hcju/isabelleps huggingface dataset:
simps_12: fixes f1 :: "agent \<Rightarrow> agent \<Rightarrow> msg \<Rightarrow> 'a"   and f2 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and f3 :: "agent \<Rightarrow> msg \<Rightarrow> 'a"   and x31 :: "agent"   and x32 :: "msg" shows "(case Notes x31 x32 of Says x xa xb \<Rightarrow> f1 x xa xb | Gets x xa \<Rightarrow> f2 x xa | Notes x xa \<Rightarrow> f3 x xa) = f3 x31 x32"
le_supI1: fixes x :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "x \<subseteq># a" shows "x \<subseteq># a \<union># b"
ad_eq: fixes x :: "'a" shows "antidomain_semiringl.ads_d a x = d x"
assms: fixes gs :: "(('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and bs :: "(('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list"   and h :: "('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and ps :: "(bool \<times> (('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) \<times> ('b \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd) list" shows "set (aps flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))"
is_lbI: fixes S :: "'a set"   and l :: "'a" assumes "\<And>x. x \<in> S \<Longrightarrow> l \<sqsubseteq> x" shows "S >| l"
filtercomap_ident: fixes F :: "'a filter" shows "filtercomap (\<lambda>x. x) F = F"
wo_rel_def: fixes r :: "('b \<times> 'b) set" shows "wo_rel r \<equiv> Well_order r"
dvd_div_iff_mult: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and c :: "'m"   and b :: "'m"   and a :: "'m" assumes "class.algebraic_semidom divide plus minus zero times one"   and "c \<noteq> zero"   and "dvd.dvd times c b" shows "dvd.dvd times a (divide b c) = dvd.dvd times (times a c) b"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>heap = heap r, locals = locals r, \<dots> = state.more r\<rparr>"
closedin_self: fixes S :: "'b set" shows "closedin (top_of_set S) S"
upright_def: fixes a :: "'c zmultiset" shows "upright a = (\<forall>t. 0 < zcount a t \<longrightarrow> supported a t)"
ss_noprimes: shows "ssubst (vquot_dbfm V' (trans_fm [i] \<alpha>)) V' F' = ssubst (vquot_dbfm Vs (trans_fm [i] \<alpha>)) Vs F"
setX_def: fixes r :: "'a point_scheme"   and n :: "nat" shows "setX r n = r\<lparr>xpos := n\<rparr>"
path_first_vertex: fixes G :: "('a, 'b) Graph_scheme"   and x :: "'a"   and xs :: "'a list" assumes "Digraph G"   and "Digraph.walk G (x # xs) \<and> distinct (x # xs)" shows "x \<notin> set xs"
obtainPrefix: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and B :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)"   and "\<alpha> \<noteq> \<tau>"   and "B \<sharp>* P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* B"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* subject \<alpha>" obtains M :: "'a" where "\<Psi> \<otimes> \<Psi>\<^sub>P \<turnstile> the (subject \<alpha>) \<leftrightarrow> M" and "B \<sharp>* M"
strict_monoidal_functors_compose: fixes B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T\<^sub>B :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha>\<^sub>B :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota>\<^sub>B :: "'b"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>C :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>C :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>C :: "'d"   and F :: "'b \<Rightarrow> 'd"   and D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and T\<^sub>D :: "'e \<times> 'e \<Rightarrow> 'e"   and \<alpha>\<^sub>D :: "'e \<times> 'e \<times> 'e \<Rightarrow> 'e"   and \<iota>\<^sub>D :: "'e"   and G :: "'d \<Rightarrow> 'e" assumes "strict_monoidal_functor B T\<^sub>B \<alpha>\<^sub>B \<iota>\<^sub>B C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C F"   and "strict_monoidal_functor C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D G" shows "strict_monoidal_functor B T\<^sub>B \<alpha>\<^sub>B \<iota>\<^sub>B D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D (G \<circ> F)"
transpos_ij_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "transpos i j i = j"
fun_upd_twist: fixes a :: "'f"   and c :: "'f"   and m :: "'f \<Rightarrow> 'g"   and b :: "'g"   and d :: "'g" assumes "a \<noteq> c" shows "m(a := b, c := d) = m(c := d, a := b)"
