hcju/isabelleps huggingface dataset:
DIVERGENCES_def: fixes P :: "('c event list \<times> 'c event set) set \<times> 'c event list set" shows "DIVERGENCES P = snd P"
mult_zero_right: fixes times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g" assumes "class.mult_zero times zero" shows "times a zero = zero"
\<psi>_1: shows "\<psi> \<in> fmla"
\<R>\<^sub>1_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> Sec option" where "c\<^sub>2 = c\<^sub>1" and "mds\<^sub>2 = mds" and "\<turnstile> \<Gamma> {c\<^sub>1} \<Gamma>'__" and "mds_consistent mds \<Gamma>" and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"
update_delete: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" shows "Mapping.update k v (Mapping.delete k m) = Mapping.update k v m"
invertible_mult_JNF: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "invertible_mat A"   and "invertible_mat B" shows "invertible_mat (A * B)"
det_four_block_mat_upper_right_zero: fixes A1 :: "'a mat"   and n :: "nat"   and A2 :: "'a mat"   and m :: "nat"   and A3 :: "'a mat"   and A4 :: "'a mat" assumes "A1 \<in> carrier_mat n n"   and "A2 = 0\<^sub>m n m"   and "A3 \<in> carrier_mat m n"   and "A4 \<in> carrier_mat m m" shows "Determinant.det (four_block_mat A1 A2 A3 A4) = Determinant.det A1 * Determinant.det A4"
lspasl_magicr_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and h2 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "\<not> sep_algebra.sep_impl plus sep_disj A B h2" shows "\<exists>h1 h0. heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> A h1 \<and> \<not> B h0"
lexordp_irreflexive': fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.order less_eq less" shows "\<not> ord.lexordp less xs xs"
sorted_ConsD: fixes y :: "'c"   and xs :: "'c list"   and x :: "'c" assumes "sorted_less (y # xs)"   and "x \<le> y" shows "x \<notin> set xs"
eq_id_iff: fixes f :: "'b \<Rightarrow> 'b" shows "(\<forall>x. f x = x) = (f = id)"
D_ndet: fixes P :: "'b process"   and Q :: "'b process" shows "D (P |-| Q) = D P \<union> D Q"
hyps_3: fixes y :: "'a" assumes "fold_graph f z B y"   and "B \<subseteq> A"   and "z \<in> A" shows "y = ya__"
small_big_antisym: fixes F :: "'a filter"   and H :: "('a \<Rightarrow> real) set"   and h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "landau_function_family F H"   and "h1 \<in> H"   and "h2 \<in> H"   and "h1 \<in> o[F](h2)"   and "h2 \<in> O[F](h1)" shows "False"
increasing: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "bourbaki_witt_fixpoint lub leq f"   and "x \<in> Field leq" shows "(x, f x) \<in> leq"
fw_canonical: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and n :: "nat" assumes "cycle_free m n" shows "canonical (FW m n) n"
simps_3: fixes uu :: "pnat" shows "pnat_plus uu pnat.PInfty = pnat.PInfty"
map_indets_power: fixes f :: "'e \<Rightarrow> 'c"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and m :: "nat" shows "map_indets f (p ^ m) = map_indets f p ^ m"
tpx: fixes x :: "'a"   and y :: "'a"   and rest1 :: "'a list" shows "t\<^sub>p [x, y] x (hd (OPT2 (x # rest1) [x, y])) = 0"
omit_branch_mem_dual: fixes i :: "'d"   and xs :: "(nat \<times> nat) set"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and p :: "('c, 'd) fm" assumes "p at i in omit_branch xs branch" shows "p at i in branch"
prod: fixes i :: "'c"   and I :: "'c set"   and j :: "'c"   and J :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "i \<in> I"   and "j \<in> J" shows "i \<otimes>\<^bsub>R\<^esub> j \<in> I \<cdot>\<^bsub>R\<^esub> J"
less_linear: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "less x y \<or> x = y \<or> less y x"
