statement stringlengths 1 3.35k | proof stringlengths 0 26.9k | type stringclasses 16
values | symbolic_name stringlengths 1 89 | library stringclasses 189
values | filename stringlengths 20 105 | imports listlengths 1 72 | deps listlengths 0 64 | docstring stringlengths 0 3.07k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
Dcuts_half_bot : Dcuts_def_bot Dcuts_half_val. | Proof.
intros r Hr q Hq.
apply X_bot with (1 := Hr).
eapply istrans_leNonnegativeRationals, plusNonnegativeRationals_lecompat_l, Hq.
now apply plusNonnegativeRationals_lecompat_r, Hq.
Qed. | Lemma | Dcuts_half_bot | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_def_bot",
"Dcuts_half_val",
"istrans_leNonnegativeRationals",
"plusNonnegativeRationals_lecompat_l",
"plusNonnegativeRationals_lecompat_r"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_half_open : Dcuts_def_open Dcuts_half_val. | Proof.
intros r Hr.
generalize (X_open _ Hr).
apply hinhfun ; intros q.
exists (pr1 q / 2)%NRat ; split.
- unfold Dcuts_half_val.
rewrite <- NQhalf_double.
exact (pr1 (pr2 q)).
- apply_pr2 (multNonnegativeRationals_ltcompat_l 2%NRat).
+ exact ispositive_twoNonnegativeRationals.
+ pattern r a... | Lemma | Dcuts_half_open | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_def_open",
"Dcuts_half_val",
"NQhalf_double",
"apply_pr2",
"hinhfun",
"isldistr_mult_plusNonnegativeRationals",
"ispositive_twoNonnegativeRationals",
"multNonnegativeRationals_ltcompat_l",
"multdivNonnegativeRationals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_half_corr : Dcuts_def_corr Dcuts_half_val. | Proof.
intros c Hc.
assert (Hc0 : (0 < c + c)%NRat)
by (now apply ispositive_plusNonnegativeRationals_l).
generalize (X_corr _ Hc0) ; apply hinhfun ; apply sumofmaps ; [intros Hx | intros r].
- left ; exact Hx.
- right.
exists (pr1 r / 2)%NRat ; split.
+ unfold Dcuts_half_val.
rewrite <- NQh... | Lemma | Dcuts_half_corr | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_def_corr",
"Dcuts_half_val",
"NQhalf_double",
"hinhfun",
"isassoc_plusNonnegativeRationals",
"iscomm_plusNonnegativeRationals",
"ispositive_plusNonnegativeRationals_l",
"isrefl_leNonnegativeRationals",
"left",
"plusNonnegativeRationals_lecompat_l",
"right",
"sumofmaps"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_half (x : Dcuts) : Dcuts | :=
make_Dcuts (Dcuts_half_val (pr1 x))
(Dcuts_half_bot (pr1 x) (is_Dcuts_bot x))
(Dcuts_half_open (pr1 x) (is_Dcuts_open x))
(Dcuts_half_corr (pr1 x) (is_Dcuts_bot x) (is_Dcuts_corr x)). | Definition | Dcuts_half | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_half_bot",
"Dcuts_half_corr",
"Dcuts_half_open",
"Dcuts_half_val",
"is_Dcuts_bot",
"is_Dcuts_corr",
"is_Dcuts_open",
"make_Dcuts"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_half_le :
∏ x : Dcuts, Dcuts_half x <= x. | Proof.
intros x.
intros r Hr.
apply is_Dcuts_bot with (1 := Hr).
now apply plusNonnegativeRationals_le_l.
Qed. | Lemma | Dcuts_half_le | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_half",
"is_Dcuts_bot",
"plusNonnegativeRationals_le_l"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isdistr_Dcuts_half_plus :
∏ x y : Dcuts, Dcuts_half (Dcuts_plus x y) = Dcuts_plus (Dcuts_half x) (Dcuts_half y). | Proof.
intros x y.
apply Dcuts_eq_is_eq.
intros r ; split.
- apply hinhfun ; apply sumofmaps ; [apply sumofmaps ; [intros Xr | intros Yr] | intros xy ].
+ left.
left.
exact Xr.
+ left.
right.
exact Yr.
+ right.
exists (pr1 (pr1 xy) / 2%NRat,, pr2 (pr1 xy)/2%NRat).
... | Lemma | isdistr_Dcuts_half_plus | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_eq_is_eq",
"Dcuts_half",
"Dcuts_half_val",
"Dcuts_plus",
"NQhalf_double",
"divNonnegativeRationals",
"hinhfun",
"isassoc_plusNonnegativeRationals",
"iscomm_plusNonnegativeRationals",
"isrdistr_mult_plusNonnegativeRationals",
"left",
"maponpaths",
"right",
"sumofmaps",
"... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_half_double :
∏ x : Dcuts, x = Dcuts_plus (Dcuts_half x) (Dcuts_half x). | Proof.
intros x.
rewrite <- isdistr_Dcuts_half_plus.
apply Dcuts_eq_is_eq ; split.
- intros Hr.
apply hinhpr ; right ; exists (r,,r).
now repeat split.
- apply hinhuniv ; apply sumofmaps ; [ apply sumofmaps | intros xy ].
+ now simple refine (Dcuts_half_le _ _).
+ now simple refine (Dcuts_hal... | Lemma | Dcuts_half_double | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_eq_is_eq",
"Dcuts_half",
"Dcuts_half_le",
"Dcuts_plus",
"apply_pr2",
"hinhpr",
"hinhuniv",
"is_Dcuts_bot",
"isdecrel_ltNonnegativeRationals",
"isdistr_Dcuts_half_plus",
"lt_leNonnegativeRationals",
"notlt_geNonnegativeRationals",
"plusNonnegativeRationals_lecompat_l",
"pl... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_half_correct :
∏ x, Dcuts_half x = Dcuts_mult x (Dcuts_inv Dcuts_two Dcuts_two_ap_zero). | Proof.
intros x.
pattern x at 2 ; rewrite (Dcuts_half_double x).
rewrite Dcuts_plus_double, iscomm_Dcuts_mult, <- isassoc_Dcuts_mult, islinv_Dcuts_inv, islunit_Dcuts_mult_one.
reflexivity.
