hcju/isabelleps huggingface dataset:
Aexp_Let_simp1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" assumes "\<not> nonrec \<Gamma>" shows "Aexp_syn' a (Terms.Let \<Gamma> e) = Afix \<Gamma>\<cdot>(Aexp_syn' a e \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>) f|` (- domA \<Gamma>)"
basis_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "X_insert.basis_in \<E> X = indep_system.basis \<E> (X_insert.indep_in \<E>) X"
basevars: fixes vs :: "state \<Rightarrow> 'a"   and c :: "'a" assumes "basevars vs" shows "\<exists>u. vs u = c"
orientable_7_implies_8: fixes x :: "'a" assumes "local.comp_inf.symmetric x \<longrightarrow> (\<exists>y. y \<squnion> id y = x \<and> y \<sqinter> id y = x \<sqinter> top)" shows "local.comp_inf.symmetric x \<and> x \<sqinter> top = bot \<longrightarrow> bounded_distrib_allegory.orientable (\<sqinter>) (\<squnion>) bot id x"
ide_dom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "ide (local.dom f)"
orthogonal_clauses_5: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and a :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner"   and "real_inner.orthogonal inner a x"   and "real_inner.orthogonal inner a y" shows "real_inner.orthogonal inner a (minus x y)"
map_add_dom_app_simps_3: fixes m :: "'d"   and l2 :: "'d \<Rightarrow> 'e option"   and l1 :: "'d \<Rightarrow> 'e option" assumes "m \<notin> dom l2" shows "(l1 ++ l2) m = l1 m"
word_of_int_2p: fixes n :: "nat" shows "word_of_int (2 ^ n) = 2 ^ n"
good: shows "qGoodAbs (pick \<circ> asAbs) (qAbs xs x (pick X)) $[[pickE rho]]"
downI_def: shows "downI = liftI (\<lambda>d l p a. downI' d l p a 0 0)"
simps_3: fixes a :: "'a"   and b :: "'a"   and \<sigma>' :: "'a list"   and x :: "'a"   and y :: "'a" shows "OPT2 (a # b # \<sigma>') [x, y] = (if a = x then (0, []) # OPT2 (b # \<sigma>') [x, y] else if b = x then (0, []) # OPT2 (b # \<sigma>') [x, y] else (1, []) # OPT2 (b # \<sigma>') [y, x])"
H2_idem2: fixes P :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" shows "H2 (H2 P) = H2 P"
normal_form_with_simplifier_def: fixes \<phi> :: "String.literal ltln" shows "normal_form_with_simplifier \<phi> \<equiv> min_dnf (simplify Slow (normal_form (simplify Slow \<phi>)))"
HOM_carrier: fixes M :: "('i, 'a, 'k) Module_scheme"   and N :: "('j, 'a, 'l) Module_scheme" shows "carrier (HOM\<^bsub>R\<^esub> M N) = mHom R M N"
euler_err_fas_nth_def: fixes X0 :: "'c"   and h :: "floatarith"   and CX :: "floatarith list"   and i :: "nat" shows "euler_err_fas_nth X0 h CX i = h ^\<^sub>e 2 / floatarith.Num 2 * ode_d_fa_nth 0 CX (ode_fa CX) i"
finite_SigmaI2: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "finite {x \<in> A. B x \<noteq> {}}"   and "\<And>a. a \<in> A \<Longrightarrow> finite (B a)" shows "finite (Sigma A B)"
mbuf2_take_add'_1: fixes f :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> 'a"   and xs :: "event_data option list set list"   and ys :: "event_data option list set list"   and buf :: "event_data option list set list \<times> event_data option list set list"   and zs :: "'a list"   and buf' :: "event_data option list set list \<times> event_data option list set list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and j :: "nat"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and P' :: "String.literal \<Rightarrow> nat option"   and j' :: "nat" assumes "mbuf2_take f (mbuf2_add xs ys buf) = (zs, buf')"   and "wf_mbuf2' \<sigma> P V j n R \<phi> \<psi> buf"   and "rel_mapping (\<le>) P P'"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) [Monitor.progress \<sigma> P \<phi> j..