.
interpretation "Disjoint union" 'plus A B = (Sum A B).
(* option *)
-inductive option (A:Type[0]) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="option"\ 6inductive option (A:Type[0]) : Type[0] ≝
None : option A
| Some : A → option A.
-definition option_map : ∀A,B:Type[0]. (A → B) → option A → option B ≝
-λA,B,f,o. match o with [ None ⇒ None B | Some a ⇒ Some B (f a) ].
+\ 5img class="anchor" src="icons/tick.png" id="option_map"\ 6definition option_map : ∀A,B:Type[0]. (A → B) → \ 5a href="cic:/matita/basics/types/option.ind(1,0,1)"\ 6option\ 5/a\ 6 A → \ 5a href="cic:/matita/basics/types/option.ind(1,0,1)"\ 6option\ 5/a\ 6 B ≝
+λA,B,f,o. match o with [ None ⇒ \ 5a href="cic:/matita/basics/types/option.con(0,1,1)"\ 6None\ 5/a\ 6 B | Some a ⇒ \ 5a href="cic:/matita/basics/types/option.con(0,2,1)"\ 6Some\ 5/a\ 6 B (f a) ].
-lemma option_map_none : ∀A,B,f,x.
- option_map A B f x = None B → x = None A.
+\ 5img class="anchor" src="icons/tick.png" id="option_map_none"\ 6lemma option_map_none : ∀A,B,f,x.
+ \ 5a href="cic:/matita/basics/types/option_map.def(1)"\ 6option_map\ 5/a\ 6 A B f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,1,1)"\ 6None\ 5/a\ 6 B → x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,1,1)"\ 6None\ 5/a\ 6 A.
#A #B #f * [ // | #a #E whd in E:(??%?); destruct ]
qed.
-lemma option_map_some : ∀A,B,f,x,v.
- option_map A B f x = Some B v → ∃y. x = Some ? y ∧ f y = v.
+\ 5img class="anchor" src="icons/tick.png" id="option_map_some"\ 6lemma option_map_some : ∀A,B,f,x,v.
+ \ 5a href="cic:/matita/basics/types/option_map.def(1)"\ 6option_map\ 5/a\ 6 A B f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,2,1)"\ 6Some\ 5/a\ 6 B v → \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6y. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,2,1)"\ 6Some\ 5/a\ 6 ? y \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 f y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 v.
#A #B #f *
[ #v normalize #E destruct
| #y #v normalize #E %{y} destruct % //
] qed.
-definition option_map_def : ∀A,B:Type[0]. (A → B) → B → option A → B ≝
+\ 5img class="anchor" src="icons/tick.png" id="option_map_def"\ 6definition option_map_def : ∀A,B:Type[0]. (A → B) → B → \ 5a href="cic:/matita/basics/types/option.ind(1,0,1)"\ 6option\ 5/a\ 6 A → B ≝
λA,B,f,d,o. match o with [ None ⇒ d | Some a ⇒ f a ].
-lemma refute_none_by_refl : ∀A,B:Type[0]. ∀P:A → B. ∀Q:B → Type[0]. ∀x:option A. ∀H:x = None ? → False.
- (∀v. x = Some ? v → Q (P v)) →
- Q (match x return λy.x = y → ? with [ Some v ⇒ λ_. P v | None ⇒ λE. match H E in False with [ ] ] (refl ? x)).
+\ 5img class="anchor" src="icons/tick.png" id="refute_none_by_refl"\ 6lemma refute_none_by_refl : ∀A,B:Type[0]. ∀P:A → B. ∀Q:B → Type[0]. ∀x:\ 5a href="cic:/matita/basics/types/option.ind(1,0,1)"\ 6option\ 5/a\ 6 A. ∀H:x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,1,1)"\ 6None\ 5/a\ 6 ? → \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
+ (∀v. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,2,1)"\ 6Some\ 5/a\ 6 ? v → Q (P v)) →
+ Q (match x return λy.x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → ? with [ Some v ⇒ λ_. P v | None ⇒ λE. match H E in False with [ ] ] (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? x)).
#A #B #P #Q *
-[ #H cases (H (refl ??))
-| #a #H #p normalize @p @refl
+[ #H cases (H (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ??))
+| #a #H #p normalize @p @\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6
] qed.
(* sigma *)
-record Sig (A:Type[0]) (f:A→Type[0]) : Type[0] ≝ {
+\ 5img class="anchor" src="icons/tick.png" id="Sig"\ 6record Sig (A:Type[0]) (f:A→Type[0]) : Type[0] ≝ {
pi1: A
; pi2: f pi1
}.
(* Prod *)
-record Prod (A,B:Type[0]) : Type[0] ≝ {
+\ 5img class="anchor" src="icons/tick.png" id="Prod"\ 6record Prod (A,B:Type[0]) : Type[0] ≝ {
fst: A
; snd: B
}.
interpretation "Quadruple construction" 'quadruple w x y z = (mk_Prod ? ? (mk_Prod ? ? w x) (mk_Prod ? ? y z)).
-theorem eq_pair_fst_snd: ∀A,B.∀p:A × B.
- p = 〈 \fst p, \snd p 〉.
+\ 5img class="anchor" src="icons/tick.png" id="eq_pair_fst_snd"\ 6theorem eq_pair_fst_snd: ∀A,B.∀p:A \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 B.
+ p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p, \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#A #B #p (cases p) // qed.
-lemma fst_eq : ∀A,B.∀a:A.∀b:B. \fst 〈a,b〉 = a.
+\ 5img class="anchor" src="icons/tick.png" id="fst_eq"\ 6lemma fst_eq : ∀A,B.∀a:A.∀b:B. \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6a,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a.
// qed.
-lemma snd_eq : ∀A,B.∀a:A.∀b:B. \snd 〈a,b〉 = b.
+\ 5img class="anchor" src="icons/tick.png" id="snd_eq"\ 6lemma snd_eq : ∀A,B.∀a:A.∀b:B. \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6a,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b.
// qed.
-notation > "hvbox('let' 〈ident x,ident y〉 ≝ t 'in' s)"
+notation > "hvbox('let' 〈ident x,ident y〉 ≝ t 'in' s)"
with precedence 10
for @{ match $t with [ mk_Prod ${ident x} ${ident y} ⇒ $s ] }.
-notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y〉 \nbsp≝ break t \nbsp 'in' \nbsp) break s)"
+notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y〉 \nbsp≝ break t \nbsp 'in' \nbsp) break s)"
with precedence 10
for @{ match $t with [ mk_Prod (${ident x}:$X) (${ident y}:$Y) ⇒ $s ] }.
(* Also extracts an equality proof (useful when not using Russell). *)
-notation > "hvbox('let' 〈ident x,ident y〉 'as' ident E ≝ t 'in' s)"
+notation > "hvbox('let' 〈ident x,ident y〉 'as' ident E ≝ t 'in' s)"
with precedence 10
for @{ match $t return λx.x = $t → ? with [ mk_Prod ${ident x} ${ident y} ⇒
λ${ident E}.$s ] (refl ? $t) }.
(* Prop sigma *)
-record PSig (A:Type[0]) (P:A→Prop) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="PSig"\ 6record PSig (A:Type[0]) (P:A→Prop) : Type[0] ≝
{elem:>A; eproof: P elem}.
interpretation "subset type" 'sigma x = (PSig ? x).
-notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y〉 \nbsp 'as'\nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
+notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y〉 \nbsp 'as'\nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
with precedence 10
for @{ match $t return λ${ident k}:$X.$eq $T $k $t → ? with [ mk_Prod (${ident x}:$U) (${ident y}:$W) ⇒
λ${ident E}:$e.$s ] ($refl $T $t) }.
-notation > "hvbox('let' 〈ident x,ident y,ident z〉 'as' ident E ≝ t 'in' s)"
+notation > "hvbox('let' 〈ident x,ident y,ident z〉 'as' ident E ≝ t 'in' s)"
with precedence 10
for @{ match $t return λx.x = $t → ? with [ mk_Prod ${fresh xy} ${ident z} ⇒
match ${fresh xy} return λx. ? = $t → ? with [ mk_Prod ${ident x} ${ident y} ⇒
λ${ident E}.$s ] ] (refl ? $t) }.
-notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y,ident z〉 \nbsp'as'\nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
+notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y,ident z〉 \nbsp'as'\nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
with precedence 10
for @{ match $t return λ${ident x}.$eq $T $x $t → $U with [ mk_Prod (${fresh xy}:$V) (${ident z}:$Z) ⇒
match ${fresh xy} return λ${ident y}. $eq $R $r $t → ? with [ mk_Prod (${ident x}:$L) (${ident y}:$I) ⇒
λ${ident E}:$J.$s ] ] ($refl $A $t) }.
-notation > "hvbox('let' 〈ident w,ident x,ident y,ident z〉 ≝ t 'in' s)"
+notation > "hvbox('let' 〈ident w,ident x,ident y,ident z〉 ≝ t 'in' s)"
with precedence 10
for @{ match $t with [ mk_Prod ${fresh wx} ${fresh yz} ⇒ match ${fresh wx} with [ mk_Prod ${ident w} ${ident x} ⇒ match ${fresh yz} with [ mk_Prod ${ident y} ${ident z} ⇒ $s ] ] ] }.
-notation > "hvbox('let' 〈ident x,ident y,ident z〉 ≝ t 'in' s)"
+notation > "hvbox('let' 〈ident x,ident y,ident z〉 ≝ t 'in' s)"
with precedence 10
for @{ match $t with [ mk_Prod ${fresh xy} ${ident z} ⇒ match ${fresh xy} with [ mk_Prod ${ident x} ${ident y} ⇒ $s ] ] }.
(* This appears to upset automation (previously provable results require greater
depth or just don't work), so use example rather than lemma to prevent it
being indexed. *)
-example contract_pair : ∀A,B.∀e:A×B. (let 〈a,b〉 ≝ e in 〈a,b〉) = e.
+\ 5img class="anchor" src="icons/tick.png" id="contract_pair"\ 6example contract_pair : ∀A,B.∀e:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. (let 〈a,b〉 ≝ e in \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6a,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.
#A #B * // qed.
-lemma extract_pair : ∀A,B,C,D. ∀u:A×B. ∀Q:A → B → C×D. ∀x,y.
-((let 〈a,b〉 ≝ u in Q a b) = 〈x,y〉) →
-∃a,b. 〈a,b〉 = u ∧ Q a b = 〈x,y〉.
-#A #B #C #D * #a #b #Q #x #y normalize #E1 %{a} %{b} % try @refl @E1 qed.
+\ 5img class="anchor" src="icons/tick.png" id="extract_pair"\ 6lemma extract_pair : ∀A,B,C,D. ∀u:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. ∀Q:A → B → C\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6D. ∀x,y.
+((let 〈a,b〉 ≝ u in Q a b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6x,y\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) →
+\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6a,b\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6a,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 u \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 Q a b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6x,y\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
+#A #B #C #D * #a #b #Q #x #y normalize #E1 %{a} %{b} % try @\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 @E1 qed.
-lemma breakup_pair : ∀A,B,C:Type[0].∀x. ∀R:C → Prop. ∀P:A → B → C.
- R (P (\fst x) (\snd x)) → R (let 〈a,b〉 ≝ x in P a b).
-#A #B #C *; normalize /2/
+\ 5img class="anchor" src="icons/tick.png" id="breakup_pair"\ 6lemma breakup_pair : ∀A,B,C:Type[0].∀x. ∀R:C → Prop. ∀P:A → B → C.
+ R (P (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 x) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 x)) → R (let 〈a,b〉 ≝ x in P a b).
+#A #B #C *; normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
qed.
-lemma pair_elim:
+\ 5img class="anchor" src="icons/tick.png" id="pair_elim"\ 6lemma pair_elim:
∀A,B,C: Type[0].
∀T: A → B → C.
∀p.
- ∀P: A×B → C → Prop.
- (∀lft, rgt. p = 〈lft,rgt〉 → P 〈lft,rgt〉 (T lft rgt)) →
- P p (let 〈lft, rgt〉 ≝ p in T lft rgt).
- #A #B #C #T * /2/
+ ∀P: A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B → C → Prop.
+ (∀lft, rgt. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6lft,rgt\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 → P \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6lft,rgt\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 (T lft rgt)) →
+ P p (let 〈lft, rgt〉 ≝ p in T lft rgt).
+ #A #B #C #T * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
qed.
-lemma pair_elim2:
+\ 5img class="anchor" src="icons/tick.png" id="pair_elim2"\ 6lemma pair_elim2:
∀A,B,C,C': Type[0].
∀T: A → B → C.
∀T': A → B → C'.
∀p.
- ∀P: A×B → C → C' → Prop.
- (∀lft, rgt. p = 〈lft,rgt〉 → P 〈lft,rgt〉 (T lft rgt) (T' lft rgt)) →
- P p (let 〈lft, rgt〉 ≝ p in T lft rgt) (let 〈lft, rgt〉 ≝ p in T' lft rgt).
- #A #B #C #C' #T #T' * /2/
-qed.
+ ∀P: A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B → C → C' → Prop.
+ (∀lft, rgt. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6lft,rgt\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 → P \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6lft,rgt\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 (T lft rgt) (T' lft rgt)) →
+ P p (let 〈lft, rgt〉 ≝ p in T lft rgt) (let 〈lft, rgt〉 ≝ p in T' lft rgt).
+ #A #B #C #C' #T #T' * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+qed.
\ No newline at end of file
include "basics/pts.ma".
-definition hint_declaration_Type0 ≝ λA:Type[0] .λa,b:A.Prop.
-definition hint_declaration_Type1 ≝ λA:Type[1].λa,b:A.Prop.
-definition hint_declaration_Type2 ≝ λa,b:Type[2].Prop.
-definition hint_declaration_CProp0 ≝ λA:CProp[0].λa,b:A.Prop.
-definition hint_declaration_CProp1 ≝ λA:CProp[1].λa,b:A.Prop.
-definition hint_declaration_CProp2 ≝ λa,b:CProp[2].Prop.
+\ 5img class="anchor" src="icons/tick.png" id="hint_declaration_Type0"\ 6definition hint_declaration_Type0 ≝ λA:Type[0] .λa,b:A.Prop.
+\ 5img class="anchor" src="icons/tick.png" id="hint_declaration_Type1"\ 6definition hint_declaration_Type1 ≝ λA:Type[1].λa,b:A.Prop.
+\ 5img class="anchor" src="icons/tick.png" id="hint_declaration_Type2"\ 6definition hint_declaration_Type2 ≝ λa,b:Type[2].Prop.
+\ 5img class="anchor" src="icons/tick.png" id="hint_declaration_CProp0"\ 6definition hint_declaration_CProp0 ≝ λA:CProp[0].λa,b:A.Prop.
+\ 5img class="anchor" src="icons/tick.png" id="hint_declaration_CProp1"\ 6definition hint_declaration_CProp1 ≝ λA:CProp[1].λa,b:A.Prop.
+\ 5img class="anchor" src="icons/tick.png" id="hint_declaration_CProp2"\ 6definition hint_declaration_CProp2 ≝ λa,b:CProp[2].Prop.
interpretation "hint_decl_Type2" 'hint_decl a b = (hint_declaration_Type2 a b).
interpretation "hint_decl_CProp2" 'hint_decl a b = (hint_declaration_CProp2 a b).
-inductive nat : Set ≝
-| O : nat
-| S : nat \ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6\ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6\ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6→\ 5/span\ 6\ 5/span\ 6\ 5/span\ 6 nat.
+(*
+ ||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_______________________________________________________________ *)
+
+(*
+ * QUICK START GUIDE
+ * =================
+ * MatitaWeb is a web interface for the Matita interactive theorem prover.
+ * Its user interface is composed of two main panes:
+ *
+ * -> the script area, whose content you are currently reading, which is
+ * used to input definitions and theorems to be checked by Matita;
+ * -> a top area with a title bar and a toolbar providing all the
+ * interactive operations apart from proof authoring.
+ *
+ * The toolbar on the top is divided in two sections:
+ * -> the large blue tiles provide the basic operations concerning the
+ * execution and checking of a proof script, including step-by-step
+ * execution
+ * -> the smaller circular buttons on the right provide script management
+ * operations (create, open, save...)
+ * Hover the mouse pointer over these elements to show tooltips explaining
+ * their purpose.
+ *
+ * If you are new to Matita and/or interactive theorem proving, we have a
+ * tutorial in 10 parts for you. To load it:
+ * 1) click on the "Open script" button in the toolbar (the second circular
+ * button, whose icon shows a folder)
+ * 2) in the dialog box, select the "tutorial/chapter1.ma" script, then
+ * press OK (scripts chapter2.ma ... chapter10.ma contain more advanced
+ * examples that the user is encouraged to try after chapter1.ma)
+ * 3) follow the instructions in the script.
+ *
+ *)
-let rec plus n m ≝
-match n with
-[O ⇒ m
-|(S p) ⇒ S (plus p m)].
\ No newline at end of file
include "basics/logic.ma".
-inductive bank: Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="bank"\ 6inductive bank: Type[0] ≝
| east : bank
| west : bank.
(* We can now define a simple function computing, for each bank of the river, the
opposite one. *)
-definition opposite ≝ λs.
+\ 5img class="anchor" src="icons/tick.png" id="opposite"\ 6definition opposite ≝ λs.
match s with
- [ east ⇒ west
- | west ⇒ east
+ [ east ⇒ \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6
+ | west ⇒ \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6
].
(* Functions are live entities, and can be actually computed. To check this, let
us state the property that the opposite bank of east is west; every lemma needs a
name for further reference, and we call it "east_to_west". *)
-lemma east_to_west : opposite east = west.
+\ 5img class="anchor" src="icons/tick.png" id="east_to_west"\ 6lemma east_to_west : \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6.
(*
\ 5h2 class="section"\ 6The goal window\ 5/h2\ 6
In this case, we avoid the unnecessary simplification step: // will take care of
it. *)
-lemma west_to_east : opposite west = east.
+\ 5img class="anchor" src="icons/tick.png" id="west_to_east"\ 6lemma west_to_east : \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6.
// qed.
(*
\ 5h2 class="section"\ 6Introduction\ 5/h2\ 6
A slightly more complex problem consists in proving that opposite is idempotent *)
-lemma idempotent_opposite : ∀x. opposite (opposite x) = x.
+\ 5img class="anchor" src="icons/tick.png" id="idempotent_opposite"\ 6lemma idempotent_opposite : ∀x. \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 x) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x.
(* we start the proof moving x from the conclusion into the context, that is a
(backward) introduction step. Matita syntax for an introduction step is simply
definition:
*)
-inductive opp : bank → bank → Prop ≝
-| east_west : opp east west
-| west_east : opp west east.
+\ 5img class="anchor" src="icons/tick.png" id="opp"\ 6inductive opp : \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6 → Prop ≝
+| east_west : opp \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6
+| west_east : opp \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6.
(* In precisely the same way as "bank" is the smallest type containing east and
west, opp is the smallest predicate containing the two sub-cases east_west and
opp a b iff a = opposite b
Let us prove it, starting from the left to right implication, first *)
-lemma opp_to_opposite: ∀a,b. opp a b → a = opposite b.
+\ 5img class="anchor" src="icons/tick.png" id="opp_to_opposite"\ 6lemma opp_to_opposite: ∀a,b. \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 a b → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 b.
(* We start the proof introducing a, b and the hypothesis opp a b, that we
call oppab. *)
\ 5h2 class="section"\ 6Rewriting\ 5/h2\ 6
Let us come to the opposite direction. *)
-lemma opposite_to_opp: ∀a,b. a = opposite b → opp a b.
+\ 5img class="anchor" src="icons/tick.png" id="opposite_to_opp"\ 6lemma opposite_to_opp: ∀a,b. a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/opposite.def(1)"\ 6opposite\ 5/a\ 6 b → \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 a b.
(* As usual, we start introducing a, b and the hypothesis (a = opposite b),
that we call eqa. *)
is to use a record.
*)
-record state : Type[0] ≝
- {goat_pos : bank;
- wolf_pos : bank;
- cabbage_pos: bank;
- boat_pos : bank}.
+\ 5img class="anchor" src="icons/tick.png" id="state"\ 6record state : Type[0] ≝
+ {goat_pos : \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6;
+ wolf_pos : \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6;
+ cabbage_pos: \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6;
+ boat_pos : \ 5a href="cic:/matita/tutorial/chapter1/bank.ind(1,0,0)"\ 6bank\ 5/a\ 6}.
(* When you define a record named foo, the system automatically defines a record
constructor named mk_foo. To construct a new record you pass as arguments to
mk_foo the values of the record fields *)
-definition start ≝ mk_state east east east east.
-definition end ≝ mk_state west west west west.
+\ 5img class="anchor" src="icons/tick.png" id="start"\ 6definition start ≝ \ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="end"\ 6definition end ≝ \ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6.
(* We must now define the possible moves. A natural way to do it is in the form
of a relation (a binary predicate) over states. *)
-inductive move : state → state → Prop ≝
-| move_goat: ∀g,g1,w,c. opp g g1 → move (mk_state g w c g) (mk_state g1 w c g1)
+\ 5img class="anchor" src="icons/tick.png" id="move"\ 6inductive move : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
+| move_goat: ∀g,g1,w,c. \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 g g1 → move (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c g) (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g1 w c g1)
(* We can move the goat from a bank g to the opposite bank g1 if and only if the
boat is on the same bank g of the goat and we move the boat along with it. *)
-| move_wolf: ∀g,w,w1,c. opp w w1 → move (mk_state g w c w) (mk_state g w1 c w1)
-| move_cabbage: ∀g,w,c,c1.opp c c1 → move (mk_state g w c c) (mk_state g w c1 c1)
-| move_boat: ∀g,w,c,b,b1. opp b b1 → move (mk_state g w c b) (mk_state g w c b1).
+| move_wolf: ∀g,w,w1,c. \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 w w1 → move (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c w) (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w1 c w1)
+| move_cabbage: ∀g,w,c,c1.\ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 c c1 → move (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c c) (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c1 c1)
+| move_boat: ∀g,w,c,b,b1. \ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 b b1 → move (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c b) (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c b1).
(* A state is safe if either the goat is on the same bank of the boat, or both
the wolf and the cabbage are on the opposite bank of the goat. *)
-inductive safe_state : state → Prop ≝
-| with_boat : ∀g,w,c.safe_state (mk_state g w c g)
-| opposite_side : ∀g,g1,b.opp g g1 → safe_state (mk_state g g1 g1 b).
+\ 5img class="anchor" src="icons/tick.png" id="safe_state"\ 6inductive safe_state : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
+| with_boat : ∀g,w,c.safe_state (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g w c g)
+| opposite_side : ∀g,g1,b.\ 5a href="cic:/matita/tutorial/chapter1/opp.ind(1,0,0)"\ 6opp\ 5/a\ 6 g g1 → safe_state (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 g g1 g1 b).
(* Finally, a state y is reachable from x if either there is a single move
leading from x to y, or there is a safe state z such that z is reachable from x
and there is a move leading from z to y *)
-inductive reachable : state → state → Prop ≝
-| one : ∀x,y.move x y → reachable x y
-| more : ∀x,z,y. \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6reachable x z → safe_state z → \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6move z y → reachable x y.
+\ 5img class="anchor" src="icons/tick.png" id="reachable"\ 6inductive reachable : \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter1/state.ind(1,0,0)"\ 6state\ 5/a\ 6 → Prop ≝
+| one : ∀x,y.\ 5a href="cic:/matita/tutorial/chapter1/move.ind(1,0,0)"\ 6move\ 5/a\ 6 x y → reachable x y
+| more : ∀x,z,y. \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6reachable x z → \ 5a href="cic:/matita/tutorial/chapter1/safe_state.ind(1,0,0)"\ 6safe_state\ 5/a\ 6 z → \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter1/move.ind(1,0,0)"\ 6move\ 5/a\ 6 z y → reachable x y.
(*
\ 5h2 class="section"\ 6Automation\ 5/h2\ 6
need a few more applications to handle reachability, and side conditions.
The magic number to let automation work is, in this case, 9. *)
-lemma problem: reachable start end.
-normalize /\ 5span class="autotactic"\ 69\ 5span class="autotrace"\ 6 trace one, more, with_boat, opposite_side, move_goat, move_wolf, move_cabbage, move_boat, east_west, west_east\ 5/span\ 6\ 5/span\ 6/ qed.
+\ 5img class="anchor" src="icons/tick.png" id="problem"\ 6lemma problem: \ 5a href="cic:/matita/tutorial/chapter1/reachable.ind(1,0,0)"\ 6reachable\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/start.def(1)"\ 6start\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/end.def(1)"\ 6end\ 5/a\ 6.
+normalize /\ 5span class="autotactic"\ 69\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,1,0)"\ 6one\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"\ 6with_boat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,2,0)"\ 6opposite_side\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"\ 6move_goat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,2,0)"\ 6move_wolf\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,3,0)"\ 6move_cabbage\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,4,0)"\ 6move_boat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"\ 6east_west\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"\ 6west_east\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(*
\ 5h2 class="section"\ 6Application\ 5/h2\ 6
step, and so on).
*)
-lemma problem1: reachable start end.
-normalize @more
+\ 5img class="anchor" src="icons/tick.png" id="problem1"\ 6lemma problem1: \ 5a href="cic:/matita/tutorial/chapter1/reachable.ind(1,0,0)"\ 6reachable\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/start.def(1)"\ 6start\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/end.def(1)"\ 6end\ 5/a\ 6.
+normalize @\ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6
(*
\ 5h2 class="section"\ 6Focusing\ 5/h2\ 6
provide the intermediate state, that is [east,west,west,east]. We can do it, by
just applying this term. *)
- @(mk_state east west west east)
+ @(\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6)
(* This application closes the goal; at present, no goal has the focus on.
In order to act on the next goal, we must focus on it using the "|" operator. In
refocus on the skipped goal, going back to a situation similar to the one we
started with. *)
- | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace move_goat, east_west\ 5/span\ 6\ 5/span\ 6/ ]
+ | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"\ 6move_goat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"\ 6east_west\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ ]
(*
\ 5h2 class="section"\ 6Implicit arguments\ 5/h2\ 6
type a question mark that stands for an implicit argument to be guessed by
the system. *)
-@(more ? (mk_state east west west west))
+@(\ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter1/state.con(0,1,0)"\ 6mk_state\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6))
(* We now get three independent subgoals, all actives, and two of them are
trivial. We\ 5span style="font-family: Verdana,sans-serif;"\ 6 \ 5/span\ 6can just apply automation to all of them, and it will close the two
trivial goals. *)
-/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace opposite_side, move_boat, east_west, west_east\ 5/span\ 6\ 5/span\ 6/
+/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,2,0)"\ 6opposite_side\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,4,0)"\ 6move_boat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"\ 6east_west\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"\ 6west_east\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
(* Let us come to the next step, that consists in moving the wolf. Suppose that
instead of specifying the next intermediate state, we prefer to specify the next
move. In the spirit of the previous example, we can do it in the following way
*)
-@(more … (move_wolf … ))
+@(\ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter1/move.con(0,2,0)"\ 6move_wolf\ 5/a\ 6 … ))
(* The dots stand here for an arbitrary number of implicit arguments, to be
guessed by the system.
the fourth subgoal, and explicitly instatiate it. Instead of repeatedly using "|",
we can perform focusing by typing "4:" as described by the following command. *)
-[4: @east] /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace with_boat, east_west\ 5/span\ 6\ 5/span\ 6/
+[4: @\ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6] /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"\ 6with_boat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"\ 6east_west\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
(* Alternatively, we can directly instantiate the bank into the move. Let
us complete the proof in this, very readable way. *)
-@(more … (move_goat west … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace with_boat, west_east\ 5/span\ 6\ 5/span\ 6/
-@(more … (move_cabbage ?? east … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace opposite_side, east_west, west_east\ 5/span\ 6\ 5/span\ 6/
-@(more … (move_boat ??? west … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace with_boat, west_east\ 5/span\ 6\ 5/span\ 6/
-@one /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace move_goat, east_west\ 5/span\ 6\ 5/span\ 6/ qed.
\ No newline at end of file
+@(\ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"\ 6move_goat\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"\ 6with_boat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"\ 6west_east\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+@(\ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter1/move.con(0,3,0)"\ 6move_cabbage\ 5/a\ 6 ?? \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,1,0)"\ 6east\ 5/a\ 6 … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,2,0)"\ 6opposite_side\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"\ 6east_west\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"\ 6west_east\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+@(\ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,2,0)"\ 6more\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter1/move.con(0,4,0)"\ 6move_boat\ 5/a\ 6 ??? \ 5a href="cic:/matita/tutorial/chapter1/bank.con(0,2,0)"\ 6west\ 5/a\ 6 … )) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/safe_state.con(0,1,0)"\ 6with_boat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,2,0)"\ 6west_east\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+@\ 5a href="cic:/matita/tutorial/chapter1/reachable.con(0,1,0)"\ 6one\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter1/move.con(0,1,0)"\ 6move_goat\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter1/opp.con(0,1,0)"\ 6east_west\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
\ No newline at end of file
(* We say that two pres 〈i_1,b_1〉 and 〈i_1,b_1〉 are {\em cofinal} if and
only if b_1 = b_2. *)
-definition cofinal ≝ λS.λp:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
+\ 5img class="anchor" src="icons/tick.png" id="cofinal"\ 6definition cofinal ≝ λS.λp:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
(* As a corollary of decidable_sem, we have that two expressions
e1 and e2 are equivalent iff for any word w the states reachable
through w are cofinal. *)
-theorem equiv_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2} \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 ∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉.
+\ 5img class="anchor" src="icons/tick.png" id="equiv_sem"\ 6theorem equiv_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 ∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#S #e1 #e2 %
[#same_sem #w
cut (∀b1,b2. \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) (b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) → (b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b2))
of S. Instead of requiring S to be finite, we may restrict the analysis
to characters occurring in the given pres. *)
-definition occ ≝ λS.λe1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|)) (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)).
+\ 5img class="anchor" src="icons/tick.png" id="occ"\ 6definition occ ≝ λS.λe1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)) (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)).
-lemma occ_enough: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
-(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉)
- →∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉.
+\ 5img class="anchor" src="icons/tick.png" id="occ_enough"\ 6lemma occ_enough: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6)
+ →∀w.\ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#S #e1 #e2 #H #w
cases (\ 5a href="cic:/matita/tutorial/chapter5/decidable_sublist.def(6)"\ 6decidable_sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2)) [@H] -H #H
>\ 5a href="cic:/matita/tutorial/chapter9/to_pit.def(10)"\ 6to_pit\ 5/a\ 6 [2: @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 #a #memba @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"\ 6sublist_unique_append_l1\ 5/a\ 6 @H1 //]
(* The following is a stronger version of equiv_sem, relative to characters
occurring the given regular expressions. *)
-lemma equiv_sem_occ: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
-(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2〉)
-→ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+\ 5img class="anchor" src="icons/tick.png" id="equiv_sem_occ"\ 6lemma equiv_sem_occ: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+(∀w.(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2))→ \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6)
+→ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #e1 #e2 #H @(\ 5a href="cic:/matita/basics/logic/proj2.def(2)"\ 6proj2\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter10/equiv_sem.def(16)"\ 6equiv_sem\ 5/a\ 6 …)) @\ 5a href="cic:/matita/tutorial/chapter10/occ_enough.def(11)"\ 6occ_enough\ 5/a\ 6 #w @H
qed.
w.r.t. moves, and all its members are cofinal.
*)
-definition sons ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.λp:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
- \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (λa.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)),\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p))〉) l.
+\ 5img class="anchor" src="icons/tick.png" id="sons"\ 6definition sons ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.λp:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
+ \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (λa.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)),\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p))\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) l.
-lemma memb_sons: ∀S,l.∀p,q:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 ? l q) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5img class="anchor" src="icons/tick.png" id="memb_sons"\ 6lemma memb_sons: ∀S,l.∀p,q:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 ? l q) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6a.(\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 q)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6
\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 q)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
#S #l elim l [#p #q normalize in ⊢ (%→?); #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
[#H @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … a) >(\P H) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ |#H @Hind @H]
qed.
-definition is_bisim ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.λalpha:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="is_bisim"\ 6definition is_bisim ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.λalpha:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S.
∀p:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter10/cofinal.def(2)"\ 6cofinal\ 5/a\ 6 ? p \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 ? alpha p) l).
(* Using lemma equiv_sem_occ it is easy to prove the following result: *)
-lemma bisim_to_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"\ 6is_bisim\ 5/a\ 6 S l (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2) → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ?\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+\ 5img class="anchor" src="icons/tick.png" id="bisim_to_sem"\ 6lemma bisim_to_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"\ 6is_bisim\ 5/a\ 6 S l (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 S e1 e2) → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #l #e1 #e2 #Hbisim #Hmemb @\ 5a href="cic:/matita/tutorial/chapter10/equiv_sem_occ.def(17)"\ 6equiv_sem_occ\ 5/a\ 6
-#w #Hsub @(\ 5a href="cic:/matita/basics/logic/proj1.def(2)"\ 6proj1\ 5/a\ 6 … (Hbisim \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e2〉 ?))
+#w #Hsub @(\ 5a href="cic:/matita/basics/logic/proj1.def(2)"\ 6proj1\ 5/a\ 6 … (Hbisim \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e1,\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 ?))
lapply Hsub @(\ 5a href="cic:/matita/basics/list/list_elim_left.def(10)"\ 6list_elim_left\ 5/a\ 6 … w) [//]
#a #w1 #Hind #Hsub >\ 5a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"\ 6moves_left\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"\ 6moves_left\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/proj2.def(2)"\ 6proj2\ 5/a\ 6 …(Hbisim …(Hind ?)))
[#x #Hx @Hsub @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"\ 6memb_append_l1\ 5/a\ 6 //
Here is the extremely simple algorithm: *)
-let rec bisim S l n (frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?) on n ≝
+\ 5img class="anchor" src="icons/tick.png" id="bisim"\ 6let rec bisim S l n (frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?) on n ≝
match n with
- [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉 (* assert false *)
+ [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 (* assert false *)
| S m ⇒
match frontier with
- [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉
+ [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
| cons hd tl ⇒
if \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 hd)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 hd)) then
- bisim S l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)))
- (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 S l hd)) tl) (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)
- else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉
+ bisim S l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6visited)))
+ (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 S l hd)) tl) (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6visited)
+ else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
]
].
The following results explicitly state the behaviour of bisim is the general
case and in some relevant instances *)
-lemma unfold_bisim: ∀S,l,n.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+\ 5img class="anchor" src="icons/tick.png" id="unfold_bisim"\ 6lemma unfold_bisim: ∀S,l,n.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
\ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
match n with
- [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉 (* assert false *)
+ [ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 (* assert false *)
| S m ⇒
match frontier with
- [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉
+ [ nil ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
| cons hd tl ⇒
if \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 hd)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 hd)) then
- \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)))
- (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 S l hd)) tl) (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)
- else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6visited)))
+ (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 S l hd)) tl) (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6visited)
+ else \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
]
].
#S #l #n cases n // qed.
-lemma bisim_never: ∀S,l.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
- \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉.
+\ 5img class="anchor" src="icons/tick.png" id="bisim_never"\ 6lemma bisim_never: ∀S,l.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6 frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#frontier #visited >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 //
qed.
-lemma bisim_end: ∀Sig,l,m.∀visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
- \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited〉.
+\ 5img class="anchor" src="icons/tick.png" id="bisim_end"\ 6lemma bisim_end: ∀Sig,l,m.∀visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#n #visisted >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 //
qed.
-lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+\ 5img class="anchor" src="icons/tick.png" id="bisim_step_true"\ 6lemma bisim_step_true: ∀Sig,l,m.∀p.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
- \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:frontier) visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)))
- (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 Sig l p)) frontier) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited).
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6frontier) visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l m (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? (λx.\ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6visited)))
+ (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 Sig l p)) frontier) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6visited).
#Sig #l #m #p #frontier #visited #test >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 whd in ⊢ (??%?); >test //
qed.
-lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+\ 5img class="anchor" src="icons/tick.png" id="bisim_step_false"\ 6lemma bisim_step_false: ∀Sig,l,m.∀p.∀frontier,visited: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 →
- \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:frontier) visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited〉.
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 Sig l (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 m) (p\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6frontier) visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,visited\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#Sig #l #m #p #frontier #visited #test >\ 5a href="cic:/matita/tutorial/chapter10/unfold_bisim.def(9)"\ 6unfold_bisim\ 5/a\ 6 whd in ⊢ (??%?); >test //
qed.
#b cases b normalize //
qed. *)
-let rec pitem_enum S (i:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i ≝
+\ 5img class="anchor" src="icons/tick.png" id="pitem_enum"\ 6let rec pitem_enum S (i:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i ≝
match i with
- [ z ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
- | e ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,2,1)"\ 6pe\ 5/a\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
- | s y ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,3,1)"\ 6ps\ 5/a\ 6 S y)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:(\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,4,1)"\ 6pp\ 5/a\ 6 S y)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
+ [ z ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"\ 6\ 5/span\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
+ | e ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,2,1)"\ 6pe\ 5/a\ 6 S)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
+ | s y ⇒ (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,3,1)"\ 6ps\ 5/a\ 6 S y)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,4,1)"\ 6pp\ 5/a\ 6 S y)\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
| o i1 i2 ⇒ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,6,1)"\ 6po\ 5/a\ 6 S) (pitem_enum S i1) (pitem_enum S i2)
| c i1 i2 ⇒ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,5,1)"\ 6pc\ 5/a\ 6 S) (pitem_enum S i1) (pitem_enum S i2)
| k i ⇒ \ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,7,1)"\ 6pk\ 5/a\ 6 S) (pitem_enum S i)
].
-lemma pitem_enum_complete : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) i (\ 5a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"\ 6pitem_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="pitem_enum_complete"\ 6lemma pitem_enum_complete : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) i (\ 5a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"\ 6pitem_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #i elim i
[1,2://
|3,4:#c normalize >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … c)) //
]
qed.
-definition pre_enum ≝ λS.λi:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
- \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉) ( \ 5a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"\ 6pitem_enum\ 5/a\ 6 S i) (\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
+\ 5img class="anchor" src="icons/tick.png" id="pre_enum"\ 6definition pre_enum ≝ λS.λi:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter10/pitem_enum.fix(0,1,3)"\ 6pitem_enum\ 5/a\ 6 S i) (\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6).
-lemma pre_enum_complete : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? e (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
-#S * #i #b @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) \ 5a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"\ 6DeqBool\ 5/a\ 6 ? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉))
+\ 5img class="anchor" src="icons/tick.png" id="pre_enum_complete"\ 6lemma pre_enum_complete : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? e (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#S * #i #b @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/DeqItem.def(6)"\ 6DeqItem\ 5/a\ 6 S) \ 5a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"\ 6DeqBool\ 5/a\ 6 ? (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6))
// cases b normalize //
qed.
-definition space_enum ≝ λS.λi1,i2: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
- \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λe1,e2.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉) ( \ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i1) (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i2).
+\ 5img class="anchor" src="icons/tick.png" id="space_enum"\ 6definition space_enum ≝ λS.λi1,i2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 ??? (λe1,e2.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i1) (\ 5a href="cic:/matita/tutorial/chapter10/pre_enum.def(4)"\ 6pre_enum\ 5/a\ 6 S i2).
-lemma space_enum_complete : ∀S.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉 ( \ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
-#S #e1 #e2 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 … (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉))
+\ 5img class="anchor" src="icons/tick.png" id="space_enum_complete"\ 6lemma space_enum_complete : ∀S.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#S #e1 #e2 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_compose.def(6)"\ 6memb_compose\ 5/a\ 6 … (λi,b.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6))
// qed.
-definition all_reachable ≝ λS.λe1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
+\ 5img class="anchor" src="icons/tick.png" id="all_reachable"\ 6definition all_reachable ≝ λS.λe1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λl: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
\ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 ? l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6
∀p. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6w.(\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p).
-definition disjoint ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="disjoint"\ 6definition disjoint ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λl1,l2.
∀p:S. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S p l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S p l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
(* We are ready to prove that bisim is correct; we use the invariant
nodes reachable from 〈e_1,e_2〉, hence it is absurd to suppose to meet a pair
which is not cofinal. *)
-lemma bisim_correct: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2} →
+\ 5img class="anchor" src="icons/tick.png" id="bisim_correct"\ 6lemma bisim_correct: ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 →
∀l,n.∀frontier,visited:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ((\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S)).
