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

import proveit
theory = proveit.Theory() # the theorem's theory
from proveit import a, b, x
from proveit.logic import InSet
from proveit.numbers import (zero, one, two, five, Add, Exp, frac, greater,
                             Integer, Neg, Real, RealPos, subtract)
# QPE common expressions
from proveit.physics.quantum.QPE import _t, _n
# QPE axioms
from proveit.physics.quantum.QPE import (
        _eps_in_interval, _n_in_natural_pos, _t_in_natural_pos, _t_req)

%proving _e_value_ge_two

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

_t_in_natural_pos

 ⊢  

_n_in_natural_pos

 ⊢  

_eps_in_interval

 ⊢  

_t_req

 ⊢  

bound_01 = _t_req.right_add_both_sides(Neg(_n))

bound_01:  ⊢  

_eps_le_one = _eps_in_interval.derive_element_upper_bound()

_eps_le_one:  ⊢  

bound_02 = bound_01.rhs.deduce_bound(_eps_le_one)

bound_02:  ⊢  

bound_03 = bound_01.apply_transitivity(bound_02)

bound_03:  ⊢  

Note that ${\rm log}_2(x) > 1$ if $x > 2$:

bound_04 = bound_03.rhs.deduce_bound(greater(frac(five, two), two))

bound_04:  ⊢  

bound_05 = bound_03.apply_transitivity(bound_04)

bound_05:  ⊢  

bound_06 = Exp(two, subtract(_t, _n)).deduce_bound(bound_05)

