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In [1]:
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
Out[1]:
 step typerequirementsstatement
0modus ponens1, 2  ⊢  
1instantiation3, 26, 133, 81, 61, 82, 4*  ⊢  
  : , : , : , : , : , : , : , :
2instantiation50, 5, 6  ⊢  
  : , : , :
3theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_distribution_over_summation
4instantiation64, 7, 8  ⊢  
  : , : , :
5instantiation46, 89, 90, 57, 9  ⊢  
  : , : , : , :
6instantiation59, 60, 61  ⊢  
  : , : , :
7instantiation10, 11, 12*, 13*  ⊢  
  : , : , :
8modus ponens14, 15  ⊢  
9modus ponens16, 17  ⊢  
10axiom  ⊢  
 proveit.logic.equality.substitution
11modus ponens18, 19  ⊢  
12instantiation20, 121  ⊢  
  : , :
13instantiation20, 121  ⊢  
  : , :
14instantiation21, 26  ⊢  
  : , : , : , :
15generalization22  ⊢  
16instantiation23, 26, 29  ⊢  
  : , : , : , : , : , :
17generalization24  ⊢  
18instantiation25, 26  ⊢  
  : , : , : , : , : , : , :
19generalization27  ⊢  
20theorem  ⊢  
 proveit.core_expr_types.conditionals.satisfied_condition_reduction
21axiom  ⊢  
 proveit.linear_algebra.addition.scalar_sum_extends_number_sum
22instantiation94, 44, 45,  ⊢  
  : , :
23theorem  ⊢  
 proveit.linear_algebra.addition.summation_closure
24instantiation28, 29, 44, 58  ⊢  
  : , : , : , :
25theorem  ⊢  
 proveit.core_expr_types.lambda_maps.general_lambda_substitution
26theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat1
27instantiation30, 31, 32, 33,  ⊢  
  : , : , : , :
28theorem  ⊢  
 proveit.linear_algebra.scalar_multiplication.scalar_mult_closure
29instantiation34, 115  ⊢  
  :
30theorem  ⊢  
 proveit.logic.equality.four_chain_transitivity
31instantiation35, 44, 133, 81, 61, 82, 40,  ⊢  
  : , : , : , : , : , :
32instantiation36, 133, 138, 44, 61, 40,  ⊢  
  : , : , : , : , :
33instantiation37, 38, 44, 39, 40, 41*,  ⊢  
  : , : , :
34theorem  ⊢  
 proveit.linear_algebra.complex_vec_set_is_vec_space
35theorem  ⊢  
 proveit.physics.quantum.algebra.scalar_mult_absorption
36theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_pulling_scalar_out_front
37theorem  ⊢  
 proveit.physics.quantum.algebra.scalar_mult_factorization
38theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat3
39instantiation87  ⊢  
  : , : , :
40instantiation50, 42, 52,  ⊢  
  : , : , :
41instantiation43, 44, 45,  ⊢  
  : , :
42instantiation46, 89, 90, 57, 58,  ⊢  
  : , : , : , :
43axiom  ⊢  
 proveit.linear_algebra.scalar_multiplication.scalar_mult_extends_number_mult
44instantiation47, 48, 49  ⊢  
  : , :
45instantiation50, 51, 52,  ⊢  
  : , : , :
46theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_op_ket_in_QmultCodomain
47theorem  ⊢  
 proveit.numbers.exponentiation.exp_complex_closure
48instantiation136, 107, 53  ⊢  
  : , : , :
49instantiation69, 54, 55  ⊢  
  : , : , :
50theorem  ⊢  
 proveit.logic.equality.sub_left_side_into
51instantiation56, 89, 90, 57, 58,  ⊢  
  : , : , : , :
52instantiation59, 60, 61  ⊢  
  : , : , :
53instantiation136, 112, 62  ⊢  
  : , : , :
54instantiation94, 72, 63  ⊢  
  : , :
55instantiation64, 65, 66  ⊢  
  : , : , :
56theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_op_ket_is_ket
57instantiation88, 89, 90, 67  ⊢  
  : , : , :
58instantiation68, 135, 121  ⊢  
  : , :
59axiom  ⊢  
 proveit.physics.quantum.algebra.multi_qmult_def
60theorem  ⊢  
 proveit.numbers.numerals.decimals.posnat2
61instantiation97  ⊢  
  : , :
62theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.e_is_real_pos
63instantiation69, 70, 71  ⊢  
  : , : , :
64axiom  ⊢  
 proveit.logic.equality.equals_transitivity
65instantiation80, 138, 73, 81, 75, 82, 72, 95, 96, 84  ⊢  
  : , : , : , : , : , :
66instantiation80, 81, 133, 73, 82, 74, 75, 85, 86, 95, 96, 84  ⊢  
  : , : , : , : , : , :
67instantiation76, 89, 90, 77, 78  ⊢  
  : , : , : , : , :
68theorem  ⊢  
 proveit.physics.quantum.algebra.num_ket_in_register_space
69theorem  ⊢  
 proveit.logic.equality.sub_right_side_into