<Monitor.progress \<sigma> P' \<phi> j'] xs"   and "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>)) [Monitor.progress \<sigma> P \<psi> j..<Monitor.progress \<sigma> P' \<psi> j'] ys"   and "j \<le> j'" shows "wf_mbuf2' \<sigma> P' V j' n R \<phi> \<psi> buf'"
recvmsg_simps_1: fixes P :: "'d \<Rightarrow> bool"   and m :: "'d" shows "recvmsg P (broadcast m) = True"
add_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b" shows "less_eq (plus c a) (plus c b)"
uniformly_summable_deriv_ln_Gamma: fixes z :: "'a"   and d :: "real" assumes "z \<noteq> (0::'a)"   and "0 < d"   and "d \<le> norm z / 2" shows "uniformly_convergent_on (ball z d) (\<lambda>k z. \<Sum>i<k. inverse (of_nat (Suc i)) - inverse (z + of_nat (Suc i)))"
pmf_le_0_iff: fixes M :: "'c pmf"   and p :: "'c" shows "(pmf M p \<le> 0) = (pmf M p = 0)"
finfun_default_def: shows "finfun_default \<equiv> map_fun finfun_apply id finfun_default_aux"
top_add: fixes x :: "'c" assumes "(0::'c) \<le> x" shows "\<top> + x = \<top>"
perp_col1: fixes C :: "'p"   and X :: "'p"   and A :: "'p"   and B :: "'p"   and D :: "'p" assumes "C \<noteq> X"   and "A B Perp C D"   and "Col C D X" shows "A B Perp C X"
maximal_independent_subset: fixes V :: "('a, 'b) vec set" obtains B :: "('a, 'b) vec set" where "B \<subseteq> V" and "vec.independent B" and "V \<subseteq> vec.span B"
rem_inclusion: fixes Cn :: "'b set \<Rightarrow> 'b set"   and B :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "Tarskian_logic Cn"   and "B \<in> Tarskian_logic.remainder Cn A \<phi>" shows "B \<subseteq> A"
fresh_finite_set_at_base: fixes S :: "'a set"   and a :: "'a" assumes "finite S" shows "atom a \<sharp> S = (a \<notin> S)"
map_pmf_def: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c pmf" shows "map_pmf f M = M \<bind> (\<lambda>x. return_pmf (f x))"
Mertens_mangoldt_versus_ln: fixes I :: "nat set"   and n :: "nat" assumes "I \<subseteq> {1..n}" shows "\<bar>(\<Sum>i\<in>I. mangoldt i / real i) - (\<Sum>p | prime p \<and> p \<in> I. ln (real p) / real p)\<bar> \<le> 3"
rOpen: fixes x :: "name"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi" assumes "cP = \<lparr>\<nu>x\<rparr>P"   and "cRs = M\<lparr>\<nu>*(xvec @ x # yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "x \<in> supp N"   and "x \<sharp> xvec"   and "x \<sharp> yvec"   and "x \<sharp> M"   and "x \<sharp> \<Psi>"   and "distinct xvec"   and "distinct yvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* yvec"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* M"   and "xvec \<sharp>* C"   and "x \<sharp> C"   and "yvec \<sharp>* C" shows "Prop"
matrix'_compose: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'c) vec, 'c) vec"   and Z :: "(('a, 'd) vec, 'd) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and g :: "('a, 'c) vec \<Rightarrow> ('a, 'd) vec" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)"   and "is_basis (set_of_vector Z)"   and "linear (*s) (*s) f"   and "linear (*s) (*s) g" shows "matrix' X Z (g \<circ> f) = matrix' Y Z g ** matrix' X Y f"
