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Expression of type Forall

from the theory of proveit.numbers.multiplication

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 i, j
from proveit.core_expr_types import a_1_to_i, b_1_to_j
from proveit.logic import Equals, Forall
from proveit.numbers import Complex, Mult, Natural, zero
In [2]:
# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [i, j], instance_expr = Forall(instance_param_or_params = [a_1_to_i, b_1_to_j], instance_expr = Equals(Mult(a_1_to_i, zero, b_1_to_j), zero), domain = Complex), domain = Natural)
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_{i, j \in \mathbb{N}}~\left[\forall_{a_{1}, a_{2}, \ldots, a_{i}, b_{1}, b_{2}, \ldots, b_{j} \in \mathbb{C}}~\left(\left(a_{1} \cdot  a_{2} \cdot  \ldots \cdot  a_{i} \cdot 0\cdot b_{1} \cdot  b_{2} \cdot  \ldots \cdot  b_{j}\right) = 0\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: 7
operand: 2
1ExprTuple2
2Lambdaparameters: 3
body: 4
3ExprTuple37, 40
4Conditionalvalue: 5
condition: 6
5Operationoperator: 7
operand: 10
6Operationoperator: 22
operands: 9
7Literal
8ExprTuple10
9ExprTuple11, 12
10Lambdaparameters: 13
body: 14
11Operationoperator: 42
operands: 15
12Operationoperator: 42
operands: 16
13ExprTuple31, 33
14Conditionalvalue: 17
condition: 18
15ExprTuple37, 19
16ExprTuple40, 19
17Operationoperator: 20
operands: 21
18Operationoperator: 22
operands: 23
19Literal
20Literal
21ExprTuple24, 32
22Literal
23ExprTuple25, 26
24Operationoperator: 27
operands: 28
25ExprRangelambda_map: 29
start_index: 39
end_index: 37
26ExprRangelambda_map: 30
start_index: 39
end_index: 40
27Literal
28ExprTuple31, 32, 33
29Lambdaparameter: 50
body: 34
30Lambdaparameter: 50
body: 35
31ExprRangelambda_map: 36
start_index: 39
end_index: 37
32Literal
33ExprRangelambda_map: 38
start_index: 39
end_index: 40
34Operationoperator: 42
operands: 41
35Operationoperator: 42
operands: 43
36Lambdaparameter: 50
body: 44
37Variable
38Lambdaparameter: 50
body: 45
39Literal
40Variable
41ExprTuple44, 46
42Literal
43ExprTuple45, 46
44IndexedVarvariable: 47
index: 50
45IndexedVarvariable: 48
index: 50
46Literal
47Variable
48Variable
49ExprTuple50
50Variable