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 |
|---|---|---|---|---|---|---|---|---|---|---|
aprel (X : UU) | := ∑ ap : hrel X, isaprel ap. | Definition | aprel | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"UU",
"ap",
"hrel",
"isaprel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
aprel_pr1 {X : UU} (ap : aprel X) : hrel X | := pr1 ap. | Definition | aprel_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"UU",
"ap",
"aprel",
"hrel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
aprel_pr1 : aprel >-> hrel. | Coercion | aprel_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"aprel",
"hrel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | ||
apSet | := ∑ X : hSet, aprel X. | Definition | apSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"aprel",
"hSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apSet_pr1 (X : apSet) : hSet | := pr1 X. | Definition | apSet_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet",
"hSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apSet_pr1 : apSet >-> hSet. | Coercion | apSet_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet",
"hSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | ||
apSet_pr2 (X : apSet) : aprel X | := pr2 X. | Definition | apSet_pr2 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet",
"aprel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x # y" | := (apSet_pr2 _ x y) : ap_scope. | Notation | x # y | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet_pr2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isirreflapSet {X : apSet} :
∏ x : X, ¬ (x # x). | Proof.
exact (pr1 (pr2 (pr2 X))).
Qed. | Lemma | isirreflapSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet"
] | Lemmas about apartness | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
issymmapSet {X : apSet} :
∏ x y : X, x # y → y # x. | Proof.
exact (pr1 (pr2 (pr2 (pr2 X)))).
Qed. | Lemma | issymmapSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
iscotransapSet {X : apSet} :
∏ x y z : X, x # z → x # y ∨ y # z. | Proof.
exact (pr2 (pr2 (pr2 (pr2 X)))).
Qed. | Lemma | iscotransapSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
istight {X : UU} (R : hrel X) | :=
∏ x y : X, ¬ (R x y) → x = y. | Definition | istight | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"UU",
"hrel"
] | ** Tight apartness | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
istightap {X : UU} (ap : hrel X) | :=
isaprel ap × istight ap. | Definition | istightap | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"UU",
"ap",
"hrel",
"isaprel",
"istight"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
tightap (X : UU) | := ∑ ap : hrel X, istightap ap. | Definition | tightap | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"UU",
"ap",
"hrel",
"istightap"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
tightap_aprel {X : UU} (ap : tightap X) : aprel X | := pr1 ap ,, (pr1 (pr2 ap)). | Definition | tightap_aprel | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"UU",
"ap",
"aprel",
"tightap"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
tightap_aprel : tightap >-> aprel. | Coercion | tightap_aprel | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"aprel",
"tightap"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | ||
tightapSet | := ∑ X : hSet, tightap X. | Definition | tightapSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"hSet",
"tightap"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
tightapSet_apSet (X : tightapSet) : apSet | := pr1 X ,, (tightap_aprel (pr2 X)). | Definition | tightapSet_apSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet",
"tightapSet",
"tightap_aprel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
tightapSet_apSet : tightapSet >-> apSet. | Coercion | tightapSet_apSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | ||
tightapSet_rel (X : tightapSet) : hrel X | := (pr1 (pr2 X)). | Definition | tightapSet_rel | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"hrel",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
"x ≠ y" | := (tightapSet_rel _ x y) (at level 70, no associativity) : tap_scope. | Notation | x ≠ y | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"tightapSet_rel"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isirrefltightapSet {X : tightapSet} :
∏ x : X, ¬ (x ≠ x). | Proof.
exact isirreflapSet.
Qed. | Lemma | isirrefltightapSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"isirreflapSet",
"tightapSet"
] | Some lemmas | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
issymmtightapSet {X : tightapSet} :
∏ x y : X, x ≠ y → y ≠ x. | Proof.
exact issymmapSet.
Qed. | Lemma | issymmtightapSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"issymmapSet",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
iscotranstightapSet {X : tightapSet} :
∏ x y z : X, x ≠ z → x ≠ y ∨ y ≠ z. | Proof.
exact iscotransapSet.
Qed. | Lemma | iscotranstightapSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"iscotransapSet",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
istighttightapSet {X : tightapSet} :
∏ x y : X, ¬ (x ≠ y) → x = y. | Proof.
exact (pr2 (pr2 (pr2 X))).
Qed. | Lemma | istighttightapSet | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
istighttightapSet_rev {X : tightapSet} :
∏ x y : X, x = y → ¬ (x ≠ y). | Proof.
intros x _ <-.
now apply isirrefltightapSet.
Qed. | Lemma | istighttightapSet_rev | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"isirrefltightapSet",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
tightapSet_dec {X : tightapSet} :
LEM → ∏ x y : X, (x != y <-> x ≠ y). | Proof.
intros Hdec x y.
destruct (Hdec (x ≠ y)) as [ Hneq | Heq ].
- split.
+ intros _ ; apply Hneq.
+ intros _ Heq.
rewrite <- Heq in Hneq.
revert Hneq.
now apply isirrefltightapSet.
