lemma_lt_succ_succ : {a : ℕ} {b : ℕ} ⊦ a < b ⟶ ⊦ `?NaturalNumbers?succ a < `?NaturalNumbers?succ b ❘# lemma_lt_succ_succ 1 2 3 ❙
lemma_le_succ_succ : {a : ℕ} {b : ℕ} ⊦ a ≤ b ⟶ ⊦ `?NaturalNumbers?succ a ≤ `?NaturalNumbers?succ b ❘ # lemma_le_succ_succ 1 2 3 ❙
// added by sw ❙
...
...
@@ -178,9 +180,12 @@ theory CompDec : base:?Logic =
include ?NaturalArithmetics ❙
include lfxcop:?LFCoprod❙
le_dec_add : {a : ℕ} {b : ℕ} ((⊦ a ≤ b) ⨁ (⊦ b < a)) ⟶ ((⊦ `?NaturalNumbers?succ a ≤ `?NaturalNumbers?succ b) ⨁ (⊦ `?NaturalNumbers?succ b < `?NaturalNumbers?succ a)) ❙
le_dec_add_l : {a : ℕ} {b : ℕ } {h : ⊦ a ≤ b} ⊦ le_dec_add a b (h ↪l (⊦ b < a)) ≐ ((lemma_le_succ_succ a b h ) ↪l (⊦ `?NaturalNumbers?succ b < `?NaturalNumbers?succ a)) ❙
le_dec_add_r : {a : ℕ} {b : ℕ } {h : ⊦ b < a} ⊦ le_dec_add a b ((⊦ a ≤ b) r↩ h) ≐ ( (⊦ `?NaturalNumbers?succ a ≤ `?NaturalNumbers?succ b) r↩ (lemma_lt_succ_succ b a h)) ❙
le_dec : {a : ℕ} {b : ℕ} ((⊦ a ≤ b) ⨁ (⊦ b < a)) ❙
le_dec_b : {n : ℕ} ⊦ le_dec 0 n ≐ ((lemma_zero_le n) ↪l (⊦ n < 0)) ❙