Qed. | Lemma | Dcuts_half_correct | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_half",
"Dcuts_half_double",
"Dcuts_inv",
"Dcuts_mult",
"Dcuts_plus_double",
"Dcuts_two",
"Dcuts_two_ap_zero",
"isassoc_Dcuts_mult",
"iscomm_Dcuts_mult",
"islinv_Dcuts_inv",
"islunit_Dcuts_mult_one"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
ispositive_Dcuts_half:
∏ x : Dcuts, (0 < x) <-> (0 < Dcuts_half x). | Proof.
intros.
rewrite Dcuts_half_correct.
pattern 0 at 2 ; rewrite <- (islabsorb_Dcuts_mult_zero (Dcuts_inv Dcuts_two Dcuts_two_ap_zero)).
split.
- intro Hx0.
apply Dcuts_mult_ltcompat_l.
+ apply Dcuts_mult_ltcompat_l' with Dcuts_two.
rewrite islabsorb_Dcuts_mult_zero, islinv_Dcuts_inv.
u... | Lemma | ispositive_Dcuts_half | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_half",
"Dcuts_half_correct",
"Dcuts_inv",
"Dcuts_mult_ltcompat_l",
"Dcuts_mult_ltcompat_l'",
"Dcuts_one",
"Dcuts_two",
"Dcuts_two_ap_zero",
"Dcuts_zero",
"isapfun_NonnegativeRationals_to_Dcuts_aux",
"islabsorb_Dcuts_mult_zero",
"islinv_Dcuts_inv",
"ispositive_oneNonnegative... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_locatedness :
∏ X : Dcuts, ∏ p q : NonnegativeRationals, (p < q)%NRat -> p ∈ X ∨ ¬ (q ∈ X). | Proof.
intros X p q Hlt.
apply ispositive_minusNonnegativeRationals in Hlt.
generalize (is_Dcuts_corr X _ Hlt).
apply_pr2_in ispositive_minusNonnegativeRationals Hlt.
apply hinhuniv ; apply sumofmaps ; [ intros Xr | ].
- apply hinhpr ; right.
intro H ; apply Xr.
apply is_Dcuts_bot with (1 := H).
... | Lemma | Dcuts_locatedness | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"NonnegativeRationals",
"apply_pr2",
"apply_pr2_in",
"hinhpr",
"hinhuniv",
"is_Dcuts_bot",
"is_Dcuts_corr",
"isassoc_plusNonnegativeRationals",
"iscomm_plusNonnegativeRationals",
"isdecrel_leNonnegativeRationals",
"ispositive_minusNonnegativeRationals",
"left",
"lt_leNonnegativeRa... | ** Locatedness | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
Dcuts_lim_cauchy_val : hsubtype NonnegativeRationals | :=
λ r : NonnegativeRationals, hexists (λ c : NonnegativeRationals, (0 < c)%NRat × ∑ N : nat, ∏ n : nat, N ≤ n -> U n (r + c)). | Definition | Dcuts_lim_cauchy_val | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals",
"hexists",
"hsubtype",
"nat"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_lim_cauchy_bot : Dcuts_def_bot Dcuts_lim_cauchy_val. | Proof.
intros r Hr q Hq.
revert Hr ; apply hinhfun ; intros c.
exists (pr1 c) ; split.
- exact (pr1 (pr2 c)).
- exists (pr1 (pr2 (pr2 c))) ; intros n Hn.
apply (U_bot n) with (1 := pr2 (pr2 (pr2 c)) n Hn).
apply plusNonnegativeRationals_lecompat_r.
exact Hq.
Qed. | Lemma | Dcuts_lim_cauchy_bot | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_def_bot",
"Dcuts_lim_cauchy_val",
"hinhfun",
"plusNonnegativeRationals_lecompat_r"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_lim_cauchy_open : Dcuts_def_open Dcuts_lim_cauchy_val. | Proof.
intros r.
apply hinhfun ; intros c.
exists (r + (pr1 c / 2))%NRat ; split.
- apply hinhpr.
exists (pr1 c / 2)%NRat ; split.
+ now apply ispositive_NQhalf, (pr1 (pr2 c)).
+ exists (pr1 (pr2 (pr2 c))) ; intros n Hn.
rewrite isassoc_plusNonnegativeRationals, <- NQhalf_double.
now app... | Lemma | Dcuts_lim_cauchy_open | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_def_open",
"Dcuts_lim_cauchy_val",
"NQhalf_double",
"hinhfun",
"hinhpr",
"isassoc_plusNonnegativeRationals",
"ispositive_NQhalf",
"plusNonnegativeRationals_lt_r"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_lim_cauchy_corr : Dcuts_def_corr Dcuts_lim_cauchy_val. | Proof.
intros c Hc.
apply ispositive_NQhalf, ispositive_NQhalf in Hc.
generalize (U_cauchy _ Hc) ; clear U_cauchy ; apply hinhuniv ; intros N.
generalize (λ n Hn, pr2 N n (pr1 N) Hn (isreflnatleh _)) ; intro Hu.
generalize (U_corr (pr1 N) _ Hc).
apply hinhuniv ; apply sumofmaps ; intros HuN.
