Expression of type And

from the theory of proveit.linear_algebra.inner_products

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 Px, Py, X, x, y
from proveit.linear_algebra import Hspace, InnerProd, VecZero
from proveit.logic import And, Equals, Forall, Implies
from proveit.numbers import zero

# build up the expression from sub-expressions
sub_expr1 = [x]
expr = And(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(Px, x), domain = X), Forall(instance_param_or_params = [y], instance_expr = Implies(Forall(instance_param_or_params = sub_expr1, instance_expr = Equals(InnerProd(x, y), zero), domain = X), Equals(Py, VecZero(Hspace))), domain = Hspace)).with_wrapping_at(2)

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())

\begin{array}{c} \left[\forall_{x \in X}~\left(P\left(x\right) = x\right)\right] \land  \\ \left[\forall_{y \in \mathcal{H}}~\left(\left[\forall_{x \in X}~\left(\left \langle x, y\right \rangle = 0\right)\right] \Rightarrow \left(P\left(y\right) = \vec{0}\left(\mathcal{H}\right)\right)\right)\right] \end{array}

stored_expr.style_options()

name	description	default	current value	related methods
operation	'infix' or 'function' style formatting	infix	infix
wrap_positions	position(s) at which wrapping is to occur; '2 n - 1' is after the nth operand, '2 n' is after the nth operation.	()	(2)	('with_wrapping_at', 'with_wrap_before_operator', 'with_wrap_after_operator', 'without_wrapping', 'wrap_positions')
justification	if any wrap positions are set, justify to the 'left', 'center', or 'right'	center	center	('with_justification',)

# display the expression information
stored_expr.expr_info()

	core type	sub-expressions	expression
0	Operation	operator: 1 operands: 2
1	Literal
2	ExprTuple	3, 4
3	Operation	operator: 21 operand: 7
4	Operation	operator: 21 operand: 8
5	ExprTuple	7
6	ExprTuple	8
7	Lambda	parameter: 45 body: 9
8	Lambda	parameter: 46 body: 10
9	Conditional	value: 11 condition: 34
10	Conditional	value: 12 condition: 13
11	Operation	operator: 36 operands: 14
12	Operation	operator: 15 operands: 16
13	Operation	operator: 38 operands: 17
14	ExprTuple	18, 45
15	Literal
16	ExprTuple	19, 20
17	ExprTuple	46, 35
18	Operation	operator: 29 operand: 45
19	Operation	operator: 21 operand: 24
20	Operation	operator: 36 operands: 23
21	Literal
22	ExprTuple	24
23	ExprTuple	25, 26
24	Lambda	parameter: 45 body: 28
25	Operation	operator: 29 operand: 46
26	Operation	operator: 31 operand: 35
27	ExprTuple	45
28	Conditional	value: 33 condition: 34
29	Variable
30	ExprTuple	46
31	Literal
32	ExprTuple	35
33	Operation	operator: 36 operands: 37
34	Operation	operator: 38 operands: 39
35	Variable
36	Literal
37	ExprTuple	40, 41
38	Literal
39	ExprTuple	45, 42
40	Operation	operator: 43 operands: 44
41	Literal
42	Variable
43	Literal
44	ExprTuple	45, 46
45	Variable
46	Variable