- split.
+ intros Hneq.
apply fromempty, Hneq.
now apply istighttightapSet.
+ intr... | Lemma | tightapSet_dec | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"LEM",
"fromempty",
"isirrefltightapSet",
"istighttightapSet",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isapunop {X : tightapSet} (op :unop X) | :=
∏ x y : X, op x ≠ op y → x ≠ y. | Definition | isapunop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"op",
"tightapSet",
"unop"
] | ** Operations and apartness | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
isaprop_isapunop {X : tightapSet} (op :unop X) :
isaprop (isapunop op). | Proof.
intros ap.
apply impred_isaprop ; intro x.
apply impred_isaprop ; intro y.
apply isapropimpl.
now apply pr2.
Qed. | Lemma | isaprop_isapunop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"ap",
"impred_isaprop",
"isaprop",
"isapropimpl",
"isapunop",
"op",
"tightapSet",
"unop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
islapbinop {X : tightapSet} (op : binop X) | :=
∏ x, isapunop (λ y, op y x). | Definition | islapbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"binop",
"isapunop",
"op",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
israpbinop {X : tightapSet} (op : binop X) | :=
∏ x, isapunop (λ y, op x y). | Definition | israpbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"binop",
"isapunop",
"op",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isapbinop {X : tightapSet} (op : binop X) | :=
(islapbinop op) × (israpbinop op). | Definition | isapbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"binop",
"islapbinop",
"israpbinop",
"op",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isaprop_islapbinop {X : tightapSet} (op : binop X) :
isaprop (islapbinop op). | Proof.
apply impred_isaprop ; intro x.
now apply isaprop_isapunop.
Qed. | Lemma | isaprop_islapbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"binop",
"impred_isaprop",
"isaprop",
"isaprop_isapunop",
"islapbinop",
"op",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isaprop_israpbinop {X : tightapSet} (op : binop X) :
isaprop (israpbinop op). | Proof.
apply impred_isaprop ; intro x.
now apply isaprop_isapunop.
Qed. | Lemma | isaprop_israpbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"binop",
"impred_isaprop",
"isaprop",
"isaprop_isapunop",
"israpbinop",
"op",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isaprop_isapbinop {X : tightapSet} (op :binop X) :
isaprop (isapbinop op). | Proof.
intros ap.
apply isapropdirprod.
now apply isaprop_islapbinop.
now apply isaprop_israpbinop.
Qed. | Lemma | isaprop_isapbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"ap",
"binop",
"isapbinop",
"isaprop",
"isaprop_islapbinop",
"isaprop_israpbinop",
"isapropdirprod",
"op",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apbinop (X : tightapSet) | := ∑ op : binop X, isapbinop op. | Definition | apbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"binop",
"isapbinop",
"op",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apbinop_pr1 {X : tightapSet} (op : apbinop X) : binop X | := pr1 op. | Definition | apbinop_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apbinop",
"binop",
"op",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apbinop_pr1 : apbinop >-> binop. | Coercion | apbinop_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apbinop",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | ||
apsetwithbinop | := ∑ X : tightapSet, apbinop X. | Definition | apsetwithbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apbinop",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apsetwithbinop_pr1 (X : apsetwithbinop) : tightapSet | := pr1 X. | Definition | apsetwithbinop_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwithbinop",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apsetwithbinop_pr1 : apsetwithbinop >-> tightapSet. | Coercion | apsetwithbinop_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwithbinop",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | ||
apsetwithbinop_setwithbinop : apsetwithbinop → setwithbinop | :=
λ X : apsetwithbinop, (apSet_pr1 (apsetwithbinop_pr1 X)),, (pr1 (pr2 X)). | Definition | apsetwithbinop_setwithbinop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet_pr1",
"apsetwithbinop",
"apsetwithbinop_pr1",
"setwithbinop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
op {X : apsetwithbinop} : binop X | := op (X := apsetwithbinop_setwithbinop X). | Definition | op | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwithbinop",
"apsetwithbinop_setwithbinop",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apsetwith2binop | := ∑ X : tightapSet, apbinop X × apbinop X. | Definition | apsetwith2binop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apbinop",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apsetwith2binop_pr1 (X : apsetwith2binop) : tightapSet | := pr1 X. | Definition | apsetwith2binop_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apsetwith2binop_pr1 : apsetwith2binop >-> tightapSet. | Coercion | apsetwith2binop_pr1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop",
"tightapSet"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | ||
apsetwith2binop_setwith2binop : apsetwith2binop → setwith2binop | :=
λ X : apsetwith2binop,
apSet_pr1 (apsetwith2binop_pr1 X),, pr1 (pr1 (pr2 X)),, pr1 (pr2 (pr2 X)). | Definition | apsetwith2binop_setwith2binop | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apSet_pr1",
"apsetwith2binop",
"apsetwith2binop_pr1",
"setwith2binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
op1 {X : apsetwith2binop} : binop X | := op1 (X := apsetwith2binop_setwith2binop X). | Definition | op1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop",
"apsetwith2binop_setwith2binop",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
op2 {X : apsetwith2binop} : binop X | := op2 (X := apsetwith2binop_setwith2binop X). | Definition | op2 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop",
"apsetwith2binop_setwith2binop",
"binop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
islapbinop_op :
∏ x x' y : X, op x y ≠ op x' y → x ≠ x'. | Proof.
intros x y y'.
now apply (pr1 (pr2 (pr2 X))).
