Expression of type Lambda

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 Conditional, Lambda, Py, X, x, y
from proveit.linear_algebra import Hspace, InnerProd, VecZero
from proveit.logic import Equals, Forall, Implies, InSet
from proveit.numbers import zero

# build up the expression from sub-expressions
expr = Lambda(y, Conditional(Implies(Forall(instance_param_or_params = [x], instance_expr = Equals(InnerProd(x, y), zero), domain = X), Equals(Py, VecZero(Hspace))), InSet(y, Hspace)))

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())

y \mapsto \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) \textrm{ if } y \in \mathcal{H}\right..

stored_expr.style_options()

no style options

# display the expression information
stored_expr.expr_info()

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