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David masters numerics #9

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677c243
Move numeric R injection into its own class
david-masters Jun 18, 2020
690df90
Made numerics use lt_bool from dyadics and made dyadic lt_bool much f…
david-masters Jun 18, 2020
cfafb2b
Merge branch 'master' of https://github.com/OUPL/OUVerT into david-ma…
david-masters Jun 18, 2020
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32 changes: 28 additions & 4 deletions banach.v
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Original file line number Diff line number Diff line change
Expand Up @@ -753,7 +753,31 @@ Module banach.
apply converge_func_correct.
Qed.


Lemma dist_converge_func_step_r : forall (f : T->R), dist f (step_f converge_func) = dist f converge_func.
Proof.
intros.
unfold max_dist.
apply mapmax_ne_ext. intros.
rewrite rec_fixpoint. auto.
Qed.

Lemma dist_fixpoint_step_ub: forall (f : T->R),
((1+ - q) * dist f converge_func <= dist f (step_f f)).
Proof.
intros.
rewrite plus_mult_distr_r.
enough(dist f converge_func <= dist f (step_f f) + q * dist f converge_func).
{
replace(_ <= _) with (1 * dist f converge_func + - q * dist f converge_func <= dist f (step_f f))%R; auto.
replace(_ <= _) with (dist f converge_func <= dist f (step_f f) + q * dist f converge_func)%R in H; auto.
lra.
}
apply le_trans with (dist f (step_f f) + dist (step_f f) converge_func).
apply dist_triangle. apply Numeric_Props_R.
apply plus_le_compat_l.
rewrite <- dist_converge_func_step_r.
apply is_contr.
Qed.

End banach_R.

Expand All @@ -780,9 +804,9 @@ Module banach.
apply H1. auto.
Qed.




Record R_Nt_relation {Nt : Type} `{Numerics.Numeric Nt} : Type :=
Record R_Nt_relation {Nt : Type} `{numeric_t : Numerics.Numeric Nt} `{@Numerics.Numeric_R_inj Nt numeric_t} : Type :=
R_Nt_relation_mk {
relation_contr : @contraction_func R _;
Nt_step : ((x_t relation_contr) -> Nt) -> ((x_t relation_contr) -> Nt);
Expand All @@ -791,7 +815,7 @@ Module banach.
}.

Section banach_Nt_R.
Context {Nt:Type} `{Numerics.Numeric Nt}.
Context {Nt:Type} `{Numerics.Numeric Nt} `{Nt_R_inj : @Numerics.Numeric_R_inj Nt H}.
Variable relation : R_Nt_relation.


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