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

import proveit
theory = proveit.Theory() # the theorem's theory
from proveit import defaults
from proveit import k, m
from proveit.numbers import Exp
from proveit.physics.quantum.QPE import (
    _alpha_m_evaluation, _ideal_phase_cond, _t_in_natural_pos, _two_pow_t_is_nat_pos, _phase_is_real)

%proving _alpha_ideal_case

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

defaults.assumptions = [_ideal_phase_cond]

defaults.assumptions:

_scaled_phase = _ideal_phase_cond.element

_scaled_phase:

Keep the $2$ in $2 \pi i$ from combining with powers of $2$:

Exp.change_simplification_directives(factor_numeric_rational=True)

_alpha_m_evaluation

 ⊢  

Reduced through auto-simplification:

_alpha_ideal_01 = _alpha_m_evaluation.instantiate({m: _scaled_phase})

_alpha_ideal_01:  ⊢  

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

%qed

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

	step type	requirements	statement
0	deduction	1	⊢
1	instantiation	2, 3, 4*	⊢
	:
2	conjecture		⊢
	proveit.physics.quantum.QPE._alpha_m_evaluation
3	assumption		⊢
4	instantiation	131, 5, 6	⊢
	: , : , :
5	instantiation	103, 7	⊢
	: , : , :
6	instantiation	140, 8, 9	⊢
	: , : , :
7	instantiation	131, 10, 11	⊢
	: , : , :
8	instantiation	140, 12, 13	⊢
	: , : , :
9	instantiation	42, 14, 15, 16	⊢
	: , : , : , :
10	instantiation	103, 17	⊢
	: , : , :
11	instantiation	18, 183, 184, 55, 19*	⊢
	: , : , :
12	instantiation	20, 55, 107, 106	⊢
	: , : , : , : , :
13	instantiation	131, 21, 22	⊢
	: , : , :
14	instantiation	103, 23	⊢
	: , : , :
15	instantiation	103, 23	⊢
	: , : , :
16	instantiation	119, 55	⊢
	:
17	modus ponens	24, 25	⊢
18	conjecture		⊢
	proveit.numbers.summation.trivial_sum
19	instantiation	131, 26, 109	⊢
	: , : , :
20	conjecture		⊢
	proveit.numbers.division.mult_frac_cancel_denom_left
21	instantiation	103, 27	⊢
	: , : , :
22	instantiation	103, 28	⊢
	: , : , :
23	instantiation	99, 55	⊢
	:
24	instantiation	29, 150	⊢
	: , : , : , : , : , :
25	generalization	30	⊢
26	instantiation	103, 31	⊢
	: , : , :
27	instantiation	103, 32	⊢
	: , : , :
28	instantiation	131, 33, 34	⊢
	: , : , :
29	axiom		⊢
	proveit.core_expr_types.lambda_maps.lambda_substitution
30	instantiation	35, 36	⊢
	: , : , :
31	instantiation	131, 37, 38	⊢
	: , : , :
32	instantiation	39, 126	⊢
	:
33	instantiation	103, 109	⊢
	: , : , :
34	instantiation	99, 126	⊢
	:
35	axiom		⊢
	proveit.core_expr_types.conditionals.conditional_substitution
36	deduction	40	⊢
37	instantiation	103, 41	⊢
	: , : , :
38	instantiation	42, 43, 44, 45	⊢
	: , : , : , :
39	conjecture		⊢
	proveit.numbers.multiplication.elim_one_left
40	instantiation	131, 46, 47	⊢
	: , : , :
41	conjecture		⊢
	proveit.numbers.negation.negated_zero
42	conjecture		⊢
	proveit.logic.equality.four_chain_transitivity
43	instantiation	48, 153, 179, 154, 52, 49, 126, 53, 50, 55	⊢
	: , : , : , : , : , :
44	instantiation	51, 179, 194, 52, 126, 53, 55	⊢
	: , : , : , :
45	instantiation	54, 55, 126, 56	⊢
	: , : , :
46	instantiation	103, 57	⊢
	: , : , :
47	instantiation	58, 89, 59, 81, 60*	⊢
	: , : , :
48	conjecture		⊢
	proveit.numbers.addition.disassociation
49	instantiation	165	⊢
	: , :
50	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.zero_is_complex
51	conjecture		⊢
	proveit.numbers.addition.elim_zero_any
52	instantiation	165	⊢
	: , :
53	instantiation	192, 169, 61	⊢
	: , : , :
54	conjecture		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
55	instantiation	192, 169, 62	⊢
	: , : , :
56	instantiation	75	⊢
	:
57	instantiation	63, 64, 65, 66, 67, 68	⊢
	: , : , :
58	conjecture		⊢
