statement stringlengths 1 4.15k | proof stringlengths 0 26.3k | type stringclasses 26
values | symbolic_name stringlengths 1 75 | library stringclasses 44
values | filename stringclasses 463
values | imports listlengths 0 23 | deps listlengths 0 64 | docstring stringlengths 0 2.4k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
le_n_0_eq_stt | := fun n Hle => eq_sym (proj1 (Nat.le_0_r n) Hle). | Definition | le_n_0_eq_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"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 | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lt_le_S_stt | := fun n m => (proj2 (Nat.le_succ_l n m)). | Definition | lt_le_S_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_succ_l"
] | Lt.lt_irrefl | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lt_n_Sm_le_stt | := fun n m => (proj1 (Nat.lt_succ_r n m)). | Definition | lt_n_Sm_le_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_succ_r"
] | Lt.lt_le_S | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_lt_n_Sm_stt | := fun n m => (proj2 (Nat.lt_succ_r n m)). | Definition | le_lt_n_Sm_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_succ_r"
] | Lt.lt_n_Sm_le | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_not_lt_stt | := fun n m => (proj1 (Nat.le_ngt n m)). | Definition | le_not_lt_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_ngt"
] | Lt.le_lt_n_Sm | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lt_not_le_stt | := fun n m => (proj1 (Nat.lt_nge n m)). | Definition | lt_not_le_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_nge"
] | Lt.le_not_lt | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
neq_0_lt_stt | := fun n Hn => proj1 (Nat.neq_0_lt_0 n) (Nat.neq_sym 0 n Hn). | Definition | neq_0_lt_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"neq_0_lt_0",
"neq_sym"
] | Lt.lt_0_Sn Lt.lt_n_0 | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lt_0_neq_stt | := fun n Hlt => Nat.neq_sym n 0 (proj2 (Nat.neq_0_lt_0 n) Hlt). | Definition | lt_0_neq_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"neq_0_lt_0",
"neq_sym"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
lt_n_S_stt | := fun n m => (proj1 (Nat.succ_lt_mono n m)). | Definition | lt_n_S_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"succ_lt_mono"
] | Lt.lt_n_Sn Lt.lt_S | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lt_S_n_stt | := fun n m => (proj2 (Nat.succ_lt_mono n m)). | Definition | lt_S_n_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"succ_lt_mono"
] | Lt.lt_n_S | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lt_pred_stt | := fun n m => proj1 (Nat.lt_succ_lt_pred n m). | Definition | lt_pred_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_succ_lt_pred"
] | Lt.lt_S_n | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lt_pred_n_n_stt | := fun n Hlt => Nat.lt_pred_l n (proj2 (Nat.neq_0_lt_0 n) Hlt). | Definition | lt_pred_n_n_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_pred_l",
"neq_0_lt_0"
] | Lt.lt_pred | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
gt_Sn_O_stt : forall n, S n > 0 | := Nat.lt_0_succ. | Definition | gt_Sn_O_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_0_succ"
] | Lt.lt_le_weak | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
gt_Sn_n_stt : forall n, S n > n | := Nat.lt_succ_diag_r. | Definition | gt_Sn_n_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_succ_diag_r"
] | Gt.gt_Sn_O | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | gt_n_S_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"succ_lt_mono"
] | Gt.gt_Sn_n | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | gt_S_n_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"succ_lt_mono"
] | Gt.gt_n_S | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | gt_pred_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_succ_lt_pred",
"pred"
] | Gt.gt_S_n | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
gt_irrefl_stt : forall n, ~ n > n | := Nat.lt_irrefl. | Definition | gt_irrefl_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_irrefl"
] | Gt.gt_pred | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
gt_asym_stt : forall n m, n > m -> ~ m > n | := fun n m => Nat.lt_asymm m n. | Definition | gt_asym_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_asymm"
] | Gt.gt_irrefl | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_not_gt_stt : forall n m, n <= m -> ~ n > m | := fun n m => proj1 (Nat.le_ngt n m). | Definition | le_not_gt_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_ngt"
] | Gt.gt_asym | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
gt_not_le_stt: forall n m, n > m -> ~ n <= m | := fun n m => proj1 (Nat.lt_nge m n). | Definition | gt_not_le_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_nge"
] | Gt.le_not_gt | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_S_gt_stt: forall n m, S n <= m -> m > n | := fun n m => proj1 (Nat.le_succ_l n m). | Definition | le_S_gt_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_succ_l"
] | Gt.gt_not_le | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
gt_S_le_stt : forall n m, S m > n -> n <= m | := fun n m => proj2 (Nat.succ_le_mono n m). | Definition | gt_S_le_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"succ_le_mono"
] | Gt.le_S_gt | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
gt_le_S_stt : forall n m, m > n -> S n <= m | := fun n m => proj2 (Nat.le_succ_l n m). | Definition | gt_le_S_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_succ_l"
] | Gt.gt_S_le | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_gt_S_stt : forall n m, n <= m -> S m > n | := fun n m => proj1 (Nat.succ_le_mono n m). | Definition | le_gt_S_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"succ_le_mono"
] | Gt.gt_le_S | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | gt_trans_S_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_le_trans",
"succ_le_mono"
] | Gt.le_gt_S | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | le_gt_trans_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_le_trans"
] | Gt.gt_trans_S | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | gt_le_trans_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_lt_trans"
] | Gt.le_gt_trans | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | plus_gt_compat_l_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"add_lt_mono_l"
] | Gt.gt_le_trans | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
plus_assoc_reverse_stt | := fun n m p => eq_sym (Nat.add_assoc n m p). | Definition | plus_assoc_reverse_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"add_assoc",
"eq_sym"
] | Plus.plus_assoc | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_plus_r_stt | := (fun n m => Nat.le_add_l m n). | Definition | le_plus_r_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_add_l"
] | Plus.plus_le_compat_r | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_plus_trans_stt | := (fun n m p Hle => Nat.le_trans n _ _ Hle (Nat.le_add_r m p)). | Definition | le_plus_trans_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_add_r",
"le_trans"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
lt_plus_trans_stt | := (fun n m p Hlt => Nat.lt_le_trans n _ _ Hlt (Nat.le_add_r m p)). | Definition | lt_plus_trans_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_add_r",
"lt_le_trans"
] | Plus.le_plus_l Plus.le_plus_r_stt Plus.le_plus_trans_stt | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
minus_n_O_stt | := fun n => eq_sym (Nat.sub_0_r n). | Definition | minus_n_O_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"eq_sym",
"sub_0_r"
] | ** [sub] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
minus_Sn_m_stt | := fun n m Hle => eq_sym (Nat.sub_succ_l m n Hle). | Definition | minus_Sn_m_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"eq_sym",
"sub_succ_l"
] | Minus.minus_n_O | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
minus_diag_reverse_stt | := fun n => eq_sym (Nat.sub_diag n). | Definition | minus_diag_reverse_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"eq_sym",
"sub_diag"
] | Minus.minus_Sn_m | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | minus_plus_simpl_l_reverse_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"add_comm",
"add_sub",
"sub_add_distr"
] | Minus.minus_diag_reverse | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
plus_minus_stt | := fun n m p Heq => eq_sym (Nat.add_sub_eq_l n m p (eq_sym Heq)). | Definition | plus_minus_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"add_sub_eq_l",
"eq_sym"
] | Minus.minus_plus_simpl_l_reverse | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
minus_plus_stt | := (fun n m => eq_ind _ (fun x => x - n = m) (Nat.add_sub m n) _ (Nat.add_comm _ _)). | Definition | minus_plus_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"add_comm",
"add_sub"
] | Minus.plus_minus | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_plus_minus_stt | := fun n m Hle => eq_sym (eq_trans (Nat.add_comm _ _) (Nat.sub_add n m Hle)). | Definition | le_plus_minus_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"add_comm",
"eq_sym",
"eq_trans",
"sub_add"
] | Minus.minus_plus | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
le_plus_minus_r_stt | := fun n m Hle => eq_trans (Nat.add_comm _ _) (Nat.sub_add n m Hle). | Definition | le_plus_minus_r_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"add_comm",
"eq_trans",
"sub_add"
] | Minus.le_plus_minus | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | lt_O_minus_lt_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_add_lt_sub_r"
] | Minus.lt_minus | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
mult_assoc_reverse_stt | := fun n m p => eq_sym (Nat.mul_assoc n m p). | Definition | mult_assoc_reverse_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"eq_sym",
"mul_assoc"
] | Mult.mult_minus_distr_l | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | mult_O_le_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"le_add_r",
"left",
"right"
] | Mult.mult_assoc_reverse Mult.mult_assoc | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | mult_S_lt_compat_l_stt | Arith | theories/Arith/Arith_base.v | [
"Stdlib",
"PeanoNat",
"Factorial",
"Between",
"Peano_dec",
"Compare_dec",
"EqNat",
"Wf_nat"
] | [
"lt_0_succ",
"mul_lt_mono_pos_l"
] | Mult.mult_le_compat_l | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
between k : nat -> Prop | :=
| bet_emp : between k k
| bet_S : forall l, between k l -> P l -> between k (S l). | Inductive | between | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [] | The [between] type expresses the concept
[forall i: nat, k <= i < l -> P i.]. | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
bet_eq : forall k l, l = k -> between k l. | Proof.
intros * ->; constructor.
Qed. | Lemma | bet_eq | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
between_le : forall k l, between k l -> k <= l. | Proof.
induction 1; auto.
Qed. | Lemma | between_le | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between",
"induction"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between",
"induction",
"nle_succ_diag_l"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between",
"between_Sk_l",
"induction"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [] | The [exists_between] type expresses the concept
[exists i: nat, k <= i < l /\ Q i]. | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"exists_between",
"induction",
"succ_le_mono"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
exists_lt : forall k l, exists_between k l -> k < l. | Proof.
exact exists_le_S.
Qed. | Lemma | exists_lt | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"exists_between",
"exists_le_S"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"exists_between"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
in_int p q r | := p <= r /\ r < q. | Definition | in_int | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
in_int_intro : forall p q r, p <= r -> r < q -> in_int p q r. | Proof.
split; assumption.
Qed. | Lemma | in_int_intro | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"in_int",
"split"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"in_int",
"le_lt_trans"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"in_int",
"lt_eq_cases",
"lt_succ_r"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"in_int"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"in_int",
"in_int_intro"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between",
"in_int",
"in_int_lt",
"in_int_p_Sq",
"induction",
"lt_irrefl"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between",
"in_int",
"induction"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"exists_between",
"in_int",
"induction"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"exists_between",
"in_int",
"induction"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between",
"exists_between",
"in_int",
"induction"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | between_not_exists | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between",
"exists_between",
"exists_in_int",
"in_int",
"in_int_exists",
"induction",
"lt_eq_cases",
"lt_irrefl",
"lt_succ_r",
"replace"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"between",
"init"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"P_nth",
"between_le",
"induction",
"init",
"le_trans"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
eventually (n:nat) | := exists2 k : nat, k <= n & Q k. | Definition | eventually | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
event_O : eventually 0 -> Q 0. | Proof.
intros (x, ?, ?).
replace 0 with x; auto.
apply Nat.le_0_r; assumption.
Qed. | Lemma | event_O | Arith | theories/Arith/Between.v | [
"Stdlib",
"PeanoNat"
] | [
"eventually",
"le_0_r",
"replace"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
notzerop n | := sumbool_not _ _ (zerop n). | 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. | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
lt_ge_dec : forall x y, {x < y} + {x >= y} | :=
fun n m => sumbool_not _ _ (le_lt_dec m n). | Definition | lt_ge_dec | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"le_lt_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
nat_lt_ge_bool x y | := bool_of_sumbool (lt_ge_dec x y). | Definition | nat_lt_ge_bool | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"lt_ge_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
nat_ge_lt_bool x y | :=
bool_of_sumbool (sumbool_not _ _ (lt_ge_dec x y)). | Definition | nat_ge_lt_bool | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"lt_ge_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
nat_le_gt_bool x y | := bool_of_sumbool (le_gt_dec x y). | Definition | nat_le_gt_bool | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"le_gt_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
nat_gt_le_bool x y | :=
bool_of_sumbool (sumbool_not _ _ (le_gt_dec x y)). | Definition | nat_gt_le_bool | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"le_gt_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
nat_eq_bool x y | := bool_of_sumbool (eq_nat_dec x y). | Definition | nat_eq_bool | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"eq_nat_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
nat_noteq_bool x y | :=
bool_of_sumbool (sumbool_not _ _ (eq_nat_dec x y)). | Definition | nat_noteq_bool | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"eq_nat_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
zerop_bool x | := bool_of_sumbool (zerop x). | Definition | zerop_bool | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"zerop"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
notzerop_bool x | := bool_of_sumbool (notzerop x). | Definition | notzerop_bool | Arith | theories/Arith/Bool_nat.v | [
"Stdlib",
"Compare_dec",
"Peano_dec",
"Sumbool"
] | [
"notzerop"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
to_nat '(x, y) : nat | :=
y + (nat_rec _ 0 (fun i m => (S i) + m) (y + x)). | Definition | to_nat | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [] | Cantor pairing [to_nat] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [] | Cantor pairing inverse [of_nat] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | cancel_of_to | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [
"add_0_r",
"add_succ_r",
"fold",
"induction",
"of_nat",
"to_nat"
] | [of_nat] is the left inverse for [to_nat] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [
"cancel_of_to",
"of_nat",
"to_nat"
] | [to_nat] is injective | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [
"add_0_r",
"add_succ_r",
"fold",
"induction",
"of_nat",
"to_nat"
] | [to_nat] is the left inverse for [of_nat] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [
"cancel_to_of",
"of_nat",
"to_nat"
] | [of_nat] is injective | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | to_nat_spec | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [
"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] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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 | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [
"div_add_l",
"div_mul",
"to_nat",
"to_nat_spec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | to_nat_non_decreasing | Arith | theories/Arith/Cantor.v | [
"Stdlib",
"PeanoNat"
] | [
"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 | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
not_eq_sym | := not_eq_sym (only parsing). | Notation | not_eq_sym | Arith | theories/Arith/Compare.v | [
"Stdlib",
"PeanoNat",
"Compare_dec",
"Wf_nat"
] | [] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
le_or_le_S | := le_le_S_dec. | Definition | le_or_le_S | Arith | theories/Arith/Compare.v | [
"Stdlib",
"PeanoNat",
"Compare_dec",
"Wf_nat"
] | [
"le_le_S_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
Pcompare | := gt_eq_gt_dec. | Definition | Pcompare | Arith | theories/Arith/Compare.v | [
"Stdlib",
"PeanoNat",
"Compare_dec",
"Wf_nat"
] | [
"gt_eq_gt_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
le_dec : forall n m, {n <= m} + {m <= n}. | Proof.
exact le_ge_dec.
Qed. | Lemma | le_dec | Arith | theories/Arith/Compare.v | [
"Stdlib",
"PeanoNat",
"Compare_dec",
"Wf_nat"
] | [
"le_ge_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
lt_or_eq n m | := {m > n} + {n = m}. | Definition | lt_or_eq | Arith | theories/Arith/Compare.v | [
"Stdlib",
"PeanoNat",
"Compare_dec",
"Wf_nat"
] | [] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
le_decide : forall n m, n <= m -> lt_or_eq n m. | Proof.
exact le_lt_eq_dec.
Qed. | Lemma | le_decide | Arith | theories/Arith/Compare.v | [
"Stdlib",
"PeanoNat",
"Compare_dec",
"Wf_nat"
] | [
"le_lt_eq_dec",
"lt_or_eq"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Compare.v | [
"Stdlib",
"PeanoNat",
"Compare_dec",
"Wf_nat"
] | [
"lt_eq_cases"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | discrete_nat | Arith | theories/Arith/Compare.v | [
"Stdlib",
"PeanoNat",
"Compare_dec",
"Wf_nat"
] | [
"add_0_r",
"add_comm",
"add_sub_assoc",
"induction",
"le_succ_l",
"lt_eq_cases",
"right",
"sub_diag"
] | By special request of G. Kahn - Used in Group Theory | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
zerop n : {n = 0} + {0 < n}. | Proof.
destruct n; [left|right]; auto.
apply Nat.lt_0_succ.
Defined. | Definition | zerop | Arith | theories/Arith/Compare_dec.v | [
"Stdlib",
"PeanoNat",
"Decidable"
] | [
"left",
"lt_0_succ",
"right"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | lt_eq_lt_dec | Arith | theories/Arith/Compare_dec.v | [
"Stdlib",
"PeanoNat",
"Decidable"
] | [
"induction",
"left",
"lt_0_succ",
"right",
"succ_lt_mono"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 | |
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 | Arith | theories/Arith/Compare_dec.v | [
"Stdlib",
"PeanoNat",
"Decidable"
] | [
"lt_eq_lt_dec"
] | https://github.com/rocq-prover/stdlib | f76a666b0b2c28c671d4fdf6dd25bcab865b9c36 |
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