berlekamp_monic_factorization: fixes f :: "'a mod_ring poly"   and d :: "nat"   and us :: "'a mod_ring poly list" assumes "square_free f"   and "berlekamp_monic_factorization d f = us"   and "\<And>g. \<lbrakk>g dvd f; degree g = d\<rbrakk> \<Longrightarrow> irreducible g"   and "0 < degree f"   and "monic f" shows "f = prod_list us \<and> (\<forall>u\<in>set us. monic u \<and> irreducible u)"
P1: shows "Bet A' x C' \<and> A B C Cong3 A' x C'"
repeat_stabilize_funpow: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c"   and v :: "'c" shows "repeat_stabilize n f v = (f ^^ n) v"
thetaParZOL1_ZObis: shows "thetaParZOL1 \<subseteq> ZObis"
approx_hrabs_add_minus_Infinitesimal: fixes e :: "real star"   and x :: "real star" assumes "e \<in> Infinitesimal" shows "\<bar>x\<bar> \<approx> \<bar>x + - e\<bar>"
uadmit_prog_adjoint: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and a :: "('sf, 'sc, 'sz) hp"   and U :: "('sz + 'sz) set"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp" assumes "PUadmit \<sigma> a U"   and "Vagree \<nu> \<omega> (- U)"   and "hpsafe a"   and "ssafe \<sigma>"   and "is_interp I" shows "prog_sem (local.adjoint I \<sigma> \<nu>) a = prog_sem (local.adjoint I \<sigma> \<omega>) a"
get_curr_win3_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state" assumes "low_equal s1 s2" shows "traps (snd (fst (get_curr_win () s1))) = traps (snd (fst (get_curr_win () s2)))"
T_delete2_def2: fixes x :: "'a"   and t :: "'a tree"   and n :: "nat"   and dl :: "nat" shows "T_delete2 x (t, n, dl) = T_del x t + (case del x t of None \<Rightarrow> 1 | Some t' \<Rightarrow> let n' = n - 1; dl' = dl + 1 in if bal_d n' dl' then 1 else T_bal_tree n' t' + 1)"
ok_commute: fixes F :: "'c program"   and G :: "'c program" shows "F ok G = G ok F"
mat_of_cblinfun_def: fixes f :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "mat_of_cblinfun f = mat (length canonical_basis) (length canonical_basis) (\<lambda>(i, j). crepresentation (set canonical_basis) (f *\<^sub>V canonical_basis ! j) (canonical_basis ! i))"
wellformed_init: fixes t :: "'t2" assumes "wf2 t" shows "wellformed2 (init2 t)"
cInt_cinsert_left_if1: fixes a :: "'f"   and C :: "'f cset"   and B :: "'f cset" assumes "cin a C" shows "cInt (cinsert a B) C = cinsert a (cInt B C)"
intro: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "finite_dimensional_vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1"   and "vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2" shows "finite_dimensional_vector_space_pair_1_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
steps_append_single: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and x :: "'c" assumes "Graph_Defs.steps E xs"   and "E (last xs) x"   and "xs \<noteq> []" shows "Graph_Defs.steps E (xs @ [x])"
det_echelon_form_of_upt_k_det: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" assumes "is_bezout_ext bezout" shows "fst (echelon_form_of_upt_k_det bezout (1::'b, A) k) * det A = det (snd (echelon_form_of_upt_k_det bezout (1::'b, A) k))"
mem_PiCI: fixes x :: "'c \<Rightarrow>\<^sub>C 'd"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd set" assumes "apply_bcontfun x \<in> Pi I X" shows "x \<in> PiC I X"
lang_verund4: fixes rs :: "nat rexp list" shows "L (set rs) = lang (verund rs)"
NSTD_lemma1: fixes c :: "nat" shows "rec_exec rec_NSTD [c] = Suc 0 \<or> rec_exec rec_NSTD [c] = 0"
assms_3: shows "AE x in M. (\<lambda>n. F n x) \<longlonglongrightarrow> f x"
m_nth: fixes ia :: "nat" assumes "ia < max_Var_floatarith fa" shows "ia < DIM('a) + length params"
lp'': shows "iszlfm (zlfm p) (real_of_int i # bs)"
inv_RBT_Impl_diag_1: fixes t :: "('c, 'd) rbt" assumes "inv1 t"   and "inv2 t" shows "inv1 (RBT_Impl_diag t)"
all_carrier_ex_sub_rel: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" assumes "rational_preference carrier relation" shows "\<forall>c\<subseteq>carrier. \<exists>r\<subseteq>relation. rational_preference c r"
vars_clause_hd_partitioned_from_tl: fixes Cs :: "('a, nat) Term.term literal multiset list" assumes "Cs \<noteq> []" shows "vars_clause (hd (subst_cls_lists Cs (renamings_apart Cs))) \<inter> vars_clause_list (tl (subst_cls_lists Cs (renamings_apart Cs))) = {}"
invL_FL_bisimilar_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and F :: "'d fs_set"   and P :: "'a"   and Q :: "'a" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.invL_FL_bisimilar satisfies transition effect_apply L F P Q \<equiv> nominal_ts.bisimilar (effect_nominal_ts.L_satisfies satisfies) (effect_nominal_ts.L_transition transition effect_apply L) (EF (F, P)) (EF (F, Q))"
funpow_shift1_degree: fixes k :: "nat"   and p :: "poly" shows "Reflected_Multivariate_Polynomial.degree ((shift1 ^^ k) p) = k + Reflected_Multivariate_Polynomial.degree p"
ide_def: fixes a :: "'a" shows "B.ide a \<equiv> a \<cdot>\<^sub>B a \<noteq> B.null \<and> (\<forall>f. (f \<cdot>\<^sub>B a \<noteq> B.null \<longrightarrow> f \<cdot>\<^sub>B a = f) \<and> (a \<cdot>\<^sub>B f \<noteq> B.null \<longrightarrow> a \<cdot>\<^sub>B f = f))"
simps: fixes m :: "nat"   and n :: "nat"   and p :: "nat" shows "addition m n p = [Dec m 4, Inc n, Inc p, Goto 0, Dec p 7, Inc m, Goto 4]"
compact_convex_collinear_segment_alt: fixes S :: "'b set" assumes "S \<noteq> {}"   and "compact S"   and "connected S"   and "collinear S" obtains a :: "'b" and b :: "'b" where "S = closed_segment a b"
succ_chain_no_vars_valuation: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "\<forall>x. x \<notin> tvars (\<T> (l ! i)) \<longrightarrow> Mapping.lookup (\<V> (l ! i)) x = Mapping.lookup (\<V> (l ! j)) x"
assms_3: fixes \<Delta>1 :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and \<Delta>2 :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" assumes "(\<Delta>1, \<Delta>2) \<in> allConsec \<Delta>s" shows "unwind_cont \<Delta>1 {\<Delta>1, \<Delta>2, \<Delta>e}"
elims_3: fixes x :: "'a"   and xa :: "'b node_action" assumes "\<not> inoclosed x xa" obtains uu_ :: "'a" and ii :: "nat set" and ni :: "nat set" and m :: "'b" where "x = uu_" and "xa = ii\<not>ni:arrive(m)" and "\<not> eq_newpkt m"   | uv_ :: "'a" and i :: "nat" and d :: "nat" and di :: "nat" where "x = uv_" and "xa = i:newpkt(d, di)"
is_iso_classI: fixes f :: "'a" assumes "ide f" shows "is_iso_class \<lbrakk>f\<rbrakk>"
simps_3: fixes n :: "nat"   and p :: "'a"   and t' :: "'b tree23" shows "dfull n (Some (p, False, t')) = full (Suc n) t'"
card_fset_Suc_case: fixes S :: "'a fset"   and T :: "'a fset" assumes "Suc (card_fset S) = card_fset T"   and "P S" shows "P T"
not_all_zero: shows "\<not> (\<forall>a. is_zero_row_upt_k a k A)"
ksup_annil: fixes f :: "'c \<Rightarrow> 'b set" shows "\<zeta> \<circ>\<^sub>K f = \<zeta>"
id_moebius_mmat_def: shows "id_moebius_mmat \<equiv> Abs_moebius_mat id_moebius_cmat"
greaterThanAtMost_borel: fixes a :: "'c"   and b :: "'c" shows "{a<..b} \<in> sets borel"
respects_labels_gallery: fixes B :: "'c set"   and \<phi> :: "'a \<Rightarrow> 'c"   and C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "label_wrt B \<phi>"   and "\<forall>v\<in>C. \<phi> (f v) = \<phi> v"   and "gallery (C # Cs @ [D])" shows "\<forall>v\<in>D. \<phi> (f v) = \<phi> v"
morphism_presheaves_of_rings_axioms_def: fixes is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" shows "morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> (\<forall>U. is_open U \<longrightarrow> ring_homomorphism (fam_morphisms U) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF>' U) (add_str' U) (mult_str' U) (zero_str' U) (one_str' U)) \<and> (\<forall>U V. is_open U \<longrightarrow> is_open V \<longrightarrow> V \<subseteq> U \<longrightarrow> (\<forall>x. x \<in> \<FF> U \<longrightarrow> (\<rho>' U V \<circ> fam_morphisms U) x = (fam_morphisms V \<circ> \<rho> U V) x))"
submonoid_nonempty: fixes G :: "('e, 'f) monoid_scheme" shows "\<not> submonoid {} G"
alpha2_def: shows "alpha2 \<equiv> $pc2 = #a \<and> pc2$ = #b \<and> $sem = Suc<sem$> \<and> unchanged (x, y, pc1)"
row_mat: fixes i :: "nat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "i < nr" shows "Matrix.row (Matrix.mat nr nc f) i = Matrix.vec nc (\<lambda>j. f (i, j))"
nth_map: fixes n :: "nat"   and xs :: "'f list"   and f :: "'f \<Rightarrow> 'g" assumes "n < length xs" shows "map f xs ! n = f (xs ! n)"
true_cls_mset_add_mset: fixes I :: "'b set"   and C :: "'b literal multiset"   and CC :: "'b literal multiset multiset" shows "(I \<Turnstile>m add_mset C CC) = (I \<Turnstile> C \<and> I \<Turnstile>m CC)"
HoHV_def: fixes \<mu> :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "HoHV \<mu> \<equiv> if VVV.arr \<mu> then (fst \<mu> \<star> fst (snd \<mu>)) \<star> snd (snd \<mu>) else null"
no_small_separations_hyp: fixes G' :: "('a, 'b) Graph_scheme" assumes "\<And>S. Separation G' v0 v1 S \<Longrightarrow> sep_size \<le> card S"   and "v0_v1_Digraph G' v0 v1" shows "\<exists>paths. DisjointPaths G' v0 v1 paths \<and> card paths = sep_size"
push_down_state_rank_configuration_None: fixes q :: "'b"   and n :: "nat" assumes "state_rank q n = None"   and "\<not> sink q" shows "configuration q n = {}"
simps_8: shows "\<infinity> * - \<infinity> = - \<infinity>"
C_eq_sorted: fixes p :: "((int \<times> int) set set, int) Combinators list"   and l :: "(int \<times> int) set set set list" assumes "distinct p"   and "all_in_list p l"   and "singleCombinators p"   and "wellformed_policy1_strong p"   and "wellformed_policy3 p"   and "allNetsDistinct p" shows "C (list2FWpolicy (FWNormalisationCore.sort p l)) = C (list2FWpolicy p)"
distinct_60: fixes pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "!pi' \<noteq> name1<name2>.pi"
finite_F_bound: fixes S :: "nat set" assumes "S \<subseteq> range mmap"   and "finite S" shows "\<exists>n. S \<subseteq> set (map Inf (F n))"
f: shows "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"
simps_2: fixes f :: "'a"   and Tl :: "'a trm list" shows "nv2T (Fn f Tl) = {}"
sum_diff1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "finite A" shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
homotopy_eqv_empty2: fixes S :: "'b set" shows "({} homotopy_eqv S) = (S = {})"
sas_plus_problem_has_serial_solution_iff_ii': fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list"   and h :: "nat" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi>"   and "length \<psi> \<le> h" shows "\<exists>\<A>. \<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<phi> prob_with_noop \<Psi>  h"
PTrans: shows "P \<longmapsto> a<\<nu>x> \<prec> P'"
lfinite_lmap: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist" shows "lfinite (lmap f xs) = lfinite xs"
strategy_winning_by_Odd_def: fixes A :: "'c llist set"   and N :: "enat"   and \<sigma>\<^sub>o :: "'c list \<Rightarrow> 'c"   and p :: "'c list" assumes "GSgame A N" shows "GSgame.strategy_winning_by_Odd A N \<sigma>\<^sub>o p \<equiv> \<forall>\<sigma>\<^sub>e. GSgame.induced_play N (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) p \<notin> A"
inj_map_inv: fixes f :: "'c \<Rightarrow> 'd option" assumes "inj_on f (dom f)" shows "inj_on (map_inv f) (ran f)"
tranclp_imp_stepfun: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b" assumes "P\<^sup>+\<^sup>+ x z" shows "\<exists>f n. f 0 = x \<and> f (Suc n) = z \<and> (\<forall>i\<le>n. P (f i) (f (Suc i)))"
surjective: fixes r :: "'a two_fields_scheme" shows "r = \<lparr>Field1 = Field1 r, Field2 = Field2 r, \<dots> = more r\<rparr>"
lift_subst_bvs1'_lt: fixes i :: "nat"   and j :: "nat"   and t :: "Core.term"   and ss :: "Core.term list" assumes "i < j + 1" shows "lift (subst_bvs1' t j ss) i = subst_bvs1' (lift t i) (j + 1) (map (\<lambda>s. lift s i) ss)"
cinner_add_left: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (plus x y) z = cinner x z + cinner y z"
seq_imp_steps: fixes ss :: "'c \<times> ('d \<times> 'c) list"   and ars :: "('c \<times> 'd \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "ss \<in> seq ars"   and "fst ss = a"   and "lst ss = b" shows "(a, b) \<in> (unlabel ars)\<^sup>*"
Abs_permutation_inverse: fixes y :: "'b \<Rightarrow> 'b" assumes "y \<in> {f. bij f}" shows "(\<rightarrow>) (Abs_permutation y) = y"
not_iFROM_iMODb_subset: fixes n' :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" shows "\<not> [n'\<dots>] \<subseteq> [ r, mod m, c ]"
frac_add_leD: fixes d :: "real"   and a :: "real"   and b :: "real" assumes "0 < d"   and "d < 1 - frac a"   and "d < 1 - frac b"   and "frac (a + d) \<le> frac (b + d)" shows "frac a \<le> frac b"
apex_def: shows "HfHgh.cod.apex \<equiv> C.dom HfHgh.cod.leg0"
unit_is_iso: shows "S.iso (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f))"
euler_char_dv: shows "euler_char = 1 + H.euler_char"
sp_generalize2: fixes F :: "'a \<Rightarrow> 'b fundef option"   and xs :: "('dyn, 'c, 'a, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr list"   and n :: "nat"   and \<Sigma>' :: "Unboxed.type option list" assumes "Subx.sp F xs (replicate n None) = Ok \<Sigma>'" shows "Subx.sp F (map Subx.generalize_instr xs) (replicate n None) = Ok (map Map.empty \<Sigma>')"
contr: shows "\<not> prio (prio_emb_pos prio t s) \<le> prio (prio_emb_pos prio (prio_emb_step prio t s) s)"
is_confluent_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "relation.is_confluent r \<equiv> relation.is_confluent_on r UNIV"
g: shows "bij_betw (\<lambda>l. (hd l + 1) # tl l) {l. length l = m \<and> sum_list l + 1 = N} {l. length l = m \<and> sum_list l = N \<and> hd l \<noteq> 0}"
interleaving_Nil_implies_eq2: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "interleavings xs ys zs"   and "ys = []" shows "xs = zs"
partition_filter2: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "snd (partition P xs) = filter (Not \<circ> P) xs"
f_at_h_nz: shows "\<forall>x. f x (h x) \<noteq> 0"
elem_of_symcl_2: fixes a :: "'a"   and b :: "'a"   and Rel :: "('a \<times> 'a) set" assumes "(a, b) \<in> Rel" shows "(b, a) \<in> symcl Rel"
rep_eq: fixes x :: "nat"   and xa :: "int"   and xb :: "int" shows "real_of_float (lapprox_rat x xa xb) = truncate_down x (real_of_int xa / real_of_int xb)"
kpower_simp: fixes f :: "'b \<Rightarrow> 'b set"   and n :: "nat"   and s :: "'b" shows "kpower f (Suc n) s = (f ; kpower f n) s"
its: fixes t' :: "(('form fset \<times> 'form) \<times> ('rule \<times> 'form) NatRule) tree" assumes "t' |\<in>| cont t" shows "local_iwf (its t') (fst (tree.root t'))"
simps_2: fixes x :: "('a, 'b) strand_step"   and S :: "('a, 'b) strand_step list" shows "to_est (x # S) = Step x # to_est S"
in_path_weight: fixes s :: "'lbl"   and loc1 :: "'vtx"   and loc2 :: "'vtx" shows "(s \<in>\<^sub>A path_weight loc1 loc2) = (s \<in> minimal_antichain {s. path_weightp loc1 loc2 s})"
to_list_insertion: fixes f :: "nat \<Rightarrow> real"   and p :: "real mpoly"   and v :: "nat" shows "insertion f p = (\<Sum>(term, i)\<leftarrow>to_list v p. insertion f term * f v ^ i)"
le_floor_iff: fixes z :: "int"   and x :: "'d" shows "(z \<le> \<lfloor>x\<rfloor>) = (of_int z \<le> x)"
assms: assumes "\<Phi>" shows "(m', m) \<in> \<langle>R\<rangle>nres_rel"
wls_Abs_vsubst_ex: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and x' :: "'var" assumes "wls s X"   and "wls s X'" shows "(Abs xs x X = Abs xs x' X') = (\<exists>y. y \<notin> {x, x'} \<and> fresh xs y X \<and> fresh xs y X' \<and> X #[y // x]_xs = X' #[y // x']_xs)"