Show the Proof

import proveit
# Automation is not needed when only showing a stored proof:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%show_proof

	step type	requirements	statement
0	instantiation	1, 2, 3	⊢
	:
1	reference	173	⊢
2	instantiation	213, 205, 7	⊢
	: , : , :
3	instantiation	4, 5	⊢
	: , :
4	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
5	instantiation	6, 34, 7, 8	⊢
	: , :
6	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
7	instantiation	9, 34, 177, 11	⊢
	: , : , :
8	instantiation	10, 34, 177, 11	⊢
	: , : , :
9	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
10	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
11	instantiation	12, 13	⊢
	:
12	theorem		⊢
	proveit.trigonometry.sine_pos_interval
13	instantiation	14, 34, 102, 15, 16	⊢
	: , : , :
14	theorem		⊢
	proveit.numbers.number_sets.real_numbers.in_IntervalOO
15	instantiation	213, 129, 20	⊢
	: , : , :
16	instantiation	17, 18, 19	⊢
	: , :
17	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
18	instantiation	89, 20	⊢
	:
19	instantiation	21, 22, 23	⊢
	: , : , :
20	instantiation	24, 128, 130	⊢
	: , :
21	axiom		⊢
	proveit.numbers.ordering.transitivity_less_less
22	instantiation	62, 25, 43	⊢
	: , : , :
23	instantiation	73, 71, 26, 27, 28*, 29*	⊢
	: , : , :
24	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
25	instantiation	30, 31, 32	⊢
	: , : , :
26	instantiation	90, 91, 34	⊢
	: , :
27	instantiation	33, 91, 34, 102, 65, 35	⊢
	: , : , :
28	instantiation	163, 36, 37	⊢
	: , : , :
29	instantiation	38, 39, 40*	⊢
	: , :
30	theorem		⊢
	proveit.numbers.ordering.transitivity_less_less_eq
31	instantiation	41, 42, 43	⊢
	: , : , :
32	instantiation	44, 102, 45, 158, 46, 47, 48*, 49*	⊢
	: , : , :
33	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_right_factor_bound
34	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
35	instantiation	50, 140	⊢
	:
36	instantiation	104, 51	⊢
	: , : , :
37	instantiation	52, 53	⊢
	:
38	theorem		⊢
	proveit.logic.equality.equals_reversal
39	instantiation	54, 121, 215, 208, 122, 55, 70, 79	⊢
	: , : , : , : , : , :
40	instantiation	163, 56, 57	⊢
	: , : , :
41	theorem		⊢
	proveit.logic.equality.sub_left_side_into
42	instantiation	58, 208, 128, 130, 59, 60*	⊢
	: , : , : , : , : , :
43	instantiation	104, 61	⊢
	: , : , :
44	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
45	instantiation	90, 153, 103	⊢
	: , :
46	instantiation	62, 63, 64	⊢
	: , : , :
47	instantiation	137, 65	⊢
	: , :
48	instantiation	66, 208, 215, 121, 67, 122, 79, 126, 80	⊢
	: , : , : , : , : , :
49	instantiation	68, 79, 70	⊢
	: , :
50	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.positive_if_in_rational_pos
51	instantiation	69, 70	⊢
	:
52	theorem		⊢
	proveit.numbers.addition.elim_zero_left
53	instantiation	213, 205, 71	⊢
	: , : , :
54	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
55	instantiation	196	⊢
	: , :
56	instantiation	104, 72	⊢
	: , : , :
57	instantiation	197, 79	⊢
	:
58	theorem		⊢
	proveit.numbers.multiplication.strong_bound_via_factor_bound
59	instantiation	73, 152, 206, 74, 75, 76*, 77*	⊢
	: , : , :
60	instantiation	78, 208, 79, 80	⊢
	: , : , : , :
61	instantiation	163, 81, 82	⊢
	: , : , :
62	theorem		⊢
	proveit.logic.equality.sub_right_side_into
63	instantiation	83, 84, 85	⊢
	: , :
64	instantiation	86, 195, 87, 126, 174, 88*	⊢
	: , :
65	instantiation	89, 128	⊢
	:
66	theorem		⊢
	proveit.numbers.multiplication.disassociation
67	instantiation	196	⊢
	: , :
68	theorem		⊢
	proveit.numbers.multiplication.commutation
69	theorem		⊢
	proveit.numbers.multiplication.mult_zero_right
70	instantiation	213, 205, 158	⊢
	: , : , :
71	instantiation	90, 91, 102	⊢
	: , :
72	instantiation	92, 204, 212, 93*	⊢
	: , : , : , :
73	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
74	instantiation	213, 209, 94	⊢
	: , : , :
75	instantiation	95, 206, 153, 177, 96, 97	⊢
	: , : , :
76	instantiation	98, 132, 199, 99	⊢
	: , : , :
77	instantiation	163, 100, 101	⊢
	: , : , :
78	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
79	instantiation	213, 205, 102	⊢
	: , : , :
80	instantiation	213, 205, 103	⊢
	: , : , :
81	instantiation	104, 105	⊢
	: , : , :
82	instantiation	106, 174	⊢
	:
83	theorem		⊢
	proveit.numbers.absolute_value.weak_upper_bound
84	instantiation	107, 135, 158, 136	⊢
	: , : , :
85	instantiation	137, 108	⊢
	: , :
86	theorem		⊢
	proveit.numbers.absolute_value.abs_prod
87	instantiation	196	⊢
	: , :
88	instantiation	109, 110	⊢
	:
89	theorem		⊢
	proveit.numbers.number_sets.real_numbers.positive_if_in_real_pos
90	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
91	instantiation	213, 209, 111	⊢
	: , : , :
92	theorem		⊢
	proveit.numbers.addition.rational_pair_addition
93	instantiation	163, 112, 113	⊢
	: , : , :
94	instantiation	213, 211, 114	⊢
	: , : , :
95	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
96	instantiation	115, 206, 177, 116, 117, 118, 119*	⊢
	: , : , :
97	theorem		⊢
	proveit.numbers.numerals.decimals.less_0_1
98	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
99	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
100	instantiation	120, 121, 215, 208, 122, 123, 126, 132, 124	⊢
	: , : , : , : , : , :
101	instantiation	125, 132, 126, 127	⊢
	: , : , :
102	instantiation	213, 129, 128	⊢
	: , : , :
103	instantiation	213, 129, 130	⊢
	: , : , :
104	axiom		⊢
	proveit.logic.equality.substitution
105	instantiation	131, 132, 174, 133*	⊢
	: , :
106	theorem		⊢
	proveit.numbers.absolute_value.abs_even
107	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_lower_bound
108	instantiation	134, 135, 158, 136	⊢
	: , : , :
109	theorem		⊢
	proveit.numbers.absolute_value.abs_non_neg_elim
110	instantiation	137, 138	⊢
	: , :
111	instantiation	213, 139, 140	⊢
	: , : , :
112	instantiation	181, 215, 141, 142, 143, 144	⊢
	: , : , : , :
113	instantiation	145, 146, 147	⊢
	:
114	instantiation	213, 214, 148	⊢
	: , : , :
115	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
116	instantiation	171, 172, 150	⊢
	: , : , :
117	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
118	instantiation	149, 150	⊢
	:
119	instantiation	151, 199	⊢
	:
120	theorem		⊢
	proveit.numbers.addition.disassociation
121	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
122	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
123	instantiation	196	⊢
	: , :
124	instantiation	213, 205, 152	⊢
	: , : , :
125	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
126	instantiation	213, 205, 153	⊢
	: , : , :
127	instantiation	154	⊢
	:
128	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pi_is_real_pos
129	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
130	instantiation	155, 156	⊢
	:
131	theorem		⊢
	proveit.numbers.multiplication.mult_neg_left
132	instantiation	213, 205, 177	⊢
	: , : , :
133	instantiation	197, 174	⊢
	:
134	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_co_upper_bound
135	instantiation	157, 158	⊢
	:
136	theorem		⊢
	proveit.physics.quantum.QPE._scaled_delta_b_round_in_interval
137	theorem		⊢
	proveit.numbers.ordering.relax_less
138	instantiation	159, 188	⊢
	:
139	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational
140	instantiation	160, 161, 162	⊢
	: , :
141	instantiation	196	⊢
	: , :
142	instantiation	196	⊢
	: , :
143	instantiation	163, 164, 165	⊢
	: , : , :
144	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
145	theorem		⊢
	proveit.numbers.division.frac_cancel_complete
146	instantiation	213, 205, 166	⊢
	: , : , :
147	instantiation	194, 167	⊢
	:
148	instantiation	168, 169, 208	⊢
	: , :
149	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_lower_bound
150	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
151	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
152	instantiation	213, 209, 170	⊢
	: , : , :
153	instantiation	171, 172, 188	⊢
	: , : , :
154	axiom		⊢
	proveit.logic.equality.equals_reflexivity
155	theorem		⊢
	proveit.numbers.absolute_value.abs_nonzero_closure
156	instantiation	173, 174, 175	⊢
	:
157	theorem		⊢
	proveit.numbers.negation.real_closure
158	instantiation	176, 177, 206, 178	⊢
	: , :
159	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
160	theorem		⊢
	proveit.numbers.division.div_rational_pos_closure
161	instantiation	213, 180, 179	⊢
	: , : , :
162	instantiation	213, 180, 195	⊢
	: , : , :
163	axiom		⊢
	proveit.logic.equality.equals_transitivity
164	instantiation	181, 215, 182, 183, 184, 185	⊢
	: , : , : , :
165	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_2
166	instantiation	213, 209, 186	⊢
	: , : , :
167	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
168	theorem		⊢
	proveit.numbers.addition.add_nat_closure_bin
169	instantiation	213, 187, 188	⊢
	: , : , :
170	instantiation	213, 211, 189	⊢
	: , : , :
171	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
172	instantiation	190, 191	⊢
	: , :
173	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
174	instantiation	213, 205, 192	⊢
	: , : , :
175	assumption		⊢
176	theorem		⊢
	proveit.numbers.division.div_real_closure
177	instantiation	213, 209, 193	⊢
	: , : , :
178	instantiation	194, 195	⊢
	:
179	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
180	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
181	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
182	instantiation	196	⊢
	: , :
183	instantiation	196	⊢
	: , :
184	instantiation	197, 199	⊢
	:
185	instantiation	198, 199	⊢
	:
186	instantiation	213, 211, 200	⊢
	: , : , :
187	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
188	theorem		⊢
	proveit.physics.quantum.QPE._two_pow_t_is_nat_pos
189	instantiation	201, 204	⊢
	:
190	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
191	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
192	instantiation	202, 203	⊢
	:
193	instantiation	213, 211, 204	⊢
	: , : , :
194	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
195	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
196	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
197	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
198	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
199	instantiation	213, 205, 206	⊢
	: , : , :
200	instantiation	213, 214, 207	⊢
	: , : , :
201	theorem		⊢
	proveit.numbers.negation.int_closure
202	theorem		⊢
	proveit.physics.quantum.QPE._delta_b_is_real
203	theorem		⊢
	proveit.physics.quantum.QPE._best_round_is_int
204	instantiation	213, 214, 208	⊢
	: , : , :
205	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
206	instantiation	213, 209, 210	⊢
	: , : , :
207	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
208	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
209	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
210	instantiation	213, 211, 212	⊢
	: , : , :
211	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
212	instantiation	213, 214, 215	⊢
	: , : , :
213	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
214	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
215	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
*equality replacement requirements