- \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|)| \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6visited|→
+ \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="natural 'less than'" href="cic:/fakeuri.def(1)"\ 6<\ 5/a\ 6 n \ 5a title="natural plus" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6visited\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6→
\ 5a href="cic:/matita/tutorial/chapter10/all_reachable.def(8)"\ 6all_reachable\ 5/a\ 6 S e1 e2 visited →
\ 5a href="cic:/matita/tutorial/chapter10/all_reachable.def(8)"\ 6all_reachable\ 5/a\ 6 S e1 e2 frontier →
\ 5a href="cic:/matita/tutorial/chapter10/disjoint.def(5)"\ 6disjoint\ 5/a\ 6 ? frontier visited →
\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#Sig #e1 #e2 #same #l #n elim n
[#frontier #visited #abs * #unique #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … abs)
- @\ 5a href="cic:/matita/arithmetics/nat/le_to_not_lt.def(8)"\ 6le_to_not_lt\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_length.def(9)"\ 6sublist_length\ 5/a\ 6 // * #e11 #e21 #membp
- cut ((\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e11| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e21| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|))
+ @\ 5a href="cic:/matita/arithmetics/nat/le_to_not_lt.def(8)"\ 6le_to_not_lt\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_length.def(6)"\ 6sublist_length\ 5/a\ 6 // * #e11 #e21 #membp
+ cut ((|\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e11\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 |\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6) \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (|\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e21\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 |\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6))
[|* #H1 #H2 <H1 <H2 @\ 5a href="cic:/matita/tutorial/chapter10/space_enum_complete.def(9)"\ 6space_enum_complete\ 5/a\ 6]
cases (H … membp) #w * #we1 #we2 <we1 <we2 % >\ 5a href="cic:/matita/tutorial/chapter9/same_kernel_moves.def(9)"\ 6same_kernel_moves\ 5/a\ 6 //
|#m #HI * [#visited #vinv #finv >\ 5a href="cic:/matita/tutorial/chapter10/bisim_end.def(10)"\ 6bisim_end\ 5/a\ 6 //]
|whd % [@\ 5a href="cic:/matita/tutorial/chapter5/unique_append_unique.def(6)"\ 6unique_append_unique\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/bool/andb_true_r.def(4)"\ 6andb_true_r\ 5/a\ 6 … u_frontier)]
@\ 5a href="cic:/matita/tutorial/chapter5/unique_append_elim.def(7)"\ 6unique_append_elim\ 5/a\ 6 #q #H
[cases (\ 5a href="cic:/matita/tutorial/chapter10/memb_sons.def(8)"\ 6memb_sons\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter5/memb_filter_memb.def(5)"\ 6memb_filter_memb\ 5/a\ 6 … H)) -H
- #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (w1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6])))
+ #a * #m1 #m2 cases rp #w1 * #mw1 #mw2 @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (w1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6[\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6)))
>\ 5a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"\ 6moves_left\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter9/moves_left.def(9)"\ 6moves_left\ 5/a\ 6 >mw1 >mw2 >m1 >m2 % //
|@r_frontier @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 //
]
|@\ 5a href="cic:/matita/tutorial/chapter5/unique_append_elim.def(7)"\ 6unique_append_elim\ 5/a\ 6 #q #H
[@\ 5a href="cic:/matita/basics/bool/injective_notb.def(4)"\ 6injective_notb\ 5/a\ 6 @(\ 5a href="cic:/matita/tutorial/chapter5/memb_filter_true.def(5)"\ 6memb_filter_true\ 5/a\ 6 … H)
- |cut ((q\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=p) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6)
+ |cut ((q=\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6p) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6)
[|#Hpq whd in ⊢ (??%?); >Hpq @disjoint @\ 5a href="cic:/matita/tutorial/chapter5/memb_cons.def(5)"\ 6memb_cons\ 5/a\ 6 //]
cases (\ 5a href="cic:/matita/basics/bool/andb_true.def(5)"\ 6andb_true\ 5/a\ 6 … u_frontier) #notp #_ @(\bf ?)
@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … \ 5a href="cic:/matita/basics/bool/not_eq_true_false.def(3)"\ 6not_eq_true_false\ 5/a\ 6) #eqqp <notp <eqqp >H //
and the sons of visited are either in visited or in the frontier; since
at the end frontier is empty, visited is hence a bisimulation. *)
-definition all_true ≝ λS.λl.∀p:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S) \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5img class="anchor" src="icons/tick.png" id="all_true"\ 6definition all_true ≝ λS.λl.∀p:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S) \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S). \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? p l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
(\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p)) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6).
-definition sub_sons ≝ λS,l,l1,l2.∀x:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S) \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
+\ 5img class="anchor" src="icons/tick.png" id="sub_sons"\ 6definition sub_sons ≝ λS,l,l1,l2.∀x:(\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S) \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S).
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? x l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/sons.def(7)"\ 6sons\ 5/a\ 6 ? l x) l2.
-lemma bisim_complete:
+\ 5img class="anchor" src="icons/tick.png" id="bisim_complete"\ 6lemma bisim_complete:
∀S,l,n.∀frontier,visited,visited_res:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 ?.
\ 5a href="cic:/matita/tutorial/chapter10/all_true.def(8)"\ 6all_true\ 5/a\ 6 S visited →
\ 5a href="cic:/matita/tutorial/chapter10/sub_sons.def(8)"\ 6sub_sons\ 5/a\ 6 S l visited (frontier\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6visited) →
- \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited_res〉 →
+ \ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 S l n frontier visited \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 〈\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,visited_res\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 →
\ 5a href="cic:/matita/tutorial/chapter10/is_bisim.def(8)"\ 6is_bisim\ 5/a\ 6 S visited_res l \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 ? (frontier\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6visited) visited_res.
#S #l #n elim n
[#fron #vis #vis_res #_ #_ >\ 5a href="cic:/matita/tutorial/chapter10/bisim_never.def(10)"\ 6bisim_never\ 5/a\ 6 #H destruct
(* frontier = hd:: tl and hd is ok *)
#H #tl #visited #visited_res #allv >(\ 5a href="cic:/matita/tutorial/chapter10/bisim_step_true.def(10)"\ 6bisim_step_true\ 5/a\ 6 … H)
(* new_visited = hd::visited are all ok *)
- cut (\ 5a href="cic:/matita/tutorial/chapter10/all_true.def(8)"\ 6all_true\ 5/a\ 6 S (hd\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited))
+ cut (\ 5a href="cic:/matita/tutorial/chapter10/all_true.def(8)"\ 6all_true\ 5/a\ 6 S (hd:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6visited))
[#p #H1 cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … H1) [#eqp >(\P eqp) @H |@allv]]
(* we now exploit the induction hypothesis *)
#allh #subH #bisim cases (Hind … allh … bisim) -bisim -Hind
#eqhdx <(\P eqhdx) #xa #membxa
(* xa is a son of x; we must distinguish the case xa
was already visited form the case xa is new *)
- cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? xa (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:visited)))
+ cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 ? xa (x:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6visited)))
[(* xa visited - trivial *) #membxa @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l2.def(5)"\ 6memb_append_l2\ 5/a\ 6 //
|(* xa new *) #membxa @\ 5a href="cic:/matita/tutorial/chapter5/memb_append_l1.def(5)"\ 6memb_append_l1\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"\ 6sublist_unique_append_l1\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter5/memb_filter_l.def(5)"\ 6memb_filter_l\ 5/a\ 6
[>membxa //|//]
prove that two expressions are equivalente if and only if they define
the same language. *)
-definition equiv ≝ λSig.λre1,re2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 Sig.
+\ 5img class="anchor" src="icons/tick.png" id="equiv"\ 6definition equiv ≝ λSig.λre1,re2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 Sig.
let e1 ≝ \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 ? re1) in
let e2 ≝ \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 ? re2) in
- let n ≝ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 Sig (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|))) in
+ let n ≝ \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/list/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/space_enum.def(5)"\ 6space_enum\ 5/a\ 6 Sig (|\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6) (|\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6))) in
let sig ≝ (\ 5a href="cic:/matita/tutorial/chapter10/occ.def(7)"\ 6occ\ 5/a\ 6 Sig e1 e2) in
- (\ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 ? sig n (\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6e1,e2〉\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]).
+ (\ 5a href="cic:/matita/tutorial/chapter10/bisim.fix(0,2,8)"\ 6bisim\ 5/a\ 6 ? sig n (〈e1,e2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6:\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6[\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) [\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6).
-theorem euqiv_sem : ∀Sig.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 Sig.
- \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? e1 e2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+\ 5img class="anchor" src="icons/tick.png" id="euqiv_sem"\ 6theorem euqiv_sem : ∀Sig.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 Sig.
+ \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? e1 e2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \sem{e1\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 =1 \sem{e2\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#Sig #re1 #re2 %
[#H @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"\ 6eqP_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"\ 6re_embedding\ 5/a\ 6] @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/tutorial/chapter8/re_embedding.def(13)"\ 6re_embedding\ 5/a\ 6]
- cut (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2)〉)
+ cut (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 〈\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? re1 re2)\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6)
[<H //] #Hcut
cases (\ 5a href="cic:/matita/tutorial/chapter10/bisim_complete.def(11)"\ 6bisim_complete\ 5/a\ 6 … Hcut)
[2,3: #p whd in ⊢ ((??%?)→?); #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
]
qed.
-definition eqbnat ≝ λn,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m.
+\ 5img class="anchor" src="icons/tick.png" id="eqbnat"\ 6definition eqbnat ≝ λn,m:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6. \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 n m.
-lemma eqbnat_true : ∀n,m. \ 5a href="cic:/matita/tutorial/chapter10/eqbnat.def(2)"\ 6eqbnat\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
+\ 5img class="anchor" src="icons/tick.png" id="eqbnat_true"\ 6lemma eqbnat_true : ∀n,m. \ 5a href="cic:/matita/tutorial/chapter10/eqbnat.def(2)"\ 6eqbnat\ 5/a\ 6 n m \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 m.
#n #m % [@\ 5a href="cic:/matita/arithmetics/nat/eqb_true_to_eq.def(6)"\ 6eqb_true_to_eq\ 5/a\ 6 | @\ 5a href="cic:/matita/arithmetics/nat/eq_to_eqb_true.def(5)"\ 6eq_to_eqb_true\ 5/a\ 6]
qed.
-definition DeqNat ≝ \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/eqbnat_true.def(7)"\ 6eqbnat_true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="DeqNat"\ 6definition DeqNat ≝ \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/eqb.fix(0,0,1)"\ 6eqb\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/eqbnat_true.def(7)"\ 6eqbnat_true\ 5/a\ 6.
-definition a ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
-definition b ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6).
-definition c ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)).
+\ 5img class="anchor" src="icons/tick.png" id="a"\ 6definition a ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="b"\ 6definition b ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6).
+\ 5img class="anchor" src="icons/tick.png" id="c"\ 6definition c ≝ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,3,1)"\ 6s\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/DeqNat.def(8)"\ 6DeqNat\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/arithmetics/nat/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.con(0,1,0)"\ 6O\ 5/a\ 6)).
-definition exp1 ≝ ((\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6).
-definition exp2 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
-definition exp4 ≝ (\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="exp1"\ 6definition exp1 ≝ ((\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6)^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6).
+\ 5img class="anchor" src="icons/tick.png" id="exp2"\ 6definition exp2 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="exp4"\ 6definition exp4 ≝ (\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
-definition exp6 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* ).
-definition exp7 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="exp6"\ 6definition exp6 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 ).
+\ 5img class="anchor" src="icons/tick.png" id="exp7"\ 6definition exp7 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/b.def(9)"\ 6b\ 5/a\ 6^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
-definition exp8 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* ).
-definition exp9 ≝ (\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="exp8"\ 6definition exp8 ≝ \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 ).
+\ 5img class="anchor" src="icons/tick.png" id="exp9"\ 6definition exp9 ≝ (\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6 \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6\ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/a.def(9)"\ 6a\ 5/a\ 6)^\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
-example ex1 : \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/exp8.def(10)"\ 6exp8\ 5/a\ 6\ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/exp9.def(10)"\ 6exp9\ 5/a\ 6) \ 5a href="cic:/matita/tutorial/chapter10/exp9.def(10)"\ 6exp9\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="ex1"\ 6example ex1 : \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter10/equiv.def(9)"\ 6equiv\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter10/exp8.def(10)"\ 6exp8\ 5/a\ 6\ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter10/exp9.def(10)"\ 6exp9\ 5/a\ 6) \ 5a href="cic:/matita/tutorial/chapter10/exp9.def(10)"\ 6exp9\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
normalize // qed.
\ No newline at end of file
generated by a constant 0 and a successor function from natural numbers to natural
numbers *)
-inductive nat : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="nat"\ 6inductive nat : Type[0] ≝
| O :nat
| S: nat →nat.
you received in input. Most mathematical functions can be naturally defined in this
way. For instance, the sum of two natural numbers can be defined as follows *)
-let rec add n m ≝
+\ 5img class="anchor" src="icons/tick.png" id="add"\ 6let rec add n m ≝
match n with
[ O ⇒ m
| S a ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (add a m)
Let us apply it to our case *)
-lemma add_0: ∀a. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a.
+\ 5img class="anchor" src="icons/tick.png" id="add_0"\ 6lemma add_0: ∀a. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a.
#a elim a
(* If you stop the computation here, you will see on the right the two subgoals
(* In a similar way, it is convenient to state a lemma about the behaviour of
add when the second argument is not zero. *)
-lemma add_S : ∀a,b. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a b).
+\ 5img class="anchor" src="icons/tick.png" id="add_S"\ 6lemma add_S : ∀a,b. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a b).
(* In the same way as before, we proceed by induction over a. *)
(* We are now in the position to prove the commutativity of the sum *)
-theorem add_comm : ∀a,b. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 b a.
+\ 5img class="anchor" src="icons/tick.png" id="add_comm"\ 6theorem add_comm : ∀a,b. \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 a b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 b a.
#a elim a normalize
(* We have two sub goals:
(* COERCIONS *)
-inductive bool : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="bool"\ 6inductive bool : Type[0] ≝
| tt : bool
| ff : bool.
-definition nat_of_bool ≝ λb. match b with
+\ 5img class="anchor" src="icons/tick.png" id="nat_of_bool"\ 6definition nat_of_bool ≝ λb. match b with
[ tt ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6
| ff ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6
].
(* Let us now define the following function: *)
-definition twice ≝ λn.\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 n n.
+\ 5img class="anchor" src="icons/tick.png" id="twice"\ 6definition twice ≝ λn.\ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 n n.
(*
\ 5h2 class="section"\ 6Existential\ 5/h2\ 6
introduce new connectives and quantifiers and, later on, to make some interesting
consideration on proofs and computations. *)
-theorem ex_half: ∀n.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6m. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 m \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 m).
+\ 5img class="anchor" src="icons/tick.png" id="ex_half"\ 6theorem ex_half: ∀n.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6m. n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 m \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 m).
#n elim n normalize
(* We proceed by induction on n, that breaks down to the following goals:
is just a sort of general purpose record, with standard fields fst and snd, called
projections.
A pair of values n and m is written (pair … m n) or \langle n,m \rangle - visually
-rendered as 〈n,m〉
+rendered as 〈n,m〉
We first write down the function, and then discuss it.*)
-let rec div2 n ≝
+\ 5img class="anchor" src="icons/tick.png" id="div2"\ 6let rec div2 n ≝
match n with
-[ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5span class="error" title="Parse error: [sym,] expected after [term level 19] (in [term])"\ 6\ 5/span\ 6,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉
+[ O ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6\ 5span class="error" title="Parse error: [sym,] expected after [term level 19] (in [term])"\ 6\ 5/span\ 6,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
| S a ⇒ \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6
let p ≝ (div2 a) in
match (\ 5a href="cic:/matita/basics/types/snd.fix(0,2,1)"\ 6snd\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL ':' or RPAREN expected after [term] (in [term])"\ 6\ 5/span\ 6 … p) with
- [ tt ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉
- | ff ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p, \ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉
+ [ tt ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | ff ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p, \ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
]
].
(* The function is computed by recursion over the input n. If n is 0, then the
quotient is 0 and the remainder is tt. If n = S a, we start computing the half
-of a, say 〈q,b〉. Then we have two cases according to the possible values of b:
-if b is tt, then we must return 〈q,ff〉, while if b = ff then we must return
-〈S q,tt〉.
+of a, say 〈q,b〉. Then we have two cases according to the possible values of b:
+if b is tt, then we must return 〈q,ff〉, while if b = ff then we must return
+〈S q,tt〉.
It is important to point out the deep, substantial analogy between the algorithm
for computing div2 and the the proof of ex_half. In particular ex_half returns a
B(n) is met. This is precisely the quotient-remainder pair returned by div2.
In both cases we proceed by recurrence (respectively, induction or recursion) over
the input argument n. In case n = 0, we conclude the proof in ex_half by providing
-the witness O and a proof of A(O); this corresponds to returning the pair 〈O,ff〉 in
+the witness O and a proof of A(O); this corresponds to returning the pair 〈O,ff〉 in
div2. Similarly, in the inductive case n = S a, we must exploit the inductive
hypothesis for a (i.e. the result of the recursive call), distinguishing two subcases
according to the the two possibilites A(a) or B(a) (i.e. the two possibile values of
Let us now prove that our div2 function has the expected behaviour.
*)
-lemma surjective_pairing: ∀A,B.∀p:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6\ 5span class="error" title="Parse error: [sym〉] or [sym,] expected after [term level 19] (in [term])"\ 6\ 5/span\ 6 … p〉.
+\ 5img class="anchor" src="icons/tick.png" id="surjective_pairing"\ 6lemma surjective_pairing: ∀A,B.∀p:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6\ 5span class="error" title="Parse error: [sym〉] or [sym,] expected after [term level 19] (in [term])"\ 6\ 5/span\ 6 … p\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#A #B * // qed.
-lemma div2SO: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="div2SO"\ 6lemma div2SO: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#n #q #H normalize >H normalize // qed.
-lemma div2S1: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="div2S1"\ 6lemma div2S1: ∀n,q. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 n) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 q,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#n #q #H normalize >H normalize // qed.
-lemma div2_ok: ∀n,q,r. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r〉 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q) (\ 5a href="cic:/matita/tutorial/chapter2/nat_of_bool.def(1)"\ 6nat_of_bool\ 5/a\ 6 r).
+\ 5img class="anchor" src="icons/tick.png" id="div2_ok"\ 6lemma div2_ok: ∀n,q,r. \ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q) (\ 5a href="cic:/matita/tutorial/chapter2/nat_of_bool.def(1)"\ 6nat_of_bool\ 5/a\ 6 r).
#n elim n
[#q #r normalize #H destruct //
|#a #Hind #q #r
- cut (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a), \ 5a href="cic:/matita/basics/types/snd.fix(0,2,1)"\ 6snd\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a)〉) [//]
+ cut (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a), \ 5a href="cic:/matita/basics/types/snd.fix(0,2,1)"\ 6snd\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a)\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) [//]
cases (\ 5a href="cic:/matita/basics/types/snd.fix(0,2,1)"\ 6snd\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 a))
[#H >(\ 5a href="cic:/matita/tutorial/chapter2/div2S1.def(3)"\ 6div2S1\ 5/a\ 6 … H) #H1 destruct @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6>\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 whd in ⊢ (???%); <\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @(Hind … H)
|#H >(\ 5a href="cic:/matita/tutorial/chapter2/div2SO.def(3)"\ 6div2SO\ 5/a\ 6 … H) #H1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @(Hind … H)
\ 5h2 class="section"\ 6Mixing proofs and computations\ 5/h2\ 6
There is still another possibility, however, namely to mix the program and its
specification into a single entity. The idea is to refine the output type of the
-div2 function: it should not be just a generic pair 〈q,r〉 of natural numbers but a
+div2 function: it should not be just a generic pair 〈q,r〉 of natural numbers but a
specific pair satisfying the specification of the function. In other words, we need
the possibility to define, for a type A and a property P over A, the subset type
{a:A|P(a)} of all elements a of type A that satisfy the property P. Subset types
language reach enough to comprise proofs among its expressions.
*)
-record Sub (A:Type[0]) (P:A → Prop) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="Sub"\ 6record Sub (A:Type[0]) (P:A → Prop) : Type[0] ≝
{witness: A;
proof: P witness}.
-definition qr_spec ≝ λn.λp.∀q,r. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r〉 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q) (\ 5a href="cic:/matita/tutorial/chapter2/nat_of_bool.def(1)"\ 6nat_of_bool\ 5/a\ 6 r).
