Datasets:

phanerozoic
/

Coq-Stdlib

fact stringlengths 14 26.5k	statement stringlengths 5 4.15k	proof stringlengths 0 26.3k	type stringclasses 26 values	kind stringclasses 9 values	symbolic_name stringlengths 0 75	library stringclasses 44 values	filename stringclasses 463 values	imports listlengths 0 25	deps listlengths 0 64	docstring stringlengths 3 2.4k ⌀	line_start int64 3 4.06k	line_end int64 5 4.06k	has_proof bool 2 classes	source_url stringclasses 1 value	commit stringclasses 1 value	content_level stringclasses 1 value
#[local] Definition le_n_0_eq_stt := fun n Hle => eq_sym (proj1 (Nat.le_0_r n) Hle).	#[local] Definition le_n_0_eq_stt	:= fun n Hle => eq_sym (proj1 (Nat.le_0_r n) Hle).	Definition	definition	le_n_0_eq_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "eq_sym", "le_0_r" ]	Le.le_n_S Le.le_n_Sn Le.le_Sn_n	43	44	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_le_S_stt := fun n m => (proj2 (Nat.le_succ_l n m)).	#[local] Definition lt_le_S_stt	:= fun n m => (proj2 (Nat.le_succ_l n m)).	Definition	definition	lt_le_S_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "le_succ_l" ]	Lt.lt_irrefl	53	54	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_n_Sm_le_stt := fun n m => (proj1 (Nat.lt_succ_r n m)).	#[local] Definition lt_n_Sm_le_stt	:= fun n m => (proj1 (Nat.lt_succ_r n m)).	Definition	definition	lt_n_Sm_le_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_succ_r" ]	Lt.lt_le_S	59	60	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_lt_n_Sm_stt := fun n m => (proj2 (Nat.lt_succ_r n m)).	#[local] Definition le_lt_n_Sm_stt	:= fun n m => (proj2 (Nat.lt_succ_r n m)).	Definition	definition	le_lt_n_Sm_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_succ_r" ]	Lt.lt_n_Sm_le	65	66	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_not_lt_stt := fun n m => (proj1 (Nat.le_ngt n m)).	#[local] Definition le_not_lt_stt	:= fun n m => (proj1 (Nat.le_ngt n m)).	Definition	definition	le_not_lt_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "le_ngt" ]	Lt.le_lt_n_Sm	71	72	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_not_le_stt := fun n m => (proj1 (Nat.lt_nge n m)).	#[local] Definition lt_not_le_stt	:= fun n m => (proj1 (Nat.lt_nge n m)).	Definition	definition	lt_not_le_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_nge" ]	Lt.le_not_lt	77	78	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition neq_0_lt_stt := fun n Hn => proj1 (Nat.neq_0_lt_0 n) (Nat.neq_sym 0 n Hn).	#[local] Definition neq_0_lt_stt	:= fun n Hn => proj1 (Nat.neq_0_lt_0 n) (Nat.neq_sym 0 n Hn).	Definition	definition	neq_0_lt_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "neq_0_lt_0", "neq_sym" ]	Lt.lt_0_Sn Lt.lt_n_0	85	86	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_0_neq_stt := fun n Hlt => Nat.neq_sym n 0 (proj2 (Nat.neq_0_lt_0 n) Hlt).	#[local] Definition lt_0_neq_stt	:= fun n Hlt => Nat.neq_sym n 0 (proj2 (Nat.neq_0_lt_0 n) Hlt).	Definition	definition	lt_0_neq_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "neq_0_lt_0", "neq_sym" ]	null	89	90	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_n_S_stt := fun n m => (proj1 (Nat.succ_lt_mono n m)).	#[local] Definition lt_n_S_stt	:= fun n m => (proj1 (Nat.succ_lt_mono n m)).	Definition	definition	lt_n_S_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "succ_lt_mono" ]	Lt.lt_n_Sn Lt.lt_S	97	98	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_S_n_stt := fun n m => (proj2 (Nat.succ_lt_mono n m)).	#[local] Definition lt_S_n_stt	:= fun n m => (proj2 (Nat.succ_lt_mono n m)).	Definition	definition	lt_S_n_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "succ_lt_mono" ]	Lt.lt_n_S	103	104	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_pred_stt := fun n m => proj1 (Nat.lt_succ_lt_pred n m).	#[local] Definition lt_pred_stt	:= fun n m => proj1 (Nat.lt_succ_lt_pred n m).	Definition	definition	lt_pred_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_succ_lt_pred" ]	Lt.lt_S_n	109	110	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_pred_n_n_stt := fun n Hlt => Nat.lt_pred_l n (proj2 (Nat.neq_0_lt_0 n) Hlt).	#[local] Definition lt_pred_n_n_stt	:= fun n Hlt => Nat.lt_pred_l n (proj2 (Nat.neq_0_lt_0 n) Hlt).	Definition	definition	lt_pred_n_n_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_pred_l", "neq_0_lt_0" ]	Lt.lt_pred	115	116	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_Sn_O_stt : forall n, S n > 0 := Nat.lt_0_succ.	#[local] Definition gt_Sn_O_stt : forall n, S n > 0	:= Nat.lt_0_succ.	Definition	definition	gt_Sn_O_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "lt_0_succ" ]	Lt.lt_le_weak	125	126	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_Sn_n_stt : forall n, S n > n := Nat.lt_succ_diag_r.	#[local] Definition gt_Sn_n_stt : forall n, S n > n	:= Nat.lt_succ_diag_r.	Definition	definition	gt_Sn_n_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "lt_succ_diag_r" ]	Gt.gt_Sn_O	131	132	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_n_S_stt : forall n m, n > m -> S n > S m := fun n m Hgt => proj1 (Nat.succ_lt_mono m n) Hgt.	#[local] Definition gt_n_S_stt : forall n m, n > m -> S n > S m	:= fun n m Hgt => proj1 (Nat.succ_lt_mono m n) Hgt.	Definition	definition	gt_n_S_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "succ_lt_mono" ]	Gt.gt_Sn_n	137	138	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_S_n_stt : forall n m, S m > S n -> m > n := fun n m Hgt => proj2 (Nat.succ_lt_mono n m) Hgt.	#[local] Definition gt_S_n_stt : forall n m, S m > S n -> m > n	:= fun n m Hgt => proj2 (Nat.succ_lt_mono n m) Hgt.	Definition	definition	gt_S_n_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "succ_lt_mono" ]	Gt.gt_n_S	143	144	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_pred_stt : forall n m, m > S n -> pred m > n := fun n m Hgt => proj1 (Nat.lt_succ_lt_pred n m) Hgt.	#[local] Definition gt_pred_stt : forall n m, m > S n -> pred m > n	:= fun n m Hgt => proj1 (Nat.lt_succ_lt_pred n m) Hgt.	Definition	definition	gt_pred_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "lt_succ_lt_pred", "pred" ]	Gt.gt_S_n	149	150	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_irrefl_stt : forall n, ~ n > n := Nat.lt_irrefl.	#[local] Definition gt_irrefl_stt : forall n, ~ n > n	:= Nat.lt_irrefl.	