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 Description: For sequences that correspond to valid integers, the adder sequence function produces the sequence for the sum. This is effectively a proof of the correctness of the ripple carry adder, implemented with logic gates corresponding to df-had 1530 and df-cad 1543. It is interesting to consider in what sense the sadd function can be said to be "adding" things outside the range of the bits function, that is, when adding sequences that are not eventually constant and so do not denote any integer. The correct interpretation is that the sequences are representations of 2-adic integers, which have a natural ring structure. (Contributed by Mario Carneiro, 9-Sep-2016.)
Assertion
Ref Expression
sadadd ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((bits‘𝐴) sadd (bits‘𝐵)) = (bits‘(𝐴 + 𝐵)))

Dummy variables 𝑘 𝑐 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bitsss 15091 . . . . . 6 (bits‘𝐴) ⊆ ℕ0
2 bitsss 15091 . . . . . 6 (bits‘𝐵) ⊆ ℕ0
3 sadcl 15127 . . . . . 6 (((bits‘𝐴) ⊆ ℕ0 ∧ (bits‘𝐵) ⊆ ℕ0) → ((bits‘𝐴) sadd (bits‘𝐵)) ⊆ ℕ0)
41, 2, 3mp2an 707 . . . . 5 ((bits‘𝐴) sadd (bits‘𝐵)) ⊆ ℕ0
54sseli 3584 . . . 4 (𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) → 𝑘 ∈ ℕ0)
65a1i 11 . . 3 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) → 𝑘 ∈ ℕ0))
7 bitsss 15091 . . . . 5 (bits‘(𝐴 + 𝐵)) ⊆ ℕ0
87sseli 3584 . . . 4 (𝑘 ∈ (bits‘(𝐴 + 𝐵)) → 𝑘 ∈ ℕ0)
98a1i 11 . . 3 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑘 ∈ (bits‘(𝐴 + 𝐵)) → 𝑘 ∈ ℕ0))
10 eqid 2621 . . . . . . . . 9 seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ (bits‘𝐴), 𝑚 ∈ (bits‘𝐵), ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1)))) = seq0((𝑐 ∈ 2𝑜, 𝑚 ∈ ℕ0 ↦ if(cadd(𝑚 ∈ (bits‘𝐴), 𝑚 ∈ (bits‘𝐵), ∅ ∈ 𝑐), 1𝑜, ∅)), (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, ∅, (𝑛 − 1))))
11 eqid 2621 . . . . . . . . 9 (bits ↾ ℕ0) = (bits ↾ ℕ0)
12 simpll 789 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℤ)
13 simplr 791 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℤ)
14 simpr 477 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
15 1nn0 11268 . . . . . . . . . . 11 1 ∈ ℕ0
1615a1i 11 . . . . . . . . . 10 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → 1 ∈ ℕ0)
1714, 16nn0addcld 11315 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝑘 + 1) ∈ ℕ0)
1810, 11, 12, 13, 17sadaddlem 15131 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^(𝑘 + 1))) = (bits‘((𝐴 + 𝐵) mod (2↑(𝑘 + 1)))))
1912, 13zaddcld 11446 . . . . . . . . 9 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℤ)
20 bitsmod 15101 . . . . . . . . 9 (((𝐴 + 𝐵) ∈ ℤ ∧ (𝑘 + 1) ∈ ℕ0) → (bits‘((𝐴 + 𝐵) mod (2↑(𝑘 + 1)))) = ((bits‘(𝐴 + 𝐵)) ∩ (0..^(𝑘 + 1))))
2119, 17, 20syl2anc 692 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (bits‘((𝐴 + 𝐵) mod (2↑(𝑘 + 1)))) = ((bits‘(𝐴 + 𝐵)) ∩ (0..^(𝑘 + 1))))
2218, 21eqtrd 2655 . . . . . . 7 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^(𝑘 + 1))) = ((bits‘(𝐴 + 𝐵)) ∩ (0..^(𝑘 + 1))))
2322eleq2d 2684 . . . . . 6 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^(𝑘 + 1))) ↔ 𝑘 ∈ ((bits‘(𝐴 + 𝐵)) ∩ (0..^(𝑘 + 1)))))
24 elin 3780 . . . . . 6 (𝑘 ∈ (((bits‘𝐴) sadd (bits‘𝐵)) ∩ (0..^(𝑘 + 1))) ↔ (𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) ∧ 𝑘 ∈ (0..^(𝑘 + 1))))
25 elin 3780 . . . . . 6 (𝑘 ∈ ((bits‘(𝐴 + 𝐵)) ∩ (0..^(𝑘 + 1))) ↔ (𝑘 ∈ (bits‘(𝐴 + 𝐵)) ∧ 𝑘 ∈ (0..^(𝑘 + 1))))
2623, 24, 253bitr3g 302 . . . . 5 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → ((𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) ∧ 𝑘 ∈ (0..^(𝑘 + 1))) ↔ (𝑘 ∈ (bits‘(𝐴 + 𝐵)) ∧ 𝑘 ∈ (0..^(𝑘 + 1)))))
27 nn0uz 11682 . . . . . . . . 9 0 = (ℤ‘0)
2814, 27syl6eleq 2708 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (ℤ‘0))
29 eluzfz2 12307 . . . . . . . 8 (𝑘 ∈ (ℤ‘0) → 𝑘 ∈ (0...𝑘))
3028, 29syl 17 . . . . . . 7 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (0...𝑘))
3114nn0zd 11440 . . . . . . . 8 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℤ)
32 fzval3 12493 . . . . . . . 8 (𝑘 ∈ ℤ → (0...𝑘) = (0..^(𝑘 + 1)))
3331, 32syl 17 . . . . . . 7 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (0...𝑘) = (0..^(𝑘 + 1)))
3430, 33eleqtrd 2700 . . . . . 6 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ (0..^(𝑘 + 1)))
3534biantrud 528 . . . . 5 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) ↔ (𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) ∧ 𝑘 ∈ (0..^(𝑘 + 1)))))
3634biantrud 528 . . . . 5 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ (bits‘(𝐴 + 𝐵)) ↔ (𝑘 ∈ (bits‘(𝐴 + 𝐵)) ∧ 𝑘 ∈ (0..^(𝑘 + 1)))))
3726, 35, 363bitr4d 300 . . . 4 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ 𝑘 ∈ ℕ0) → (𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) ↔ 𝑘 ∈ (bits‘(𝐴 + 𝐵))))
3837ex 450 . . 3 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑘 ∈ ℕ0 → (𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) ↔ 𝑘 ∈ (bits‘(𝐴 + 𝐵)))))
396, 9, 38pm5.21ndd 369 . 2 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑘 ∈ ((bits‘𝐴) sadd (bits‘𝐵)) ↔ 𝑘 ∈ (bits‘(𝐴 + 𝐵))))
4039eqrdv 2619 1 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((bits‘𝐴) sadd (bits‘𝐵)) = (bits‘(𝐴 + 𝐵)))