c_refusals: fixes X :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and xps :: "('e \<times> 'f) list" shows "(X \<in> refusals (c_process step out s\<^sub>0) xps) = ((xps, X) \<in> c_failures step out s\<^sub>0)"
funC_ne: fixes m :: "C"   and n :: "nat" shows "funC m (Suc n) \<noteq> []"
split_face_edges_f_vs1_vs2: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list" assumes "pre_split_face f ram1 ram2 vs"   and "between (vertices f) ram1 ram2 = []"   and "between (vertices f) ram2 ram1 = []" shows "\<E> f = {(ram2, ram1), (ram1, ram2)}"
n_mult_glbI': fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "!z \<le> !x"   and "!z \<le> !y" shows "!z \<le> !x \<cdot> !y"
ticonc': fixes xs :: "int list" assumes "xs \<noteq> []" shows "\<exists>ys. xs = take_incr xs @ ys \<and> (ys \<noteq> [] \<longrightarrow> \<not> last (take_incr xs) < hd ys)"
lg_cong_lg: fixes l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "QCong l"   and "l A B"   and "Cong A B C D" shows "l C D"
strategy_extends_VVp: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v0 \<in> VV p"   and "\<not> deadend v0"   and "strategy p \<sigma>"   and "winning_strategy p \<sigma> v0" shows "winning_strategy p \<sigma> (\<sigma> v0)"
cyclic_order_is_ord: fixes g :: "int" assumes "g \<in> carrier (Z 1)" shows "ord g = order (subgroup_generated (Z 1) {g})"
lexLessState_def: fixes F0 :: "Literal list list"   and decisionVars :: "nat set" shows "lexLessState F0 decisionVars \<equiv> {(stateA, stateB). (getM stateA, getM stateB) \<in> lexLessRestricted (vars F0 \<union> decisionVars)}"
filter_IFS_no_violations_correct: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and edgesList :: "('b \<times> 'b) list" assumes "valid_reqs (get_IFS M)"   and "wf_graph G"   and "all_security_requirements_fulfilled (get_IFS M) G"   and "set edgesList \<subseteq> edges G" shows "all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set (filter_IFS_no_violations G M edgesList)\<rparr>)"
PDG_cdep_edge: fixes n :: "'node"   and n' :: "'node" assumes "n weakly controls n'" shows "PDG.PDG_edge sourcenode targetnode valid_edge Def Use weak_control_dependence n None n'"
simps: fixes a :: "ed_ou_witness set" shows "open a = (a = {} \<or> a = {ed_ou_witness.a} \<or> a = {ed_ou_witness.b} \<or> a = {e} \<or> a = {ed_ou_witness.a, ed_ou_witness.c} \<or> a = {ed_ou_witness.b, d} \<or> a = {ed_ou_witness.a, ed_ou_witness.c, e} \<or> a = {ed_ou_witness.a, ed_ou_witness.b} \<or> a = {ed_ou_witness.a, e} \<or> a = {ed_ou_witness.b, e} \<or> a = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c} \<or> a = {ed_ou_witness.a, ed_ou_witness.b, d} \<or> a = {ed_ou_witness.a, ed_ou_witness.b, e} \<or> a = {ed_ou_witness.b, d, e} \<or> a = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, d} \<or> a = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, e} \<or> a = {ed_ou_witness.a, ed_ou_witness.b, d, e} \<or> a = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, d, e})"
seq_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1 ;; c\<^sub>2, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w"   and "c\<^sub>1 \<noteq> Stop" shows "\<exists>c\<^sub>1'. \<langle>c\<^sub>1, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c\<^sub>1', mds', mem'\<rangle>\<^sub>w \<and> c' = c\<^sub>1' ;; c\<^sub>2"
edge_preserving_atomic: fixes h1 :: "('d \<times> 'e) set"   and e1 :: "('f \<times> 'd \<times> 'd) set"   and e2 :: "('f \<times> 'e \<times> 'e) set"   and v1 :: "'d"   and v1' :: "'e"   and v2 :: "'d"   and v2' :: "'e"   and k :: "'f" assumes "edge_preserving h1 e1 e2"   and "(v1, v1') \<in> h1"   and "(v2, v2') \<in> h1"   and "(k, v1, v2) \<in> e1" shows "(k, v1', v2') \<in> e2"
simps_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "nat \<Rightarrow> 'b" shows "foldseq_transposed f s 0 = s 0"
HFinite_Infinitesimal_not_zero: fixes x :: "'a star" assumes "x \<in> HFinite - Infinitesimal" shows "x \<noteq> 0"
branch_inv_Leaf2: fixes T :: "tree" shows "(T = Leaf) = anybranch T (\<lambda>b. b = [])"
