hcju/isabelleps huggingface dataset:
dvd_add_triv_left_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus a b) = dvd.dvd times a b"
monomial1_def: fixes pp :: "'a" shows "monomial1 pp = monomial (1::'b) pp"
H_face_cycle_succ_neq_uv: fixes a :: "'b" assumes "a \<notin> {uv, rev_G uv}" shows "H.face_cycle_succ a \<notin> {uv, rev_G uv}"
index_col: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "col A j $ i = A $$ (i, j)"
get_ACS_def: fixes M :: "'b SecurityInvariant_configured list" shows "get_ACS M \<equiv> filter (\<lambda>m. \<not> c_isIFS m) M"
uw\<theta>: shows "ide chine \<and> \<guillemotleft>the_\<theta> : s\<^sub>0 \<star> chine \<Rightarrow> r\<^sub>0\<guillemotright>"
bound_insert: fixes x :: "'b"   and X :: "'b set"   and r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c" shows "bound (insert x X) r b = (r x b \<and> bound X r b)"
lt_row_to_poly_pivot_fun_less: fixes S :: "'t set"   and A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and i1 :: "nat"   and i2 :: "nat" assumes "card S = dim_col A"   and "pivot_fun A f (dim_col A)"   and "i1 < i2"   and "i2 < dim_row A"   and "f i1 < dim_col A"   and "f i2 < dim_col A" shows "pps_to_list S ! f i2 \<prec>\<^sub>t pps_to_list S ! f i1"
inv_fg_def: fixes l :: "(bool \<times> 'b) list" shows "inv_fg l = rev (map inv1 l)"
applied_rule_rev_def: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and a :: "'e"   and x :: "'d list" shows "applied_rule_rev C a x = applied_rule C a (rev x)"
coeff_eq_zero_iff: fixes c :: "'a mpoly"   and p :: "nat \<Rightarrow>\<^sub>0 nat" shows "(MPoly_Type.coeff c p = (0::'a)) = (p \<notin> monomials c)"
Diff_mono: fixes A :: "'d set"   and C :: "'d set"   and D :: "'d set"   and B :: "'d set" assumes "A \<subseteq> C"   and "D \<subseteq> B" shows "A - B \<subseteq> C - D"
res_to_cong_simps_1: fixes m :: "int"   and x :: "int"   and y :: "int" assumes "residues m" shows "x mod m \<oplus>\<^bsub>residue_ring m\<^esub> y mod m = (x + y) mod m"
LLq_def2: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "yy \<notin> FvarsT t1" shows "LLq t1 t2 = subst (subst Lq t1 zz) t2 yy"
Pd_range': fixes x :: "'a"   and d :: "'b" assumes "x \<in> Field (Pd'_tax d)" shows "Xd x = d"
simps: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and x :: "'b"   and t :: "'b tree" assumes "RBTi1 TYPE('b) bal_i" shows "RBTi1.insert bal_i x t = (case RBTi1.ins bal_i (size t) 0 x t of Same \<Rightarrow> t | Bal t' \<Rightarrow> t')"
Grp_iff: fixes B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" shows "BNF_Def.Grp B g x y = (y = g x \<and> x \<in> B)"
assms_2: fixes i :: "nat" assumes "i < n" shows "f $ i = (0::'a)"
is_lub_upper: fixes P :: "'a prog"   and U :: "ty"   and V :: "ty"   and T :: "ty" assumes "P \<turnstile> lub(U, V) = T" shows "P \<turnstile> U \<le> T \<and> P \<turnstile> V \<le> T"
i_join_UNIV: fixes f :: "nat \<Rightarrow> 'b" shows "f \<Join>\<^sub> UNIV = f"
mult_scalar_minus_mult_right: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "p \<odot> - q = - (p \<odot> q)"
entt_disjI2': fixes A :: "assn"   and C :: "assn"   and B :: "assn" assumes "A \<Longrightarrow>\<^sub>t C" shows "A \<Longrightarrow>\<^sub>t B \<or>\<^sub>A C"
nn_integral_indicator_singleton: fixes y :: "'c"   and M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "{y} \<in> sets M" shows "set_nn_integral M {y} f = f y * emeasure M {y}"
