hcju/isabelleps huggingface dataset:
oclosed_oreachable_inclosed: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and \<zeta> :: "'e net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p :: "net_tree"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool" assumes "(\<sigma>, \<zeta>) \<in> oreachable (oclosed (opnet np p)) (\<lambda>_ _ _. True) U" shows "(\<sigma>, \<zeta>) \<in> oreachable (opnet np p) (otherwith (=) (net_tree_ips p) inoclosed) U"
axioms_1: fixes V :: "'c set"   and f :: "'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real" assumes "Function_Norm.continuous V f norm" shows "linearform V f"
l13_var: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> y) = - x \<sqinter> - - y"
fi_map_None: fixes u :: "'v" shows "(fi_map u = None) = (u \<notin> f ` V)"
finite_if_finite_subsets_card_bdd: fixes F :: "'b set"   and C :: "nat" assumes "\<And>G. \<lbrakk>G \<subseteq> F; finite G\<rbrakk> \<Longrightarrow> card G \<le> C" shows "finite F \<and> card F \<le> C"
simps_2: fixes uv :: "'c"   and uw :: "'d"   and \<Pi> :: "('c, 'd) chain" shows "chainlen (\<lbrace> uv \<rbrace> \<cdot> uw \<cdot> \<Pi>) = 1 + chainlen \<Pi>"
sup_idem: fixes x :: "'b \<times> 'c" shows "sup_prod x x = x"
disjoint_sets_measure_Cesaro_tendsto_zero: fixes A :: "nat \<Rightarrow> 'a set"   and P :: "'a measure" assumes "\<And>n. A n \<in> sets M"   and "disjoint_family A"   and "absolutely_continuous M P"   and "sets P = sets M"   and "emeasure P (space M) \<noteq> \<infinity>" shows "(\<lambda>n. (\<Sum>i<n. Sigma_Algebra.measure P (space M \<inter> (T ^^ i) -` A i)) / real n) \<longlonglongrightarrow> 0"
boundedI: fixes b :: "'b::semilattice_sup"   and a :: "'b::semilattice_sup"   and c :: "'b::semilattice_sup" assumes "(b::'b::semilattice_sup) \<le> (a::'b::semilattice_sup)"   and "(c::'b::semilattice_sup) \<le> (a::'b::semilattice_sup)" shows "(b::'b::semilattice_sup) \<squnion> (c::'b::semilattice_sup) \<le> (a::'b::semilattice_sup)"
assms_7: shows "g \<circ> h \<circ> u \<le>\<le> u \<circ> h \<circ> g"
bounded_proj: fixes k :: "'b" assumes "k \<in> basis" shows "bounded ((\<lambda>x. x proj k) ` range f)"
ranking_def: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and f :: "nat \<times> 'd \<Rightarrow> nat" shows "ranking A w f \<equiv> (\<forall>v\<in>gunodes A w. f v \<le> 2 * card (nodes A)) \<and> (\<forall>v\<in>gunodes A w. \<forall>u\<in>gusuccessors A w v. f u \<le> f v) \<and> (\<forall>v\<in>gunodes A w. gaccepting A v \<longrightarrow> even (f v)) \<and> (\<forall>v\<in>gunodes A w. \<forall>r k. gurun A w r v \<longrightarrow> smap f (gtrace r v) = sconst k \<longrightarrow> odd k)"
summable_imp_convergent_sum_upto: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" obtains c :: "'a" where "(sum_upto f \<longlongrightarrow> c) at_top"
lnth_lmap_ldropn_Suc: fixes n :: "nat"   and xs :: "'c llist"   and f :: "'c \<Rightarrow> 'd" assumes "enat (Suc n) < llength xs" shows "lmap f (ldropn n xs) $ Suc 0 = lmap f xs $ Suc n"
r_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier R"   and "x \<in> carrier R" shows "H #> x \<subseteq> carrier R"
select_convs_1: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and more :: "'d" shows "counter \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = counter"
cases: fixes a1 :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a2 :: "'a expr \<times> String.literal list \<times> 'a OCL_Types.type option \<times> 'a expr"   and a3 :: "'a OCL_Types.type \<times> 'a OCL_Types.type \<times> 'a OCL_Types.type" assumes "a1 \<turnstile>\<^sub>I a2 : a3" obtains \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" and src :: "'a expr" and \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and its_ty :: "'a OCL_Types.type" and its :: "String.literal list" and body :: "'a expr" and \<rho> :: "'a OCL_Types.type" where "a1 = \<Gamma>" and "a2 = (src, its, Some its_ty, body)" and "a3 = (\<tau>, \<sigma>, \<rho>)" and "\<Gamma> \<turnstile>\<^sub>E src : \<tau>" and "element_type \<tau> \<sigma>" and "\<sigma> \<le> its_ty" and "\<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, its_ty)) its) \<turnstile>\<^sub>E body : \<rho>"
HIGH_CALL: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" assumes "({twiddle} \<union> G) \<rhd> body : twiddle" shows "G \<rhd> Call : twiddle"
map_M_ok_I: fixes xs :: "'a list"   and h :: "'b"   and f :: "'a \<Rightarrow> ('b, 'c, 'd) prog" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<turnstile> ok f x"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (f x) h" shows "h \<turnstile> ok map_M f xs"
divide_p_def: fixes p :: "int"   and x :: "int"   and y :: "int" shows "divide_p p x y = mult_p p x (inverse_p p y)"
add_zero: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and a :: "'b" assumes "comm_monoid_add_ow S pls z"   and "a \<in> S" shows "pls z a = a"
Int_insert_right_if1: fixes a :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "a \<in> A" shows "A \<inter> insert a B = insert a (A \<inter> B)"
two_comparisons_into_case_order_6: fixes x :: "'a"   and y :: "'a"   and Q :: "'c"   and R :: "'c"   and P :: "'c" shows "(if lt x y then Q else if lt y x then R else P) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
right_lang_Reverse: fixes q :: "hf" shows "nfa.right_lang (Reverse_nfa M) q = rev ` left_lang q"
sup_absorb1: fixes sup :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and y :: "'a::type"   and x :: "'a::type" assumes "class.semilattice_sup (sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)"   and "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (y::'a::type) (x::'a::type)" shows "(sup::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (x::'a::type) (y::'a::type) = x"
cfun_arg_cong: fixes x :: "'e"   and y :: "'e"   and f :: "'e \<rightarrow> 'f" assumes "x = y" shows "f\<cdot>x = f\<cdot>y"
F\<^sub>i\<^sub>n\<^sub>v_def: fixes \<alpha> :: "nat \<times> nat"   and \<tau> :: "nat"   and y :: "nat" shows "F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y = y ^ \<tau> mod fst \<alpha>"
intros: fixes f :: "real \<Rightarrow> real"   and cs :: "real msstream"   and f' :: "real \<Rightarrow> real"   and c :: "real" assumes "local.asymp_powser f cs"   and "f' \<in> O[F](\<lambda>_. 1)"   and "\<forall>\<^sub>F x in F. f' x = c + f x * h x" shows "local.asymp_powser f' (MSSCons c cs)"
valuation_for_operator_variables_is: fixes \<Pi> :: "'b strips_problem"   and \<pi> :: "'b strips_operator list list"   and k :: "nat"   and op :: "'b strips_operator" assumes "is_parallel_solution_for_problem \<Pi> \<pi>"   and "k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1"   and "op \<in> set (\<Pi>\<^sub>\<O>)" shows "valuation_for_operator_variables \<Pi> \<pi> (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"
disconnected_nodes_eq2_h: fixes doc_ptr :: "(_) document_ptr" assumes "doc_ptr \<noteq> new_document_ptr" shows "|h \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h' \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
rel_optionI: fixes x :: "'d option"   and y :: "'e option"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "Option.is_none x = Option.is_none y"   and "\<lbrakk>\<not> Option.is_none x; \<not> Option.is_none y\<rbrakk> \<Longrightarrow> P (the x) (the y)" shows "rel_option P x y"
events_before_exist: fixes x :: "('a \<times> 'b) event"   and i :: "nat" assumes "x \<in> set (history i)" shows "\<exists>pre. pre @ [x] prefix of i"
assms: fixes B :: "'a tensor" assumes "B \<in> set Bs" shows "cprank_max1 B"
of_rat_of_int_poly: fixes p :: "int poly" shows "map_poly of_rat (of_int_poly p) = of_int_poly p"
elims_2: fixes x :: "'v"   and xa :: "'v list" assumes "adj_path x xa" obtains v :: "'v" where "x = va__" and "xa = []" and "va__ \<in> V"   | v :: "'v" and u :: "'v" and us :: "'v list" where "x = v" and "xa = u # us" and "adjacent v u \<and> adj_path u us"
apply_pdevs_prod_of_pdevs: fixes x :: "'d pdevs"   and y :: "'e pdevs"   and i :: "nat" shows "pdevs_apply (prod_of_pdevs x y) i = (pdevs_apply x i, pdevs_apply y i)"
add_del_block_inv: fixes bl :: "'a set" assumes "bl \<subseteq> bl" shows "incidence_system.del_block (derived_incidence_sys.add_block bl) bl = \<B>\<^sup>D"
injI: fixes f :: "'q \<Rightarrow> 'r" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
subcls1_induct: fixes G :: "prog"   and P :: "qtname \<Rightarrow> bool"   and a :: "qtname" assumes "ws_prog G"   and "\<And>x. \<forall>y. G\<turnstile>x\<prec>\<^sub>C1y \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
one_dim_onorm: fixes f :: "'c \<Rightarrow> 'd" assumes "clinear f" shows "onorm f = norm (f (1::'c))"
hyps: fixes bs :: "'b list" shows "sum_list (map2 (*s) qs bs) \<in> local.span (set bs)"
simps_9: fixes f :: "'c"   and b :: "'a set"   and c :: "('a, 'b, 'c) com" shows "redexes (Guard f b c) = {Guard f b c}"
ennreal_cong: fixes x :: "real"   and y :: "real" assumes "x = y" shows "ennreal x = ennreal y"
Hadjicostas_Chapman_formula_real: fixes z :: "real" assumes "- 1 < z" shows "Hadjicostas_nn_integral z = ennreal (Gamma (z + 2) * (Re (zeta (complex_of_real (z + 2))) - 1 / (z + 1)))"
mset_neg_plus: fixes A :: "'a zmultiset"   and B :: "'a zmultiset" shows "mset_neg (A + B) = mset_neg A - mset_pos B + (mset_neg B - mset_pos A)"
simps_5: fixes v :: "('a, 'b) form"   and va :: "('a, 'b) form" shows "compl (And v va) = Neg (And v va)"
flowsto_def: fixes X0 :: "('a \<times> 'b \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and CX :: "('a \<times> 'b \<Rightarrow>\<^sub>L 'a) set"   and X1 :: "('a \<times> 'b \<Rightarrow>\<^sub>L 'a) set" shows "flowsto X0 T CX X1 = (\<forall>(x0, d0)\<in>X0. \<exists>h\<in>T. h \<in> existence_ivl0 x0 \<and> (flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1 \<and> (\<forall>h'\<in>{0<--<h}. (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX))"
rotate9: fixes A9 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and A5 :: "fmla"   and A6 :: "fmla"   and A7 :: "fmla"   and A8 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A9 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H)))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 H)))))))) \<turnstile> B"
subspace_linear_fun_on: fixes S :: "'c set" assumes "subspace S" shows "subspace {E. linear_fun_on S E}"
t1: shows "term_ok \<Theta> (subst_bv (Fv x \<tau>') s)"
subspaceI: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale"   and "(0::'d) \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> scale c x \<in> S" shows "module.subspace scale S"
wf_prog_def: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" shows "wf_prog wf_md P = (wf_syscls P \<and> distinct_fst (classes P) \<and> (\<forall>c\<in>set (classes P). wf_cdecl wf_md P c))"
r_into_trancl: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
mult_ac_1: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "a * b * c = a * (b * c)"
comp_def: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "f \<circ> g = (\<lambda>x. f (g x))"
constT_ss_objT: shows "constT \<subseteq>\<^sub>m objT"
asm: shows "\<xi> \<Turnstile>\<^sub>c \<phi> R\<^sub>c \<psi> and\<^sub>c F\<^sub>c \<phi>"
Ide_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "Ide (Dom t)"
result_vmsaux_def: shows "result_vmsaux = (\<lambda>args (cur, auxlist). foldr (\<union>) (concat (map (\<lambda>(t, rel). if left (args_ivl args) \<le> cur - t then [rel] else []) auxlist)) {})"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "([:x:] = [:y:]) = (x = y)"
integrate_insert_commute: fixes M :: "('a, 'b) message set"   and m :: "('a, 'b) message"   and s :: "('a, 'b option) insert_message list" assumes "consistent (M \<union> {m})"   and "is_delete m \<or> m \<notin> M"   and "is_associated_string M s" shows "is_certified_associated_string (M \<union> {m}) (integrate s m)"
finite_funs\<^sub>s\<^sub>t\<^sub>p: fixes x :: "('c, 'd) strand_step" shows "finite (funs\<^sub>s\<^sub>t\<^sub>p x)"
usubstappf_and_usubstappp_det_2: 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 \<alpha> :: "game"   and V :: "variable set" assumes "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg"   and "snd (usubstappp \<sigma> V \<alpha>) \<noteq> undefg" shows "snd (usubstappp \<sigma> U \<alpha>) = snd (usubstappp \<sigma> V \<alpha>)"
insert_union_subset_iff: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(add_mset a A \<subset># B) = (a \<in># B \<and> A \<subset># B - {#a#})"
Par1B: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "x \<sharp> Q" shows "P \<parallel> Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P' \<parallel> Q"
exist_conjugateI: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "\<exists>g. conjugation_pair f g"
lu: shows "([\<theta>]<>\<nu>) = (l1, u1)"
simps_1: fixes M :: "(('fun, 'c, 'sets) prot_fun, 'd) Term.term set"   and TI :: "('sets set \<times> 'sets set) list"   and x :: "'d" shows "intruder_synth_mod_eq_timpls M TI (Var x) = (Var x \<in> M)"
le_diff_imp_gt_prod: fixes x :: "real"   and y :: "real"   and z :: "real"   and w :: "real" assumes "\<bar>x - y\<bar> < \<bar>z - w\<bar>"   and "x + y = z + w" shows "z * w < x * y"
geq_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<succeq> y"   and "y \<succeq> z"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<succeq> z"
Ifg: fixes t :: "real"   and s :: "real" assumes "0 \<le> t"   and "t \<le> s"   and "s \<le> dist x y" shows "dist (f s) (f t) = s - t"
coreE: fixes x :: "'a"   and s :: "'a set" assumes "x \<in> core s" obtains m :: "'a set" where "m open" and "m \<subseteq> s" and "x \<in> m"
minus_eq: fixes x :: "'c"   and y :: "'c" shows "x \<ominus>\<^bsub>M\<^esub> y = x \<oplus>\<^bsub>M\<^esub> \<ominus>\<^bsub>M\<^esub> y"
f_less2: fixes p :: "nat \<times> nat"   and q :: "nat \<times> nat" assumes "p \<noteq> q"   and "p \<in> UU"   and "q \<in> UU" shows "f {p, q} < Suc (Suc 0)"
Swap: fixes X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and Y :: "('a, 'b, 'c, 'd, 'e) qTerm"   and zs :: "'c"   and x :: "'d"   and y :: "'d" assumes "(X, Y) \<in> qSwapped" shows "(X, Y #[[x \<and> y]]_zs) \<in> qSwapped"
INF_image: fixes g :: "'c \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "Inf (g ` f ` A) = Inf ((g \<circ> f) ` A)"
disjoint_add_mset_2: fixes A :: "'f multiset"   and b :: "'f"   and B :: "'f multiset" shows "({#} = A \<inter># add_mset b B) = (b \<notin># A \<and> {#} = A \<inter># B)"
nonzero_mult_divide_mult_cancel_right: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times b c) = divide a b"
rebase_id: fixes x :: "'p mod_ring" shows "@@x = @x"
lt_of_comp_of_ords: fixes le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lt :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "lt_of_comp (comp_of_ords le lt) = lt"
mem4: shows "rotate (n - 1) ls = rotate (|ls| - 1) (rotate m ms)"
non_exec_assign: fixes \<sigma> :: "'a control_state_scheme"   and f :: "'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "\<not> exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> _bind_SE _ (assign f) M) = (f \<sigma> \<Turnstile> M)"
exhaust: fixes y :: "'b negation_type" obtains x1 :: "'b" where "y = Pos x1"   | x2 :: "'b" where "y = Neg x2"
dehomogenize_dehomogenize: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "dehomogenize x (dehomogenize x p) = dehomogenize x p"
lincomb_list_as_mat_mult: fixes ws :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "\<forall>w\<in>set ws. dim_vec w = nr" shows "nr.lincomb_list c ws = mat_of_cols nr ws *\<^sub>v vec (length ws) c"
dense_linorder_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "class.dense_linorder less_eq less \<equiv> class.dense_order less_eq less \<and> class.linorder less_eq less"
not_eq_pos_or_neg_iff_1: fixes lb :: "real"   and ub :: "real"   and p :: "real poly" shows "(\<forall>z. lb < z \<and> z \<le> ub \<longrightarrow> poly p z \<noteq> 0) = ((\<forall>z. lb < z \<and> z \<le> ub \<longrightarrow> 0 < poly p z) \<or> (\<forall>z. lb < z \<and> z \<le> ub \<longrightarrow> poly p z < 0))"
progress_e1: fixes \<S> :: "s_context"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and ts :: "t list"   and k :: "nat"   and lholed :: "Lholed" assumes "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts" shows "\<not> Lfilled k lholed [$Return] es"
intros_1: fixes \<pi> :: "char list \<Rightarrow> com option"   and cs :: "com \<times> (char list \<Rightarrow> int \<Rightarrow> int)" shows "small_steps \<pi> cs (Some cs)"
simps_2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and v :: "'c"   and va :: "'c list" shows "filter2 P (v # va) [] = ([], [])"
intros_8: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F" shows "\<Gamma> \<turnstile> F \<^bold>\<or> G"
empty_zero: shows "(\<chi>i. frechet I (local.empty i) (fst \<nu>) (snd \<nu>)) = 0"
urel64_inj': fixes x :: "uint64"   and z :: "int"   and y :: "uint64" assumes "urel64 x z"   and "urel64 y z" shows "x = y"
space_uniform_measure: fixes M :: "'c measure"   and A :: "'c set" shows "space (uniform_measure M A) = space M"
pow_eq_div2: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and m :: "nat"   and n :: "nat" assumes "Group.group G"   and "x \<in> carrier G"   and "x [^]\<^bsub>G\<^esub> m = x [^]\<^bsub>G\<^esub> n" shows "x [^]\<^bsub>G\<^esub> (m - n) = \<one>\<^bsub>G\<^esub>"
rem_implicit_pres_ops_in\<delta>D: fixes op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "op \<in> set (ast_problem.ast\<delta> (rem_implicit_pres_ops prob))" shows "\<exists>op'. op' \<in> set (ast_problem.ast\<delta> prob) \<and> op = rem_implicit_pres op'"
equality: fixes r :: "('c, 'd, 'e) graph_scheme"   and r' :: "('c, 'd, 'e) graph_scheme" assumes "nodes r = nodes r'"   and "edges r = edges r'"   and "more r = more r'" shows "r = r'"
some_equality: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(SOME x. P x) = a"
finite_measure_axioms_def: fixes M :: "'b measure" shows "finite_measure_axioms M \<equiv> emeasure M (space M) \<noteq> \<top>"
parSym: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi" assumes "\<And>R S. (R \<parallel> S, S \<parallel> R) \<in> Rel"   and "\<And>R S x. (R, S) \<in> Rel \<Longrightarrow> (<\<nu>x>R, <\<nu>x>S) \<in> Rel" shows "P \<parallel> Q \<leadsto>[Rel] Q \<parallel> P"
bex_one_point2: fixes A :: "'a set"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>A. a = x \<and> P x) = (a \<in> A \<and> P a)"
integral_mono: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and g :: "'b \<Rightarrow> real" assumes "integrable f M"   and "integrable g M"   and "f \<le> g" shows "\<integral> f \<partial>M \<le> \<integral> g \<partial>M"
nfa_accepting_help: fixes as :: "bool list"   and q :: "bool list" shows "nfa_accepting' as q = (\<exists>i<length as. i < length q \<and> as ! i \<and> q ! i)"
contradiction: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_y_neq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post" shows "False"
add_carrier_mat': fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A + B \<in> carrier_mat nr nc"
atLeastSucAtMost_greaterThanAtMost: fixes l :: "nat"   and u :: "nat" shows "{Suc l..u} = {l<..u}"
set_membership_equality_technicalD: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" assumes "{x} \<union> set xs = {y} \<union> set ys" shows "x = y \<or> y \<in> set xs"
inf_sup_ord_3: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and y :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) x (y::'b::type))"
assms_3: shows "enat ts + right (args_ivl args) < enat dt"
cases: fixes a1 :: "'a list"   and a2 :: "'b sequent" assumes "(a1, a2) \<in> Ax" obtains i :: "nat" where "a1 = []" and "a2 = ( \<LM> At i  \<RM> \<Rightarrow>* At i)"   | "a1 = []" and "a2 = ( \<LM> ff  \<RM> \<Rightarrow>* Em)"
simps_3: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and z :: "Vars.var"   and body :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* (y, z)" shows "(Terms.Let \<Gamma> body)[y::=z] = Terms.Let \<Gamma>[y::h=z] body[y::=z]"
A1: shows "(cl, f) \<in> (SIGMA cl:CL. {f. f \<in> pset cl \<inter> cl})"