Expression of type And

from the theory of proveit.numbers.exponentiation

import proveit
# Automation is not needed when building an expression:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%load_expr # Load the stored expression as 'stored_expr'
# import Expression classes needed to build the expression
from proveit import a, a_star, b, b_star
from proveit.logic import And, Equals, Exists, FALSE, Forall, Implies
from proveit.numbers import Abs, GCD, NaturalPos, frac, one, sqrt, two

# build up the expression from sub-expressions
sub_expr1 = Abs(sqrt(two))
expr = And(Exists(instance_param_or_params = [a, b], instance_expr = And(Equals(sub_expr1, frac(a, b)), Equals(GCD(a, b), one)), domain = NaturalPos), Forall(instance_param_or_params = [a_star, b_star], instance_expr = Implies(And(Equals(sub_expr1, frac(a_star, b_star)), Equals(GCD(a_star, b_star), one)), FALSE), domain = NaturalPos))

expr:

# check that the built expression is the same as the stored expression
assert expr == stored_expr
assert expr._style_id == stored_expr._style_id
print("Passed sanity check: expr matches stored_expr")

Passed sanity check: expr matches stored_expr

# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\left[\exists_{a, b \in \mathbb{N}^+}~\left(\left(\left|\sqrt{2}\right| = \frac{a}{b}\right) \land \left(gcd\left(a, b\right) = 1\right)\right)\right] \land \left[\forall_{a^*, b^* \in \mathbb{N}^+}~\left(\left(\left(\left|\sqrt{2}\right| = \frac{a^*}{b^*}\right) \land \left(gcd\left(a^*, b^*\right) = 1\right)\right) \Rightarrow \bot\right)\right]

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	()	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 33 operands: 1
1	ExprTuple	2, 3
2	Operation	operator: 4 operand: 8
3	Operation	operator: 6 operand: 9
4	Literal
5	ExprTuple	8
6	Literal
7	ExprTuple	9
8	Lambda	parameters: 43 body: 10
9	Lambda	parameters: 55 body: 11
10	Conditional	value: 12 condition: 13
11	Conditional	value: 14 condition: 15
12	Operation	operator: 33 operands: 16
13	Operation	operator: 33 operands: 17
14	Operation	operator: 18 operands: 19
15	Operation	operator: 33 operands: 20
16	ExprTuple	21, 22
17	ExprTuple	23, 24
18	Literal
19	ExprTuple	25, 26
20	ExprTuple	27, 28
21	Operation	operator: 45 operands: 29
22	Operation	operator: 45 operands: 30
23	Operation	operator: 36 operands: 31
24	Operation	operator: 36 operands: 32
25	Operation	operator: 33 operands: 34
26	Literal
27	Operation	operator: 36 operands: 35
28	Operation	operator: 36 operands: 37
29	ExprTuple	49, 38
30	ExprTuple	39, 64
31	ExprTuple	47, 42
32	ExprTuple	48, 42
33	Literal
34	ExprTuple	40, 41
35	ExprTuple	57, 42
36	Literal
37	ExprTuple	58, 42
38	Operation	operator: 62 operands: 43
39	Operation	operator: 54 operands: 43
40	Operation	operator: 45 operands: 44
41	Operation	operator: 45 operands: 46
42	Literal
43	ExprTuple	47, 48
44	ExprTuple	49, 50
45	Literal
46	ExprTuple	51, 64
47	Variable
48	Variable
49	Operation	operator: 52 operand: 56
50	Operation	operator: 62 operands: 55
51	Operation	operator: 54 operands: 55
52	Literal
53	ExprTuple	56
54	Literal
55	ExprTuple	57, 58
56	Operation	operator: 59 operands: 60
57	Variable
58	Variable
59	Literal
60	ExprTuple	65, 61
61	Operation	operator: 62 operands: 63
62	Literal
63	ExprTuple	64, 65
64	Literal
65	Literal