Expression of type Forall

from the theory of proveit.core_expr_types.tuples

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 f, g, i, j, k
from proveit.core_expr_types import f_i_to_j, fk, g_i_to_j, gk
from proveit.logic import Equals, Forall, Implies
from proveit.numbers import Integer, Interval

# build up the expression from sub-expressions
expr = Forall(instance_param_or_params = [f, g], instance_expr = Forall(instance_param_or_params = [i, j], instance_expr = Implies(Forall(instance_param_or_params = [k], instance_expr = Equals(fk, gk), domain = Interval(i, j)), Equals([f_i_to_j], [g_i_to_j])), domain = Integer))

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_{f, g}~\left[\forall_{i, j \in \mathbb{Z}}~\left(\left[\forall_{k \in \{i~\ldotp \ldotp~j\}}~\left(f\left(k\right) = g\left(k\right)\right)\right] \Rightarrow \left(\left(f\left(i\right), f\left(i + 1\right), \ldots, f\left(j\right)\right) = \left(g\left(i\right), g\left(i + 1\right), \ldots, g\left(j\right)\right)\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: 18 operand: 2
1	ExprTuple	2
2	Lambda	parameters: 3 body: 4
3	ExprTuple	44, 45
4	Operation	operator: 18 operand: 6
5	ExprTuple	6
6	Lambda	parameters: 48 body: 7
7	Conditional	value: 8 condition: 9
8	Operation	operator: 10 operands: 11
9	Operation	operator: 12 operands: 13
10	Literal
11	ExprTuple	14, 15
12	Literal
13	ExprTuple	16, 17
14	Operation	operator: 18 operand: 23
15	Operation	operator: 34 operands: 20
16	Operation	operator: 36 operands: 21
17	Operation	operator: 36 operands: 22
18	Literal
19	ExprTuple	23
20	ExprTuple	24, 25
21	ExprTuple	51, 26
22	ExprTuple	52, 26
23	Lambda	parameter: 50 body: 27
24	ExprTuple	28
25	ExprTuple	29
26	Literal
27	Conditional	value: 30 condition: 31
28	ExprRange	lambda_map: 32 start_index: 51 end_index: 52
29	ExprRange	lambda_map: 33 start_index: 51 end_index: 52
30	Operation	operator: 34 operands: 35
31	Operation	operator: 36 operands: 37
32	Lambda	parameter: 49 body: 38
33	Lambda	parameter: 49 body: 39
34	Literal
35	ExprTuple	40, 41
36	Literal
37	ExprTuple	50, 42
38	Operation	operator: 44 operand: 49
39	Operation	operator: 45 operand: 49
40	Operation	operator: 44 operand: 50
41	Operation	operator: 45 operand: 50
42	Operation	operator: 47 operands: 48
43	ExprTuple	49
44	Variable
45	Variable
46	ExprTuple	50
47	Literal
48	ExprTuple	51, 52
49	Variable
50	Variable
51	Variable
52	Variable