Qed. | Lemma | islapbinop_op | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"op",
"x'",
"y'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
israpbinop_op :
∏ x y y' : X, op x y ≠ op x y' → y ≠ y'. | Proof.
intros x y y'.
now apply (pr2 (pr2 (pr2 X))).
Qed. | Lemma | israpbinop_op | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"op",
"y'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isapbinop_op :
∏ x x' y y' : X, op x y ≠ op x' y' → x ≠ x' ∨ y ≠ y'. | Proof.
intros x x' y y' Hop.
apply (iscotranstightapSet _ (op x' y)) in Hop.
revert Hop ; apply hinhfun ; intros [Hop | Hop].
- left ; revert Hop.
now apply islapbinop_op.
- right ; revert Hop.
now apply israpbinop_op.
Qed. | Lemma | isapbinop_op | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"hinhfun",
"iscotranstightapSet",
"islapbinop_op",
"israpbinop_op",
"left",
"op",
"right",
"x'",
"y'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apsetwith2binop_apsetwithbinop1 : apsetwithbinop | :=
(pr1 X) ,, (pr1 (pr2 X)). | Definition | apsetwith2binop_apsetwithbinop1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwithbinop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
apsetwith2binop_apsetwithbinop2 : apsetwithbinop | :=
(pr1 X) ,, (pr2 (pr2 X)). | Definition | apsetwith2binop_apsetwithbinop2 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwithbinop"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
islapbinop_op1 :
∏ x x' y : X, op1 x y ≠ op1 x' y → x ≠ x'. | Proof.
exact (islapbinop_op (X := apsetwith2binop_apsetwithbinop1)).
Qed. | Lemma | islapbinop_op1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop_apsetwithbinop1",
"islapbinop_op",
"op1",
"x'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
israpbinop_op1 :
∏ x y y' : X, op1 x y ≠ op1 x y' → y ≠ y'. | Proof.
exact (israpbinop_op (X := apsetwith2binop_apsetwithbinop1)).
Qed. | Lemma | israpbinop_op1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop_apsetwithbinop1",
"israpbinop_op",
"op1",
"y'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isapbinop_op1 :
∏ x x' y y' : X, op1 x y ≠ op1 x' y' → x ≠ x' ∨ y ≠ y'. | Proof.
exact (isapbinop_op (X := apsetwith2binop_apsetwithbinop1)).
Qed. | Lemma | isapbinop_op1 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop_apsetwithbinop1",
"isapbinop_op",
"op1",
"x'",
"y'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
islapbinop_op2 :
∏ x x' y : X, op2 x y ≠ op2 x' y → x ≠ x'. | Proof.
exact (islapbinop_op (X := apsetwith2binop_apsetwithbinop2)).
Qed. | Lemma | islapbinop_op2 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop_apsetwithbinop2",
"islapbinop_op",
"op2",
"x'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
israpbinop_op2 :
∏ x y y' : X, op2 x y ≠ op2 x y' → y ≠ y'. | Proof.
exact (israpbinop_op (X := apsetwith2binop_apsetwithbinop2)).
Qed. | Lemma | israpbinop_op2 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop_apsetwithbinop2",
"israpbinop_op",
"op2",
"y'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isapbinop_op2 :
∏ x x' y y' : X, op2 x y ≠ op2 x' y' → x ≠ x' ∨ y ≠ y'. | Proof.
exact (isapbinop_op (X := apsetwith2binop_apsetwithbinop2)).
Qed. | Lemma | isapbinop_op2 | Algebra | UniMath/Algebra/Apartness.v | [
"UniMath.Algebra.BinaryOperations",
"UniMath.Foundations.Propositions",
"UniMath.MoreFoundations.Sets",
"UniMath.MoreFoundations.Tactics",
"UniMath.MoreFoundations.DecidablePropositions"
] | [
"apsetwith2binop_apsetwithbinop2",
"isapbinop_op",
"op2",
"x'",
"y'"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
natmult {X : monoid} (n : nat) (x : X) : X | :=
match n with
| O => 0%addmonoid
| S O => x
| S m => (x + natmult m x)%addmonoid
end. | Fixpoint | natmult | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"monoid",
"nat"
] | ** The standard function from the natural numbers to a monoid | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
nattorig {X : rig} (n : nat) : X | :=
natmult (X := rigaddabmonoid X) n 1%rig. | Definition | nattorig | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"nat",
"natmult",
"rig",
"rigaddabmonoid"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
nattoring {X : ring} (n : nat) : X | :=
nattorig (X := ringtorig X) n. | Definition | nattoring | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"nat",
"nattorig",
"ring",
"ringtorig"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
natmultS :
∏ {X : monoid} (n : nat) (x : X),
natmult (S n) x = (x + natmult n x)%addmonoid. | Proof.
intros X n x.
induction n as [|n].
