Quick cryptid generator

Quick cryptid generator or QCG for short is an esolang that defines a recursive function

Definition

start with a number at the top of your code this is the number of inputs $ will have (#i is a free variable) you use recursion like this:

$(expr_1(#1),expr_2(#2),...,expr_n(#n)) = $(f_1(#1,#2,...,#n),f_2(#1,#2,...,#n),f_n(#1,#2,...,#n))

(no expr_i(x) or f_i(...) may use $) and base cases:

$(expr_1(#1),expr_2(#2),...,expr(#n)) = f(#1,#2,...,#n)

(no expr_i(x) or f(...) can use $) you can use +, -, *, () and any natural number (@ means user input)

$(x_1,x_2,...,x_n)

means evaluate $(x_1,x_2,...,x_n) and print the result

example:

2
$(2*#1,#2) = $(#1,#2+1)
$(2*#1+1,#2) = $(3*#1+2,#2+1)
$(0,#2) = #2
$(@,0)

returns https://oeis.org/search?q=0%2C1%2C7%2C2%2C5%2C8&language=english&go=Search (input)

Programs

not a

1
$(0) = 1
$(1) = 0
$(@)

a or b

2
$(0,0) = 0
$(1,0) = 1
$(0,1) = 1
$(1,1) = 1
$(@,@)

a and b

2
$(0,0) = 0
$(1,0) = 0
$(0,1) = 0
$(1,1) = 1
$(@,@)

a+b

2
$(#1,#2) = #1+#2
$(@,@)

a*b

2
$(#1,#2) = #1*#2
$(@,@)

a^b

3
$(#1,#2,#3) = $(#1,#2-1,#3*#1)
$(#1,0,#3) = #3
$(@,@,1)

a!

2
$(#1,#2) = $(#1-1,#2*#1)
$(0,#2) = #2
$(@,1)