Expression of type Implies

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 = Implies(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)), FALSE)

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(\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]\right) \Rightarrow \bot

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',)
direction	Direction of the relation (normal or reversed)	normal	normal	('with_direction_reversed', 'is_reversed')

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 20 operands: 1
1	ExprTuple	2, 28
2	Operation	operator: 35 operands: 3
3	ExprTuple	4, 5
4	Operation	operator: 6 operand: 10
5	Operation	operator: 8 operand: 11
6	Literal
7	ExprTuple	10
8	Literal
9	ExprTuple	11
10	Lambda	parameters: 45 body: 12
11	Lambda	parameters: 57 body: 13
12	Conditional	value: 14 condition: 15
13	Conditional	value: 16 condition: 17
14	Operation	operator: 35 operands: 18
15	Operation	operator: 35 operands: 19
16	Operation	operator: 20 operands: 21
17	Operation	operator: 35 operands: 22
18	ExprTuple	23, 24
19	ExprTuple	25, 26
20	Literal
21	ExprTuple	27, 28
22	ExprTuple	29, 30
23	Operation	operator: 47 operands: 31
24	Operation	operator: 47 operands: 32
25	Operation	operator: 38 operands: 33
26	Operation	operator: 38 operands: 34
27	Operation	operator: 35 operands: 36
28	Literal
29	Operation	operator: 38 operands: 37
30	Operation	operator: 38 operands: 39
31	ExprTuple	51, 40
32	ExprTuple	41, 66
33	ExprTuple	49, 44
34	ExprTuple	50, 44
35	Literal
36	ExprTuple	42, 43
37	ExprTuple	59, 44
38	Literal
39	ExprTuple	60, 44
40	Operation	operator: 64 operands: 45
41	Operation	operator: 56 operands: 45
42	Operation	operator: 47 operands: 46
43	Operation	operator: 47 operands: 48
44	Literal
45	ExprTuple	49, 50
46	ExprTuple	51, 52
47	Literal
48	ExprTuple	53, 66
49	Variable
50	Variable
51	Operation	operator: 54 operand: 58
52	Operation	operator: 64 operands: 57
53	Operation	operator: 56 operands: 57
54	Literal
55	ExprTuple	58
56	Literal
57	ExprTuple	59, 60
58	Operation	operator: 61 operands: 62
59	Variable
60	Variable
61	Literal
62	ExprTuple	67, 63
63	Operation	operator: 64 operands: 65
64	Literal
65	ExprTuple	66, 67
66	Literal
67	Literal