- apply hinhp... | Lemma | Dcuts_lim_cauchy_corr | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_def_corr",
"Dcuts_lim_cauchy_val",
"NQhalf_double",
"apply_pr2",
"eps",
"fromempty",
"hinhpr",
"hinhuniv",
"isassoc_plusNonnegativeRationals",
"iscomm_plusNonnegativeRationals",
"isdecrel_leNonnegativeRationals",
"ispositive_NQhalf",
"isreflnatleh",
"istrans_leNonnegativeRationals",... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_Cauchy_seq (u : nat -> Dcuts) : hProp | := make_hProp (∏ eps : Dcuts,
0 < eps ->
hexists
(λ N : nat,
∏ n m : nat, N ≤ n -> N ≤ m -> u n < Dcuts_plus (u m) eps × u m < Dcuts_plus (u n) eps))
(impred_isaprop _ (λ _, isapropimpl _ _ (pr2 _))). | Definition | Dcuts_Cauchy_seq | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_plus",
"eps",
"hProp",
"hexists",
"impred_isaprop",
"isapropimpl",
"make_hProp",
"nat"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
is_Dcuts_lim_seq (u : nat -> Dcuts) (l : Dcuts) : hProp | := make_hProp (∏ eps : Dcuts,
0 < eps ->
hexists
(λ N : nat,
∏ n : nat, N ≤ n -> u n < Dcuts_plus l eps × l < Dcuts_plus (u n) eps))
(impred_isaprop _ (λ _, isapropimpl _ _ (pr2 _))). | Definition | is_Dcuts_lim_seq | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_plus",
"eps",
"hProp",
"hexists",
"impred_isaprop",
"isapropimpl",
"make_hProp",
"nat"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_lim_cauchy_seq (U : nat → Dcuts) (HU : Dcuts_Cauchy_seq U) : Dcuts. | Proof.
exists (Dcuts_lim_cauchy_val (λ n, pr1 (U n))).
repeat split.
- apply Dcuts_lim_cauchy_bot.
intro ; now apply is_Dcuts_bot.
- apply Dcuts_lim_cauchy_open.
- apply Dcuts_lim_cauchy_corr.
+ intro ; now apply is_Dcuts_bot.
+ intro ; now apply is_Dcuts_corr.
+ intros eps Heps.
assert ... | Definition | Dcuts_lim_cauchy_seq | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_Cauchy_seq",
"Dcuts_lim_cauchy_bot",
"Dcuts_lim_cauchy_corr",
"Dcuts_lim_cauchy_open",
"Dcuts_lim_cauchy_val",
"Dcuts_lt_le_rel",
"HU",
"Hm",
"NonnegativeRationals_to_Dcuts",
"apply_pr2",
"eps",
"hinhfun",
"is_Dcuts_bot",
"is_Dcuts_corr",
"isapfun_NonnegativeRationals_t... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_Cauchy_seq_impl_ex_lim_seq (U : nat → Dcuts) (HU : Dcuts_Cauchy_seq U) :
is_Dcuts_lim_seq U (Dcuts_lim_cauchy_seq U HU). | Proof.
intros eps.
apply hinhuniv ; intros c'.
generalize (is_Dcuts_open _ _ (pr2 (pr2 c'))).
apply hinhuniv ; intros c.
assert (Hc0 : (0 < pr1 c)%NRat).
{ eapply istrans_le_lt_ltNonnegativeRationals, (pr2 (pr2 c)).
now apply isnonnegative_NonnegativeRationals. }
apply ispositive_NQhalf in Hc0.
gene... | Lemma | Dcuts_Cauchy_seq_impl_ex_lim_seq | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_Cauchy_seq",
"Dcuts_half",
"Dcuts_half_double",
"Dcuts_half_val",
"Dcuts_lim_cauchy_seq",
"Dcuts_lt_le_rel",
"Dcuts_plus",
"Dcuts_plus_lecompat_l",
"Dcuts_plus_lecompat_r",
"Dcuts_plus_ltcompat_l",
"HU",
"Hm",
"NQhalf_double",
"NonnegativeRationals_to_Dcuts",
"apply_pr2... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts_val : NonnegativeRationals → hProp | :=
λ r : NonnegativeRationals, ∃ X : Dcuts, (E X) × (r ∈ X). | Definition | Dcuts_of_Dcuts_val | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"NonnegativeRationals",
"hProp"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts_bot :
∏ (x : NonnegativeRationals),
Dcuts_of_Dcuts_val x -> ∏ y : NonnegativeRationals, (y <= x)%NRat -> Dcuts_of_Dcuts_val y. | Proof.
intros r Xr n Xn.
revert Xr ; apply hinhfun ; intros X.
exists (pr1 X) ; split.
- exact (pr1 (pr2 X)).
- apply is_Dcuts_bot with r.
+ exact (pr2 (pr2 X)).
+ exact Xn.
Qed. | Lemma | Dcuts_of_Dcuts_bot | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_of_Dcuts_val",
"NonnegativeRationals",
"hinhfun",
"is_Dcuts_bot"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts_open :
∏ (x : NonnegativeRationals),
Dcuts_of_Dcuts_val x ->
hexists (λ y : NonnegativeRationals, (Dcuts_of_Dcuts_val y) × (x < y)%NRat). | Proof.
intros r.
apply hinhuniv ; intros X.
generalize (is_Dcuts_open _ _ (pr2 (pr2 X))).
apply hinhfun ; intros n.
exists (pr1 n) ; split.
- apply hinhpr.
exists (pr1 X) ; split.
+ exact (pr1 (pr2 X)).
+ exact (pr1 (pr2 n)).
- exact (pr2 (pr2 n)).
Qed. | Lemma | Dcuts_of_Dcuts_open | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_of_Dcuts_val",
"NonnegativeRationals",
"hexists",
"hinhfun",
"hinhpr",
"hinhuniv",
"is_Dcuts_open"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts_corr:
Dcuts_def_corr Dcuts_of_Dcuts_val. | Proof.
intros c Hc.
apply ispositive_NQhalf in Hc.
apply (pr2 (isapfun_NonnegativeRationals_to_Dcuts_aux _ _)) in Hc.
generalize (E_corr _ Hc).
apply isapfun_NonnegativeRationals_to_Dcuts_aux in Hc.
apply hinhuniv ; apply sumofmaps ; [intros He | ].
- apply hinhpr ; left.
use factor_through_squash.