- now rewrite runax.
- reflexivity.
Qed. | Lemma | natmultS | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"monoid",
"nat",
"natmult",
"runax"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
nattorigS {X : rig} :
∏ (n : nat), nattorig (X := X) (S n) = (1 + nattorig n)%rig. | Proof.
intros.
now apply (natmultS (X := rigaddabmonoid X)).
Qed. | Lemma | nattorigS | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"nat",
"natmultS",
"nattorig",
"rig",
"rigaddabmonoid"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
nattorig_natmult :
∏ {X : rig} (n : nat) (x : X),
(nattorig n * x)%rig = natmult (X := rigaddabmonoid X) n x. | Proof.
intros.
induction n as [|n IHn].
- now apply rigmult0x.
- rewrite nattorigS, natmultS.
now rewrite rigrdistr, IHn, riglunax2.
Qed. | Lemma | nattorig_natmult | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"nat",
"natmult",
"natmultS",
"nattorig",
"nattorigS",
"rig",
"rigaddabmonoid",
"riglunax2",
"rigmult0x",
"rigrdistr"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
natmult_plus :
∏ {X : monoid} (n m : nat) (x : X),
natmult (n + m) x = (natmult n x + natmult m x)%addmonoid. | Proof.
induction n as [|n IHn] ; intros m x.
- rewrite lunax.
reflexivity.
- change (S n + m)%nat with (S (n + m))%nat.
rewrite !natmultS, IHn, assocax.
reflexivity.
Qed. | Lemma | natmult_plus | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"assocax",
"lunax",
"monoid",
"nat",
"natmult",
"natmultS"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
nattorig_plus :
∏ {X : rig} (n m : nat),
(nattorig (n + m) : X) = (nattorig n + nattorig m)%rig. | Proof.
intros X n m.
apply (natmult_plus (X := rigaddabmonoid X)).
Qed. | Lemma | nattorig_plus | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"nat",
"natmult_plus",
"nattorig",
"rig",
"rigaddabmonoid"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
natmult_mult :
∏ {X : monoid} (n m : nat) (x : X),
natmult (n * m) x = (natmult n (natmult m x))%addmonoid. | Proof.
induction n as [|n IHn] ; intros m x.
- reflexivity.
- simpl (_ * _)%nat.
assert (H : S n = (n + 1)%nat).
{ rewrite <- plus_n_Sm, natplusr0.
reflexivity. }
rewrite H ; clear H.
rewrite !natmult_plus, IHn.
reflexivity.
Qed. | Lemma | natmult_mult | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"monoid",
"nat",
"natmult",
"natmult_plus",
"natplusr0",
"plus_n_Sm"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
nattorig_mult :
∏ {X : rig} (n m : nat),
(nattorig (n * m) : X) = (nattorig n * nattorig m)%rig. | Proof.
intros X n m.
unfold nattorig.
rewrite (natmult_mult (X := rigaddabmonoid X)), <- nattorig_natmult.
reflexivity.
Qed. | Lemma | nattorig_mult | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"nat",
"natmult_mult",
"nattorig",
"nattorig_natmult",
"rig",
"rigaddabmonoid"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
natmult_op {X : monoid} :
∏ (n : nat) (x y : X),
(x + y = y + x)%addmonoid
→ natmult n (x + y)%addmonoid = (natmult n x + natmult n y)%addmonoid. | Proof.
intros.
induction n as [|n IHn].
- rewrite lunax.
reflexivity.
- rewrite natmultS, assocax, IHn, <- (assocax _ y).
assert (X1 : (y + natmult n x = natmult n x + y)%addmonoid).
{ clear IHn.
induction n as [|n IHn].
- rewrite lunax, runax.
reflexivity.
- rewrite !natmu... | Lemma | natmult_op | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"assocax",
"lunax",
"monoid",
"nat",
"natmult",
"natmultS",
"runax"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
natmult_binophrel {X : monoid} (R : hrel X) :
istrans R → isbinophrel R →
∏ (n : nat) (x y : X), R x y → R (natmult (S n) x) (natmult (S n) y). | Proof.
intros Hr Hop n x y H.
induction n as [|n IHn].
exact H.
rewrite !(natmultS (S _)).
eapply Hr.
apply (pr1 Hop).
exact IHn.
apply (pr2 Hop).
exact H.
