Expression of type Forall

from the theory of proveit.trigonometry

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 a, b, r
from proveit.logic import Equals, Forall
from proveit.numbers import Abs, Exp, Mult, Real, e, frac, i, subtract, two
from proveit.trigonometry import Sin

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [a, b], instance_expr = Equals(Abs(subtract(Mult(r, Exp(e, Mult(i, a))), Mult(r, Exp(e, Mult(i, b))))), Mult(two, r, Sin(frac(Abs(subtract(a, b)), two)))), domain = Real)

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_{a, b \in \mathbb{R}}~\left(\left|\left(r \cdot \mathsf{e}^{\mathsf{i} \cdot a}\right) - \left(r \cdot \mathsf{e}^{\mathsf{i} \cdot b}\right)\right| = \left(2 \cdot r \cdot \sin{\frac{\left|a - b\right|}{2}}\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: 1 operand: 3
1	Literal
2	ExprTuple	3
3	Lambda	parameters: 4 body: 5
4	ExprTuple	53, 60
5	Conditional	value: 6 condition: 7
6	Operation	operator: 8 operands: 9
7	Operation	operator: 10 operands: 11
8	Literal
9	ExprTuple	12, 13
10	Literal
11	ExprTuple	14, 15
12	Operation	operator: 40 operand: 21
13	Operation	operator: 55 operands: 17
14	Operation	operator: 19 operands: 18
15	Operation	operator: 19 operands: 20
16	ExprTuple	21
17	ExprTuple	37, 43, 22
18	ExprTuple	53, 23
19	Literal
20	ExprTuple	60, 23
21	Operation	operator: 49 operands: 24
22	Operation	operator: 25 operand: 29
23	Literal
24	ExprTuple	27, 28
25	Literal
26	ExprTuple	29
27	Operation	operator: 55 operands: 30
28	Operation	operator: 57 operand: 35
29	Operation	operator: 32 operands: 33
30	ExprTuple	43, 34
31	ExprTuple	35
32	Literal
33	ExprTuple	36, 37
34	Operation	operator: 47 operands: 38
35	Operation	operator: 55 operands: 39
36	Operation	operator: 40 operand: 45
37	Literal
38	ExprTuple	51, 42
39	ExprTuple	43, 44
40	Literal
41	ExprTuple	45
42	Operation	operator: 55 operands: 46
43	Variable
44	Operation	operator: 47 operands: 48
45	Operation	operator: 49 operands: 50
46	ExprTuple	59, 53
47	Literal
48	ExprTuple	51, 52
49	Literal
50	ExprTuple	53, 54
51	Literal
52	Operation	operator: 55 operands: 56
53	Variable
54	Operation	operator: 57 operand: 60
55	Literal
56	ExprTuple	59, 60
57	Literal
58	ExprTuple	60
59	Literal
60	Variable