From: Claudio Sacerdoti Coen <claudio.sacerdoticoen@unibo.it>
Date: Mon, 26 Nov 2007 12:42:51 +0000 (+0000)
Subject: Notation improved.
X-Git-Tag: make_still_working~5777
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=87bfe7fa6c7b906646a0094180e69c0b0373ce10;p=helm.git

Notation improved.
---

diff --git a/helm/software/matita/dama_didactic/deriv.ma b/helm/software/matita/dama_didactic/deriv.ma
index f4de77580..201471c45 100644
--- a/helm/software/matita/dama_didactic/deriv.ma
+++ b/helm/software/matita/dama_didactic/deriv.ma
@@ -32,16 +32,16 @@ interpretation "function mult" 'mult x y =
 interpretation "function compositon" 'compose x y =
  (cic:/matita/didactic/deriv/fcomp.con x y).
 
-notation "hvbox(a break ⊕ b)"
+notation "hvbox(a break + b)"
   right associative with precedence 45
 for @{ 'oplus $a $b }.
 
-interpretation "function plus" 'oplus x y =
+interpretation "function plus" 'plus x y =
  (cic:/matita/didactic/deriv/fplus.con x y).
 
 axiom i:R→F. (*mappatura  R in F*)
 coercion cic:/matita/didactic/deriv/i.con.
-axiom i_comm_plus: ∀x,y:R. (i (x+y)) = (i x) ⊕ (i y).
+axiom i_comm_plus: ∀x,y:R. (i (x+y)) = (i x) + (i y).
 axiom i_comm_mult: ∀x,y:R. (i (Rmult x y)) = (i x) · (i y).
 
 axiom freflex:F.
@@ -77,22 +77,22 @@ let rec felev (f:F) (n:nat) on n: F ≝
 
 (* Proprietà *)
 
-axiom fplus_commutative: ∀ f,g:F. f ⊕ g = g ⊕ f.
-axiom fplus_associative: ∀ f,g,h:F. f ⊕ (g ⊕ h) = (f ⊕ g) ⊕ h.
-axiom fplus_neutral: ∀f:F. (i R0) ⊕ f = f.
+axiom fplus_commutative: ∀ f,g:F. f + g = g + f.
+axiom fplus_associative: ∀ f,g,h:F. f + (g + h) = (f + g) + h.
+axiom fplus_neutral: ∀f:F. (i R0) + f = f.
 axiom fmult_commutative: ∀ f,g:F. f · g = g · f.
 axiom fmult_associative: ∀ f,g,h:F. f · (g · h) = (f · g) · h.
 axiom fmult_neutral: ∀f:F. (i R1) · f = f.
 axiom fmult_assorb: ∀f:F. (i R0) · f = (i R0).
-axiom fdistr: ∀ f,g,h:F. (f ⊕ g) · h = (f · h) ⊕ (g · h).
+axiom fdistr: ∀ f,g,h:F. (f + g) · h = (f · h) + (g · h).
 axiom fcomp_associative: ∀ f,g,h:F. f ∘ (g ∘ h) = (f ∘ g) ∘ h.
 
-axiom fcomp_distr1: ∀ f,g,h:F. (f ⊕ g) ∘ h = (f ∘ h) ⊕ (g ∘ h).
+axiom fcomp_distr1: ∀ f,g,h:F. (f + g) ∘ h = (f ∘ h) + (g ∘ h).
 axiom fcomp_distr2: ∀ f,g,h:F. (f · g) ∘ h = (f ∘ h) · (g ∘ h).
 
