matita/matita/contribs/lambdadelta/ground/lib/exteq.ma

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  14
  15 include "ground/notation/relations/circled_eq_4.ma".
  16 include "ground/lib/relations.ma".
  17
  18 (* EXTENSIONAL EQUIVALENCE FOR FUNCTIONS ************************************)
  19
  20 definition exteq (A,B:Type[0]): relation (A → B) ≝
  21            λf1,f2. ∀a. f1 a = f2 a.
  22
  23 interpretation
  24   "extensional equivalence"
  25   'CircledEq A B f1 f2 = (exteq A B f1 f2).
  26
  27 (* Basic constructions ******************************************************)
  28
  29 lemma exteq_refl (A) (B):
  30       reflexive … (exteq A B).
  31 // qed.
  32
  33 lemma exteq_repl (A) (B):
  34       replace_2 … (exteq A B) (exteq A B) (exteq A B).
  35 // qed-.
  36
  37 lemma exteq_sym (A) (B):
  38       symmetric … (exteq A B).
  39 /2 width=1 by exteq_repl/ qed-.
  40
  41 lemma exteq_trans (A) (B):
  42       Transitive … (exteq A B).
  43 /2 width=1 by exteq_repl/ qed-.
  44
  45 lemma exteq_canc_sn (A) (B):
  46       left_cancellable … (exteq A B).
  47 /2 width=1 by exteq_repl/ qed-.
  48
  49 lemma exteq_canc_dx (A) (B):
  50       right_cancellable … (exteq A B).
  51 /2 width=1 by exteq_repl/ qed-.
  52
  53 (* Constructions with compose ***********************************************)
  54
  55 lemma compose_repl_fwd_dx (A) (B) (C) (g) (f1) (f2):
  56       f1 ⊜{A,B} f2 → g ∘ f1 ⊜{A,C} g ∘ f2.
  57 #A #B #C #g #f1 #f2 #Hf12 #a
  58 whd in ⊢ (??%%); //
  59 qed.
  60
  61 lemma compose_repl_bak_dx (A) (B) (C) (g) (f1) (f2):
  62       f1 ⊜{A,B} f2 → g ∘ f2 ⊜{A,C} g ∘ f1.
  63 /3 width=1 by compose_repl_fwd_dx, exteq_sym/
  64 qed.
  65
  66 lemma compose_repl_fwd_sn (A) (B) (C) (g1) (g2) (f):
  67       g1 ⊜{B,C} g2 → g1 ∘ f ⊜{A,C} g2 ∘ f.
  68 #A #B #C #g1 #g2 #f #Hg12 #a
  69 whd in ⊢ (??%%); //
  70 qed.
  71
  72 lemma compose_repl_bak_sn (A) (B) (C) (g1) (g2) (f):
  73       g1 ⊜{B,C} g2 → g2 ∘ f ⊜{A,C} g1 ∘ f.
  74 /3 width=1 by compose_repl_fwd_sn, exteq_sym/
  75 qed.
  76
  77 lemma compose_assoc (A1) (A2) (A3) (A4) (f3:A3→A4) (f2:A2→A3) (f1:A1→A2):
  78       f3 ∘ (f2 ∘ f1) ⊜ f3 ∘ f2 ∘ f1.
  79 // qed.