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Diffstat (limited to 'test-suite/bugs/closed/shouldsucceed/1507.v')
| -rw-r--r-- | test-suite/bugs/closed/shouldsucceed/1507.v | 121 |
1 files changed, 121 insertions, 0 deletions
diff --git a/test-suite/bugs/closed/shouldsucceed/1507.v b/test-suite/bugs/closed/shouldsucceed/1507.v new file mode 100644 index 00000000..b484c7dc --- /dev/null +++ b/test-suite/bugs/closed/shouldsucceed/1507.v @@ -0,0 +1,121 @@ +(* + Implementing reals a la Stolzenberg + + Danko Ilik, March 2007 + svn revision: $Id: 1507.v 10068 2007-08-10 12:06:59Z notin $ + + XField.v -- (unfinished) axiomatisation of the theories of real and + rational intervals. +*) + +Definition associative (A:Type)(op:A->A->A) := + forall x y z:A, op (op x y) z = op x (op y z). + +Definition commutative (A:Type)(op:A->A->A) := + forall x y:A, op x y = op y x. + +Definition trichotomous (A:Type)(R:A->A->Prop) := + forall x y:A, R x y \/ x=y \/ R y x. + +Definition relation (A:Type) := A -> A -> Prop. +Definition reflexive (A:Type)(R:relation A) := forall x:A, R x x. +Definition transitive (A:Type)(R:relation A) := + forall x y z:A, R x y -> R y z -> R x z. +Definition symmetric (A:Type)(R:relation A) := forall x y:A, R x y -> R y x. + +Record interval (X:Set)(le:X->X->Prop) : Set := + interval_make { + interval_left : X; + interval_right : X; + interval_nonempty : le interval_left interval_right + }. + +Record I (grnd:Set)(le:grnd->grnd->Prop) : Type := Imake { + Icar := interval grnd le; + Iplus : Icar -> Icar -> Icar; + Imult : Icar -> Icar -> Icar; + Izero : Icar; + Ione : Icar; + Iopp : Icar -> Icar; + Iinv : Icar -> Icar; + Ic : Icar -> Icar -> Prop; (* consistency *) + (* monoids *) + Iplus_assoc : associative Icar Iplus; + Imult_assoc : associative Icar Imult; + (* abelian groups *) + Iplus_comm : commutative Icar Iplus; + Imult_comm : commutative Icar Imult; + Iplus_0_l : forall x:Icar, Ic (Iplus Izero x) x; + Iplus_0_r : forall x:Icar, Ic (Iplus x Izero) x; + Imult_0_l : forall x:Icar, Ic (Imult Ione x) x; + Imult_0_r : forall x:Icar, Ic (Imult x Ione) x; + Iplus_opp_r : forall x:Icar, Ic (Iplus x (Iopp x)) (Izero); + Imult_inv_r : forall x:Icar, ~(Ic x Izero) -> Ic (Imult x (Iinv x)) Ione; + (* distributive laws *) + Imult_plus_distr_l : forall x x' y y' z z' z'', + Ic x x' -> Ic y y' -> Ic z z' -> Ic z z'' -> + Ic (Imult (Iplus x y) z) (Iplus (Imult x' z') (Imult y' z'')); + (* order and lattice structure *) + Ilt : Icar -> Icar -> Prop; + Ilc := fun (x y:Icar) => Ilt x y \/ Ic x y; + Isup : Icar -> Icar -> Icar; + Iinf : Icar -> Icar -> Icar; + Ilt_trans : transitive _ lt; + Ilt_trich : forall x y:Icar, Ilt x y \/ Ic x y \/ Ilt y x; + Isup_lub : forall x y z:Icar, Ilc x z -> Ilc y z -> Ilc (Isup x y) z; + Iinf_glb : forall x y z:Icar, Ilc x y -> Ilc x z -> Ilc x (Iinf y z); + (* order preserves operations? *) + (* properties of Ic *) + Ic_refl : reflexive _ Ic; + Ic_sym : symmetric _ Ic +}. + +Definition interval_set (X:Set)(le:X->X->Prop) := + (interval X le) -> Prop. (* can be Set as well *) +Check interval_set. +Check Ic. +Definition consistent (X:Set)(le:X->X->Prop)(TI:I X le)(p:interval_set X le) := + forall I J:interval X le, p I -> p J -> (Ic X le TI) I J. +Check consistent. +(* define 'fine' *) + +Record N (grnd:Set)(le:grnd->grnd->Prop)(grndI:I grnd le) : Type := Nmake { + Ncar := interval_set grnd le; + Nplus : Ncar -> Ncar -> Ncar; + Nmult : Ncar -> Ncar -> Ncar; + Nzero : Ncar; + None : Ncar; + Nopp : Ncar -> Ncar; + Ninv : Ncar -> Ncar; + Nc : Ncar -> Ncar -> Prop; (* Ncistency *) + (* monoids *) + Nplus_assoc : associative Ncar Nplus; + Nmult_assoc : associative Ncar Nmult; + (* abelian groups *) + Nplus_comm : commutative Ncar Nplus; + Nmult_comm : commutative Ncar Nmult; + Nplus_0_l : forall x:Ncar, Nc (Nplus Nzero x) x; + Nplus_0_r : forall x:Ncar, Nc (Nplus x Nzero) x; + Nmult_0_l : forall x:Ncar, Nc (Nmult None x) x; + Nmult_0_r : forall x:Ncar, Nc (Nmult x None) x; + Nplus_opp_r : forall x:Ncar, Nc (Nplus x (Nopp x)) (Nzero); + Nmult_inv_r : forall x:Ncar, ~(Nc x Nzero) -> Nc (Nmult x (Ninv x)) None; + (* distributive laws *) + Nmult_plus_distr_l : forall x x' y y' z z' z'', + Nc x x' -> Nc y y' -> Nc z z' -> Nc z z'' -> + Nc (Nmult (Nplus x y) z) (Nplus (Nmult x' z') (Nmult y' z'')); + (* order and lattice structure *) + Nlt : Ncar -> Ncar -> Prop; + Nlc := fun (x y:Ncar) => Nlt x y \/ Nc x y; + Nsup : Ncar -> Ncar -> Ncar; + Ninf : Ncar -> Ncar -> Ncar; + Nlt_trans : transitive _ lt; + Nlt_trich : forall x y:Ncar, Nlt x y \/ Nc x y \/ Nlt y x; + Nsup_lub : forall x y z:Ncar, Nlc x z -> Nlc y z -> Nlc (Nsup x y) z; + Ninf_glb : forall x y z:Ncar, Nlc x y -> Nlc x z -> Nlc x (Ninf y z); + (* order preserves operations? *) + (* properties of Nc *) + Nc_refl : reflexive _ Nc; + Nc_sym : symmetric _ Nc +}. + |