phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	source_url stringclasses 1 value	commit stringclasses 1 value
Nnat_spec (i : N) (n : nat) : N -> nat -> bool -> Set	:= \| Nnat_spec_false : Nnat_spec i n i n false \| Nnat_spec_N0 : i = N0 -> n = 0%N -> Nnat_spec i n N0 0%N true \| Nnat_spec_Npos : forall p', i = Npos p' -> pos_nat p' n -> Nnat_spec i n (Npos p') n true.	Variant	Nnat_spec	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "nat", "pos_nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NnatP i n : Nnat_spec i n i n (Nnat i n).	Proof. case: (boolP (Nnat i n)) => /eqP Nin; last exact: Nnat_spec_false. by case: i n Nin => [\|p] n <- /=; [exact: Nnat_spec_N0\|exact: Nnat_spec_Npos]. Qed.	Lemma	NnatP	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Nnat", "Nnat_spec", "last" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Nnat0 : Nnat N0 0.	Proof. by []. Qed.	Lemma	Nnat0	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Nnat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Nnat_pos p n : Nnat (Npos p) n = pos_nat p n.	Proof. by []. Qed.	Lemma	Nnat_pos	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Nnat", "pos_nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NnatE	:= (Nnat0, Nnat_pos).	Definition	NnatE	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Nnat0", "Nnat_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NnatD i n (Nin : Nnat i n) i' n' (Ni'n' : Nnat i' n') : Nnat (N.add i i') (n + n').	Proof. case: NnatP Nin => [//\|/[!add0n]//\| {i}p _ pn _]. case: NnatP Ni'n' => [//\|/[!addn0]//\| {i'}p' _ p'n' _]. by rewrite Nnat_pos pos_natD. Qed.	Lemma	NnatD	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Nnat", "NnatP", "Nnat_pos", "add", "add0n", "addn0", "n'", "pos_natD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NnatB i n (Nin : Nnat i n) i' n' (Ni'n' : Nnat i' n') : Nnat (N.sub i i') (n - n').	Proof. case: NnatP Nin => [//\|//\|{}i _ pin _]. case: NnatP Ni'n' => [//\|_ _ _\|{}i' _ pi'n' _ /=]; first by rewrite subn0. case: (ltnP n n') => nn'; rewrite /Nnat eq_sym. by rewrite Pos_sub_mask_Neg ?(eqP pin) ?(eqP pi'n')// subn_eq0 ltnW. by case: Pos.sub_mask (mask_natB pin pi'n' nn') => //= ? /eqP->. Qed.	Lemma	NnatB	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Nnat", "NnatP", "Pos_sub_mask_Neg", "eq_sym", "ltnP", "ltnW", "mask_natB", "n'", "sub", "subn0", "subn_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NnatM i n (Nin : Nnat i n) i' n' (Ni'n' : Nnat i' n') : Nnat (N.mul i i') (n * n').	Proof. case: NnatP Nin => [//\|/[!mul0n]//\| {i}p _ pn _]. case: NnatP Ni'n' => [//\|/[!muln0]//\| {i'}p' _ p'n' _]. by rewrite Nnat_pos pos_natM. Qed.	Lemma	NnatM	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Nnat", "NnatP", "Nnat_pos", "mul", "mul0n", "muln0", "n'", "pos_natM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Nnat_eq i n (Nin : Nnat i n) i' n' (Ni'n' : Nnat i' n') : N.eqb i i' = (n == n').	Proof. case: NnatP Nin => [//\|_ _ _\| {i}p _ pn _]. by case: NnatP Ni'n' => [//\|//\| {i'}p' _ + _] => /pos_nat_exS[{}n'->]. case: NnatP Ni'n' => [//\|_ _ _\| {i'}p' _ p'n' _]. by move: pn => /pos_nat_exS[{}n->]. exact: (pos_nat_eq pn p'n'). Qed.	Lemma	Nnat_eq	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Nnat", "NnatP", "eqb", "n'", "pos_nat_eq", "pos_nat_exS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
N_to_natI : injective N.to_nat.	Proof. by case=> [\|i] [\|i'] => [//\|/esym/eqP\|/eqP\|/Pos_to_natI->//] /[!Pos_to_nat0F]. Qed.	Lemma	N_to_natI	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Pos_to_nat0F", "Pos_to_natI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
N_to_natB i j : N.to_nat (N.sub i j) = (N.to_nat i - N.to_nat j)%N.	Proof. exact: eqP (NnatB (Nnat_N_to_nat i) (Nnat_N_to_nat j)). Qed.	Lemma	N_to_natB	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NnatB", "Nnat_N_to_nat", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
int_of_Z (i : Z) : int	:= match i with \| Z0 => Posz 0 \| Zpos p => Posz (Pos.to_nat p) \| Zneg p => Negz (Pos.to_nat p).-1 end.	Definition	int_of_Z	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Posz", "int" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint (i : Z) (n : int)	:= int_of_Z i == n.	Definition	Zint	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "int", "int_of_Z" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_int_of_Z i : Zint i (int_of_Z i).	Proof. exact/eqP. Qed.	Lemma	Zint_int_of_Z	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Zint", "int_of_Z" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_spec (i : Z) (n : int) : Z -> int -> bool -> Set	:= \| Zint_spec_false : Zint_spec i n i n false \| Zint_spec_Z0 : i = Z0 -> n = Posz 0 -> Zint_spec i n Z0 (Posz 0) true \| Zint_spec_Zpos : forall p' n', i = Zpos p' -> n = Posz n' -> pos_nat p' n' -> Zint_spec i n (Zpos p') (Posz n') true \| Zint_spec_Zneg : forall p' n', i = Zneg p' -> n = Ne...	Variant	Zint_spec	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Posz", "int", "n'", "pos_nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ZintP i n : Zint_spec i n i n (Zint i n).	Proof. case: (boolP (Zint i n)) => /eqP Zin; last exact: Zint_spec_false. case: i n Zin => [\|p\|p] n <- /=; [exact: Zint_spec_Z0\|exact: Zint_spec_Zpos\|]. by apply: Zint_spec_Zneg; rewrite ?prednK ?Pos_to_nat_gt0. Qed.	Lemma	ZintP	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Pos_to_nat_gt0", "Zint", "Zint_spec", "apply", "last", "prednK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint0 : Zint Z0 0.	Proof. by []. Qed.	Lemma	Zint0	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Zint" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_pos p n : Zint (Zpos p) (Posz n) = pos_nat p n.	Proof. by []. Qed.	Lemma	Zint_pos	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Posz", "Zint", "pos_nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_neg p n : Zint (Zneg p) (Negz n) = pos_nat p n.+1.	Proof. by apply/idP/idP; [case: ZintP => // _ _ [<-] [<-]\|rewrite /Zint/= => /eqP->]. Qed.	Lemma	Zint_neg	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Zint", "ZintP", "apply", "pos_nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ZintE	:= (Zint0, Zint_pos, Zint_neg).	Definition	ZintE	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Zint0", "Zint_neg", "Zint_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_double i n : Zint i n -> Zint (Z.double i) (2 * n).	Proof. case: ZintP => // {i}p {}n _ _ pn _ /=; first by rewrite ZintE mul2n pos_natE. by rewrite -Negz_doubleS ZintE -doubleS pos_natE. Qed.	Lemma	Zint_double	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Negz_doubleS", "Zint", "ZintE", "ZintP", "double", "doubleS", "mul2n", "pos_natE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_succ_double i n : Zint i n -> Zint (Z.succ_double i) (2 * n + 1).	Proof. case: ZintP => // {i}p {}n _ _ pn _ /=. by rewrite ZintE mul2n addn1 pos_natE. by rewrite -Negz_doubleS NegzS addrK ZintE -predn_doubleS pos_nat_pred_double. Qed.	Lemma	Zint_succ_double	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzS", "Negz_doubleS", "Zint", "ZintE", "ZintP", "addn1", "addrK", "mul2n", "pos_natE", "pos_nat_pred_double", "predn_doubleS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_pred_double i n : Zint i n -> Zint (Z.pred_double i) (2 * n - 1).	Proof. case: ZintP => // {i}p {}n _ _/= => [/pos_nat_exS[{}n-> pn] \| pn] _; last first. by rewrite -Negz_doubleS -NegzS ZintE -doubleS pos_natE. rewrite -PoszM mul2n doubleS -addn1 PoszD addrK. by rewrite ZintE -predn_doubleS pos_nat_pred_double. Qed.	Lemma	Zint_pred_double	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzS", "Negz_doubleS", "PoszD", "PoszM", "Zint", "ZintE", "ZintP", "addn1", "addrK", "doubleS", "last", "mul2n", "pos_natE", "pos_nat_exS", "pos_nat_pred_double", "predn_doubleS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_pos_sub p n p' n' : pos_nat p n -> pos_nat p' n' -> Zint (Z.pos_sub p p') (Posz n - Posz n').	Proof. move=> pn p'n'; elim/pos_nat_ind: p'n' p n pn => [p n\|\|]; [\|move=> {}p' {}n' p'n' IH p n..]. - case: pos_natP => // {}p {}n _ _ + _ /=; last first. by rewrite -addn1 PoszD addrK ZintE pos_natE. move=> /pos_nat_exS[{}n-> pn]; rewrite doubleS -addn1 PoszD addrK. by rewrite ZintE -predn_doubleS pos_nat_...	Lemma	Zint_pos_sub	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzE", "Posz", "PoszD", "PoszM", "Zint", "ZintE", "Zint_double", "Zint_pred_double", "Zint_succ_double", "add1n", "addn1", "addrA", "addrAC", "addrK", "addrKA", "doubleS", "last", "mul2n", "mulrBr", "n'", "opprB", "opprD", "pos_nat", "pos_natE", "pos_natP", "pos_n...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ZintD i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Zint (Z.add i i') (n + n').	Proof. case: ZintP Zin => [//\|\|\|]; [\|move=> {i}p {}n _ _ pn _..]. - by rewrite add0r. - case: ZintP Zi'n' => [//\|\|\|]; [\|move=> {i'}p' {}n' _ _ p'n' _ /=..]. + by rewrite addr0. + by rewrite ZintE pos_natD. + by rewrite NegzE Zint_pos_sub. - case: ZintP Zi'n' => [//\|\|\|]; [\|move=> {i'}p' {}n' _ _ p'n' _ /=..]. + ...	Lemma	ZintD	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzE", "Posz", "PoszD", "Zint", "ZintE", "ZintP", "Zint_pos_sub", "add", "add0r", "addSn", "addr0", "addrC", "n'", "opprD", "pos_natD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ZintN i n : Zint i n -> Zint (Z.opp i) (- n).	Proof. by case: ZintP => // {i}p {}n _ _ /pos_nat_exS[{}n-> pn]; rewrite -NegzE ZintE. Qed.	Lemma	ZintN	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzE", "Zint", "ZintE", "ZintP", "opp", "pos_nat_exS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ZintB i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Zint (Z.sub i i') (n - n').	Proof. by apply: ZintD => //; apply: ZintN. Qed.	Lemma	ZintB	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Zint", "ZintD", "ZintN", "apply", "n'", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ZintM i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Zint (Z.mul i i') (n * n').	Proof. case: ZintP Zin => [//\|\|\|]; [\|move=> {i}p {}n _ _ pn _..]. - by rewrite mul0r. - case: ZintP Zi'n' => [//\|\|\|] /=; [\|move=> {i'}p' {}n' _ _ p'n' _..]. + by rewrite mulr0. + by rewrite ZintE pos_natM. + move: pn => /pos_nat_exS[{}n-> pn]; rewrite NegzE mulrN -PoszM. by rewrite mulnS addSn -NegzE ZintE -a...	Lemma	ZintM	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzE", "PoszM", "Zint", "ZintE", "ZintP", "addSn", "mul", "mul0r", "mulNr", "mulnS", "mulr0", "mulrN", "mulrNN", "n'", "pos_natM", "pos_nat_exS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_pow_pos i n (Zin : Zint i n) p m (pm : pos_nat p m) : Zint (Z.pow_pos i p) (n ^+ m).	Proof. rewrite /Z.pow_pos -[n ^+ m]mul1r; have : Zint (Zpos xH) 1 by []. elim/pos_nat_ind: pm i n Zin (Zpos xH) 1 => [\| {}p {}m pm IH \| {}p {}m pm IH] i n Zin j k Zjk /=. - by rewrite expr1 mulrC; apply: ZintM. - by rewrite -addnn exprD mulrA; apply: (IH) => //; apply: IH. - rewrite -addn1 -addnn !exprD expr1 !mulr...	Lemma	Zint_pow_pos	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Zint", "ZintM", "addn1", "addnn", "apply", "expr1", "exprD", "mul1r", "mulrA", "mulrC", "pos_nat", "pos_nat_ind", "pow_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_eq i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Z.eqb i i' = (n == n').	Proof. case: ZintP Zin => [//\|\|\|]; [\|move=> {i}p {}n _ _ pn _..]. - case: ZintP Zi'n' => [//\|//\|\|//] /= {i'}p' {}n' _ _ p'n' _ _ _ _. by apply/esym/negbTE; rewrite eq_sym eqz_nat -(eqP p'n') Pos_to_nat0F. - case: ZintP Zi'n' => [//\|_ _ _\|{i'}p' {}n' _ _ p'n' _\|//] /=. + by apply/esym/negbTE; rewrite eqz_nat -(eqP p...	Lemma	Zint_eq	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzE", "Pos_to_nat0F", "Zint", "ZintP", "apply", "eq_sym", "eqb", "eqr_opp", "eqz_nat", "n'", "pos_nat_eq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zint_le i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Z.leb i i' = (n <= n').	Proof. case: ZintP Zin => [//\|\|\|]; [\|move=> {i}p {}n _ _ pn _..]. - by case: ZintP Zi'n'. - case: ZintP Zi'n' => [//\|_ _ _\|{i'}p' {}n' _ _ p'n' _\|//] /=. + by apply/esym/negbTE; rewrite lez0_nat -(eqP pn) Pos_to_nat0F. + exact: pos_nat_le. - case: ZintP Zi'n' => [//\|//\|//\|{i'}p' {}n' _ _ p'n' _ /=]. rewrite !Negz...	Lemma	Zint_le	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzE", "Pos_to_nat0F", "Zint", "ZintP", "apply", "eqSS", "leqNgt", "leqnn", "lerN2", "lez0_nat", "lez_nat", "ltnP", "ltnS", "n'", "pos_nat_compare", "pos_nat_le" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rat_of_Q (q : Q) : rat	:= (int_of_Z (Qnum q))%:~R / (Pos.to_nat (Qden q))%:R.	Definition	rat_of_Q	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "int_of_Z", "rat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat (q : Q) (r : rat)	:= rat_of_Q q == r.	Definition	Qrat	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "rat", "rat_of_Q" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat_rat_of_Q q : Qrat q (rat_of_Q q).	Proof. exact/eqP. Qed.	Lemma	Qrat_rat_of_Q	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat", "rat_of_Q" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat_spec (q : Q) (r : rat) : Prop	:= Qrat_spec_Qmake n d of Zint (Qnum q) n & pos_nat (Qden q) d & r = n%:~R / d%:R.	Variant	Qrat_spec	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Zint", "pos_nat", "rat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
QratP q r : reflect (Qrat_spec q r) (Qrat q r).	Proof. apply/(iffP idP) => [/eqP<-{r}\|]; first exact: Qrat_spec_Qmake. by case=> _ _ /eqP<- /eqP<- ->. Qed.	Lemma	QratP	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat", "Qrat_spec", "apply" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat_spec_Q_to_rat q : Qrat_spec q (rat_of_Q q).	Proof. exact/QratP. Qed.	Lemma	Qrat_spec_Q_to_rat	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "QratP", "Qrat_spec", "rat_of_Q" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat0 : Qrat Q0 0.	Proof. by []. Qed.	Lemma	Qrat0	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat1 : Qrat Q1 1.	Proof. by []. Qed.	Lemma	Qrat1	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat_Qmake i n p d : Zint i n -> pos_nat p d -> Qrat (Qmake i p) (n%:~R / d%:R).	Proof. by move=> /eqP<- /eqP<-; rewrite /Qrat /rat_of_Q/=. Qed.	Lemma	Qrat_Qmake	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat", "Zint", "pos_nat", "rat_of_Q" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
intr_pos_nat_neq0 {R : numDomainType} {p n} : pos_nat p n -> n%:R != 0 :> R.	Proof. by move=> /eqP<-; rewrite lt0r_neq0// ltr0n Pos_to_nat_gt0. Qed.	Lemma	intr_pos_nat_neq0	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Pos_to_nat_gt0", "lt0r_neq0", "ltr0n", "pos_nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
QratD q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qrat (Qplus q q') (r + r').	Proof. move: qr q'r' => /QratP[n d qn qd ->] /QratP[n' d' qn' qd' ->] {r r'}. rewrite addf_div ?(intr_pos_nat_neq0 qd) ?(intr_pos_nat_neq0 qd')//. rewrite !pmulrn -!rmorphM -rmorphD/= -pmulrn Qrat_Qmake ?pos_natM//. by rewrite ZintD ?ZintM. Qed.	Lemma	QratD	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat", "QratP", "Qrat_Qmake", "ZintD", "ZintM", "addf_div", "intr_pos_nat_neq0", "n'", "pmulrn", "pos_natM", "rmorphD", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
QratM q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qrat (Qmult q q') (r * r').	Proof. move: qr q'r' => /QratP[n d qn qd ->] /QratP[n' d' qn' qd' ->] {r r'}. by rewrite mulf_div -!rmorphM/= Qrat_Qmake ?ZintM ?pos_natM. Qed.	Lemma	QratM	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat", "QratP", "Qrat_Qmake", "ZintM", "mulf_div", "n'", "pos_natM", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
QratN q r : Qrat q r -> Qrat (Qopp q) (- r).	Proof. by move=> /QratP[n d qn dn ->] {r}; rewrite -mulNr -rmorphN/= Qrat_Qmake ?ZintN. Qed.	Lemma	QratN	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat", "QratP", "Qrat_Qmake", "ZintN", "mulNr", "rmorphN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
QratB q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qrat (Qminus q q') (r - r').	Proof. by rewrite QratD ?QratN. Qed.	Lemma	QratB	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat", "QratD", "QratN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
QratV q r : Qrat q r -> Qrat (Qinv q) (r^-1).	Proof. move=> /QratP[n d qn dn ->] {r}; rewrite invf_div /Qinv/=. case: ZintP qn => [//\|_ _ _\|{}p {}n _ _ pn _\|{}p {}n _ _ pn _]/=. - by rewrite invr0 mulr0. - by rewrite pmulrn -[n%:~R]pmulrn Qrat_Qmake. - rewrite NegzE mulrNz divrN -mulNr nmulrn Qrat_Qmake//. by move: dn => /pos_nat_exS[{}d-> qd]; rewrite -NegzE Zi...	Lemma	QratV	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "NegzE", "Qrat", "QratP", "Qrat_Qmake", "ZintP", "Zint_neg", "divrN", "invf_div", "invr0", "mulNr", "mulr0", "mulrNz", "nmulrn", "pmulrn", "pos_nat_exS" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat_eq q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qeq_bool q q' = (r == r').	Proof. move: qr q'r' => /QratP[n d qn qd ->] /QratP[n' d' qn' qd' ->] {r r'}. rewrite eqr_div ?(intr_pos_nat_neq0 qd) ?(intr_pos_nat_neq0 qd')//. by rewrite !pmulrn -!rmorphM/= eqr_int; apply: Zint_eq; apply: ZintM. Qed.	Lemma	Qrat_eq	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Qrat", "QratP", "ZintM", "Zint_eq", "apply", "eqr_div", "eqr_int", "intr_pos_nat_neq0", "n'", "pmulrn", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Qrat_le q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qle_bool q q' = (r <= r').	Proof. move: qr q'r' => /QratP[n d qn qd ->] /QratP[n' d' qn' qd' ->] {r r'}. rewrite ler_pdivrMr 1?mulrAC; first by rewrite ltr0n -(eqP qd) Pos_to_nat_gt0. rewrite ler_pdivlMr; first by rewrite ltr0n -(eqP qd') Pos_to_nat_gt0//. by rewrite !pmulrn -!rmorphM/= ler_int; apply: Zint_le; apply: ZintM. Qed.	Lemma	Qrat_le	algebra	algebra/binnums.v	[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]	[ "Pos_to_nat_gt0", "Qrat", "QratP", "ZintM", "Zint_le", "apply", "ler_int", "ler_pdivlMr", "ler_pdivrMr", "ltr0n", "mulrAC", "n'", "pmulrn", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
SemiRing R	:= (NzSemiRing R) (only parsing).	Notation	SemiRing	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort	:= (NzSemiRing.sort) (only parsing).	Notation	sort	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
on R	:= (NzSemiRing.on R) (only parsing).	Notation	on	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
copy T U	:= (NzSemiRing.copy T U) (only parsing).	Notation	copy	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ring R	:= (NzRing R) (only parsing).	Notation	Ring	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort	:= (NzRing.sort) (only parsing).	Notation	sort	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
on R	:= (NzRing.on R) (only parsing).	Notation	on	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
copy T U	:= (NzRing.copy T U) (only parsing).	Notation	copy	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ComSemiRing R	:= (ComNzSemiRing R) (only parsing).	Notation	ComSemiRing	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort	:= (ComNzSemiRing.sort) (only parsing).	Notation	sort	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
on R	:= (ComNzSemiRing.on R) (only parsing).	Notation	on	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
copy T U	:= (ComNzSemiRing.copy T U) (only parsing).	Notation	copy	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ComRing R	:= (ComNzRing R) (only parsing).	Notation	ComRing	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort	:= (ComNzRing.sort) (only parsing).	Notation	sort	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
on R	:= (ComNzRing.on R) (only parsing).	Notation	on	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
copy T U	:= (ComNzRing.copy T U) (only parsing).	Notation	copy	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
countSemiRingType	:= (countNzSemiRingType) (only parsing).	Notation	countSemiRingType	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
countRingType	:= (countNzRingType) (only parsing).	Notation	countRingType	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
countComSemiRingType	:= (countComNzSemiRingType) (only parsing).	Notation	countComSemiRingType	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
