Use z3 at-most and at-least when comparing a cardinalty constraint with a fixed number
For the following specification:
```idp
vocabulary {
type Node := {0..61}
edge: Node * Node -> Bool
dom: Node -> Bool
lt: -> Int
}
theory {
!x in Node: dom(x) | ?y in Node: dom(y) & (edge(x, y) | edge(y, x)).
#{x in Node: dom(x)} < lt().
}
```
with the following structure:
<details>
<summary>b2sat lt13</summary>
```idp
structure {
lt := 13.
edge := {
(0, 1), (1, 0), (1, 2), (1, 3), (1, 4),
(2, 1), (2, 3), (2, 4), (2, 5), (2, 6),
(2, 8), (3, 1), (3, 2), (3, 6), (3, 7),
(4, 1), (4, 2), (4, 8), (4, 9), (5, 2),
(5, 6), (5, 8), (5, 10), (5, 11), (5, 14),
(6, 2), (6, 3), (6, 5), (6, 7), (6, 11),
(6, 12), (7, 3), (7, 6), (7, 12), (7, 13),
(8, 2), (8, 4), (8, 5), (8, 9), (8, 14),
(8, 15), (9, 4), (9, 8), (9, 15), (9, 16),
(10, 5), (10, 11), (10, 14), (10, 17), (10, 18),
(10, 22), (11, 5), (11, 6), (11, 10), (11, 12),
(11, 18), (11, 19), (12, 6), (12, 7), (12, 11),
(12, 13), (12, 19), (12, 20), (13, 7), (13, 12),
(13, 20), (13, 21), (14, 5), (14, 8), (14, 10),
(14, 15), (14, 22), (14, 23), (15, 8), (15, 9),
(15, 14), (15, 16), (15, 23), (15, 24), (16, 9),
(16, 15), (16, 24), (16, 25), (17, 10), (17, 18),
(17, 22), (17, 26), (17, 27), (17, 31), (18, 10),
(18, 11), (18, 17), (18, 19), (18, 27), (18, 28),
(19, 11), (19, 12), (19, 18), (19, 20), (19, 28),
(19, 29), (20, 12), (20, 13), (20, 19), (20, 21),
(20, 29), (20, 30), (21, 13), (21, 20), (21, 30),
(22, 10), (22, 14), (22, 17), (22, 23), (22, 31),
(22, 32), (23, 14), (23, 15), (23, 22), (23, 24),
(23, 32), (23, 33), (24, 15), (24, 16), (24, 23),
(24, 25), (24, 33), (24, 34), (25, 16), (25, 24),
(25, 34), (26, 17), (26, 27), (26, 31), (26, 35),
(26, 36), (26, 40), (27, 17), (27, 18), (27, 26),
(27, 28), (27, 36), (27, 37), (28, 18), (28, 19),
(28, 27), (28, 29), (28, 37), (28, 38), (29, 19),
(29, 20), (29, 28), (29, 30), (29, 38), (29, 39),
(30, 20), (30, 21), (30, 29), (30, 39), (31, 17),
(31, 22), (31, 26), (31, 32), (31, 40), (31, 41),
(32, 22), (32, 23), (32, 31), (32, 33), (32, 41),
(32, 42), (33, 23), (33, 24), (33, 32), (33, 34),
(33, 42), (33, 43), (34, 24), (34, 25), (34, 33),
(34, 43), (35, 26), (35, 36), (35, 40), (35, 44),
(35, 45), (35, 49), (36, 26), (36, 27), (36, 35),
(36, 37), (36, 45), (36, 46), (37, 27), (37, 28),
(37, 36), (37, 38), (37, 46), (37, 47), (38, 28),
(38, 29), (38, 37), (38, 39), (38, 47), (38, 48),
(39, 29), (39, 30), (39, 38), (39, 48), (40, 26),
(40, 31), (40, 35), (40, 41), (40, 49), (40, 50),
(41, 31), (41, 32), (41, 40), (41, 42), (41, 50),
(41, 51), (42, 32), (42, 33), (42, 41), (42, 43),
(42, 51), (42, 52), (43, 33), (43, 34), (43, 42),
(43, 52), (44, 35), (44, 45), (44, 49), (44, 53),
(44, 54), (44, 58), (45, 35), (45, 36), (45, 44),
(45, 46), (45, 54), (45, 55), (46, 36), (46, 37),
(46, 45), (46, 47), (46, 55), (46, 56), (47, 37),
(47, 38), (47, 46), (47, 48), (47, 56), (47, 57),
(48, 38), (48, 39), (48, 47), (48, 57), (49, 35),
(49, 40), (49, 44), (49, 50), (49, 58), (49, 59),
(50, 40), (50, 41), (50, 49), (50, 51), (50, 59),
(50, 60), (51, 41), (51, 42), (51, 50), (51, 52),
(51, 60), (51, 61), (52, 42), (52, 43), (52, 51),
(52, 61), (53, 44), (53, 54), (53, 58), (54, 44),
(54, 45), (54, 53), (54, 55), (55, 45), (55, 46),
(55, 54), (55, 56), (56, 46), (56, 47), (56, 55),
(56, 57), (57, 47), (57, 48), (57, 56), (58, 44),
(58, 49), (58, 53), (58, 59), (59, 49), (59, 50),
(59, 58), (59, 60), (60, 50), (60, 51), (60, 59),
(60, 61), (61, 51), (61, 52), (61, 60)
}.
}
```
</details>
Solving this instance takes ~15s.
The following is the generated smt:
<details>
<summary> Generated SMTLIB2 </summary>
```smt
(declare-fun dom (Int) Bool)
(assert (let ((a!1 (> 13
(+ (ite (dom 0) 1 0)
(ite (dom 1) 1 0)
(ite (dom 2) 1 0)
(ite (dom 3) 1 0)
(ite (dom 4) 1 0)
(ite (dom 5) 1 0)
(ite (dom 6) 1 0)
(ite (dom 7) 1 0)
(ite (dom 8) 1 0)
(ite (dom 9) 1 0)
(ite (dom 10) 1 0)
(ite (dom 11) 1 0)
(ite (dom 12) 1 0)
(ite (dom 13) 1 0)
(ite (dom 14) 1 0)
(ite (dom 15) 1 0)
(ite (dom 16) 1 0)
(ite (dom 17) 1 0)
(ite (dom 18) 1 0)
(ite (dom 19) 1 0)
(ite (dom 20) 1 0)
(ite (dom 21) 1 0)
(ite (dom 22) 1 0)
(ite (dom 23) 1 0)
(ite (dom 24) 1 0)
(ite (dom 25) 1 0)
(ite (dom 26) 1 0)
(ite (dom 27) 1 0)
(ite (dom 28) 1 0)
(ite (dom 29) 1 0)
(ite (dom 30) 1 0)
(ite (dom 31) 1 0)
(ite (dom 32) 1 0)
(ite (dom 33) 1 0)
(ite (dom 34) 1 0)
(ite (dom 35) 1 0)
(ite (dom 36) 1 0)
(ite (dom 37) 1 0)
(ite (dom 38) 1 0)
(ite (dom 39) 1 0)
(ite (dom 40) 1 0)
(ite (dom 41) 1 0)
(ite (dom 42) 1 0)
(ite (dom 43) 1 0)
(ite (dom 44) 1 0)
(ite (dom 45) 1 0)
(ite (dom 46) 1 0)
(ite (dom 47) 1 0)
(ite (dom 48) 1 0)
(ite (dom 49) 1 0)
(ite (dom 50) 1 0)
(ite (dom 51) 1 0)
(ite (dom 52) 1 0)
(ite (dom 53) 1 0)
(ite (dom 54) 1 0)
(ite (dom 55) 1 0)
(ite (dom 56) 1 0)
(ite (dom 57) 1 0)
(ite (dom 58) 1 0)
(ite (dom 59) 1 0)
(ite (dom 60) 1 0)
(ite (dom 61) 1 0)))))
(and (or (dom 0) (dom 1))
(or (dom 1) (dom 0) (dom 2) (dom 3) (dom 4))
(or (dom 2) (dom 1) (dom 3) (dom 4) (dom 5) (dom 6) (dom 8))
(or (dom 3) (dom 1) (dom 2) (dom 6) (dom 7))
(or (dom 4) (dom 1) (dom 2) (dom 8) (dom 9))
(or (dom 5) (dom 2) (dom 6) (dom 8) (dom 10) (dom 11) (dom 14))
(or (dom 6) (dom 2) (dom 3) (dom 5) (dom 7) (dom 11) (dom 12))
(or (dom 7) (dom 3) (dom 6) (dom 12) (dom 13))
(or (dom 8) (dom 2) (dom 4) (dom 5) (dom 9) (dom 14) (dom 15))
(or (dom 9) (dom 4) (dom 8) (dom 15) (dom 16))
(or (dom 10) (dom 5) (dom 11) (dom 14) (dom 17) (dom 18) (dom 22))
(or (dom 11) (dom 5) (dom 6) (dom 10) (dom 12) (dom 18) (dom 19))
(or (dom 12) (dom 6) (dom 7) (dom 11) (dom 13) (dom 19) (dom 20))
(or (dom 13) (dom 7) (dom 12) (dom 20) (dom 21))
(or (dom 14) (dom 5) (dom 8) (dom 10) (dom 15) (dom 22) (dom 23))
(or (dom 15) (dom 8) (dom 9) (dom 14) (dom 16) (dom 23) (dom 24))
(or (dom 16) (dom 9) (dom 15) (dom 24) (dom 25))
(or (dom 17) (dom 10) (dom 18) (dom 22) (dom 26) (dom 27) (dom 31))
(or (dom 18) (dom 10) (dom 11) (dom 17) (dom 19) (dom 27) (dom 28))
(or (dom 19) (dom 11) (dom 12) (dom 18) (dom 20) (dom 28) (dom 29))
(or (dom 20) (dom 12) (dom 13) (dom 19) (dom 21) (dom 29) (dom 30))
(or (dom 21) (dom 13) (dom 20) (dom 30))
(or (dom 22) (dom 10) (dom 14) (dom 17) (dom 23) (dom 31) (dom 32))
(or (dom 23) (dom 14) (dom 15) (dom 22) (dom 24) (dom 32) (dom 33))
(or (dom 24) (dom 15) (dom 16) (dom 23) (dom 25) (dom 33) (dom 34))
(or (dom 25) (dom 16) (dom 24) (dom 34))
(or (dom 26) (dom 17) (dom 27) (dom 31) (dom 35) (dom 36) (dom 40))
(or (dom 27) (dom 17) (dom 18) (dom 26) (dom 28) (dom 36) (dom 37))
(or (dom 28) (dom 18) (dom 19) (dom 27) (dom 29) (dom 37) (dom 38))
(or (dom 29) (dom 19) (dom 20) (dom 28) (dom 30) (dom 38) (dom 39))
(or (dom 30) (dom 20) (dom 21) (dom 29) (dom 39))
(or (dom 31) (dom 17) (dom 22) (dom 26) (dom 32) (dom 40) (dom 41))
(or (dom 32) (dom 22) (dom 23) (dom 31) (dom 33) (dom 41) (dom 42))
(or (dom 33) (dom 23) (dom 24) (dom 32) (dom 34) (dom 42) (dom 43))
(or (dom 34) (dom 24) (dom 25) (dom 33) (dom 43))
(or (dom 35) (dom 26) (dom 36) (dom 40) (dom 44) (dom 45) (dom 49))
(or (dom 36) (dom 26) (dom 27) (dom 35) (dom 37) (dom 45) (dom 46))
(or (dom 37) (dom 27) (dom 28) (dom 36) (dom 38) (dom 46) (dom 47))
(or (dom 38) (dom 28) (dom 29) (dom 37) (dom 39) (dom 47) (dom 48))
(or (dom 39) (dom 29) (dom 30) (dom 38) (dom 48))
(or (dom 40) (dom 26) (dom 31) (dom 35) (dom 41) (dom 49) (dom 50))
(or (dom 41) (dom 31) (dom 32) (dom 40) (dom 42) (dom 50) (dom 51))
(or (dom 42) (dom 32) (dom 33) (dom 41) (dom 43) (dom 51) (dom 52))
(or (dom 43) (dom 33) (dom 34) (dom 42) (dom 52))
(or (dom 44) (dom 35) (dom 45) (dom 49) (dom 53) (dom 54) (dom 58))
(or (dom 45) (dom 35) (dom 36) (dom 44) (dom 46) (dom 54) (dom 55))
(or (dom 46) (dom 36) (dom 37) (dom 45) (dom 47) (dom 55) (dom 56))
(or (dom 47) (dom 37) (dom 38) (dom 46) (dom 48) (dom 56) (dom 57))
(or (dom 48) (dom 38) (dom 39) (dom 47) (dom 57))
(or (dom 49) (dom 35) (dom 40) (dom 44) (dom 50) (dom 58) (dom 59))
(or (dom 50) (dom 40) (dom 41) (dom 49) (dom 51) (dom 59) (dom 60))
(or (dom 51) (dom 41) (dom 42) (dom 50) (dom 52) (dom 60) (dom 61))
(or (dom 52) (dom 42) (dom 43) (dom 51) (dom 61))
(or (dom 53) (dom 44) (dom 54) (dom 58))
(or (dom 54) (dom 44) (dom 45) (dom 53) (dom 55))
(or (dom 55) (dom 45) (dom 46) (dom 54) (dom 56))
(or (dom 56) (dom 46) (dom 47) (dom 55) (dom 57))
(or (dom 57) (dom 47) (dom 48) (dom 56))
(or (dom 58) (dom 44) (dom 49) (dom 53) (dom 59))
(or (dom 59) (dom 49) (dom 50) (dom 58) (dom 60))
(or (dom 60) (dom 50) (dom 51) (dom 59) (dom 61))
(or (dom 61) (dom 51) (dom 52) (dom 60))
a!1)))
```
</details>
This SMTLIB2 program contains many if then else statements that are summed up and then compared with 13.
Z3 has an [extension for cardinality constraints](https://microsoft.github.io/z3guide/docs/logic/propositional-logic#cardinality-constraints) with `at-least` and `at-most`.
If we replace the if then else chain with an `at-most` where the at most number is 12 (1 less than 13 since comparison is strictly less than 13).
We get the following smt program:
<details>
<summary>
`at-most` smt program
</summary>
```smt
(declare-fun dom (Int) Bool)
(assert (let ((a!1
((_ at-most 12)
(dom 0)
(dom 1)
(dom 2)
(dom 3)
(dom 4)
(dom 5)
(dom 6)
(dom 7)
(dom 8)
(dom 9)
(dom 10)
(dom 11)
(dom 12)
(dom 13)
(dom 14)
(dom 15)
(dom 16)
(dom 17)
(dom 18)
(dom 19)
(dom 20)
(dom 21)
(dom 22)
(dom 23)
(dom 24)
(dom 25)
(dom 26)
(dom 27)
(dom 28)
(dom 29)
(dom 30)
(dom 31)
(dom 32)
(dom 33)
(dom 34)
(dom 35)
(dom 36)
(dom 37)
(dom 38)
(dom 39)
(dom 40)
(dom 41)
(dom 42)
(dom 43)
(dom 44)
(dom 45)
(dom 46)
(dom 47)
(dom 48)
(dom 49)
(dom 50)
(dom 51)
(dom 52)
(dom 53)
(dom 54)
(dom 55)
(dom 56)
(dom 57)
(dom 58)
(dom 59)
(dom 60)
(dom 61))))
(and (or (dom 0) (dom 1))
(or (dom 1) (dom 0) (dom 2) (dom 3) (dom 4))
(or (dom 2) (dom 1) (dom 3) (dom 4) (dom 5) (dom 6) (dom 8))
(or (dom 3) (dom 1) (dom 2) (dom 6) (dom 7))
(or (dom 4) (dom 1) (dom 2) (dom 8) (dom 9))
(or (dom 5) (dom 2) (dom 6) (dom 8) (dom 10) (dom 11) (dom 14))
(or (dom 6) (dom 2) (dom 3) (dom 5) (dom 7) (dom 11) (dom 12))
(or (dom 7) (dom 3) (dom 6) (dom 12) (dom 13))
(or (dom 8) (dom 2) (dom 4) (dom 5) (dom 9) (dom 14) (dom 15))
(or (dom 9) (dom 4) (dom 8) (dom 15) (dom 16))
(or (dom 10) (dom 5) (dom 11) (dom 14) (dom 17) (dom 18) (dom 22))
(or (dom 11) (dom 5) (dom 6) (dom 10) (dom 12) (dom 18) (dom 19))
(or (dom 12) (dom 6) (dom 7) (dom 11) (dom 13) (dom 19) (dom 20))
(or (dom 13) (dom 7) (dom 12) (dom 20) (dom 21))
(or (dom 14) (dom 5) (dom 8) (dom 10) (dom 15) (dom 22) (dom 23))
(or (dom 15) (dom 8) (dom 9) (dom 14) (dom 16) (dom 23) (dom 24))
(or (dom 16) (dom 9) (dom 15) (dom 24) (dom 25))
(or (dom 17) (dom 10) (dom 18) (dom 22) (dom 26) (dom 27) (dom 31))
(or (dom 18) (dom 10) (dom 11) (dom 17) (dom 19) (dom 27) (dom 28))
(or (dom 19) (dom 11) (dom 12) (dom 18) (dom 20) (dom 28) (dom 29))
(or (dom 20) (dom 12) (dom 13) (dom 19) (dom 21) (dom 29) (dom 30))
(or (dom 21) (dom 13) (dom 20) (dom 30))
(or (dom 22) (dom 10) (dom 14) (dom 17) (dom 23) (dom 31) (dom 32))
(or (dom 23) (dom 14) (dom 15) (dom 22) (dom 24) (dom 32) (dom 33))
(or (dom 24) (dom 15) (dom 16) (dom 23) (dom 25) (dom 33) (dom 34))
(or (dom 25) (dom 16) (dom 24) (dom 34))
(or (dom 26) (dom 17) (dom 27) (dom 31) (dom 35) (dom 36) (dom 40))
(or (dom 27) (dom 17) (dom 18) (dom 26) (dom 28) (dom 36) (dom 37))
(or (dom 28) (dom 18) (dom 19) (dom 27) (dom 29) (dom 37) (dom 38))
(or (dom 29) (dom 19) (dom 20) (dom 28) (dom 30) (dom 38) (dom 39))
(or (dom 30) (dom 20) (dom 21) (dom 29) (dom 39))
(or (dom 31) (dom 17) (dom 22) (dom 26) (dom 32) (dom 40) (dom 41))
(or (dom 32) (dom 22) (dom 23) (dom 31) (dom 33) (dom 41) (dom 42))
(or (dom 33) (dom 23) (dom 24) (dom 32) (dom 34) (dom 42) (dom 43))
(or (dom 34) (dom 24) (dom 25) (dom 33) (dom 43))
(or (dom 35) (dom 26) (dom 36) (dom 40) (dom 44) (dom 45) (dom 49))
(or (dom 36) (dom 26) (dom 27) (dom 35) (dom 37) (dom 45) (dom 46))
(or (dom 37) (dom 27) (dom 28) (dom 36) (dom 38) (dom 46) (dom 47))
(or (dom 38) (dom 28) (dom 29) (dom 37) (dom 39) (dom 47) (dom 48))
(or (dom 39) (dom 29) (dom 30) (dom 38) (dom 48))
(or (dom 40) (dom 26) (dom 31) (dom 35) (dom 41) (dom 49) (dom 50))
(or (dom 41) (dom 31) (dom 32) (dom 40) (dom 42) (dom 50) (dom 51))
(or (dom 42) (dom 32) (dom 33) (dom 41) (dom 43) (dom 51) (dom 52))
(or (dom 43) (dom 33) (dom 34) (dom 42) (dom 52))
(or (dom 44) (dom 35) (dom 45) (dom 49) (dom 53) (dom 54) (dom 58))
(or (dom 45) (dom 35) (dom 36) (dom 44) (dom 46) (dom 54) (dom 55))
(or (dom 46) (dom 36) (dom 37) (dom 45) (dom 47) (dom 55) (dom 56))
(or (dom 47) (dom 37) (dom 38) (dom 46) (dom 48) (dom 56) (dom 57))
(or (dom 48) (dom 38) (dom 39) (dom 47) (dom 57))
(or (dom 49) (dom 35) (dom 40) (dom 44) (dom 50) (dom 58) (dom 59))
(or (dom 50) (dom 40) (dom 41) (dom 49) (dom 51) (dom 59) (dom 60))
(or (dom 51) (dom 41) (dom 42) (dom 50) (dom 52) (dom 60) (dom 61))
(or (dom 52) (dom 42) (dom 43) (dom 51) (dom 61))
(or (dom 53) (dom 44) (dom 54) (dom 58))
(or (dom 54) (dom 44) (dom 45) (dom 53) (dom 55))
(or (dom 55) (dom 45) (dom 46) (dom 54) (dom 56))
(or (dom 56) (dom 46) (dom 47) (dom 55) (dom 57))
(or (dom 57) (dom 47) (dom 48) (dom 56))
(or (dom 58) (dom 44) (dom 49) (dom 53) (dom 59))
(or (dom 59) (dom 49) (dom 50) (dom 58) (dom 60))
(or (dom 60) (dom 50) (dom 51) (dom 59) (dom 61))
(or (dom 61) (dom 51) (dom 52) (dom 60))
a!1)))
(check-sat)
```
</details>
Running this with `z3 at-most.smt` produces sat in 20ms. This is 75000% faster.
Whilst grounding when we encounter a comparison operator we can check if one side contains a cardinality expression and the other a number, if this is the case we can use `at-least` and `at-most` when grounding.
issue
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