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, RealNonNeg, e, frac, i, subtract, two
from proveit.trigonometry import Sin

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [r], instance_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), domain = RealNonNeg)

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_{r \in \mathbb{R}^{\ge 0}}~\left[\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)\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: 51 body: 4
3	ExprTuple	51
4	Conditional	value: 5 condition: 6
5	Operation	operator: 7 operand: 10
6	Operation	operator: 27 operands: 9
7	Literal
8	ExprTuple	10
9	ExprTuple	51, 11
10	Lambda	parameters: 12 body: 13
11	Literal
12	ExprTuple	61, 68
13	Conditional	value: 14 condition: 15
14	Operation	operator: 16 operands: 17
15	Operation	operator: 18 operands: 19
16	Literal
17	ExprTuple	20, 21
18	Literal
19	ExprTuple	22, 23
20	Operation	operator: 48 operand: 29
21	Operation	operator: 63 operands: 25
22	Operation	operator: 27 operands: 26
23	Operation	operator: 27 operands: 28
24	ExprTuple	29
25	ExprTuple	45, 51, 30
26	ExprTuple	61, 31
27	Literal
28	ExprTuple	68, 31
29	Operation	operator: 57 operands: 32
30	Operation	operator: 33 operand: 37
31	Literal
32	ExprTuple	35, 36
33	Literal
34	ExprTuple	37
35	Operation	operator: 63 operands: 38
36	Operation	operator: 65 operand: 43
37	Operation	operator: 40 operands: 41
38	ExprTuple	51, 42
39	ExprTuple	43
40	Literal
41	ExprTuple	44, 45
42	Operation	operator: 55 operands: 46
43	Operation	operator: 63 operands: 47
44	Operation	operator: 48 operand: 53
45	Literal
46	ExprTuple	59, 50
47	ExprTuple	51, 52
48	Literal
49	ExprTuple	53
50	Operation	operator: 63 operands: 54
51	Variable
52	Operation	operator: 55 operands: 56
53	Operation	operator: 57 operands: 58
54	ExprTuple	67, 61
55	Literal
56	ExprTuple	59, 60
57	Literal
58	ExprTuple	61, 62
59	Literal
60	Operation	operator: 63 operands: 64
61	Variable
62	Operation	operator: 65 operand: 68
63	Literal
64	ExprTuple	67, 68
65	Literal
66	ExprTuple	68
67	Literal
68	Variable