countComRingType	:= (countComNzRingType) (only parsing).	Notation	countComRingType	algebra	algebra/countalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PEeval	:= (PEeval rO rI radd rmul rsub ropp Cpow_of_N rpow R_of_C (env_nth rO)).	Notation	PEeval	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "env_nth" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PCond	:= (PCond true negb andb rO rI radd rmul rsub ropp Cpow_of_N rpow req R_of_C (env_nth rO)).	Notation	PCond	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "env_nth" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PCond_cons l c1 c2 : PCond l (c1 :: c2) = ~~ (req (PEeval l c1) rO) && PCond l c2.	Proof. by case: c2 => [\|//]; rewrite andbT. Qed.	Lemma	PCond_cons	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "PCond", "PEeval", "c1", "c2" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PCond_app l c1 c2 : PCond l (app c1 c2) = PCond l c1 && PCond l c2.	Proof. by elim: c1 c2 => [//\|c c1 IH] c2; rewrite !PCond_cons IH andbA. Qed.	Lemma	PCond_app	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "PCond", "PCond_cons", "c1", "c2" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub	:= (fun x y => x - y).	Notation	sub	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PEeval	:= (PEeval 0 1 +%R *%R sub -%R N.to_nat (@GRing.exp R) R_of_C (env_nth 0)).	Notation	PEeval	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "env_nth", "exp", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PEeval_eqs	:= (PEeval_eqs true andb 0 1 +%R *%R sub -%R N.to_nat (@GRing.exp R) eq_op R_of_C (env_nth 0)).	Notation	PEeval_eqs	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "env_nth", "exp", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
FEeval	:= (FEeval 0 1 +%R *%R sub -%R (fun x y => x / y) (fun x => x^-1) N.to_nat (@GRing.exp R) R_of_C (env_nth 0)).	Notation	FEeval	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "env_nth", "exp", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PCond	:= (PCond true negb andb 0 1 +%R *%R sub -%R N.to_nat (@GRing.exp R) eq_op R_of_C (env_nth 0)).	Notation	PCond	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "env_nth", "exp", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NPEadd	:= (NPEadd 0 +%R eq_op).	Notation	NPEadd	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NPEsub	:= (NPEsub 0 sub eq_op).	Notation	NPEsub	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NPEopp	:= (NPEopp -%R).	Notation	NPEopp	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pow_pos	:= (fun x n => x ^+ Pos.to_nat n).	Notation	pow_pos	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NPEpow	:= (NPEpow 0 1 pow_pos eq_op).	Notation	NPEpow	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
NPEmul	:= (NPEmul 0 1 *%R pow_pos eq_op).	Notation	NPEmul	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PExpr_eq	:= (PExpr_eq eq_op).	Notation	PExpr_eq	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
default_isIn	:= (default_isIn 0 1 pow_pos eq_op).	Notation	default_isIn	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isIn	:= (isIn 0 1 *%R pow_pos eq_op).	Notation	isIn	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
split_aux	:= (split_aux 0 1 *%R pow_pos eq_op).	Notation	split_aux	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
split	:= (split 0 1 *%R pow_pos eq_op).	Notation	split	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fnorm	:= (Fnorm 0 1 +%R *%R sub -%R pow_pos eq_op).	Notation	Fnorm	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_checker	:= (field_checker 0 1 +%R *%R sub -%R pow_pos eq_op cdiv).	Notation	field_checker	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fcons0	:= (Fcons0 0 1 +%R *%R sub -%R eq_op).	Notation	Fcons0	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fcons00	:= (Fcons00 0 1 +%R *%R sub -%R eq_op).	Notation	Fcons00	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fcons1	:= (Fcons1 0 1 +%R *%R sub -%R eq_op).	Notation	Fcons1	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Fcons2	:= (Fcons2 0 1 +%R *%R sub -%R pow_pos eq_op).	Notation	Fcons2	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "pow_pos", "sub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PEeval_NPEadd l e1 e2 : PEeval l (NPEadd e1 e2) = PEeval l (PEadd e1 e2).	Proof. by case: e1 => [\|\|c\|i\|e1 e1'\|e1 e1'\|e1 e1'\|e1\|e1 n]; case: e2 => [\|\|c'\|i'\|e2 e2'\|e2 e2'\|e2 e2'\|e2\|e2 n']; do ?[reflexivity] => /=; do ?[by case: eqP => [->\|//]; rewrite rmorph0 (addr0, add0r)]; rewrite rmorphD. Qed.	Lemma	PEeval_NPEadd	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "NPEadd", "PEeval", "add0r", "addr0", "n'", "rmorph0", "rmorphD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PEeval_NPEsub l e1 e2 : PEeval l (NPEsub e1 e2) = PEeval l (PEsub e1 e2).	Proof. by case: e1 => [\|\|c\|i\|e1 e1'\|e1 e1'\|e1 e1'\|e1\|e1 n]; case: e2 => [\|\|c'\|i'\|e2 e2'\|e2 e2'\|e2 e2'\|e2\|e2 n']; do ?[reflexivity] => /=; do ?[by case: eqP => [->\|//]; rewrite rmorph0 (subr0, add0r)//= ?oppr0]; rewrite rmorphB. Qed.	Lemma	PEeval_NPEsub	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "NPEsub", "PEeval", "add0r", "n'", "oppr0", "rmorph0", "rmorphB", "subr0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PEeval_NPEopp l e : PEeval l (NPEopp e) = PEeval l (PEopp e).	Proof. by case: e => [\|\|c\|i\|e e'\|e e'\|e e'\|e\|e n]//=; rewrite rmorphN. Qed.	Lemma	PEeval_NPEopp	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "NPEopp", "PEeval", "e'", "rmorphN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PEeval_NPEpow l e n : PEeval l (NPEpow e n) = PEeval l (PEpow e n).	Proof. case: n => [\| p]; rewrite /NPEpow; first by rewrite /= ?rmorph1 ?expr0. case: Pos.eqb (pos_nat_eq (pos_nat_Pos_to_nat p) pos_nat1) => [\|_]. by move/esym/eqP; rewrite -[1%N]/(Pos.to_nat xH) => /Pos_to_natI->. case: e => //= c; case: eqP => [->\|_]; rewrite /= ?rmorph1 ?expr1n//. by case: eqP => [->\|_]; rewrite /...	Lemma	PEeval_NPEpow	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "NPEpow", "PEeval", "Pos_to_nat0F", "Pos_to_natI", "eqb", "expr0", "expr0n", "expr1n", "pos_nat1", "pos_nat_Pos_to_nat", "pos_nat_eq", "rmorph0", "rmorph1", "rmorphXn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PEeval_NPEmul l e1 e2 : PEeval l (NPEmul e1 e2) = PEeval l (PEmul e1 e2).	Proof. elim: e1 e2 => [\|\|c\|i\|e1 IH e1' IH'\|e1 IH e1' IH'\|e1 IH e1' IH'\|e1 IH\|e1 IH n]; case=> [\|\|c'\|i'\|e2 e2'\|e2 e2'\|e2 e2'\|e2\|e2 n']; do ?[reflexivity] => /=; do ?[by case: eqP => [->\|_]; rewrite ?rmorph1 ?mul1r?mulr1; do ?[reflexivity]; case: eqP => [->\|_]; rewrite ?rmorph0 /= ?mul0r ?mulr0 ?mul1r ?mulr1]....	Lemma	PEeval_NPEmul	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "NPEmul", "N_to_natI", "Nnat_N_to_nat", "Nnat_eq", "PEeval", "PEeval_NPEpow", "eqb", "exprMn", "mul0r", "mul1r", "mulr0", "mulr1", "n'", "rmorph0", "rmorph1", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PExpr_eqP (e1 e2 : PExpr C) : reflect (e1 = e2) (PExpr_eq e1 e2).	Proof. apply/(iffP idP) => [\|<-]; last first. elim: e1 => /=[//\|//\|//\|p\|?->?->//\|?->?->//\|?->?->//\|//\|e1 IH n]. - by rewrite (pos_nat_eq (pos_nat_Pos_to_nat p) (pos_nat_Pos_to_nat p)). - by rewrite (Nnat_eq (Nnat_N_to_nat n) (Nnat_N_to_nat n)) eqxx IH. elim: e1 e2 => [\|\|c1\|i1\|e1 IH e1' IH'\|e1 IH e1' IH'\|e1 IH e1'...	Lemma	PExpr_eqP	algebra	algebra/field_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...	[ "N_to_natI", "Nnat_N_to_nat", "Nnat_eq", "PExpr_eq", "Pos_to_natI", "apply", "c1", "c2", "eqb", "eqxx", "last", "n'", "pos_nat_Pos_to_nat", "pos_nat_eq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

