hcju/isabelleps huggingface dataset:
assms: shows "\<forall>A n. A \<in> carrier_mat n n \<and> isDiagonal_mat A \<longrightarrow> (\<exists>P Q. P \<in> carrier_mat n n \<and> Q \<in> carrier_mat n n \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q))"
carrier_mat_triv: fixes m :: "'a mat" shows "m \<in> carrier_mat (dim_row m) (dim_col m)"
vars: shows "free_vars (IF\<^sub>c CReal 0 <\<^sub>c CVar 0 THEN (f \<circ>\<^sub>c ln\<^sub>c (CVar 0)) /\<^sub>c CVar 0 ELSE CReal 0) \<subseteq> shift_var_set (set vs')"
admI: fixes P :: "'d \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>chain Y; \<forall>i. P (Y i)\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)" shows "adm P"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" shows "A.arr f"
\<sigma>2_2: fixes v :: "'a" assumes "v \<in> V\<^bsub>G'\<^esub>" shows "G'.winning_strategy p \<sigma>2 v"
idem_succs_of_clause: fixes l :: "lit" shows "comp_fun_idem (succs_of_clause l)"
jointly_exhaustive: fixes p :: "'a"   and q :: "'a" assumes "\<I> p"   and "\<I> q" shows "(p, q) \<in> b \<or> (p, q) \<in> m \<or> (p, q) \<in> ov \<or> (p, q) \<in> s \<or> (p, q) \<in> d \<or> (p, q) \<in> f\<inverse> \<or> (p, q) \<in> e \<or> (p, q) \<in> f \<or> (p, q) \<in> s\<inverse> \<or> (p, q) \<in> d\<inverse> \<or> (p, q) \<in> ov\<inverse> \<or> (p, q) \<in> m\<inverse> \<or> (p, q) \<in> b\<inverse>"
finite_Invs: fixes xs :: "'c list"   and ys :: "'c list" shows "finite (Inv xs ys)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat" assumes "\<And>a b. P a b 0"   and "\<And>a b. P a b (Suc 0)"   and "\<And>a b n. P b (a + b) (Suc n) \<Longrightarrow> P a b (Suc (Suc n))" shows "P a0 a1 a2"
uus: shows "uus \<equiv> getFrN (x # map snd txs) (Var y # t # map fst txs) [] (Suc (length txs))"
case_resumption_Linear: fixes t :: "'g \<Rightarrow> 'f"   and l :: "'h \<Rightarrow> ('g, 'h, 'i, 'j) resumption \<Rightarrow> 'f"   and br :: "'i \<Rightarrow> ('j \<Rightarrow> ('g, 'h, 'i, 'j) resumption) \<Rightarrow> 'f"   and b :: "'h"   and r :: "('g, 'h, 'i, 'j) resumption" shows "(case Linear b r of Terminal x \<Rightarrow> t x | Linear x xa \<Rightarrow> l x xa | Branch x xa \<Rightarrow> br x xa) = l b r"
gauss_cnj_dvd_right_iff: fixes a :: "gauss_int"   and b :: "gauss_int" shows "(a dvd gauss_cnj b) = (gauss_cnj a dvd b)"
ordered_fmap_keys: fixes S :: "('a \<times> 'b) fset" assumes "is_fmap S" shows "map fst (ordered_fmap S) = sorted_list_of_fset (fst |`| S)"
plus_pres_lens_indep: fixes X :: "'d \<Longrightarrow> 'e"   and Z :: "'f \<Longrightarrow> 'e"   and Y :: "'g \<Longrightarrow> 'e" assumes "X \<bowtie> Z"   and "Y \<bowtie> Z" shows "X +\<^sub>L Y \<bowtie> Z"
to_of_name: fixes a :: "name" shows "to_name (of_name a) = a"
cases: fixes x :: "'a atom formula" obtains a :: "predicate" and vs :: "'a list" where "x = Atom (predAtm a vs)"   | va :: "'a" and vb :: "'a" where "x = Atom (Eq va__ vb)"   | "x = \<bottom>"   | v :: "'a atom formula" where "x = \<^bold>\<not> v__"   | v :: "'a atom formula" and va :: "'a atom formula" where "x = vc__ \<^bold>\<and> vaa__"   | v :: "'a atom formula" and va :: "'a atom formula" where "x = vd__ \<^bold>\<or> vab__"   | v :: "'a atom formula" and va :: "'a atom formula" where "x = v \<^bold>\<rightarrow> va"
C_def: shows "C = - (complex_of_real a powr - s / 2)"
entailsD: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real"   and s :: "'c" assumes "P \<tturnstile> Q" shows "P s \<le> Q s"
gen_enumerate_def: fixes tol :: "'f \<Rightarrow> 'g list"   and upd :: "'g \<Rightarrow> nat \<Rightarrow> 'e \<Rightarrow> 'e"   and emp :: "'e"   and S :: "'f" shows "gen_enumerate tol upd emp S \<equiv> snd (fold (\<lambda>x (k, m). (Suc k, upd x k m)) (tol S) (0, emp))"
odd_coeff_sign_changes_imp_pos_roots_aux: fixes p :: "real poly" assumes "p \<noteq> 0"   and "poly p 0 \<noteq> 0"   and "odd (sign_changes (coeffs p))" obtains x :: "real" where "0 < x" and "poly p x = 0"
raw_db_subst_def: fixes i :: "nat"   and x :: "'b" shows "raw_db_subst i x = (\<lambda>y. Hd (if y = x then Sym (db i) else Var y))"
simps_1: fixes bs :: "nat \<Rightarrow> 'a"   and c :: "'a" shows "insertion bs (c)\<^sub>p = c"
Aik: shows "A $ i $ mod_type_class.from_nat k \<noteq> (0::'a)"
bounded_linear_mult_right: fixes x :: "'d" shows "bounded_linear ((*) x)"
lex_ext_stri_imp_nstri: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> bool \<times> bool"   and m :: "nat"   and xs :: "'c list"   and ys :: "'c list" assumes "fst (lex_ext f m xs ys)" shows "snd (lex_ext f m xs ys)"
eq_Fun_iff_eq_corestr: fixes f :: "'s"   and f' :: "'s" assumes "span f f'" shows "(Fun f = Fun f') = (corestr f = corestr f')"
leading_monomial_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'd" shows "p = monomial (lc p) (lt p) + tail p"
assms_4: shows "s2' = s2(V := \<lbrakk>e\<rbrakk>s2)"
simps_5: fixes M :: "('fun, 'var) Term.term set"   and X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c' = (\<lambda>\<I>. ineq_model \<I> X F \<and> \<lbrakk>M; S\<rbrakk>\<^sub>c' \<I>)"
tiny_dg_Cod_vrange: shows "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
finished_step: fixes s :: "'state"   and s' :: "'state" assumes "s \<rightarrow> s'" shows "\<not> finished (\<rightarrow>) s"
chain_implies_mono: fixes C :: "nat \<Rightarrow> 'b set" assumes "chain C" shows "mono C"
arr_char: fixes f :: "'c \<times> 'c" shows "C.VV.arr f = (C.arr (fst f) \<and> C.arr (snd f) \<and> C'.trg (fst f) = C'.src (snd f))"
maxinch_is_thelub: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "class.cpo below"   and "po.chain below Y" shows "po.max_in_chain i Y = (po.lub below (range Y) = Y i)"
prv_eql_subst: fixes x :: "'var"   and y :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "y \<in> var"   and "\<phi> \<in> fmla" shows "prv (imp (eql (Var x) (Var y)) (imp \<phi> (subst \<phi> (Var y) x)))"
eq2: fixes n :: "nat" assumes "degree F < n" shows "coeff (reduce_fps_poly F) n = (0::'a)"
ln_float_interval_eqI: fixes p :: "nat"   and X :: "float interval"   and IVL :: "float interval"   and x :: "real" assumes "ln_float_interval p X = Some IVL"   and "x \<in>\<^sub>r X" shows "ln x \<in>\<^sub>r IVL"
oStrictLimit_induct: fixes P :: "ordinal \<Rightarrow> bool"   and a :: "ordinal" assumes "P oZero"   and "\<And>f. \<forall>n. P (f n) \<Longrightarrow> P (oStrictLimit f)" shows "P a"
F_def: shows "F = (\<lambda>x. x powr (1 - s) / (1 - s))"
rel_gpv_Pause: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and out :: "'c"   and c :: "('a, 'c, 'e) rpv"   and out' :: "'d"   and c' :: "('b, 'd, 'e) rpv" shows "rel_gpv A C (Generative_Probabilistic_Value.Pause out c) (Generative_Probabilistic_Value.Pause out' c') = (C out out' \<and> (\<forall>x. rel_gpv A C (c x) (c' x)))"
