Expression of type Forall

from the theory of proveit.core_expr_types.conditionals

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 Conditional, ConditionalSet, ExprRange, IndexedVar, Variable, a, b, c, m, n
from proveit.core_expr_types import a_1_to_m, c_1_to_n
from proveit.logic import Equals, FALSE, Forall
from proveit.numbers import Natural, one

# build up the expression from sub-expressions
sub_expr1 = Variable("_a", latex_format = r"{_{-}a}")
expr = Forall(instance_param_or_params = [m, n], instance_expr = Forall(instance_param_or_params = [a_1_to_m, b, c_1_to_n], instance_expr = Equals(ConditionalSet(ExprRange(sub_expr1, Conditional(IndexedVar(a, sub_expr1), FALSE), one, m), b, ExprRange(sub_expr1, Conditional(IndexedVar(c, sub_expr1), FALSE), one, n)), b)), 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_{m, n \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{m}, b, c_{1}, c_{2}, \ldots, c_{n}}~\left(\left\{\begin{array}{ccc}a_{1} \textrm{ if } \bot \\ a_{2} \textrm{ if } \bot \\  \vdots \\ a_{m} \textrm{ if } \bot \\ b \\ c_{1} \textrm{ if } \bot \\ c_{2} \textrm{ if } \bot \\  \vdots \\ c_{n} \textrm{ if } \bot\end{array}\right.. = b\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	parameters: 3 body: 4
3	ExprTuple	33, 36
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	12, 13
11	Lambda	parameters: 14 body: 15
12	Operation	operator: 17 operands: 16
13	Operation	operator: 17 operands: 18
14	ExprTuple	19, 30, 20
15	Operation	operator: 21 operands: 22
16	ExprTuple	33, 23
17	Literal
18	ExprTuple	36, 23
19	ExprRange	lambda_map: 24 start_index: 35 end_index: 33
20	ExprRange	lambda_map: 25 start_index: 35 end_index: 36
21	Literal
22	ExprTuple	26, 30
23	Literal
24	Lambda	parameter: 45 body: 39
25	Lambda	parameter: 45 body: 40
26	Operation	operator: 27 operands: 28
27	Literal
28	ExprTuple	29, 30, 31
29	ExprRange	lambda_map: 32 start_index: 35 end_index: 33
30	Variable
31	ExprRange	lambda_map: 34 start_index: 35 end_index: 36
32	Lambda	parameter: 45 body: 37
33	Variable
34	Lambda	parameter: 45 body: 38
35	Literal
36	Variable
37	Conditional	value: 39 condition: 41
38	Conditional	value: 40 condition: 41
39	IndexedVar	variable: 42 index: 45
40	IndexedVar	variable: 43 index: 45
41	Literal
42	Variable
43	Variable
44	ExprTuple	45
45	Variable