70instantiation94, 79, 84  ⊢  
  : , :
71instantiation80, 81, 133, 138, 82, 83, 95, 96, 84  ⊢  
  : , : , : , : , : , :
72instantiation94, 85, 86  ⊢  
  : , :
73theorem  ⊢  
 proveit.numbers.numerals.decimals.nat3
74instantiation97  ⊢  
  : , :
75instantiation87  ⊢  
  : , : , :
76theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_op_op_is_op
77instantiation88, 89, 90, 91  ⊢  
  : , : , :
78instantiation92, 115, 93  ⊢  
  : , : , :
79instantiation94, 95, 96  ⊢  
  : , :
80theorem  ⊢  
 proveit.numbers.multiplication.disassociation
81axiom  ⊢  
 proveit.numbers.number_sets.natural_numbers.zero_in_nats
82theorem  ⊢  
 proveit.core_expr_types.tuples.tuple_len_0_typical_eq
83instantiation97  ⊢  
  : , :
84instantiation136, 107, 98  ⊢  
  : , : , :
85instantiation136, 107, 99  ⊢  
  : , : , :
86instantiation136, 107, 100  ⊢  
  : , : , :
87theorem  ⊢  
 proveit.numbers.numerals.decimals.tuple_len_3_typical_eq
88theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_op_is_linmap
89instantiation101, 115  ⊢  
  :
90theorem  ⊢  
 proveit.linear_algebra.inner_products.complex_set_is_hilbert_space
91instantiation102, 135, 103  ⊢  
  : , :
92theorem  ⊢  
 proveit.physics.quantum.algebra.qmult_matrix_is_linmap
93instantiation104, 105, 106  ⊢  
  : , : , :
94theorem  ⊢  
 proveit.numbers.multiplication.mult_complex_closure_bin
95theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.i_is_complex
96instantiation136, 107, 108  ⊢  
  : , : , :
97theorem  ⊢  
 proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
98instantiation136, 110, 109  ⊢  
  : , : , :
99instantiation136, 110, 111  ⊢  
  : , : , :
100instantiation136, 112, 113  ⊢  
  : , : , :
101theorem  ⊢  
 proveit.linear_algebra.inner_products.complex_vec_set_is_hilbert_space
102theorem  ⊢  
 proveit.physics.quantum.algebra.num_bra_is_lin_map
103assumption  ⊢  
104theorem  ⊢  
 proveit.logic.sets.inclusion.unfold_subset_eq
105instantiation114, 115  ⊢  
  :
106instantiation116, 135  ⊢  
  :
107theorem  ⊢  
 proveit.numbers.number_sets.complex_numbers.real_within_complex
108theorem  ⊢  
 proveit.physics.quantum.QPE._phase_is_real
109instantiation136, 118, 117  ⊢  
  : , : , :
110theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.rational_within_real
111instantiation136, 118, 129  ⊢  
  : , : , :
112theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.real_pos_within_real
113theorem  ⊢  
 proveit.numbers.number_sets.real_numbers.pi_is_real_pos
114theorem  ⊢  
 proveit.linear_algebra.matrices.unitaries_are_matrices
115instantiation119, 133, 130  ⊢  
  : , :
116theorem  ⊢  
 proveit.physics.quantum.QFT.invFT_is_unitary
117instantiation136, 120, 121  ⊢  
  : , : , :
118theorem  ⊢  
 proveit.numbers.number_sets.rational_numbers.int_within_rational
119theorem  ⊢  
 proveit.numbers.exponentiation.exp_natpos_closure
120instantiation122, 123, 124  ⊢  
  : , :
121assumption  ⊢  
122theorem  ⊢  
 proveit.numbers.number_sets.integers.int_interval_within_int
123theorem  ⊢  
 proveit.numbers.number_sets.integers.zero_is_int
124instantiation125, 126, 127  ⊢  
  : , :
125theorem  ⊢  
 proveit.numbers.addition.add_int_closure_bin
126instantiation128, 129, 130  ⊢  
  : , :
127instantiation131, 132  ⊢  
  :
128theorem  ⊢  
 proveit.numbers.exponentiation.exp_int_closure
129instantiation136, 137, 133  ⊢  
  : , : , :
130instantiation136, 134, 135  ⊢  
  : , : , :
131theorem  ⊢  
 proveit.numbers.negation.int_closure
132instantiation136, 137, 138  ⊢  
  : , : , :
133theorem  ⊢  
 proveit.numbers.numerals.decimals.nat2
134theorem  ⊢  
 proveit.numbers.number_sets.natural_numbers.nat_pos_within_nat
135axiom  ⊢  
 proveit.physics.quantum.QPE._t_in_natural_pos
136theorem  ⊢  
 proveit.logic.sets.inclusion.superset_membership_from_proper_subset
137theorem  ⊢  
 proveit.numbers.number_sets.integers.nat_within_int
138theorem  ⊢  
 proveit.numbers.numerals.decimals.nat1
*equality replacement requirements