isom_tgch_unitsTr0: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and h :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and f :: "nat \<Rightarrow> nat" assumes "Ugp E"   and "card {i. i \<le> n \<and> g i \<cong> E} = card {i. i \<le> n \<and> h i \<cong> E}"   and "Gchain (Suc n) g \<and> Gchain (Suc n) h \<and> Gch_bridge (Suc n) g h f"   and "f (Suc n) = Suc n" shows "card {i. i \<le> Suc n \<and> g i \<cong> E} = card {i. i \<le> Suc n \<and> h i \<cong> E}"
rel_inject_1: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R None None"
invfun_l1: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and a :: "'e" assumes "f \<in> A \<rightarrow> B"   and "bij_to f A B"   and "a \<in> A" shows "(f\<inverse>\<^bsub>B,A\<^esub>) (f a) = a"
cartesian_product: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and B :: "'e set"   and A :: "'d set" shows "F (\<lambda>x. F (g x) B) A = F (\<lambda>(x, y). g x y) (A \<times> B)"
divides_prod_l: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a \<in> carrier (poly_ring R)"   and "b \<in> carrier (poly_ring R)"   and "c \<in> carrier (poly_ring R)"   and "a divides\<^bsub>poly_ring R\<^esub> b" shows "a divides\<^bsub>poly_ring R\<^esub> c \<otimes>\<^bsub>poly_ring R\<^esub> b"
simps: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and xs :: "('a \<times> 'c) list"   and ko :: "'o" shows "update_by_fun_raw k f (xs, ko) = (update_by_fun_pair ko k f xs, ko)"
c: shows "\<forall>I \<omega> X. (\<omega> \<in> game_sem I \<alpha> X) = (\<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> (M \<union> N)))"
a1: shows "setbydecision TYPE('pkt_ext) iface rs FinalAllow \<subseteq> ipcidr_union_set (set (the (ipassmt iface)))"
induct: fixes P :: "term \<Rightarrow> bool"   and "term" :: "term" assumes "\<And>x. P (Const x)"   and "\<And>x. P (Free x)"   and "\<And>x. P x \<Longrightarrow> P (\<Lambda> x)"   and "\<And>x. P (Bound x)"   and "\<And>x1a x2a. \<lbrakk>P x1a; P x2a\<rbrakk> \<Longrightarrow> P (x1a $ x2a)" shows "P term"
product_nn_integral_insert: fixes I :: "'i set"   and i :: "'i"   and f :: "('i \<Rightarrow> 'a) \<Rightarrow> ennreal" assumes "finite I"   and "i \<notin> I"   and "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)" shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x(i := y)) \<partial>M i \<partial>Pi\<^sub>M I M"
Inl_inject: fixes x :: "'g"   and y :: "'g" assumes "Inl x = Inl y" shows "x = y"
scast_scast_id_2: fixes y :: "'b signed word" shows "scast (scast y) = y"
rep_eq: fixes x :: "Linear_Aggregation_Algebras.unit"   and xa :: "Linear_Aggregation_Algebras.unit" shows "Linear_Aggregation_Algebras.unit.Rep_unit (x + xa) = local.tl (Linear_Aggregation_Algebras.unit.Rep_unit x) (Linear_Aggregation_Algebras.unit.Rep_unit xa)"
assms_5: shows "ptr \<notin> set |h \<turnstile> get_scdom_component ptr'|\<^sub>r"
tappend_inf: fixes xs :: "('f, 'g) tllist"   and f :: "'g \<Rightarrow> ('f, 'h) tllist" assumes "\<not> tfinite xs" shows "tappend xs f = tcast xs"
MOST_finite_Ball_distrib: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "finite A" shows "(\<forall>\<^sub>\<infinity>y. \<forall>x\<in>A. P x y) = (\<forall>x\<in>A. \<forall>\<^sub>\<infinity>y. P x y)"
cong_mult: fixes b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a"   and e :: "'a" assumes "[b = c] (mod a)"   and "[d = e] (mod a)" shows "[b * d = c * e] (mod a)"
CX_p2D: fixes x :: "'a \<times> 'b \<Rightarrow>\<^sub>L 'a" assumes "x \<in> CX" shows "fst x \<notin> p2"
wrm_less_rm: fixes t :: "'b Tree"   and h :: "'b \<Rightarrow> int" shows "t \<noteq> Tip \<and> sortedTree h t \<longrightarrow> (\<forall>l\<in>setOf (wrm h t). h l < h (rm h t))"
