hcju/isabelleps huggingface dataset:
Nsub_normN: fixes y :: "int \<times> int"   and x :: "int \<times> int" assumes "isnormNum y" shows "isnormNum (x -\<^sub>N y)"
raw_converter_invariant_run_colossless_converter: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>" shows "raw_converter_invariant \<I> \<I>' run_converter (\<lambda>conv. colossless_converter \<I> \<I>' conv \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>)"
put\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a_def: fixes character_data_ptr :: "(_) character_data_ptr"   and character_data :: "'h option RCharacterData_scheme" shows "put character_data_ptr character_data = put (cast character_data_ptr) (cast character_data)"
WhileTrue: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate"   and t :: "('a, 'c) xstate" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> \<Rightarrow> t"
ev: shows "\<forall>\<^sub>F s in nhds 1. s \<in> {s. 0 < Re s}"
coinduct_strong: fixes R :: "('g, 'h) tllist \<Rightarrow> ('g, 'h) tllist \<Rightarrow> bool"   and tllist :: "('g, 'h) tllist"   and tllist' :: "('g, 'h) tllist" assumes "R tllist tllist'"   and "\<And>tllist tllist'. R tllist tllist' \<Longrightarrow> is_TNil tllist = is_TNil tllist' \<and> (is_TNil tllist \<longrightarrow> is_TNil tllist' \<longrightarrow> terminal tllist = terminal tllist') \<and> (\<not> is_TNil tllist \<longrightarrow> \<not> is_TNil tllist' \<longrightarrow> thd tllist = thd tllist' \<and> (R (ttl tllist) (ttl tllist') \<or> ttl tllist = ttl tllist'))" shows "tllist = tllist'"
\<beta>2_le_\<beta>1: shows "\<beta>2 \<le> \<beta>1"
square_integrable_iff_lspace: fixes S :: "real set"   and f :: "real \<Rightarrow> real" assumes "S \<in> sets lebesgue" shows "(f square_integrable S) = (f \<in> lspace (lebesgue_on S) 2)"
reduce_wf_prog_to_body: fixes p :: "prog" assumes "vars p = vars\<^sub>0" shows "wfp p = wfs \<Gamma>\<^sub>0 (body p)"
assms_4: shows "InvariantWatchListsUniq (getWatchList state)"
quasi_geodesic_made_lipschitz: fixes lambda :: "real"   and C :: "real"   and a :: "real"   and b :: "real"   and c :: "real \<Rightarrow> 'a" assumes "lambda C -quasi_isometry_on {a..b} c"   and "2 * C \<le> dist (c a) (c b)" shows "\<exists>d. continuous_on {a..b} d \<and> d a = c a \<and> d b = c b \<and> (\<forall>x\<in>{a..b}. dist (c x) (d x) \<le> 4 * C) \<and> lambda (4 * C) -quasi_isometry_on {a..b} d \<and> (2 * lambda)-lipschitz_on {a..b} d \<and> hausdorff_distance (c ` {a..b}) (d ` {a..b}) \<le> 2 * C"
add_mult_distrib: fixes m :: "nat"   and n :: "nat"   and k :: "nat" shows "(m + n) * k = m * k + n * k"
lookup_def: fixes destr :: "'d \<Rightarrow> 'c"   and n :: "'e"   and s :: "'e \<Rightarrow> 'd" shows "lookup destr n s = destr (s n)"
content_ff_smult: fixes a :: "'b fract"   and p :: "'b fract poly" shows "content_ff (smult a p) =dff a * content_ff p"
rel_cases: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d list"   and b :: "'e list" assumes "list_all2 R a b" obtains "a = []" and "b = []"   | x1 :: "'d" and x2 :: "'d list" and y1 :: "'e" and y2 :: "'e list" where "a = x1 # x2" and "b = y1 # y2" and "R x1 y1" and "list_all2 R x2 y2"
nat_term_compare_inv_alt: fixes to :: "'a nat_term_order"   and x :: "'a"   and y :: "'a" shows "nat_term_compare_inv to x y = nat_term_compare to y x"
reachable_step: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and s :: "'c"   and a :: "'d"   and s' :: "'c" assumes "reachablep A I s"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "reachablep A I s'"
subst_dbfm_graph_aux_def: shows "subst_dbfm_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>subst_dbfm_aux u i t1 t2. x1 = (u, i, DBMem t1 t2) \<and> x2 = DBMem (subst_dbtm u i t1) (subst_dbtm u i t2)) \<or> (\<exists>subst_dbfm_aux u i t1 t2. x1 = (u, i, DBEq t1 t2) \<and> x2 = DBEq (subst_dbtm u i t1) (subst_dbtm u i t2)) \<or> (\<exists>subst_dbfm_aux u i A1 A2. x1 = (u, i, DBDisj A1 A2) \<and> x2 = DBDisj (((\<lambda>uu_. uu_) \<circ> subst_dbfm_aux) (u, i, A1)) (((\<lambda>uu_. uu_) \<circ> subst_dbfm_aux) (u, i, A2)) \<and> p (u, i, A1) (((\<lambda>uu_. uu_) \<circ> subst_dbfm_aux) (u, i, A1)) \<and> p (u, i, A2) (((\<lambda>uu_. uu_) \<circ> subst_dbfm_aux) (u, i, A2))) \<or> (\<exists>subst_dbfm_aux u i A. x1 = (u, i, DBNeg A) \<and> x2 = DBNeg (((\<lambda>uu_. uu_) \<circ> subst_dbfm_aux) (u, i, A)) \<and> p (u, i, A) (((\<lambda>uu_. uu_) \<circ> subst_dbfm_aux) (u, i, A))) \<or> (\<exists>subst_dbfm_aux u i A. x1 = (u, i, DBEx A) \<and> x2 = DBEx (((\<lambda>uu_. uu_) \<circ> subst_dbfm_aux) (u, i, A)) \<and> p (u, i, A) (((\<lambda>uu_. uu_) \<circ> subst_dbfm_aux) (u, i, A))))"
neg_bounds_compare_6: fixes c :: "'a"   and b :: "'a option" assumes "\<not> c \<ge>\<^sub>l\<^sub>b b" shows "c <\<^sub>l\<^sub>b b"
igConsIPresIGWls_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.igConsIPresIGWls Delta MOD \<equiv> FixSyn.igVarIPresIGWls Delta MOD \<and> FixSyn.igAbsIPresIGWls Delta MOD \<and> FixSyn.igOpIPresIGWls Delta MOD"
ss_fm_imp_Sigma_fm: fixes A :: "fm" assumes "ss_fm A" shows "Sigma_fm A"
1: fixes y :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "y \<in> set xs" shows "y \<in> ideal F \<inter> P[X]"
dg_GRPH_components_4: fixes \<alpha> :: "V" shows "dg_GRPH \<alpha>\<lparr>Cod\<rparr> = (\<lambda>\<FF>\<in>\<^sub>\<circ>all_dghms \<alpha>. \<FF>\<lparr>HomCod\<rparr>)"
assms_1: shows "K\<^bsub>m,n\<^esub> G"
M_plus_2: fixes m :: "int"   and x :: "int"   and y :: "int" shows "poly_mod.M m (x + poly_mod.M m y) = poly_mod.M m (x + y)"
simps_1: shows "bheight \<langle>\<rangle> = 0"
less_of_match_field_def: fixes a :: "of_match_field"   and b :: "of_match_field" shows "(a < b) = (a \<noteq> b \<and> less_eq_of_match_field1 a b)"
minus_closed: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "x \<ominus>\<^bsub>K [X]\<^esub> y \<in> carrier (K [X])"
nn_integral_le_const: fixes c :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and p :: "'c spmf" assumes "0 \<le> c"   and "AE x in measure_spmf p. f x \<le> c" shows "integral\<^sup>N (measure_spmf p) f \<le> c"
accs_is_b_accessible: fixes \<delta> :: "('a, 'b) ta_rule set"   and t :: "'b tree"   and q :: "'a" assumes "accs \<delta> t q" shows "q \<in> b_accessible \<delta>"
f7: shows "invara aa \<and> invar b \<and> \<alpha>' aa \<subseteq> a (\<alpha> b)"
Hausdorff_imp_t1_space: fixes X :: "'c topology" assumes "Hausdorff_space X" shows "t1_space X"
call_ctermsl_empty: fixes p :: "('a, 'b, 'c, 'd) seqp"   and p' :: "('a, 'b, 'c, 'd) seqp" assumes "p \<in> ctermsl p'"   and "not_call p" shows "not_call p'"
F_def: shows "F = Pascal_Property.inter (line A' C') (line R Q)"
set_integral_null_delta: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "integrable M f"   and "A \<in> sets M"   and "B \<in> sets M"   and "A \<Delta> B \<in> null_sets M" shows "set_lebesgue_integral M A f = set_lebesgue_integral M B f"
assms_1: shows "d (pp_of_term u) \<le> dgrad_max d"
preserves_dom: fixes f :: "'t" assumes "S'.arr f" shows "S.dom (\<Psi> f) = \<Psi> (S'.dom f)"
is_acc_run_def: fixes ma :: "'a igb_graph_rec \<Rightarrow> 'd"   and m :: "'c"   and r :: "nat \<Rightarrow> 'a \<times> nat" shows "gb_graph.is_acc_run (b_graph.to_gbg_ext (degeneralize_ext ma) m) r \<equiv> graph_defs.is_run (b_graph.to_gbg_ext (degeneralize_ext ma) m) r \<and> gb_graph.is_acc (b_graph.to_gbg_ext (degeneralize_ext ma) m) r"
one_and_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and one a = modulo a (numeral.numeral one plus (num.Bit0 num.One))"
map_ran_simps_2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'b"   and v :: "'c"   and ps :: "('b \<times> 'c) list" shows "map_ran f ((k, v) # ps) = (k, f k v) # map_ran f ps"
le_lt_convs_5: fixes x :: "'a"   and y :: "'a" shows "(comp x y = Eq) = (x = y)"
\<I>_full_sel_1: shows "outs_\<I> \<I>_full = UNIV"
length_nonempty: fixes x :: "'d list" assumes "x \<noteq> []" shows "Suc 0 \<le> length x"
R_O_Id: fixes R :: "('c \<times> 'd) set" shows "R ; Id = R"
resume_def: fixes resumption :: "('e, 'd, 'f) resumption" shows "resume resumption = (case resumption of Pause x21 x22 \<Rightarrow> x22)"
distinct_25: fixes x6 :: "'a fm" shows "FalseF \<noteq> Neg x6"
max_exists: fixes X :: "nat set" assumes "finite X" shows "X \<noteq> {} \<longrightarrow> (\<exists>x. x \<in> X \<and> (\<forall>y. y \<in> X \<longrightarrow> y \<le> x))"
root_term_def: fixes S :: "'a eclause set"   and t :: "'a trm" shows "root_term S t = (trm_rep t S = get_min t (set_of_candidate_values S t))"
bdd_above_def: fixes A :: "'c multiset set" shows "subset_mset.bdd_above A = (\<exists>M. \<forall>x\<in>A. x \<subseteq># M)"
envAction_def: shows "envAction \<equiv> \<lambda>_. [()]"
rev_upt_aux_def: fixes i :: "nat"   and j :: "nat"   and js :: "nat list" shows "rev_upt_aux i j js = [i>..j] @ js"
jointgd_tool2: fixes i :: "nat" assumes "0 < i" shows "i = Suc (i - Suc 0)"
echelon_form_upt_k_interchange: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and i :: "nat"   and m :: "'f" assumes "echelon_form_upt_k A k"   and "is_zero_row_upt_k (mod_type_class.from_nat i) k A"   and "A $ m $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "mod_type_class.from_nat i \<le> m"   and "k < ncols A" shows "echelon_form_upt_k (interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'d) \<and> mod_type_class.from_nat i \<le> n)) k"
pow_int_mult: fixes x :: "'b"   and y :: "'b"   and a :: "int" shows "pow_int (x * y) a = pow_int x a * pow_int y a"
fps_dvd1_trivial_unit_factor: fixes f :: "'b fps" assumes "f dvd 1" shows "unit_factor f = f"
measurable_T1: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a"   and M :: "'b measure" shows "MC_syntax.T K s \<rightarrow>\<^sub>M M = stream_space (count_space UNIV) \<rightarrow>\<^sub>M M"
identify_blocksD_3: fixes k :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and A :: "'b mat"   and l :: "nat" assumes "k < n"   and "(i, j) \<in> set (??.Jordan_Normal_Form_Existence.identify_blocks A k)"   and "i \<le> l"   and "l < j" shows "A $$ (l, Suc l) = (1::'b)"
prems: shows "n' \<in> sum_SDG_slice2 n''"
Mapping_lookup_filter_not_None: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "('b, 'a) mapping"   and k :: "'b" assumes "Mapping.lookup (Mapping.filter P m) k \<noteq> None" shows "Mapping.lookup (Mapping.filter P m) k = Mapping.lookup m k"
concat_sing: fixes s :: "'b list list" assumes "s = [hd s]" shows "concat s = hd s"
simple_distributed: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real" assumes "simple_distributed M X Px" shows "distributed M (count_space (X ` space M)) X (\<lambda>x. ennreal (Px x))"
rel_S_witness'_def: shows "rel_S_witness' = (id ---> id ---> id ---> id ---> id ---> id ---> id ---> id ---> id ---> id ---> map_prod Rep_S Rep_S ---> Abs_S) rel_G_witness"
project_set_Int_subset: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and A :: "'f set"   and B :: "'f set" shows "project_set h (A \<inter> B) \<subseteq> project_set h A \<inter> project_set h B"
fps_divide_X_power': fixes f :: "'a fps"   and n :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "f / fps_X ^ n = fps_shift n f"
pq: fixes x :: "'a option"   and y :: "'a option" assumes "(x, y) \<in> set_pmf pq" shows "ord_option (=) x y"
sim_rD: fixes f :: "'f \<Rightarrow> 'e"   and u :: "'f"   and v' :: "'e"   and M' :: "('g, 'h, 'e) KripkeStructure"   and a :: "'g"   and M :: "('g, 'h, 'f) KripkeStructure" assumes "(f u, v') \<in> relations M' a"   and "sim M M' f"   and "u \<in> worlds M" shows "\<exists>v. (u, v) \<in> relations M a \<and> f v = v'"
carrier_stalk_def: shows "stk1.carrier_stalk \<equiv> dlim \<FF>1 \<rho>1 (stk1.neighborhoods x)"
Supq_distr: fixes X :: "'a set"   and y :: "'a" shows "\<Squnion> X \<cdot> y = (\<Squnion>x\<in>X. x \<cdot> y)"
cut_bound: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and m :: "nat" assumes "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n"   and "F, \<Gamma> \<Rightarrow> \<Delta> \<down> m" shows "\<Gamma> \<Rightarrow> \<Delta> \<down> cut_bound n m F"
remove_vertex_walk_add: fixes x :: "'a"   and xs :: "'a list" assumes "Digraph.walk (H.remove_vertex x) xs" shows "H.walk xs"
ge_0_def: fixes x :: "real" shows "ge_0 x = (0 \<le> x)"
subset: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme" assumes "submonoid H G" shows "H \<subseteq> carrier G"
simps: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" shows "integrable M f = (\<exists>x. has_bochner_integral M f x)"
is_acc_run_limit_alt: fixes r :: "nat \<Rightarrow> 'Q" assumes "finite reachable" shows "is_acc_run r = (is_run r \<and> (\<forall>A\<in>F. limit r \<inter> A \<noteq> {}))"
B: shows "next_dom_events P D u (ipurge_tr_rev I D u xs) = next_dom_events P D u xs"
rel_pmf_measureI: fixes p :: "'a spmf"   and q :: "'b spmf"   and R :: "'a option \<Rightarrow> 'b option \<Rightarrow> bool" assumes "\<And>A. measure_pmf.prob p A \<le> measure_pmf.prob q {y. \<exists>x\<in>A. R x y}" shows "rel_pmf R p q"
submodule_asubg: fixes H :: "'a set" assumes "submodule R M H" shows "M +> H"
ComposeC: fixes T :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term list"   and f :: "('fun, 'atom, 'sets) prot_fun"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term set" assumes "length T = arity f"   and "public f"   and "\<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t" shows "M \<turnstile>\<^sub>c Fun f T"
permutation_mat_row: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "p permutes {..<n}"   and "i < n" shows "Matrix.row (permutation_mat n p) i = unit_vec n (inv p i)"
that_1: shows "\<forall>x\<in>set_spmf p. trace_eq_simcl S (f x) q"
take_hd_drop: fixes n :: "nat"   and xs :: "'e list" assumes "n < length xs" shows "take n xs @ [hd (drop n xs)] = take (Suc n) xs"
wf_R: shows "wf {(f, g). f \<in> local.ground_heads \<zeta> \<and> g \<in> local.ground_heads \<zeta> \<and> wt_sym f + \<delta>\<^sub>h * arity_sym\<^sub>h f < wt_sym g + \<delta>\<^sub>h * arity_sym\<^sub>h g}"
Int_empty_left: fixes B :: "'g set" shows "{} \<inter> B = {}"
annot_def: fixes t :: "('d, 'c) FingerTreeStruc" shows "annot t = gmft t"
inf_disjS_def: fixes IdSet :: "'d set"   and nS :: "'d \<Rightarrow> nat \<Rightarrow> 'e list" shows "inf_disjS IdSet nS \<equiv> \<forall>t i j. i \<in> IdSet \<and> j \<in> IdSet \<and> nS i t \<noteq> [] \<longrightarrow> nS j t = []"
lookup_eqvt: fixes pi :: "(name \<times> name) list"   and \<theta> :: "(name \<times> trm) list"   and X :: "name" shows "pi \<bullet> lookup \<theta> X = lookup (pi \<bullet> \<theta>) (pi \<bullet> X)"
OF_spm3_noa_none: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'e) flow_entry_match list"   and p :: "'d" assumes "no_overlaps \<gamma> ft"   and "OF_priority_match \<gamma> ft p = NoAction" shows "\<forall>e\<in>set ft. \<not> \<gamma> (ofe_fields e) p"
sel_4: fixes x4 :: "'a" shows "occurs_on (Snapshot x4) = x4"
t'_def: shows "t' \<equiv> \<langle>lb1, u, \<langle>lb2, b, A'\<rangle>\<rangle>"
finite_remove_induct: fixes B :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite B"   and "P {}"   and "\<And>A. \<lbrakk>finite A; A \<noteq> {}; A \<subseteq> B; \<And>x. x \<in> A \<Longrightarrow> P (A - {x})\<rbrakk> \<Longrightarrow> P A" shows "P B"
ctor1_dtor1: fixes x :: "'d IF1" shows "ctor1 (dtor1 x) = x"
sum_list_take_pdevs_val_eq: fixes d :: "nat"   and xs :: "'a list" shows "sum_list (take d xs) = pdevs_val (\<lambda>i. if i < d then 1 else 0) (pdevs_of_list xs)"
is_iso_imp_same_card: fixes H :: "('b, 'c) monoid_scheme"   and G :: "('d, 'e) monoid_scheme" assumes "H \<cong> G" shows "order H = order G"
simps_1: fixes A :: "ty"   and B :: "ty" shows "Vf (A \<rightarrow> B) = [(V A, V B)]"
integrable_max: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. max (f x) (g x))"
filter_empty_mset: fixes P :: "'b \<Rightarrow> bool" shows "filter_mset P {#} = {#}"
x2: shows "item_dot x = item_dot (inc_dot (length (item_\<alpha> x)) (init_item (item_nonterminal x, item_rhs x) 0))"
mono_fun_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" shows "mono_fun R = (\<forall>i. mono (R i))"
satisfies_bounded_norm_ACI: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b \<langle>\<phi>\<rangle>) = (\<AA> \<Turnstile>\<^sub>b \<phi>)"
domains_def: fixes f :: "'b" shows "C'.domains f \<equiv> {a. C'.ide a \<and> C' f a \<noteq> C'.null}"
unit_closed: shows "\<one>\<^bsub>V\<^esub> \<in> \<FF> V"
sset_sinterleave: fixes s1 :: "'a stream"   and s2 :: "'a stream" shows "i.R (sinterleave s1 s2) = i.R s1 \<union> i.R s2"
fine_sum_list_eq_diff: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and D :: "(real \<times> real \<times> real) list"   and f :: "real \<Rightarrow> real" assumes "fine \<delta> (a, b) D" shows "(\<Sum>(u, x, v)\<leftarrow>D. f v - f u) = f b - f a"
mk_globally_def: fixes x :: "'a ltln" shows "mk_globally x \<equiv> case x of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> G\<^sub>n remove_weak_ops x"
trans_Refl: fixes c :: "('test, 'atom) com"   and s :: "'state" shows "(c, s) \<rightarrow>*c (c, s)"
nsum_mem: fixes A :: "('d, 'e) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "aGroup A"   and "\<forall>j\<le>n. f j \<in> carrier A" shows "\<Sigma>\<^sub>e A f n \<in> carrier A"
that: fixes e' :: "nat \<Rightarrow> real" assumes "aform_vals e X = aform_vals e' (summarize_aforms p C d X)"   and "\<And>i. i < d \<Longrightarrow> e i = e' i"   and "e' \<in> UNIV \<rightarrow> {- 1..1}" shows "thesis"
absolutely_integrable_measurable_real: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "S \<in> sets lebesgue" shows "(f absolutely_integrable_on S) = (f \<in> borel_measurable (lebesgue_on S) \<and> integrable (lebesgue_on S) (\<lambda>x. \<bar>f x\<bar>))"
differential_zero_maxmin_component: fixes k :: "'a"   and e :: "real"   and x :: "'b"   and f :: "'b \<Rightarrow> 'a" assumes "k \<in> Basis"   and "0 < e"   and "(\<forall>y\<in>ball x e. f y \<bullet> k \<le> f x \<bullet> k) \<or> (\<forall>y\<in>ball x e. f x \<bullet> k \<le> f y \<bullet> k)"   and "f differentiable at x" shows "(\<Sum>j\<in>Basis. (frechet_derivative f (at x) j \<bullet> k) *\<^sub>R j) = (0::'b)"
simps_2: fixes h :: "'a \<times> 'b"   and t :: "('a \<times> 'b) list" shows "|(h # t)| = Suc |(delete t (fst h))|"
FCD1: shows "FCD \<Gamma>1 \<V>1 Tr\<^bsub>ES1\<^esub>"
path_entry_uncons: fixes G :: "nat graph_rec"   and E :: "(nat \<times> nat) set"   and u' :: "nat"   and l :: "nat list"   and w :: "nat" assumes "cfg_doms G"   and "path_entry E (u' # l) w" obtains "(u', w) \<in> E" and "path_entry E l u'"
prems: shows "P,E,h \<turnstile> es [:] T' # Ts'"
right_unique_conversep: fixes A :: "'g \<Rightarrow> 'f \<Rightarrow> bool" shows "right_unique A\<inverse>\<inverse> = left_unique A"
check_poly_ge: fixes p :: "('c monom \<times> 'd) list"   and q :: "('c monom \<times> 'd) list" assumes "check_poly_ge p q" shows "p \<ge>p q"
induct: fixes P :: "('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'b list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and a1 :: "'b list"   and a2 :: "'c list" assumes "\<And>f x xs y ys. P f xs ys \<Longrightarrow> P f (x # xs) (y # ys)"   and "\<And>f uv_. P f [] uv_"   and "\<And>f uu_. P f uu_ []" shows "P a0 a1 a2"
iso_in_carr: fixes f :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme"   and x :: "'e" assumes "f \<in> Group.iso G H"   and "x \<in> carrier G" shows "f x \<in> carrier H"
master1: fixes p' :: "real" assumes "g \<in> O(\<lambda>x. real x powr p')"   and "1 < (\<Sum>i<k. as ! i * bs ! i powr p')"   and "\<forall>\<^sub>F x in sequentially. 0 < f x" shows "f \<in> \<Theta>(\<lambda>x. real x powr p)"
set_mset_fold: fixes A :: "'c multiset" shows "set_mset A = fold_mset insert {} A"
basic_def: fixes ins :: "Instr" shows "basic ins = ((\<exists>c. ins = const c) \<or> ins = dup \<or> ins = pop \<or> ins = swap \<or> (\<exists>x. ins = load x) \<or> (\<exists>y. ins = store y) \<or> (\<exists>f. ins = binop f) \<or> (\<exists>g. ins = unop g) \<or> (\<exists>c1 F1. ins = getfield c1 F1) \<or> (\<exists>c2 F2. ins = putfield c2 F2) \<or> (\<exists>c3. ins = checkcast c3))"
inj_imp_inv_eq: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "inj f"   and "\<forall>x. f (g x) = x" shows "inv f = g"
lvl_update_sngl: fixes t :: "(('c \<times> 'd) \<times> nat) tree"   and x :: "'c"   and y :: "'d" assumes "invar t"   and "sngl t" shows "lvl (update x y t) = lvl t"
injective_iff_kernel_unit: shows "inj_on \<eta> R = (additive.Ker = {\<zero>})"
weakly_preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "D.isomorphic (G (trg\<^sub>C \<mu>)) (trg\<^sub>D (G \<mu>))"
floor_le_ceiling: fixes x :: "'a" shows "\<lfloor>x\<rfloor> \<le> \<lceil>x\<rceil>"
sublinear_wp_Seq: fixes x :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and y :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sublinear (wp x)"   and "sublinear (wp y)"   and "healthy (wp x)"   and "healthy (wp y)" shows "sublinear (wp (x ;; y))"
assms_3: fixes c :: "'a" assumes "\<And>b. b \<in> A \<Longrightarrow> c dvd b" shows "c dvd a"
dvd_condition: shows "\<forall>a b. to_nat a \<in> insert i (set xs) \<and> to_nat a = to_nat b \<longrightarrow> A $ from_nat c $ from_nat c dvd A $ a $ b"
bisim: shows "P,blocks1 0 (Class D # Ts) body,h \<turnstile> (e, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
HomFtorContraDom: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" shows "CatDom (Hom\<^bsub>C\<^esub>[\<midarrow>,X]) = Op C"
lconjseq_append: fixes xs :: "'a list"   and ys :: "'a list" shows "lconjseq (xs @ ys) = lconjseq xs @ map (lconjby (sum_list xs)) (lconjseq ys)"
Var\<^sub>1: fixes ls :: "val list"   and i :: "nat"   and v :: "val"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "ls ! i = v"   and "i < length ls" shows "P \<turnstile>\<^sub>1 \<langle>Var i,(h, ls)\<rangle> \<Rightarrow> \<langle>Val v,(h, ls)\<rangle>"
Suc_1: fixes fs :: "int vec list" assumes "LLL_bound_invariant False True i fs"   and "\<mu>_small_row i fs j"   and "\<mu>_bound_row_inner fs i j"   and "basis_reduction_add_rows_loop i fs j = fs'"   and "i < m"   and "j \<le> i" shows "LLL_bound_invariant True False i fs'"
zero_dom_zero: fixes i :: "nat" shows "local.dom i 0 = (i = 0)"