+\ 5img class="anchor" src="icons/tick.png" id="qr_spec"\ 6definition qr_spec ≝ λn.λp.∀q,r. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6q,r\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 → n \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 q) (\ 5a href="cic:/matita/tutorial/chapter2/nat_of_bool.def(1)"\ 6nat_of_bool\ 5/a\ 6 r).
(* We can now construct a function from n to {p|qr_spec n p} by composing the objects
we already have *)
-definition div2P: ∀n. \ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5a href="cic:/matita/tutorial/chapter2/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6\ 5/span\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/qr_spec.def(3)"\ 6qr_spec\ 5/a\ 6 n) ≝ λn.
+\ 5img class="anchor" src="icons/tick.png" id="div2P"\ 6definition div2P: ∀n. \ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5a href="cic:/matita/tutorial/chapter2/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6\ 5/span\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/qr_spec.def(3)"\ 6qr_spec\ 5/a\ 6 n) ≝ λn.
\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter2/div2.fix(0,0,2)"\ 6div2\ 5/a\ 6 n) (\ 5a href="cic:/matita/tutorial/chapter2/div2_ok.def(4)"\ 6div2_ok\ 5/a\ 6 n).
(* But we can also try do directly build such an object *)
-definition div2Pagain : ∀n.\ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/qr_spec.def(3)"\ 6qr_spec\ 5/a\ 6 n).
+\ 5img class="anchor" src="icons/tick.png" id="div2Pagain"\ 6definition div2Pagain : ∀n.\ 5a href="cic:/matita/tutorial/chapter2/Sub.ind(1,0,2)"\ 6Sub\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6\ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter2/qr_spec.def(3)"\ 6qr_spec\ 5/a\ 6 n).
#n elim n
- [@(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉) normalize #q #r #H destruct //
+ [@(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) normalize #q #r #H destruct //
|#a * #p #qrspec
- cut (p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p, \ 5a href="cic:/matita/basics/types/snd.fix(0,2,1)"\ 6snd\ 5/a\ 6 … p〉) [//]
+ cut (p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p, \ 5a href="cic:/matita/basics/types/snd.fix(0,2,1)"\ 6snd\ 5/a\ 6 … p\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) [//]
cases (\ 5a href="cic:/matita/basics/types/snd.fix(0,2,1)"\ 6snd\ 5/a\ 6 … p)
- [#H @(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉) whd #q #r #H1 destruct @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6>\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6
+ [#H @(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) whd #q #r #H1 destruct @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6>\ 5/span\ 6\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6
whd in ⊢ (???%); <\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @(qrspec … H)
- |#H @(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉) whd #q #r #H1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @(qrspec … H)
+ |#H @(\ 5a href="cic:/matita/tutorial/chapter2/Sub.con(0,1,2)"\ 6mk_Sub\ 5/a\ 6 … \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/types/fst.fix(0,2,1)"\ 6fst\ 5/a\ 6 … p,\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) whd #q #r #H1 destruct >\ 5a href="cic:/matita/tutorial/chapter2/add_S.def(2)"\ 6add_S\ 5/a\ 6 @\ 5a href="cic:/matita/basics/logic/eq_f.def(3)"\ 6eq_f\ 5/a\ 6 @(qrspec … H)
]
qed.
-example quotient7: \ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"\ 6witness\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2Pagain.def(4)"\ 6div2Pagain\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6)))))))) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6)),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="quotient7"\ 6example quotient7: \ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"\ 6witness\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2Pagain.def(4)"\ 6div2Pagain\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6)))))))) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6)),\ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,1,0)"\ 6tt\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
-example quotient8: \ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"\ 6witness\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2Pagain.def(4)"\ 6div2Pagain\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6))))))
- \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6))), \ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="quotient8"\ 6example quotient8: \ 5a href="cic:/matita/tutorial/chapter2/witness.fix(0,2,1)"\ 6witness\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter2/div2Pagain.def(4)"\ 6div2Pagain\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6))))))
+ \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/twice.def(2)"\ 6twice\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6))), \ 5a href="cic:/matita/tutorial/chapter2/bool.con(0,2,0)"\ 6ff\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
-\ 5pre\ 6\ 5pre\ 6 \ 5/pre\ 6\ 5/pre\ 6
+\ 5pre\ 6\ 5pre\ 6 \ 5/pre\ 6\ 5/pre\ 6
\ No newline at end of file
(* Matita supports polymorphic data types. The most typical case are polymorphic
lists, parametric in the type of their elements: *)
-inductive list (A:Type[0]) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="list"\ 6inductive list (A:Type[0]) : Type[0] ≝
| nil: list A
| cons: A -> list A -> list A.
If not othewise specified, recursion is supposed to act on the first argument of the
function.*)
-let rec append A (l1: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) l2 on l1 ≝
+\ 5img class="anchor" src="icons/tick.png" id="append"\ 6let rec append A (l1: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) l2 on l1 ≝
match l1 with
[ nil ⇒ l2
- | cons hd tl ⇒ hd \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5span class="error" title="Parse error: [sym:] expected after [sym:] (in [term])"\ 6\ 5/span\ 6: append A tl l2 ].
+ | cons hd tl ⇒ hd \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5span class="error" title="Parse error: [sym:] expected after [sym:] (in [term])"\ 6\ 5/span\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 append A tl l2 ].
interpretation "append" 'append l1 l2 = (append ? l1 l2).
(* As usual, the function is executable. For instance, (append A nil l) reduces to
l, as shown by the following example: *)
-example nil_append: ∀A.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A. \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym[] (in [term])"\ 6\ 5/span\ 6] \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
+\ 5img class="anchor" src="icons/tick.png" id="nil_append"\ 6example nil_append: ∀A.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A. \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym[] (in [term])"\ 6\ 5/span\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
#A #l normalize // qed.
(* Proving that l @ [] = l is just a bit more complex. The situation is exactly
defined by recutsion over the first argument, the computation of l @ [] is stuck,
and we must proceed by induction on l *)
-lemma append_nil: ∀A.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6\ 5span class="error" title="Parse error: [term level 46] expected after [sym@] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: [term level 46] expected after [sym@] (in [term])"\ 6\ 5/span\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
+\ 5img class="anchor" src="icons/tick.png" id="append_nil"\ 6lemma append_nil: ∀A.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6\ 5span class="error" title="Parse error: [term level 46] expected after [sym@] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: [term level 46] expected after [sym@] (in [term])"\ 6\ 5/span\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l.
#A #l (elim l) normalize // qed.
(* similarly, we can define the two functions head and tail. Since we can only define
For tl, it is natural to return the empty list; for hd, we take in input a default
element d of type A to return in this case. *)
-definition head ≝ λA.λl: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.λd:A.
+\ 5img class="anchor" src="icons/tick.png" id="head"\ 6definition head ≝ λA.λl: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.λd:A.
match l with [ nil ⇒ d | cons a _ ⇒ a].
-definition tail ≝ λA.λl: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
- match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] | cons hd tl ⇒ tl].
+\ 5img class="anchor" src="icons/tick.png" id="tail"\ 6definition tail ≝ λA.λl: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 | cons hd tl ⇒ tl].
-example ex_head: ∀A.∀a,d,l. \ 5a href="cic:/matita/tutorial/chapter3/head.def(1)"\ 6head\ 5/a\ 6 A (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) d \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6 a.
+\ 5img class="anchor" src="icons/tick.png" id="ex_head"\ 6example ex_head: ∀A.∀a,d,l. \ 5a href="cic:/matita/tutorial/chapter3/head.def(1)"\ 6head\ 5/a\ 6 A (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) d \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6 a.
#A #a #d #l normalize // qed.
-example ex_tail: \ 5a href="cic:/matita/tutorial/chapter3/tail.def(1)"\ 6tail\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6].
+\ 5img class="anchor" src="icons/tick.png" id="ex_tail"\ 6example ex_tail: \ 5a href="cic:/matita/tutorial/chapter3/tail.def(1)"\ 6tail\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
normalize // qed.
-theorem associative_append:
+\ 5img class="anchor" src="icons/tick.png" id="associative_append"\ 6theorem associative_append:
∀A.∀l1,l2,l3: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A. (l1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l2) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l3 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 l1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 (l2 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l3).
#A #l1 #l2 #l3 (elim l1) normalize // qed.
(* Problemi con la notazione *)
-lemma a_append: ∀A.∀a.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A. (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l.
+\ 5img class="anchor" src="icons/tick.png" id="a_append"\ 6lemma a_append: ∀A.∀a.∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A. (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l.
// qed.
-theorem append_cons:
-∀A.∀a:A.∀l,l1: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l1)\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? a \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6])) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l1.
+\ 5img class="anchor" src="icons/tick.png" id="append_cons"\ 6theorem append_cons:
+∀A.∀a:A.∀l,l1: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.l\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l1)\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (l \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? a \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6)) \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 l1.
// qed.
(* Other typical functions over lists are those computing the length
of a list, and the function returning the nth element *)
-let rec length (A:Type[0]) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l ≝
+\ 5img class="anchor" src="icons/tick.png" id="length"\ 6let rec length (A:Type[0]) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l ≝
match l with
[ nil ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6
| cons a tl ⇒ \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 (length A tl)].
-let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (d:A) ≝
+\ 5img class="anchor" src="icons/tick.png" id="nth"\ 6let rec nth n (A:Type[0]) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) (d:A) ≝
match n with
[O ⇒ \ 5a href="cic:/matita/tutorial/chapter3/head.def(1)"\ 6head\ 5/a\ 6 A l d
|S m ⇒ nth m A (\ 5a href="cic:/matita/tutorial/chapter3/tail.def(1)"\ 6tail\ 5/a\ 6 A l) d].
-example ex_length: \ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="ex_length"\ 6example ex_length: \ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
normalize // qed.
-example ex_nth: \ 5a href="cic:/matita/tutorial/chapter3/nth.fix(0,0,2)"\ 6nth\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym[] (in [term])"\ 6\ 5/span\ 6])) \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="ex_nth"\ 6example ex_nth: \ 5a href="cic:/matita/tutorial/chapter3/nth.fix(0,0,2)"\ 6nth\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) ? (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? (\ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,2,0)"\ 6S\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6) (\ 5a href="cic:/matita/tutorial/chapter3/list.con(0,2,1)"\ 6cons\ 5/a\ 6 ? \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym[] (in [term])"\ 6\ 5/span\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6)) \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/nat.con(0,1,0)"\ 6O\ 5/a\ 6.
normalize // qed.
(* Proving that the length of l1@l2 is the sum of the lengths of l1
and l2 just requires a trivial induction on the first list. *)
- lemma length_add: ∀A.∀l1,l2:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
+ \ 5img class="anchor" src="icons/tick.png" id="length_add"\ 6lemma length_add: ∀A.∀l1,l2:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.
\ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? (l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter2/add.fix(0,0,1)"\ 6add\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? l1) (\ 5a href="cic:/matita/tutorial/chapter3/length.fix(0,1,1)"\ 6length\ 5/a\ 6 ? l2).
#A #l1 elim l1 normalize // qed.
of the kind (a::l)? We start defining a simple predicate stating if a list is
empty or not. The predicate is computed by inspection over the list *)
-definition is_nil: ∀A:Type[0].\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A → Prop ≝
-λA.λl.match l with [ nil ⇒ l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] | cons hd tl ⇒ (l \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6])].
+\ 5img class="anchor" src="icons/tick.png" id="is_nil"\ 6definition is_nil: ∀A:Type[0].\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A → Prop ≝
+λA.λl.match l with [ nil ⇒ l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 | cons hd tl ⇒ (l \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6)].
(* Next we need a simple result about negation: if you wish to prove ¬P you are
authorized to add P to your hypothesis: *)
-lemma neg_aux : ∀P:Prop. (P → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6P) → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6P.
+\ 5img class="anchor" src="icons/tick.png" id="neg_aux"\ 6lemma neg_aux : ∀P:Prop. (P → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6P) → \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6P.
#P #PtonegP % /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-theorem diff_cons_nil:
-∀A:Type[0].∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a:A. a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6].
+\ 5img class="anchor" src="icons/tick.png" id="diff_cons_nil"\ 6theorem diff_cons_nil:
+∀A:Type[0].∀l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a:A. a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
#A #l #a @\ 5a href="cic:/matita/tutorial/chapter3/neg_aux.def(3)"\ 6neg_aux\ 5/a\ 6 #Heq
(* we start assuming the new hypothesis Heq of type a::l = [] using neg_aux.
Next we use the change tactic to pass from the current goal a::l≠ [] to the
expression is_nil a::l, convertible with it. *)
-(change with (\ 5a href="cic:/matita/tutorial/chapter3/is_nil.def(1)"\ 6is_nil\ 5/a\ 6 ? (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l)))
+(change with (\ 5a href="cic:/matita/tutorial/chapter3/is_nil.def(1)"\ 6is_nil\ 5/a\ 6 ? (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l)))
(* Now, we rewrite with Heq, obtaining (is_nil A []), that reduces to the trivial
goal [] = [] *)
>Heq // qed.
then absurd: ∀A:Prop. A → ¬A → False to reduce to the contradictory cases.
Usually, you may invoke automation to take care to solve the absurd case. *)
-lemma nil_to_nil: ∀A.∀l1,l2:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6A.
- l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] → l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6].
+\ 5img class="anchor" src="icons/tick.png" id="nil_to_nil"\ 6lemma nil_to_nil: ∀A.∀l1,l2:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6A.
+ l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 → l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
#A #l1 cases l1 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ #a #tl #l2 #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
(*
f:A → B, a list l = [a1; a2; ... ; an] and returning the list
[f a1; f a2; ... ; f an]. *)
-let rec map (A,B:Type[0]) (f: A → B) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 B ≝
- match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] | cons x tl ⇒ f x \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6: (map A B f tl)].
+\ 5img class="anchor" src="icons/tick.png" id="map"\ 6let rec map (A,B:Type[0]) (f: A → B) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l: \ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 B ≝
+ match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 | cons x tl ⇒ f x \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 (map A B f tl)].
(* Another major example is the fold function, that taken a list
l = [a1; a2; ... ;an], a base value b:B, and a function f: A → B → B returns
(f a1 (f a2 (... (f an b)...))). *)
-let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l :B ≝
+\ 5img class="anchor" src="icons/tick.png" id="foldr"\ 6let rec foldr (A,B:Type[0]) (f:A → B → B) (b:B) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l :B ≝
match l with [ nil ⇒ b | cons a l ⇒ f a (foldr A B f b l)].
(* As an example of application of foldr, let us use it to define a filter
the function that taken x and l returns x::l, if x satisfies the test, and l
otherwise. We use an if_then_else function included from bool.ma to this purpose. *)
-definition filter ≝
+\ 5img class="anchor" src="icons/tick.png" id="filter"\ 6definition filter ≝
λT.λp:T → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
- \ 5a href="cic:/matita/tutorial/chapter3/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 T (\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 T) (λx,l0. if p x then x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l0 else l0) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6].
+ \ 5a href="cic:/matita/tutorial/chapter3/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 T (\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 T) (λx,l0. if p x then x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l0 else l0) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6.
(* Here are a couple of simple lemmas on the behaviour of the filter function.
It is often convenient to state such lemmas, in order to be able to use rewriting
as an alternative to reduction in proofs: reduction is a bit difficult to control.
*)
-lemma filter_true : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
- \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6: \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
+\ 5img class="anchor" src="icons/tick.png" id="filter_true"\ 6lemma filter_true : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+ \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a \ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
#A #l #a #p #pa (elim l) normalize >pa // qed.
-lemma filter_false : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 →
- \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
+\ 5img class="anchor" src="icons/tick.png" id="filter_false"\ 6lemma filter_false : ∀A,l,a,p. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 →
+ \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p l.
#A #l #a #p #pa (elim l) normalize >pa normalize // qed.
(* As another example, let us redefine the map function using foldr. The
of type A → list B → list B is the function mapping a and l to (f a)::l.
*)
-definition map_again ≝ λA,B,f,l. \ 5a href="cic:/matita/tutorial/chapter3/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 A (\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 B) (λa,l.f a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6] l.
+\ 5img class="anchor" src="icons/tick.png" id="map_again"\ 6definition map_again ≝ λA,B,f,l. \ 5a href="cic:/matita/tutorial/chapter3/foldr.fix(0,4,1)"\ 6foldr\ 5/a\ 6 A (\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 B) (λa,l.f a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 l.
(*
\ 5h2 class="section"\ 6Extensional equality\ 5/h2\ 6
programs behave in the same way: this is a different, extensional equality
that can be defined in the following way. *)
-definition ExtEq ≝ λA,B:Type[0].λf,g:A→B.∀a:A.f a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g a.
+\ 5img class="anchor" src="icons/tick.png" id="ExtEq"\ 6definition ExtEq ≝ λA,B:Type[0].λf,g:A→B.∀a:A.f a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 g a.
(* Proving that map and map_again are extentionally equal in the
previous sense can be proved by a trivial structural induction on the list *)
-lemma eq_maps: ∀A,B,f. \ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"\ 6ExtEq\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f) (\ 5a href="cic:/matita/tutorial/chapter3/map_again.def(2)"\ 6map_again\ 5/a\ 6 A B f).
+\ 5img class="anchor" src="icons/tick.png" id="eq_maps"\ 6lemma eq_maps: ∀A,B,f. \ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"\ 6ExtEq\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B f) (\ 5a href="cic:/matita/tutorial/chapter3/map_again.def(2)"\ 6map_again\ 5/a\ 6 A B f).
#A #B #f #n (elim n) normalize // qed.
(* Let us make another remark about extensional equality. It is clear that,
if f is extensionally equal to g, then (map A B f) is extensionally equal to
(map A B g). Let us prove it. *)
-theorem eq_map : ∀A,B,f,g. \ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"\ 6ExtEq\ 5/a\ 6 A B f g → \ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"\ 6ExtEq\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"\ 6map\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6A B f) (\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B g).
+\ 5img class="anchor" src="icons/tick.png" id="eq_map"\ 6theorem eq_map : ∀A,B,f,g. \ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"\ 6ExtEq\ 5/a\ 6 A B f g → \ 5a href="cic:/matita/tutorial/chapter3/ExtEq.def(1)"\ 6ExtEq\ 5/a\ 6 ?? (\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"\ 6map\ 5/a\ 6 \ 5span style="text-decoration: underline;"\ 6\ 5/span\ 6A B f) (\ 5a href="cic:/matita/tutorial/chapter3/map.fix(0,3,1)"\ 6map\ 5/a\ 6 A B g).
#A #B #f #g #eqfg
(* the relevant point is that we cannot proceed by rewriting f with g via
that essentially allow you to iterate on every subset of a given enumerated
(finite) type, represented as a list. *)
- let rec fold (A,B:Type[0]) (op:B→B→B) (b:B) (p:A→\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"\ 6bool\ 5/a\ 6) (f:A→B) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l:B ≝
+ \ 5img class="anchor" src="icons/tick.png" id="fold"\ 6let rec fold (A,B:Type[0]) (op:B→B→B) (b:B) (p:A→\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)" title="null"\ 6bool\ 5/a\ 6) (f:A→B) (l:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l:B ≝
match l with
[ nil ⇒ b
| cons a l ⇒ if p a then op (f a) (fold A B op b p f l) else
interpretation "\fold" 'fold op nil p f l = (fold ? ? op nil p f l).
-theorem fold_true:
+\ 5img class="anchor" src="icons/tick.png" id="fold_true"\ 6theorem fold_true:
∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A→B. p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
- \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l| p i} (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- op (f a) \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i} (f i).
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ op (f a) \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
#A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
-theorem fold_false:
+\ 5img class="anchor" src="icons/tick.png" id="fold_false"\ 6theorem fold_false:
∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f.
-p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l| p i} (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i} (f i).
+p a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
#A #B #a #l #p #op #nil #f #pa normalize >pa // qed.
-theorem fold_filter:
+\ 5img class="anchor" src="icons/tick.png" id="fold_filter"\ 6theorem fold_filter:
∀A,B.∀a:A.∀l.∀p.∀op:B→B→B.∀nil.∀f:A →B.
- \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i} (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ (\ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p l)} (f i).