Nnat_spec (i : N) (n : nat) : N -> nat -> bool -> Set

:= | Nnat_spec_false : Nnat_spec i n i n false | Nnat_spec_N0 : i = N0 -> n = 0%N -> Nnat_spec i n N0 0%N true | Nnat_spec_Npos : forall p', i = Npos p' -> pos_nat p' n -> Nnat_spec i n (Npos p') n true.

Variant

Nnat_spec

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "nat", "pos_nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NnatP i n : Nnat_spec i n i n (Nnat i n).

Proof. case: (boolP (Nnat i n)) => /eqP Nin; last exact: Nnat_spec_false. by case: i n Nin => [|p] n <- /=; [exact: Nnat_spec_N0|exact: Nnat_spec_Npos]. Qed.

Lemma

NnatP

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Nnat", "Nnat_spec", "last" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Nnat0 : Nnat N0 0.

Proof. by []. Qed.

Lemma

Nnat0

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Nnat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Nnat_pos p n : Nnat (Npos p) n = pos_nat p n.

Proof. by []. Qed.

Lemma

Nnat_pos

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Nnat", "pos_nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NnatE

:= (Nnat0, Nnat_pos).

Definition

NnatE

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Nnat0", "Nnat_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NnatD i n (Nin : Nnat i n) i' n' (Ni'n' : Nnat i' n') : Nnat (N.add i i') (n + n').

Proof. case: NnatP Nin => [//|/[!add0n]//| {i}p _ pn _]. case: NnatP Ni'n' => [//|/[!addn0]//| {i'}p' _ p'n' _]. by rewrite Nnat_pos pos_natD. Qed.

Lemma

NnatD

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Nnat", "NnatP", "Nnat_pos", "add", "add0n", "addn0", "n'", "pos_natD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NnatB i n (Nin : Nnat i n) i' n' (Ni'n' : Nnat i' n') : Nnat (N.sub i i') (n - n').

Proof. case: NnatP Nin => [//|//|{}i _ pin _]. case: NnatP Ni'n' => [//|_ _ _|{}i' _ pi'n' _ /=]; first by rewrite subn0. case: (ltnP n n') => nn'; rewrite /Nnat eq_sym. by rewrite Pos_sub_mask_Neg ?(eqP pin) ?(eqP pi'n')// subn_eq0 ltnW. by case: Pos.sub_mask (mask_natB pin pi'n' nn') => //= ? /eqP->. Qed.

Lemma

NnatB

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Nnat", "NnatP", "Pos_sub_mask_Neg", "eq_sym", "ltnP", "ltnW", "mask_natB", "n'", "sub", "subn0", "subn_eq0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NnatM i n (Nin : Nnat i n) i' n' (Ni'n' : Nnat i' n') : Nnat (N.mul i i') (n * n').

Proof. case: NnatP Nin => [//|/[!mul0n]//| {i}p _ pn _]. case: NnatP Ni'n' => [//|/[!muln0]//| {i'}p' _ p'n' _]. by rewrite Nnat_pos pos_natM. Qed.

Lemma

NnatM

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Nnat", "NnatP", "Nnat_pos", "mul", "mul0n", "muln0", "n'", "pos_natM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Nnat_eq i n (Nin : Nnat i n) i' n' (Ni'n' : Nnat i' n') : N.eqb i i' = (n == n').

Proof. case: NnatP Nin => [//|_ _ _| {i}p _ pn _]. by case: NnatP Ni'n' => [//|//| {i'}p' _ + _] => /pos_nat_exS[{}n'->]. case: NnatP Ni'n' => [//|_ _ _| {i'}p' _ p'n' _]. by move: pn => /pos_nat_exS[{}n->]. exact: (pos_nat_eq pn p'n'). Qed.

Lemma

Nnat_eq

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Nnat", "NnatP", "eqb", "n'", "pos_nat_eq", "pos_nat_exS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

N_to_natI : injective N.to_nat.

Proof. by case=> [|i] [|i'] => [//|/esym/eqP|/eqP|/Pos_to_natI->//] /[!Pos_to_nat0F]. Qed.

Lemma

N_to_natI

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Pos_to_nat0F", "Pos_to_natI" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

N_to_natB i j : N.to_nat (N.sub i j) = (N.to_nat i - N.to_nat j)%N.

Proof. exact: eqP (NnatB (Nnat_N_to_nat i) (Nnat_N_to_nat j)). Qed.

Lemma

N_to_natB

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NnatB", "Nnat_N_to_nat", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

int_of_Z (i : Z) : int

:= match i with | Z0 => Posz 0 | Zpos p => Posz (Pos.to_nat p) | Zneg p => Negz (Pos.to_nat p).-1 end.

Definition

int_of_Z

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Posz", "int" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint (i : Z) (n : int)

:= int_of_Z i == n.

Definition

Zint

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "int", "int_of_Z" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_int_of_Z i : Zint i (int_of_Z i).

Proof. exact/eqP. Qed.

Lemma

Zint_int_of_Z

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Zint", "int_of_Z" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_spec (i : Z) (n : int) : Z -> int -> bool -> Set

:= | Zint_spec_false : Zint_spec i n i n false | Zint_spec_Z0 : i = Z0 -> n = Posz 0 -> Zint_spec i n Z0 (Posz 0) true | Zint_spec_Zpos : forall p' n', i = Zpos p' -> n = Posz n' -> pos_nat p' n' -> Zint_spec i n (Zpos p') (Posz n') true | Zint_spec_Zneg : forall p' n', i = Zneg p' -> n = Ne...

Variant

Zint_spec

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Posz", "int", "n'", "pos_nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ZintP i n : Zint_spec i n i n (Zint i n).

Proof. case: (boolP (Zint i n)) => /eqP Zin; last exact: Zint_spec_false. case: i n Zin => [|p|p] n <- /=; [exact: Zint_spec_Z0|exact: Zint_spec_Zpos|]. by apply: Zint_spec_Zneg; rewrite ?prednK ?Pos_to_nat_gt0. Qed.

Lemma

ZintP

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Pos_to_nat_gt0", "Zint", "Zint_spec", "apply", "last", "prednK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint0 : Zint Z0 0.

Proof. by []. Qed.

Lemma

Zint0

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Zint" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_pos p n : Zint (Zpos p) (Posz n) = pos_nat p n.

Proof. by []. Qed.

Lemma

Zint_pos

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Posz", "Zint", "pos_nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_neg p n : Zint (Zneg p) (Negz n) = pos_nat p n.+1.

Proof. by apply/idP/idP; [case: ZintP => // _ _ [<-] [<-]|rewrite /Zint/= => /eqP->]. Qed.

Lemma

Zint_neg

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Zint", "ZintP", "apply", "pos_nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ZintE

:= (Zint0, Zint_pos, Zint_neg).

Definition

ZintE

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Zint0", "Zint_neg", "Zint_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_double i n : Zint i n -> Zint (Z.double i) (2 * n).

Proof. case: ZintP => // {i}p {}n _ _ pn _ /=; first by rewrite ZintE mul2n pos_natE. by rewrite -Negz_doubleS ZintE -doubleS pos_natE. Qed.

Lemma

Zint_double

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Negz_doubleS", "Zint", "ZintE", "ZintP", "double", "doubleS", "mul2n", "pos_natE" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_succ_double i n : Zint i n -> Zint (Z.succ_double i) (2 * n + 1).

Proof. case: ZintP => // {i}p {}n _ _ pn _ /=. by rewrite ZintE mul2n addn1 pos_natE. by rewrite -Negz_doubleS NegzS addrK ZintE -predn_doubleS pos_nat_pred_double. Qed.

Lemma

Zint_succ_double

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzS", "Negz_doubleS", "Zint", "ZintE", "ZintP", "addn1", "addrK", "mul2n", "pos_natE", "pos_nat_pred_double", "predn_doubleS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_pred_double i n : Zint i n -> Zint (Z.pred_double i) (2 * n - 1).

