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Expression of type Forall

from the theory of proveit.trigonometry

In [1]:
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
In [2]:
# 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:
In [3]:
# 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
In [4]:
# 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)
In [5]:
stored_expr.style_options()
namedescriptiondefaultcurrent valuerelated methods
with_wrappingIf 'True', wrap the Expression after the parametersNoneNone/False('with_wrapping',)
condition_wrappingWrap 'before' or 'after' the condition (or None).NoneNone/False('with_wrap_after_condition', 'with_wrap_before_condition')
wrap_paramsIf 'True', wraps every two parameters AND wraps the Expression after the parametersNoneNone/False('with_params',)
justificationjustify to the 'left', 'center', or 'right' in the array cellscentercenter('with_justification',)
In [6]:
# display the expression information
stored_expr.expr_info()
 core typesub-expressionsexpression
0Operationoperator: 1
operand: 3
1Literal
2ExprTuple3
3Lambdaparameters: 4
body: 5
4ExprTuple53, 60
5Conditionalvalue: 6
condition: 7
6Operationoperator: 8
operands: 9
7Operationoperator: 10
operands: 11
8Literal
9ExprTuple12, 13
10Literal
11ExprTuple14, 15
12Operationoperator: 40
operand: 21
13Operationoperator: 55
operands: 17
14Operationoperator: 19
operands: 18
15Operationoperator: 19
operands: 20
16ExprTuple21
17ExprTuple37, 43, 22
18ExprTuple53, 23
19Literal
20ExprTuple60, 23
21Operationoperator: 49
operands: 24
22Operationoperator: 25
operand: 29
23Literal
24ExprTuple27, 28
25Literal
26ExprTuple29
27Operationoperator: 55
operands: 30
28Operationoperator: 57
operand: 35
29Operationoperator: 32
operands: 33
30ExprTuple43, 34
31ExprTuple35
32Literal
33ExprTuple36, 37
34Operationoperator: 47
operands: 38
35Operationoperator: 55
operands: 39
36Operationoperator: 40
operand: 45
37Literal
38ExprTuple51, 42
39ExprTuple43, 44
40Literal
41ExprTuple45
42Operationoperator: 55
operands: 46
43Variable
44Operationoperator: 47
operands: 48
45Operationoperator: 49
operands: 50
46ExprTuple59, 53
47Literal
48ExprTuple51, 52
49Literal
50ExprTuple53, 54
51Literal
52Operationoperator: 55
operands: 56
53Variable
54Operationoperator: 57
operand: 60
55Literal
56ExprTuple59, 60
57Literal
58ExprTuple60
59Literal
60Variable