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ l| p i\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ (\ 5a href="cic:/matita/tutorial/chapter3/filter.def(2)"\ 6filter\ 5/a\ 6 A p l)\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
#A #B #a #l #p #op #nil #f elim l //
#a #tl #Hind cases(\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (p a)) #pa
[ >\ 5a href="cic:/matita/tutorial/chapter3/filter_true.def(3)"\ 6filter_true\ 5/a\ 6 // > \ 5a href="cic:/matita/tutorial/chapter3/fold_true.def(3)"\ 6fold_true\ 5/a\ 6 // >\ 5a href="cic:/matita/tutorial/chapter3/fold_true.def(3)"\ 6fold_true\ 5/a\ 6 //
| >\ 5a href="cic:/matita/tutorial/chapter3/filter_false.def(3)"\ 6filter_false\ 5/a\ 6 // >\ 5a href="cic:/matita/tutorial/chapter3/fold_false.def(3)"\ 6fold_false\ 5/a\ 6 // ]
qed.
-record Aop (A:Type[0]) (nil:A) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="Aop"\ 6record Aop (A:Type[0]) (nil:A) : Type[0] ≝
{op :2> A → A → A;
nill:∀a. op nil a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a;
nilr:∀a. op a nil \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a;
assoc: ∀a,b,c.op a (op b c) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 op (op a b) c
}.
-theorem fold_sum: ∀A,B. ∀I,J:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀nil.∀op:\ 5a href="cic:/matita/tutorial/chapter3/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f:A → B.
- op (\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ I} (f i)) (\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ J} (f i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
- \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ (I\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6J)} (f i).
+\ 5img class="anchor" src="icons/tick.png" id="fold_sum"\ 6theorem fold_sum: ∀A,B. ∀I,J:\ 5a href="cic:/matita/tutorial/chapter3/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀nil.∀op:\ 5a href="cic:/matita/tutorial/chapter3/Aop.ind(1,0,2)"\ 6Aop\ 5/a\ 6 B nil.∀f:A → B.
+ op (\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ I\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i)) (\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ J\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6
+ \ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6\fold\ 5/a\ 6[op,nil]_{i ∈ (I\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6J)\ 5a title="\fold" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (f i).
#A #B #I #J #nil #op #f (elim I) normalize
[>\ 5a href="cic:/matita/tutorial/chapter3/nill.fix(0,2,2)"\ 6nill\ 5/a\ 6//|#a #tl #Hind <\ 5a href="cic:/matita/tutorial/chapter3/assoc.fix(0,2,2)"\ 6assoc\ 5/a\ 6 //]
qed.
\ No newline at end of file
A→Prop.
For instance the empty set is defined by the always false function: *)
-definition empty_set ≝ λA:Type[0].λa:A.\ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="empty_set"\ 6definition empty_set ≝ λA:Type[0].λa:A.\ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
notation "\emptyv" non associative with precedence 90 for @{'empty_set}.
interpretation "empty set" 'empty_set = (empty_set ?).
(* Similarly, a singleton set contaning containing an element a, is defined
by by the characteristic function asserting equality with a *)
-definition singleton ≝ λA.λx,a:A.x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6a.
+\ 5img class="anchor" src="icons/tick.png" id="singleton"\ 6definition singleton ≝ λA.λx,a:A.x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6a.
(* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}. *)
interpretation "singleton" 'singl x = (singleton ? x).
are easily defined in terms of the propositional connectives of dijunction,
conjunction and negation *)
-definition union : ∀A:Type[0].∀P,Q.A → Prop ≝ λA,P,Q,a.P a \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 Q a.
+\ 5img class="anchor" src="icons/tick.png" id="union"\ 6definition union : ∀A:Type[0].∀P,Q.A → Prop ≝ λA,P,Q,a.P a \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 Q a.
interpretation "union" 'union a b = (union ? a b).
-definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym∧] (in [term])"\ 6\ 5/span\ 6 Q a.
+\ 5img class="anchor" src="icons/tick.png" id="intersection"\ 6definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym∧] (in [term])"\ 6\ 5/span\ 6 Q a.
interpretation "intersection" 'intersects a b = (intersection ? a b).
-definition complement ≝ λU:Type[0].λA:U → Prop.λw.\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 A w.
+\ 5img class="anchor" src="icons/tick.png" id="complement"\ 6definition complement ≝ λU:Type[0].λA:U → Prop.λw.\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 A w.
interpretation "complement" 'not a = (complement ? a).
-definition substraction := λU:Type[0].λA,B:U → Prop.λw.A w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 B w.
+\ 5img class="anchor" src="icons/tick.png" id="substraction"\ 6definition substraction := λU:Type[0].λA,B:U → Prop.λw.A w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 B w.
interpretation "substraction" 'minus a b = (substraction ? a b).
(* Finally, we use implication to define the inclusion relation between
sets *)
-definition subset: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝ λA,P,Q.∀a:A.(P a → Q a).
+\ 5img class="anchor" src="icons/tick.png" id="subset"\ 6definition subset: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝ λA,P,Q.∀a:A.(P a → Q a).
interpretation "subset" 'subseteq a b = (subset ? a b).
(*
Two sets are equals if and only if they have the same elements, that is,
if the two characteristic functions are extensionally equivalent: *)
-definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym↔] (in [term])"\ 6\ 5/span\ 6 Q a.
+\ 5img class="anchor" src="icons/tick.png" id="eqP"\ 6definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym↔] (in [term])"\ 6\ 5/span\ 6 Q a.
notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
interpretation "extensional equality" 'eqP a b = (eqP ? a b).
functions; the fact it defines an equivalence relation must be explicitly
proved: *)
-lemma eqP_sym: ∀U.∀A,B:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqP_sym"\ 6lemma eqP_sym: ∀U.∀A,B:U →Prop.
A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 B → B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A.
#U #A #B #eqAB #a @\ 5a href="cic:/matita/basics/logic/iff_sym.def(2)"\ 6iff_sym\ 5/a\ 6 @eqAB qed.
-lemma eqP_trans: ∀U.∀A,B,C:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqP_trans"\ 6lemma eqP_trans: ∀U.∀A,B,C:U →Prop.
A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 B → B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C.
#U #A #B #C #eqAB #eqBC #a @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 // qed.
(* For the same reason, we must also prove that all the operations behave well
with respect to eqP: *)
-lemma eqP_union_r: ∀U.∀A,B,C:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqP_union_r"\ 6lemma eqP_union_r: ∀U.∀A,B,C:U →Prop.
A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: NUMBER '1' or [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 61 C → A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B.
#U #A #B #C #eqAB #a @\ 5a href="cic:/matita/basics/logic/iff_or_r.def(2)"\ 6iff_or_r\ 5/a\ 6 @eqAB qed.
-lemma eqP_union_l: ∀U.∀A,B,C:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqP_union_l"\ 6lemma eqP_union_l: ∀U.∀A,B,C:U →Prop.
B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 C.
#U #A #B #C #eqBC #a @\ 5a href="cic:/matita/basics/logic/iff_or_l.def(2)"\ 6iff_or_l\ 5/a\ 6 @eqBC qed.
-lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqP_intersect_r"\ 6lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B.
#U #A #B #C #eqAB #a @\ 5a href="cic:/matita/basics/logic/iff_and_r.def(2)"\ 6iff_and_r\ 5/a\ 6 @eqAB qed.
-lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqP_intersect_l"\ 6lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym∩] (in [term])"\ 6\ 5/span\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C.
#U #A #B #C #eqBC #a @\ 5a href="cic:/matita/basics/logic/iff_and_l.def(2)"\ 6iff_and_l\ 5/a\ 6 @eqBC qed.
-lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqP_substract_r"\ 6lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 B.
#U #A #B #C #eqAB #a @\ 5a href="cic:/matita/basics/logic/iff_and_r.def(2)"\ 6iff_and_r\ 5/a\ 6 @eqAB qed.
-lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eqP_substract_l"\ 6lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 C.
#U #A #B #C #eqBC #a @\ 5a href="cic:/matita/basics/logic/iff_and_l.def(2)"\ 6iff_and_l\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/iff_not.def(4)"\ 6iff_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
In particular, union is commutative and associative, and the empty set is an
identity element: *)
-lemma union_empty_r: ∀U.∀A:U→Prop.
+\ 5img class="anchor" src="icons/tick.png" id="union_empty_r"\ 6lemma union_empty_r: ∀U.∀A:U→Prop.
A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A.
#U #A #w % [* // normalize #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
qed.
-lemma union_comm : ∀U.∀A,B:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="union_comm"\ 6lemma union_comm : ∀U.∀A,B:U →Prop.
A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 B \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 A.
#U #A #B #a % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma union_assoc: ∀U.∀A,B,C:U → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="union_assoc"\ 6lemma union_assoc: ∀U.∀A,B,C:U → Prop.
A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 C \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (B \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 C).
#S #A #B #C #w % [* [* /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ ] | * [/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
qed.
(* In the same way we prove commutativity and associativity for set
interesection *)
-lemma cap_comm : ∀U.∀A,B:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="cap_comm"\ 6lemma cap_comm : ∀U.∀A,B:U →Prop.
A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 B \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 A.
#U #A #B #a % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma cap_assoc: ∀U.∀A,B,C:U→Prop.
+\ 5img class="anchor" src="icons/tick.png" id="cap_assoc"\ 6lemma cap_assoc: ∀U.∀A,B,C:U→Prop.
A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 (B \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C) \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 (A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B) \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C.
#U #A #B #C #w % [ * #Aw * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ \ 5span class="autotactic"\ 6\ 5span class="autotrace"\ 6\ 5/span\ 6\ 5/span\ 6| * * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ ]
qed.
(* We can also easily prove idempotency for union and intersection *)
-lemma union_idemp: ∀U.∀A:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="union_idemp"\ 6lemma union_idemp: ∀U.∀A:U →Prop.
A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A.
#U #A #a % [* // | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/] qed.
-lemma cap_idemp: ∀U.∀A:U →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="cap_idemp"\ 6lemma cap_idemp: ∀U.∀A:U →Prop.
A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A.
#U #A #a % [* // | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/] qed.
(* We conclude our examples with a couple of distributivity theorems, and a
characterization of substraction in terms of interesection and complementation. *)
-lemma distribute_intersect : ∀U.∀A,B,C:U→Prop.
+\ 5img class="anchor" src="icons/tick.png" id="distribute_intersect"\ 6lemma distribute_intersect : ∀U.∀A,B,C:U→Prop.
(A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B) \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 (A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C) \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (B \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C).
#U #A #B #C #w % [* * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | * * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
qed.
-lemma distribute_substract : ∀U.∀A,B,C:U→Prop.
+\ 5img class="anchor" src="icons/tick.png" id="distribute_substract"\ 6lemma distribute_substract : ∀U.∀A,B,C:U→Prop.
(A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B) \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 C \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 (A \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 C) \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (B \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 C).
#U #A #B #C #w % [* * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | * * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
qed.
-lemma substract_def:∀U.∀A,B:U→Prop. A\ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 \ 5a title="complement" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6B.
+\ 5img class="anchor" src="icons/tick.png" id="substract_def"\ 6lemma substract_def:∀U.∀A,B:U→Prop. A\ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 \ 5a title="complement" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6B.
#U #A #B #w normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
for any pair of elements a and b in U, (eqb x y) is true if and only if x=y.
A set equipped with such an equality is called a DeqSet: *)
-record DeqSet : Type[1] ≝ { carr :> Type[0];
+\ 5img class="anchor" src="icons/tick.png" id="DeqSet"\ 6record DeqSet : Type[1] ≝ { carr :> Type[0];
eqb: carr → carr → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6;
eqb_true: ∀x,y. (eqb x y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 (x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y)
}.
statement asserts that we can reflect a proof that eqb a b is false into
a proof of the proposition a ≠ b. *)
-lemma eqb_false: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a,b:S.
+\ 5img class="anchor" src="icons/tick.png" id="eqb_false"\ 6lemma eqb_false: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a,b:S.
(\ 5a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 ? a b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 a \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b.
(* We start the proof introducing the hypothesis, and then split the "if" and
(* The following statement proves that propositional equality in a
DeqSet is decidable in the traditional sense, namely either a=b or a≠b *)
- lemma dec_eq: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a,b:S. a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 a \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b.
+ \ 5img class="anchor" src="icons/tick.png" id="dec_eq"\ 6lemma dec_eq: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a,b:S. a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 a \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b.
#S #a #b cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 ? a b)) #H
[%1 @(\P H) | %2 @(\Pf H)]
qed.
beqb b1 b2 is true if and only if b1=b2, and finally build the type DeqBool by
instantiating the DeqSet record with the previous information *)
-definition beqb ≝ λb1,b2.
+\ 5img class="anchor" src="icons/tick.png" id="beqb"\ 6definition beqb ≝ λb1,b2.
match b1 with [ true ⇒ b2 | false ⇒ \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b2].
notation < "a == b" non associative with precedence 45 for @{beqb $a $b }.
-lemma beqb_true: ∀b1,b2. \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 b1 b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) (b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b2).
+\ 5img class="anchor" src="icons/tick.png" id="beqb_true"\ 6lemma beqb_true: ∀b1,b2. \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 b1 b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) (b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b2).
#b1 #b2 cases b1 cases b2 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-definition DeqBool ≝ \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"\ 6beqb_true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="DeqBool"\ 6definition DeqBool ≝ \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"\ 6beqb\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"\ 6beqb_true\ 5/a\ 6.
(* At this point, we would expect to be able to prove things like the
following: for any boolean b, if b==false is true then b=false.
DeqBool (change the type in the following statement and see what
happens). *)
-example exhint: ∀b:\ 5a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"\ 6DeqBool\ 5/a\ 6. (b\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="exhint"\ 6example exhint: ∀b:\ 5a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"\ 6DeqBool\ 5/a\ 6. (b\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
#b #H @(\P H)
qed.
(* After having provided the previous hints, we may rewrite example exhint
declaring b of type bool. *)
-example exhint1: ∀b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. (b \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6= \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="exhint1"\ 6example exhint1: ∀b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. (b \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
#b #H @(\P H)
qed.
this, we must as usual define the boolen equality function, and prove
it correctly reflects propositional equality. *)
-definition eq_pairs ≝
- λA,B:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λp1,p2:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B.(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p1 \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6= \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p2) \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p1 \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6= \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p2).
+\ 5img class="anchor" src="icons/tick.png" id="eq_pairs"\ 6definition eq_pairs ≝
+ λA,B:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λp1,p2:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B.(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p1 \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p2) \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p1 \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p2).
-lemma eq_pairs_true: ∀A,B:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀p1,p2:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B.
+\ 5img class="anchor" src="icons/tick.png" id="eq_pairs_true"\ 6lemma eq_pairs_true: ∀A,B:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀p1,p2:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B.
\ 5a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"\ 6eq_pairs\ 5/a\ 6 A B p1 p2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 p1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 p2.
#A #B * #a1 #b1 * #a2 #b2 %
[#H cases (\ 5a href="cic:/matita/basics/bool/andb_true.def(5)"\ 6andb_true\ 5/a\ 6 …H) normalize #eqa #eqb >(\P eqa) >(\P eqb) //
]
qed.
-definition DeqProd ≝ λA,B:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="DeqProd"\ 6definition DeqProd ≝ λA,B:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 (A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B) (\ 5a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"\ 6eq_pairs\ 5/a\ 6 A B) (\ 5a href="cic:/matita/tutorial/chapter4/eq_pairs_true.def(6)"\ 6eq_pairs_true\ 5/a\ 6 A B).
(* Having an unification problem of the kind T1×T2 = carr X, what kind
(* ---------------------------------------- *) ⊢
\ 5a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"\ 6eq_pairs\ 5/a\ 6 T1 T2 p1 p2 ≡ \ 5a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 X p1 p2.
-example hint2: ∀b1,b2.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,b2〉\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,b2〉.
+\ 5img class="anchor" src="icons/tick.png" id="hint2"\ 6example hint2: ∀b1,b2.
+ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,b2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,b2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#b1 #b2 #H @(\P H).
\ No newline at end of file
between an element x and a list l. Its definition is a straightforward recursion on
l.*)
-let rec memb (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (x:S) (l: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6\ 5span class="error" title="Parse error: RPAREN expected after [term] (in [arg])"\ 6\ 5/span\ 6 S) on l ≝
+\ 5img class="anchor" src="icons/tick.png" id="memb"\ 6let rec memb (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (x:S) (l: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6\ 5span class="error" title="Parse error: RPAREN expected after [term] (in [arg])"\ 6\ 5/span\ 6 S) on l ≝
match l with
[ nil ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6
- | cons a tl ⇒ (x \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6= a) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 memb S x tl
+ | cons a tl ⇒ (x \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 memb S x tl
]\ 5span class="error" title="Parse error: NUMBER '1' or [term] or [sym=] expected after [sym=] (in [term])"\ 6\ 5/span\ 6\ 5span class="error" title="No choices for ID nil"\ 6\ 5/span\ 6.
notation < "\memb x l" non associative with precedence 90 for @{'memb $x $l}.
(op a b) is a member of (compose op l1 l2)
*)
-lemma memb_hd: ∀S,a,l. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="memb_hd"\ 6lemma memb_hd: ∀S,a,l. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #a #l normalize >(\ 5a href="cic:/matita/basics/logic/proj2.def(2)"\ 6proj2\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter4/eqb_true.fix(0,0,4)"\ 6eqb_true\ 5/a\ 6 S …) (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 S a)) //
qed.
-lemma memb_cons: ∀S,a,b,l.
- \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6\ 5/span\ 6 S a (b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
-#S #a #b #l normalize cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize //
+\ 5img class="anchor" src="icons/tick.png" id="memb_cons"\ 6lemma memb_cons: ∀S,a,b,l.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6\ 5/span\ 6 S a (b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+#S #a #b #l normalize cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b) normalize //
qed.
-lemma memb_single: ∀S,a,x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x.
-#S #a #x normalize cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x)) #H
+\ 5img class="anchor" src="icons/tick.png" id="memb_single"\ 6lemma memb_single: ∀S,a,x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 x.
+#S #a #x normalize cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x)) #H
[#_ >(\P H) // |>H normalize #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
qed.
-lemma memb_append: ∀S,a,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="memb_append"\ 6lemma memb_append: ∀S,a,l1,l2.
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6\ 5span class="error" title="Parse error: [term level 46] expected after [sym@] (in [term])"\ 6\ 5/span\ 6l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #a #l1\ 5span class="error" title="Parse error: illegal begin of statement"\ 6\ 5/span\ 6\ 5span class="error" title="Parse error: illegal begin of statement"\ 6\ 5/span\ 6 elim l1 normalize [#l2 #H %2 //]
-#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-lemma memb_append_l1: ∀S,a,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="memb_append_l1"\ 6lemma memb_append_l1: ∀S,a,l1,l2.
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #a #l1 elim l1 normalize
[normalize #le #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
- |#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+ |#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
]
qed.
-lemma memb_append_l2: ∀S,a,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="memb_append_l2"\ 6lemma memb_append_l2: ∀S,a,l1,l2.
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l2\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #a #l1 elim l1 normalize //
-#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+#b #tl #Hind #l2 cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
qed.
-lemma memb_exists: ∀S,a,l.\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6\ 5/span\ 6 → \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6l1,l2.l\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l2).
+\ 5img class="anchor" src="icons/tick.png" id="memb_exists"\ 6lemma memb_exists: ∀S,a,l.\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6\ 5/span\ 6 → \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6l1,l2\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6l\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6l1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l2).
#S #a #l elim l [normalize #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
#b #tl #Hind #H cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … H)
[#eqba @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (\ 5a href="cic:/matita/basics/list/list.con(0,1,1)"\ 6nil\ 5/a\ 6 S)) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … tl) >(\P eqba) //
|#mem_tl cases (Hind mem_tl) #l1 * #l2 #eqtl
- @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l1)) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … l2) >eqtl //
+ @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l1)) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … l2) >eqtl //
]
qed.
-lemma not_memb_to_not_eq: ∀S,a,b,l.
- \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S b l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
-#S #a #b #l cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b)) //
+\ 5img class="anchor" src="icons/tick.png" id="not_memb_to_not_eq"\ 6lemma not_memb_to_not_eq: ∀S,a,b,l.
+ \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S b l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+#S #a #b #l cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b)) //
#eqab >(\P eqab) #H >H #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-lemma memb_map: ∀S1,S2,f,a,l. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S1 a l\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5img class="anchor" src="icons/tick.png" id="memb_map"\ 6lemma memb_map: ∀S1,S2,f,a,l. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S1 a l\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S2 (f a) (\ 5a href="cic:/matita/basics/list/map.fix(0,3,1)"\ 6map\ 5/a\ 6 … f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S1 #S2 #f #a #l elim l normalize [//]
-#x #tl #memba cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x))
+#x #tl #memba cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x))
[#eqx >eqx >(\P eqx) >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … (f x))) normalize //
- |#eqx >eqx cases (f a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=f x) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+ |#eqx >eqx cases (f a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6f x) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
]
qed.
-lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="memb_compose"\ 6lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S1 a1 l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S2 a2 l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S3 (op a1 a2) (\ 5a href="cic:/matita/basics/list/compose.def(2)"\ 6compose\ 5/a\ 6 S1 S2 S3 op l1 l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S1 #S2 #S3 #op #a1 #a2 #l1 elim l1 [normalize //]
to avoid duplications of elements. The following uniqueb check this property.
*)
-let rec uniqueb (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) l on l : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
+\ 5img class="anchor" src="icons/tick.png" id="uniqueb"\ 6let rec uniqueb (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) l on l : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
match l with
[ nil ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6
| cons a tl ⇒ \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a tl) \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 uniqueb S tl
(* unique_append l1 l2 add l1 in fornt of l2, but preserving unicity *)
-let rec unique_append (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l1,l2: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S) on l1 ≝
+\ 5img class="anchor" src="icons/tick.png" id="unique_append"\ 6let rec unique_append (S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l1,l2: \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 S) on l1 ≝
match l1 with
[ nil ⇒ l2
| cons a tl ⇒
let r ≝ unique_append S tl l2 in
- if \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a r then r else a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:r
+ if \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a r then r else a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6r
].
-lemma memb_unique_append: ∀S,a,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="memb_unique_append"\ 6lemma memb_unique_append: ∀S,a,l1,l2.
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l1\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #a #l1 elim l1 normalize [#l2 #H %2 //]
-#b #tl #Hind #l2 cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b)) #Hab >Hab normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+#b #tl #Hind #l2 cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b)) #Hab >Hab normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
cases (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S b (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)) normalize
[@Hind | >Hab normalize @Hind]
qed.
-lemma unique_append_elim: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀P: S → Prop.∀l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="unique_append_elim"\ 6lemma unique_append_elim: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀P: S → Prop.∀l1,l2.
(∀x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: NUMBER '1' or [term] or [sym=] expected after [sym=] (in [term])"\ 6\ 5/span\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → P x) → (∀x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → P x) →
∀x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → P x.
#S #P #l1 #l2 #Hl1 #Hl2 #x #membx cases (\ 5a href="cic:/matita/tutorial/chapter5/memb_unique_append.def(6)"\ 6memb_unique_append\ 5/a\ 6\ 5span class="error" title="No choices for ID memb_unique_append"\ 6\ 5/span\ 6 … membx) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
qed.
-lemma unique_append_unique: ∀S,l1,l2. \ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 S l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5img class="anchor" src="icons/tick.png" id="unique_append_unique"\ 6lemma unique_append_unique: ∀S,l1,l2. \ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 S l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
\ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 S (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #l1 elim l1 normalize // #a #tl #Hind #l2 #uniquel2
cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)))
(*
\ 5h2 class="section"\ 6Sublists\ 5/h2\ 6
*)
-definition sublist ≝
+\ 5img class="anchor" src="icons/tick.png" id="sublist"\ 6definition sublist ≝
λS,l1,l2.∀a. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
-lemma sublist_length: ∀S,l1,l2.
- \ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 S l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 l2 → \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l1| \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l2|.
+\ 5img class="anchor" src="icons/tick.png" id="sublist_length"\ 6lemma sublist_length: ∀S,l1,l2.
+ \ 5a href="cic:/matita/tutorial/chapter5/uniqueb.fix(0,1,5)"\ 6uniqueb\ 5/a\ 6 S l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 l2 → \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l1\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="natural 'less or equal to'" href="cic:/fakeuri.def(1)"\ 6≤\ 5/a\ 6 \ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6l2\ 5a title="norm" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.
#S #l1 elim l1 //
#a #tl #Hind #l2 #unique #sub
-cut (\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6l3,l4.l2\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6l3\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l4)) [@\ 5a href="cic:/matita/tutorial/chapter5/memb_exists.def(5)"\ 6memb_exists\ 5/a\ 6 @sub //]
+cut (\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6l3,l4\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6l2\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6l3\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l4)) [@\ 5a href="cic:/matita/tutorial/chapter5/memb_exists.def(5)"\ 6memb_exists\ 5/a\ 6 @sub //]
* #l3 * #l4 #eql2 >eql2 >\ 5a href="cic:/matita/basics/list/length_append.def(2)"\ 6length_append\ 5/a\ 6 normalize
applyS \ 5a href="cic:/matita/arithmetics/nat/le_S_S.def(2)"\ 6le_S_S\ 5/a\ 6 <\ 5a href="cic:/matita/basics/list/length_append.def(2)"\ 6length_append\ 5/a\ 6 @Hind [@(\ 5a href="cic:/matita/basics/bool/andb_true_r.def(4)"\ 6andb_true_r\ 5/a\ 6 … unique)]
>eql2 in sub; #sub #x #membx
]
qed.
-lemma sublist_unique_append_l1:
+\ 5img class="anchor" src="icons/tick.png" id="sublist_unique_append_l1"\ 6lemma sublist_unique_append_l1:
∀S,l1,l2. \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2).
#S #l1 elim l1 normalize [#l2 #S #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
#x #tl #Hind #l2 #a
-normalize cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x)) #eqax >eqax
+normalize cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 … (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x)) #eqax >eqax
[<(\P eqax) cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)))
[#H >H normalize // | #H >H normalize >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … a)) //]
|cases (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)) normalize
]
qed.
-lemma sublist_unique_append_l2:
+\ 5img class="anchor" src="icons/tick.png" id="sublist_unique_append_l2"\ 6lemma sublist_unique_append_l2:
∀S,l1,l2. \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l2 (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S l1 l2).
#S #l1 elim l1 [normalize //] #x #tl #Hind normalize
#l2 #a cases (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 S tl l2)) normalize
-[@Hind | cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x) normalize // @Hind]
+[@Hind | cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x) normalize // @Hind]
qed.
-lemma decidable_sublist:∀S,l1,l2.
+\ 5img class="anchor" src="icons/tick.png" id="decidable_sublist"\ 6lemma decidable_sublist:∀S,l1,l2.
(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 l2) \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S l1 l2).
#S #l1 #l2 elim l1
[%1 #a normalize in ⊢ (%→?); #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
(*\ 5h2 class="section"\ 6Filtering\ 5/h2\ 6*)
-lemma memb_filter_true: ∀S,f,a,l.
+\ 5img class="anchor" src="icons/tick.png" id="memb_filter_true"\ 6lemma memb_filter_true: ∀S,f,a,l.
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 S f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → f a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #f #a #l elim l [normalize #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
#b #tl #Hind cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (f b)) #H
normalize >H normalize [2:@Hind]
-cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b)) #eqab
+cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b)) #eqab
[#_ >(\P eqab) // | >eqab normalize @Hind]
qed.
-lemma memb_filter_memb: ∀S,f,a,l.
+\ 5img class="anchor" src="icons/tick.png" id="memb_filter_memb"\ 6lemma memb_filter_memb: ∀S,f,a,l.
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 S f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #f #a #l elim l [normalize //]
#b #tl #Hind normalize (cases (f b)) normalize
-cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b) normalize // @Hind
+cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b) normalize // @Hind
qed.
-lemma memb_filter: ∀S,f,l,x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5img class="anchor" src="icons/tick.png" id="memb_filter"\ 6lemma memb_filter: ∀S,f,l,x. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6).
/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter5/memb_filter_memb.def(5)"\ 6memb_filter_memb\ 5/a\ 6, \ 5a href="cic:/matita/tutorial/chapter5/memb_filter_true.def(5)"\ 6memb_filter_true\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
-lemma memb_filter_l: ∀S,f,x,l. (f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5img class="anchor" src="icons/tick.png" id="memb_filter_l"\ 6lemma memb_filter_l: ∀S,f,x,l. (f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) → \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x l \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S x (\ 5a href="cic:/matita/basics/list/filter.def(2)"\ 6filter\ 5/a\ 6 ? f l) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S #f #x #l #fx elim l normalize //
-#b #tl #Hind cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (x\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=b)) #eqxb
+#b #tl #Hind cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (x\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6b)) #eqxb
[<(\P eqxb) >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … x)) >fx normalize >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … x)) normalize //
|>eqxb cases (f b) normalize [>eqxb normalize @Hind| @Hind]
]
\ 5h2 class="section"\ 6Exists\ 5/h2\ 6
*)
-let rec exists (A:Type[0]) (p:A → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
+\ 5img class="anchor" src="icons/tick.png" id="exists"\ 6let rec exists (A:Type[0]) (p:A → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6) (l:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A) on l : \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≝
match l with
[ nil ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6
| cons h t ⇒ \ 5a href="cic:/matita/basics/bool/orb.def(1)"\ 6orb\ 5/a\ 6 (p h) (exists A p t)
include "tutorial/chapter5.ma".
(* A word (or string) over an alphabet S is just a list of elements of S.*)
-definition word ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6\ 5/span\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="word"\ 6definition word ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '.' expected after [grafite_ncommand] (in [executable])"\ 6\ 5/span\ 6 S.
(* For any alphabet there is only one word of length 0, the \ 5i\ 6empty word\ 5/i\ 6, which is
denoted by ϵ .*)
A · B of two languages A and B, the so called Kleene's star A* of A and the
derivative of a language A w.r.t. a given character a. *)
-definition cat : ∀S,l1,l2,w.Prop ≝
- λS.λl1,l2.λw:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6\ 5span class="error" title="Parse error: [sym_] or [ident] expected after [sym∃] (in [term])"\ 6\ 5/span\ 6w1,w2.w1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 w2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l1 w1 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym∧] (in [term])"\ 6\ 5/span\ 6 l2 w2.
+\ 5img class="anchor" src="icons/tick.png" id="cat"\ 6definition cat : ∀S,l1,l2,w.Prop ≝
+ λS.λl1,l2.λw:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6\ 5span class="error" title="Parse error: [sym_] or [ident] expected after [sym∃] (in [term])"\ 6\ 5/span\ 6w1,w2\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6w1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 w2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 l1 w1 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym∧] (in [term])"\ 6\ 5/span\ 6 l2 w2.
notation "a · b" non associative with precedence 60 for @{ 'middot $a $b}.
interpretation "cat lang" 'middot a b = (cat ? a b).
We need to define the latter operations. The following flatten function takes in
input a list of words and concatenates them together. *)
-let rec flatten (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l : \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S)) on l : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S ≝
-match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] | cons w tl ⇒ w \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 flatten ? tl ].
+\ 5img class="anchor" src="icons/tick.png" id="flatten"\ 6let rec flatten (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l : \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S)) on l : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S ≝
+match l with [ nil ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 | cons w tl ⇒ w \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 flatten ? tl ].
(* Given a list of words l and a language r, (conjunct l r) is true if and only if
all words in l are in r, that is for every w in l, r w holds. *)
-let rec conjunct (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l : \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6\ 5span class="error" title="Parse error: RPAREN expected after [term] (in [arg])"\ 6\ 5/span\ 6 (\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S)) (r : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop) on l: Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="conjunct"\ 6let rec conjunct (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l : \ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6\ 5span class="error" title="Parse error: RPAREN expected after [term] (in [arg])"\ 6\ 5/span\ 6 (\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S)) (r : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop) on l: Prop ≝
match l with [ nil ⇒ \ 5a href="cic:/matita/basics/logic/True.ind(1,0,0)"\ 6True\ 5/a\ 6 | cons w tl ⇒ r w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 conjunct ? tl r ].
(* We are ready to give the formal definition of the Kleene's star of l:
a word w belongs to l* is and only if there exists a list of strings
lw such that (conjunct lw l) and l = flatten lw. *)
-definition star ≝ λS.λl.λw:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6lw.\ 5a href="cic:/matita/tutorial/chapter6/flatten.fix(0,1,4)"\ 6flatten\ 5/a\ 6 ? lw \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter6/conjunct.fix(0,1,4)"\ 6conjunct\ 5/a\ 6 ? lw l.
+\ 5img class="anchor" src="icons/tick.png" id="star"\ 6definition star ≝ λS.λl.λw:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S.\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6lw.\ 5a href="cic:/matita/tutorial/chapter6/flatten.fix(0,1,4)"\ 6flatten\ 5/a\ 6 ? lw \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter6/conjunct.fix(0,1,4)"\ 6conjunct\ 5/a\ 6 ? lw l.
notation "a ^ *" non associative with precedence 90 for @{ 'star $a}.
interpretation "star lang" 'star l = (star ? l).
(* The derivative of a language A with respect to a character a is the set of
all strings w such that aw is in A. *)
-definition deriv ≝ λS.λA:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.λa,w. A (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w).
+\ 5img class="anchor" src="icons/tick.png" id="deriv"\ 6definition deriv ≝ λS.λA:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.λa,w. A (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w).
(*
\ 5h2 class="section"\ 6Language equalities\ 5/h2\ 6
Equality between languages is just the usual extensional equality between
sets. The operation of concatenation behaves well with respect to this equality. *)
-lemma cat_ext_l: ∀S.∀A,B,C:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="cat_ext_l"\ 6lemma cat_ext_l: ∀S.∀A,B,C:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 B.
#S #A #B #C #H #w % * #w1 * #w2 * * #eqw #inw1 #inw2
cases (H w1) /\ 5span class="autotactic"\ 66\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-lemma cat_ext_r: ∀S.∀A,B,C:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="cat_ext_r"\ 6lemma cat_ext_r: ∀S.∀A,B,C:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 C.
#S #A #B #C #H #w % * #w1 * #w2 * * #eqw #inw1 #inw2
cases (H w2) /\ 5span class="autotactic"\ 66\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
(* Concatenating a language with the empty language results in the
empty language. *)
-lemma cat_empty_l: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S→Prop. \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="cat_empty_l"\ 6lemma cat_empty_l: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S→Prop. \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6.
#S #A #w % [|*] * #w1 * #w2 * * #_ *
qed.
empty string, results in the language l; that is {ϵ} is a left and right
unit with respect to concatenation. *)
-lemma epsilon_cat_r: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
- A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A.
+\ 5img class="anchor" src="icons/tick.png" id="epsilon_cat_r"\ 6lemma epsilon_cat_r: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
+ A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A.
#S #A #w %
[* #w1 * #w2 * * #eqw #inw1 normalize #eqw2 <eqw //
- |#inA @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … w) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ]) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ |#inA @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … w) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
]
qed.
-lemma epsilon_cat_l: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
- \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6} \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A.
+\ 5img class="anchor" src="icons/tick.png" id="epsilon_cat_l"\ 6lemma epsilon_cat_l: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
+ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A.
#S #A #w %
[* #w1 * #w2 * * #eqw normalize #eqw2 <eqw <eqw2 //
|#inA @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … w) /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
(* Concatenation is distributive w.r.t. union. *)
-lemma distr_cat_r: ∀S.∀A,B,C:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
+\ 5img class="anchor" src="icons/tick.png" id="distr_cat_r"\ 6lemma distr_cat_r: ∀S.∀A,B,C:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
(A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 C \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 C \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 C.
#S #A #B #C #w %
[* #w1 * #w2 * * #eqw * /\ 5span class="autotactic"\ 66\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ |* * #w1 * #w2 * * /\ 5span class="autotactic"\ 66\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
qed.
-lemma distr_cat_r_eps: ∀S.∀A,C:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
- (A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 C \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 C \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 C.
+\ 5img class="anchor" src="icons/tick.png" id="distr_cat_r_eps"\ 6lemma distr_cat_r_eps: ∀S.∀A,C:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S →Prop.
+ (A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 C \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 C \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 C.
#S #A #C @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"\ 6distr_cat_r\ 5/a\ 6 |@\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"\ 6eqP_union_l\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter6/epsilon_cat_l.def(5)"\ 6epsilon_cat_l\ 5/a\ 6]
qed.
(* The following is a major property of derivatives *)
-lemma deriv_middot: ∀S,A,B,a. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 A \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter6/deriv.def(4)"\ 6deriv\ 5/a\ 6 S (A\ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6B) a \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 (\ 5a href="cic:/matita/tutorial/chapter6/deriv.def(4)"\ 6deriv\ 5/a\ 6 S A a) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 B.
+\ 5img class="anchor" src="icons/tick.png" id="deriv_middot"\ 6lemma deriv_middot: ∀S,A,B,a. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 A \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 → \ 5a href="cic:/matita/tutorial/chapter6/deriv.def(4)"\ 6deriv\ 5/a\ 6 S (A\ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6B) a \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 (\ 5a href="cic:/matita/tutorial/chapter6/deriv.def(4)"\ 6deriv\ 5/a\ 6 S A a) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 B.
#S #A #B #a #noteps #w normalize %
[* #w1 cases w1
[* #w2 * * #_ #Aeps @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|#b #w2 * #w3 * * whd in ⊢ ((??%?)→?); #H destruct
#H #H1 @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … w2) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … w3) % // % //
]
- |* #w1 * #w2 * * #H #H1 #H2 @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w1))
+ |* #w1 * #w2 * * #H #H1 #H2 @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w1))
@(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … w2) % // % normalize //
]
qed.
We conclude this section with some important properties of Kleene's
star that will be used in the following chapters. *)
-lemma espilon_in_star: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
- A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6.
-#S #A @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ]) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/True.con(0,1,0)"\ 6I\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+\ 5img class="anchor" src="icons/tick.png" id="espilon_in_star"\ 6lemma espilon_in_star: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
+ A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6.
+#S #A @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/True.con(0,1,0)"\ 6I\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-lemma cat_to_star:∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
- ∀w1,w2. A w1 → A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* w2 → A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* (w1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6w2).
-#S #A #w1 #w2 #Aw * #l * #H #H1 @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (w1\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l))
+\ 5img class="anchor" src="icons/tick.png" id="cat_to_star"\ 6lemma cat_to_star:∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
+ ∀w1,w2. A w1 → A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 w2 → A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 (w1\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6w2).
+#S #A #w1 #w2 #Aw * #l * #H #H1 @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (w1\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l))
% normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-lemma fix_star: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
- A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}.
+\ 5img class="anchor" src="icons/tick.png" id="fix_star"\ 6lemma fix_star: ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
+ A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #A #w %
[* #l generalize in match w; -w cases l [normalize #w * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
#w1 #tl #w * whd in ⊢ ((??%?)→?); #eqw whd in ⊢ (%→?); *
]
qed.
-lemma star_fix_eps : ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
- A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 (A \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}.
+\ 5img class="anchor" src="icons/tick.png" id="star_fix_eps"\ 6lemma star_fix_eps : ∀S.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
+ A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 (A \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #A #w %
[* #l elim l
[* whd in ⊢ ((??%?)→?); #eqw #_ %2 <eqw //
|* [#tl #Hind * #H * #_ #H2 @Hind % [@H | //]
|#a #w1 #tl #Hind * whd in ⊢ ((??%?)→?); #H1 * #H2 #H3 %1
- @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w1)) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter6/flatten.fix(0,1,4)"\ 6flatten\ 5/a\ 6 S tl)) %
+ @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w1)) @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter6/flatten.fix(0,1,4)"\ 6flatten\ 5/a\ 6 S tl)) %
[% [@H1 | normalize % /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/] |whd @(\ 5a href="cic:/matita/basics/logic/ex.con(0,1,2)"\ 6ex_intro\ 5/a\ 6 … tl) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
]
]
]
qed.
-lemma star_epsilon: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
- A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="star_epsilon"\ 6lemma star_epsilon: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀A:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
+ A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
#S #A #w % /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * //
qed.
\ No newline at end of file
(* The type re of regular expressions over an alphabet $S$ is the smallest
collection of objects generated by the following constructors: *)
-inductive re (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="re"\ 6inductive re (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) : Type[0] ≝
z: re S (* empty: ∅ *)
| e: re S (* epsilon: ϵ *)
| s: S → re S (* symbol: a *)
(* The language sem{e} associated with the regular expression e is inductively
defined by the following function: *)
-let rec in_l (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6\ 5span class="error" title="Parse error: RPAREN expected after [term] (in [arg])"\ 6\ 5/span\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '≝' expected (in [let_defs])"\ 6\ 5/span\ 6 S → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="in_l"\ 6let rec in_l (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6\ 5span class="error" title="Parse error: RPAREN expected after [term] (in [arg])"\ 6\ 5/span\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '≝' expected (in [let_defs])"\ 6\ 5/span\ 6 S → Prop ≝
match r with
[ z ⇒ \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
-| e ⇒ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}
-| s x ⇒ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5span class="error" title="Parse error: [ident] or [term level 19] expected after [sym{] (in [term])"\ 6\ 5/span\ 6 (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) }
+| e ⇒ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6
+| s x ⇒ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5span class="error" title="Parse error: [ident] or [term level 19] expected after [sym{] (in [term])"\ 6\ 5/span\ 6 (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6
| c r1 r2 ⇒ (in_l ? r1) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (in_l ? r2)
| o r1 r2 ⇒ (in_l ? r1) \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (in_l ? r2)
-| k r1 ⇒ (in_l ? r1) \ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*].
+| k r1 ⇒ (in_l ? r1) \ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6].
notation "\sem{term 19 E}" non associative with precedence 75 for @{'in_l $E}.
interpretation "in_l" 'in_l E = (in_l ? E).
interpretation "in_l mem" 'mem w l = (in_l ? l w).
-lemma rsem_star : ∀S.∀r: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S. \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{r\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{r}\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="rsem_star"\ 6lemma rsem_star : ∀S.∀r: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S. \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{r\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{r\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
// qed.
enough to mark positions preceding individual characters, so we shall have two kinds
of characters •a (pp a) and a (ps a) according to the case a is pointed or not. *)
-inductive pitem (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) : Type[0] ≝
+\ 5img class="anchor" src="icons/tick.png" id="pitem"\ 6inductive pitem (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) : Type[0] ≝
pz: pitem S (* empty *)
| pe: pitem S (* epsilon *)
| ps: S → pitem S (* symbol *)
understood as states of a DFA, and the boolean indicates if
the state is final or not. *)
-definition pre ≝ λS.\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="pre"\ 6definition pre ≝ λS.\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6.
interpretation "pitem star" 'star a = (pk ? a).
interpretation "pitem or" 'plus a b = (po ? a b).
removing all the points. Similarly, the carrier of a pointed regular expression
is the carrier of its item. *)
-let rec forget (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l : \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on l: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S ≝
+\ 5img class="anchor" src="icons/tick.png" id="forget"\ 6let rec forget (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (l : \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on l: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S ≝
match l with
[ pz ⇒ \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,1,1)"\ 6z\ 5/a\ 6 ? (* `∅ *)
| pe ⇒ \ 5a title="re epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6
| pp x ⇒ \ 5a title="atom" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6x
| pc E1 E2 ⇒ (forget ? E1) \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (forget ? E2)
| po E1 E2 ⇒ (forget ? E1) \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym+] (in [term])"\ 6\ 5/span\ 6 (forget ? E2)
- | pk E ⇒ (forget ? E)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* ].
+ | pk E ⇒ (forget ? E)\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 ].
(* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
interpretation "forget" 'norm a = (forget ? a).
-lemma erase_dot : ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6e1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 e2| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,4,1)"\ 6c\ 5/a\ 6 ? (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6e1|) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6e2|).
+\ 5img class="anchor" src="icons/tick.png" id="erase_dot"\ 6lemma erase_dot : ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6e1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/re.con(0,4,1)"\ 6c\ 5/a\ 6 ? (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6e1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6) (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6).
// qed.
-lemma erase_plus : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i1| \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2|.
+\ 5img class="anchor" src="icons/tick.png" id="erase_plus"\ 6lemma erase_plus : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="re or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.
// qed.
-lemma erase_star : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="erase_star"\ 6lemma erase_star : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
// qed.
(*
beqitem_true that it refects propositional equality, enriching the set (pitem S)
to a DeqSet. *)
-let rec beqitem S (i1,i2: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on i1 ≝
+\ 5img class="anchor" src="icons/tick.png" id="beqitem"\ 6let rec beqitem S (i1,i2: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on i1 ≝
match i1 with
[ pz ⇒ match i2 with [ pz ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 | _ ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6]
| pe ⇒ match i2 with [ pe ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 | _ ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6]
- | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=y2 | _ ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6]
- | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=y2 | _ ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6]
+ | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y2 | _ ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6]
+ | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6y2 | _ ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6]
| po i11 i12 ⇒ match i2 with
[ po i21 i22 ⇒ beqitem S i11 i21 \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 beqitem S i12 i22
| _ ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5span class="error" title="Parse error: SYMBOL '|' or SYMBOL ']' expected (in [term])"\ 6\ 5/span\ 6]
| pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6]
].
-lemma beqitem_true: ∀S,i1,i2. \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"\ 6beqitem\ 5/a\ 6 S i1 i2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) (i1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 i2).
+\ 5img class="anchor" src="icons/tick.png" id="beqitem_true"\ 6lemma beqitem_true: ∀S,i1,i2. \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"\ 6beqitem\ 5/a\ 6 S i1 i2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) (i1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 i2).
#S #i1 elim i1
[#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
|#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
]
qed.
-definition DeqItem ≝ λS.
+\ 5img class="anchor" src="icons/tick.png" id="DeqItem"\ 6definition DeqItem ≝ λS.
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) (\ 5a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"\ 6beqitem\ 5/a\ 6 S) (\ 5a href="cic:/matita/tutorial/chapter7/beqitem_true.def(5)"\ 6beqitem_true\ 5/a\ 6 S).
(* We also add a couple of unification hints to allow the type inference system
we should start reading the regular expression. The language associated
to a pre is the union of the languages associated with its points. *)
-let rec in_pl (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop ≝
+\ 5img class="anchor" src="icons/tick.png" id="in_pl"\ 6let rec in_pl (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (r : \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on r : \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop ≝
match r with
[ pz ⇒ \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
| pe ⇒ \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
| ps _ ⇒ \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6
-| pp x ⇒ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6 (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]) }
-| pc r1 r2 ⇒ (in_pl ? r1) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter7/forget.fix(0,1,3)"\ 6forget\ 5/a\ 6 ? r2} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym∪] (in [term])"\ 6\ 5/span\ 6 (in_pl ? r2)
+| pp x ⇒ \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6 (x\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6) \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6
+| pc r1 r2 ⇒ (in_pl ? r1) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter7/forget.fix(0,1,3)"\ 6forget\ 5/a\ 6 ? r2\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym∪] (in [term])"\ 6\ 5/span\ 6 (in_pl ? r2)
| po r1 r2 ⇒ (in_pl ? r1) \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (in_pl ? r2)
-| pk r1 ⇒ (in_pl ? r1) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter7/forget.fix(0,1,3)"\ 6forget\ 5/a\ 6 ? r1}\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* ].
+| pk r1 ⇒ (in_pl ? r1) \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter7/forget.fix(0,1,3)"\ 6forget\ 5/a\ 6 ? r1\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 ].
interpretation "in_pl" 'in_l E = (in_pl ? E).
interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
-definition in_prl ≝ λS : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λp:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- if (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p) then \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6} else \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p}.
+\ 5img class="anchor" src="icons/tick.png" id="in_prl"\ 6definition in_prl ≝ λS : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λp:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ if (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p) then \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 else \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
interpretation "in_prl" 'in_l E = (in_prl ? E).
(* The following, trivial lemmas are only meant for rewriting purposes. *)
-lemma sem_pre_true : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_pre_true"\ 6lemma sem_pre_true : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
// qed.
-lemma sem_pre_false : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_pre_false"\ 6lemma sem_pre_false : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
// qed.
-lemma sem_cat: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1} \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2|} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_cat"\ 6lemma sem_cat: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
// qed.
-lemma sem_cat_w: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w.
- \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2} w \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 ((\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1} \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2|}) w \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2} w).
+\ 5img class="anchor" src="icons/tick.png" id="sem_cat_w"\ 6lemma sem_cat_w: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w.
+ \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 ((\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6) w \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w).
// qed.
-lemma sem_plus: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_plus"\ 6lemma sem_plus: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
// qed.
-lemma sem_plus_w: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w.
- \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2} w \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1} w \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2} w).
+\ 5img class="anchor" src="icons/tick.png" id="sem_plus_w"\ 6lemma sem_plus_w: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w.
+ \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i1\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w).
// qed.
-lemma sem_star : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|}\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="sem_star"\ 6lemma sem_star : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
// qed.
-lemma sem_star_w : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w.
- \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*} w \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6w1,w2.w1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 w2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} w1 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|}\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* w2).
+\ 5img class="anchor" src="icons/tick.png" id="sem_star_w"\ 6lemma sem_star_w : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w.
+ \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6w1,w2\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6w1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 w2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w1 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 w2).
// qed.
(* Below are a few, simple, semantic properties of items. In particular:
The first property is proved by a simple induction on $i$; the other
results are easy corollaries. We need an auxiliary lemma first. *)
-lemma append_eq_nil : ∀S.∀w1,w2:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. w1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 w2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 → w1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="append_eq_nil"\ 6lemma append_eq_nil : ∀S.∀w1,w2:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. w1 \ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6 w2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 → w1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6.
#S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
-lemma not_epsilon_lp : ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 (\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 \ 5a title="in_pl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e).
+\ 5img class="anchor" src="icons/tick.png" id="not_epsilon_lp"\ 6lemma not_epsilon_lp : ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 (\ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 \ 5a title="in_pl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e).
#S #e elim e normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Not.con(0,1,1)"\ 6nmk\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
[#r1 #r2 * #n1 #n2 % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * #w1 * #w2 * * #H
>(\ 5a href="cic:/matita/tutorial/chapter7/append_eq_nil.def(4)"\ 6append_eq_nil\ 5/a\ 6 …H…) /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
]
qed.
-lemma epsilon_to_true : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e → \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="epsilon_to_true"\ 6lemma epsilon_to_true : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e → \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6.
#S * #i #b cases b // normalize #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
qed.
-lemma true_to_epsilon : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e.
+\ 5img class="anchor" src="icons/tick.png" id="true_to_epsilon"\ 6lemma true_to_epsilon : ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → \ 5a title="epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6 \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e.
#S * #i #b #btrue normalize in btrue; >btrue %2 //
qed.
-lemma minus_eps_item: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i}\ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ]}.
+\ 5img class="anchor" src="icons/tick.png" id="minus_eps_item"\ 6lemma minus_eps_item: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #i #w %
[#H whd % // normalize @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter7/not_epsilon_lp.def(8)"\ 6not_epsilon_lp\ 5/a\ 6 …i)) //
|* //
]
qed.
-lemma minus_eps_pre: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6\ 5span class="error" title="Parse error: [sym{] expected after [sym\sem ] (in [term])"\ 6\ 5/span\ 6{\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e}\ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ]}.
+\ 5img class="anchor" src="icons/tick.png" id="minus_eps_pre"\ 6lemma minus_eps_pre: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6\ 5span class="error" title="Parse error: [sym{] expected after [sym\sem ] (in [term])"\ 6\ 5/span\ 6{\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S * #i *
[>\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"\ 6sem_pre_true\ 5/a\ 6 normalize in ⊢ (??%?); #w %
[/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | * * // #H1 #H2 @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 …H1 H2)]
then, we just have •(i1+i2) = •(i1)⊕ •(i2).
*)
-definition lo ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λa,b:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 a \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 b,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 a \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 b〉.
+\ 5img class="anchor" src="icons/tick.png" id="lo"\ 6definition lo ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λa,b:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 a \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 b,\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 a \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
interpretation "oplus" 'oplus a b = (lo ? a b).
-lemma lo_def: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b1,b2. \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,b1〉\ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i2,b2〉\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1\ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6i2,b1\ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6b2〉.
+\ 5img class="anchor" src="icons/tick.png" id="lo_def"\ 6lemma lo_def: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b1,b2. \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,b1\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i2,b2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1\ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6i2,b1\ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6b2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
(*
Let us come to the formalized definitions:
*)
-definition pre_concat_r ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λi:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- match e with [ mk_Prod i1 b ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i1, b〉].
+\ 5img class="anchor" src="icons/tick.png" id="pre_concat_r"\ 6definition pre_concat_r ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λi:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ match e with [ mk_Prod i1 b ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i1, b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6].
notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
-lemma eq_to_ex_eq: ∀S.∀A,B:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
+\ 5img class="anchor" src="icons/tick.png" id="eq_to_ex_eq"\ 6lemma eq_to_ex_eq: ∀S.∀A,B:\ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S → Prop.
A \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 B → A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 B.
#S #A #B #H >H #x % // qed.
(* The behaviour of ◃ is summarized by the following, easy lemma: *)
-lemma sem_pre_concat_r : ∀S,i.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i \ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"\ 6◃\ 5/a\ 6 e} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_pre_concat_r"\ 6lemma sem_pre_concat_r : ∀S,i.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i \ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"\ 6◃\ 5/a\ 6 e\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #i * #i1 #b1 cases b1 [2: @\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"\ 6eq_to_ex_eq\ 5/a\ 6 //]
>\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"\ 6sem_pre_true\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"\ 6sem_cat\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"\ 6sem_pre_true\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
qed.
we abstract pre_concat_l with respect to an input bcast function from items to
pres. *)
-definition pre_concat_l ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λbcast:∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S → \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λe1:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λi2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="pre_concat_l"\ 6definition pre_concat_l ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λbcast:∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S → \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λe1:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λi2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
match e1 with
[ mk_Prod i1 b1 ⇒ match b1 with
[ true ⇒ (i1 \ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"\ 6◃\ 5/a\ 6 (bcast ? i2))
- | false ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉
+ | false ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
]
].
notation "•" non associative with precedence 60 for @{eclose ?}.
-let rec eclose (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (i: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on i : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝
+\ 5img class="anchor" src="icons/tick.png" id="eclose"\ 6let rec eclose (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (i: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on i : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝
match i with
- [ pz ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 ?, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
- | pe ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 〉
- | ps x ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem pp" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6.x, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉
- | pp x ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem pp" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6.x, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
+ [ pz ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 ?, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | pe ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | ps x ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem pp" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6\ 5a title="pitem pp" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6x, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | pp x ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem pp" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6\ 5a title="pitem pp" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6x, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
| po i1 i2 ⇒ •i1 \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 •i2
| pc i1 i2 ⇒ •i1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2
- | pk i ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (•i))\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉].
+ | pk i ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (•i))\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6].
notation "• x" non associative with precedence 60 for @{'eclose $x}.
interpretation "eclose" 'eclose x = (eclose ? x).
(* Here are a few simple properties of ▹ and •(-) *)
-lemma pcl_true : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 i1 \ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"\ 6◃\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2).
+\ 5img class="anchor" src="icons/tick.png" id="pcl_true"\ 6lemma pcl_true : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 i1 \ 5a title="pre_concat_r" href="cic:/fakeuri.def(1)"\ 6◃\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2).
// qed.
-lemma pcl_true_bis : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2), \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2)〉.
+\ 5img class="anchor" src="icons/tick.png" id="pcl_true_bis"\ 6lemma pcl_true_bis : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2), \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2)\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#S #i1 #i2 normalize cases (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2) // qed.
-lemma pcl_false: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="pcl_false"\ 6lemma pcl_false: ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
-lemma eclose_plus: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="eclose_plus"\ 6lemma eclose_plus: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i1 \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2.
// qed.
-lemma eclose_dot: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="eclose_dot"\ 6lemma eclose_dot: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2.
// qed.
-lemma eclose_star: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6(\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i))\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="eclose_star"\ 6lemma eclose_star: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6(\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i))\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
(* The definition of •(-) (eclose) can then be lifted from items to pres
in the obvious way. *)
-definition lift ≝ λS.λf:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S →\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="lift"\ 6definition lift ≝ λS.λf:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S →\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
match e with
- [ mk_Prod i b ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (f i), \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (f i) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 b〉].
+ [ mk_Prod i b ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (f i), \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (f i) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6].
-definition preclose ≝ λS. \ 5a href="cic:/matita/tutorial/chapter8/lift.def(2)"\ 6lift\ 5/a\ 6 S (\ 5a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"\ 6eclose\ 5/a\ 6 S).
+\ 5img class="anchor" src="icons/tick.png" id="preclose"\ 6definition preclose ≝ λS. \ 5a href="cic:/matita/tutorial/chapter8/lift.def(2)"\ 6lift\ 5/a\ 6 S (\ 5a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"\ 6eclose\ 5/a\ 6 S).
interpretation "preclose" 'eclose x = (preclose ? x).
(* Obviously, broadcasting does not change the carrier of the item,
as it is easily proved by structural induction. *)
-lemma erase_bull : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i)| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|.
+\ 5img class="anchor" src="icons/tick.png" id="erase_bull"\ 6lemma erase_bull : ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i)\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.
#S #i elim i //
[ #i1 #i2 #IH1 #IH2 >\ 5a href="cic:/matita/tutorial/chapter7/erase_dot.def(4)"\ 6erase_dot\ 5/a\ 6 <IH1 >\ 5a href="cic:/matita/tutorial/chapter8/eclose_dot.def(5)"\ 6eclose_dot\ 5/a\ 6
cases (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i1) #i11 #b1 cases b1 // <IH2 >\ 5a href="cic:/matita/tutorial/chapter8/pcl_true_bis.def(5)"\ 6pcl_true_bis\ 5/a\ 6 //
The proof of sem_oplus is straightforward. *)
-lemma sem_oplus: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 e2} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_oplus"\ 6lemma sem_oplus: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 e2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S * #i1 #b1 * #i2 #b2 #w %
[cases b1 cases b2 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|cases b1 cases b2 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
Then, using the previous result, we prove sem_bullet by induction
on i. Finally, sem_pcl_aux and sem_bullet give sem_pcl. *)
-lemma LcatE : ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 e2} \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6e2|} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+\ 5img class="anchor" src="icons/tick.png" id="LcatE"\ 6lemma LcatE : ∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 e2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
// qed.
-lemma sem_pcl_aux : ∀S.∀e1:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.∀i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2|} →
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1} \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2|} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_pcl_aux"\ 6lemma sem_pcl_aux : ∀S.∀e1:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.∀i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 →
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i2\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S * #i1 #b1 #i2 cases b1
[2:#th >\ 5a href="cic:/matita/tutorial/chapter8/pcl_false.def(5)"\ 6pcl_false\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"\ 6sem_pre_false\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"\ 6sem_pre_false\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"\ 6sem_cat\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"\ 6eq_to_ex_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|#H >\ 5a href="cic:/matita/tutorial/chapter8/pcl_true.def(5)"\ 6pcl_true\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"\ 6sem_pre_true\ 5/a\ 6 @(\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 … (\ 5a href="cic:/matita/tutorial/chapter8/sem_pre_concat_r.def(10)"\ 6sem_pre_concat_r\ 5/a\ 6 …))
]
qed.
-lemma minus_eps_pre_aux: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀A.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 A → \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (A \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ]}).
+\ 5img class="anchor" src="icons/tick.png" id="minus_eps_pre_aux"\ 6lemma minus_eps_pre_aux: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀A.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 A → \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (A \ 5a title="substraction" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6\ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6).
#S #e #i #A #seme
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter7/minus_eps_pre.def(10)"\ 6minus_eps_pre\ 5/a\ 6]
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"\ 6eqP_union_r\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"\ 6eqP_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter7/minus_eps_item.def(9)"\ 6minus_eps_item\ 5/a\ 6]]
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_substract_r.def(3)"\ 6eqP_substract_r\ 5/a\ 6 //
qed.
-theorem sem_bull: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6. ∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_bull"\ 6theorem sem_bull: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6. ∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #e elim e
[#w normalize % [/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | * //]
|/\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"\ 6eq_to_ex_eq\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
(blank e) is obviously the empty language: from the point of view of the automaton,
it corresponds with the pit state. *)
-let rec blank (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (i: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i :\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S ≝
+\ 5img class="anchor" src="icons/tick.png" id="blank"\ 6let rec blank (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (i: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i :\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S ≝
match i with
[ z ⇒ \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 ?
| e ⇒ \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6
| s y ⇒ \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y
| o e1 e2 ⇒ (blank S e1) \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 (blank S e2)
| c e1 e2 ⇒ (blank S e1) \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (blank S e2)
- | k e ⇒ (blank S e)\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* ].
+ | k e ⇒ (blank S e)\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 ].
-lemma forget_blank: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S e| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.
+\ 5img class="anchor" src="icons/tick.png" id="forget_blank"\ 6lemma forget_blank: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.
#S #e elim e normalize //
qed.
-lemma sem_blank: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S e} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6.
+\ 5img class="anchor" src="icons/tick.png" id="sem_blank"\ 6lemma sem_blank: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S e\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6.
