Expression of type Forall

from the theory of proveit.core_expr_types.expr_arrays

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 VertExprArray, b, d, i
from proveit.core_expr_types import a_1_to_i, c_1_to_i
from proveit.logic import And, Equals, Forall
from proveit.numbers import Natural

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b, c_1_to_i, d], instance_expr = Equals(Equals(VertExprArray(a_1_to_i, b), VertExprArray(c_1_to_i, d)), And(Equals([a_1_to_i], [c_1_to_i]), Equals(b, d))).with_wrapping_at(2)), 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_{i \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b, c_{1}, c_{2}, \ldots, c_{i}, d}~\left(\begin{array}{c} \begin{array}{l} \left(\left(\begin{array}{ccccc} 
 \multirow{1}{*}{$\begin{array}{c} \uparrow \\ a_{1} \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ a_{2} \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ \cdots \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ a_{i} \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ b \\ \downarrow \end{array}$} \\
\end{array}
\right) = \left(\begin{array}{ccccc} 
 \multirow{1}{*}{$\begin{array}{c} \uparrow \\ c_{1} \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ c_{2} \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ \cdots \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ c_{i} \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ d \\ \downarrow \end{array}$} \\
\end{array}
\right)\right) =  \\ \left(\left(\left(a_{1}, a_{2}, \ldots, a_{i}\right) = \left(c_{1}, c_{2}, \ldots, c_{i}\right)\right) \land \left(b = d\right)\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: 37 body: 4
3	ExprTuple	37
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	37, 12
11	Lambda	parameters: 13 body: 14
12	Literal
13	ExprTuple	32, 30, 33, 31
14	Operation	operator: 26 operands: 15
15	ExprTuple	16, 17
16	Operation	operator: 26 operands: 18
17	Operation	operator: 19 operands: 20
18	ExprTuple	21, 22
19	Literal
20	ExprTuple	23, 24
21	ExprTuple	32, 30
22	ExprTuple	33, 31
23	Operation	operator: 26 operands: 25
24	Operation	operator: 26 operands: 27
25	ExprTuple	28, 29
26	Literal
27	ExprTuple	30, 31
28	ExprTuple	32
29	ExprTuple	33
30	Variable
31	Variable
32	ExprRange	lambda_map: 34 start_index: 36 end_index: 37
33	ExprRange	lambda_map: 35 start_index: 36 end_index: 37
34	Lambda	parameter: 43 body: 38
35	Lambda	parameter: 43 body: 39
36	Literal
37	Variable
38	IndexedVar	variable: 40 index: 43
39	IndexedVar	variable: 41 index: 43
40	Variable
41	Variable
42	ExprTuple	43
43	Variable