Expression of type Forall

from the theory of proveit.core_expr_types.operations

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 f, n
from proveit.core_expr_types import f__x_1_to_n, f__y_1_to_n, x_1_to_n, x_eq_y__1_to_n, y_1_to_n
from proveit.logic import Equals, Forall
from proveit.numbers import Natural

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [n], instance_expr = Forall(instance_param_or_params = [f, x_1_to_n, y_1_to_n], instance_expr = Equals(f__x_1_to_n, f__y_1_to_n).with_wrapping_at(2), condition = x_eq_y__1_to_n), domain = Natural)

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_{n \in \mathbb{N}}~\left[\forall_{f, x_{1}, x_{2}, \ldots, x_{n}, y_{1}, y_{2}, \ldots, y_{n}~|~\left(x_{1} = y_{1}\right), \left(x_{2} = y_{2}\right), \ldots, \left(x_{n} = y_{n}\right)}~\left(\begin{array}{c} \begin{array}{l} f\left(x_{1}, x_{2}, \ldots, x_{n}\right) =  \\ f\left(y_{1}, y_{2}, \ldots, y_{n}\right) \end{array} \end{array}\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: 7 operand: 2
1	ExprTuple	2
2	Lambda	parameter: 33 body: 4
3	ExprTuple	33
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 11
6	Operation	operator: 9 operands: 10
7	Literal
8	ExprTuple	11
9	Literal
10	ExprTuple	33, 12
11	Lambda	parameters: 13 body: 14
12	Literal
13	ExprTuple	24, 27, 28
14	Conditional	value: 15 condition: 16
15	Operation	operator: 34 operands: 17
16	Operation	operator: 18 operands: 19
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	22
20	Operation	operator: 24 operands: 23
21	Operation	operator: 24 operands: 25
22	ExprRange	lambda_map: 26 start_index: 32 end_index: 33
23	ExprTuple	27
24	Variable
25	ExprTuple	28
26	Lambda	parameter: 41 body: 29
27	ExprRange	lambda_map: 30 start_index: 32 end_index: 33
28	ExprRange	lambda_map: 31 start_index: 32 end_index: 33
29	Operation	operator: 34 operands: 35
30	Lambda	parameter: 41 body: 36
31	Lambda	parameter: 41 body: 37
32	Literal
33	Variable
34	Literal
35	ExprTuple	36, 37
36	IndexedVar	variable: 38 index: 41
37	IndexedVar	variable: 39 index: 41
38	Variable
39	Variable
40	ExprTuple	41
41	Variable