#S #e elim e
[1,2:@\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"\ 6eq_to_ex_eq\ 5/a\ 6 //
|#s @\ 5a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"\ 6eq_to_ex_eq\ 5/a\ 6 //
]
qed.
-theorem re_embedding: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S e)} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e}.
+\ 5img class="anchor" src="icons/tick.png" id="re_embedding"\ 6theorem re_embedding: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6(\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S e)\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #e @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter8/sem_bull.def(12)"\ 6sem_bull\ 5/a\ 6] >\ 5a href="cic:/matita/tutorial/chapter8/forget_blank.def(4)"\ 6forget_blank\ 5/a\ 6
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"\ 6eqP_union_r\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter8/sem_blank.def(9)"\ 6sem_blank\ 5/a\ 6]]
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"\ 6union_comm\ 5/a\ 6] @\ 5a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"\ 6union_empty_r\ 5/a\ 6.
Plus and bullet have been already lifted from items to pres. We can now
do a similar job for concatenation ⊙ and Kleene's star ⊛.*)
-definition lifted_cat ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="lifted_cat"\ 6definition lifted_cat ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
\ 5a href="cic:/matita/tutorial/chapter8/lift.def(2)"\ 6lift\ 5/a\ 6 S (\ 5a href="cic:/matita/tutorial/chapter8/pre_concat_l.def(3)"\ 6pre_concat_l\ 5/a\ 6 S \ 5a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"\ 6eclose\ 5/a\ 6 e).
notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
-lemma odot_true_b : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i2,b〉 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2)),\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 b〉.
+\ 5img class="anchor" src="icons/tick.png" id="odot_true_b"\ 6lemma odot_true_b : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
+ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i2,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2)),\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#S #i1 #i2 #b normalize in ⊢ (??%?); cases (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i2) //
qed.
-lemma odot_false_b : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i2,b〉 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2 ,b〉.
+\ 5img class="anchor" src="icons/tick.png" id="odot_false_b"\ 6lemma odot_false_b : ∀S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
+ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i2,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2 ,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
//
qed.
-lemma erase_odot:∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 e2)| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1| \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|).
+\ 5img class="anchor" src="icons/tick.png" id="erase_odot"\ 6lemma erase_odot:∀S.∀e1,e2:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 e2)\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e1\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="re cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6).
#S * #i1 * * #i2 #b2 // >\ 5a href="cic:/matita/tutorial/chapter8/odot_true_b.def(6)"\ 6odot_true_b\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/erase_dot.def(4)"\ 6erase_dot\ 5/a\ 6 //
qed.
(* Let us come to the star operation: *)
-definition lk ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="lk"\ 6definition lk ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
match e with
[ mk_Prod i1 b1 ⇒
match b1 with
- [true ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"\ 6eclose\ 5/a\ 6 ? i1))\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*, \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉
- |false ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉
+ [true ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"\ 6eclose\ 5/a\ 6 ? i1))\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ |false ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i1\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
]
].
interpretation "lk" 'lk a = (lk ? a).
notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
-lemma ostar_true: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛ \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i))\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*, \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="ostar_true"\ 6lemma ostar_true: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a title="eclose" href="cic:/fakeuri.def(1)"\ 6•\ 5/a\ 6i))\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
-lemma ostar_false: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛ \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="ostar_false"\ 6lemma ostar_false: ∀S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
// qed.
-lemma erase_ostar: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛)| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="erase_ostar"\ 6lemma erase_ostar: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6)\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="re star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
#S * #i * // qed.
-lemma sem_odot_true: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.∀i.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] }.
+\ 5img class="anchor" src="icons/tick.png" id="sem_odot_true"\ 6lemma sem_odot_true: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.∀i.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6{\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 \ 5a title="singleton" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #e1 #i
-cut (e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i), \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6(e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉) [//]
+cut (e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i), \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6(e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) [//]
#H >H cases (e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i) #i1 #b1 cases b1
[>\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"\ 6sem_pre_true\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [||@\ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"\ 6eqP_sym\ 5/a\ 6 @\ 5a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"\ 6union_assoc\ 5/a\ 6]
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"\ 6eqP_union_l\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"\ 6eqP_sym\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
]
qed.
-lemma eq_odot_false: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.∀i.
- e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i.
+\ 5img class="anchor" src="icons/tick.png" id="eq_odot_false"\ 6lemma eq_odot_false: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e1:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.∀i.
+ e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i.
#S #e1 #i
-cut (e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i), \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6(e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉) [//]
+cut (e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i), \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6(e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i) \ 5a title="boolean or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) [//]
cases (e1 \ 5a title="item-pre concat" href="cic:/fakeuri.def(1)"\ 6▹\ 5/a\ 6 i) #i1 #b1 cases b1 #H @H
qed.
(* We conclude this section with the proof of the main semantic properties
of ⊙ and ⊛. *)
-lemma sem_odot:
- ∀S.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 e2} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1}\ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2|} \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2}.
+\ 5img class="anchor" src="icons/tick.png" id="sem_odot"\ 6lemma sem_odot:
+ ∀S.∀e1,e2: \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1 \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 e2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e1\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e2\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e2\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6.
#S #e1 * #i2 *
[>\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"\ 6sem_pre_true\ 5/a\ 6
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter8/sem_odot_true.def(10)"\ 6sem_odot_true\ 5/a\ 6]
]
qed.
-theorem sem_ostar: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛} \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e} \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|}\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6*.
+\ 5img class="anchor" src="icons/tick.png" id="sem_ostar"\ 6theorem sem_ostar: ∀S.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 \ 5a title="cat lang" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 \ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="in_l" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="star lang" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6.
#S * #i #b cases b
[>\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"\ 6sem_pre_true\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"\ 6sem_pre_true\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"\ 6sem_star\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"\ 6erase_bull\ 5/a\ 6
@\ 5a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"\ 6eqP_trans\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"\ 6eqP_union_r\ 5/a\ 6[|@\ 5a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"\ 6cat_ext_l\ 5/a\ 6 [|@\ 5a href="cic:/matita/tutorial/chapter8/minus_eps_pre_aux.def(11)"\ 6minus_eps_pre_aux\ 5/a\ 6 //]]]
include "tutorial/chapter8.ma".
-let rec move (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (x:S) (E: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on E : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝
+\ 5img class="anchor" src="icons/tick.png" id="move"\ 6let rec move (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (x:S) (E: \ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S) on E : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝
match E with
- [ pz ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 S, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
- | pe ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
- | ps y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 〉
- | pp y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, x \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6= y 〉
+ [ pz ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"\ 6pz\ 5/a\ 6 S, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | pe ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem epsilon" href="cic:/fakeuri.def(1)"\ 6ϵ\ 5/a\ 6, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | ps y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
+ | pp y ⇒ \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pitem ps" href="cic:/fakeuri.def(1)"\ 6`\ 5/a\ 6y, x \ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6
| po e1 e2 ⇒ (move ? x e1) \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 (move ? x e2)
| pc e1 e2 ⇒ (move ? x e1) \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 (move ? x e2)
- | pk e ⇒ (move ? x e)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛ ].
+ | pk e ⇒ (move ? x e)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6 ].
-lemma move_plus: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="move_plus"\ 6lemma move_plus: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (i1 \ 5a title="pitem or" href="cic:/fakeuri.def(1)"\ 6+\ 5/a\ 6 i2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i1) \ 5a title="oplus" href="cic:/fakeuri.def(1)"\ 6⊕\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i2).
// qed.
-lemma move_cat: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+\ 5img class="anchor" src="icons/tick.png" id="move_cat"\ 6lemma move_cat: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i1,i2:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (i1 \ 5a title="pitem cat" href="cic:/fakeuri.def(1)"\ 6·\ 5/a\ 6 i2) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i1) \ 5a title="lifted cat" href="cic:/fakeuri.def(1)"\ 6⊙\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i2).
// qed.
-lemma move_star: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6* \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6⊛.
+\ 5img class="anchor" src="icons/tick.png" id="move_star"\ 6lemma move_star: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x i\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="pitem star" href="cic:/fakeuri.def(1)"\ 6*\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i)\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6^\ 5/a\ 6\ 5a title="lk" href="cic:/fakeuri.def(1)"\ 6⊛\ 5/a\ 6.
// qed.
(*
point stops in front of the final b, while the other one broadcast inside (b^*a + b)b,
so
- move((•a + ϵ)((•b)*•a + •b)b,a) = 〈(a + ϵ)((•b)^*•a + •b)•b, false〉
+ move((•a + ϵ)((•b)*•a + •b)b,a) = 〈(a + ϵ)((•b)^*•a + •b)•b, false〉
For b, we have two positions too. The innermost point stops in front of the final b too,
while the other point reaches the end of b* and must go back through b*a:
- move((•a + ϵ)((•b)*•a + •b)b ,b) = 〈(a + ϵ)((•b)*•a + b)•b, false〉
+ move((•a + ϵ)((•b)*•a + •b)b ,b) = 〈(a + ϵ)((•b)*•a + b)•b, false〉
*)
-definition pmove ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λx:S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).
+\ 5img class="anchor" src="icons/tick.png" id="pmove"\ 6definition pmove ≝ λS:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λx:S.λe:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S. \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).
-lemma pmove_def : ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
- \ 5a href="cic:/matita/tutorial/chapter9/pmove.def(7)"\ 6pmove\ 5/a\ 6 ? x \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b〉 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i.
+\ 5img class="anchor" src="icons/tick.png" id="pmove_def"\ 6lemma pmove_def : ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀x:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀b.
+ \ 5a href="cic:/matita/tutorial/chapter9/pmove.def(7)"\ 6pmove\ 5/a\ 6 ? x \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6i,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? x i.
// qed.
-lemma eq_to_eq_hd: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a,b.
- a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:l2 → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b.
+\ 5img class="anchor" src="icons/tick.png" id="eq_to_eq_hd"\ 6lemma eq_to_eq_hd: ∀A.∀l1,l2:\ 5a href="cic:/matita/basics/list/list.ind(1,0,1)"\ 6list\ 5/a\ 6 A.∀a,b.
+ a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6l2 → a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b.
#A #l1 #l2 #a #b #H destruct //
qed.
(* Obviously, a move does not change the carrier of the item, as one can easily
prove by induction on the item. *)
-lemma same_kernel: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
- \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i)| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|.
+\ 5img class="anchor" src="icons/tick.png" id="same_kernel"\ 6lemma same_kernel: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.
+ \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i)\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.
#S #a #i elim i //
[#i1 #i2 #H1 #H2 >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter8/erase_odot.def(7)"\ 6erase_odot\ 5/a\ 6 //
|#i1 #i2 #H1 #H2 >\ 5a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"\ 6move_plus\ 5/a\ 6 whd in ⊢ (??%%); //
(* Here is our first, major result, stating the correctness of the
move operation. The proof is a simple induction on i. *)
-theorem move_ok:
+\ 5img class="anchor" src="icons/tick.png" id="move_ok"\ 6theorem move_ok:
∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S.
- \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i} w \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i} (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w).
+ \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 ? a i\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{i\ 5a title="in_pl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w).
#S #a #i elim i
[normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
- |normalize #x #w cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x)) #H >H normalize
+ |normalize #x #w cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x)) #H >H normalize
[>(\P H) % [* // #bot @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 //| #H1 destruct /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
|% [@\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 |#H1 cases (\Pf H) #H2 @H2 destruct //]
]
notation > "x ↦* E" non associative with precedence 60 for @{moves ? $x $E}.
-let rec moves (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) w e on w : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝
+\ 5img class="anchor" src="icons/tick.png" id="moves"\ 6let rec moves (S : \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) w e on w : \ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S ≝
match w with
[ nil ⇒ e
| cons x w' ⇒ w' ↦* (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S x (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e))].
-lemma moves_empty: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ] e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.
+\ 5img class="anchor" src="icons/tick.png" id="moves_empty"\ 6lemma moves_empty: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6 e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.
// qed.
-lemma moves_cons: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀w.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w) e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e)).
+\ 5img class="anchor" src="icons/tick.png" id="moves_cons"\ 6lemma moves_cons: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀w.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? (a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w) e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w (\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e)).
// qed.
-lemma moves_left : ∀S,a,w,e.
- \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S (w\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6])) e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e)).
+\ 5img class="anchor" src="icons/tick.png" id="moves_left"\ 6lemma moves_left : ∀S,a,w,e.
+ \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S (w\ 5a title="append" href="cic:/fakeuri.def(1)"\ 6@\ 5/a\ 6(a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6)) e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e)).
#S #a #w elim w // #x #tl #Hind #e >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 //
qed.
-lemma not_epsilon_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. ∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 ((a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w) \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e) ((a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:w) \ 5a title="in_pl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).
+\ 5img class="anchor" src="icons/tick.png" id="not_epsilon_sem"\ 6lemma not_epsilon_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a:S.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. ∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a href="cic:/matita/basics/logic/iff.def(1)"\ 6iff\ 5/a\ 6 ((a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w) \ 5a title="in_prl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 e) ((a\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6w) \ 5a title="in_pl mem" href="cic:/fakeuri.def(1)"\ 6∈\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).
#S #a #w * #i #b cases b normalize
[% /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ * // #H destruct |% normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/]
qed.
-lemma same_kernel_moves: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀w.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e)| \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|.
+\ 5img class="anchor" src="icons/tick.png" id="same_kernel_moves"\ 6lemma same_kernel_moves: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀w.∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e)\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6.
#S #w elim w //
qed.
-theorem decidable_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. ∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
- (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e} w.
+\ 5img class="anchor" src="icons/tick.png" id="decidable_sem"\ 6theorem decidable_sem: ∀S:\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀w: \ 5a href="cic:/matita/tutorial/chapter6/word.def(3)"\ 6word\ 5/a\ 6 S. ∀e:\ 5a href="cic:/matita/tutorial/chapter7/pre.def(1)"\ 6pre\ 5/a\ 6 S.
+ (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 ? w e) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 \ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6\sem\ 5/a\ 6{e\ 5a title="in_prl" href="cic:/fakeuri.def(1)"\ 6}\ 5/a\ 6 w.
#S #w elim w
[* #i #b >\ 5a href="cic:/matita/tutorial/chapter9/moves_empty.def(8)"\ 6moves_empty\ 5/a\ 6 cases b % /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/tutorial/chapter7/true_to_epsilon.def(9)"\ 6true_to_epsilon\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/\ 5span class="error" title="error location"\ 6\ 5/span\ 6 #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|#a #w1 #Hind #e >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6
qed.
(* It is now clear that we can build a DFA D_e for e by taking pre as states,
-and move as transition function; the initial state is •(e) and a state 〈i,b〉 is
+and move as transition function; the initial state is •(e) and a state 〈i,b〉 is
final if and only if b is true. The fact that states in D_e are finite is obvious:
in fact, their cardinality is at most 2^{n+1} where n is the number of symbols in
e. This is one of the advantages of pointed regular expressions w.r.t. derivatives,
The graphical description of the automaton is the traditional one, with nodes for
states and labelled arcs for transitions. Unreachable states are not shown.
-Final states are emphasized by a double circle: since a state 〈e,b〉 is final if and
+Final states are emphasized by a double circle: since a state 〈e,b〉 is final if and
only if b is true, we may just label nodes with the item.
The automaton is not minimal: it is easy to see that the two states corresponding to
the items (a•c +bc)* and (ac+b•c)* are equivalent (a way to prove it is to observe
We conclude this chapter with a few properties of the move opertions in relation
with the pit state. *)
-definition pit_pre ≝ λS.λi.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|), \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6〉.
+\ 5img class="anchor" src="icons/tick.png" id="pit_pre"\ 6definition pit_pre ≝ λS.λi.\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6\ 5a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"\ 6blank\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6), \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
(* The following function compute the list of characters occurring in a given
item i. *)
-let rec occur (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (i: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i ≝
+\ 5img class="anchor" src="icons/tick.png" id="occur"\ 6let rec occur (S: \ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6) (i: \ 5a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"\ 6re\ 5/a\ 6 S) on i ≝
match i with
- [ z ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ]
- | e ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 ]
- | s y ⇒ y\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6:\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6]
+ [ z ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
+ | e ⇒ \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6 \ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
+ | s y ⇒ y\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="cons" href="cic:/fakeuri.def(1)"\ 6:\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6[\ 5/a\ 6\ 5a title="nil" href="cic:/fakeuri.def(1)"\ 6]\ 5/a\ 6
| o e1 e2 ⇒ \ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (occur S e1) (occur S e2)
| c e1 e2 ⇒ \ 5a href="cic:/matita/tutorial/chapter5/unique_append.fix(0,1,5)"\ 6unique_append\ 5/a\ 6 ? (occur S e1) (occur S e2)
| k e ⇒ occur S e].
(* If a symbol a does not occur in i, then move(i,a) gets to the
pit state. *)
-lemma not_occur_to_pit: ∀S,a.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i|)) \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
+\ 5img class="anchor" src="icons/tick.png" id="not_occur_to_pit"\ 6lemma not_occur_to_pit: ∀S,a.∀i:\ 5a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"\ 6pitem\ 5/a\ 6 S. \ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6i\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)) \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 →
\ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a i \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i.
#S #a #i elim i //
- [#x normalize cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=x) normalize // #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+ [#x normalize cases (a\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6x) normalize // #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
|#i1 #i2 #Hind1 #Hind2 #H >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6
>Hind1 [2:@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l1.def(6)"\ 6sublist_unique_append_l1\ 5/a\ 6 //]
>Hind2 [2:@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 @\ 5a href="cic:/matita/tutorial/chapter5/sublist_unique_append_l2.def(6)"\ 6sublist_unique_append_l2\ 5/a\ 6 //] //
(* We cannot escape form the pit state. *)
-lemma move_pit: ∀S,a,i. \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i.
+\ 5img class="anchor" src="icons/tick.png" id="move_pit"\ 6lemma move_pit: ∀S,a,i. \ 5a href="cic:/matita/tutorial/chapter9/move.fix(0,2,6)"\ 6move\ 5/a\ 6 S a (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i)) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i.
#S #a #i elim i //
[#i1 #i2 #Hind1 #Hind2 >\ 5a href="cic:/matita/tutorial/chapter9/move_cat.def(7)"\ 6move_cat\ 5/a\ 6 >Hind1 >Hind2 //
|#i1 #i2 #Hind1 #Hind2 >\ 5a href="cic:/matita/tutorial/chapter9/move_plus.def(7)"\ 6move_plus\ 5/a\ 6 >Hind1 >Hind2 //
]
qed.
-lemma moves_pit: ∀S,w,i. \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i.
+\ 5img class="anchor" src="icons/tick.png" id="moves_pit"\ 6lemma moves_pit: ∀S,w,i. \ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S i.
#S #w #i elim w //
qed.
(* If any character in w does not occur in i, then moves(i,w) gets
to the pit state. *)
-lemma to_pit: ∀S,w,e. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|)) →
+\ 5img class="anchor" src="icons/tick.png" id="to_pit"\ 6lemma to_pit: ∀S,w,e. \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter5/sublist.def(5)"\ 6sublist\ 5/a\ 6 S w (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6)) →
\ 5a href="cic:/matita/tutorial/chapter9/moves.fix(0,1,7)"\ 6moves\ 5/a\ 6 S w e \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/tutorial/chapter9/pit_pre.def(4)"\ 6pit_pre\ 5/a\ 6 S (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e).
#S #w elim w
[#e * #H @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @H normalize #a #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
- |#a #tl #Hind #e #H cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e|))))
+ |#a #tl #Hind #e #H cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/tutorial/chapter5/memb.fix(0,2,4)"\ 6memb\ 5/a\ 6 S a (\ 5a href="cic:/matita/tutorial/chapter9/occur.fix(0,1,6)"\ 6occur\ 5/a\ 6 S (\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 e\ 5a title="forget" href="cic:/fakeuri.def(1)"\ 6|\ 5/a\ 6))))
[#Htrue >\ 5a href="cic:/matita/tutorial/chapter9/moves_cons.def(8)"\ 6moves_cons\ 5/a\ 6 whd in ⊢ (???%); <(\ 5a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"\ 6same_kernel\ 5/a\ 6 … a)
@Hind >\ 5a href="cic:/matita/tutorial/chapter9/same_kernel.def(8)"\ 6same_kernel\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … H) #H1 #b #memb cases (\ 5a href="cic:/matita/basics/bool/orb_true_l.def(2)"\ 6orb_true_l\ 5/a\ 6 … memb)
[#H2 >(\P H2) // |#H2 @H1 //]
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