Definition	definition	gt_irrefl_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_irrefl" ]	Gt.gt_pred	155	156	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_asym_stt : forall n m, n > m -> ~ m > n := fun n m => Nat.lt_asymm m n.	#[local] Definition gt_asym_stt : forall n m, n > m -> ~ m > n	:= fun n m => Nat.lt_asymm m n.	Definition	definition	gt_asym_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_asymm" ]	Gt.gt_irrefl	161	162	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_not_gt_stt : forall n m, n <= m -> ~ n > m := fun n m => proj1 (Nat.le_ngt n m).	#[local] Definition le_not_gt_stt : forall n m, n <= m -> ~ n > m	:= fun n m => proj1 (Nat.le_ngt n m).	Definition	definition	le_not_gt_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "le_ngt" ]	Gt.gt_asym	167	168	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_not_le_stt: forall n m, n > m -> ~ n <= m := fun n m => proj1 (Nat.lt_nge m n).	#[local] Definition gt_not_le_stt: forall n m, n > m -> ~ n <= m	:= fun n m => proj1 (Nat.lt_nge m n).	Definition	definition	gt_not_le_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_nge" ]	Gt.le_not_gt	173	174	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_S_gt_stt: forall n m, S n <= m -> m > n := fun n m => proj1 (Nat.le_succ_l n m).	#[local] Definition le_S_gt_stt: forall n m, S n <= m -> m > n	:= fun n m => proj1 (Nat.le_succ_l n m).	Definition	definition	le_S_gt_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "le_succ_l" ]	Gt.gt_not_le	179	180	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_S_le_stt : forall n m, S m > n -> n <= m := fun n m => proj2 (Nat.succ_le_mono n m).	#[local] Definition gt_S_le_stt : forall n m, S m > n -> n <= m	:= fun n m => proj2 (Nat.succ_le_mono n m).	Definition	definition	gt_S_le_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "succ_le_mono" ]	Gt.le_S_gt	185	186	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_le_S_stt : forall n m, m > n -> S n <= m := fun n m => proj2 (Nat.le_succ_l n m).	#[local] Definition gt_le_S_stt : forall n m, m > n -> S n <= m	:= fun n m => proj2 (Nat.le_succ_l n m).	Definition	definition	gt_le_S_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "le_succ_l" ]	Gt.gt_S_le	191	192	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_gt_S_stt : forall n m, n <= m -> S m > n := fun n m => proj1 (Nat.succ_le_mono n m).	#[local] Definition le_gt_S_stt : forall n m, n <= m -> S m > n	:= fun n m => proj1 (Nat.succ_le_mono n m).	Definition	definition	le_gt_S_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "succ_le_mono" ]	Gt.gt_le_S	197	198	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_trans_S_stt : forall n m p, S n > m -> m > p -> n > p := fun n m p Hgt1 Hgt2 => Nat.lt_le_trans p m n Hgt2 (proj2 (Nat.succ_le_mono _ _) Hgt1).	#[local] Definition gt_trans_S_stt : forall n m p, S n > m -> m > p -> n > p	:= fun n m p Hgt1 Hgt2 => Nat.lt_le_trans p m n Hgt2 (proj2 (Nat.succ_le_mono _ _) Hgt1).	Definition	definition	gt_trans_S_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "lt_le_trans", "p", "succ_le_mono" ]	Gt.le_gt_S	203	205	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_gt_trans_stt : forall n m p, m <= n -> m > p -> n > p := fun n m p Hle Hgt => Nat.lt_le_trans p m n Hgt Hle.	#[local] Definition le_gt_trans_stt : forall n m p, m <= n -> m > p -> n > p	:= fun n m p Hle Hgt => Nat.lt_le_trans p m n Hgt Hle.	Definition	definition	le_gt_trans_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_le_trans", "p" ]	Gt.gt_trans_S	210	212	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition gt_le_trans_stt : forall n m p, n > m -> p <= m -> n > p := fun n m p Hgt Hle => Nat.le_lt_trans p m n Hle Hgt.	#[local] Definition gt_le_trans_stt : forall n m p, n > m -> p <= m -> n > p	:= fun n m p Hgt Hle => Nat.le_lt_trans p m n Hle Hgt.	Definition	definition	gt_le_trans_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "le_lt_trans", "p" ]	Gt.le_gt_trans	217	219	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition plus_gt_compat_l_stt : forall n m p, n > m -> p + n > p + m := fun n m p Hgt => proj1 (Nat.add_lt_mono_l m n p) Hgt.	#[local] Definition plus_gt_compat_l_stt : forall n m p, n > m -> p + n > p + m	:= fun n m p Hgt => proj1 (Nat.add_lt_mono_l m n p) Hgt.	Definition	definition	plus_gt_compat_l_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "add_lt_mono_l", "p" ]	Gt.gt_le_trans	224	226	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition plus_assoc_reverse_stt := fun n m p => eq_sym (Nat.add_assoc n m p).	#[local] Definition plus_assoc_reverse_stt	:= fun n m p => eq_sym (Nat.add_assoc n m p).	Definition	definition	plus_assoc_reverse_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "add_assoc", "eq_sym", "p" ]	Plus.plus_assoc	237	238	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_plus_r_stt := (fun n m => Nat.le_add_l m n).	#[local] Definition le_plus_r_stt	:= (fun n m => Nat.le_add_l m n).	Definition	definition	le_plus_r_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "le_add_l" ]	Plus.plus_le_compat_r	247	248	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_plus_trans_stt := (fun n m p Hle => Nat.le_trans n _ _ Hle (Nat.le_add_r m p)).	#[local] Definition le_plus_trans_stt	:= (fun n m p Hle => Nat.le_trans n _ _ Hle (Nat.le_add_r m p)).	Definition	definition	le_plus_trans_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "le_add_r", "le_trans", "p" ]	null	249	250	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_plus_trans_stt := (fun n m p Hlt => Nat.lt_le_trans n _ _ Hlt (Nat.le_add_r m p)).	#[local] Definition lt_plus_trans_stt	:= (fun n m p Hlt => Nat.lt_le_trans n _ _ Hlt (Nat.le_add_r m p)).	Definition	definition	lt_plus_trans_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "le_add_r", "lt_le_trans", "p" ]	Plus.le_plus_l Plus.le_plus_r_stt Plus.le_plus_trans_stt	256	257	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition minus_n_O_stt := fun n => eq_sym (Nat.sub_0_r n).	#[local] Definition minus_n_O_stt	:= fun n => eq_sym (Nat.sub_0_r n).	Definition	definition	minus_n_O_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "eq_sym", "sub_0_r" ]	** [sub]	269	270	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition minus_Sn_m_stt := fun n m Hle => eq_sym (Nat.sub_succ_l m n Hle).	#[local] Definition minus_Sn_m_stt	:= fun n m Hle => eq_sym (Nat.sub_succ_l m n Hle).	