Qed. | Lemma | natmult_binophrel | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"isbinophrel",
"istrans",
"monoid",
"nat",
"natmult",
"natmultS"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
setquot_aux {X : monoid} (R : hrel X) : hrel X | :=
λ x y : X, ∃ c : X, R (x + c)%addmonoid (y + c)%addmonoid. | Definition | setquot_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"monoid"
] | ** relation | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
istrans_setquot_aux {X : abmonoid} (R : hrel X) :
istrans R → isbinophrel R → istrans (setquot_aux R). | Proof.
intros Hr Hop.
intros x y z.
apply hinhfun2.
intros (c1,Hc1) (c2,Hc2).
exists (c1 + c2)%addmonoid.
eapply Hr.
rewrite <- assocax.
apply (pr2 Hop).
exact Hc1.
rewrite assocax, (commax _ c1), <- !assocax.
apply (pr2 Hop).
exact Hc2.
Qed. | Lemma | istrans_setquot_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abmonoid",
"assocax",
"c1",
"commax",
"hinhfun2",
"hrel",
"isbinophrel",
"istrans",
"setquot_aux"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isbinophrel_setquot_aux {X : abmonoid} (R : hrel X) :
isbinophrel R → isbinophrel (setquot_aux R). | Proof.
intros Hop.
split.
- intros x y z.
apply hinhfun.
intros (c,Hc).
exists c.
rewrite !assocax.
apply (pr1 Hop).
exact Hc.
- intros x y z.
apply hinhfun.
intros (c,Hc).
exists c.
rewrite !assocax, (commax _ z), <- !assocax.
apply (pr2 Hop).
exact Hc.
Qed. | Lemma | isbinophrel_setquot_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abmonoid",
"assocax",
"commax",
"hinhfun",
"hrel",
"isbinophrel",
"setquot_aux"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isequiv_setquot_aux {X : abmonoid} (R : hrel X) :
isinvbinophrel R →
∏ x y : X, (setquot_aux R) x y <-> R x y. | Proof.
intros H x y.
split.
apply hinhuniv.
intros (c,H').
generalize H'; clear H'.
apply (pr2 H).
intros H1.
apply hinhpr.
exists 0%addmonoid.
rewrite !runax.
exact H1.
Qed. | Lemma | isequiv_setquot_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"H'",
"H1",
"abmonoid",
"hinhpr",
"hinhuniv",
"hrel",
"isinvbinophrel",
"runax",
"setquot_aux"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchmonoid {X : abmonoid} (R : hrel X) | :=
∏ x y1 y2 : X,
R y1 y2 →
(∃ n : nat, R (natmult n y1 + x)%addmonoid (natmult n y2))
× (∃ n : nat, R (natmult n y1) (natmult n y2 + x)%addmonoid). | Definition | isarchmonoid | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abmonoid",
"hrel",
"nat",
"natmult",
"y1",
"y2"
] | ** Archimedean property in a monoid | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
isarchmonoid_1 {X : abmonoid} (R : hrel X) :
isarchmonoid R →
∏ x y1 y2 : X,
R y1 y2 →
∃ n : nat, R (natmult n y1 + x)%addmonoid (natmult n y2) | :=
λ H x y1 y2 Hy, (pr1 (H x y1 y2 Hy)). | Definition | isarchmonoid_1 | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abmonoid",
"hrel",
"isarchmonoid",
"nat",
"natmult",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchmonoid_2 {X : abmonoid} (R : hrel X) :
isarchmonoid R →
∏ x y1 y2 : X,
R y1 y2 →
∃ n : nat, R (natmult n y1) (natmult n y2 + x)%addmonoid | :=
λ H x y1 y2 Hy, (pr2 (H x y1 y2 Hy)). | Definition | isarchmonoid_2 | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abmonoid",
"hrel",
"isarchmonoid",
"nat",
"natmult",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchgr {X : abgr} (R : hrel X) | :=
∏ x y1 y2 : X,
R y1 y2 →
∃ n : nat, R (natmult n y1 + x)%addmonoid (natmult n y2). | Definition | isarchgr | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abgr",
"hrel",
"nat",
"natmult",
"y1",
"y2"
] | ** Archimedean property in a group | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
isarchgr_isarchmonoid_aux {X : abgr} (R : hrel X) :
isbinophrel R →
∏ (n : nat) (x y1 y2 : X),
R (natmult n y1 * grinv X x)%multmonoid (natmult n y2) → R (natmult n y1) (natmult n y2 * x)%multmonoid. | Proof.
intros Hop.
intros n x y1 y2 Hy.
apply (pr2 (isinvbinophrelgr _ Hop) _ _ (grinv X x)).
rewrite assocax, (grrinvax X), runax.
exact Hy.
Qed. | Lemma | isarchgr_isarchmonoid_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abgr",
"assocax",
"grinv",
"grrinvax",
"hrel",
"isbinophrel",
"isinvbinophrelgr",
"nat",
"natmult",
"runax",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchgr_isarchmonoid {X : abgr} (R : hrel X) :
isbinophrel R →
isarchgr R →
isarchmonoid (X := abgrtoabmonoid X) R. | Proof.
intros Hop H x y1 y2 Hy.
split.