Proof. case: ZintP => // {i}p {}n _ _/= => [/pos_nat_exS[{}n-> pn] | pn] _; last first. by rewrite -Negz_doubleS -NegzS ZintE -doubleS pos_natE. rewrite -PoszM mul2n doubleS -addn1 PoszD addrK. by rewrite ZintE -predn_doubleS pos_nat_pred_double. Qed.

Lemma

Zint_pred_double

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzS", "Negz_doubleS", "PoszD", "PoszM", "Zint", "ZintE", "ZintP", "addn1", "addrK", "doubleS", "last", "mul2n", "pos_natE", "pos_nat_exS", "pos_nat_pred_double", "predn_doubleS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_pos_sub p n p' n' : pos_nat p n -> pos_nat p' n' -> Zint (Z.pos_sub p p') (Posz n - Posz n').

Proof. move=> pn p'n'; elim/pos_nat_ind: p'n' p n pn => [p n||]; [|move=> {}p' {}n' p'n' IH p n..]. - case: pos_natP => // {}p {}n _ _ + _ /=; last first. by rewrite -addn1 PoszD addrK ZintE pos_natE. move=> /pos_nat_exS[{}n-> pn]; rewrite doubleS -addn1 PoszD addrK. by rewrite ZintE -predn_doubleS pos_nat_...

Lemma

Zint_pos_sub

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzE", "Posz", "PoszD", "PoszM", "Zint", "ZintE", "Zint_double", "Zint_pred_double", "Zint_succ_double", "add1n", "addn1", "addrA", "addrAC", "addrK", "addrKA", "doubleS", "last", "mul2n", "mulrBr", "n'", "opprB", "opprD", "pos_nat", "pos_natE", "pos_natP", "pos_n...

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ZintD i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Zint (Z.add i i') (n + n').

Proof. case: ZintP Zin => [//|||]; [|move=> {i}p {}n _ _ pn _..]. - by rewrite add0r. - case: ZintP Zi'n' => [//|||]; [|move=> {i'}p' {}n' _ _ p'n' _ /=..]. + by rewrite addr0. + by rewrite ZintE pos_natD. + by rewrite NegzE Zint_pos_sub. - case: ZintP Zi'n' => [//|||]; [|move=> {i'}p' {}n' _ _ p'n' _ /=..]. + ...

Lemma

ZintD

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzE", "Posz", "PoszD", "Zint", "ZintE", "ZintP", "Zint_pos_sub", "add", "add0r", "addSn", "addr0", "addrC", "n'", "opprD", "pos_natD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ZintN i n : Zint i n -> Zint (Z.opp i) (- n).

Proof. by case: ZintP => // {i}p {}n _ _ /pos_nat_exS[{}n-> pn]; rewrite -NegzE ZintE. Qed.

Lemma

ZintN

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzE", "Zint", "ZintE", "ZintP", "opp", "pos_nat_exS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ZintB i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Zint (Z.sub i i') (n - n').

Proof. by apply: ZintD => //; apply: ZintN. Qed.

Lemma

ZintB

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Zint", "ZintD", "ZintN", "apply", "n'", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ZintM i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Zint (Z.mul i i') (n * n').

Proof. case: ZintP Zin => [//|||]; [|move=> {i}p {}n _ _ pn _..]. - by rewrite mul0r. - case: ZintP Zi'n' => [//|||] /=; [|move=> {i'}p' {}n' _ _ p'n' _..]. + by rewrite mulr0. + by rewrite ZintE pos_natM. + move: pn => /pos_nat_exS[{}n-> pn]; rewrite NegzE mulrN -PoszM. by rewrite mulnS addSn -NegzE ZintE -a...

Lemma

ZintM

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzE", "PoszM", "Zint", "ZintE", "ZintP", "addSn", "mul", "mul0r", "mulNr", "mulnS", "mulr0", "mulrN", "mulrNN", "n'", "pos_natM", "pos_nat_exS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_pow_pos i n (Zin : Zint i n) p m (pm : pos_nat p m) : Zint (Z.pow_pos i p) (n ^+ m).

Proof. rewrite /Z.pow_pos -[n ^+ m]mul1r; have : Zint (Zpos xH) 1 by []. elim/pos_nat_ind: pm i n Zin (Zpos xH) 1 => [| {}p {}m pm IH | {}p {}m pm IH] i n Zin j k Zjk /=. - by rewrite expr1 mulrC; apply: ZintM. - by rewrite -addnn exprD mulrA; apply: (IH) => //; apply: IH. - rewrite -addn1 -addnn !exprD expr1 !mulr...

Lemma

Zint_pow_pos

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Zint", "ZintM", "addn1", "addnn", "apply", "expr1", "exprD", "mul1r", "mulrA", "mulrC", "pos_nat", "pos_nat_ind", "pow_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_eq i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Z.eqb i i' = (n == n').

Proof. case: ZintP Zin => [//|||]; [|move=> {i}p {}n _ _ pn _..]. - case: ZintP Zi'n' => [//|//||//] /= {i'}p' {}n' _ _ p'n' _ _ _ _. by apply/esym/negbTE; rewrite eq_sym eqz_nat -(eqP p'n') Pos_to_nat0F. - case: ZintP Zi'n' => [//|_ _ _|{i'}p' {}n' _ _ p'n' _|//] /=. + by apply/esym/negbTE; rewrite eqz_nat -(eqP p...

Lemma

Zint_eq

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzE", "Pos_to_nat0F", "Zint", "ZintP", "apply", "eq_sym", "eqb", "eqr_opp", "eqz_nat", "n'", "pos_nat_eq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zint_le i n (Zin : Zint i n) i' n' (Zi'n' : Zint i' n') : Z.leb i i' = (n <= n').

Proof. case: ZintP Zin => [//|||]; [|move=> {i}p {}n _ _ pn _..]. - by case: ZintP Zi'n'. - case: ZintP Zi'n' => [//|_ _ _|{i'}p' {}n' _ _ p'n' _|//] /=. + by apply/esym/negbTE; rewrite lez0_nat -(eqP pn) Pos_to_nat0F. + exact: pos_nat_le. - case: ZintP Zi'n' => [//|//|//|{i'}p' {}n' _ _ p'n' _ /=]. rewrite !Negz...

Lemma

Zint_le

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzE", "Pos_to_nat0F", "Zint", "ZintP", "apply", "eqSS", "leqNgt", "leqnn", "lerN2", "lez0_nat", "lez_nat", "ltnP", "ltnS", "n'", "pos_nat_compare", "pos_nat_le" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rat_of_Q (q : Q) : rat

:= (int_of_Z (Qnum q))%:~R / (Pos.to_nat (Qden q))%:R.

Definition

rat_of_Q

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "int_of_Z", "rat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat (q : Q) (r : rat)

:= rat_of_Q q == r.

Definition

Qrat

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "rat", "rat_of_Q" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat_rat_of_Q q : Qrat q (rat_of_Q q).

Proof. exact/eqP. Qed.

Lemma

Qrat_rat_of_Q

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat", "rat_of_Q" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat_spec (q : Q) (r : rat) : Prop

:= Qrat_spec_Qmake n d of Zint (Qnum q) n & pos_nat (Qden q) d & r = n%:~R / d%:R.

Variant

Qrat_spec

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Zint", "pos_nat", "rat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

QratP q r : reflect (Qrat_spec q r) (Qrat q r).

Proof. apply/(iffP idP) => [/eqP<-{r}|]; first exact: Qrat_spec_Qmake. by case=> _ _ /eqP<- /eqP<- ->. Qed.

Lemma

QratP

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat", "Qrat_spec", "apply" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat_spec_Q_to_rat q : Qrat_spec q (rat_of_Q q).

Proof. exact/QratP. Qed.

Lemma

Qrat_spec_Q_to_rat

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "QratP", "Qrat_spec", "rat_of_Q" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat0 : Qrat Q0 0.

Proof. by []. Qed.

Lemma

Qrat0

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat1 : Qrat Q1 1.

Proof. by []. Qed.

Lemma

Qrat1

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat_Qmake i n p d : Zint i n -> pos_nat p d -> Qrat (Qmake i p) (n%:~R / d%:R).

Proof. by move=> /eqP<- /eqP<-; rewrite /Qrat /rat_of_Q/=. Qed.

Lemma

Qrat_Qmake

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat", "Zint", "pos_nat", "rat_of_Q" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

intr_pos_nat_neq0 {R : numDomainType} {p n} : pos_nat p n -> n%:R != 0 :> R.

Proof. by move=> /eqP<-; rewrite lt0r_neq0// ltr0n Pos_to_nat_gt0. Qed.

Lemma

intr_pos_nat_neq0

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Pos_to_nat_gt0", "lt0r_neq0", "ltr0n", "pos_nat" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

QratD q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qrat (Qplus q q') (r + r').

Proof. move: qr q'r' => /QratP[n d qn qd ->] /QratP[n' d' qn' qd' ->] {r r'}. rewrite addf_div ?(intr_pos_nat_neq0 qd) ?(intr_pos_nat_neq0 qd')//. rewrite !pmulrn -!rmorphM -rmorphD/= -pmulrn Qrat_Qmake ?pos_natM//. by rewrite ZintD ?ZintM. Qed.

Lemma

QratD

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat", "QratP", "Qrat_Qmake", "ZintD", "ZintM", "addf_div", "intr_pos_nat_neq0", "n'", "pmulrn", "pos_natM", "rmorphD", "rmorphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

QratM q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qrat (Qmult q q') (r * r').

Proof. move: qr q'r' => /QratP[n d qn qd ->] /QratP[n' d' qn' qd' ->] {r r'}. by rewrite mulf_div -!rmorphM/= Qrat_Qmake ?ZintM ?pos_natM. Qed.

Lemma

QratM

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat", "QratP", "Qrat_Qmake", "ZintM", "mulf_div", "n'", "pos_natM", "rmorphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

QratN q r : Qrat q r -> Qrat (Qopp q) (- r).

Proof. by move=> /QratP[n d qn dn ->] {r}; rewrite -mulNr -rmorphN/= Qrat_Qmake ?ZintN. Qed.

Lemma

QratN

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat", "QratP", "Qrat_Qmake", "ZintN", "mulNr", "rmorphN" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

QratB q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qrat (Qminus q q') (r - r').

Proof. by rewrite QratD ?QratN. Qed.

Lemma

QratB

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat", "QratD", "QratN" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

QratV q r : Qrat q r -> Qrat (Qinv q) (r^-1).

Proof. move=> /QratP[n d qn dn ->] {r}; rewrite invf_div /Qinv/=. case: ZintP qn => [//|_ _ _|{}p {}n _ _ pn _|{}p {}n _ _ pn _]/=. - by rewrite invr0 mulr0. - by rewrite pmulrn -[n%:~R]pmulrn Qrat_Qmake. - rewrite NegzE mulrNz divrN -mulNr nmulrn Qrat_Qmake//. by move: dn => /pos_nat_exS[{}d-> qd]; rewrite -NegzE Zi...

Lemma

QratV

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "NegzE", "Qrat", "QratP", "Qrat_Qmake", "ZintP", "Zint_neg", "divrN", "invf_div", "invr0", "mulNr", "mulr0", "mulrNz", "nmulrn", "pmulrn", "pos_nat_exS" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat_eq q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qeq_bool q q' = (r == r').

Proof. move: qr q'r' => /QratP[n d qn qd ->] /QratP[n' d' qn' qd' ->] {r r'}. rewrite eqr_div ?(intr_pos_nat_neq0 qd) ?(intr_pos_nat_neq0 qd')//. by rewrite !pmulrn -!rmorphM/= eqr_int; apply: Zint_eq; apply: ZintM. Qed.

Lemma

Qrat_eq

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Qrat", "QratP", "ZintM", "Zint_eq", "apply", "eqr_div", "eqr_int", "intr_pos_nat_neq0", "n'", "pmulrn", "rmorphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Qrat_le q r (qr : Qrat q r) q' r' (q'r' : Qrat q' r') : Qle_bool q q' = (r <= r').

Proof. move: qr q'r' => /QratP[n d qn qd ->] /QratP[n' d' qn' qd' ->] {r r'}. rewrite ler_pdivrMr 1?mulrAC; first by rewrite ltr0n -(eqP qd) Pos_to_nat_gt0. rewrite ler_pdivlMr; first by rewrite ltr0n -(eqP qd') Pos_to_nat_gt0//. by rewrite !pmulrn -!rmorphM/= ler_int; apply: Zint_le; apply: ZintM. Qed.

Lemma

Qrat_le

algebra

algebra/binnums.v

[ "micromega_plugin", "PosDef", "NatDef", "Corelib", "IntDef", "RatDef", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "ssralg", "ssrnum", "ssrint", "rat", "GRing.Theory", "Num.Theory" ]

[ "Pos_to_nat_gt0", "Qrat", "QratP", "ZintM", "Zint_le", "apply", "ler_int", "ler_pdivlMr", "ler_pdivrMr", "ltr0n", "mulrAC", "n'", "pmulrn", "rmorphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

SemiRing R

:= (NzSemiRing R) (only parsing).

Notation

SemiRing

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort

:= (NzSemiRing.sort) (only parsing).

Notation

sort

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

on R

:= (NzSemiRing.on R) (only parsing).

Notation

on

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

copy T U

:= (NzSemiRing.copy T U) (only parsing).

Notation

copy

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ring R

:= (NzRing R) (only parsing).

Notation

Ring

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort

:= (NzRing.sort) (only parsing).

Notation

sort

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

on R

:= (NzRing.on R) (only parsing).

Notation

on

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

copy T U

:= (NzRing.copy T U) (only parsing).

Notation

copy

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ComSemiRing R

:= (ComNzSemiRing R) (only parsing).

Notation

ComSemiRing

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort

:= (ComNzSemiRing.sort) (only parsing).

Notation

sort

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

on R

:= (ComNzSemiRing.on R) (only parsing).

Notation

on

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

copy T U

:= (ComNzSemiRing.copy T U) (only parsing).

Notation

copy

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ComRing R

:= (ComNzRing R) (only parsing).

Notation

ComRing

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort

:= (ComNzRing.sort) (only parsing).

Notation

sort

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

on R

:= (ComNzRing.on R) (only parsing).

Notation

on

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

copy T U

:= (ComNzRing.copy T U) (only parsing).

Notation

copy

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

countSemiRingType

:= (countNzSemiRingType) (only parsing).

Notation

countSemiRingType

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

countRingType

:= (countNzRingType) (only parsing).

Notation

countRingType

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

countComSemiRingType

:= (countComNzSemiRingType) (only parsing).

Notation

countComSemiRingType

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

countComRingType

:= (countComNzRingType) (only parsing).

Notation

countComRingType

algebra

algebra/countalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "decfield", "GRing.Theory", "CodeSeq", "CountRing" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PEeval

:= (PEeval rO rI radd rmul rsub ropp Cpow_of_N rpow R_of_C (env_nth rO)).

Notation

PEeval

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "env_nth" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PCond

:= (PCond true negb andb rO rI radd rmul rsub ropp Cpow_of_N rpow req R_of_C (env_nth rO)).

Notation

PCond

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "env_nth" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PCond_cons l c1 c2 : PCond l (c1 :: c2) = ~~ (req (PEeval l c1) rO) && PCond l c2.

Proof. by case: c2 => [|//]; rewrite andbT. Qed.