Definition	definition	minus_Sn_m_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "eq_sym", "sub_succ_l" ]	Minus.minus_n_O	275	276	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition minus_diag_reverse_stt := fun n => eq_sym (Nat.sub_diag n).	#[local] Definition minus_diag_reverse_stt	:= fun n => eq_sym (Nat.sub_diag n).	Definition	definition	minus_diag_reverse_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "eq_sym", "sub_diag" ]	Minus.minus_Sn_m	281	282	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Lemma minus_plus_simpl_l_reverse_stt n m p : n - m = p + n - (p + m). Proof. now rewrite Nat.sub_add_distr, Nat.add_comm, Nat.add_sub. Qed.	#[local] Lemma minus_plus_simpl_l_reverse_stt n m p : n - m = p + n - (p + m).	Proof. now rewrite Nat.sub_add_distr, Nat.add_comm, Nat.add_sub. Qed.	Lemma	theorem	minus_plus_simpl_l_reverse_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "add_comm", "add_sub", "p", "sub_add_distr" ]	Minus.minus_diag_reverse	287	291	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition plus_minus_stt := fun n m p Heq => eq_sym (Nat.add_sub_eq_l n m p (eq_sym Heq)).	#[local] Definition plus_minus_stt	:= fun n m p Heq => eq_sym (Nat.add_sub_eq_l n m p (eq_sym Heq)).	Definition	definition	plus_minus_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "add_sub_eq_l", "eq_sym", "p" ]	Minus.minus_plus_simpl_l_reverse	295	296	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition minus_plus_stt := (fun n m => eq_ind _ (fun x => x - n = m) (Nat.add_sub m n) _ (Nat.add_comm _ _)).	#[local] Definition minus_plus_stt	:= (fun n m => eq_ind _ (fun x => x - n = m) (Nat.add_sub m n) _ (Nat.add_comm _ _)).	Definition	definition	minus_plus_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "add_comm", "add_sub" ]	Minus.plus_minus	301	302	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_plus_minus_stt := fun n m Hle => eq_sym (eq_trans (Nat.add_comm _ _) (Nat.sub_add n m Hle)).	#[local] Definition le_plus_minus_stt	:= fun n m Hle => eq_sym (eq_trans (Nat.add_comm _ _) (Nat.sub_add n m Hle)).	Definition	definition	le_plus_minus_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "add_comm", "eq_sym", "eq_trans", "sub_add" ]	Minus.minus_plus	307	308	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition le_plus_minus_r_stt := fun n m Hle => eq_trans (Nat.add_comm _ _) (Nat.sub_add n m Hle).	#[local] Definition le_plus_minus_r_stt	:= fun n m Hle => eq_trans (Nat.add_comm _ _) (Nat.sub_add n m Hle).	Definition	definition	le_plus_minus_r_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "add_comm", "eq_trans", "sub_add" ]	Minus.le_plus_minus	313	314	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition lt_O_minus_lt_stt : forall n m, 0 < n - m -> m < n := fun n m => proj2 (Nat.lt_add_lt_sub_r 0 n m).	#[local] Definition lt_O_minus_lt_stt : forall n m, 0 < n - m -> m < n	:= fun n m => proj2 (Nat.lt_add_lt_sub_r 0 n m).	Definition	definition	lt_O_minus_lt_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "lt_add_lt_sub_r" ]	Minus.lt_minus	321	323	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition mult_assoc_reverse_stt := fun n m p => eq_sym (Nat.mul_assoc n m p).	#[local] Definition mult_assoc_reverse_stt	:= fun n m p => eq_sym (Nat.mul_assoc n m p).	Definition	definition	mult_assoc_reverse_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "eq_sym", "mul_assoc", "p" ]	Mult.mult_minus_distr_l	340	341	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Lemma mult_O_le_stt n m : m = 0 \/ n <= m * n. Proof. destruct m; [left\|right]; simpl; trivial using Nat.le_add_r. Qed.	#[local] Lemma mult_O_le_stt n m : m = 0 \/ n <= m * n.	Proof. destruct m; [left\|right]; simpl; trivial using Nat.le_add_r. Qed.	Lemma	theorem	mult_O_le_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "le_add_r", "left", "right" ]	Mult.mult_assoc_reverse Mult.mult_assoc	346	350	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[local] Definition mult_S_lt_compat_l_stt := (fun n m p Hlt => proj1 (Nat.mul_lt_mono_pos_l (S n) m p (Nat.lt_0_succ n)) Hlt).	#[local] Definition mult_S_lt_compat_l_stt	:= (fun n m p Hlt => proj1 (Nat.mul_lt_mono_pos_l (S n) m p (Nat.lt_0_succ n)) Hlt).	Definition	definition	mult_S_lt_compat_l_stt	Arith	theories/Arith/Arith_base.v	[ "Stdlib", "Factorial", "Between", "Peano_dec", "Compare_dec", "EqNat", "Wf_nat" ]	[ "S", "lt_0_succ", "mul_lt_mono_pos_l", "p" ]	Mult.mult_le_compat_l	356	357	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Inductive between k : nat -> Prop := \| bet_emp : between k k \| bet_S : forall l, between k l -> P l -> between k (S l).	Inductive between k : nat -> Prop	:= \| bet_emp : between k k \| bet_S : forall l, between k l -> P l -> between k (S l).	Inductive	inductive	between	Arith	theories/Arith/Between.v	[]	[ "P", "S" ]	The [between] type expresses the concept [forall i: nat, k <= i < l -> P i.].	22	24	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma bet_eq : forall k l, l = k -> between k l. Proof. intros * ->; constructor. Qed.	Lemma bet_eq : forall k l, l = k -> between k l.	Proof. intros * ->; constructor. Qed.	Lemma	theorem	bet_eq	Arith	theories/Arith/Between.v	[]	[ "between" ]	null	29	32	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma between_le : forall k l, between k l -> k <= l. Proof. induction 1; auto. Qed.	Lemma between_le : forall k l, between k l -> k <= l.	Proof. induction 1; auto. Qed.	Lemma	theorem	between_le	Arith	theories/Arith/Between.v	[]	[ "between", "induction" ]	null	37	40	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma between_Sk_l : forall k l, between k l -> S k <= l -> between (S k) l. Proof. induction 1 as [\|* [\|]]; auto. - intros Hle; exfalso; apply (Nat.nle_succ_diag_l _ Hle). - intros Hle; inversion Hle; constructor; auto. Qed.	Lemma between_Sk_l : forall k l, between k l -> S k <= l -> between (S k) l.	Proof. induction 1 as [\|* [\|]]; auto. - intros Hle; exfalso; apply (Nat.nle_succ_diag_l _ Hle). - intros Hle; inversion Hle; constructor; auto. Qed.	Lemma	theorem	between_Sk_l	Arith	theories/Arith/Between.v	[]	[ "S", "between", "induction", "nle_succ_diag_l" ]	null	44	49	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma between_restr : forall k l (m:nat), k <= l -> l <= m -> between k m -> between l m. Proof. induction 1; auto. intros; auto. apply between_Sk_l; auto. apply IHle; auto. transitivity (S m0); auto. Qed.	Lemma between_restr : forall k l (m:nat), k <= l -> l <= m -> between k m -> between l m.	Proof. induction 1; auto. intros; auto. apply between_Sk_l; auto. apply IHle; auto. transitivity (S m0); auto. Qed.	Lemma	theorem	between_restr	Arith	theories/Arith/Between.v	[]	[ "S", "between", "between_Sk_l", "induction" ]	null	53	61	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Inductive exists_between k : nat -> Prop := \| exists_S : forall l, exists_between k l -> exists_between k (S l) \| exists_le : forall l, k <= l -> Q l -> exists_between k (S l).	Inductive exists_between k : nat -> Prop	:= \| exists_S : forall l, exists_between k l -> exists_between k (S l) \| exists_le : forall l, k <= l -> Q l -> exists_between k (S l).	Inductive	inductive	exists_between	Arith	theories/Arith/Between.v	[]	[ "Q", "S" ]	The [exists_between] type expresses the concept [exists i: nat, k <= i < l /\ Q i].	65	67	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma exists_le_S : forall k l, exists_between k l -> S k <= l. Proof. induction 1; auto. apply -> Nat.succ_le_mono; assumption. Qed.	Lemma exists_le_S : forall k l, exists_between k l -> S k <= l.	Proof. induction 1; auto. apply -> Nat.succ_le_mono; assumption. Qed.	Lemma	theorem	exists_le_S	Arith	theories/Arith/Between.v	[]	[ "S", "exists_between", "induction", "succ_le_mono" ]	null	72	76	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma exists_lt : forall k l, exists_between k l -> k < l. Proof. exact exists_le_S. Qed.	Lemma exists_lt : forall k l, exists_between k l -> k < l.	Proof. exact exists_le_S. Qed.	Lemma	theorem	exists_lt	Arith	theories/Arith/Between.v	[]	[ "exists_between", "exists_le_S" ]	null	78	81	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma exists_S_le : forall k l, exists_between k (S l) -> k <= l. Proof. intros; apply le_S_n; auto. Qed.	Lemma exists_S_le : forall k l, exists_between k (S l) -> k <= l.	Proof. intros; apply le_S_n; auto. Qed.	Lemma	theorem	exists_S_le	Arith	theories/Arith/Between.v	[]	[ "S", "exists_between" ]	null	85	88	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition in_int p q r := p <= r /\ r < q.	Definition in_int p q r	:= p <= r /\ r < q.	Definition	definition	in_int	Arith	theories/Arith/Between.v	[]	[ "p", "r" ]	null	92	92	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma in_int_intro : forall p q r, p <= r -> r < q -> in_int p q r. Proof. split; assumption. Qed.	Lemma in_int_intro : forall p q r, p <= r -> r < q -> in_int p q r.	Proof. split; assumption. Qed.	Lemma	theorem	in_int_intro	Arith	theories/Arith/Between.v	[]	[ "in_int", "p", "r", "split" ]	null	94	97	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma in_int_lt : forall p q r, in_int p q r -> p < q. Proof. intros * []. eapply Nat.le_lt_trans; eassumption. Qed.	Lemma in_int_lt : forall p q r, in_int p q r -> p < q.	Proof. intros * []. eapply Nat.le_lt_trans; eassumption. Qed.	Lemma	theorem	in_int_lt	Arith	theories/Arith/Between.v	[]	[ "in_int", "le_lt_trans", "p", "r" ]	null	101	105	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma in_int_p_Sq : forall p q r, in_int p (S q) r -> in_int p q r \/ r = q. Proof. intros p q r []. destruct (proj1 (Nat.lt_eq_cases r q)); auto. apply Nat.lt_succ_r; assumption. Qed.	Lemma in_int_p_Sq : forall p q r, in_int p (S q) r -> in_int p q r \/ r = q.	Proof. intros p q r []. destruct (proj1 (Nat.lt_eq_cases r q)); auto. apply Nat.lt_succ_r; assumption. Qed.	Lemma	theorem	in_int_p_Sq	Arith	theories/Arith/Between.v	[]	[ "S", "in_int", "lt_eq_cases", "lt_succ_r", "p", "r" ]	null	107	113	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma in_int_S : forall p q r, in_int p q r -> in_int p (S q) r. Proof. intros * []; auto. Qed.	Lemma in_int_S : forall p q r, in_int p q r -> in_int p (S q) r.	Proof. intros * []; auto. Qed.	Lemma	theorem	in_int_S	Arith	theories/Arith/Between.v	[]	[ "S", "in_int", "p", "r" ]	null	115	118	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma in_int_Sp_q : forall p q r, in_int (S p) q r -> in_int p q r. Proof. intros * []; auto. apply in_int_intro; auto. transitivity (S p); auto. Qed.	Lemma in_int_Sp_q : forall p q r, in_int (S p) q r -> in_int p q r.	Proof. intros * []; auto. apply in_int_intro; auto. transitivity (S p); auto. Qed.	Lemma	theorem	in_int_Sp_q	Arith	theories/Arith/Between.v	[]	[ "S", "in_int", "in_int_intro", "p", "r" ]	null	122	127	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma between_in_int : forall k l, between k l -> forall r, in_int k l r -> P r. Proof. intro k; induction 1 as [\|l]; intros r ?. - absurd (k < k). { apply Nat.lt_irrefl. } eapply in_int_lt; eassumption. - destruct (in_int_p_Sq k l r) as [\| ->]; auto. Qed.	Lemma between_in_int : forall k l, between k l -> forall r, in_int k l r -> P r.	Proof. intro k; induction 1 as [\|l]; intros r ?. - absurd (k < k). { apply Nat.lt_irrefl. } eapply in_int_lt; eassumption. - destruct (in_int_p_Sq k l r) as [\| ->]; auto. Qed.	Lemma	theorem	between_in_int	Arith	theories/Arith/Between.v	[]	[ "P", "between", "in_int", "in_int_lt", "in_int_p_Sq", "induction", "lt_irrefl", "r" ]	null	131	138	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma in_int_between : forall k l, k <= l -> (forall r, in_int k l r -> P r) -> between k l. Proof. induction 1; auto. Qed.	Lemma in_int_between : forall k l, k <= l -> (forall r, in_int k l r -> P r) -> between k l.	Proof. induction 1; auto. Qed.	Lemma	theorem	in_int_between	Arith	theories/Arith/Between.v	[]	[ "P", "between", "in_int", "induction", "r" ]	null	140	144	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma exists_in_int : forall k l, exists_between k l -> exists2 m : nat, in_int k l m & Q m. Proof. induction 1 as [* ? (p, ?, ?)\|l]. - exists p; auto. - exists l; auto. Qed.	Lemma exists_in_int : forall k l, exists_between k l -> exists2 m : nat, in_int k l m & Q m.	Proof. induction 1 as [* ? (p, ?, ?)\|l]. - exists p; auto. - exists l; auto. Qed.	Lemma	theorem	exists_in_int	Arith	theories/Arith/Between.v	[]	[ "Q", "exists_between", "in_int", "induction", "p" ]	null	146	152	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma in_int_exists : forall k l r, in_int k l r -> Q r -> exists_between k l. Proof. intros * (?, lt_r_l) ?. induction lt_r_l; auto. Qed.	Lemma in_int_exists : forall k l r, in_int k l r -> Q r -> exists_between k l.	Proof. intros * (?, lt_r_l) ?. induction lt_r_l; auto. Qed.	Lemma	theorem	in_int_exists	Arith	theories/Arith/Between.v	[]	[ "Q", "exists_between", "in_int", "induction", "r" ]	null	154	158	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma between_or_exists : forall k l, k <= l -> (forall n:nat, in_int k l n -> P n \/ Q n) -> between k l \/ exists_between k l. Proof. induction 1 as [\|m ? IHle]. - auto. - intros P_or_Q. destruct IHle; auto. destruct (P_or_Q m); auto. Qed.	Lemma between_or_exists : forall k l, k <= l -> (forall n:nat, in_int k l n -> P n \/ Q n) -> between k l \/ exists_between k l.	Proof. induction 1 as [\|m ? IHle]. - auto. - intros P_or_Q. destruct IHle; auto. destruct (P_or_Q m); auto. Qed.	Lemma	theorem	between_or_exists	Arith	theories/Arith/Between.v	[]	[ "P", "Q", "between", "exists_between", "in_int", "induction" ]	null	160	171	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma between_not_exists : forall k l, between k l -> (forall n:nat, in_int k l n -> P n -> ~ Q n) -> ~ exists_between k l. Proof. intro k; induction 1 as [\|l]; red; intros. - absurd (k < k). { apply Nat.lt_irrefl. } auto. - absurd (Q l). { auto. } destruct (exists_in_int k (S l)) as...	Lemma between_not_exists : forall k l, between k l -> (forall n:nat, in_int k l n -> P n -> ~ Q n) -> ~ exists_between k l.	Proof. intro k; induction 1 as [\|l]; red; intros. - absurd (k < k). { apply Nat.lt_irrefl. } auto. - absurd (Q l). { auto. } destruct (exists_in_int k (S l)) as (l',[],?). + auto. + replace l with l'. { trivial. } destruct (proj1 (Nat.lt_eq_cases l' l)); auto. * apply Nat.l...	Lemma	theorem	between_not_exists	Arith	theories/Arith/Between.v	[]	[ "P", "Q", "S", "between", "exists_between", "exists_in_int", "in_int", "in_int_exists", "induction", "lt_eq_cases", "lt_irrefl", "lt_succ_r", "replace" ]	null	173	188	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Inductive P_nth (init:nat) : nat -> nat -> Prop := \| nth_O : P_nth init init 0 \| nth_S : forall k l (n:nat), P_nth init k n -> between (S k) l -> Q l -> P_nth init l (S n).	Inductive P_nth (init:nat) : nat -> nat -> Prop	:= \| nth_O : P_nth init init 0 \| nth_S : forall k l (n:nat), P_nth init k n -> between (S k) l -> Q l -> P_nth init l (S n).	Inductive	inductive	P_nth	Arith	theories/Arith/Between.v	[]	[ "Q", "S", "between", "init" ]	null	190	194	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma nth_le : forall (init:nat) l (n:nat), P_nth init l n -> init <= l. Proof. induction 1 as [\|a b c H0 H1 H2 H3]. - auto. - eapply Nat.le_trans; eauto. apply between_le in H2. transitivity (S a); auto. Qed.	Lemma nth_le : forall (init:nat) l (n:nat), P_nth init l n -> init <= l.	Proof. induction 1 as [\|a b c H0 H1 H2 H3]. - auto. - eapply Nat.le_trans; eauto. apply between_le in H2. transitivity (S a); auto. Qed.	Lemma	theorem	nth_le	Arith	theories/Arith/Between.v	[]	[ "P_nth", "S", "between_le", "induction", "init", "le_trans" ]	null	196	203	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition eventually (n:nat) := exists2 k : nat, k <= n & Q k.	Definition eventually (n:nat)	:= exists2 k : nat, k <= n & Q k.	Definition	definition	eventually	Arith	theories/Arith/Between.v	[]	[ "Q" ]	null	205	205	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma event_O : eventually 0 -> Q 0. Proof. intros (x, ?, ?). replace 0 with x; auto. apply Nat.le_0_r; assumption. Qed.	Lemma event_O : eventually 0 -> Q 0.	Proof. intros (x, ?, ?). replace 0 with x; auto. apply Nat.le_0_r; assumption. Qed.	Lemma	theorem	event_O	Arith	theories/Arith/Between.v	[]	[ "Q", "eventually", "le_0_r", "replace" ]	null	207	212	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use Coq.Arith.Compare_dec.zerop instead")] Definition notzerop n := sumbool_not _ _ (zerop n).	#[deprecated(since="8.20", note="Use Coq.Arith.Compare_dec.zerop instead")] Definition notzerop n	:= sumbool_not _ _ (zerop n).	Definition	definition	notzerop	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "zerop" ]	The decidability of equality and order relations over type [nat] give some boolean functions with the adequate specification.	23	24	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use Arith.Compare_dec.lt_dec and PeanoNat.Nat.nlt_ge instead")] Definition lt_ge_dec : forall x y, {x < y} + {x >= y} := fun n m => sumbool_not _ _ (le_lt_dec m n).	#[deprecated(since="8.20", note="Use Arith.Compare_dec.lt_dec and PeanoNat.Nat.nlt_ge instead")] Definition lt_ge_dec : forall x y, {x < y} + {x >= y}	:= fun n m => sumbool_not _ _ (le_lt_dec m n).	Definition	definition	lt_ge_dec	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "and", "le_lt_dec", "lt_dec", "nlt_ge" ]	null	26	29	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use PeanoNat.Nat.ltb instead")] Definition nat_lt_ge_bool x y := bool_of_sumbool (lt_ge_dec x y).	#[deprecated(since="8.20", note="Use PeanoNat.Nat.ltb instead")] Definition nat_lt_ge_bool x y	:= bool_of_sumbool (lt_ge_dec x y).	Definition	definition	nat_lt_ge_bool	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "lt_ge_dec", "ltb" ]	null	31	32	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use PeanoNat.Nat.leb instead")] Definition nat_ge_lt_bool x y := bool_of_sumbool (sumbool_not _ _ (lt_ge_dec x y)).	#[deprecated(since="8.20", note="Use PeanoNat.Nat.leb instead")] Definition nat_ge_lt_bool x y	:= bool_of_sumbool (sumbool_not _ _ (lt_ge_dec x y)).	Definition	definition	nat_ge_lt_bool	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "leb", "lt_ge_dec" ]	null	34	36	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use PeanoNat.Nat.leb instead")] Definition nat_le_gt_bool x y := bool_of_sumbool (le_gt_dec x y).	#[deprecated(since="8.20", note="Use PeanoNat.Nat.leb instead")] Definition nat_le_gt_bool x y	:= bool_of_sumbool (le_gt_dec x y).	Definition	definition	nat_le_gt_bool	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "le_gt_dec", "leb" ]	null	38	39	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use PeanoNat.Nat.ltb instead")] Definition nat_gt_le_bool x y := bool_of_sumbool (sumbool_not _ _ (le_gt_dec x y)).	#[deprecated(since="8.20", note="Use PeanoNat.Nat.ltb instead")] Definition nat_gt_le_bool x y	:= bool_of_sumbool (sumbool_not _ _ (le_gt_dec x y)).	Definition	definition	nat_gt_le_bool	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "le_gt_dec", "ltb" ]	null	41	43	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use PeanoNat.Nat.eqb instead")] Definition nat_eq_bool x y := bool_of_sumbool (eq_nat_dec x y).	#[deprecated(since="8.20", note="Use PeanoNat.Nat.eqb instead")] Definition nat_eq_bool x y	:= bool_of_sumbool (eq_nat_dec x y).	Definition	definition	nat_eq_bool	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "eq_nat_dec", "eqb" ]	null	45	46	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use PeanoNat.Nat.eqb instead")] Definition nat_noteq_bool x y := bool_of_sumbool (sumbool_not _ _ (eq_nat_dec x y)).	#[deprecated(since="8.20", note="Use PeanoNat.Nat.eqb instead")] Definition nat_noteq_bool x y	:= bool_of_sumbool (sumbool_not _ _ (eq_nat_dec x y)).	Definition	definition	nat_noteq_bool	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "eq_nat_dec", "eqb" ]	null	48	50	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use Coq.Arith.Compare_dec.zerop instead")] Definition zerop_bool x := bool_of_sumbool (zerop x).	#[deprecated(since="8.20", note="Use Coq.Arith.Compare_dec.zerop instead")] Definition zerop_bool x	:= bool_of_sumbool (zerop x).	Definition	definition	zerop_bool	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "zerop" ]	null	52	53	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
#[deprecated(since="8.20", note="Use Coq.Arith.Compare_dec.zerop instead")] Definition notzerop_bool x := bool_of_sumbool (notzerop x).	#[deprecated(since="8.20", note="Use Coq.Arith.Compare_dec.zerop instead")] Definition notzerop_bool x	:= bool_of_sumbool (notzerop x).	Definition	definition	notzerop_bool	Arith	theories/Arith/Bool_nat.v	[ "Stdlib", "Compare_dec", "Peano_dec", "Sumbool" ]	[ "notzerop", "zerop" ]	null	55	56	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition to_nat '(x, y) : nat := y + (nat_rec _ 0 (fun i m => (S i) + m) (y + x)).	Definition to_nat '(x, y) : nat	:= y + (nat_rec _ 0 (fun i m => (S i) + m) (y + x)).	Definition	definition	to_nat	Arith	theories/Arith/Cantor.v	[]	[ "S" ]	Cantor pairing [to_nat]	19	20	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition of_nat (n : nat) : nat * nat := nat_rec _ (0, 0) (fun _ '(x, y) => match x with \| S x => (x, S y) \| _ => (S y, 0) end) n.	Definition of_nat (n : nat) : nat * nat	:= nat_rec _ (0, 0) (fun _ '(x, y) => match x with \| S x => (x, S y) \| _ => (S y, 0) end) n.	Definition	definition	of_nat	Arith	theories/Arith/Cantor.v	[]	[ "S" ]	Cantor pairing inverse [of_nat]	24	26	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma cancel_of_to p : of_nat (to_nat p) = p. Proof. enough (H : forall n p, to_nat p = n -> of_nat n = p) by now apply H. intro n. induction n as [\|n IHn]. - now intros [[\|?] [\|?]]. - intros [x [\|y]]. + destruct x as [\|x]; [discriminate\|]. intros [=H]. cbn. fold (of_nat n). rewrite (IHn (0, x))...	Lemma cancel_of_to p : of_nat (to_nat p) = p.	Proof. enough (H : forall n p, to_nat p = n -> of_nat n = p) by now apply H. intro n. induction n as [\|n IHn]. - now intros [[\|?] [\|?]]. - intros [x [\|y]]. + destruct x as [\|x]; [discriminate\|]. intros [=H]. cbn. fold (of_nat n). rewrite (IHn (0, x)); [reflexivity\|]. rewrite <- H. cbn. now...	Lemma	theorem	cancel_of_to	Arith	theories/Arith/Cantor.v	[]	[ "S", "add_0_r", "add_succ_r", "fold", "induction", "of_nat", "p", "to_nat" ]	[of_nat] is the left inverse for [to_nat]	30	43	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Corollary to_nat_inj p q : to_nat p = to_nat q -> p = q. Proof. intros H %(f_equal of_nat). now rewrite ?cancel_of_to in H. Qed.	Corollary to_nat_inj p q : to_nat p = to_nat q -> p = q.	Proof. intros H %(f_equal of_nat). now rewrite ?cancel_of_to in H. Qed.	Corollary	theorem	to_nat_inj	Arith	theories/Arith/Cantor.v	[]	[ "cancel_of_to", "of_nat", "p", "to_nat" ]	[to_nat] is injective	47	50	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma cancel_to_of n : to_nat (of_nat n) = n. Proof. induction n as [\|n IHn]; [reflexivity\|]. cbn. fold (of_nat n). destruct (of_nat n) as [[\|x] y]. - rewrite <- IHn. cbn. now rewrite PeanoNat.Nat.add_0_r. - rewrite <- IHn. cbn. now rewrite (Nat.add_succ_r y x). Qed.	Lemma cancel_to_of n : to_nat (of_nat n) = n.	Proof. induction n as [\|n IHn]; [reflexivity\|]. cbn. fold (of_nat n). destruct (of_nat n) as [[\|x] y]. - rewrite <- IHn. cbn. now rewrite PeanoNat.Nat.add_0_r. - rewrite <- IHn. cbn. now rewrite (Nat.add_succ_r y x). Qed.	Lemma	theorem	cancel_to_of	Arith	theories/Arith/Cantor.v	[]	[ "add_0_r", "add_succ_r", "fold", "induction", "of_nat", "to_nat" ]	[to_nat] is the left inverse for [of_nat]	54	60	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Corollary of_nat_inj n m : of_nat n = of_nat m -> n = m. Proof. intros H %(f_equal to_nat). now rewrite ?cancel_to_of in H. Qed.	Corollary of_nat_inj n m : of_nat n = of_nat m -> n = m.	Proof. intros H %(f_equal to_nat). now rewrite ?cancel_to_of in H. Qed.	Corollary	theorem	of_nat_inj	Arith	theories/Arith/Cantor.v	[]	[ "cancel_to_of", "of_nat", "to_nat" ]	[of_nat] is injective	64	67	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma to_nat_spec x y : to_nat (x, y) * 2 = y * 2 + (y + x) * S (y + x). Proof. cbn; induction (y + x) as [\|n IHn]; cbn; [now rewrite !Nat.add_0_r\|]. rewrite <-plus_Sn_m, Nat.add_assoc, (Nat.add_comm y), <-Nat.add_assoc. rewrite Nat.mul_add_distr_r, IHn, Nat.add_comm, <-Nat.add_assoc. apply f_equal2; [reflexi...	Lemma to_nat_spec x y : to_nat (x, y) * 2 = y * 2 + (y + x) * S (y + x).	Proof. cbn; induction (y + x) as [\|n IHn]; cbn; [now rewrite !Nat.add_0_r\|]. rewrite <-plus_Sn_m, Nat.add_assoc, (Nat.add_comm y), <-Nat.add_assoc. rewrite Nat.mul_add_distr_r, IHn, Nat.add_comm, <-Nat.add_assoc. apply f_equal2; [reflexivity\|]. rewrite Nat.mul_comm, <-Nat.mul_add_distr_l. rewrite <-!plus_Sn...	Lemma	theorem	to_nat_spec	Arith	theories/Arith/Cantor.v	[]	[ "S", "add_0_r", "add_1_r", "add_assoc", "add_comm", "induction", "mul_1_l", "mul_add_distr_l", "mul_add_distr_r", "mul_comm", "to_nat" ]	Polynomial specifications of [to_nat]	71	82	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma to_nat_spec2 x y : to_nat (x, y) = y + (y + x) * S (y + x) / 2. Proof. now rewrite <- Nat.div_add_l, <- to_nat_spec, Nat.div_mul. Qed.	Lemma to_nat_spec2 x y : to_nat (x, y) = y + (y + x) * S (y + x) / 2.	Proof. now rewrite <- Nat.div_add_l, <- to_nat_spec, Nat.div_mul. Qed.	Lemma	theorem	to_nat_spec2	Arith	theories/Arith/Cantor.v	[]	[ "S", "div_add_l", "div_mul", "to_nat", "to_nat_spec" ]	null	84	88	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma to_nat_non_decreasing x y : y + x <= to_nat (x, y). Proof. pose proof (to_nat_spec x y). rewrite (Nat.mul_le_mono_pos_r _ _ 2 Nat.lt_0_2), H. rewrite Nat.mul_add_distr_r, <-Nat.add_le_mono_l. case x as [\|x]; [now rewrite Nat.mul_0_l; apply le_0_n\|]. rewrite Nat.mul_add_distr_r, <-(Nat.add_0_l (S x * 2))...	Lemma to_nat_non_decreasing x y : y + x <= to_nat (x, y).	Proof. pose proof (to_nat_spec x y). rewrite (Nat.mul_le_mono_pos_r _ _ 2 Nat.lt_0_2), H. rewrite Nat.mul_add_distr_r, <-Nat.add_le_mono_l. case x as [\|x]; [now rewrite Nat.mul_0_l; apply le_0_n\|]. rewrite Nat.mul_add_distr_r, <-(Nat.add_0_l (S x * 2)); apply Nat.add_le_mono. now apply le_0_n. apply Nat...	Lemma	theorem	to_nat_non_decreasing	Arith	theories/Arith/Cantor.v	[]	[ "S", "add_0_l", "add_1_r", "add_assoc", "add_le_mono", "add_le_mono_l", "le_add_l", "lt_0_2", "mul_0_l", "mul_add_distr_r", "mul_le_mono_l", "mul_le_mono_pos_r", "proof", "to_nat", "to_nat_spec" ]	[to_nat] is non-decreasing in (the sum of) pair components	92	103	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Notation not_eq_sym := not_eq_sym (only parsing).	Notation not_eq_sym	:= not_eq_sym (only parsing).	Notation	notation	not_eq_sym	Arith	theories/Arith/Compare.v	[ "Stdlib", "PeanoNat", "Compare_dec", "Wf_nat" ]	[]	null	15	15	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition le_or_le_S := le_le_S_dec.	Definition le_or_le_S	:= le_le_S_dec.	Definition	definition	le_or_le_S	Arith	theories/Arith/Compare.v	[ "Stdlib", "PeanoNat", "Compare_dec", "Wf_nat" ]	[ "le_le_S_dec" ]	null	21	21	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition Pcompare := gt_eq_gt_dec.	Definition Pcompare	:= gt_eq_gt_dec.	Definition	definition	Pcompare	Arith	theories/Arith/Compare.v	[ "Stdlib", "PeanoNat", "Compare_dec", "Wf_nat" ]	[ "gt_eq_gt_dec" ]	null	23	23	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma le_dec : forall n m, {n <= m} + {m <= n}. Proof. exact le_ge_dec. Qed.	Lemma le_dec : forall n m, {n <= m} + {m <= n}.	Proof. exact le_ge_dec. Qed.	Lemma	theorem	le_dec	Arith	theories/Arith/Compare.v	[ "Stdlib", "PeanoNat", "Compare_dec", "Wf_nat" ]	[ "le_ge_dec" ]	null	25	28	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition lt_or_eq n m := {m > n} + {n = m}.	Definition lt_or_eq n m	:= {m > n} + {n = m}.	Definition	definition	lt_or_eq	Arith	theories/Arith/Compare.v	[ "Stdlib", "PeanoNat", "Compare_dec", "Wf_nat" ]	[]	null	30	30	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma le_decide : forall n m, n <= m -> lt_or_eq n m. Proof. exact le_lt_eq_dec. Qed.	Lemma le_decide : forall n m, n <= m -> lt_or_eq n m.	Proof. exact le_lt_eq_dec. Qed.	Lemma	theorem	le_decide	Arith	theories/Arith/Compare.v	[ "Stdlib", "PeanoNat", "Compare_dec", "Wf_nat" ]	[ "le_lt_eq_dec", "lt_or_eq" ]	null	32	35	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma le_le_S_eq : forall n m, n <= m -> S n <= m \/ n = m. Proof. exact (fun n m Hle => proj1 (Nat.lt_eq_cases n m) Hle). Qed.	Lemma le_le_S_eq : forall n m, n <= m -> S n <= m \/ n = m.	Proof. exact (fun n m Hle => proj1 (Nat.lt_eq_cases n m) Hle). Qed.	Lemma	theorem	le_le_S_eq	Arith	theories/Arith/Compare.v	[ "Stdlib", "PeanoNat", "Compare_dec", "Wf_nat" ]	[ "S", "lt_eq_cases" ]	null	37	40	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Lemma discrete_nat : forall n m, n < m -> S n = m \/ (exists r : nat, m = S (S (n + r))). Proof. intros m n H. lapply (proj1 (Nat.le_succ_l m n)); auto. intro H'; lapply (proj1 (Nat.lt_eq_cases (S m) n)); auto. induction 1; auto. right; exists (n - S (S m)); simpl. rewrite (Nat.add_comm m (n - S (S m))). ...	Lemma discrete_nat : forall n m, n < m -> S n = m \/ (exists r : nat, m = S (S (n + r))).	Proof. intros m n H. lapply (proj1 (Nat.le_succ_l m n)); auto. intro H'; lapply (proj1 (Nat.lt_eq_cases (S m) n)); auto. induction 1; auto. right; exists (n - S (S m)); simpl. rewrite (Nat.add_comm m (n - S (S m))). rewrite (plus_n_Sm (n - S (S m)) m). rewrite (plus_n_Sm (n - S (S m)) (S m)). rewrite ...	Lemma	theorem	discrete_nat	Arith	theories/Arith/Compare.v	[ "Stdlib", "PeanoNat", "Compare_dec", "Wf_nat" ]	[ "S", "add_0_r", "add_comm", "add_sub_assoc", "induction", "le_succ_l", "lt_eq_cases", "r", "right", "sub_diag" ]	By special request of G. Kahn - Used in Group Theory	43	60	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition zerop n : {n = 0} + {0 < n}. Proof. destruct n; [left\|right]; auto. apply Nat.lt_0_succ. Defined.	Definition zerop n : {n = 0} + {0 < n}.	Proof. destruct n; [left\|right]; auto. apply Nat.lt_0_succ. Defined.	Definition	definition	zerop	Arith	theories/Arith/Compare_dec.v	[]	[ "left", "lt_0_succ", "right" ]	null	17	21	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition lt_eq_lt_dec n m : {n < m} + {n = m} + {m < n}. Proof. induction n as [\|n IHn] in m \|- *; destruct m as [\|m]; auto. - left; left; apply Nat.lt_0_succ. - right; apply Nat.lt_0_succ. - destruct (IHn m) as [[H\|H]\|H]; auto. + left; left; now apply Nat.succ_lt_mono in H. + right; now apply Nat.suc...	Definition lt_eq_lt_dec n m : {n < m} + {n = m} + {m < n}.	Proof. induction n as [\|n IHn] in m \|- *; destruct m as [\|m]; auto. - left; left; apply Nat.lt_0_succ. - right; apply Nat.lt_0_succ. - destruct (IHn m) as [[H\|H]\|H]; auto. + left; left; now apply Nat.succ_lt_mono in H. + right; now apply Nat.succ_lt_mono in H. Defined.	Definition	definition	lt_eq_lt_dec	Arith	theories/Arith/Compare_dec.v	[]	[ "induction", "left", "lt_0_succ", "right", "succ_lt_mono" ]	null	23	31	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof
Definition gt_eq_gt_dec n m : {m > n} + {n = m} + {n > m}. Proof. now apply lt_eq_lt_dec. Defined.	Definition gt_eq_gt_dec n m : {m > n} + {n = m} + {n > m}.	Proof. now apply lt_eq_lt_dec. Defined.	Definition	definition	gt_eq_gt_dec	Arith	theories/Arith/Compare_dec.v	[]	[ "lt_eq_lt_dec" ]	null	33	36	true	https://github.com/rocq-prover/stdlib	f76a666b0b2c28c671d4fdf6dd25bcab865b9c36	statement+proof

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Coq-Stdlib

Structured dataset of definitions and theorems from the Coq standard library.

Source

Repository: https://github.com/rocq-prover/stdlib
Commit: f76a666b0b2c28c671d4fdf6dd25bcab865b9c36
Files: 583
License: lgpl-2.1

Schema

Column	Type	Description
fact	string	Verbatim declaration: statement followed by proof where present
statement	string	Verbatim statement (keyword through the closing period)
proof	string	Verbatim proof block (`Proof. ... Qed.`/`Defined.`), empty if none
type	string	Raw declaration keyword
kind	string	Normalized kind
symbolic_name	string	Declaration identifier
library	string	Sub-library
filename	string	Repository-relative source path
imports	list[string]	File-level `Require`/`Import` modules
deps	list[string]	Intra-corpus identifiers referenced
docstring	string	Preceding documentation comment, null if absent
line_start	int	First source line
line_end	int	Last source line
has_proof	bool	Whether a proof block was captured
source_url	string	Upstream repository
commit	string	Upstream commit extracted
content_level	string	`statement+proof`

Statistics

Entries: 19,736
With proof: 19,011 (96.3%)
With docstring: 3,080 (15.6%)
Libraries: 44

By type

Type	Count
Lemma	10,586
Definition	2,880
Notation	1,716
Theorem	1,398
Instance	770
Ltac	662
Fixpoint	545
Parameter	364
Inductive	185
Hypothesis	133
Axiom	132
Let	116
Record	60
Class	45
Example	42
Corollary	24
Fact	17
Coercion	16
Scheme	15
Remark	11
Parameters	6
CoFixpoint	5
Variant	3
CoInductive	3
Proposition	1
Structure	1

Example

#[local]
Lemma minus_plus_simpl_l_reverse_stt n m p : n - m = p + n - (p + m).
Proof.
 now rewrite Nat.sub_add_distr, Nat.add_comm, Nat.add_sub.
Qed.

kind: theorem | symbolic_name: minus_plus_simpl_l_reverse_stt | theories/Arith/Arith_base.v:287

Use

Statement and proof are available both joined (fact) and split (statement, proof) for proof-term modeling, autoformalization, retrieval, and dependency analysis via deps.

Citation

@misc{coq_stdlib_dataset,
  title  = {Coq-Stdlib},
  author = {Norton, Charles},
  year   = {2026},
  note   = {Extracted from https://github.com/rocq-prover/stdlib, commit f76a666b0b2c},
  url    = {https://huggingface.co/datasets/phanerozoic/Coq-Stdlib}
}

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