list_e_eq_lel_1: fixes e :: "'c"   and l1 :: "'c list"   and e' :: "'c"   and l2 :: "'c list" shows "([e] = l1 @ e' # l2) = (l1 = [] \<and> e' = e \<and> l2 = [])"
euclerian_cycle_ex: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_Eulerian_circuit v ps v'"   and "finite V"   and "finite E" shows "\<forall>v\<in>V. even (degree v G)"
pOp_assocTr43: fixes A :: "('d, 'e) aGroup_scheme"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> (c \<plusminus>\<^bsub>A\<^esub> d) = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c) \<plusminus>\<^bsub>A\<^esub> d"
guessed_frame_Init_xgnx: fixes R :: "fid_t" assumes "role (guessed_runs R) = Init" shows "sklvl1.guessed_frame R xgnx = Some (Exp Gen (NonceF (R $ nx)))"
scc_root_unique_is_scc: fixes v :: "'v"   and scc :: "'v set" assumes "scc_root s' v scc" shows "scc_root s' v (scc_of E v)"
qSwap_preserves_qFresh: fixes ys :: "'d"   and zs :: "'d"   and z1 :: "'e"   and z2 :: "'e"   and y :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "qFresh ys (y @ys[z1 \<and> z2]_zs) (X #[[z1 \<and> z2]]_zs) = qFresh ys y X"
nextl_state: fixes M :: "'b dfa"   and q :: "hf"   and xs :: "'b list" assumes "dfa M"   and "q \<in> dfa.states M" shows "dfa.nextl M q xs \<in> dfa.states M"
retract_of_empty_2: fixes S :: "'c set" shows "(S retract_of {}) = (S = {})"
xy: shows "(x, y) \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"
relation_subsumption_impl_def: fixes r :: "'b \<Rightarrow> 'c"   and subsumes :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and set_of :: "'c \<Rightarrow> 'b set"   and sel :: "'c \<Rightarrow> 'b \<times> 'c"   and un :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and size :: "'c \<Rightarrow> nat" shows "relation_subsumption_impl r subsumes set_of sel un size \<equiv> subsumption_impl r subsumes set_of sel un size \<and> relation_subsumption_impl_axioms r set_of"
repeatable_both_self: fixes t :: "'b ttree" assumes "repeatable t" shows "t \<otimes>\<otimes> t = t"
C3: fixes R :: "ccs"   and S :: "ccs" assumes "(R \<parallel> !R, S) \<in> Rel'" shows "(!R, S) \<in> Rel'"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs" shows "l_delete_shadow_root_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
control_spec: shows "\<forall>s d aseqs. case control s d aseqs of (a, aseqs', s') \<Rightarrow> thread_empty aseqs \<and> (a, aseqs') = (None, []) \<or> aseqs \<noteq> [] \<and> hd aseqs \<noteq> [] \<and> \<not> aborting s' d (the a) \<and> \<not> waiting s' d (the a) \<and> (a, aseqs') = (Some (hd (hd aseqs)), tl (hd aseqs) # tl aseqs) \<or> aseqs \<noteq> [] \<and> hd aseqs \<noteq> [] \<and> waiting s' d (the a) \<and> (a, aseqs', s') = (Some (hd (hd aseqs)), aseqs, s) \<or> (a, aseqs') = (None, tl aseqs)"
hoare_hadamard_n: fixes P :: "complex mat"   and k :: "nat" assumes "is_quantum_predicate P"   and "k < n" shows "\<turnstile>\<^sub>p {adjoint (exexH_k k) * P * exexH_k k} hadamard_n (Suc k) {P}"
setdist_sing_in_set: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> S" shows "setdist {x} S = 0"
less_eq_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le> ys \<equiv> prefix xs ys"
im_lconjby: fixes x :: "'g"   and g :: "'g" assumes "x \<in> UNIV"   and "g \<in> UNIV" shows "T (lconjby x g) = lconjby (T x) (T g)"
a_star_redu_OrL_elim: fixes x :: "name"   and M :: "trm"   and y :: "name"   and N :: "trm"   and z :: "name"   and R :: "trm" assumes "OrL x.M y.N z \<longrightarrow>\<^sub>a* R" shows "\<exists>M' N'. R = OrL x.M' y.N' z \<and> M \<longrightarrow>\<^sub>a* M' \<and> N \<longrightarrow>\<^sub>a* N'"