fst_spp_of: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" shows "fst (spp_of r) = lt r"
mwb_lens_axioms_def: fixes x :: "'d \<Longrightarrow> 'e" shows "mwb_lens_axioms x \<equiv> \<forall>\<sigma> v u. put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u"
error_pure: fixes e :: "'b"   and h :: "'a" shows "pure (error e) h"
trm_simp_1: fixes x :: "'b"   and y :: "'b" assumes "Var x = Var y" shows "x = y"
last_ConsL: fixes xs :: "'d list"   and x :: "'d" assumes "xs = []" shows "last (x # xs) = x"
real_cond_exp_int_1: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "integrable M f" shows "integrable M (real_cond_exp M F f)"
thetaSeqWD_Wretr: shows "thetaSeqWD \<subseteq> Wretr (thetaSeqWD \<union> Wbis)"
swap_cs_Recv_Snapshot: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "cs e i = cs e' i"
offs_pred_ub_aux: fixes ns :: "nat list"   and ub :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and j :: "nat" assumes "offs_pred ns ub xs index key mi ma"   and "i < length ns"   and "j < length ns"   and "i \<le> j"   and "0 < offs_num (length ns) xs index key mi ma j" shows "ns ! j + offs_num (length ns) xs index key mi ma j \<le> ub"
orthogonal_clauses_6: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner zero a"
cases: fixes a1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and a2 :: "'d"   and a3 :: "('c \<Rightarrow> real) set"   and a4 :: "'d"   and a5 :: "('c \<Rightarrow> real) set" assumes "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>a4, a5\<rangle>" obtains A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)" and l :: "'d" and Z :: "('c \<Rightarrow> real) set" where "a1 = Aa__" and "a2 = la__" and "a3 = Za__" and "a4 = la__" and "a5 = Za__"   | A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)" and l :: "'d" and Z :: "('c \<Rightarrow> real) set" and l' :: "'d" and Z' :: "('c \<Rightarrow> real) set" and l'' :: "'d" and Z'' :: "('c \<Rightarrow> real) set" where "a1 = A" and "a2 = l" and "a3 = Z" and "a4 = l''" and "a5 = Z''" and "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l', Z'\<rangle>" and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<alpha> \<langle>l'', Z''\<rangle>"
L1: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> Rel" shows "P \<leadsto>\<^sup>^<Rel> Q"
proj3_def: shows "\<pi>\<^sub>3 \<equiv> \<lambda>(uu_, dsk, uu_, uu_, uu_, uu_). dsk"
elims_2: fixes x :: "'c arctic_delta"   and xa :: "'c arctic_delta" assumes "weak_gt_arctic_delta x xa" obtains uu_ :: "'c arctic_delta" where "x = uu_" and "xa = MinInfty_delta"   | xb :: "'c" and y :: "'c" where "x = Num_arc_delta xb" and "xa = Num_arc_delta y" and "y < xb"
y_in_2: fixes z :: "\<nu>"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" shows "[z\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> x) \<^bold>\<rightarrow> \<phi> z in dw]"
Src_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "Src (Cod t) = Src t"
Card_def: fixes i :: "V" shows "Card i \<equiv> i = vcard i"
Suc: fixes f :: "nat \<Rightarrow> 'a" shows "map f [0..<n + m] = map f [0..<n] @ map (\<lambda>i. f (i + n)) [0..<m]"
size_replicate_mset: fixes n :: "nat"   and M :: "'b" shows "size (replicate_mset n M) = n"
Leaf_in_tree_sigma: fixes M :: "'d measure" shows "{\<langle>\<rangle>} \<in> sets (tree_sigma M)"
weakly_step_consistent: shows "\<forall>s t u a. vpeq u s t \<and> vpeq (current s) s t \<and> kprecondition s a \<and> kprecondition t a \<and> current s = current t \<longrightarrow> vpeq u (kstep s a) (kstep t a)"
assms: fixes i :: "'a" assumes "i \<le> n" shows "X i \<in> M \<rightarrow>\<^sub>M N"
perfect_number_def: fixes n :: "nat" shows "perfect_number n \<equiv> n = aliquot_sum n"
comp_4: shows "insert a_ (insert b_ S_) \<inter> B = {}"
assms_1: shows "convert_poly var p (xs' @ x # xs) = Some p'"
inf_set_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B = Collect (inf (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
l9_41_2: 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 P :: "'a"   and R :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TSP Bet A B C P R"   and "Tarski_neutral_dimensionless.OSP Bet A B C P Q" shows "Tarski_neutral_dimensionless.TSP Bet A B C Q R"
bind_returns_result_E: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and y :: "'c" assumes "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y" obtains x :: "'d" and h' :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>r y"
iMin_mono2: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set" assumes "mono f"   and "I \<noteq> {}" shows "iMin (f ` I) = f (iMin I)"
inside_inside_compact_connected: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "compact T"   and "connected T"   and "S \<subseteq> inside T" shows "inside S \<subseteq> inside T"
basis_to_substdbasis_subspace_isomorphism: fixes B :: "'a set" assumes "independent B" shows "\<exists>f d. card d = card B \<and> linear f \<and> f ` B = d \<and> f ` span B = {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} \<and> inj_on f (span B) \<and> d \<subseteq> Basis"
br_if: fixes i :: "nat"   and \<C> :: "t_context"   and ts :: "t list" assumes "i < length (label \<C>)"   and "label \<C> ! i = ts" shows "\<C> \<turnstile> [Br_if i] : ts @ [T_i32] _> ts"
cart2_proj2: fixes v :: "(real, 2) vec" shows "cart2_pt (proj2_pt v) = v"
equalityI: fixes aeq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. aeq x y = (x = y)" shows "equality aeq"
sum_content_area_over_thin_division: fixes \<D> :: "'a set set"   and S :: "'a set"   and a :: "'a"   and b :: "'a"   and i :: "'a"   and c :: "real" assumes "\<D> division_of S"   and "S \<subseteq> cbox a b"   and "i \<in> Basis"   and "a \<bullet> i \<le> c"   and "c \<le> b \<bullet> i"   and "\<And>K. K \<in> \<D> \<Longrightarrow> K \<inter> {x. x \<bullet> i = c} \<noteq> {}" shows "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<le> 2 * content (cbox a b)"
Image_def: fixes r :: "('b \<times> 'a) set"   and s :: "'b set" shows "r `` s = rel_ext (\<lambda>y. \<exists>x\<in>s. (x, y) \<in> r)"
ground_fmla_aux_simps_3: fixes A :: "fmla"   and B :: "fmla"   and S :: "atom set" shows "ground_fmla_aux (A OR B) S = (ground_fmla_aux A S \<and> ground_fmla_aux B S)"
D2: shows "D2 r (\<upsilon>, \<tau>', \<tau>'', \<upsilon>')"
ivp_solsD_3: fixes X :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" assumes "X \<in> Sols f U S t\<^sub>0 s" shows "X \<in> U s \<rightarrow> S"
nemp_pos_len: fixes r :: "'c list" assumes "r \<noteq> \<epsilon>" shows "1 \<le> \<^bold>|r\<^bold>|"
is_span: shows "C.span \<epsilon>.cod.leg0 \<epsilon>.cod.leg1"
xor_left_inverse: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<oplus> x \<oplus> y = y"
closed_sum_right_subset: fixes A :: "'d set"   and B :: "'d set" assumes "(0::'d) \<in> A" shows "B \<subseteq> A +\<^sub>M B"
T_def_spec: fixes P :: "'b process" shows "T P = {tr. \<exists>a. a \<in> F P \<and> tr = fst a}"
f_def: fixes i :: "'a" shows "f i = (cmod (Rep_ell2 \<psi> i))\<^sup>2"
dist_metric_completion_limit: fixes x :: "'b metric_completion"   and y :: "'b metric_completion" shows "(\<lambda>n. dist (rep_metric_completion x n) (rep_metric_completion y n)) \<longlonglongrightarrow> dist x y"
antisymmetric_axioms: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "asymmetric A less" shows "antisymmetric A less"
simps_1: fixes uu :: "'a tree" shows "mrbst_diff \<langle>\<rangle> uu = return_pmf \<langle>\<rangle>"
efficient_map_def: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b list" shows "efficient_map f l \<equiv> rev (Efficient_Distinct.map_tailrec f l [])"
dX_empty_or_singleton: fixes X :: "'x set" assumes "inj_on Xd X" shows "\<forall>d. dX X d = {} \<or> (\<exists>x. dX X d = {x})"
less_add_same_cancel2: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus b a) = less zero b"
wlog_rotation_to_positive_y_axis: fixes u :: "complex_homo"   and P :: "complex_homo \<Rightarrow> bool" assumes "u \<in> unit_disc"   and "u \<noteq> 0\<^sub>h"   and "\<And>\<phi> u. \<lbrakk>u \<in> unit_disc; u \<noteq> 0\<^sub>h; P (moebius_pt (moebius_rotation \<phi>) u)\<rbrakk> \<Longrightarrow> P u"   and "\<And>x. \<lbrakk>is_imag x; 0 < Im x; Im x < 1\<rbrakk> \<Longrightarrow> P (of_complex x)" shows "P u"
Rep_eqD: fixes x :: "nat"   and y :: "nat" assumes "int x = int y" shows "x = y"
stable_rank_succeed: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat"   and x :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "infinite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i)"   and "x \<in> mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i"   and "q\<^sub>0 \<notin> F" shows "\<not> semi_mojmir.stable_rank \<Sigma> \<delta> q\<^sub>0 w x i"
max_ge_x: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.non_strict_order less_eq" shows "less_eq x (ord.max less_eq x y)"
simps: fixes n :: "nat" shows "ups n = map (\<lambda>k. [0..<k + 1]) [0..<n + 1]"
rel_def: shows "rel \<equiv> rel_asset_i \<union> rel_owner_ii \<union> rel_asset_ii \<union> rel_owner_iii \<union> rel_asset_iii \<union> rel_owner_iv \<union> rel_asset_iv \<union> rel_owner_v \<union> rel_asset_v \<union> rel_prik \<union> rel_pubk \<union> rel_sesk \<union> rel_fact \<union> rel_mult \<union> rel_hash \<union> rel_dec \<union> rel_enc \<union> rel_sep \<union> rel_con \<union> rel_id_agent \<union> rel_id_invk \<union> rel_id_sesk \<union> rel_id_fact \<union> rel_id_mult \<union> rel_id_hash \<union> rel_id_crypt \<union> rel_id_sep \<union> rel_id_con"
induct: fixes x1 :: "exp \<times> Vars.var \<times> Vars.var + (Vars.var \<times> exp) list \<times> Vars.var \<times> Vars.var"   and x2 :: "exp + (Vars.var \<times> exp) list"   and P :: "exp \<times> Vars.var \<times> Vars.var + (Vars.var \<times> exp) list \<times> Vars.var \<times> Vars.var \<Rightarrow> exp + (Vars.var \<times> exp) list \<Rightarrow> bool" assumes "subst_subst_heap_graph x1 x2"   and "\<And>subst_subst_heap_sum x y z. P (Inl (Var x, y, z)) (Inl (Var x[y::v=z]))"   and "\<And>subst_subst_heap_sum e v y z. \<lbrakk>subst_subst_heap_graph (Inl (e, y, z)) (subst_subst_heap_sum (Inl (e, y, z))); P (Inl (e, y, z)) (subst_subst_heap_sum (Inl (e, y, z)))\<rbrakk> \<Longrightarrow> P (Inl (App e v, y, z)) (Inl (App (projl (subst_subst_heap_sum (Inl (e, y, z)))) v[y::v=z]))"   and "\<And>subst_subst_heap_sum \<Gamma> y z body. \<lbrakk>atom ` domA \<Gamma> \<sharp>* (y, z); subst_subst_heap_graph (Inr (\<Gamma>, y, z)) (subst_subst_heap_sum (Inr (\<Gamma>, y, z))); P (Inr (\<Gamma>, y, z)) (subst_subst_heap_sum (Inr (\<Gamma>, y, z))); subst_subst_heap_graph (Inl (body, y, z)) (subst_subst_heap_sum (Inl (body, y, z))); P (Inl (body, y, z)) (subst_subst_heap_sum (Inl (body, y, z)))\<rbrakk> \<Longrightarrow> P (Inl (Let \<Gamma> body, y, z)) (Inl (Let (projr (subst_subst_heap_sum (Inr (\<Gamma>, y, z)))) (projl (subst_subst_heap_sum (Inl (body, y, z))))))"   and "\<And>subst_subst_heap_sum x y z e. \<lbrakk>atom x \<sharp> (y, z); subst_subst_heap_graph (Inl (e, y, z)) (subst_subst_heap_sum (Inl (e, y, z))); P (Inl (e, y, z)) (subst_subst_heap_sum (Inl (e, y, z)))\<rbrakk> \<Longrightarrow> P (Inl (Lam [x]. e, y, z)) (Inl (Lam [x]. projl (subst_subst_heap_sum (Inl (e, y, z)))))"   and "\<And>subst_subst_heap_sum b y z. P (Inl (Bool b, y, z)) (Inl (Bool b))"   and "\<And>subst_subst_heap_sum scrut e1 e2 y z. \<lbrakk>subst_subst_heap_graph (Inl (scrut, y, z)) (subst_subst_heap_sum (Inl (scrut, y, z))); P (Inl (scrut, y, z)) (subst_subst_heap_sum (Inl (scrut, y, z))); subst_subst_heap_graph (Inl (e1, y, z)) (subst_subst_heap_sum (Inl (e1, y, z))); P (Inl (e1, y, z)) (subst_subst_heap_sum (Inl (e1, y, z))); subst_subst_heap_graph (Inl (e2, y, z)) (subst_subst_heap_sum (Inl (e2, y, z))); P (Inl (e2, y, z)) (subst_subst_heap_sum (Inl (e2, y, z)))\<rbrakk> \<Longrightarrow> P (Inl (scrut ? e1 : e2, y, z)) (Inl (projl (subst_subst_heap_sum (Inl (scrut, y, z))) ? projl (subst_subst_heap_sum (Inl (e1, y, z))) : projl (subst_subst_heap_sum (Inl (e2, y, z)))))"   and "\<And>subst_subst_heap_sum y z. P (Inr ([], y, z)) (Inr [])"   and "\<And>subst_subst_heap_sum v e \<Gamma> y z. \<lbrakk>subst_subst_heap_graph (Inl (e, y, z)) (subst_subst_heap_sum (Inl (e, y, z))); P (Inl (e, y, z)) (subst_subst_heap_sum (Inl (e, y, z))); subst_subst_heap_graph (Inr (\<Gamma>, y, z)) (subst_subst_heap_sum (Inr (\<Gamma>, y, z))); P (Inr (\<Gamma>, y, z)) (subst_subst_heap_sum (Inr (\<Gamma>, y, z)))\<rbrakk> \<Longrightarrow> P (Inr ((v, e) # \<Gamma>, y, z)) (Inr ((v, projl (subst_subst_heap_sum (Inl (e, y, z)))) # projr (subst_subst_heap_sum (Inr (\<Gamma>, y, z)))))" shows "P x1 x2"
monotone_if_bot: fixes g :: "'c llist \<Rightarrow> 'd"   and bound :: "'c llist"   and bot :: "'d"   and f :: "'c llist \<Rightarrow> 'd"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. g x = (if x \<sqsubseteq> bound then bot else f x)"   and "\<And>x y. \<lbrakk>x \<sqsubseteq> y; \<not> x \<sqsubseteq> bound\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>x. \<not> x \<sqsubseteq> bound \<Longrightarrow> ord bot (f x)"   and "ord bot bot" shows "monotone (\<sqsubseteq>) ord g"
fib_Cassini_nat: fixes n :: "nat" shows "fib (Suc (Suc n)) * fib n = (if even n then (fib (Suc n))\<^sup>2 - 1 else (fib (Suc n))\<^sup>2 + 1)"