# from the theory of proveit.core_expr_types.expr_arrays¶

In [1]:
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.
# import Expression classes needed to build the expression
from proveit import VertExprArray, i, j
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import Forall, NotEquals
from proveit.numbers import Natural

In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j], instance_expr = NotEquals(VertExprArray(a_1_to_i), VertExprArray(b_1_to_j))), domain = Natural, condition = NotEquals(i, j))

expr:
In [3]:
# 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

In [4]:
# Show the LaTeX representation of the expression for convenience if you need it.
print(stored_expr.latex())

\forall_{i, j \in \mathbb{N}~|~i \neq j}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j}}~\left(\left(\begin{array}{cccc}
\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}{*}{$\begin{array}{c} \uparrow \\ a_{i} \\ \downarrow \end{array}$} \\
\end{array}
\right) \neq \left(\begin{array}{cccc}
\multirow{1}{*}{$\begin{array}{c} \uparrow \\ b_{1} \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ b_{2} \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ \cdots \\ \downarrow \end{array}$} & \multirow{1}{*}{$\begin{array}{c} \uparrow \\ b_{j} \\ \downarrow \end{array}$} \\
\end{array}
\right)\right)\right]

In [5]:
stored_expr.style_options()

namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()

core typesub-expressionsexpression
0Operationoperator: 6
operand: 2
1ExprTuple2
2Lambdaparameters: 19
body: 3
3Conditionalvalue: 4
condition: 5
4Operationoperator: 6
operand: 10
5Operationoperator: 8
operands: 9
6Literal
7ExprTuple10
8Literal
9ExprTuple11, 12, 13
10Lambdaparameters: 14
body: 15
11Operationoperator: 17
operands: 16
12Operationoperator: 17
operands: 18
13Operationoperator: 20
operands: 19
14ExprTuple25, 26
15Operationoperator: 20
operands: 21
16ExprTuple28, 22
17Literal
18ExprTuple31, 22
19ExprTuple28, 31
20Literal
21ExprTuple23, 24
22Literal
23ExprTuple25
24ExprTuple26
25ExprRangelambda_map: 27
start_index: 30
end_index: 28
26ExprRangelambda_map: 29
start_index: 30
end_index: 31
27Lambdaparameter: 37
body: 32
28Variable
29Lambdaparameter: 37
body: 33
30Literal
31Variable
32IndexedVarvariable: 34
index: 37
33IndexedVarvariable: 35
index: 37
34Variable
35Variable
36ExprTuple37
37Variable