	proveit.numbers.exponentiation.product_of_complex_powers
59	instantiation	69, 108	⊢
	:
60	instantiation	131, 70, 71	⊢
	: , : , :
61	instantiation	192, 174, 72	⊢
	: , : , :
62	instantiation	192, 174, 73	⊢
	: , : , :
63	conjecture		⊢
	proveit.core_expr_types.tuples.tuple_eq_via_elem_eq
64	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat2
65	instantiation	165	⊢
	: , :
66	instantiation	165	⊢
	: , :
67	instantiation	103, 74	⊢
	: , : , :
68	instantiation	75	⊢
	:
69	conjecture		⊢
	proveit.numbers.negation.complex_closure
70	instantiation	103, 76	⊢
	: , : , :
71	instantiation	77, 78	⊢
	:
72	instantiation	192, 177, 187	⊢
	: , : , :
73	instantiation	192, 177, 191	⊢
	: , : , :
74	instantiation	103, 79	⊢
	: , : , :
75	axiom		⊢
	proveit.logic.equality.equals_reflexivity
76	instantiation	80, 108, 81, 82	⊢
	: , :
77	conjecture		⊢
	proveit.numbers.exponentiation.exp_zero_eq_one
78	instantiation	192, 169, 83	⊢
	: , : , :
79	instantiation	131, 84, 85	⊢
	: , : , :
80	conjecture		⊢
	proveit.numbers.addition.subtraction.add_cancel_reverse
81	instantiation	140, 86, 87	⊢
	: , : , :
82	instantiation	120, 144, 194, 153, 88, 154, 157, 158, 163, 164, 156	⊢
	: , : , : , : , : , : , :
83	instantiation	192, 172, 89	⊢
	: , : , :
84	instantiation	103, 90	⊢
	: , : , :
85	instantiation	131, 91, 92	⊢
	: , : , :
86	instantiation	162, 143, 93	⊢
	: , :
87	instantiation	131, 94, 95	⊢
	: , : , :
88	instantiation	159	⊢
	: , : , :
89	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.e_is_real_pos
90	instantiation	152, 121, 179, 153, 122, 96, 154, 157, 158, 163, 164, 126, 156	⊢
	: , : , : , : , : , :
91	instantiation	131, 97, 98	⊢
	: , : , :
92	instantiation	99, 108	⊢
	:
93	instantiation	140, 100, 101	⊢
	: , : , :
94	instantiation	152, 194, 144, 153, 102, 154, 143, 163, 156, 164	⊢
	: , : , : , : , : , :
95	instantiation	152, 153, 179, 144, 154, 145, 102, 157, 158, 163, 156, 164	⊢
	: , : , : , : , : , :
96	instantiation	165	⊢
	: , :
97	instantiation	103, 104	⊢
	: , : , :
98	instantiation	105, 106, 107, 108, 109*	⊢
	: , : , :
99	conjecture		⊢
	proveit.numbers.division.frac_one_denom
100	instantiation	162, 110, 164	⊢
	: , :
101	instantiation	152, 153, 179, 194, 154, 111, 163, 156, 164	⊢
	: , : , : , : , : , :
102	instantiation	159	⊢
	: , : , :
103	axiom		⊢
	proveit.logic.equality.substitution
104	instantiation	131, 112, 113	⊢
	: , : , :
105	conjecture		⊢
	proveit.numbers.division.frac_cancel_left
106	instantiation	192, 115, 114	⊢
	: , : , :
107	instantiation	192, 115, 116	⊢
	: , : , :
108	instantiation	140, 117, 118	⊢
	: , : , :
109	instantiation	119, 126	⊢
	:
110	instantiation	162, 163, 156	⊢
	: , :
111	instantiation	165	⊢
	: , :
112	instantiation	120, 153, 121, 194, 154, 122, 157, 158, 163, 164, 126, 156	⊢
	: , : , : , : , : , : , :
113	instantiation	123, 194, 124, 153, 125, 154, 126, 157, 158, 163, 164, 156	⊢
	: , : , : , : , : , :
114	instantiation	192, 128, 127	⊢
	: , : , :
115	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
116	instantiation	192, 128, 129	⊢
	: , : , :
117	instantiation	162, 143, 130	⊢
	: , :
118	instantiation	131, 132, 133	⊢
	: , : , :
119	conjecture		⊢
	proveit.numbers.multiplication.elim_one_right
120	conjecture		⊢
	proveit.numbers.multiplication.leftward_commutation
121	conjecture		⊢
	proveit.numbers.numerals.decimals.nat4
122	instantiation	134	⊢
	: , : , : , :
123	conjecture		⊢
	proveit.numbers.multiplication.association
124	conjecture		⊢
	proveit.numbers.numerals.decimals.nat5
125	instantiation	135	⊢
	: , : , : , : , :
126	instantiation	192, 169, 136	⊢
	: , : , :
127	instantiation	192, 138, 137	⊢
	: , : , :
128	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
