hcju/isabelleps huggingface dataset:
linear_span_image: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec"   and S :: "('c, 'd) vec set" assumes "Vector_Spaces.linear (*s) (*s) f" shows "vec.span (f ` S) = f ` vec.span S"
UNF_on_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "UNF_on r A = (\<forall>a\<in>A. \<forall>b c. (a, b) \<in> r\<^sup>! \<and> (a, c) \<in> r\<^sup>! \<longrightarrow> b = c)"
project_extend_constrains_I: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set"   and C :: "'c set" assumes "F \<in> A co B" shows "project h C (extend h F) \<in> A co B"
simps: fixes f :: "int poly"   and us :: "int poly list" shows "LLL_reconstruction f us = (let d = degree f; u = choose_u us; g = LLL_short_polynomial pl d u; f2 = gcd f g; deg = degree f2 in if deg = 0 \<or> d \<le> deg then [f] else let f1 = f div f2; (us1, us2) = List.partition (\<lambda>gi. p.dvdm gi f1) us in LLL_reconstruction f1 us1 @ LLL_reconstruction f2 us2)"
le_comp_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x * z \<le> y * z"
simps_3: fixes v :: "'b" shows "cycle_of_list [v] = id"
inv_closed: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "\<ominus>\<^bsub>poly_ring R\<^esub> x \<in> carrier (poly_ring R)"
invertible_def: fixes u :: "'a \<Rightarrow> 'a" assumes "u \<in> S \<rightarrow>\<^sub>E S" shows "invertible u = (\<exists>v\<in>S \<rightarrow>\<^sub>E S. compose S u v = identity S \<and> compose S v u = identity S)"
subspace_is_vs: fixes W :: "'a mod_ring poly set" assumes "subspace class_ring W local.poly_abelian_monoid" shows "vectorspace class_ring (vector_space_poly.vs W)"
aux: shows "y\<^sup>T ; 1 \<cdot> - (x ; 1) = y\<^sup>T ; 1"
assms_1: fixes x :: "'t \<Rightarrow>\<^sub>0 'b"   and y :: "'t \<Rightarrow>\<^sub>0 'b" shows "f (x + y) = f x + f y"
equiv_imp_le_alt: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans u t"
LeastI_ex: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (LEAST x. P x)"
adj_mat_3: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "adj_mat A * A = det A \<cdot>\<^sub>m 1\<^sub>m n"
less_add_1_iff_le_hmset: fixes m :: "hmultiset"   and n :: "hmultiset" shows "(m < n + 1) = (m \<le> n)"
euclidean_size_eq_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "(euclidean_size b = 0) = (b = zero)"
Inter_lower: fixes B :: "'c set"   and A :: "'c set set" assumes "B \<in> A" shows "\<Inter> A \<subseteq> B"
union_mset_add_mset_left: fixes a :: "'f"   and A :: "'f multiset"   and B :: "'f multiset" shows "add_mset a A + B = add_mset a (A + B)"
simple_term_inject: fixes x :: "('d, 'e) trm"   and y :: "('d, 'e) trm" assumes "x \<in> {\<theta>. dfree \<theta>}"   and "y \<in> {\<theta>. dfree \<theta>}" shows "(simple_term x = simple_term y) = (x = y)"
divides_prod2: fixes A :: "'b set"   and c :: "'b" assumes "finite A"   and "\<forall>a\<in>A. a dvd c"   and "\<forall>a1 a2. a1 \<in> A \<and> a2 \<in> A \<and> a1 \<noteq> a2 \<longrightarrow> comm_monoid_mult_class.coprime a1 a2" shows "\<Prod>A dvd c"
simps_1: fixes n :: "nat" shows "boundslt n T = True"
rel_refl_strong: fixes x :: "'g list"   and Ra :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>z. z \<in> set x \<Longrightarrow> Ra z z" shows "list_all2 Ra x x"
bits_of_Suc_0: shows "bits_of (Suc 0) = [True]"
bdd_below_Ioi: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.preorder less_eq less" shows "preorder.bdd_below less_eq (ord.greaterThan less a)"
l_null: fixes x :: "'a" shows "(0::'a) * x = (0::'a)"
length_greater_0_conv: fixes xs :: "'f list" shows "(0 < length xs) = (xs \<noteq> [])"
igSwapIPresIGWls_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSwapIPresIGWls MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> igWls MOD s (igSwap MOD zs z1 z2 X)"
med01ia_def: fixes t :: "l1_state" shows "med01ia t \<equiv> \<lparr>a0n_state.signals = l1_state.signals t\<rparr>"
assoc_props_6: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and h :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f"   and "partial_magma.ide (span_vertical_category.vcomp C) g"   and "partial_magma.ide (span_vertical_category.vcomp C) h"   and "span_vertical_category.src C f = span_vertical_category.trg C g"   and "span_vertical_category.src C g = span_vertical_category.trg C h" shows "partial_magma.in_hom (span_vertical_category.vcomp C) \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' C prj0 prj1 f g h, Dom = Cod (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h)), Cod = Dom (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h)\<rparr> (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h)) (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h)"
elem_setmonotone: fixes f :: "'a set \<Rightarrow> 'a set"   and x :: "'a"   and X :: "'a set" assumes "setmonotone f"   and "x \<in> X" shows "x \<in> f X"
indicator_eq_1_iff: fixes A :: "'b set"   and x :: "'b" shows "(chi A x = (1::'a)) = (x \<in> A)"
exists_fresh': fixes x :: "'b" assumes "finite (supp x)" shows "\<exists>c. c \<sharp> x"
assms_1: shows "(x, y) \<in> terminationLessState1 Vbl"
cases: fixes x :: "real_alg_2" obtains x :: "rat" where "xb__ = Rational x"   | n :: "nat" and p :: "int poly" and l :: "rat" and r :: "rat" where "xb__ = Irrational n (p, l, r)"
disj_ac_2: fixes a :: "'b"   and b :: "'b" shows "a OR b = b OR a"
trelPS: shows "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
IH: fixes B :: "'a set" assumes "B \<subset> A" shows "random_bst B = map_pmf bst_of_list (pmf_of_set (permutations_of_set B))"
assms_1: shows "n \<in> set (\<alpha>n g)"
Sleator_Tarjan': fixes n :: "nat" shows "T_mtf n \<le> 2 * T_A n - int n"
M: shows "M \<equiv> mult_row A i (ass (A $ i $ n) div A $ i $ n)"
ENR_rel_frontier_convex: fixes S :: "'a set" assumes "bounded S"   and "convex S" shows "ENR (rel_frontier S)"
rec_quo_def: shows "rec_quo = (let lhs = CN S [recf.Id 3 0]; rhs = CN rec_mult [recf.Id 3 2, CN S [recf.Id 3 1]]; cond = CN rec_eq [lhs, rhs] in Let (CN rec_if [cond, CN S [recf.Id 3 1], recf.Id 3 1]) (PR Z))"
index_minus_mat_3: fixes A :: "'c mat"   and B :: "'c mat" shows "dim_col (A - B) = dim_col B"
continuous_injective_image_subspace_dim_le: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "subspace S"   and "subspace T"   and "continuous_on S f"   and "f ` S \<subseteq> T"   and "inj_on f S" shows "dim S \<le> dim T"
subst_graph_aux_def: shows "subst_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>subst_aux i x. x1 = (i, x, Zero) \<and> x2 = Zero) \<or> (\<exists>subst_aux i x k. x1 = (i, x, Var k) \<and> x2 = (if i = k then x else Var k)) \<or> (\<exists>subst_aux i x t u. x1 = (i, x, Eats t u) \<and> x2 = Eats (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, t)) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, u)) \<and> p (i, x, t) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, t)) \<and> p (i, x, u) (((\<lambda>uu_. uu_) \<circ> subst_aux) (i, x, u))))"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "hom.par f f'"   and "trg \<mu> \<star> f \<star> e = trg \<mu> \<star> f' \<star> e" shows "f = f'"
simps: fixes x :: "'b"   and hp :: "'b tree" shows "Pairing_Heap_Tree.insert x hp = merge \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> hp"
subset_eq_diff_conv: fixes A :: "'a multiset"   and C :: "'a multiset"   and B :: "'a multiset" shows "(A - C \<subseteq># B) = (A \<subseteq># B + C)"
in_out_arcs_conv: fixes e :: "'e"   and G :: "('f, 'e) pre_digraph"   and v :: "'f" shows "(e \<in> out_arcs G v) = (e \<in> arcs G \<and> tail G e = v)"
char_poly_transpose_mat: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "char_poly A\<^sup>T = char_poly A"
no_free_return_seq: fixes r1 :: "'a rule list"   and m :: "'a match_expr"   and r2 :: "'a rule list"   and ta :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r1 @ Rule m Return # r2, Undecided\<rangle> \<Rightarrow> ta"   and "matches \<gamma> m p"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r1, Undecided\<rangle> \<Rightarrow> Undecided" shows "False"
test_bit_1: fixes n :: "nat" shows "1 !! n = (n = 0)"
is_continuation_increase_u: fixes k :: "nat"   and u :: "nat"   and q :: "(symbol \<times> character list) list"   and ts :: "(symbol \<times> character list) list"   and v :: "nat" assumes "is_continuation k u q ts"   and "u \<le> v" shows "is_continuation k v q ts"
finite_induct: fixes F :: "'d set"   and P :: "'d set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
intros_2: fixes A :: "'b \<Rightarrow> bool"   and l :: "'b llist"   and a :: "'b" assumes "alllstsp A l"   and "A a" shows "alllstsp A (a ## l)"
C: fixes x :: "real" shows "norm (pbernpoly n x) \<le> C"
hyps: fixes X :: "nat set"   and Y :: "nat set" assumes "a \<notin> all_shared sb"   and "a \<notin> X"   and "a \<notin> acquired True sb X"   and "a \<notin> Y" shows "a \<notin> acquired True sb Y"
repa_repb_nc: shows "\<forall>pt. pt \<noteq> node \<longrightarrow> repa pt = repb pt"
normalize_eigen_vector_2: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and v :: "('b, 'c) Finite_Cartesian_Product.vec"   and ev :: "'b" assumes "eigen_vector A v ev" shows "norm1 ((1 / norm1 v) *\<^sub>R v) = 1"
starr_applied_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and F :: "'a \<Rightarrow> bool" assumes "class.heap_sep_algebra plus zero sep_disj" shows "heap_sep_algebra.starr_applied plus sep_disj h1 h2 h0 F \<equiv> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> \<not> F h0"
\<tau>_def: shows "\<tau> \<equiv> \<tau>J \<psi>"
Inr_Inl_False: fixes x :: "'c"   and y :: "'b" shows "(Inr x = Inl y) = False"
left_unique_pcr_spmf_gpv: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "left_unique A" shows "left_unique (spmf_to_gpv.pcr_spmf_gpv A)"
simps_1: fixes t :: "nat"   and v :: "nat" shows "issue_expr t (Const v) = []"
rtranclp_rtrancl_eq: fixes r :: "('c \<times> 'c) set" shows "(\<lambda>x xa. (x, xa) \<in> r)\<^sup>*\<^sup>* = (\<lambda>x xa. (x, xa) \<in> r\<^sup>*)"
zero_less_diff: fixes n :: "nat"   and m :: "nat" shows "(0 < (n::nat) - (m::nat)) = (m < n)"
trg_dom: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg\<^sub>B (B.dom \<mu>) = trg\<^sub>B \<mu>"
mem_context_val_w32_def: fixes asi :: "word8"   and addr :: "word36"   and m :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option" shows "mem_context_val_w32 asi addr m \<equiv> let addr' = addr AND 68719476732; addr0 = addr' OR 0; addr1 = addr' OR 1; addr2 = addr' OR 2; addr3 = addr' OR 3; r0 = mem_context_val asi addr0 m; r1 = mem_context_val asi addr1 m; r2 = mem_context_val asi addr2 m; r3 = mem_context_val asi addr3 m in if r0 = None \<or> r1 = None \<or> r2 = None \<or> r3 = None then None else let byte0 = case r0 of Some v \<Rightarrow> v; byte1 = case r1 of Some v \<Rightarrow> v; byte2 = case r2 of Some v \<Rightarrow> v; byte3 = case r3 of Some v \<Rightarrow> v in Some ((((ucast byte0 << 24) OR (ucast byte1 << 16)) OR (ucast byte2 << 8)) OR ucast byte3)"
det3_pos_transitive_coll: fixes t :: "real \<times> real"   and s :: "real \<times> real"   and p :: "real \<times> real"   and r :: "real \<times> real"   and q :: "real \<times> real" assumes "0 < det3 t s p"   and "0 \<le> det3 t s r"   and "0 \<le> det3 t p q"   and "0 < det3 t q r"   and "coll t s q" shows "0 < det3 t p r"
poly_mapping_eqI: fixes f :: "'b \<Rightarrow>\<^sub>0 'c"   and g :: "'b \<Rightarrow>\<^sub>0 'c" assumes "\<And>k. poly_mapping.lookup f k = poly_mapping.lookup g k" shows "f = g"
y_2: shows "tendsto_in\<^sub>N N (\<lambda>n. sum u {0..<n}) y"
subspace_def: fixes S :: "'b set" shows "local.subspace S = ((0::'b) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. scale c x \<in> S))"
path_end: fixes ps :: "('v \<times> 'w \<times> 'v) list"   and v :: "'v"   and v' :: "'v" assumes "ps \<noteq> []"   and "is_path v ps v'" shows "v' = snd (snd (last ps))"
floorlog_mono: fixes x :: "nat"   and y :: "nat"   and b :: "nat" assumes "x \<le> y" shows "floorlog b x \<le> floorlog b y"
init_imp_inv_implications_nonneg: fixes c :: "('a, 'b, 'c) configuration_scheme" assumes "init_config c" shows "inv_implications_nonneg c"
clop_range_Fix: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and cl_op :: "'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.clattice_with_clop less_eq less cl_op Inf Sup inf sup bot top" shows "range cl_op = Fix cl_op"
FAddSA_IsCompFun: fixes SA :: "('d, 'e, 'f) seqauto"   and F :: "('d, 'e, 'f) seqauto set"   and S :: "'d"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" assumes "States SA \<inter> \<Union> (States ` F) = {}"   and "S \<in> \<Union> (States ` F)"   and "IsCompFun F G" shows "IsCompFun (insert SA F) (G [f+] (S, SA))"
map_cong: fixes x :: "('d, 'e, 'f) generat"   and ya :: "('d, 'e, 'f) generat"   and f1 :: "'d \<Rightarrow> 'g"   and g1 :: "'d \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'h"   and g2 :: "'e \<Rightarrow> 'h"   and f3 :: "'f \<Rightarrow> 'i"   and g3 :: "'f \<Rightarrow> 'i" assumes "x = ya"   and "\<And>z1. z1 \<in> generat_pures ya \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> generat_outs ya \<Longrightarrow> f2 z2 = g2 z2"   and "\<And>z3. z3 \<in> generat_conts ya \<Longrightarrow> f3 z3 = g3 z3" shows "map_generat f1 f2 f3 x = map_generat g1 g2 g3 ya"
phiArg_def: fixes \<alpha>n :: "'c \<Rightarrow> 'd list"   and defs :: "'c \<Rightarrow> 'd \<Rightarrow> 'e set"   and phis :: "'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> 'e list option"   and g :: "'c"   and v :: "'e"   and v' :: "'e" shows "CFG_SSA_wf_base.phiArg \<alpha>n defs phis g v v' \<equiv> \<exists>vs. CFG_SSA_wf_base.phi \<alpha>n defs phis g v = Some vs \<and> v' \<in> set vs"
nsqn\<^sub>r_val: fixes dsn :: "nat"   and dsk :: "k"   and hops :: "nat"   and nhip :: "nat" shows "nsqn\<^sub>r (dsn, dsk, val, hops, nhip) = dsn"
select_convs_1: fixes states :: "hf set"   and init :: "hf set"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf set"   and eps :: "(hf \<times> hf) set"   and more :: "'c" shows "nfa.states \<lparr>nfa.states = states, init = init, final = final, nxt = nxt, eps = eps, \<dots> = more\<rparr> = states"
power_radical: fixes a :: "'a fps"   and r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat" assumes "fps_nth a 0 \<noteq> (0::'a)" shows "(r (Suc k) (fps_nth a 0) ^ Suc k = fps_nth a 0) = (fps_radical r (Suc k) a ^ Suc k = a)"
path_valid_node_2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "CFG.valid_node sourcenode targetnode valid_edge n'"
b0': shows "inj_map (inv_map (Some \<circ> \<pi>))"
calc_ideal_point1_sq: fixes z1 :: "complex"   and z2 :: "complex"   and A :: "complex"   and B :: "complex" assumes "(z1, z2) = calc_ideal_point1_cvec A B"   and "is_real A"   and "(cmod A)\<^sup>2 < (cmod B)\<^sup>2" shows "z1 * cnj z1 + z2 * cnj z2 = 2 * (B * cnj B)\<^sup>2"
fmap_minus_fmap: fixes F :: "'h \<Rightarrow> 'g option"   and x :: "'h \<times> 'g"   and a :: "'h"   and b :: "'g" assumes "(F -- x) a = Some b" shows "F a = Some b"
has_vector_derivative_componentwise_within: fixes f :: "real \<Rightarrow> 'c"   and f' :: "'c"   and a :: "real"   and S :: "real set" shows "(f has_vector_derivative f') (at a within S) = (\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) has_vector_derivative f' \<bullet> i) (at a within S))"
one_dvd: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.comm_monoid_mult times one" shows "dvd.dvd times one a"
Frct_less: fixes a :: "int"   and b :: "num"   and c :: "int"   and d :: "num" shows "(Frct (a, numeral b) < Frct (c, numeral d)) = (a * numeral d < c * numeral b)"
elims_3: fixes x :: "'b poly" assumes "\<not> isconstant x" obtains c :: "'b poly" and p :: "'b poly" where "x = CN c 0 p"
rel2pD: fixes R :: "('a \<times> 'b) set"   and a :: "'a"   and b :: "'b" assumes "rel2p R a b" shows "(a, b) \<in> R"
domIff: fixes a :: "'b"   and m :: "'b \<Rightarrow> 'c option" shows "(a \<in> dom m) = (m a \<noteq> \<bottom>)"
init_args_def: fixes I :: "\<I>"   and n :: "nat"   and L :: "nat set"   and R :: "nat set"   and pos :: "bool" shows "init_args I n L R pos = \<lparr>args_ivl = I, args_n = n, args_L = L, args_R = R, args_pos = pos\<rparr>"
tranclp_unfold: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" shows "r\<^sup>+\<^sup>+ a b \<equiv> (a, b) \<in> {(x, y). r x y}\<^sup>+"
prob_short_count_le: shows "\<forall>\<^sup>\<infinity>n. probGn p n (\<lambda>es. real n / 2 \<le> real (short_count (edge_space.edge_ugraph n es))) \<le> real (2 * (k - 2)) * real n powr (\<epsilon> * real k - 1)"
map_def: fixes f :: "'c" shows "G\<epsilon>'_\<eta>G.map f = (if C.arr f then (G \<circ> \<epsilon>'.map) (C.cod f) \<cdot>\<^sub>D (\<eta> \<circ> G) f else D.null)"
R_1: shows "[u]\<^sub>\<R> \<in> \<R>"
rep_Gromov_completion_to_Gromov_completion: fixes y :: "'b" shows "rep_Gromov_completion (to_Gromov_completion y) = (\<lambda>n. y)"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_step_subst: fixes \<delta> :: "'c \<Rightarrow> ('b, 'a) Term.term"   and F :: "(('b, 'c) Term.term \<times> ('b, 'c) Term.term) list" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F) = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)"
unfold_\<S>_eq: fixes \<G> :: "'b ltl set"   and \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "Only_G \<G>"   and "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>"   and "mojmir_def.accept \<up>af\<^sub>G (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q}" shows "\<forall>\<^sub>\<infinity>n. mojmir_def.\<S> \<Sigma> \<up>af\<^sub>G (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q} (Suc n) = (\<lambda>q. \<up>step q (w n)) ` mojmir_def.\<S> \<Sigma> \<up>af\<^sub>G\<^sub>\<UU> (Abs (Unf\<^sub>G \<phi>)) w {q. \<G> \<Turnstile>\<^sub>P Rep q} n \<union> {Abs \<phi>} \<union> {q. \<G> \<Turnstile>\<^sub>P Rep q}"
assms_2: fixes r1 :: "'a"   and r2 :: "'b" assumes "P x (r1, r2)" shows "P x' (r1, r2)"
from_hma\<^sub>m_def: fixes a :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>m a = Matrix.mat CARD('f) CARD('e) (\<lambda>(i, j). a $h mod_type_class.from_nat i $h mod_type_class.from_nat j)"
coerce_ssum: shows "COERCE('e \<oplus> 'f, 'g \<oplus> 'h) = ssum_map\<cdot>COERCE('e, 'g)\<cdot>COERCE('f, 'h)"
at_within_Icc_at: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and x :: "'a"   and b :: "'a" assumes "class.order_topology less_eq less open"   and "less a x"   and "less x b" shows "topological_space.at_within open x (ord.atLeastAtMost less_eq a b) = topological_space.at_within open x UNIV"
compl_le_swap2: fixes y :: "'a"   and x :: "'a" assumes "- y \<le> x" shows "- x \<le> y"
trans: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and u :: "('a, 'b) Term.term" assumes "s \<sqsubseteq> t"   and "t \<sqsubseteq> u" shows "s \<sqsubseteq> u"
sset_streams: fixes s :: "'b stream"   and A :: "'b set" assumes "sset s \<subseteq> A" shows "s \<in> streams A"
qe: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bbs bs (qe p) = Ifm bbs bs (E p)"
cases: fixes x :: "'a word \<times> 'a word" obtains ip_start :: "'a word" and ip_end :: "'a word" where "x = (ip_start, ip_end)"
intra_path_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.intra_path sourcenode targetnode kind valid_edge n as n' \<equiv> CFG.path sourcenode targetnode valid_edge n as n' \<and> (\<forall>a\<in>set as. intra_kind (kind a))"
swaps_ins_sws: fixes k :: "'b \<Rightarrow> 'c"   and x :: "'b"   and xs :: "'b list" shows "swaps (ins_sws k x xs) (x # xs) = insort_key k x xs"
matches_DeMorgan: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and m1 :: "'a match_expr"   and m2 :: "'a match_expr"   and a :: "action"   and p :: "'b" shows "matches \<gamma> (MatchNot (MatchAnd m1 m2)) a p = (matches \<gamma> (MatchNot m1) a p \<or> matches \<gamma> (MatchNot m2) a p)"
Or: fixes n :: "nat"   and p :: "pf"   and q :: "pf" shows "dfa_of_pf n (Or p q) = or_dfa (dfa_of_pf n p) (dfa_of_pf n q)"
multrow_mat: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat"   and k :: "nat"   and a :: "'a" assumes "A \<in> carrier_mat n nc" shows "multrow k a A = multrow_mat n k a * A"
simps_2: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "Matrix_Tensor.mult id f" shows "mult.vec_vec_Tensor f (x # xs) ys = mult.times f x ys @ mult.vec_vec_Tensor f xs ys"
normalized_cscalar_prod: fixes v :: "complex vec"   and n :: "nat"   and w :: "complex vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "inner_prod w v = vec_norm v * vec_norm w * inner_prod (vec_normalize w) (vec_normalize v)"
append_eq_map_conv: fixes ys :: "'b list"   and zs :: "'b list"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "(ys @ zs = map f xs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
AtDynCom: fixes x :: "'d"   and r :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and ad :: "('d, 'e, 'f) ann"   and f :: "'d \<Rightarrow> ('d, 'e, 'f) com"   and a :: "'d set \<times> ('d, 'e, 'f) com" assumes "x \<in> r"   and "atomicsR \<Gamma> \<Theta> ad (f x) a" shows "atomicsR \<Gamma> \<Theta> (AnnRec r ad) (DynCom f) a"