Lemma

PCond_cons

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "PCond", "PEeval", "c1", "c2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PCond_app l c1 c2 : PCond l (app c1 c2) = PCond l c1 && PCond l c2.

Proof. by elim: c1 c2 => [//|c c1 IH] c2; rewrite !PCond_cons IH andbA. Qed.

Lemma

PCond_app

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "PCond", "PCond_cons", "c1", "c2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub

:= (fun x y => x - y).

Notation

sub

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PEeval

:= (PEeval 0 1 +%R *%R sub -%R N.to_nat (@GRing.exp R) R_of_C (env_nth 0)).

Notation

PEeval

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "env_nth", "exp", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PEeval_eqs

:= (PEeval_eqs true andb 0 1 +%R *%R sub -%R N.to_nat (@GRing.exp R) eq_op R_of_C (env_nth 0)).

Notation

PEeval_eqs

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "env_nth", "exp", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

FEeval

:= (FEeval 0 1 +%R *%R sub -%R (fun x y => x / y) (fun x => x^-1) N.to_nat (@GRing.exp R) R_of_C (env_nth 0)).

Notation

FEeval

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "env_nth", "exp", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PCond

:= (PCond true negb andb 0 1 +%R *%R sub -%R N.to_nat (@GRing.exp R) eq_op R_of_C (env_nth 0)).

Notation

PCond

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "env_nth", "exp", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NPEadd

:= (NPEadd 0 +%R eq_op).

Notation

NPEadd

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NPEsub

:= (NPEsub 0 sub eq_op).

Notation

NPEsub

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NPEopp

:= (NPEopp -%R).

Notation

NPEopp

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pow_pos

:= (fun x n => x ^+ Pos.to_nat n).

Notation

pow_pos

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NPEpow

:= (NPEpow 0 1 pow_pos eq_op).

Notation

NPEpow

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

NPEmul

:= (NPEmul 0 1 *%R pow_pos eq_op).

Notation

NPEmul

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PExpr_eq

:= (PExpr_eq eq_op).

Notation

PExpr_eq

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

default_isIn

:= (default_isIn 0 1 pow_pos eq_op).

Notation

default_isIn

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isIn

:= (isIn 0 1 *%R pow_pos eq_op).

Notation

isIn

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

split_aux

:= (split_aux 0 1 *%R pow_pos eq_op).

Notation

split_aux

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

split

:= (split 0 1 *%R pow_pos eq_op).

Notation

split

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fnorm

:= (Fnorm 0 1 +%R *%R sub -%R pow_pos eq_op).

Notation

Fnorm

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_checker

:= (field_checker 0 1 +%R *%R sub -%R pow_pos eq_op cdiv).

Notation

field_checker

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fcons0

:= (Fcons0 0 1 +%R *%R sub -%R eq_op).

Notation

Fcons0

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fcons00

:= (Fcons00 0 1 +%R *%R sub -%R eq_op).

Notation

Fcons00

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fcons1

:= (Fcons1 0 1 +%R *%R sub -%R eq_op).

Notation

Fcons1

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Fcons2

:= (Fcons2 0 1 +%R *%R sub -%R pow_pos eq_op).

Notation

Fcons2

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "pow_pos", "sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PEeval_NPEadd l e1 e2 : PEeval l (NPEadd e1 e2) = PEeval l (PEadd e1 e2).

Proof. by case: e1 => [||c|i|e1 e1'|e1 e1'|e1 e1'|e1|e1 n]; case: e2 => [||c'|i'|e2 e2'|e2 e2'|e2 e2'|e2|e2 n']; do ?[reflexivity] => /=; do ?[by case: eqP => [->|//]; rewrite rmorph0 (addr0, add0r)]; rewrite rmorphD. Qed.

Lemma

PEeval_NPEadd

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "NPEadd", "PEeval", "add0r", "addr0", "n'", "rmorph0", "rmorphD" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PEeval_NPEsub l e1 e2 : PEeval l (NPEsub e1 e2) = PEeval l (PEsub e1 e2).

Proof. by case: e1 => [||c|i|e1 e1'|e1 e1'|e1 e1'|e1|e1 n]; case: e2 => [||c'|i'|e2 e2'|e2 e2'|e2 e2'|e2|e2 n']; do ?[reflexivity] => /=; do ?[by case: eqP => [->|//]; rewrite rmorph0 (subr0, add0r)//= ?oppr0]; rewrite rmorphB. Qed.

Lemma

PEeval_NPEsub

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "NPEsub", "PEeval", "add0r", "n'", "oppr0", "rmorph0", "rmorphB", "subr0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PEeval_NPEopp l e : PEeval l (NPEopp e) = PEeval l (PEopp e).

Proof. by case: e => [||c|i|e e'|e e'|e e'|e|e n]//=; rewrite rmorphN. Qed.

Lemma

PEeval_NPEopp

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "NPEopp", "PEeval", "e'", "rmorphN" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PEeval_NPEpow l e n : PEeval l (NPEpow e n) = PEeval l (PEpow e n).

Proof. case: n => [| p]; rewrite /NPEpow; first by rewrite /= ?rmorph1 ?expr0. case: Pos.eqb (pos_nat_eq (pos_nat_Pos_to_nat p) pos_nat1) => [|_]. by move/esym/eqP; rewrite -[1%N]/(Pos.to_nat xH) => /Pos_to_natI->. case: e => //= c; case: eqP => [->|_]; rewrite /= ?rmorph1 ?expr1n//. by case: eqP => [->|_]; rewrite /...

Lemma

PEeval_NPEpow

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "NPEpow", "PEeval", "Pos_to_nat0F", "Pos_to_natI", "eqb", "expr0", "expr0n", "expr1n", "pos_nat1", "pos_nat_Pos_to_nat", "pos_nat_eq", "rmorph0", "rmorph1", "rmorphXn" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PEeval_NPEmul l e1 e2 : PEeval l (NPEmul e1 e2) = PEeval l (PEmul e1 e2).

Proof. elim: e1 e2 => [||c|i|e1 IH e1' IH'|e1 IH e1' IH'|e1 IH e1' IH'|e1 IH|e1 IH n]; case=> [||c'|i'|e2 e2'|e2 e2'|e2 e2'|e2|e2 n']; do ?[reflexivity] => /=; do ?[by case: eqP => [->|_]; rewrite ?rmorph1 ?mul1r?mulr1; do ?[reflexivity]; case: eqP => [->|_]; rewrite ?rmorph0 /= ?mul0r ?mulr0 ?mul1r ?mulr1]....

Lemma

PEeval_NPEmul

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "NPEmul", "N_to_natI", "Nnat_N_to_nat", "Nnat_eq", "PEeval", "PEeval_NPEpow", "eqb", "exprMn", "mul0r", "mul1r", "mulr0", "mulr1", "n'", "rmorph0", "rmorph1", "rmorphM" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PExpr_eqP (e1 e2 : PExpr C) : reflect (e1 = e2) (PExpr_eq e1 e2).

Proof. apply/(iffP idP) => [|<-]; last first. elim: e1 => /=[//|//|//|p|?->?->//|?->?->//|?->?->//|//|e1 IH n]. - by rewrite (pos_nat_eq (pos_nat_Pos_to_nat p) (pos_nat_Pos_to_nat p)). - by rewrite (Nnat_eq (Nnat_N_to_nat n) (Nnat_N_to_nat n)) eqxx IH. elim: e1 e2 => [||c1|i1|e1 IH e1' IH'|e1 IH e1' IH'|e1 IH e1'...

Lemma

PExpr_eqP

algebra

algebra/field_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "field_checker", "field_eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "order", "ssralg", "ssrnum", "ssrint", "...

[ "N_to_natI", "Nnat_N_to_nat", "Nnat_eq", "PExpr_eq", "Pos_to_natI", "apply", "c1", "c2", "eqb", "eqxx", "last", "n'", "pos_nat_Pos_to_nat", "pos_nat_eq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d