-axiom demult: ∀ f,g:F. (f · g) ' = (f ' · g) ⊕ (f · g ').
+axiom demult: ∀ f,g:F. (f · g) ' = (f ' · g) + (f · g ').
 axiom decomp: ∀ f,g:F. (f ∘ g) ' = (f ' ∘ g) · g '.
-axiom deplus: ∀ f,g:F. (f ⊕ g) ' = (f ') ⊕ (g ').
+axiom deplus: ∀ f,g:F. (f + g) ' = (f ') + (g ').
 
 axiom cost_assorb: ∀x:R. ∀f:F. (i x) ∘ f = i x.
 axiom cost_deriv: ∀x:R. (i x) ' = i R0.
diff --git a/helm/software/matita/dama_didactic/ex_deriv.ma b/helm/software/matita/dama_didactic/ex_deriv.ma
index 32e1ee8a5..59e881592 100644
--- a/helm/software/matita/dama_didactic/ex_deriv.ma
+++ b/helm/software/matita/dama_didactic/ex_deriv.ma
@@ -16,18 +16,18 @@ set "baseuri" "cic:/matita/didactic/ex_deriv".
 
 include "deriv.ma".
 
-theorem p_plus_p_p: ∀f:F. ∀g:F. (fpari f ∧ fpari g) → fpari (f ⊕ g).
+theorem p_plus_p_p: ∀f:F. ∀g:F. (fpari f ∧ fpari g) → fpari (f + g).
  assume f:F.
  assume g:F.
  suppose (fpari f ∧ fpari g) (h).
  by h we have (fpari f) (H) and (fpari g) (K).
- we need to prove (fpari (f ⊕ g))
- or equivalently ((f ⊕ g) = (f ⊕ g) ∘ ρ).
+ we need to prove (fpari (f + g))
+ or equivalently ((f + g) = (f + g) ∘ ρ).
  conclude
-    (f ⊕ g)
-  = (f ⊕ (g ∘ ρ)) by _.
-  = ((f ∘ ρ) ⊕ (g ∘ ρ)) by _.
-  = ((f ⊕ g) ∘ ρ) by _
+    (f + g)
+  = (f + (g ∘ ρ)) by _.
+  = ((f ∘ ρ) + (g ∘ ρ)) by _.
+  = ((f + g) ∘ ρ) by _
  done.
 qed.
 
@@ -46,23 +46,23 @@ theorem p_mult_p_p: ∀f:F. ∀g:F. (fpari f ∧ fpari g) → fpari (f · g).
  done.
 qed.
 
-theorem d_plus_d_d: ∀f:F. ∀g:F. (fdispari f ∧ fdispari g) → fdispari (f ⊕ g).
+theorem d_plus_d_d: ∀f:F. ∀g:F. (fdispari f ∧ fdispari g) → fdispari (f + g).
  assume f:F.
  assume g:F.
  suppose (fdispari f ∧ fdispari g) (h).
  by h we have (fdispari f) (H) and (fdispari g) (K).
- we need to prove (fdispari (f ⊕ g))
- or equivalently ((f ⊕ g) = (ρ ∘ ((f ⊕ g) ∘ ρ))).
+ we need to prove (fdispari (f + g))
+ or equivalently ((f + g) = (ρ ∘ ((f + g) ∘ ρ))).
  conclude
-    (f ⊕ g)
-  = (f ⊕ (ρ ∘ (g ∘ ρ))) by _.
-  = ((ρ ∘ (f ∘ ρ)) ⊕ (ρ ∘ (g ∘ ρ))) by _.
-  = (((-R1) · (f ∘ ρ)) ⊕ (ρ ∘ (g ∘ ρ))) by _.
-  = (((i (-R1)) · (f ∘ ρ)) ⊕ ((i (-R1)) · (g ∘ ρ))) by _.
-  = (((f ∘ ρ) · (i (-R1))) ⊕ ((g ∘ ρ) · (i (-R1)))) by _.
-  = (((f ∘ ρ) ⊕ (g ∘ ρ)) · (i (-R1))) by _.
-  = ((i (-R1)) · ((f ⊕ g) ∘ ρ)) by _.
-  = (ρ ∘ ((f ⊕ g) ∘ ρ)) by _
+    (f + g)
+  = (f + (ρ ∘ (g ∘ ρ))) by _.
+  = ((ρ ∘ (f ∘ ρ)) + (ρ ∘ (g ∘ ρ))) by _.
+  = (((-R1) · (f ∘ ρ)) + (ρ ∘ (g ∘ ρ))) by _.
+  = (((i (-R1)) · (f ∘ ρ)) + ((i (-R1)) · (g ∘ ρ))) by _.
+  = (((f ∘ ρ) · (i (-R1))) + ((g ∘ ρ) · (i (-R1)))) by _.
+  = (((f ∘ ρ) + (g ∘ ρ)) · (i (-R1))) by _.
+  = ((i (-R1)) · ((f + g) ∘ ρ)) by _.
+  = (ρ ∘ ((f + g) ∘ ρ)) by _
  done.
 qed.
 
@@ -106,12 +106,12 @@ theorem p_mult_d_p: ∀f:F. ∀g:F. (fpari f ∧ fdispari g) → fdispari (f ·
  done.
 qed.
 
-theorem p_plus_c_p: ∀f:F. ∀x:R. fpari f → fpari (f ⊕ (i x)).
+theorem p_plus_c_p: ∀f:F. ∀x:R. fpari f → fpari (f + (i x)).
  assume f:F.
  assume x:R.
  suppose (fpari f) (h).
- we need to prove (fpari (f ⊕ (i x)))
- or equivalently (f ⊕ (i x) = (f ⊕ (i x)) ∘ ρ).
+ we need to prove (fpari (f + (i x)))
+ or equivalently (f + (i x) = (f + (i x)) ∘ ρ).
  by _ done.
 qed.
 
@@ -183,54 +183,51 @@ theorem dispari_in_pari: ∀ f:F. fdispari f → fpari f '.
  done.
 qed.
 
-(*
-alias id "plus" = "cic:/matita/nat/plus/plus.con".
+axiom tech1: ∀n,m. F_OF_nat n + F_OF_nat m = F_OF_nat (n + m).
+
+alias symbol "plus" = "natural plus".
+alias num (instance 0) = "natural number".
 theorem de_prodotto_funzioni:
- ∀ n:nat. (id ^ (plus n (S O))) ' = (i (n + (S O))) · (id ^ n).
+ ∀ n:nat. (id ^ (n + 1)) ' = ((n + 1)) · (id ^ n).
  assume n:nat.
  we proceed by induction on n to prove
-   ((id ^ (plus n (S O))) ' = (i (n + (S O))) · (id ^ n)).
+   ((id ^ (n + 1)) ' = (i (n + 1)) · (id ^ n)).
  case O.
-  we need to prove ((id ^ (plus O (S O))) ' = (i (S O)) · (id ^ O)).
+  we need to prove ((id ^ (0 + 1)) ' = (i 1) · (id ^ 0)).
   conclude
-     ((id ^ (plus O (S O))) ')
-   = ((id ^ (S O)) ') by _.
-   = ((id · (id ^ O)) ') by _.
-   = ((id · (i R1)) ') by _.
+     ((id ^ (0 + 1)) ')
+   = ((id ^ 1) ') by _.
+   = ((id · (id ^ 0)) ') by _.
+   = ((id · R1) ') by _.
    = (id ') by _.
    = (i R1) by _.
-   = (i R1 · i R1) by _.
+   = (i R1 · R1) by _.
    = (i (R0 + R1) · R1) by _.
-   = ((i (S O)) · (id ^ O)) by _
+   = (1 · (id ^ 0)) by _
  done.
  case S (n:nat).
   by induction hypothesis we know
-     ((id ^ (plus n (S O))) ' = (i (n + (S O))) · (id ^ n)) (H).
+     ((id ^ (n + 1)) ' = ((n + 1)) · (id ^ n)) (H).
   we need to prove
-     ((id ^ (plus (plus n (S O)) (S O))) '
-   = (i (plus (plus n (S O)) (S O))) · (id ^ (plus n (S O)))).
+     ((id ^ ((n + 1)+1)) '
+   = (i ((n + 1)+1)) · (id ^ (n+1))).
   conclude
-     ((id ^ ((n + (S O)) + (S O))) ')
-   = ((id ^ ((n + (S (S O))))) ') by _.
-   = ((id ^ (S (n + S O))) ') by _.
-   = ((id · (id ^ (n + (S O)))) ') by _.
-   = ((id ' · (id ^ (n + (S O)))) ⊕ (id · (id ^ (n + (S O))) ')) by _.
-   alias symbol "plus" (instance 4) = "natural plus".
-   = ((R1 · (id ^ (n + (S O)))) ⊕ (id · ((i (n + S O)) · (id ^ n)))) by _.
-   = ((R1 · (id ^ (n + (S O)))) ⊕ (((i (n + S O)) · id · (id ^ n)))) by _.
-   alias symbol "plus" (instance 8) = "natural plus".
-   = ((R1 · (id ^ (n + (S O)))) ⊕ ((i (n + S O)) · (id ^ (n + (S O))))) by _.
-   = ((i R1 · (id ^ (n + (S O)))) ⊕ ((i (n + S O)) · (id ^ (n + (S O))))) by _.
-   alias symbol "plus" = "natural plus".
- cut (((i R1 · (id ^ (n + (S O)))) ⊕ ((i (n + S O)) · (id ^ (n + (S O))))) =
-       (((i R1 ⊕ (i (plus n (S O)))) · (id ^ (n + (S O))))));[| by _ done;]
- unfold F_OF_nat in Hcut;
- rewrite > Hcut;
-   =   (((i R1 ⊕ (i (plus n (S O)))) · (id ^ (n + (S O))))) by _.
-   = ((i (n + S O + S O)) · (id ^ (n + S O))) by _
+     ((id ^ ((n + 1) + 1)) ')
+   = ((id ^ ((n + (S 1)))) ') by _.
+   = ((id ^ (S (n + 1))) ') by _.
+   = ((id · (id ^ (n + 1))) ') by _.
+   = ((id ' · (id ^ (n + 1))) + (id · (id ^ (n + 1)) ')) by _.
+   alias symbol "plus" (instance 1) = "function plus".
+   = ((R1 · (id ^ (n + 1))) + (id · (((n + 1)) · (id ^ n)))) by _.
+   = ((R1 · (id ^ (n + 1))) + (((n + 1) · id · (id ^ n)))) by _.
+   = ((R1 · (id ^ (n + 1))) + ((n + 1) · (id ^ (1 + n)))) by _.
+   = ((R1 · (id ^ (n + 1))) + ((n + 1) · (id ^ (n + 1)))) by _.
+   alias symbol "plus" (instance 2) = "function plus".
+   = (((R1 + (n + 1))) · (id ^ (n + 1))) by _.
+   = ((1 + (n + 1)) · (id ^ (n + 1))) by _;
+   = ((n + 1 + 1) · (id ^ (n + 1))) by _
   done.
 qed.
-*)
 
 let rec times (n:nat) (x:R) on n: R ≝
  match n with
@@ -239,8 +236,8 @@ let rec times (n:nat) (x:R) on n: R ≝
  ].
 
 axiom invS: nat→R.
-axiom invS1: ∀n:nat. Rmult (invS n) (real_of_nat (n + S O)) = R1.
-axiom invS2: invS (S O) + invS (S O) = R1. (*forse*)
+axiom invS1: ∀n:nat. Rmult (invS n) (real_of_nat (n + 1)) = R1.
+axiom invS2: invS 1 + invS 1 = R1. (*forse*)
 
 axiom e:F.
 axiom deriv_e: e ' = e.
@@ -248,5 +245,5 @@ axiom e1: e · (e ∘ ρ) = R1.
 
 (*
 theorem decosh_senh:
-   (invS (S O) · (e ⊕ (e ∘ ρ)))' = (invS (S O) · (e ⊕ (ρ ∘ (e ∘ ρ)))).
-*)
\ No newline at end of file
+   (invS 1 · (e + (e ∘ ρ)))' = (invS 1 · (e + (ρ ∘ (e ∘ ρ)))).
+*)