Proof of proveit.physics.quantum.QPE._alpha_m_mod_as_geometric_sum theorem

import proveit
theory = proveit.Theory() # the theorem's theory
from proveit import defaults
from proveit.numbers import two, i, pi, Mult, Exp
from proveit.physics.quantum.QPE import (
        _alpha_m_mod_evaluation, _phase_is_real, _two_pow_t_is_nat_pos)

%proving _alpha_m_mod_as_geometric_sum

With these allowed/disallowed theorem/theory presumptions (e.g., to avoid circular dependencies), we begin our proof of
_alpha_m_mod_as_geometric_sum:
(see dependencies)

defaults.assumptions = _alpha_m_mod_as_geometric_sum.all_conditions()

defaults.assumptions:

_alpha_m_mod_evaluation

 ⊢  

# Don't combine exponents during auto-simplification for the following steps.
Mult.change_simplification_directives(combine_all_exponents = False)

_alpha_m_mod_evaluation_inst = _alpha_m_mod_evaluation.instantiate()

_alpha_m_mod_evaluation_inst:  ⊢  

the_summation = _alpha_m_mod_evaluation_inst.rhs.factors[1]

the_summation:

k_domain = the_summation.domain

k_domain:

from proveit import k
from proveit.logic import InSet
k_in_k_domain = InSet(k, k_domain)

k_in_k_domain:

defaults.assumptions = defaults.assumptions + (k_in_k_domain,)

defaults.assumptions:

Rewriting the summand, step by step, to get

$\exp{(-\frac{2 \pi i k \ell}{2^t})}\exp{(2\pi i \varphi k)} = (\exp{(-\frac{2 \pi i k \ell}{2^t})}\exp{(2\pi i \varphi k)})^k = (e^{2\pi i (\varphi - \frac{\ell}{2^t})})^k$

(so we can eventually evaluate as a geometric sum under the right circumstances in further theorems).

the_summand_01 = the_summation.instance_expr

the_summand_01:

# need these for the common_power_extraction() to work
display(_phase_is_real)
display(_two_pow_t_is_nat_pos)

 ⊢  

 ⊢  

the_summand_02 = the_summand_01.inner_expr().common_power_extraction(exp_factor=k)

the_summand_02: ,  ⊢  

# # Now we can combine exponents, but let's preserve the left side as an exception.
Mult.change_simplification_directives(combine_all_exponents = True)
Exp.change_simplification_directives(reduce_double_exponent = False)
defaults.preserved_exprs = set([the_summand_02.lhs])

defaults.preserved_exprs:

the_summand_03 = the_summand_02.inner_expr().rhs.base.simplify()

the_summand_03: ,  ⊢  

the_summand_04 = the_summand_03.inner_expr().rhs.base.exponent.commute()

the_summand_04: ,  ⊢  

the_summand_05 = (
        the_summand_04.inner_expr().rhs.base.exponent.factor(
                Mult(two, pi, i), auto_simplify=False))

the_summand_05: ,  ⊢  

And a quick tinkering with the grouping in the exponent:

the_summand_06 = the_summand_05.inner_expr().rhs.base.exponent.disassociate(0)

the_summand_06: ,  ⊢  

Now we can substitute the modified summand expression back into the original summation:

(_alpha_m_mod_evaluation_inst.inner_expr().
        rhs.factors[1].summand.substitute(the_summand_06.rhs))

 ⊢  

_alpha_m_mod_as_geometric_sum may now be readily provable (assuming required theorems are usable).  Simply execute "%qed".

%qed

proveit.physics.quantum.QPE._alpha_m_mod_as_geometric_sum has been proven.

	step type	requirements	statement
0	generalization	1	⊢
1	instantiation	2, 3, 4	⊢
	: , :
2	theorem		⊢
	proveit.logic.equality.rhs_via_equality
3	instantiation	5, 186	⊢
	:
4	instantiation	85, 6, 7*, 8*	⊢
	: , : , :
5	conjecture		⊢
	proveit.physics.quantum.QPE._alpha_m_mod_evaluation
6	modus ponens	9, 10	⊢
7	instantiation	11, 181	⊢
	: , :
8	instantiation	11, 181	⊢
	: , :
9	instantiation	12, 174	⊢
	: , : , : , : , : , : , :
10	generalization	13	⊢
11	conjecture		⊢
	proveit.core_expr_types.conditionals.satisfied_condition_reduction
12	conjecture		⊢
	proveit.core_expr_types.lambda_maps.general_lambda_substitution
13	instantiation	128, 14, 15	,  ⊢
	: , : , :
14	instantiation	128, 16, 17	,  ⊢
	: , : , :
15	instantiation	156, 157, 123, 202, 158, 124, 169, 170, 161, 18	⊢
	: , : , : , : , : , :
16	instantiation	128, 19, 20	,  ⊢
	: , : , :
17	instantiation	31, 21, 22, 23	⊢
	: , : , : , :
18	instantiation	24, 142, 40	⊢
	: , :
19	instantiation	128, 25, 26	,  ⊢
	: , : , :
20	instantiation	27, 64, 75	⊢
	: , :
21	instantiation	85, 28	⊢
	: , : , :
22	instantiation	85, 29	⊢
	: , : , :
23	instantiation	87, 30	⊢
	: , :
24	conjecture		⊢
	proveit.numbers.addition.add_complex_closure_bin
25	instantiation	31, 32, 33, 34	,  ⊢
	: , : , : , :
26	instantiation	35, 119, 64, 75	⊢
	: , : , :
27	conjecture		⊢
	proveit.numbers.addition.commutation
28	instantiation	125, 157, 123, 202, 158, 124, 169, 170, 161, 142	⊢
	: , : , : , : , : , :
29	instantiation	146, 36, 37	⊢
	: , : , :
30	instantiation	38, 202, 195, 157, 39, 158, 101, 142, 40	⊢
	: , : , : , : , : , :
31	conjecture		⊢
	proveit.logic.equality.four_chain_transitivity
32	instantiation	85, 41	,  ⊢
	: , : , :
33	instantiation	85, 42	⊢
	: , : , :
34	instantiation	87, 43	,  ⊢
	: , :
35	conjecture		⊢
	proveit.numbers.exponentiation.product_of_complex_powers
36	instantiation	85, 44	⊢
	: , : , :
37	instantiation	87, 45	⊢
	: , :
38	conjecture		⊢
	proveit.numbers.multiplication.distribute_through_sum
39	instantiation	171	⊢
	: , :
40	instantiation	76, 58	⊢
	:
41	instantiation	146, 46, 47	,  ⊢
	: , : , :
42	instantiation	146, 48, 49	⊢
	: , : , :
43	instantiation	50, 51, 52, 152, 53, 54	,  ⊢
	: , : , :
44	instantiation	146, 55, 56	⊢
	: , : , :
45	instantiation	57, 101, 58	⊢
	: , :
46	instantiation	85, 59	,  ⊢
	: , : , :
47	instantiation	87, 60	,  ⊢
	: , :
48	instantiation	85, 61	⊢
	: , : , :
49	instantiation	87, 62	⊢
	: , :
50	conjecture		⊢
	proveit.numbers.exponentiation.real_power_of_product
51	instantiation	63, 79, 64	⊢
	: , :
52	instantiation	63, 79, 75	⊢
	: , :
53	instantiation	78, 79, 64, 81	⊢
	: , :
54	instantiation	65, 66, 67	⊢
	: , : , :
55	instantiation	85, 68	⊢
	: , : , :
56	instantiation	87, 69	⊢
	: , :
57	conjecture		⊢
	proveit.numbers.multiplication.mult_neg_right
58	instantiation	93, 162, 137, 94	⊢
	: , :
59	instantiation	146, 70, 71	,  ⊢
	: , : , :
60	instantiation	72, 79, 77, 81	,  ⊢
	: , :
61	instantiation	125, 157, 126, 202, 158, 73, 169, 170, 161, 142, 135	⊢
	: , : , : , : , : , :
62	instantiation	74, 79, 75, 81	⊢
	: , :
63	conjecture		⊢
	proveit.numbers.exponentiation.exp_complex_closure
64	instantiation	76, 77	⊢
	:
65	theorem		⊢
	proveit.logic.equality.sub_left_side_into
66	instantiation	78, 79, 80, 81	⊢
	: , :
67	instantiation	85, 82	⊢
	: , : , :
68	instantiation	125, 157, 123, 202, 158, 124, 169, 170, 161, 162	⊢
	: , : , : , : , : , :
69	instantiation	105, 101, 162, 108, 107, 83*, 84*	⊢
	: , : , : , :
70	instantiation	85, 86	,  ⊢
	: , : , :
71	instantiation	87, 88	,  ⊢
	: , :
72	instantiation	89, 175	⊢
	:
73	instantiation	144	⊢
	: , : , : , :
74	instantiation	90, 175	⊢
	:
75	instantiation	128, 91, 92	⊢
	: , : , :
76	conjecture		⊢
	proveit.numbers.negation.complex_closure
77	instantiation	93, 106, 137, 94	⊢
	: , :
78	conjecture		⊢
	proveit.numbers.exponentiation.exp_not_eq_zero
79	instantiation	200, 177, 95	⊢
	: , : , :
80	instantiation	128, 96, 97	⊢
	: , : , :
81	instantiation	98, 99	⊢
	:
82	instantiation	100, 202, 169, 170, 161, 142	⊢
	: , : , : , : , : , : , :
83	instantiation	134, 101	⊢
	:
84	instantiation	102, 137	⊢
	:
85	axiom		⊢
	proveit.logic.equality.substitution
86	instantiation	146, 103, 104	,  ⊢
	: , : , :
87	theorem		⊢
	proveit.logic.equality.equals_reversal
88	instantiation	105, 106, 135, 107, 108, 109*, 110*	,  ⊢
	: , : , : , :
89	conjecture		⊢
	proveit.numbers.exponentiation.int_exp_of_neg_exp
90	conjecture		⊢
	proveit.numbers.exponentiation.int_exp_of_exp
91	instantiation	168, 155, 111	⊢
	: , :
92	instantiation	146, 112, 113	⊢
	: , : , :
93	conjecture		⊢
	proveit.numbers.division.div_complex_closure
94	instantiation	114, 197	⊢
	:
95	instantiation	200, 184, 119	⊢
	: , : , :
96	instantiation	168, 139, 115	⊢
	: , :
97	instantiation	146, 116, 117	⊢
	: , : , :
98	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
99	instantiation	200, 118, 119	⊢
	: , : , :
100	conjecture		⊢
	proveit.numbers.multiplication.leftward_commutation
101	instantiation	128, 120, 121	⊢
	: , : , :
102	conjecture		⊢
	proveit.numbers.multiplication.elim_one_left
103	instantiation	122, 123, 202, 157, 124, 158, 169, 170, 161, 135, 162	,  ⊢
	: , : , : , : , : , : , :
104	instantiation	125, 157, 126, 202, 158, 127, 169, 170, 161, 162, 135	,  ⊢
	: , : , : , : , : , :
105	conjecture		⊢
	proveit.numbers.division.prod_of_fracs
106	instantiation	128, 129, 130	⊢
	: , : , :
107	instantiation	200, 132, 131	⊢
	: , : , :
108	instantiation	200, 132, 133	⊢
	: , : , :
109	instantiation	134, 135	⊢
	:
110	instantiation	136, 137	⊢
	:
111	instantiation	168, 161, 142	⊢
	: , :
112	instantiation	156, 202, 195, 157, 138, 158, 155, 161, 142	⊢
	: , : , : , : , : , :
113	instantiation	156, 157, 195, 158, 159, 138, 169, 170, 161, 142	⊢
	: , : , : , : , : , :
114	conjecture		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
115	instantiation	168, 170, 142	⊢
	: , :
116	instantiation	156, 202, 195, 157, 141, 158, 139, 170, 142	⊢
	: , : , : , : , : , :
117	instantiation	156, 157, 195, 158, 140, 141, 169, 161, 170, 142	⊢
	: , : , : , : , : , :
118	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
119	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
120	instantiation	168, 155, 161	⊢
	: , :
121	instantiation	156, 157, 195, 202, 158, 159, 169, 170, 161	⊢
	: , : , : , : , : , :
122	conjecture		⊢
	proveit.numbers.multiplication.rightward_commutation
123	conjecture		⊢
	proveit.numbers.numerals.decimals.nat3
124	instantiation	143	⊢
	: , : , :
125	conjecture		⊢
	proveit.numbers.multiplication.association
126	conjecture		⊢
	proveit.numbers.numerals.decimals.nat4
127	instantiation	144	⊢
	: , : , : , :
128	theorem		⊢
	proveit.logic.equality.sub_right_side_into
129	instantiation	168, 155, 145	⊢
	: , :
130	instantiation	146, 147, 148	⊢
	: , : , :
131	instantiation	200, 150, 149	⊢
	: , : , :
132	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
133	instantiation	200, 150, 151	⊢
	: , : , :
134	conjecture		⊢
	proveit.numbers.division.frac_one_denom
135	instantiation	200, 177, 152	⊢
	: , : , :
136	conjecture		⊢
	proveit.numbers.multiplication.elim_one_right
137	instantiation	200, 177, 153	⊢
	: , : , :
138	instantiation	171	⊢
	: , :
139	instantiation	168, 169, 161	⊢
	: , :
140	instantiation	171	⊢
	: , :
141	instantiation	171	⊢
	: , :
142	instantiation	200, 177, 154	⊢
	: , : , :
143	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
144	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_4_typical_eq
145	instantiation	168, 161, 162	⊢
	: , :
146	axiom		⊢
	proveit.logic.equality.equals_transitivity
147	instantiation	156, 202, 195, 157, 160, 158, 155, 161, 162	⊢
	: , : , : , : , : , :
148	instantiation	156, 157, 195, 158, 159, 160, 169, 170, 161, 162	⊢
	: , : , : , : , : , :
149	instantiation	200, 164, 163	⊢
	: , : , :
150	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
151	instantiation	200, 164, 165	⊢
	: , : , :
152	instantiation	200, 182, 166	⊢
	: , : , :
153	instantiation	200, 182, 167	⊢
	: , : , :
154	conjecture		⊢
	proveit.physics.quantum.QPE._phase_is_real
155	instantiation	168, 169, 170	⊢
	: , :
156	conjecture		⊢
	proveit.numbers.multiplication.disassociation
157	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
158	conjecture		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
159	instantiation	171	⊢
	: , :
160	instantiation	171	⊢
	: , :
161	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
162	instantiation	200, 177, 172	⊢
	: , : , :
163	instantiation	200, 173, 197	⊢
	: , : , :
164	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
165	instantiation	200, 173, 174	⊢
	: , : , :
166	instantiation	200, 190, 175	⊢
	: , : , :
167	instantiation	200, 190, 193	⊢
	: , : , :
168	conjecture		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
169	instantiation	200, 177, 176	⊢
	: , : , :
170	instantiation	200, 177, 178	⊢
	: , : , :
171	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
172	instantiation	200, 182, 179	⊢
	: , : , :
173	conjecture		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
174	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat1
175	instantiation	200, 180, 181	⊢
	: , : , :
176	instantiation	200, 182, 183	⊢
	: , : , :
177	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
178	instantiation	200, 184, 185	⊢
	: , : , :
179	instantiation	200, 190, 186	⊢
	: