hcju/isabelleps huggingface dataset:
linear_sub_cfs: fixes f :: "nat \<Rightarrow> 'a"   and d :: "'a"   and g :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "f \<in> carrier (UP R)"   and "d \<in> carrier R"   and "g = Cring_Poly.compose R f (monom (UP R) d 1)" shows "g i = d [^] i \<otimes> f i"
all_ren: fixes y :: "name"   and \<phi> :: "fmla"   and x :: "name" assumes "y \<notin> Fvars \<phi>" shows "all x \<phi> = all y (\<phi>(x::=Var y))"
indhyp: shows "(\<Sum> F \<le> z) = (\<forall>a\<in>F. a \<le> z)"
concat_replicate_trivial: fixes i :: "nat" shows "concat (replicate i []) = []"
distinct_7: fixes x1 :: "bool"   and x51 :: "'b gexp"   and x52 :: "'b gexp" shows "Bc x1 \<noteq> Nor x51 x52"
empty_map_add: fixes m :: "'g \<Rightarrow> 'h option" shows "Map.empty ++ m = m"
not_SN_onE: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "\<not> SN_on r A" obtains f :: "nat \<Rightarrow> 'b" where "f 0 \<in> A" and "\<forall>i. (f i, f (Suc i)) \<in> r"
conv_complement_sub_inf: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>T * - (x * y) \<sqinter> y = bot"
upper_triangular_mult: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" assumes "upper_triangular A"   and "upper_triangular B" shows "upper_triangular (A ** B)"
SeqStTermP_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm"   and s :: "tm"   and s' :: "tm"   and k :: "tm"   and k' :: "tm"   and v :: "tm"   and i :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'"   and "H \<turnstile> s EQ s'"   and "H \<turnstile> k EQ k'" shows "H \<turnstile> SeqStTermP v i t u s k IFF SeqStTermP v i t' u' s' k'"
le_numeral_extra_4: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one one"
lex_pair_def: shows "lex_pair \<equiv> less_than <*lex*> less_than"
diff_add_eq_diff_diff_swap: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a - (b + c) = a - c - b"
measure_spmf_return_pmf_None: shows "measure_spmf (return_pmf None) = null_measure (count_space UNIV)"
list_emb_Cons_iff2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "\<not> P x y" shows "list_emb P (x # xs) (y # ys) = list_emb P (x # xs) ys"
a_parent_child_rel_def: fixes h :: "(_) heap" shows "local.CD.a_parent_child_rel h = {(parent, child). parent |\<in>| object_ptr_kinds h \<and> child \<in> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ` set |h \<turnstile> get_child_nodes parent|\<^sub>r}"
fres_interchange: fixes z :: "'b \<times> 'c"   and x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod (times_prod' z (prod.fres x y)) (prod.fres (times_prod' z x) y)"
weight: shows "\<forall>u\<in>U. 0 < w u \<and> w u \<le> real c"
idinfo_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and X :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "\<langle>n, X\<rangle> \<in> spied s" shows "X \<in> spied s"
npClose: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup> \<in> carrier R"
Rep_uexpr_inverse: fixes x :: "('a, 'b) uexpr" shows "mk\<^sub>e \<lbrakk>x\<rbrakk>\<^sub>e = x"
subgroupE_4: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "subgroup H (G\<lparr>carrier := S\<rparr>)"   and "a \<in> H"   and "b \<in> H" shows "a \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> b \<in> H"
Imp_triv_I: fixes H :: "fm set"   and B :: "fm"   and A :: "fm" assumes "H \<turnstile> B" shows "H \<turnstile> A IMP B"
Fpow_Pow_finite: fixes A :: "'a set" shows "Fpow A = Pow A \<inter> {A. finite A}"
diagseq_fold_reduce: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and n :: "nat" assumes "Diagonal_Subsequence.subseqs P" shows "subseqs.diagseq P n = subseqs.fold_reduce P 0 n n"
diff_le_0_iff_le: fixes a :: "'b"   and b :: "'b" shows "(a - b \<le> (0::'b)) = (a \<le> b)"
star_trans: fixes x :: "'a" shows "transitive (x\<^sup>\<star>)"
GroupHom: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "VectorSpaceHom smult V smult' T" shows "GroupHom V T"
x_y_in_carrier: fixes x :: "'a"   and y :: "'a" assumes "x \<succeq>[relation] y" shows "x \<in> carrier \<and> y \<in> carrier"
finite_total_preorder_of_weak_ranking: fixes xs :: "'b set list"   and A :: "'b set" assumes "\<Union> (set xs) = A"   and "is_finite_weak_ranking xs" shows "finite_total_preorder_on A (of_weak_ranking xs)"
eq_lcs2: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "a \<diamondsuit> H = b \<diamondsuit> H" shows "a \<in> b \<diamondsuit> H"
wp1_prec: fixes c1 :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int"   and x :: "char list"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" assumes "wp\<^sub>1 c1 Q l s"   and "l x = prec c1 e s" shows "wp\<^sub>1 c1 (\<lambda>l s. Q l s \<and> e s = l x) l s"
6: shows "j - length M2 < length M * length M2 \<longrightarrow> j < length (v # M) * length M2"
smpl: shows "Sample_L req dt a1 stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st buffer"
altc_single: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'm" shows "altc (csingle x) f = f x"
oconf_init_obj: fixes G :: "prog"   and obj :: "obj"   and r :: "oref"   and s :: "st" assumes "wf_prog G"   and "case r of Inl a \<Rightarrow> is_type G (obj_ty obj) | Inr C \<Rightarrow> is_class G C" shows "G,s\<turnstile>obj\<lparr>values := init_vals (var_tys G (tag obj) r)\<rparr>\<Colon>\<preceq>\<surd>r"
base_enabled: fixes vs :: "state \<Rightarrow> 'b"   and A :: "state \<times> state \<Rightarrow> bool"   and s :: "state" assumes "basevars vs"   and "\<exists>c. \<forall>u. vs u = c \<longrightarrow> A (s, u)" shows "s \<Turnstile> Enabled A"
closure_bounded_linear_image_subset_eq: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "bounded_linear f" shows "closure (f ` closure S) = closure (f ` S)"
proj2_Col_iff_euclid_cart2: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "z_non_zero p"   and "z_non_zero q"   and "z_non_zero r" shows "proj2_Col p q r = real_euclid.Col (cart2_pt p) (cart2_pt q) (cart2_pt r)"
class_fields_f_def: fixes cld :: "cld" shows "class_fields_f cld = (case cld of cld_def dcl cl fds mds \<Rightarrow> fds)"
eq_one_str: fixes U :: "'b set" assumes "U \<subseteq> X" shows "ind_one_str U = one_str U"
fact_prod_rev: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero n = semiring_1.of_nat one plus zero (prod ((-) n) {0..<n})"
terminates_on_conv_dom_terminates_within: fixes g :: "'e \<Rightarrow> ('f, 'e) step" shows "terminates_on g = dom (terminates_within g)"
order_iff: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a \<le>s b) = (a = signed.min a b)"
homeomorphic_minimal: fixes s :: "'a set"   and t :: "'b set" shows "s homeomorphic t = (\<exists>f g. (\<forall>x\<in>s. f x \<in> t \<and> g (f x) = x) \<and> (\<forall>y\<in>t. g y \<in> s \<and> f (g y) = y) \<and> continuous_on s f \<and> continuous_on t g)"
elims: fixes x :: "interval_tree"   and y :: "nat idx_interval list" assumes "in_traverse x = y" obtains "x = Tip" and "y = []"   | l :: "interval_tree" and it :: "nat idx_interval" and m :: "nat" and r :: "interval_tree" where "x = Node l it m r" and "y = in_traverse l @ it # in_traverse r"
gcsrp_top_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "gcsrp_top G H \<equiv> \<lambda>x\<in>gcsrp_map G H ` set_rcs G H. \<lambda>y\<in>gcsrp_map G H ` set_rcs G H. gcsrp_map G H (c_top G H ((gcsrp_map G H\<inverse>\<^bsub>gcsrp_map G H ` set_rcs G H,set_rcs G H\<^esub>) x) ((gcsrp_map G H\<inverse>\<^bsub>gcsrp_map G H ` set_rcs G H,set_rcs G H\<^esub>) y))"
suc: shows "(\<pi> \<guillemotleft> j) (Suc k) \<noteq> (\<pi>' \<guillemotleft> j') (Suc k')"
dy_fake_msg_mono_bad: fixes bad :: "agent set"   and bad' :: "agent set"   and I :: "msg set"   and C :: "chan set" assumes "bad \<subseteq> bad'" shows "dy_fake_msg bad I C \<subseteq> dy_fake_msg bad' I C"
healthy_sound: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "healthy t"   and "sound P" shows "sound (t P)"
eq_iff: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "(a = b) = (a \<preceq> b \<and> b \<preceq> a)"
ilist_Previous_def: fixes f :: "nat \<Rightarrow> 'd"   and init :: "'d"   and n :: "nat" shows "f\<^bsup>\<leftarrow> init\<^esup> n \<equiv> case n of 0 \<Rightarrow> init | Suc n' \<Rightarrow> f n'"
inject: fixes x1 :: "'b"   and x2 :: "'c \<Rightarrow> 'c"   and y1 :: "'b"   and y2 :: "'c \<Rightarrow> 'c" shows "(sAbs x1 x2 = sAbs y1 y2) = (x1 = y1 \<and> x2 = y2)"
geq_my_U_imp_weak_preference: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> local.\<P>"   and "q \<in> local.\<P>"   and "local.some_best \<succeq>[\<R>] local.some_worst \<and> (local.some_worst, local.some_best) \<notin> \<R>"   and "local.my_U q \<le> local.my_U p" shows "p \<succeq>[\<R>] q"
input_sizes_remove_weights: fixes m :: "real mat convnet" shows "input_sizes m = input_sizes (remove_weights m)"
distinct_disc_4: fixes "term" :: "'b local.term" assumes "is_Hcomp term" shows "\<not> is_Prim\<^sub>0 term"
u_PiE: shows "u \<in> carrier_direct_lim \<rightarrow>\<^sub>E A"
in_realizable_in_HRB_slice: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "realizable n ns n'"   and "n'' \<in> set ns"   and "n' \<in> S" shows "n'' \<in> HRB_slice S"
has_sort_code': fixes \<Theta> :: "exetheory"   and T :: "typ"   and S :: "String.literal set" assumes "exe_wf_theory' \<Theta>" shows "has_sort (osig (sig (translate_theory \<Theta>))) T S = exehas_sort (exesorts (exesig \<Theta>)) T S"
IH: assumes "set p1 \<subseteq> K"   and "set p2 \<subseteq> K" shows "polynomial K (poly_mult p1 p2)"
AReturn_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('c, 'd) chan"   and p :: "PrIds"   and v :: "state \<Rightarrow> 'd" shows "Return ch p v \<equiv> $Calling ch p \<and> rbit<ch!p>$ = $cbit<ch!p> \<and> res<ch!p>$ = $v"
removeAll_append: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" shows "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
functor_axioms: shows "functor F.C.CCC.comp B F.C.T.ToCT"
digraphD_7: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
eval_monom_1_conv: fixes a :: "real \<times> real list"   and basis :: "(real \<Rightarrow> real) list" shows "eval_monom a basis = (\<lambda>x. fst a * eval_monom (1, snd a) basis x)"
disc_14: fixes x4 :: "'a" shows "\<not> isRecv (Snapshot x4)"
times_ordinal_def: shows "(*) = (\<lambda>x. ordinal_rec 0 (\<lambda>p w. w + x))"
insertion_add: fixes f :: "nat \<Rightarrow> 'b"   and p :: "'b mpoly"   and q :: "'b mpoly" shows "insertion f (p + q) = insertion f p + insertion f q"
91: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> (z \<squnion> x)) = y \<squnion> (z \<squnion> x)"
pos_imaginary_parts_3: fixes p :: "int poly"   and x :: "complex" assumes "p \<noteq> 0"   and "poly (complex_of_int_poly p) x = 0"   and "0 < Im x" shows "Im x \<in> real_of_3 ` set (pos_imaginary_parts_3 p)"
less_iff_Suc_add: fixes m :: "nat"   and n :: "nat" shows "(m < n) = (\<exists>k. n = Suc (m + k))"
params_nth: fixes i :: "nat"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and cf :: "'var \<Rightarrow> 'val option" assumes "i < length fs" shows "params fs cf ! i = (fs ! i) cf"
bisim1Sync2: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and v :: "'addr val"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (sync\<^bsub>V\<^esub> (Val v) e2, xs) \<leftrightarrow> ([v, v], xs, Suc (length (compE2 e1)), None)"
strong_cases: fixes a1 :: "lam"   and a2 :: "lam"   and x :: "name" assumes "a1 \<rightarrow> a2" obtains a :: "name" where "a1 = Var a" and "a2 = Var a"   | t1 :: "lam" and t2 :: "lam" where "a1 = App t1 t2" and "a2 = App t1 t2"   | t :: "lam" and t' :: "lam" where "\<lbrakk>x \<sharp> a1; x \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = Lam [x].t \<and> a2 = t' \<and> t \<rightarrow> t'"
ins_queue_invar: fixes t :: "('c, 'd) SkewBinomialTree"   and q :: "('c, 'd) SkewBinomialTree list" assumes "tree_invar t"   and "queue_invar q" shows "queue_invar (ins t q)"
simps: fixes A :: "'b mat"   and bs :: "'b mat list" shows "partition_ev_blocks A bs = (let n = dim_row A in if n = 0 then bs else case lookup_other_ev (A $$ (n - 1, n - 1)) (n - 1) A of None \<Rightarrow> A # bs | Some i \<Rightarrow> case split_block A (Suc i) (Suc i) of (UL, x, xa, LR) \<Rightarrow> partition_ev_blocks UL (LR # bs))"
backwards_wdr_res_stab: fixes ts :: "traffic"   and d :: "cars"   and n :: "nat"   and ts' :: "traffic"   and v :: "view"   and c :: "cars" shows "(ts \<^bold>\<midarrow>wdr( d, n ) \<^bold>\<rightarrow> ts') \<and> 0 < \<parallel>ext v\<parallel> \<and> len v ts' c = ext v \<and> restrict v (res ts') c = lan v \<and> |lan v| = 1 \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (res ts) c = lan v \<and> |lan v| = 1"
rbt_insertw_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "ord.rbt_insert_with less f = ord.rbt_insert_with_key less (\<lambda>_. f)"
word_zero_le: fixes y :: "'a word" shows "0 \<le> y"
keys_sum: fixes b :: "'e \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'd"   and I :: "'e set" shows "keys (sum b I) \<subseteq> (\<Union>i\<in>I. keys (b i))"
return_def: fixes a :: "'f" shows "State_Monad.return a = State (Pair a)"
inverse_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse one = one"
andci_def: fixes e1 :: "nat"   and e2 :: "nat"   and s :: "bddi" shows "andci e1 e2 s \<equiv> fci s \<bind> (\<lambda>(f, s). iteci_lu e1 e2 f s)"
c0: shows "ennreal (1 - d) < emeasure M {x \<in> G. real_cond_exp M Invariants f x / (real c0 + 1) < d}"
usubstappt_times_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and \<eta> :: "trm" assumes "usubstappt \<sigma> U (Times \<theta> \<eta>) \<noteq> undeft" shows "usubstappt \<sigma> U \<theta> \<noteq> undeft \<and> usubstappt \<sigma> U \<eta> \<noteq> undeft"
has_field_derivative_imp_has_derivative: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and F :: "'b filter" assumes "(f has_field_derivative D) F" shows "(f has_derivative (*) D) F"
lossless_cond_spmf: fixes p :: "'h spmf"   and A :: "'h set" shows "lossless_spmf (cond_spmf p A) = (set_spmf p \<inter> A \<noteq> {})"
induct: fixes P :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list \<times> 'a rose_tree \<Rightarrow> bool"   and a0 :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list \<times> 'a rose_tree" assumes "\<And>p t. (\<And>x. x \<in> set (zipper_children (p, t)) \<Longrightarrow> P x) \<Longrightarrow> P (p, t)" shows "P a0"
of_nat_eq_iff: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and m :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = semiring_1.of_nat one plus zero n) = (m = n)"
sim1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q" shows "\<tau>.P \<leadsto><Rel> Q"
col_per_perp: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> C"   and "Col B C D"   and "Per A B C" shows "C D Perp A B"
rep_eq: fixes x :: "'a" shows "mapping_of (Const x) = Const\<^sub>0 x"
finitesum_inc_zero: fixes f :: "'f \<Rightarrow> 'a set"   and I :: "'f set" assumes "f \<in> I \<rightarrow> {X. submodule R M X}"   and "I \<noteq> {}" shows "\<zero> \<in> finitesum M I f"
pr_b_least2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec2" shows "b_least2 f \<in> PrimRec2"
eq_refl: fixes x :: "'c list"   and y :: "'c list" assumes "x = y" shows "suffix x y"
OclOr_null1: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" assumes "X \<tau> \<noteq> true \<tau>"   and "X \<tau> \<noteq> UML_Types.bot_class.bot \<tau>" shows "(null or X) \<tau> = null \<tau>"
independence_symmetric: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "traces ind"   and "ind a b" shows "ind b a"
units_eq: fixes e1 :: "'c \<times> 'd"   and e2 :: "'c \<times> 'd" assumes "e1 \<in> pm_prod_E"   and "e2 \<in> pm_prod_E"   and "ps_prod_D e1 e2" shows "e1 = e2"
simps_8: fixes a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Max a b) vs = max (interpret_floatarith a vs) (interpret_floatarith b vs)"
conf_NT: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" shows "(P,h \<turnstile> v :\<le> NT) = (v = Null)"
mono_simple_extension: fixes K :: "'a set"   and K' :: "'a set"   and x :: "'a" assumes "K \<subseteq> K'" shows "simple_extension K x \<subseteq> simple_extension K' x"
intros_2: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and nat_rule :: "'d \<Rightarrow> 'b \<Rightarrow> ('b, 'a) antecedent fset \<Rightarrow> bool"   and rule :: "'d"   and c :: "'b"   and ants :: "('b, 'a) antecedent fset"   and \<Gamma> :: "'b fset"   and a :: "nat"   and s :: "'c" assumes "ND_Rules_Inst freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"   and "nat_rule rule c ants"   and "\<And>ant f. \<lbrakk>ant |\<in>| ants; f |\<in>| \<Gamma>\<rbrakk> \<Longrightarrow> freshenLC a ` a_fresh ant \<inter> lconsts f = {}"   and "\<And>ant. ant |\<in>| ants \<Longrightarrow> freshenLC a ` a_fresh ant \<inter> subst_lconsts s = {}" shows "ND_Rules_Inst.eff freshenLC renameLCs lconsts subst subst_lconsts nat_rule (NatRule rule) (\<Gamma> \<turnstile> subst s (Abstract_Formulas.freshen freshenLC renameLCs a c)) ((\<lambda>ant. (\<lambda>p. subst s (Abstract_Formulas.freshen freshenLC renameLCs a p)) |`| a_hyps ant |\<union>| \<Gamma> \<turnstile> subst s (Abstract_Formulas.freshen freshenLC renameLCs a (a_conc ant))) |`| ants)"
divide: fixes e :: "'b \<times> 'b"   and H :: "'b pair_pre_digraph"   and w :: "'b"   and G :: "'b pair_pre_digraph" assumes "e \<in> parcs H"   and "w \<notin> pverts H"   and "??.Subdivision.pair_sd G H" shows "??.Subdivision.pair_sd G (subdivide H e w)"
Pair_le: fixes a :: "'c"   and b :: "'d"   and c :: "'c"   and d :: "'d" shows "((a, b) \<le> (c, d)) = (a \<le> c \<and> b \<le> d)"
ordLeq_reflexive: fixes r :: "('c \<times> 'c) set" assumes "Well_order r" shows "r \<le>o r"
Shift_def: fixes Kl :: "'c list set"   and k :: "'c" shows "BNF_Greatest_Fixpoint.Shift Kl k = {kl. k # kl \<in> Kl}"
rotate1_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "rotate1 (map f xs) = map f (rotate1 xs)"
ap_list_def: fixes fs :: "('e \<Rightarrow> 'd) list"   and xs :: "'e list" shows "fs \<diamondop> xs = List.bind fs (\<lambda>f. List.bind xs (\<lambda>x. [f x]))"
extend_fun: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and B1 :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "B \<subseteq> B1" shows "f \<in> A \<rightarrow> B1"
ins_bal: fixes t :: "'a btree"   and k :: "nat"   and x :: "'a" assumes "bal t" shows "bal_up\<^sub>i (ins k x t)"
False: shows "(\<Pi>\<^sub>E i\<in>I. topspace (X i)) \<noteq> {}"
transfers_slice_kinds_slice_edges: fixes S :: "'node set"   and as :: "'edge list"   and s :: "'state" shows "transfers (slice_kinds S (slice_edges S as)) s = transfers (slice_kinds S as) s"
is_extensional: fixes f :: "'a" assumes "\<not> C.arr f" shows "C.map f = C.null"
wt_rOfFax: fixes f :: "'fsym" assumes "wtFsym f" shows "TE.wt (rOfFax f)"
L_def: shows "L = fst ` \<Union> ((edges \<circ> snd) ` transl_rules T) \<union> {S_Bot, S_Top, S_Idt} \<union> S_Const ` C"
WTrtLAss: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T' :: "ty" assumes "E V = \<lfloor>T\<rfloor>"   and "P,E,h \<turnstile> e : T'"   and "P \<turnstile> T' \<le> T" shows "P,E,h \<turnstile> V:=e : T"
C_def: fixes p :: "nat"   and a :: "int" assumes "GAUSS p a" shows "GAUSS.C p a = (\<lambda>x. x mod int p) ` GAUSS.B p a"
neqE: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less"   and "x \<noteq> y" obtains "less x y"   | "less y x"
2: shows "w\<^sub>1 @ u\<^sub>1 =\<^sub>F v\<^sub>2"
lea_right_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C F E D"
conf_hext: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and v :: "val"   and T :: "ty" assumes "h \<unlhd> h'"   and "P,h \<turnstile> v :\<le> T" shows "P,h' \<turnstile> v :\<le> T"
lappend_code_1: fixes ys :: "'b llist" shows "LNil @@ ys = ys"