Expression of type Forall

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 H, K, x, y, z
from proveit.linear_algebra import InnerProd, InnerProdSpaces, VecAdd
from proveit.logic import Equals, Forall

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [H], instance_expr = Forall(instance_param_or_params = [x, y, z], instance_expr = Equals(InnerProd(x, VecAdd(y, z)), VecAdd(InnerProd(x, y), InnerProd(x, z))), domain = H), domain = InnerProdSpaces(K))

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_{H \underset{{\scriptscriptstyle c}}{\in} \textrm{InnerProdSpaces}\left(K\right)}~\left[\forall_{x, y, z \in H}~\left(\left \langle x, y + z\right \rangle = \left(\left \langle x, y\right \rangle + \left \langle x, z\right \rangle\right)\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: 38 body: 4
3	ExprTuple	38
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	38, 12
11	Lambda	parameters: 13 body: 14
12	Operation	operator: 15 operand: 19
13	ExprTuple	45, 44, 46
14	Conditional	value: 17 condition: 18
15	Literal
16	ExprTuple	19
17	Operation	operator: 20 operands: 21
18	Operation	operator: 22 operands: 23
19	Variable
20	Literal
21	ExprTuple	24, 25
22	Literal
23	ExprTuple	26, 27, 28
24	Operation	operator: 42 operands: 29
25	Operation	operator: 39 operands: 30
26	Operation	operator: 33 operands: 31
27	Operation	operator: 33 operands: 32
28	Operation	operator: 33 operands: 34
29	ExprTuple	45, 35
30	ExprTuple	36, 37
31	ExprTuple	45, 38
32	ExprTuple	44, 38
33	Literal
34	ExprTuple	46, 38
35	Operation	operator: 39 operands: 40
36	Operation	operator: 42 operands: 41
37	Operation	operator: 42 operands: 43
38	Variable
39	Literal
40	ExprTuple	44, 46
41	ExprTuple	45, 44
42	Literal
43	ExprTuple	45, 46
44	Variable
45	Variable
46	Variable