... | Lemma | Dcuts_of_Dcuts_corr | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_def_corr",
"Dcuts_lt_le_rel",
"Dcuts_of_Dcuts_val",
"He",
"NQhalf_double",
"apply_pr2",
"factor_through_squash",
"fromempty",
"hinhfun",
"hinhpr",
"hinhuniv",
"is_Dcuts_bot",
"is_Dcuts_corr",
"isapfun_NonnegativeRationals_to_Dcuts_aux",
"isapropempty",
"isassoc_plusNonnegativeRa... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts (E : hsubtype Dcuts) E_bot E_corr : Dcuts | :=
make_Dcuts (Dcuts_of_Dcuts_val E) (Dcuts_of_Dcuts_bot E) (Dcuts_of_Dcuts_open E) (Dcuts_of_Dcuts_corr E E_bot E_corr). | Definition | Dcuts_of_Dcuts | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_of_Dcuts_bot",
"Dcuts_of_Dcuts_corr",
"Dcuts_of_Dcuts_open",
"Dcuts_of_Dcuts_val",
"hsubtype",
"make_Dcuts"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts'_val : hsubtype Dcuts | :=
λ x : Dcuts, ∃ r : NonnegativeRationals, (¬ (r ∈ x)) × E r. | Definition | Dcuts_of_Dcuts'_val | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"NonnegativeRationals",
"hsubtype"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts'_bot :
∏ (x : Dcuts),
Dcuts_of_Dcuts'_val x -> ∏ y : Dcuts, (y <= x) -> Dcuts_of_Dcuts'_val y. | Proof.
intros r Xr n Xn.
revert Xr.
apply hinhfun.
intros q.
exists (pr1 q).
split.
- intros Nq.
apply (pr1 (pr2 q)).
now simple refine (Xn _ _).
- exact (pr2 (pr2 q)).
Qed. | Lemma | Dcuts_of_Dcuts'_bot | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_of_Dcuts'_val",
"hinhfun"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts'_open :
∏ (x : Dcuts),
Dcuts_of_Dcuts'_val x ->
hexists (λ y : Dcuts, (Dcuts_of_Dcuts'_val y) × (x < y)). | Proof.
intros r.
apply hinhuniv.
intros q.
generalize (E_open _ (pr2 (pr2 q))).
apply hinhfun.
intros s.
exists (NonnegativeRationals_to_Dcuts (pr1 s)).
split.
- apply hinhpr.
exists (pr1 s).
split.
+ simpl.
now apply isirrefl_ltNonnegativeRationals.
+ exact (pr1 (pr2 s)).
- ap... | Lemma | Dcuts_of_Dcuts'_open | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_of_Dcuts'_val",
"NonnegativeRationals_to_Dcuts",
"hexists",
"hinhfun",
"hinhpr",
"hinhuniv",
"isirrefl_ltNonnegativeRationals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts'_corr:
∏ c : Dcuts, 0 < c -> (¬ Dcuts_of_Dcuts'_val c) ∨ (hexists (λ P, Dcuts_of_Dcuts'_val P × ¬ Dcuts_of_Dcuts'_val (Dcuts_plus P c))). | Proof.
intros C HC.
assert (∃ c : NonnegativeRationals, c ∈ C × (0 < c)%NRat).
{ revert HC ; apply hinhuniv ; intro d.
generalize (is_Dcuts_open _ _ (pr2 (pr2 d))).
apply hinhfun.
intro c.
exists (pr1 c).
split.
- exact (pr1 (pr2 c)).
- eapply istrans_le_lt_ltNonnegativeRationals, (pr2... | Lemma | Dcuts_of_Dcuts'_corr | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_of_Dcuts'_val",
"Dcuts_plus",
"HC",
"NonnegativeRationals",
"NonnegativeRationals_to_Dcuts",
"apply_pr2",
"factor_through_squash",
"hexists",
"hinhfun",
"hinhpr",
"hinhuniv",
"is_Dcuts_bot",
"is_Dcuts_open",
"isapropempty",
"isdecrel_ltNonnegativeRationals",
"isirrefl... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts_bij :
∏ x : Dcuts, Dcuts_of_Dcuts (Dcuts_of_Dcuts'_val (pr1 x)) (Dcuts_of_Dcuts'_bot (pr1 x)) (Dcuts_of_Dcuts'_corr (pr1 x) (is_Dcuts_bot x) (is_Dcuts_corr x)) = x. | Proof.
intros x.
apply Dcuts_eq_is_eq.
intros r.
split.
- apply hinhuniv.
intros y.
generalize (pr1 (pr2 y)).
apply hinhuniv.
intros q.
apply is_Dcuts_bot with (1 := pr2 (pr2 q)).
apply lt_leNonnegativeRationals, notge_ltNonnegativeRationals.
intro H.
apply (pr1 (pr2 q)).
n... | Lemma | Dcuts_of_Dcuts_bij | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_eq_is_eq",
"Dcuts_of_Dcuts",
"Dcuts_of_Dcuts'_bot",
"Dcuts_of_Dcuts'_corr",
"Dcuts_of_Dcuts'_val",
"NonnegativeRationals_to_Dcuts",
"hinhfun",
"hinhpr",
"hinhuniv",
"is_Dcuts_bot",
"is_Dcuts_corr",
"is_Dcuts_open",
"isirrefl_ltNonnegativeRationals",
"lt_leNonnegativeRatio... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
Dcuts_of_Dcuts_bij' :
∏ E : hsubtype Dcuts, ∏ (E_bot : ∏ x : Dcuts, E x -> ∏ y : Dcuts, y <= x -> E y) (E_open : ∏ x : Dcuts, E x -> ∃ y : Dcuts, x < y × E y),
Dcuts_of_Dcuts'_val (Dcuts_of_Dcuts_val E) = E. | Proof.
intros.
apply funextfun.
intros x.
apply hPropUnivalence.
- apply hinhuniv.
simpl pr1.
intros r ; generalize (pr2 (pr2 r)).
apply hinhuniv.
intros X.
apply E_bot with (1 := pr1 (pr2 X)).
apply Dcuts_lt_le_rel.
apply hinhpr.
exists (pr1 r).
split.
+ exact (pr1 (pr... | Lemma | Dcuts_of_Dcuts_bij' | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_lt_le_rel",
"Dcuts_of_Dcuts'_val",
"Dcuts_of_Dcuts_val",
"funextfun",
"hPropUnivalence",
"hinhfun",
"hinhpr",
"hinhuniv",
"hsubtype"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isub_Dcuts_of_Dcuts (E : hsubtype Dcuts) E_bot E_corr :
isUpperBound (X := PreorderedSetEffectiveOrder eo_Dcuts) E (Dcuts_of_Dcuts E E_bot E_corr). | Proof.
intros ;
intros x Ex r Hr.
apply hinhpr.
now exists x.
Qed. | Lemma | isub_Dcuts_of_Dcuts | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_of_Dcuts",
"PreorderedSetEffectiveOrder",
"eo_Dcuts",
"hinhpr",
"hsubtype",
"isUpperBound"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
islbub_Dcuts_of_Dcuts (E : hsubtype Dcuts) E_bot E_corr :
isSmallerThanUpperBounds (X := PreorderedSetEffectiveOrder eo_Dcuts) E (Dcuts_of_Dcuts E E_bot E_corr). | Proof.
intros.
intros x Hx ; simpl.
intros r ; apply hinhuniv ;
intros y.
generalize (Hx _ (pr1 (pr2 y))).
intros H ; simple refine (H _ _).
exact (pr2 (pr2 y)).
Qed. | Lemma | islbub_Dcuts_of_Dcuts | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts",
"Dcuts_of_Dcuts",
"PreorderedSetEffectiveOrder",
"eo_Dcuts",
"hinhuniv",
"hsubtype",
"isSmallerThanUpperBounds"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
islub_Dcuts_of_Dcuts (E : hsubtype eo_Dcuts) E_bot E_corr :
isLeastUpperBound (X := PreorderedSetEffectiveOrder eo_Dcuts) E (Dcuts_of_Dcuts E E_bot E_corr). | Proof.
split.
- exact (isub_Dcuts_of_Dcuts E E_bot E_corr).
- exact (islbub_Dcuts_of_Dcuts E E_bot E_corr).
Qed. | Lemma | islub_Dcuts_of_Dcuts | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_of_Dcuts",
"PreorderedSetEffectiveOrder",
"eo_Dcuts",
"hsubtype",
"isLeastUpperBound",
"islbub_Dcuts_of_Dcuts",
"isub_Dcuts_of_Dcuts"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeReals : ConstructiveCommutativeDivisionRig | := Dcuts_ConstructiveCommutativeDivisionRig. | Definition | NonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"ConstructiveCommutativeDivisionRig",
"Dcuts_ConstructiveCommutativeDivisionRig"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
EffectivelyOrdered_NonnegativeReals : EffectivelyOrderedSet. | Proof.
exists NonnegativeReals.
apply (pairEffectiveOrder Dcuts_le_rel Dcuts_lt_rel iseo_Dcuts_le_lt_rel).
Defined. | Definition | EffectivelyOrdered_NonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_le_rel",
"Dcuts_lt_rel",
"EffectivelyOrderedSet",
"NonnegativeReals",
"iseo_Dcuts_le_lt_rel",
"pairEffectiveOrder"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apNonnegativeReals : hrel NonnegativeReals | := CCDRap. | Definition | apNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"CCDRap",
"NonnegativeReals",
"hrel"
] | ** Relations | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
leNonnegativeReals : po NonnegativeReals | := EOle (X := EffectivelyOrdered_NonnegativeReals). | Definition | leNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EOle",
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"po"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
geNonnegativeReals : po NonnegativeReals | := EOge (X := EffectivelyOrdered_NonnegativeReals). | Definition | geNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EOge",
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"po"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
ltNonnegativeReals : StrongOrder NonnegativeReals | := EOlt (X := EffectivelyOrdered_NonnegativeReals). | Definition | ltNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EOlt",
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"StrongOrder"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
gtNonnegativeReals : StrongOrder NonnegativeReals | := EOgt (X := EffectivelyOrdered_NonnegativeReals). | Definition | gtNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EOgt",
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"StrongOrder"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x ≠ y" | := (apNonnegativeReals x y) (at level 70, no associativity) : NR_scope. | Notation | x ≠ y | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"apNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x <= y" | := (EOle_rel (X := EffectivelyOrdered_NonnegativeReals) x y) : NR_scope. | Notation | x <= y | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EOle_rel",
"EffectivelyOrdered_NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x >= y" | := (EOge_rel (X := EffectivelyOrdered_NonnegativeReals) x y) : NR_scope. | Notation | x >= y | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EOge_rel",
"EffectivelyOrdered_NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x < y" | := (EOlt_rel (X := EffectivelyOrdered_NonnegativeReals) x y) : NR_scope. | Notation | x < y | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EOlt_rel",
"EffectivelyOrdered_NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x > y" | := (EOgt_rel (X := EffectivelyOrdered_NonnegativeReals) eo_Dcuts x y) : NR_scope. | Notation | x > y | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EOgt_rel",
"EffectivelyOrdered_NonnegativeReals",
"eo_Dcuts"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
zeroNonnegativeReals : NonnegativeReals | := CCDRzero. | Definition | zeroNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"CCDRzero",
"NonnegativeReals"
] | ** Constants and Functions | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
oneNonnegativeReals : NonnegativeReals | := CCDRone. | Definition | oneNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"CCDRone",
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
twoNonnegativeReals : NonnegativeReals | := Dcuts_two. | Definition | twoNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_two",
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
plusNonnegativeReals : binop NonnegativeReals | := CCDRplus. | Definition | plusNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"CCDRplus",
"NonnegativeReals",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
multNonnegativeReals : binop NonnegativeReals | := CCDRmult. | Definition | multNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"CCDRmult",
"NonnegativeReals",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeRationals_to_NonnegativeReals (r : NonnegativeRationals) : NonnegativeReals | :=
NonnegativeRationals_to_Dcuts r. | Definition | NonnegativeRationals_to_NonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals",
"NonnegativeRationals_to_Dcuts",
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
nat_to_NonnegativeReals (n : nat) : NonnegativeReals | :=
NonnegativeRationals_to_NonnegativeReals (nat_to_NonnegativeRationals n). | Definition | nat_to_NonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals_to_NonnegativeReals",
"NonnegativeReals",
"nat",
"nat_to_NonnegativeRationals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"0" | := zeroNonnegativeReals : NR_scope. | Notation | 0 | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"zeroNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"1" | := oneNonnegativeReals : NR_scope. | Notation | 1 | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"oneNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"2" | := twoNonnegativeReals : NR_scope. | Notation | 2 | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"twoNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x + y" | := (plusNonnegativeReals x y) (at level 50, left associativity) : NR_scope. | Notation | x + y | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"left",
"plusNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x * y" | := (multNonnegativeReals x y) (at level 40, left associativity) : NR_scope. | Notation | x * y | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"left",
"multNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
invNonnegativeReals (x : NonnegativeReals) (Hx0 : x ≠ 0) : NonnegativeReals | :=
CCDRinv x Hx0. | Definition | invNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"CCDRinv",
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
divNonnegativeReals (x y : NonnegativeReals) (Hy0 : y ≠ 0) : NonnegativeReals | :=
x * (invNonnegativeReals y Hy0). | Definition | divNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"invNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeReals_to_hsubtypeNonnegativeRationals :
NonnegativeReals → (hsubtype NonnegativeRationals) | := pr1. | Definition | NonnegativeReals_to_hsubtypeNonnegativeRationals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals",
"NonnegativeReals",
"hsubtype"
] | ** Special functions | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
hsubtypeNonnegativeRationals_to_NonnegativeReals
(X : NonnegativeRationals -> hProp)
(Xbot : ∏ x : NonnegativeRationals,
X x -> ∏ y : NonnegativeRationals, (y <= x)%NRat -> X y)
(Xopen : ∏ x : NonnegativeRationals,
X x ->
hexists (λ y : NonnegativeRationals, (X y) × (x < y)%N... | :=
make_Dcuts X Xbot Xopen Xtop. | Definition | hsubtypeNonnegativeRationals_to_NonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_def_corr",
"NonnegativeRationals",
"NonnegativeReals",
"hProp",
"hexists",
"make_Dcuts"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
minusNonnegativeReals : binop NonnegativeReals | := Dcuts_minus. | Definition | minusNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_minus",
"NonnegativeReals",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
halfNonnegativeReals : unop NonnegativeReals | := Dcuts_half. | Definition | halfNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_half",
"NonnegativeReals",
"unop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
maxNonnegativeReals : binop NonnegativeReals | := Dcuts_max. | Definition | maxNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_max",
"NonnegativeReals",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
minNonnegativeReals : binop NonnegativeReals | := Dcuts_min. | Definition | minNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_min",
"NonnegativeReals",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x - y" | := (minusNonnegativeReals x y) (at level 50, left associativity) : NR_scope. | Notation | x - y | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"left",
"minusNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x / 2" | := (halfNonnegativeReals x) (at level 35, no associativity) : NR_scope. | Notation | x / 2 | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"halfNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeRationals_to_NonnegativeReals_lt :
∏ x y : NonnegativeRationals,
(x < y)%NRat <->
NonnegativeRationals_to_NonnegativeReals x < NonnegativeRationals_to_NonnegativeReals y. | Proof.
intros x y ; split.
- intros Hxy.
apply hinhpr.
exists x.
split ; simpl.
+ now apply isirrefl_ltNonnegativeRationals.
+ exact Hxy.
- apply hinhuniv ; simpl ; intros q.
eapply istrans_le_lt_ltNonnegativeRationals, (pr2 (pr2 q)).
apply notlt_geNonnegativeRationals.
exact (pr1 ... | Lemma | NonnegativeRationals_to_NonnegativeReals_lt | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals",
"NonnegativeRationals_to_NonnegativeReals",
"hinhpr",
"hinhuniv",
"isirrefl_ltNonnegativeRationals",
"istrans_le_lt_ltNonnegativeRationals",
"notlt_geNonnegativeRationals"
] | ** Compatibility with NonnegativeRationals | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
NonnegativeRationals_to_NonnegativeReals_le :
∏ x y : NonnegativeRationals,
(x <= y)%NRat <->
NonnegativeRationals_to_NonnegativeReals x <= NonnegativeRationals_to_NonnegativeReals y. | Proof.
intros x y ; split.
- intros H.
apply Dcuts_nlt_ge.
intro H0.
revert H.
apply_pr2 notge_ltNonnegativeRationals.
apply_pr2 NonnegativeRationals_to_NonnegativeReals_lt.
exact H0.
- intros H.
apply notlt_geNonnegativeRationals.
intros H0.
revert H.
apply Dcuts_gt_nle.
... | Lemma | NonnegativeRationals_to_NonnegativeReals_le | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_gt_nle",
"Dcuts_nlt_ge",
"NonnegativeRationals",
"NonnegativeRationals_to_NonnegativeReals",
"NonnegativeRationals_to_NonnegativeReals_lt",
"apply_pr2",
"notge_ltNonnegativeRationals",
"notlt_geNonnegativeRationals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeRationals_to_NonnegativeReals_zero :
NonnegativeRationals_to_NonnegativeReals 0%NRat = 0. | Proof.
reflexivity.
Qed. | Lemma | NonnegativeRationals_to_NonnegativeReals_zero | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals_to_NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeRationals_to_NonnegativeReals_one :
NonnegativeRationals_to_NonnegativeReals 1%NRat = 1. | Proof.
reflexivity.
Qed. | Lemma | NonnegativeRationals_to_NonnegativeReals_one | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals_to_NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeRationals_to_NonnegativeReals_plus :
∏ x y : NonnegativeRationals, NonnegativeRationals_to_NonnegativeReals (x + y)%NRat = NonnegativeRationals_to_NonnegativeReals x + NonnegativeRationals_to_NonnegativeReals y. | Proof.
intros x y.
apply Dcuts_eq_is_eq.
intros r.
split.
- intros Hr.
generalize (eq0orgt0NonnegativeRationals y) ; apply sumofmaps ; intros Hy.
2: generalize (eq0orgt0NonnegativeRationals x) ; apply sumofmaps ; intros Hx.
+ rewrite Hy in Hr |- * ; clear y Hy.
rewrite isrunit_zeroNonnegativ... | Lemma | NonnegativeRationals_to_NonnegativeReals_plus | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_eq_is_eq",
"NonnegativeRationals",
"NonnegativeRationals_to_NonnegativeReals",
"divNonnegativeRationals",
"eq0orgt0NonnegativeRationals",
"hinhpr",
"hinhuniv",
"isassoc_multNonnegativeRationals",
"iscomm_multNonnegativeRationals",
"isldistr_mult_plusNonnegativeRationals",
"islunit_Dcuts_p... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeRationals_to_NonnegativeReals_minus :
∏ x y : NonnegativeRationals, NonnegativeRationals_to_NonnegativeReals (x - y)%NRat = NonnegativeRationals_to_NonnegativeReals x - NonnegativeRationals_to_NonnegativeReals y. | Proof.
intros x y.
generalize (isdecrel_leNonnegativeRationals x y) ; apply sumofmaps ; intros Hxy.
- rewrite minusNonnegativeRationals_eq_zero, Dcuts_minus_eq_zero.
+ reflexivity.
+ apply NonnegativeRationals_to_NonnegativeReals_le.
exact Hxy.
+ exact Hxy.
- apply Dcuts_minus_correct_r.
r... | Lemma | NonnegativeRationals_to_NonnegativeReals_minus | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_minus_correct_r",
"Dcuts_minus_eq_zero",
"NonnegativeRationals",
"NonnegativeRationals_to_NonnegativeReals",
"NonnegativeRationals_to_NonnegativeReals_le",
"NonnegativeRationals_to_NonnegativeReals_plus",
"isdecrel_leNonnegativeRationals",
"lt_leNonnegativeRationals",
"minusNonnegativeRationa... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeRationals_to_NonnegativeReals_mult :
∏ x y : NonnegativeRationals, NonnegativeRationals_to_NonnegativeReals (x * y)%NRat = NonnegativeRationals_to_NonnegativeReals x * NonnegativeRationals_to_NonnegativeReals y. | Proof.
intros x y.
generalize (eq0orgt0NonnegativeRationals x) ; apply sumofmaps ; [intros -> | intros Hx].
- rewrite islabsorb_zero_multNonnegativeRationals, islabsorb_Dcuts_mult_zero.
reflexivity.
- rewrite <- (Dcuts_NQmult_mult _ _ Hx).
apply Dcuts_eq_is_eq.
intros r.
split.
+ simpl ; int... | Lemma | NonnegativeRationals_to_NonnegativeReals_mult | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_NQmult_mult",
"Dcuts_eq_is_eq",
"NonnegativeRationals",
"NonnegativeRationals_to_NonnegativeReals",
"eq0orgt0NonnegativeRationals",
"hinhpr",
"hinhuniv",
"isassoc_multNonnegativeRationals",
"iscomm_multNonnegativeRationals",
"islabsorb_Dcuts_mult_zero",
"islabsorb_zero_multNonnegativeRati... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
NonnegativeRationals_to_NonnegativeReals_nattorig :
∏ n : nat, NonnegativeRationals_to_NonnegativeReals (nattorig n) = nattorig n. | Proof.
induction n as [|n IHn].
- reflexivity.
- rewrite !nattorigS.
rewrite NonnegativeRationals_to_NonnegativeReals_plus, IHn.
reflexivity.
Qed. | Lemma | NonnegativeRationals_to_NonnegativeReals_nattorig | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals_to_NonnegativeReals",
"NonnegativeRationals_to_NonnegativeReals_plus",
"nat",
"nattorig",
"nattorigS"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
nat_to_NonnegativeReals_O :
nat_to_NonnegativeReals O = 0. | Proof.
unfold nat_to_NonnegativeReals.
rewrite nat_to_NonnegativeRationals_O.
reflexivity.
Qed. | Lemma | nat_to_NonnegativeReals_O | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"nat_to_NonnegativeRationals_O",
"nat_to_NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
nat_to_NonnegativeReals_Sn :
∏ n : nat, nat_to_NonnegativeReals (S n) = nat_to_NonnegativeReals n + 1. | Proof.
intros n.
unfold nat_to_NonnegativeReals.
rewrite nat_to_NonnegativeRationals_Sn.
rewrite NonnegativeRationals_to_NonnegativeReals_plus.
reflexivity.
Qed. | Lemma | nat_to_NonnegativeReals_Sn | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeRationals_to_NonnegativeReals_plus",
"nat",
"nat_to_NonnegativeRationals_Sn",
"nat_to_NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
istrans_leNonnegativeReals :
∏ x y z : NonnegativeReals, x <= y -> y <= z -> x <= z | := istrans_EOle (X := EffectivelyOrdered_NonnegativeReals). | Definition | istrans_leNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"istrans_EOle"
] | Order, apartness, and equality | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
isrefl_leNonnegativeReals :
∏ x : NonnegativeReals, x <= x | := isrefl_EOle (X := EffectivelyOrdered_NonnegativeReals). | Definition | isrefl_leNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"isrefl_EOle"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isantisymm_leNonnegativeReals :
∏ x y : NonnegativeReals, x <= y × y <= x <-> x = y. | Proof.
intros x y ; split.
- intros H.
apply Dcuts_le_ge_eq.
+ now apply (pr1 H).
+ now apply (pr2 H).
- intros ->.
split ; apply isrefl_leNonnegativeReals.
Qed. | Lemma | isantisymm_leNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_le_ge_eq",
"NonnegativeReals",
"isrefl_leNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
eqNonnegativeReals_le :
∏ x y : NonnegativeReals, x = y -> x <= y. | Proof.
intros x y ->.
apply isrefl_leNonnegativeReals.
Qed. | Lemma | eqNonnegativeReals_le | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"isrefl_leNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
istrans_ltNonnegativeReals :
∏ x y z : NonnegativeReals, x < y -> y < z -> x < z | := istrans_EOlt (X := EffectivelyOrdered_NonnegativeReals). | Definition | istrans_ltNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"istrans_EOlt"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
iscotrans_ltNonnegativeReals :
∏ x y z : NonnegativeReals, x < z -> x < y ∨ y < z | := iscotrans_Dcuts_lt_rel. | Definition | iscotrans_ltNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"iscotrans_Dcuts_lt_rel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isirrefl_ltNonnegativeReals :
∏ x : NonnegativeReals, ¬ (x < x) | := isirrefl_EOlt (X := EffectivelyOrdered_NonnegativeReals). | Definition | isirrefl_ltNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"isirrefl_EOlt"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
istrans_lt_le_ltNonnegativeReals :
∏ x y z : NonnegativeReals, x < y -> y <= z -> x < z | := istrans_EOlt_le (X := EffectivelyOrdered_NonnegativeReals). | Definition | istrans_lt_le_ltNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"istrans_EOlt_le"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
istrans_le_lt_ltNonnegativeReals :
∏ x y z : NonnegativeReals, x <= y -> y < z -> x < z | := istrans_EOle_lt (X := EffectivelyOrdered_NonnegativeReals). | Definition | istrans_le_lt_ltNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"EffectivelyOrdered_NonnegativeReals",
"NonnegativeReals",
"istrans_EOle_lt"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
lt_leNonnegativeReals :
∏ x y : NonnegativeReals, x < y -> x <= y. | Proof.
exact Dcuts_lt_le_rel.
Qed. | Lemma | lt_leNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_lt_le_rel",
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
notlt_leNonnegativeReals :
∏ x y : NonnegativeReals, ¬ (x < y) <-> (y <= x). | Proof.
exact Dcuts_nlt_ge.
Qed. | Lemma | notlt_leNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_nlt_ge",
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isnonnegative_NonnegativeReals :
∏ x : NonnegativeReals, 0 <= x. | Proof.
intros x.
now apply Dcuts_ge_0.
Qed. | Lemma | isnonnegative_NonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_ge_0",
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isnonnegative_NonnegativeReals' :
∏ x : NonnegativeReals, ¬ (x < 0). | Proof.
intros x.
now apply Dcuts_notlt_0.
Qed. | Lemma | isnonnegative_NonnegativeReals' | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_notlt_0",
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
le0_NonnegativeReals :
∏ x : NonnegativeReals, (x <= 0) <-> (x = 0). | Proof.
intros x ; split ; intros Hx.
- apply isantisymm_leNonnegativeReals.
split.
+ exact Hx.
+ apply isnonnegative_NonnegativeReals.
- rewrite Hx.
apply isrefl_leNonnegativeReals.
Qed. | Lemma | le0_NonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"isantisymm_leNonnegativeReals",
"isnonnegative_NonnegativeReals",
"isrefl_leNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
ap_ltNonnegativeReals :
∏ x y : NonnegativeReals, x ≠ y <-> (x < y) ⨿ (y < x). | Proof.
now intros x y ; split.
Qed. | Lemma | ap_ltNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isirrefl_apNonnegativeReals :
∏ x : NonnegativeReals, ¬ (x ≠ x) | := isirrefl_Dcuts_ap_rel. | Definition | isirrefl_apNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"isirrefl_Dcuts_ap_rel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
issymm_apNonnegativeReals :
∏ x y : NonnegativeReals, x ≠ y -> y ≠ x | := issymm_Dcuts_ap_rel. | Definition | issymm_apNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"issymm_Dcuts_ap_rel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
iscotrans_apNonnegativeReals :
∏ x y z : NonnegativeReals, x ≠ z -> x ≠ y ∨ y ≠ z | := iscotrans_Dcuts_ap_rel. | Definition | iscotrans_apNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"iscotrans_Dcuts_ap_rel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
istight_apNonnegativeReals:
∏ x y : NonnegativeReals, (¬ (x ≠ y)) <-> (x = y). | Proof.
intros x y.
split.
- now apply istight_Dcuts_ap_rel.
- intros ->.
now apply isirrefl_Dcuts_ap_rel.
Qed. | Lemma | istight_apNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"isirrefl_Dcuts_ap_rel",
"istight_Dcuts_ap_rel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
ispositive_apNonnegativeReals :
∏ x : NonnegativeReals, x ≠ 0 <-> 0 < x. | Proof.
intros X ; split.
- apply sumofmaps ; [ | intros Hlt ].
+ apply hinhuniv ; intros x.
apply fromempty.
now apply (Dcuts_zero_empty _ (pr2 (pr2 x))).
+ exact Hlt.
- intros Hx.
now right.
Qed. | Lemma | ispositive_apNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"Dcuts_zero_empty",
"NonnegativeReals",
"fromempty",
"hinhuniv",
"right",
"sumofmaps"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isnonzeroNonnegativeReals: 1 ≠ 0 | := isnonzeroCCDR (X := NonnegativeReals). | Definition | isnonzeroNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"isnonzeroCCDR"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
ispositive_oneNonnegativeReals: 0 < 1. | Proof.
apply ispositive_apNonnegativeReals.
exact isnonzeroNonnegativeReals.
Qed. | Lemma | ispositive_oneNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"isnonzeroNonnegativeReals",
"ispositive_apNonnegativeReals"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
ap_plusNonnegativeReals:
∏ x x' y y' : NonnegativeReals,
x + y ≠ x' + y' -> x ≠ x' ∨ y ≠ y' | := apCCDRplus (X := NonnegativeReals). | Definition | ap_plusNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"apCCDRplus",
"x'",
"y'"
] | addition | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
islunit_zero_plusNonnegativeReals:
∏ x : NonnegativeReals, 0 + x = x | := islunit_CCDRzero_CCDRplus (X := NonnegativeReals). | Definition | islunit_zero_plusNonnegativeReals | RealNumbers | UniMath/RealNumbers/NonnegativeReals.v | [
"UniMath.MoreFoundations.Orders",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.Notations",
"UniMath.RealNumbers.Sets",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.OrderTheory.Lattice.Lattice",
"UniMath.RealNumbers.Prelim",
"UniMath.RealNumbers.NonnegativeRationals"
] | [
"NonnegativeReals",
"islunit_CCDRzero_CCDRplus"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
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