sbelow_halfspaces_def: fixes sctns :: "'b sctn set" shows "sbelow_halfspaces sctns = \<Inter> (sbelow_halfspace ` sctns)"
card_edge_as_set_Mi_twice_card_partitions: fixes L :: "'b \<Rightarrow> nat"   and E :: "('b \<times> 'b) set"   and V :: "'b set"   and M :: "('b \<times> 'b) set"   and i :: "nat" assumes "OSC L E \<and> matching V E M \<and> 1 < i" shows "2 * card (edge_as_set ` matching_i i V E M L) = card (V_i i V E M L)"
a: shows "[prod id A * (- 1) ^ card E = (- 1) ^ card E * a ^ card A * prod id A * (- 1) ^ card E] (mod int p)"
hemptyE: fixes a :: "hf"   and R :: "bool" assumes "a \<^bold>\<in> 0" shows "R"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "wave \<Gamma> f" obtains "separating \<Gamma> (TER\<^bsub>\<Gamma>\<^esub> f)" and "\<And>x. x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<Longrightarrow> d_OUT f x = 0"
sifum_refinement_axioms: shows "sifum_refinement dma dma\<^sub>C \<C>_vars \<C>_vars\<^sub>C A.eval\<^sub>w C.eval\<^sub>w Eg2_var\<^sub>C_of_Eg1"
R_skip: fixes p :: "('b \<times> 'b) set" shows "Id \<subseteq> rel_R p p"
weakSimTauChain: fixes Q :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'"   and "(\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> R \<leadsto><Rel> S" obtains P' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" and "(\<Psi>, P', Q') \<in> Rel"
eq_iff: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" shows "(x = y) = (x \<sqsubseteq> y \<and> y \<sqsubseteq> x)"
greaterThanLessThan_eq: fixes less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" shows "ord.greaterThanLessThan less a b = ord.greaterThan less a \<inter> ord.lessThan less b"
permute_int_def: fixes p :: "perm"   and i :: "int" shows "p \<bullet> i = i"
Map_dom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Map (local.dom f) = (\<lambda>x\<in>Dom f. x)"
same_fst_def: fixes P :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> ('b \<times> 'b) set" shows "same_fst P R = {((x', y'), x, y). x' = x \<and> P x \<and> (y', y) \<in> R x}"
notxty: shows "\<forall>w. P w (\<midarrow> (x \<otimes> y)) = (\<not> O w (x \<otimes> y))"
sign_simps_6: fixes bh :: "'b"   and ah :: "'b"   and cg :: "'b" shows "bh * (ah * cg) = ah * (bh * cg)"
Suc_7: shows "snd (cs (S t i) cid) = Recording"
Rep_p_point_inject: fixes x :: "p_point"   and y :: "p_point" shows "(Rep_p_point x = Rep_p_point y) = (x = y)"
invertible_inverse_invertible: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> G" shows "sub.invertible (sub.inverse u)"
prime_primeideal: fixes p :: "int" assumes "prime p" shows "primeideal (Idl\<^bsub>\<Z>\<^esub> {p}) \<Z>"
OR2_parts_knows_Spy: fixes B :: "agent"   and N :: "msg"   and A :: "agent"   and c :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>N, Agent A, Agent B, c\<rbrace> \<in> set evs"   and "evs \<in> otway" shows "c \<in> parts (knows Spy evs)"
leg1_uniquely_isomorphic_2: shows "\<exists>!\<nu>. \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> chine\<guillemotright>"
path_equiv_implies_nf_1: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> s \<leftrightarrow> t : T" shows "s \<leadsto>|"
iMODb_div_unique: fixes k :: "nat"   and m :: "nat"   and c :: "nat"   and r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and r :: "nat" assumes "0 < k"   and "k \<le> m"   and "k \<le> c"   and "[ r', mod m', c' ] = [ r, mod m, c ] \<oslash> k" shows "r' = r div k \<and> m' = m div k \<and> c' = c"
mojmir_accept_to_Acc: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "mojmir_def.accept \<up>af\<^sub>G (Abs \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"   and "\<pi> (G \<psi>) = mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}" shows "accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (\<A>.Acc\<^sub>\<A> \<Sigma> \<pi> (G \<psi>)) w"
inj_map_add: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "inj_on f (dom f)"   and "inj_on g (dom g)"   and "ran f \<inter> ran g = {}" shows "inj_on (f ++ g) (dom f \<union> dom g)"
that: fixes vs' :: "'val list" assumes "step.liveVal' g (v # vs')" shows "thesis"
is_lasso_prpl_pre_def: fixes m :: "'a"   and prpl :: "'v list \<times> 'v list" shows "graph.is_lasso_prpl_pre (to_gbg_ext m) prpl \<equiv> case prpl of (pr, pl) \<Rightarrow> \<exists>v0 va. v0 \<in> g_V0 (to_gbg_ext m) \<and> pl \<noteq> [] \<and> path (g_E (to_gbg_ext m)) v0 pr va \<and> path (g_E (to_gbg_ext m)) va pl va"
binrelchain_trans_Cons_snoc: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "\<And>x y z. \<lbrakk>P x y; P y z\<rbrakk> \<Longrightarrow> P x z"   and "binrelchain P (x # xs @ [y])" shows "P x y"
execute_plan_action_def: fixes \<pi> :: "plan_action"   and M :: "object atom formula set" shows "execute_plan_action \<pi> M = apply_effect (ground_action.effect (resolve_instantiate \<pi>)) M"
Bit_div2: fixes w :: "int"   and b :: "bool" shows "w BIT b div 2 = w"
add_zero_expansion_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "plus zero_expansion F = F"
multiset_cases: fixes M :: "'c multiset" obtains "M = {#}"   | x :: "'c" and N :: "'c multiset" where "M = add_mset x N"
filter_M_distinct: fixes h :: "'a"   and P :: "'c \<Rightarrow> ('a, 'b, bool) prog"   and xs :: "'c list"   and ys :: "'c list" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"   and "distinct xs" shows "distinct ys"
map_option_is_None: fixes f :: "'c \<Rightarrow> 'b"   and opt :: "'c option" shows "(map_option f opt = None) = (opt = None)"
at_top_to_right: shows "at_top = filtermap inverse (at_right 0)"
axioms_1: fixes xs :: "V" assumes "vfsequence xs" shows "vsv xs"
\<Delta>''_sel_1: shows "edge \<Delta>'' = edg"
subst_2: fixes y :: "name"   and t' :: "lam"   and t1 :: "lam"   and t2 :: "lam" shows "App t1 t2[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
map_def: fixes f :: "'j" shows "A'.map f = (if J.arr f then a' else C.null)"
Zfun: fixes f :: "'e \<Rightarrow> 'a"   and F :: "'e filter"   and g :: "'e \<Rightarrow> 'b" assumes "Zfun f F"   and "Zfun g F" shows "Zfun (\<lambda>x. f x ** g x) F"
opp_half_apartment_subset_complex: shows "order.greater_eq X folding.Y"
eventually_at_top_linorder: fixes P :: "'d \<Rightarrow> bool" shows "eventually P at_top = (\<exists>N. \<forall>n\<ge>N. P n)"
card_insert_if: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "card (x \<triangleright> A) = (if x \<in> A then card A else Suc (card A))"
nat_to_set_upper_bound2: fixes u :: "nat" shows "nat_to_set u \<subseteq> {i. i < u}"
d\<^sub>1_def: shows "Semantics.d\<^sub>1 \<equiv> map_fun eval\<Pi>\<^sub>1 id Some"
integrable_mult_left: fixes c :: "'d"   and M :: "'e measure"   and f :: "'e \<Rightarrow> 'd" assumes "c \<noteq> (0::'d) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. f x * c)"
less_eq_MonoTran_def: shows "(\<le>) \<equiv> map_fun Rep_MonoTran (map_fun Rep_MonoTran id) (\<le>)"
negligible_imp_measure0: fixes S :: "'b set" assumes "negligible S" shows "Sigma_Algebra.measure lebesgue S = 0"
perp_up_incident: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident b l" shows "proj2_incident (perp_up a l) (drop_perp a l)"
extremum_unique: fixes a :: "'a" shows "(a \<sqsubseteq> \<bottom>) = (a = \<bottom>)"
subset_image_iff: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(B \<subseteq> f ` A) = (\<exists>AA\<subseteq>A. B = f ` AA)"
monoid_mult_hom_axioms: shows "monoid_mult_hom to_fract"
steps_redex': fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and r :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and r' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate"   and c :: "('a, 'b, 'c) com" assumes "\<Gamma>\<turnstile> (r, s) \<rightarrow>\<^sup>+ (r', s')" shows "\<Gamma>\<turnstile> (subst_redex c r, s) \<rightarrow>\<^sup>+ (subst_redex c r', s')"
not_vertex: fixes x :: "'v" assumes "x \<notin> A \<Gamma>"   and "x \<notin> B \<Gamma>" shows "\<not> vertex \<Gamma> x"
commutative_squareE: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and h :: "'a \<times> 'a"   and k :: "'a \<times> 'a" assumes "B.VV.commutative_square f g h k" obtains "B.VV.arr f" and "B.VV.arr g" and "B.VV.arr h" and "B.VV.arr k" and "B.VV.cod f = B.VV.cod g" and "B.VV.dom h = B.VV.dom k" and "B.VV.dom f = B.VV.cod h" and "B.VV.dom g = B.VV.cod k" and "B.VV.comp f h = B.VV.comp g k"
some_geodesic_bounded: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" shows "bounded {x--S--y}"
rad_of_inverse: fixes w :: "real" shows "deg_of (rad_of w) = w"
insert_DiffM: fixes x :: "'k"   and M :: "'k multiset" assumes "x \<in># M" shows "add_mset x (M - {#x#}) = M"
pr_strict_total: fixes weights :: "'a weights"   and fi :: "'a \<times> nat"   and gj :: "'a \<times> nat" assumes "class.weighted weights" shows "fi = gj \<or> pr_strict weights fi gj \<or> pr_strict weights gj fi"
ground_wfP: shows "wfP (\<lambda>s t. ground t \<and> t >\<^sub>t s)"
simps_2: fixes x :: "'g" shows "remdups_adj [x] = [x]"
ordstNS_def: fixes X :: "char list"   and Y :: "char list" shows "ordstNS X Y \<equiv> X = Y \<or> ordst X Y"
simps_23: fixes vc :: "'a acom"   and vd :: "'a acom"   and v :: "char list"   and va :: "aexp"   and vb :: "'a" shows "(vc;; vd \<le> v ::= va {vb}) = False"
diff_strict_right_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less a b" shows "less (minus a c) (minus b c)"
StaysEnabledHelp: fixes cfg :: "('p, 'v, 's) configuration"   and msg :: "('p, 'v) message"   and n :: "nat" assumes "enabled cfg msg"   and "cfg = execConf n"   and "n < length exec" shows "\<forall>i. n \<le> i \<and> i < length exec - 1 \<and> enabled (execConf i) msg \<longrightarrow> msg = execMsg i \<or> enabled (execConf (i + 1)) msg"
big_refl: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "f \<in> \<Omega>[F](f)"
oassert_iff_2: fixes \<Phi> :: "bool" shows "(oassert \<Phi> = None) = (\<not> \<Phi>)"
abs_eq: fixes xa :: "int"   and x :: "int" shows "Word.Word xa sdiv Word.Word x = Word.Word (signed_take_bit (LENGTH('b) - Suc 0) xa sdiv signed_take_bit (LENGTH('b) - Suc 0) x)"
poly_y_x_monom: fixes a :: "'c poly"   and n :: "nat" shows "poly_y_x (monom a n) = smult (monom (1::'c) n) (poly_lift a)"
is_parallel_solution_for_problem_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list list" shows "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> \<psi> \<equiv> let G = \<Psi>\<^sub>G\<^sub>+; I = \<Psi>\<^sub>I\<^sub>+; Ops = \<Psi>\<^sub>\<O>\<^sub>+ in G \<subseteq>\<^sub>m execute_parallel_plan_sas_plus I \<psi> \<and> list_all (list_all (\<lambda>op. ListMem op Ops)) \<psi>"
nn_integral_real_affine: fixes f :: "real \<Rightarrow> ennreal"   and c :: "real"   and t :: "real" assumes "f \<in> borel_measurable borel"   and "c \<noteq> 0" shows "integral\<^sup>N lborel f = ennreal \<bar>c\<bar> * \<integral>\<^sup>+ x. f (t + c * x) \<partial>lborel"
is_class_def: fixes P :: "'a prog"   and C :: "char list" shows "is_class P C \<equiv> class P C \<noteq> None"
deadlock_free_def: fixes P :: "'d process" shows "deadlock_free P \<equiv> DF UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
exchg: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<or> G" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H G \<^bold>\<or> F"
X4: fixes vs :: "nat \<Rightarrow> nat" shows "\<exists>vs'. g (vs 0) (vs 1) = 0 \<and> ((2, vs), 0, vs') \<in> (step X)\<^sup>* \<or> h (vs 0) (vs 1) = 0 \<and> ((2, vs), 1, vs') \<in> (step X)\<^sup>* \<or> g (vs 0) (vs 1) \<noteq> 0 \<and> h (vs 0) (vs 1) \<noteq> 0 \<and> vs' 0 = vs 0 \<and> vs' 1 = vs 1 + 1 \<and> ((2, vs), 2, vs') \<in> (step X)\<^sup>+"
takeWhile_eq_all_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "(takeWhile P xs = xs) = (\<forall>x\<in>set xs. P x)"
union_left_cancel: fixes K :: "'a multiset"   and M :: "'a multiset"   and N :: "'a multiset" shows "(K + M = K + N) = (M = N)"
filtermap_def: fixes f :: "'b \<Rightarrow> 'a"   and F :: "'b filter" shows "filtermap f F = Abs_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x))"
add_less_mono: fixes i :: "nat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i < j"   and "k < l" shows "i + k < j + l"
cloBS: shows "closedin (top_of_set B) ({0..1} \<times> S)"
butterfly_def: fixes s :: "'b"   and t :: "'a" shows "butterfly s t = vector_to_cblinfun s o\<^sub>C\<^sub>L vector_to_cblinfun t*"
lq_rq_reassoc_suf: fixes u :: "'b list"   and z :: "'b list"   and w :: "'b list" assumes "u \<le>p z"   and "u \<le>s w" shows "w \<cdot> u\<inverse>\<^sup>>z = w\<^sup><\<inverse>u  \<cdot> z"
pentagon: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "ide a"   and "ide b"   and "ide c"   and "ide d" shows "(a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d] \<cdot> (\<a>[a, b, c] \<otimes> d) = \<a>[a, b, c \<otimes> d] \<cdot> \<a>[a \<otimes> b, c, d]"
sorted_addmult_spvec: fixes a :: "(nat \<times> 'a) list"   and b :: "(nat \<times> 'a) list"   and y :: "'a" assumes "sorted_spvec a"   and "sorted_spvec b" shows "sorted_spvec (addmult_spvec y a b)"
count_it_determ: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and n :: "nat"   and n' :: "nat" assumes "count_it \<pi> b c s n"   and "count_it \<pi> b c s n'" shows "n' = n"
filter_empty_ivls: fixes A :: "('c \<times> 'd) set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "('c \<times> 'c) list"   and X :: "'d set" assumes "PREFER single_valued A"   and "GEN_OP le (\<le>) (A \<rightarrow> A \<rightarrow> bool_rel)"   and "(xs, X) \<in> clw_rel (\<langle>A\<rangle>ivl_rel)" shows "(filter_empty_ivls_impl le xs, filter_empty_ivls $ X) \<in> clw_rel (\<langle>A\<rangle>ivl_rel)"
simps_4: fixes \<A> :: "'b ltln set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<A> \<Turnstile>\<^sub>P \<phi> or\<^sub>n \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<or> \<A> \<Turnstile>\<^sub>P \<psi>)"
rep_eq: shows "pmf local.poisson_pmf = (\<lambda>k. rate ^ k / fact k * exp (- rate))"
components_are_iso: fixes a :: "'c" assumes "C.ide a" shows "C.iso (\<epsilon>'.map a)"
F_nonneg: fixes y :: "real" shows "0 \<le> F y"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map (G' \<circ> G \<circ> F \<circ> F') G'\<eta>F'o\<eta>'.map"
ordLeq_refl: fixes r :: "('b \<times> 'b) set" assumes "Card_order r" shows "r \<le>o r"
assms_1: shows "(subst_domain \<theta> \<union> range_vars \<theta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"
is_equality_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "is_equality R = (R = (=))"
cases: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr"   and a1 :: "addr_loc"   and a2 :: "ty" assumes "P,h \<turnstile> a@a1 : a2" obtains U :: "htype" and F :: "String.literal" and T :: "ty" and fm :: "fmod" and D :: "String.literal" where "a1 = CField D F" and "a2 = Ta__" and "typeof_addr h a = \<lfloor>U\<rfloor>" and "P \<turnstile> class_type_of U has F:Ta__ (fm) in D"   | T :: "ty" and n' :: "nat" and n :: "nat" where "a1 = ACell n" and "a2 = T" and "typeof_addr h a = \<lfloor>Array_type T n'\<rfloor>" and "n < n'"
mc_def: shows "mc = Mapping.combine max_tstp m_upd m'"
FtorCompM: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f) Functor"   and G :: "('b, 'g, 'd, 'h, 'f, 'i) Functor" assumes "F \<approx>>;;; G" shows "FunctorM (F ;;: G)"
proves_descend_abs_rule: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and \<tau>' :: "typ"   and x :: "variable"   and s :: "Core.term"   and t :: "Core.term" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau>' s) (Abs_fv x \<tau>' t)"   and "is_closed s"   and "is_closed t"   and "(x, \<tau>') \<notin> FV \<Gamma>"   and "typ_ok \<Theta> \<tau>'"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq s t"
inc_dot_rhs: fixes d :: "nat"   and x :: "item" shows "item_rhs (inc_dot d x) = item_rhs x"
lprefix_lSup_revert_1: shows "lSup = Sup"
set_iterator_rev_linord_def: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S0 :: "'b set" shows "set_iterator_rev_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>x y. y \<le> x)"
star_denest_var_0: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = (x\<^sup>\<star> \<cdot> y)\<^sup>\<star> \<cdot> x\<^sup>\<star>"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" obtains "a = min a b"
trans_def: fixes r :: "('a \<times> 'a) set" shows "trans r = (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
preserves_dom: fixes f :: "J.Arr" assumes "J.arr f" shows "C.dom (\<chi> f) = \<chi>.A.map (J.dom f)"
map_of_None2: fixes ps :: "('c \<times> 'd) list"   and x :: "'c" assumes "sorted (map fst ps @ [x])" shows "map_of ps x = None"
language_def: shows "ST.language \<equiv> {xs. ST.nextl (nfa.init \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr>) xs \<inter> nfa.final \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr> \<noteq> {}}"
erf_real_gt_neg1: fixes x :: "real" shows "- 1 < erf x"
contr: shows "continuous_on (W0 \<union> (S \<union> T)) r"
assms: shows "vfinite (A \<union>\<^sub>\<circ> B)"
mapi_branch_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> ('c, 'd) fm \<Rightarrow> ('c, 'd) fm"   and branch :: "(('c, 'd) fm list \<times> 'd) list" shows "mapi_branch f branch \<equiv> mapi (\<lambda>v. mapi_block (f v)) branch"