- now apply H.
- generalize (H (grinv X x) _ _ Hy).
apply hinhfun.
intros (n,Hn).
exists n.
apply isarchgr_isarchmonoid_aux.
exact Hop.
exact Hn.
Defined. | Lemma | isarchgr_isarchmonoid | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abgr",
"abgrtoabmonoid",
"grinv",
"hinhfun",
"hrel",
"isarchgr",
"isarchgr_isarchmonoid_aux",
"isarchmonoid",
"isbinophrel",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchmonoid_isarchgr {X : abgr} (R : hrel X) :
isarchmonoid (X := abgrtoabmonoid X) R →
isarchgr R. | Proof.
intros H x y1 y2 Hy.
apply (isarchmonoid_1 _ H x y1 y2 Hy).
Defined. | Lemma | isarchmonoid_isarchgr | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abgr",
"abgrtoabmonoid",
"hrel",
"isarchgr",
"isarchmonoid",
"isarchmonoid_1",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchabgrdiff_aux {X : abmonoid} (R : hrel X)
(Hr : isbinophrel R)
(Hr' : istrans R)
(y1 y2 x : abmonoiddirprod X X)
(n1 : nat)
(Hn1 : setquot_aux R (natmult n1 (pr1 y1 * pr2 y2) * pr1 x)%multmonoid
(natmult n1 (pr1 y2 * pr2 y1)%multmonoid))
(n2 : nat)
... | Proof.
intros.
assert (H0 : ∏ n y, natmult (X := abgrdiff X) n (setquotpr (binopeqrelabgrdiff X) y) = setquotpr (binopeqrelabgrdiff X) (natmult n (pr1 y) ,, natmult n (pr2 y))).
{ intros n y.
induction n as [|n IHn].
reflexivity.
rewrite !natmultS, IHn.
reflexivity. }
rewrite !H0.
revert Hn1 H... | Lemma | isarchabgrdiff_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abgrdiff",
"abgrdiffrel",
"abmonoid",
"abmonoiddirprod",
"assocax",
"binopeqrelabgrdiff",
"c1",
"commax",
"hinhfun2",
"hrel",
"isbinophrel",
"istrans",
"maponpaths",
"nat",
"natmult",
"natmultS",
"natmult_op",
"natmult_plus",
"setquot_aux",
"setquotpr",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchabgrdiff {X : abmonoid} (R : hrel X) (Hr : isbinophrel R) :
istrans R →
isarchmonoid (setquot_aux R) →
isarchgr (X := abgrdiff X) (abgrdiffrel X (L := R) Hr). | Proof.
intros Hr' H.
simple refine (setquotuniv3prop _ (λ x y1 y2, (abgrdiffrel X Hr y1 y2 →
∃ n : nat, abgrdiffrel X Hr (natmult (X := abgrdiff X) n y1 * x)%multmonoid (natmult (X := abgrdiff X) n y2)) ,, _) _).
abstract apply isapropimpl, propproperty.
intros x y1 y2 Hy.
eapply hinhfun2.
2: apply (isa... | Lemma | isarchabgrdiff | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"abgrdiff",
"abgrdiffrel",
"abmonoid",
"hinhfun2",
"hrel",
"isapropimpl",
"isarchabgrdiff_aux",
"isarchgr",
"isarchmonoid",
"isarchmonoid_1",
"isarchmonoid_2",
"isbinophrel",
"istrans",
"nat",
"natmult",
"propproperty",
"setquot_aux",
"setquotuniv3prop",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchrig {X : rig} (R : hrel X) | :=
(∏ y1 y2 : X, R y1 y2 → ∃ n : nat, R (nattorig n * y1)%rig (1 + nattorig n * y2)%rig)
× (∏ x : X, ∃ n : nat, R (nattorig n) x)
× (∏ x : X, ∃ n : nat, R (nattorig n + x)%rig 0%rig). | Definition | isarchrig | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"nat",
"nattorig",
"rig",
"y1",
"y2"
] | ** Archimedean property in a rig | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
isarchrig_diff {X : rig} (R : hrel X) :
isarchrig R →
∏ y1 y2 : X, R y1 y2 → ∃ n : nat, R (nattorig n * y1)%rig (1 + nattorig n * y2)%rig | :=
pr1. | Definition | isarchrig_diff | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"isarchrig",
"nat",
"nattorig",
"rig",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchrig_gt {X : rig} (R : hrel X) :
isarchrig R →
∏ x : X, ∃ n : nat, R (nattorig n) x | :=
λ H, (pr1 (pr2 H)). | Definition | isarchrig_gt | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"isarchrig",
"nat",
"nattorig",
"rig"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchrig_pos {X : rig} (R : hrel X) :
isarchrig R →
∏ x : X, ∃ n : nat, R (nattorig n + x)%rig 0%rig | :=
λ H, (pr2 (pr2 H)). | Definition | isarchrig_pos | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"isarchrig",
"nat",
"nattorig",
"rig"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchrig_setquot_aux {X : rig} (R : hrel X) :
isinvbinophrel (X := rigaddabmonoid X) R
→ isarchrig R
→ isarchrig (setquot_aux (X := rigaddabmonoid X) R). | Proof.
intros Hr H.
split ; [ | split].
- intros y1 y2.
apply hinhuniv.
intros Hy.
generalize (isarchrig_diff R H y1 y2 (pr2 Hr _ _ _ (pr2 Hy))).
apply hinhfun.
intros n.
exists (pr1 n).
apply hinhpr.
exists 0%rig.
rewrite runax, runax.
exact (pr2 n).
- intros x.
gene... | Lemma | isarchrig_setquot_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hinhfun",
"hinhpr",
"hinhuniv",
"hrel",
"isarchrig",
"isarchrig_diff",
"isarchrig_gt",
"isarchrig_pos",
"isinvbinophrel",
"rig",
"rigaddabmonoid",
"runax",
"setquot_aux",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchrig_isarchmonoid_1_aux {X : rig} (R : hrel X)
(Hr1 : R 1%rig 0%rig)
(Hr : istrans R)
(Hop1 : isbinophrel (X := rigaddabmonoid X) R)
(x y1 y2 : rigaddabmonoid X)
(m : nat)
(Hm : R (nattorig m * y1)%ring (1%rig + nattorig m * y2)%ring)
(n : nat)
(Hn : R (nattorig n + ... | Proof.
intros.
rewrite !nattorig_natmult in Hm.
destruct n.
+ rewrite riglunax1 in Hn.
eapply Hr.
apply (pr1 Hop1).
exact Hn.
rewrite runax.
simpl.
rewrite <- (lunax _ (natmult m y2)).
eapply Hr.
exact Hm.
apply (pr2 Hop1).
exact Hr1.
+ rewrite natmult_mult.
simpl m... | Lemma | isarchrig_isarchmonoid_1_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"Hm",
"assocax",
"hrel",
"isbinophrel",
"istrans",
"lunax",
"max",
"nat",
"natmult",
"natmult_binophrel",
"natmult_mult",
"natmult_op",
"nattorig",
"nattorig_natmult",
"op",
"op1",
"rig",
"rigaddabmonoid",
"rigcomm1",
"riglunax1",
"ring",
"runax",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchrig_isarchmonoid_2_aux {X : rig} (R : hrel X)
(Hr1 : R 1%rig 0%rig)
(Hr : istrans R)
(Hop1 : isbinophrel (X := rigaddabmonoid X) R)
(x y1 y2 : rigaddabmonoid X)
(m : nat)
(Hm : R (nattorig m * y1)%ring (1%rig + nattorig m * y2)%ring)
(n : nat)
(Hn : R (nattorig n) x... | Proof.
intros.
rewrite !nattorig_natmult in Hm.
destruct n.
+ eapply Hr.
exact Hm.
rewrite rigcomm1.
apply (pr1 Hop1).
eapply Hr.
exact Hr1.
exact Hn.
+ rewrite natmult_mult.
eapply Hr.
simpl max.
apply natmult_binophrel.
exact Hr.
exact Hop1.
exact Hm.
rewr... | Lemma | isarchrig_isarchmonoid_2_aux | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"Hm",
"hrel",
"isbinophrel",
"istrans",
"max",
"nat",
"natmult",
"natmult_binophrel",
"natmult_mult",
"natmult_op",
"nattorig",
"nattorig_natmult",
"op",
"op1",
"rig",
"rigaddabmonoid",
"rigcomm1",
"ring",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchrig_isarchmonoid {X : rig} (R : hrel X) :
R 1%rig 0%rig →
istrans R → isbinophrel (X := rigaddabmonoid X) R →
isarchrig R → isarchmonoid (X := rigaddabmonoid X) R. | Proof.
intros Hr1 Hr Hop1 H x y1 y2 Hy.
split.
- generalize (isarchrig_diff _ H _ _ Hy) (isarchrig_pos _ H x).
apply hinhfun2.
intros m n.
exists (max 1 (pr1 n) * (pr1 m))%nat.
apply isarchrig_isarchmonoid_1_aux.
exact Hr1.
exact Hr.
exact Hop1.
exact (pr2 m).
exact (pr2 n).
... | Lemma | isarchrig_isarchmonoid | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hinhfun2",
"hrel",
"isarchmonoid",
"isarchrig",
"isarchrig_diff",
"isarchrig_gt",
"isarchrig_isarchmonoid_1_aux",
"isarchrig_isarchmonoid_2_aux",
"isarchrig_pos",
"isbinophrel",
"istrans",
"max",
"nat",
"rig",
"rigaddabmonoid",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchmonoid_isarchrig {X : rig} (R : hrel X) :
(R 1%rig 0%rig)
→ isarchmonoid (X := rigaddabmonoid X) R
→ isarchrig R. | Proof.
intros H01 H.
repeat split.
- intros y1 y2 Hy.
generalize (isarchmonoid_2 _ H 1%rig y1 y2 Hy).
apply hinhfun.
intros n.
exists (pr1 n).
abstract (rewrite !nattorig_natmult, rigcomm1 ;
exact (pr2 n)).
- intros x.
generalize (isarchmonoid_2 _ H x _ _ H01).
apply h... | Lemma | isarchmonoid_isarchrig | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hinhfun",
"hrel",
"isarchmonoid",
"isarchmonoid_1",
"isarchmonoid_2",
"isarchrig",
"nattorig",
"nattorig_natmult",
"rig",
"rigaddabmonoid",
"rigcomm1",
"riglunax1",
"rigmultx0",
"y1",
"y2"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchring {X : ring} (R : hrel X) | :=
(∏ x : X, R x 0%ring → ∃ n : nat, R (nattoring n * x)%ring 1%ring)
× (∏ x : X, ∃ n : nat, R (nattoring n) x). | Definition | isarchring | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"nat",
"nattoring",
"ring"
] | ** Archimedean property in a ring | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
isarchring_1 {X : ring} (R : hrel X) :
isarchring R →
∏ x : X, R x 0%ring → ∃ n : nat, R (nattoring n * x)%ring 1%ring | := pr1. | Definition | isarchring_1 | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"isarchring",
"nat",
"nattoring",
"ring"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchring_2 {X : ring} (R : hrel X) :
isarchring R →
∏ x : X, ∃ n : nat, R (nattoring n) x | := pr2. | Definition | isarchring_2 | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"isarchring",
"nat",
"nattoring",
"ring"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchring_isarchrig {X : ring} (R : hrel X) :
isbinophrel (X := rigaddabmonoid X) R →
isarchring R → isarchrig (X := ringtorig X) R. | Proof.
intros Hop1 H.
repeat split.
- intros y1 y2 Hy.
assert (X0 : R (y1 - y2)%ring 0%ring).
abstract (apply (pr2 (isinvbinophrelgr X Hop1)) with y2 ;
change BinaryOperations.op with (@BinaryOperations.op1 X) ;
rewrite ringassoc1, ringlinvax1, ringlunax1, ringrunax1 ;
... | Lemma | isarchring_isarchrig | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hinhfun",
"hrel",
"isarchrig",
"isarchring",
"isarchring_1",
"isarchring_2",
"isbinophrel",
"isinvbinophrelgr",
"nattorig",
"op",
"op1",
"rig",
"rigaddabmonoid",
"ring",
"ringassoc1",
"ringldistr",
"ringlinvax1",
"ringlunax1",
"ringrmultminus",
"ringrunax1",
"ringtorig",
"... | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchrig_isarchring {X : ring} (R : hrel X) :
isbinophrel (X := rigaddabmonoid X) R →
isarchrig (X := ringtorig X) R → isarchring R. | Proof.
intros Hr H.
split.
- intros x Hx.
generalize (isarchrig_diff _ H _ _ Hx).
apply hinhfun.
intros (n,Hn).
exists n.
rewrite <- (ringrunax1 _ 1%ring), <- (ringmultx0 _ (nattoring n)).
exact Hn.
- apply (isarchrig_gt (X := ringtorig X)).
exact H.
Defined. | Lemma | isarchrig_isarchring | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hinhfun",
"hrel",
"isarchrig",
"isarchrig_diff",
"isarchrig_gt",
"isarchring",
"isbinophrel",
"nattoring",
"rigaddabmonoid",
"ring",
"ringmultx0",
"ringrunax1",
"ringtorig"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 | |
isarchring_isarchgr {X : ring} (R : hrel X) :
R 1%ring 0%ring →
istrans R →
isbinophrel (X := X) R →
isarchring R → isarchgr (X := X) R. | Proof.
intros Hr0 Hr Hop1 H.
apply isarchmonoid_isarchgr.
apply (isarchrig_isarchmonoid (X := X)).
exact Hr0.
exact Hr.
exact Hop1.
now apply isarchring_isarchrig.
Defined. | Lemma | isarchring_isarchgr | Algebra | UniMath/Algebra/Archimedean.v | [
"UniMath.Foundations.NaturalNumbers",
"UniMath.Algebra.RigsAndRings",
"UniMath.Algebra.Groups",
"UniMath.Algebra.DivisionRig",
"UniMath.Algebra.Domains_and_Fields",
"UniMath.Algebra.ConstructiveStructures",
"UniMath.MoreFoundations.Tactics"
] | [
"hrel",
"isarchgr",
"isarchmonoid_isarchgr",
"isarchrig_isarchmonoid",
"isarchring",
"isarchring_isarchrig",
"isbinophrel",
"istrans",
"ring"
] | https://github.com/UniMath/UniMath | fa8f7d65ac96baddca8614f1c7bdfa73f043aef5 |
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