129	instantiation	192, 138, 139	⊢
	: , : , :
130	instantiation	140, 141, 142	⊢
	: , : , :
131	axiom		⊢
	proveit.logic.equality.equals_transitivity
132	instantiation	152, 194, 144, 153, 146, 154, 143, 163, 164, 156	⊢
	: , : , : , : , : , :
133	instantiation	152, 153, 179, 144, 154, 145, 146, 157, 158, 163, 164, 156	⊢
	: , : , : , : , : , :
134	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_4_typical_eq
135	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_5_typical_eq
136	instantiation	147, 148, 189	⊢
	: , : , :
137	instantiation	192, 149, 189	⊢
	: , : , :
138	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
139	instantiation	192, 149, 150	⊢
	: , : , :
140	theorem		⊢
	proveit.logic.equality.sub_right_side_into
141	instantiation	162, 151, 156	⊢
	: , :
142	instantiation	152, 153, 179, 194, 154, 155, 163, 164, 156	⊢
	: , : , : , : , : , :
143	instantiation	162, 157, 158	⊢
	: , :
144	conjecture		⊢
	proveit.numbers.numerals.decimals.nat3
145	instantiation	165	⊢
	: , :
146	instantiation	159	⊢
	: , : , :
147	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
148	instantiation	160, 161	⊢
	: , :
149	conjecture		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
150	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat1
151	instantiation	162, 163, 164	⊢
	: , :
152	conjecture		⊢
	proveit.numbers.multiplication.disassociation
153	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
154	conjecture		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
155	instantiation	165	⊢
	: , :
156	instantiation	192, 169, 166	⊢
	: , : , :
157	instantiation	192, 169, 167	⊢
	: , : , :
158	instantiation	192, 169, 168	⊢
	: , : , :
159	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
160	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
161	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
162	conjecture		⊢
	proveit.numbers.multiplication.mult_complex_closure_bin
163	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.i_is_complex
164	instantiation	192, 169, 170	⊢
	: , : , :
165	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
166	conjecture		⊢
	proveit.physics.quantum.QPE._phase_is_real
167	instantiation	192, 174, 171	⊢
	: , : , :
168	instantiation	192, 172, 173	⊢
	: , : , :
169	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
170	instantiation	192, 174, 175	⊢
	: , : , :
171	instantiation	192, 177, 176	⊢
	: , : , :
172	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
173	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
174	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
175	instantiation	192, 177, 178	⊢
	: , : , :
176	instantiation	192, 193, 179	⊢
	: , : , :
177	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
178	instantiation	192, 180, 181	⊢
	: , : , :
179	conjecture		⊢
	proveit.numbers.numerals.decimals.nat2
180	instantiation	182, 183, 184	⊢
	: , :
181	assumption		⊢
182	conjecture		⊢
	proveit.numbers.number_sets.integers.int_interval_within_int
183	conjecture		⊢
	proveit.numbers.number_sets.integers.zero_is_int
184	instantiation	185, 186, 187	⊢
	: , :
185	conjecture		⊢
	proveit.numbers.addition.add_int_closure_bin
186	instantiation	192, 188, 189	⊢
	: , : , :
187	instantiation	190, 191	⊢
	:
188	conjecture		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
189	conjecture		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
190	conjecture		⊢
	proveit.numbers.negation.int_closure
191	instantiation	192, 193, 194	⊢
	: , : , :
192	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
193	conjecture		⊢
	proveit.numbers.number_sets.integers.nat_within_int
194	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements