Expression of type Forall

from the theory of proveit.numbers.divisibility

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, b, p
from proveit.logic import And, Equals, Forall, Not
from proveit.numbers import Divides, GCD, NaturalPos, greater, one

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a, b], instance_expr = Forall(instance_param_or_params = [p], instance_expr = Not(And(Divides(p, a), Divides(p, b))), domain = NaturalPos, condition = greater(p, one)), domain = NaturalPos, condition = Equals(GCD(a, b), one))

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())

\forall_{a, b \in \mathbb{N}^+~|~gcd\left(a, b\right) = 1}~\left[\forall_{p \in \mathbb{N}^+~|~p > 1}~(\lnot \left(\left(p \rvert a\right) \land \left(p \rvert b\right)\right))\right]

stored_expr.style_options()

name	description	default	current value	related methods
with_wrapping	If 'True', wrap the Expression after the parameters	None	None/False	('with_wrapping',)
condition_wrapping	Wrap 'before' or 'after' the condition (or None).	None	None/False	('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_params	If 'True', wraps every two parameters AND wraps the Expression after the parameters	None	None/False	('with_params',)
justification	justify to the 'left', 'center', or 'right' in the array cells	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 6 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 26 body: 3
3	Conditional	value: 4 condition: 5
4	Operation	operator: 6 operand: 9
5	Operation	operator: 30 operands: 8
6	Literal
7	ExprTuple	9
8	ExprTuple	10, 11, 12
9	Lambda	parameter: 44 body: 14
10	Operation	operator: 32 operands: 15
11	Operation	operator: 32 operands: 16
12	Operation	operator: 17 operands: 18
13	ExprTuple	44
14	Conditional	value: 19 condition: 20
15	ExprTuple	43, 38
16	ExprTuple	45, 38
17	Literal
18	ExprTuple	21, 39
19	Operation	operator: 22 operand: 27
20	Operation	operator: 30 operands: 24
21	Operation	operator: 25 operands: 26
22	Literal
23	ExprTuple	27
24	ExprTuple	28, 29
25	Literal
26	ExprTuple	43, 45
27	Operation	operator: 30 operands: 31
28	Operation	operator: 32 operands: 33
29	Operation	operator: 34 operands: 35
30	Literal
31	ExprTuple	36, 37
32	Literal
33	ExprTuple	44, 38
34	Literal
35	ExprTuple	39, 44
36	Operation	operator: 41 operands: 40
37	Operation	operator: 41 operands: 42
38	Literal
39	Literal
40	ExprTuple	44, 43
41	Literal
42	ExprTuple	44, 45
43	Variable
44	Variable
45	Variable