phis_disj_1: fixes g :: "'g"   and n :: "'node"   and v :: "'val"   and vs :: "'val list"   and n' :: "'node"   and vs' :: "'val list" assumes "phis g (n, v) = Some vs"   and "phis g (n', v) = Some vs'" shows "n = n'"
pos_divide_le_eq: fixes c :: "'d"   and b :: "'d"   and a :: "'d" assumes "(0::'d) < c" shows "(b / c \<le> a) = (b \<le> a * c)"
antisym: fixes b :: "'b::order"   and a :: "'b::order" assumes "(b::'b::order) \<le> (a::'b::order)"   and "(a::'b::order) \<le> (b::'b::order)" shows "(a::'b::order) = (b::'b::order)"
equivp_transp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "equivp R"   and "R x y"   and "R y z" shows "R x z"
permutation_def: fixes p :: "'b \<Rightarrow> 'b" shows "permutation p = (\<exists>n. swapidseq n p)"
independent_insertI: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and S :: "'e set" assumes "vector_space scale"   and "a \<notin> module.span scale S"   and "\<not> module.dependent scale S" shows "\<not> module.dependent scale (insert a S)"
dense_repr_normalize: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R" shows "ring.dense_repr R (ring.normalize R p) = ring.dense_repr R p"
inf_Int_eq: fixes R :: "'c set"   and S :: "'c set" shows "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
All_E': fixes H :: "fm set"   and i :: "name"   and A :: "fm"   and x :: "tm"   and B :: "fm" assumes "H \<turnstile> SyntaxN.All i A"   and "insert (A(i::=x)) H \<turnstile> B" shows "H \<turnstile> B"
continuous_within_tendsto_compose: fixes a :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'e \<Rightarrow> 'c"   and F :: "'e filter" assumes "continuous (at a within s) f"   and "\<forall>\<^sub>F n in F. x n \<in> s"   and "(x \<longlongrightarrow> a) F" shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
reachable_path_entry: fixes G :: "nat graph_rec"   and n :: "nat" assumes "cfg_doms G"   and "cfg_doms.reachable G n" shows "\<exists>p. path_entry (g_E G) p n"
s_in: shows "s \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
ReflectL_def: fixes P' :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p" shows "P' P ReflectL A B \<equiv> (\<exists>X. X Midpoint P P' \<and> Col A B X) \<and> (A B Perp P P' \<or> P = P')"
Heap_lub_def: shows "Heap_lub = img_lub execute Heap option.lub_fun"
axioms_5: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "inverse_transformation A B F G \<tau>" shows "natural_isomorphism A B F G \<tau>"
iteration_star: fixes x :: "'a" shows "(x\<^sup>\<infinity>)\<^sup>\<star> = x\<^sup>\<infinity>"
int_not_neg_numeral: fixes n :: "num" shows "NOT (- numeral n) = neg_numeral_class.sub n num.One"
tm_append_first_step_eq: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and A :: "(action \<times> nat) list"   and off :: "nat"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list"   and B :: "(action \<times> nat) list" assumes "step (s, l, r) (A, off) = (s', l', r')"   and "s' \<noteq> 0" shows "step (s, l, r) (A @ B, off) = (s', l', r')"
assms_9: shows "InvariantQCharacterization False (getQ state) (getF state) (getM state)"
init_rev_wrap_3: fixes P :: "'b list \<Rightarrow> bool" shows "\<forall>y. P y \<equiv> All_rev_wrap P"
continuous_on_of_real_id: fixes S :: "real set" shows "continuous_on S of_real"
assms: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
untame_notame: shows "untame (\<lambda>g. Invariants.inv g \<and> notame7 g)"
WhileTRedsThrow: fixes b :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and c :: "char list exp"   and e :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool" assumes "P \<turnstile> \<langle>b,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>c,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>while (b) c,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw e,s\<^sub>2,b\<^sub>2\<rangle>"
hseq_char: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "horizontal_composition V H src trg" shows "partial_magma.arr V (H \<nu> \<mu>) = (partial_magma.arr V \<mu> \<and> partial_magma.arr V \<nu> \<and> src \<nu> = trg \<mu>)"
ts_c_def_4: shows "\<C>' = s_inst \<S> ! i"
filterlim_at_bot_int_iff_filterlim_real: fixes f :: "'a \<Rightarrow> int"   and F :: "'a filter" shows "filterlim f at_bot F = (LIM x F. real_of_int (f x) :> at_bot)"
in_angle_reverse: fixes A' :: "'p"   and B :: "'p"   and A :: "'p"   and C :: "'p"   and D :: "'p" assumes "A' \<noteq> B"   and "Bet A B A'"   and "C InAngle A B D" shows "D InAngle A' B C"
Int_Diff: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'a measure"   and B :: "'b set" assumes "finite A" shows "sup_measure.F g A = sup (sup_measure.F g (A \<inter> B)) (sup_measure.F g (A - B))"
simps_6: fixes f1 :: "'f \<Rightarrow> 'e"   and f2 :: "'g \<Rightarrow> 'e"   and x2 :: "'g" shows "(case Inr x2 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f2 x2"
ln_powr: fixes x :: "real"   and y :: "real" assumes "x \<noteq> 0" shows "ln (x powr y) = y * ln x"
elims_3: fixes x :: "ereal"   and xa :: "ereal" assumes "\<not> x < xa" obtains x :: "real" and y :: "real" where "xa__ = ereal x" and "xa = ereal y" and "\<not> x < y"   | a :: "ereal" where "xa__ = \<infinity>" and "xa = a__"   | a :: "ereal" where "xa__ = a" and "xa = - \<infinity>"
simps_3: fixes c :: "nat"   and n :: "C"   and ns :: "C list"   and ms :: "C list" shows "stepC c (C (C (n # ns) # ms)) = stepC c (C (funC (C (n # ns)) (Suc (Suc c)) @ ms))"
R: shows "R \<in> \<R> \<and> u \<in> R"
toList_fromList: fixes xs :: "'c list" shows "toList (fromList xs) = msort xs"
AE_restrict_space_iff: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and P :: "'a \<Rightarrow> bool" assumes "\<Omega> \<inter> space M \<in> sets M" shows "almost_everywhere (restrict_space M \<Omega>) P = set_almost_everywhere \<Omega> M P"
Upper_memD: fixes u :: "'c"   and L :: "('c, 'd) gorder_scheme"   and A :: "'c set"   and x :: "'c" assumes "u \<in> Upper L A"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> u \<and> u \<in> carrier L"
apply_inverse: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and u :: "'b"   and P :: "'c \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'c" assumes "f x = u"   and "\<And>x. P x \<Longrightarrow> g (f x) = x"   and "P x" shows "x = g u"
sum_mult_subR: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier R"   and "B \<subseteq> carrier R" shows "sum_mult R A B \<subseteq> carrier R"
build_domintros3: fixes k :: "'a"   and s :: "real"   and ks :: "'a list"   and x :: "(real, 'a) vec"   and y :: "(real, 'a) vec"   and zs :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list" assumes "(k, s) = widest_spread ks (x # y # zs)"   and "(l, m, r) = partition_by_median k (x # y # zs)"   and "build_dom (ks, l)"   and "build_dom (ks, r)" shows "build_dom (ks, x # y # zs)"
Disj_commute: fixes H :: "fm set"   and B :: "fm"   and A :: "fm" assumes "H \<turnstile> B OR A" shows "H \<turnstile> A OR B"
star_n_le: fixes X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a" shows "(star_n X \<le> star_n Y) = (\<forall>\<^sub>F n in \<U>. X n \<le> Y n)"
jumpF_pos_has_sgnx: fixes f :: "real \<Rightarrow> real"   and F :: "real filter" assumes "0 < jumpF f F" shows "(f has_sgnx 1) F"
x_y_split: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> nc) \<cdot> y + x \<cdot> (0::'a) = x \<cdot> y"
idem: fixes a :: "'b" shows "sup_class.sup a a = a"
rbt_comp_bulkload_def: fixes xs :: "('a \<times> 'c) list" shows "local.rbt_comp_bulkload xs = foldr (\<lambda>(x, y). local.rbt_comp_insert x y) xs rbt.Empty"
lowerPoly\<^sub>f_def: shows "lowerPoly\<^sub>f \<equiv> map_fun id (map_fun id (map_fun fmlookup Abs_fmap)) (\<lambda>b i mp mon. mp (lowerPowers b i mon))"
set_suffixes_eq: fixes xs :: "'a list" shows "set (suffixes xs) = {ys. suffix ys xs}"
Union_iff: fixes A :: "'b"   and C :: "'b set set" shows "((A::'b) \<in> \<Union> (C::'b set set)) = (\<exists>X::'b set\<in>C. A \<in> X)"
assms_2: fixes c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>*c (c', s')"   and "(c', s') \<rightarrow>t s''" shows "phi"
fcomp_fimage: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<circ>\<^sub>\<bullet> s) `\<^sub>\<bullet> A = r `\<^sub>\<bullet> s `\<^sub>\<bullet> A"
finite_inf_closed: fixes A :: "'a set" assumes "finite A"   and "A \<subseteq> carrier L"   and "A \<noteq> {}" shows "\<Sqinter>A \<in> carrier L"
Un: fixes a :: "'d set"   and M :: "'d measure"   and b :: "'d set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a \<union> b \<in> sets M"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "Right_a.in_hom f a b = (Right_a.arr a \<and> Right_a.arr b \<and> Right_a.arr f \<and> \<guillemotleft>f : a \<Rightarrow> b\<guillemotright>)"
card_Int: fixes A :: "'a set"   and B :: "'a set" assumes "finite A" shows "card (A \<inter> B) = card A - card (A - B)"
bit_xor_int_iff: fixes k :: "int"   and l :: "int"   and n :: "nat" shows "bit (k XOR l) n = (bit k n \<noteq> bit l n)"
A_authenticates_B_temporal: fixes K :: "nat"   and Ta :: "nat"   and evs :: "event list"   and A :: "agent"   and Tk :: "nat"   and B :: "agent"   and X :: "msg" assumes "Crypt K (Number Ta) \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredK Tk evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos" shows "Says B A (Crypt K (Number Ta)) \<in> set evs"
algebra_intersection: fixes \<Omega> :: "'a set"   and A :: "'a set set"   and B :: "'a set set" assumes "algebra \<Omega> A"   and "algebra \<Omega> B" shows "algebra \<Omega> (A \<inter> B)"
lv_ivl_def: fixes xrs :: "'e list"   and yrs :: "'e list" shows "lv_ivl xrs yrs = {zrs. list_all2 (\<le>) xrs zrs \<and> list_all2 (\<le>) zrs yrs}"
unl_lift3: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b"   and x :: "'f \<Rightarrow> 'c"   and y :: "'f \<Rightarrow> 'd"   and z :: "'f \<Rightarrow> 'e"   and w :: "'f" shows "(f<x, y, z>) w \<equiv> f (x w) (y w) (z w)"
lideal: fixes r :: "'a"   and a :: "'a" assumes "r \<in> A"   and "a \<in> I" shows "multA r a \<in> I"
ks: shows "k ` ({0..1} \<times> {0..1}) \<subseteq> S"
S\<^sub>1_def: shows "rs.S\<^sub>1 = \<lbrakk>\<lbrakk>tab\<^sub>1 s\<rbrakk>\<rbrakk>"
span_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "vs1.span A \<subseteq> vs1.span B"
path_split_1: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node"   and as' :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n (as @ a # as') n'" shows "While_CFG.path prog n as (sourcenode a)"
factorgroup_is_group: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "H \<lhd> G" shows "Group.group (G Mod H)"
le_cases3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" obtains "x \<preceq> y" and "y \<preceq> z"   | "y \<preceq> x" and "x \<preceq> z"   | "x \<preceq> z" and "z \<preceq> y"   | "z \<preceq> y" and "y \<preceq> x"   | "y \<preceq> z" and "z \<preceq> x"   | "z \<preceq> x" and "x \<preceq> y"
aux_42: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry"   and z :: "'c DBMEntry" assumes "x \<prec> y"   and "\<not> dbm_add z x \<prec> dbm_add z y" shows "dbm_add z x = dbm_add z y"
gbinomial_1: fixes a :: "'a" shows "a gchoose 1 = a"
cross_ratio_commute_24: fixes z :: "complex_homo"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" shows "cross_ratio z u v w = reciprocal (cross_ratio z w v u)"
vector_3_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "vector [x, y, z] $ 3 = z"
row_length_mat: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m1 :: "'a list list"   and m2 :: "'a list list" assumes "Matrix_Tensor.mult id f" shows "mult.row_length (mult.Tensor f m1 m2) = mult.row_length m1 * mult.row_length m2"
conversion_converse: fixes A :: "('b \<times> 'b) set" shows "(A\<^sup>\<leftrightarrow>\<^sup>*)\<inverse> = A\<^sup>\<leftrightarrow>\<^sup>*"
RHS_def: shows "RHS = ((t \<star> (u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> \<a>[t, t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine] \<cdot> (\<tau> \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine)) \<cdot> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1]"
le_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Le Bet Cong A B C D \<or> Tarski_neutral_dimensionless.Le Bet Cong C D A B"
homotopic_paths_eq: fixes p :: "real \<Rightarrow> 'c"   and s :: "'c set"   and q :: "real \<Rightarrow> 'c" assumes "path p"   and "path_image p \<subseteq> s"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t" shows "homotopic_paths s p q"
pdom_res_pdom: fixes f :: "('a, 'b) pfun" shows "pdom f \<lhd>\<^sub>p f = f"
sorted_wrt_map: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "sorted_wrt R (map f xs) = sorted_wrt (\<lambda>x y. R (f x) (f y)) xs"