bound_06:  ⊢  

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

%qed

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

	step type	requirements	statement
0	instantiation	1, 2, 3, 4, 5, 6*, 7*	⊢
	: , : , :
1	reference	88	⊢
2	reference	113	⊢
3	reference	17	⊢
4	instantiation	112, 26, 242	⊢
	: , :
5	instantiation	55, 17, 26, 242, 8, 9	⊢
	: , : , :
6	instantiation	10, 185, 11, 12	⊢
	: , : , :
7	instantiation	172, 13, 14	⊢
	: , : , :
8	instantiation	15, 196, 56, 222, 57, 16*	⊢
	: , : , :
9	instantiation	184, 250	⊢
	:
10	conjecture		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
11	instantiation	248, 207, 17	⊢
	: , : , :
12	conjecture		⊢
	proveit.numbers.numerals.decimals.add_2_2
13	instantiation	130, 131, 221, 250, 132, 18, 21, 199, 79	⊢
	: , : , : , : , : , :
14	instantiation	19, 250, 221, 131, 20, 132, 21, 199, 79, 22*	⊢
	: , : , : , : , : , :
15	conjecture		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
16	instantiation	65, 23, 24, 189	⊢
	: , : , : , :
17	instantiation	248, 244, 25	⊢
	: , : , :
18	instantiation	197	⊢
	: , :
19	conjecture		⊢
	proveit.numbers.addition.association
20	instantiation	197	⊢
	: , :
21	instantiation	248, 207, 26	⊢
	: , : , :
22	instantiation	27, 199, 185, 71	⊢
	: , : , :
23	instantiation	28, 238, 185	⊢
	: , :
24	instantiation	29, 221, 30, 31, 32	⊢
	: , : , : , :
25	instantiation	248, 246, 33	⊢
	: , : , :
26	instantiation	34, 196, 35	⊢
	: , :
27	conjecture		⊢
	proveit.numbers.addition.subtraction.subtract_from_add_reversed
28	conjecture		⊢
	proveit.numbers.exponentiation.exp_nat_pos_expansion
29	conjecture		⊢
	proveit.core_expr_types.operations.operands_substitution_via_tuple
30	instantiation	36, 221	⊢
	: , :
31	instantiation	197	⊢
	: , :
32	instantiation	37	⊢
	:
33	instantiation	248, 249, 38	⊢
	: , : , :
34	conjecture		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
35	instantiation	39, 40, 41	⊢
	:
36	conjecture		⊢
	proveit.core_expr_types.tuples.range_from1_len_typical_eq
37	conjecture		⊢
	proveit.numbers.numerals.decimals.reduce_2_repeats
38	conjecture		⊢
	proveit.numbers.numerals.decimals.nat4
39	conjecture		⊢
	proveit.numbers.number_sets.integers.nonneg_int_is_natural
40	instantiation	42, 43, 44	⊢
	: , :
41	instantiation	45, 46	⊢
	: , :
42	conjecture		⊢
	proveit.numbers.addition.add_int_closure_bin
43	instantiation	248, 47, 139	⊢
	: , : , :
44	instantiation	248, 48, 49	⊢
	: , : , :
45	conjecture		⊢
	proveit.numbers.addition.subtraction.nonneg_difference
46	instantiation	88, 92, 196, 50, 51, 52*, 53*	⊢
	: , : , :
47	conjecture		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
48	conjecture		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
49	instantiation	54, 193	⊢
	:
50	instantiation	112, 56, 196	⊢
	: , :
51	instantiation	55, 196, 56, 57, 169	⊢
	: , : , :
52	instantiation	172, 58, 59	⊢
	: , : , :
53	instantiation	172, 60, 61	⊢
	: , : , :
54	conjecture		⊢
	proveit.numbers.negation.int_neg_closure
55	conjecture		⊢
	proveit.numbers.ordering.less_eq_add_right
56	instantiation	112, 114, 157	⊢
	: , :
57	instantiation	72, 62, 63	⊢
	: , : , :
58	instantiation	130, 250, 221, 131, 78, 132, 185, 136, 79	⊢
	: , : , : , : , : , :
59	instantiation	135, 185, 136, 97	⊢
	: , : , :
60	instantiation	144, 64	⊢
	: , : , :
61	instantiation	65, 66, 67, 68	⊢
	: , : , : , :
62	instantiation	69, 247, 86, 70, 71*	⊢
	: , :
63	instantiation	72, 73, 74	⊢
	: , : , :
64	instantiation	130, 131, 221, 250, 132, 96, 99, 134, 185	⊢
	: , : , : , : , : , :
65	conjecture		⊢
	proveit.logic.equality.four_chain_transitivity
66	instantiation	130, 131, 76, 250, 132, 77, 99, 134, 185, 75	⊢
	: , : , : , : , : , :
67	instantiation	130, 76, 221, 131, 77, 78, 132, 99, 134, 185, 136, 79	⊢
	: , : , : , : , : , :
68	instantiation	172, 80, 81	⊢
	: , : , :
69	conjecture		⊢
	proveit.numbers.rounding.ceil_of_real_above_int
70	instantiation	82, 230, 108, 83, 222, 84*	⊢
	: , : , :
71	conjecture		⊢
	proveit.numbers.numerals.decimals.add_1_1
72	conjecture		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
73	instantiation	85, 86, 203, 87	⊢
	: , :
74	instantiation	88, 157, 89, 114, 90, 91*	⊢
	: , : , :
75	instantiation	248, 207, 92	⊢
	: , : , :
76	conjecture		⊢
	proveit.numbers.numerals.decimals.nat3
77	instantiation	93	⊢
	: , : , :
78	instantiation	197	⊢
	: , :
79	instantiation	94, 185	⊢
	:
80	instantiation	95, 221, 250, 131, 96, 132, 99, 134, 185, 136, 97	⊢
	: , : , : , : , : , : , : , :
81	instantiation	98, 136, 99, 138	⊢
	: , : , :
82	conjecture		⊢
	proveit.numbers.logarithms.log_increasing_less
83	instantiation	100, 196, 101, 102, 103, 104*, 105*	⊢
	: , : , :
84	instantiation	106, 230	⊢
	:
85	conjecture		⊢
	proveit.numbers.rounding.ceil_increasing_less_eq
86	instantiation	208, 230, 108, 210	⊢
	: , :
87	instantiation	107, 230, 108, 209, 109, 222	⊢
	: , : , :
88	conjecture		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
89	instantiation	112, 177, 158	⊢
	: , :
90	axiom		⊢
	proveit.physics.quantum.QPE._t_req
91	instantiation	172, 110, 111	⊢
	: , : , :
92	instantiation	112, 177, 113	⊢
	: , :
93	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
94	conjecture		⊢
	proveit.numbers.negation.complex_closure
95	conjecture		⊢
	proveit.numbers.addition.subtraction.add_cancel_general
96	instantiation	197	⊢
	: , :
97	instantiation	159	⊢
	:
98	conjecture		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_32
99	instantiation	248, 207, 114	⊢
	: , : , :
100	conjecture		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
101	instantiation	115, 179, 241	⊢
	: , :
102	instantiation	248, 244, 116	⊢
	: , : , :
103	instantiation	117, 179, 241, 242, 118, 119	⊢
	: , : , :
104	instantiation	172, 120, 121	⊢
	: , : , :
105	instantiation	172, 122, 123	⊢
	: , : , :
106	conjecture		⊢
	proveit.numbers.logarithms.log_eq_1
107	conjecture		⊢
	proveit.numbers.logarithms.log_increasing_less_eq
108	instantiation	217, 124, 230, 125	⊢
	: , :
109	instantiation	126, 196, 179, 127, 128, 129*	⊢
	: , : , :
110	instantiation	130, 131, 221, 250, 132, 133, 136, 137, 134	⊢
	: , : , : , : , : , :
111	instantiation	135, 136, 137, 138	⊢
	: , : , :
112	conjecture		⊢
	proveit.numbers.addition.add_real_closure_bin
113	instantiation	176, 196	⊢
	:
114	instantiation	191, 192, 139	⊢
	: , : , :
115	conjecture		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
116	instantiation	140, 195, 245	⊢
	: , :
117	conjecture		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
118	conjecture		⊢
	proveit.numbers.numerals.decimals.less_0_1
119	instantiation	141, 205	⊢
	:
120	instantiation	144, 142	⊢
	: , : , :
121	instantiation	143, 185	⊢
	:
122	instantiation	144, 145	⊢
	: , : , :
123	instantiation	153, 247, 212, 154*, 146*, 170*	⊢
	: , : , : , :
124	instantiation	248, 232, 147	⊢
	: , : , :
125	instantiation	148, 238	⊢
	:
126	conjecture		⊢
	proveit.numbers.addition.weak_bound_via_right_term_bound
127	instantiation	248, 149, 214	⊢
	: , : , :
128	instantiation	150, 151, 228, 230, 152	⊢
	: , : , :
129	instantiation	153, 212, 247, 154*, 155*, 156*	⊢
	: , : , : , :
130	conjecture		⊢
	proveit.numbers.addition.disassociation
131	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
132	conjecture		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
133	instantiation	197	⊢
	: , :
134	instantiation	248, 207, 157	⊢
	: , : , :
135	conjecture		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
136	instantiation	248, 207, 177	⊢
	: , : , :
137	instantiation	248, 207, 158	⊢
	: , : , :
138	instantiation	159	⊢
	:
139	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
140	conjecture		⊢
	proveit.numbers.multiplication.mult_rational_closure_bin
141	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
142	instantiation	160, 161	⊢
	:
143	conjecture		⊢
	proveit.numbers.addition.elim_zero_left
144	axiom		⊢
	proveit.logic.equality.substitution
145	instantiation	198, 161	⊢
	:
146	instantiation	172, 162, 163	⊢
	: , : , :
147	instantiation	248, 237, 164	⊢
	: , : , :
148	conjecture		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
149	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
150	conjecture		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
151	instantiation	248, 165, 166	⊢
	: , : , :
152	instantiation	167, 196, 235, 242, 168, 169, 170*	⊢
	: , : , :
153	conjecture		⊢
	proveit.numbers.addition.rational_pair_addition
154	instantiation	171, 185	⊢
	:
155	instantiation	172, 173, 174	⊢
	: , : , :
156	instantiation	175, 185	⊢
	:
157	instantiation	176, 177	⊢
	:
158	instantiation	248, 244, 178	⊢
	: , : , :
159	axiom		⊢
	proveit.logic.equality.equals_reflexivity
160	conjecture		⊢
	proveit.numbers.multiplication.mult_zero_right
161	instantiation	248, 207, 179	⊢
	: , : , :
162	instantiation	186, 221, 180, 181, 190, 189	⊢
	: , : , : , :
163	conjecture		⊢
	proveit.numbers.numerals.decimals.add_1_4
164	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat5
165	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
166	instantiation	248, 182, 250	⊢
	: , : , :
167	conjecture		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
168	instantiation	183, 241, 242, 243	⊢
	: , : , :
169	instantiation	184, 221	⊢
	:
170	instantiation	198, 185	⊢
	:
171	conjecture		⊢
	proveit.numbers.division.frac_one_denom
172	axiom		⊢
	proveit.logic.equality.equals_transitivity
173	instantiation	186, 221, 187, 188, 189, 190	⊢
	: , : , : , :
174	conjecture		⊢
	proveit.numbers.numerals.decimals.add_4_1
175	conjecture		⊢
	proveit.numbers.multiplication.elim_one_left
176	conjecture		⊢
	proveit.numbers.negation.real_closure
177	instantiation	191, 192, 193	⊢
	: , : , :
178	instantiation	248, 246, 194	⊢
	: , : , :
179	instantiation	248, 244, 195	⊢
	: , : , :
180	instantiation	197	⊢
	: , :
181	instantiation	197	⊢
	: , :
182	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
183	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_upper_bound
184	conjecture		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
185	instantiation	248, 207, 196	⊢
	: , : , :
186	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
187	instantiation	197	⊢
	: , :
188	instantiation	197	⊢
	: , :
189	conjecture		⊢
	proveit.numbers.numerals.decimals.mult_2_2
190	instantiation	198, 199	⊢
	:
191	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
192	instantiation	200, 201	⊢
	: , :
193	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
194	instantiation	202, 203	⊢
	:
195	instantiation	248, 204, 205	⊢
	: , : , :
196	instantiation	248, 244, 206	⊢
	: , : , :
197	conjecture		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
198	conjecture		⊢
	proveit.numbers.multiplication.elim_one_right
199	instantiation	248, 207, 242	⊢
	: , : , :
200	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
201	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
202	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
203	instantiation	208, 230, 209, 210	⊢
	: , :
204	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
205	instantiation	211, 223, 233	⊢
	: , :
206	instantiation	248, 246, 212	⊢
	: , : , :
207	conjecture		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
208	conjecture		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
209	instantiation	213, 230, 214	⊢
	: , :
210	instantiation	215, 216	⊢
	: , :
211	conjecture		⊢
	proveit.numbers.division.div_rational_pos_closure
212	instantiation	248, 249, 221	⊢
	: , : , :
213	conjecture		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
214	instantiation	217, 218, 228, 219	⊢
	: , :
215	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
216	instantiation	220, 250, 221, 222	⊢
	: , :
217	conjecture		⊢
	proveit.numbers.division.div_real_pos_closure
218	instantiation	248, 232, 223	⊢
	: , : , :
219	instantiation	224, 225	⊢
	:
220	conjecture		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
221	conjecture		⊢
	proveit.numbers.numerals.decimals.nat2
222	conjecture		⊢
	proveit.numbers.numerals.decimals.less_1_2
223	instantiation	248, 237, 226	⊢
	: , : , :
224	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
225	instantiation	248, 227, 228	⊢
	: , : , :
226	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat1
227	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
228	instantiation	229, 230, 231	⊢
	: , :
229	conjecture		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
230	instantiation	248, 232, 233	⊢
	: , : , :
231	instantiation	234, 235, 236	⊢
	:
232	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
233	instantiation	248, 237, 238	⊢
	: , : , :
234	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
235	instantiation	239, 241, 242, 243	⊢
	: , : , :
236	instantiation	240, 241, 242, 243	⊢
	: , : , :
237	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
238	conjecture		⊢
	proveit.numbers.numerals.decimals.posnat2
239	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
240	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
241	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
242	instantiation	248, 244, 245	⊢
	: , : , :
243	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
244	conjecture		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
245	instantiation	248, 246, 247	⊢
	: , : , :
246	conjecture		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
247	instantiation	248, 249, 250	⊢
	: , : , :
248	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
249	conjecture		⊢
	proveit.numbers.number_sets.integers.nat_within_int
250	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements