name: math-comp_eval
num_files: 70
language: COQ
few_shot_data_path_for_retrieval: null
few_shot_metadata_filename_for_retrieval: null
dfs_data_path_for_retrieval: null
dfs_metadata_filename_for_retrieval: local.meta.json
theorem_cnt: 729
datasets:
- project: <path-to-repo>/math-comp/
  files:
  - path: mathcomp/solvable/frobenius.v
    theorems:
    - Frobenius_semiregularP
    - Frobenius_cent1_ker
    - prime_FrobeniusP
    - Frobenius_coprime_quotient
    - stab_semiprime
    - card_support_normedTI
    - injm_Frobenius
  - path: mathcomp/solvable/abelian.v
    theorems:
    - pnElemS
    - abelemS
    - pdiv_p_elt
    - exponent_morphim
    - rank_witness
    - Ohm_cont
    - quotient_rank_abelian
    - morphim_Mho
    - pnElem_prime
    - Mho_p_cycle
    - abelem_abelian
    - Mho0
    - injm_Ohm
    - extend_cyclic_Mho
  - path: mathcomp/ssreflect/order.v
    theorems:
    - max_idPl
    - max_minr
    - eq_meetl
    - lexx
    - idfun_is_bottom_morphism
    - diff_eq0
    - complI
    - le_sorted_leq_nth
    - anti
    - leUr
    - joins_sup_seq
    - comparable_ltP
    - subset_display
    - joinIBC
    - le_wval
    - sort_le_id
    - bigminIl
    - diffErcompl
    - lex1
    - le_path_pairwise
    - contra_leN
    - enum_rankK
    - bigmaxUr
    - lt_path_mask
    - diffIK
    - leUx
    - eq_ltLR
    - lt_sorted_ltn_nth
    - le0x
    - le_refl
    - compl1
    - lteif_andb
    - le_Rank
    - comparable_maxC
    - opred0
    - incomparable_ltF
    - ge_min_id
    - ltNge
    - enum_set1
    - lt_def
    - leBKU
    - lteifxx
    - lcmnn
    - bigmax_sup
    - contraTle
    - tnth_rcompl
    - minAC
    - maxxK
  - path: mathcomp/algebra/ssralg.v
    theorems:
    - subrI
    - mulIr
    - valM
    - mulVf
    - mulf_div
    - and_dnfP
    - natr_div
    - invr_neq0
    - pair_mul0r
    - rpredBr
    - ffun_mul_addr
    - commrV
    - mulrb
    - exprAC
    - opB
    - rpredN
    - divIf
    - eval_If
    - prodr_const_nat
    - holds_fsubst
    - divfI
    - can2_rmorphism
    - fieldP
    - rmorphMsign
    - Frobenius_aut_is_multiplicative
    - prodf_div
    - Frobenius_aut1
    - mulrBr
    - mulr_sumr
    - subrKA
  - path: mathcomp/algebra/finalg.v
    theorems:
    - unit_inv_proof
  - path: mathcomp/field/algC.v
    theorems:
    - floorC_itv
    - floorCpK
    - aut_Cint
    - CintEge0
    - algC_invaut_is_rmorphism
    - Cint1
    - conjL_K
    - rpred_Cint
    - mulVf
    - aut_Crat
    - mul1
    - eqCmodMr
  - path: mathcomp/ssreflect/prime.v
    theorems:
    - partn_part
    - logn_div
    - max_pdiv_leq
    - totient_count_coprime
    - divn_count_dvd
    - pnatM
    - partnX
    - eq_in_partn
    - dvdn_prime2
  - path: mathcomp/algebra/ssrnum.v
    theorems:
    - sgr_ge0
    - ger_real
    - nneg_addr_closed
    - leif_mean_square
    - normrX
    - pmulrn_rle0
    - ge0_cp
    - ltrDr
    - a1
    - real_ltr_distlBl
    - ReM
    - subr_ge0
    - ltrNl
    - leN_total
    - psumr_eq0
    - ltrn1
    - ler0P
    - comparabler0
    - negrE
    - sub_ge0
    - divC_Crect
    - ltr_nwDr
    - realBC
    - deg_le2_poly_le0
    - normC_Re_Im
    - unitf_gt0
    - ler_norm_sum
    - ltr_distlCBl
    - ltr_distlC
    - monic_Cauchy_bound
    - leif_AGM
    - invf_nle
    - ler_sum_nat
    - gerB_real
    - lerB_normD
    - ltr_pdivrMl
    - real_normK
    - Im_rootC_ge0
    - ReMil
    - realn_mono
    - addr_minr
    - subr_ge0
    - lt0N
    - ltr_sqrt
    - ltr_pwDl
    - real_oppr_closed
    - oppr_max
    - poly_ivt
    - normfV
    - root0C
    - mulr_sg_norm
    - real_arg_maxP
    - oppr_ge0
    - lt_trans
    - exprn_odd_lt0
    - Im_i
    - real_eqr_norm2
    - deg2_poly_min
    - psumr_neq0
    - conjCK
    - comparablerE
    - realE
    - nmulr_rle0
  - path: mathcomp/ssreflect/fintype.v
    theorems:
    - negb_exists
    - eq_liftF
    - ordS_inj
    - injF_onto
    - extremum_inP
    - card_bool
    - pickP
    - eq_proper_r
    - subset_catr
    - split_subproof
    - ltn_unsplit
    - proper_sub
    - cast_ord_comp
    - eq_proper
    - canF_eq
    - fintype_le1P
    - lshift_subproof
    - unsplitK
    - exists_eqP
    - arg_maxnP
    - forall_inPn
    - exists_inPP
    - proper_irrefl
    - cardUI
  - path: mathcomp/algebra/matrix.v
    theorems:
    - tr_submxrow
    - mxtrace_block
    - mul_mxrow_mxdiag
    - mxOver_mul_subproof
    - map_col'
    - block_mxEur
    - diag_mx_is_diag
    - comm_mx1
    - perm_mxEsub
    - drsubmxEsub
    - mulmx_rsub
    - diag_mx_is_semi_additive
    - col_mxEd
    - mxcolP
    - const_mx_is_semi_additive
    - det_scalar
    - is_scalar_mxP
    - map2_xrow
    - submxcolK
    - mul_mxblock_mxdiag
    - mul_delta_mx_0
    - tr_diag_mx
    - scalemx_eq0
    - mulmx1_min
    - col_mxrow
    - invmx1
    - tr_tperm_mx
    - mxrow_recl
  - path: mathcomp/ssreflect/ssrnat.v
    theorems:
    - leq_pmul2r
    - ltnS
    - ltn_half_double
    - ltnW_nhomo
    - addBnAC
    - subSn
    - uphalf_half
    - minnE
    - double_pred
    - leq_mono
    - leq_add2r
    - eqn_exp2l
    - multE
    - leqW_nmono
    - ex_maxn_subproof
    - ltn_ind
    - leqW_mono
    - leq_psubRL
    - addnBC
    - addnCB
    - halfK
    - geq_uphalf_double
    - odd_halfK
    - maxn_idPl
    - contra_not_ltn
    - nat_irrelevance
    - leqn0
    - ubnPgeq
    - leq_subCr
    - ltn_sub2lE
    - leq_sub2l
    - leq_add
    - addSn
  - path: mathcomp/character/mxrepresentation.v
    theorems:
    - card_linear_irr
    - rfix_mxP
    - mx_rsim_refl
    - mxtrace_sub_fact_mod
    - sum_mxsimple_direct_compl
    - irr_mx_sum
    - rstabs_group_set
    - val_genJmx
    - rstab_map
    - Clifford_simple
    - mx_iso_simple
    - gen_satP
    - in_submod_eq0
    - rcent_sub
    - in_gen0
    - subSocle_semisimple
    - rfix_morphim
    - morphpre_mx_irr
    - mxmodule_quo
    - principal_comp_key
    - irr_reprK
    - dom_hom_invmx
    - submod_mx_faithful
    - eqg_mx_irr
    - gen_mulDr
    - val_factmod_module
    - mxval_sub
    - section_module
    - rker_linear
    - gen_mx_faithful
    - rconj_mxE
    - gen_mulA
    - mx_rsim_in_submod
    - simple_Socle
  - path: mathcomp/algebra/poly.v
    theorems:
    - lreg_size
    - map_poly_eq0
    - deg2_poly_factor
    - leq_sizeP
    - derivnN
    - scale_polyE
    - size_poly_leq0P
    - comm_poly0
    - coef_prod_XsubC
    - coefN
    - root1
    - size_poly_eq0
    - coefMX
    - drop_poly_eq0
    - add_polyC
    - map_poly_eq0_id0
    - hornerM
    - map_comp_poly
    - size_poly_leq0
    - comp_polyA
    - deriv_is_linear
    - drop_polyD
    - coef_sumMXn
    - horner_coef
    - lead_coefXaddC
    - closed_nonrootP
    - coef_add_poly
    - size_Msign
    - closed_field_poly_normal
    - polyC_exp
    - take_polyMXn
    - derivZ
    - lead_coef_map_id0
    - polyOverP
    - rreg_lead0
    - deg2_poly_root2
    - size_mulX
  - path: mathcomp/solvable/gseries.v
    theorems:
    - quotient_maximal
    - subnormalEr
    - maximal_eqJ
    - minnormal_maxnormal
    - simpleP
    - sub_setIgr
    - minnormal_exists
    - morphpre_maximal_eq
  - path: mathcomp/character/mxabelem.v
    theorems:
    - sub_rowg_mx
    - rowg_group_set
    - rowg1
    - rVabelemJmx
  - path: mathcomp/solvable/sylow.v
    theorems:
    - card_Syl_mod
    - nilpotent_pcore_Hall
    - Sylow_Jsub
  - path: mathcomp/character/character.v
    theorems:
    - cfMod_lin_char
    - cfRepr_map
    - cfDprodl_irr
    - xcfun_rE
    - Cnat_cfdot_char
    - eq_sum_nth_irr
    - cfker_Ind_irr
    - dprod_IirrC
    - cfun_irr_sum
    - dprodl_Iirr_eq0
    - inv_dprod_IirrK
    - cfdotC_char
    - max_cfRepr_norm_scalar
    - tprod1
    - sdprod_Iirr_inj
    - constt_Ind_Res
    - cfdot_aut_irr
    - irr_repr_lin_char
    - xcfun_annihilate
    - sum_norm_irr_quo
    - pgroup_cyclic_faithful
  - path: mathcomp/algebra/polydiv.v
    theorems:
    - reducible_cubic_root
    - dvd0pP
    - mup_geq
    - eqpW
    - lc_expn_scalp_neq0
    - gdcop_recP
    - gcd0p
    - divp_addl_mul_small
    - divp_eq0
    - gdcop_rec_map
    - eqp_rtrans
    - modp_eq0P
    - ltn_rmodp
    - eq_rdvdp
    - eqp_sym
    - gcdp_mulr
    - eqp_root
    - dvdpNr
    - size_gcd1p
    - cubic_irreducible
  - path: mathcomp/character/inertia.v
    theorems:
    - constt_Inertia_bijection
    - cfConjg_irr
    - inertia_mod_pre
    - inertia_quo
    - cfConjgMod_norm
    - prime_invariant_irr_extendible
    - inertia_bigdprod
  - path: mathcomp/algebra/intdiv.v
    theorems:
    - eqz_div
    - coprimez_pexpr
    - gcdzz
    - modz0
    - gcdz0
    - zprimitive_eq0
    - eqz_modDl
    - dvdz_exp2l
    - dvdz_zmod_closed
    - dvdz_lcml
    - modzMr
    - lcmz0
    - ltz_mod
    - coprimezXl
    - dvdz_contents
    - gcd1z
    - mulz_divCA_gcd
  - path: mathcomp/fingroup/morphism.v
    theorems:
    - morphpreK
    - isog_sym
    - morphpreT
    - trivial_isog
    - morphim0
    - morphicP
    - factmE
    - morphpre_subcent
    - ker_trivg_morphim
    - misom_isog
    - ker_trivm
    - morphimK
    - morphpre_gen
    - injm_ifactm
    - sub_morphpre_im
    - eq_homgl
    - morphimGI
  - path: mathcomp/solvable/burnside_app.v
    theorems:
    - F_s23
    - Sd1_inj
    - r05_inv
    - S3_inv
    - sh_inv
    - S1_inv
    - rot_r1
  - path: mathcomp/algebra/ssrint.v
    theorems:
    - pmulrz_rlt0
    - nexpIrz
    - PoszD
    - exp0rz
    - mulrz_eq0
    - coefMrz
    - intP
    - leqifD_dist
    - mulzS
    - mulz2
    - addSnz
    - scaler_int
    - distnDr
    - expN1r
    - mulrzBl_nat
    - rpredXsign
  - path: mathcomp/field/fieldext.v
    theorems:
    - module_baseVspace
    - divp_polyOver
    - subfx_scaler1r
    - mem_vspaceOver
  - path: mathcomp/solvable/commutator.v
    theorems:
    - derg0
    - der_char
    - commMG
    - quotient_cents2r
  - path: mathcomp/algebra/rat.v
    theorems:
    - mul1q
    - mul_subdefA
    - rat_field_theory
    - ltrq0
    - QintP
  - path: mathcomp/solvable/maximal.v
    theorems:
    - nilpotent_Fitting
    - card_p3group_extraspecial
    - Fitting_max
    - FittingJ
    - Aut_extraspecial_full
  - path: mathcomp/field/galois.v
    theorems:
    - dim_fixed_galois
    - kHomP
    - normalFieldP
    - kHom_inv
    - kHom1
    - normalFieldf
    - kAHomP
  - path: mathcomp/ssreflect/div.v
    theorems:
    - coprime_sym
    - edivnP
    - modnXm
    - modn_dvdm
    - dvdn_sub
    - leq_trunc_div
    - eqn_mod_dvd
    - divnBr
    - dvdn_subr
    - coprime_pexpl
    - modnB
    - leq_divRL
    - dvdn_lcml
    - lcmnC
  - path: mathcomp/solvable/primitive_action.v
    theorems:
    - n_act_is_action
  - path: mathcomp/fingroup/automorphism.v
    theorems:
    - char1
    - autE
    - charY
    - injm_Aut
    - injm_conj
    - Aut_group_set
    - morphic_aut
  - path: mathcomp/algebra/mxpoly.v
    theorems:
    - integral_horner
    - horner_mx_diag
    - char_poly_trace
    - map_horner_mx
    - diagonalizable_forPex
    - map_char_poly_mx
    - algebraic_inv
    - algebraic_add
    - comm_horner_mx2
    - conjmxM
    - eval_mxvec
    - conjumx
    - horner_mx_uconjC
    - mx_inv_horner0
    - simmxW
    - conj0mx
    - coef_rVpoly
    - split_diagA
  - path: mathcomp/field/falgebra.v
    theorems:
    - sub1_agenv
    - cent_centerv
    - subX_agenv
    - prodvDr
    - amull_is_linear
    - lfun_mulRVr
    - algidr
    - centerv_sub
  - path: mathcomp/ssreflect/path.v
    theorems:
    - subseq_path
    - iota_sorted
    - sorted_ltn_index_in
    - rot_to_arc
    - eq_fpath
    - trajectS
    - path_min_sorted
    - shortenP
    - sort_stable
    - perm_merge
    - mem2lr_splice
    - sorted_map
    - homo_sort_map
    - cycle_prev
    - map_path
    - take_path
    - sorted_sort
    - prev_rotr
    - mem2rf
    - allrel_merge
    - arc_rot
    - sorted_subseq_sort_in
  - path: mathcomp/ssreflect/bigop.v
    theorems:
    - big_rec2
    - big_nseq_cond
    - big_distrr
    - big_nat_recl
    - big_orE
    - big_nat1_eq
    - big_ord1_eq
    - sig_big_dep_idem
    - big_ord_narrow_cond_leq
    - big_enum_rank_cond
    - le_big_ord_cond
    - reindex_omap
    - big_nat
    - big_cat
    - addm0
  - path: mathcomp/ssreflect/choice.v
    theorems:
    - tagged_hasChoice
    - seq_hasChoice
  - path: mathcomp/algebra/mxalgebra.v
    theorems:
    - mulmx_max_rank
    - row_base0
    - rowV0Pn
    - row_freePn
    - qidmx_eq1
    - addmx_sub
    - summx_sub_sums
    - proj_mx_proj
    - eqmx_rowsub_comp
    - genmxE
    - genmxP
    - binary_mxsum_proof
    - kermx_eq0
    - rV_eqP
    - stableNmx
    - mulmx_sub
  - path: mathcomp/ssreflect/fingraph.v
    theorems:
    - finv_inj_in
    - fcycle_undup
    - sym_connect_sym
    - roots_root
    - findex0
    - orbit_rot_cycle
    - cycle_orbit_in
    - adjunction_n_comp
    - connect0
    - eq_froot
  - path: mathcomp/solvable/hall.v
    theorems:
    - coprime_comm_pcore
    - Hall_Jsub
  - path: mathcomp/solvable/extraspecial.v
    theorems:
    - card_isog8_extraspecial
    - pX1p2_pgroup
  - path: mathcomp/solvable/extremal.v
    theorems:
    - def_q
    - generators_2dihedral
    - normal_rank1_structure
    - odd_not_extremal2
  - path: mathcomp/algebra/vector.v
    theorems:
    - memv_add
    - span_bigcat
    - zero_lfunE
    - mul_mxof
    - comp_lfunDl
    - seq1_free
    - lker0_compfV
    - cat_free
    - lfunP
    - seq1_basis
    - lfun_addN
    - vspace_modr
    - capvA
  - path: mathcomp/algebra/ring_quotient.v
    theorems:
    - prime_idealrM
  - path: mathcomp/solvable/cyclic.v
    theorems:
    - quotient_generator
    - ker_eltm
    - prime_cyclic
  - path: mathcomp/fingroup/gproduct.v
    theorems:
    - extprod_mul1g
    - cprod_card_dprod
    - injm_cprodm
    - bigcprod_card_dprod
    - pprodmEl
    - morphim_coprime_sdprod
    - dprodW
    - sdprodmEr
    - subcent_dprod
    - divgr_id
    - morphim_bigcprod
    - injm_dprodm
    - pprod1g
    - im_sdprodm1
  - path: mathcomp/field/separable.v
    theorems:
    - base_separable
    - separable_prod_XsubC
    - base_inseparable
    - separable_inseparable_element
    - separable_generator_maximal
    - separable_add
    - dvdp_separable
  - path: mathcomp/ssreflect/generic_quotient.v
    theorems:
    - enc_mod_rel_is_equiv
    - pi_morph2
    - pi_mono1
  - path: mathcomp/fingroup/action.v
    theorems:
    - dvdn_orbit
    - actbyE
    - actby_is_groupAction
    - morph_astabs
    - perm_faithful
    - astabsD
    - actpermK
    - ract_is_groupAction
    - actK
    - modact_coset_astab
    - actsP
    - orbit_inv_in
    - gacentE
    - act1
    - atransR
    - reindex_acts
    - gactM
    - subgacent1E
    - card_orbit_in_stab
    - gactJ
    - atrans_acts_card
  - path: mathcomp/character/integral_char.v
    theorems:
    - irr_gring_center
  - path: mathcomp/solvable/alt.v
    theorems:
    - rfd_morph
    - simple_Alt5
    - card_Alt
  - path: mathcomp/algebra/fraction.v
    theorems:
    - equivf_sym
  - path: mathcomp/solvable/center.v
    theorems:
    - center_ncprod
    - center_prod
  - path: mathcomp/algebra/interval.v
    theorems:
    - leBSide
    - itv_joinKI
    - itv_leEmeet
    - itv_split1U
  - path: mathcomp/ssreflect/binomial.v
    theorems:
    - fact_split
    - ffact_gt0
    - ffactnSr
    - bin_factd
  - path: mathcomp/algebra/qpoly.v
    theorems:
    - poly_of_size_mod
    - qpoly_scale1l
    - qpoly_scaleDl
    - irreducibleP
  - path: mathcomp/field/algnum.v
    theorems:
    - Aint_subring_exists
    - Qn_aut_exists
  - path: mathcomp/solvable/nilpotent.v
    theorems:
    - lcn_subS
    - ucn_id
    - TI_center_nil
    - meet_center_nil
    - injm_nil
    - quotient_nil
  - path: mathcomp/algebra/zmodp.v
    theorems:
    - Zp_mulVz
    - Zp_group_set
    - Fp_Zcast
  - path: mathcomp/field/closed_field.v
    theorems:
    - rmulpT
  - path: mathcomp/algebra/polyXY.v
    theorems:
    - swapXY_X
  - path: mathcomp/field/qfpoly.v
    theorems:
    - qlogp_lt
  - path: mathcomp/solvable/finmodule.v
    theorems:
    - fmodM
    - fmod_addrC
    - fmvalK
    - mulg_exp_card_rcosets
  - path: mathcomp/character/vcharacter.v
    theorems:
    - vcharP
    - zcharD1E
    - orthogonal_span
    - dirr_inj
  - path: mathcomp/solvable/jordanholder.v
    theorems:
    - acompsP
    - maxainv_proper
    - maxainv_asimple_quo
  - path: mathcomp/algebra/archimedean.v
    theorems:
    - lt_succ_floor
    - natr_norm_int
    - trunc1
    - ceil_floor
    - norm_natr
    - floorN
    - prod_truncK
    - int_num0
  - path: mathcomp/field/finfield.v
    theorems:
    - primeChar_abelem
    - primeChar_scale1
    - primeChar_scaleAl
  - path: mathcomp/fingroup/perm.v
    theorems:
    - porbitP
    - porbit_setP
    - preim_permV
  - path: mathcomp/fingroup/quotient.v
    theorems:
    - rcoset_kercosetP
  - path: mathcomp/ssreflect/ssrbool.v
    theorems:
    - if_implyb
  - path: mathcomp/ssreflect/finfun.v
    theorems:
    - supportE