matita/tests/demodulation_coq.ma

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  14
  15
  16
  17 include "coq.ma".
  18
  19 alias num = "Coq natural number".
  20 alias symbol "times" = "Coq's natural times".
  21 alias symbol "plus" = "Coq's natural plus".
  22 alias symbol "eq" = "Coq's leibnitz's equality".
  23 alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
  24 alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
  25 alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
  26
  27 theorem p0 : \forall m:nat. m+O = m.
  28 intro. demodulate.reflexivity.
  29 qed.
  30
  31 theorem p: \forall m.1*m = m.
  32 intros.demodulate.reflexivity.
  33 qed.
  34
  35 theorem p2: \forall x,y:nat.(S x)*y = (y+x*y).
  36 intros.demodulate.reflexivity.
  37 qed.
  38
  39 theorem p1: \forall x,y:nat.(S ((S x)*y+x))=(S x)+(y*x+y).
  40 intros.demodulate.reflexivity.
  41 qed.
  42
  43 theorem p3: \forall x,y:nat. (x+y)*(x+y) = x*x + 2*(x*y) + (y*y).
  44 intros.demodulate.reflexivity.
  45 qed.
  46
  47 theorem p4: \forall x:nat. (x+1)*(x-1)=x*x - 1.
  48 intro.
  49 apply (nat_case x)
  50 [simplify.reflexivity
  51 |intro.demodulate.reflexivity]
  52 qed.
  53