execs_pres_termi: fixes c :: "com"   and s :: "state"   and c' :: "com"   and cs' :: "com list"   and s' :: "state" assumes "([c], s) \<rightarrow>\<^sup>* (c' # cs', s')"   and "c \<down> s" shows "c' \<down> s'"
hyps: fixes ts :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option" assumes "redT_updTs ts TAS t = \<lfloor>(x, w)\<rfloor>"   and "thread_oks ts TAS"   and "ts t = None" shows "\<exists>m. NewThread t x m \<in> set TAS \<and> w = no_wait_locks"
Range_def: fixes f :: "ZF" shows "HOLZF.Range f \<equiv> Replacement f (\<lambda>p. if isOpair p then Some (Snd p) else None)"
assms_2: fixes formula :: "('a, 'b, 'c) formula"   and c :: "'d" assumes "\<And>c. P c formula" shows "P c (Formula.Not formula)"
timpls_transformable_to'_iff_in_timpl_closure: fixes TI :: "('b set \<times> 'b set) list"   and s :: "(('c, 'd, 'b) prot_fun, 'e) Term.term"   and t :: "(('c, 'd, 'b) prot_fun, 'e) Term.term" shows "timpls_transformable_to' TI s t = (t \<in> timpl_closure s (set TI))"
lconstT: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "lconst a \<in> A\<^sup>\<omega>"
l3_inv3_def: shows "l3_inv3 \<equiv> {s. analz (ik s) \<subseteq> synth (analz (ik s \<inter> payload \<union> ik s \<inter> valid \<union> ik s \<inter> range LtK \<union> Tags))}"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Macron of Acute \<Rightarrow> f1 | Grave \<Rightarrow> f2 | Macron \<Rightarrow> f3) = f3"
g_def: shows "g \<equiv> \<lambda>x. if x = z then 0 else inverse (f x)"
set_of_list_singletonE: fixes xs :: "'a list"   and x :: "'a" assumes "set xs = {x}"   and "distinct xs" shows "xs = [x]"
finite_AF: fixes k :: "nat"   and i :: "nat" shows "finite (AF k i)"
fps_catalan_recurrence: shows "fps_catalan = 1 + fps_X * fps_catalan\<^sup>2"
dim_le_card: fixes V :: "'b set"   and W :: "'b set" assumes "V \<subseteq> span W"   and "finite W" shows "dim V \<le> card W"
integral_iff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<otimes> b = \<zero>) = (a = \<zero> \<or> b = \<zero>)"
pre_subdivFace_def: fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and vOptionList :: "nat option list" shows "pre_subdivFace g f v' vOptionList \<equiv> pre_subdivFace_face f v' vOptionList \<and> \<not> invalidVertexList g f vOptionList"
update_trie_Nil: fixes v :: "'d"   and vo :: "'d option"   and ts :: "('c \<times> ('c, 'd) trie) list" shows "update_trie [] v (Trie vo ts) = Trie (Some v) ts"
ex_complex_sqrt: fixes z :: "complex" shows "\<exists>s. s * s = z"
bx: shows "Lyndon_rec (a # w) \<^bold>|w\<^bold>| = (\<forall>n. n < \<^bold>|a # w\<^bold>| \<and> 0 < n \<longrightarrow> Lyndon_rec (a # w) n)"
funpower_consume_function: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'd"   and I :: "'c"   and n :: "nat" assumes "\<And>X. P X \<Longrightarrow> f (g X) = h (f X) \<and> P (g X)"   and "P I" shows "P (funpower g n I) \<and> f (funpower g n I) = funpower h n (f I)"
edgeDecls_def: fixes e :: "edge" shows "edgeDecls e = (case effect e of EEDecl p \<Rightarrow> {p} | _ \<Rightarrow> {})"
emeasure_lborel_singleton: fixes x :: "'a" shows "emeasure lborel {x} = 0"
m_gt_one: fixes m :: "int" assumes "residues m" shows "1 < m"
ennreal_add_diff_cancel: fixes b :: "ennreal"   and a :: "ennreal" assumes "b \<noteq> \<infinity>" shows "a + b - b = a"
diff_neutralizes: fixes H :: "'a set"   and R :: "'a set"   and r1 :: "'a"   and r2 :: "'a" assumes "subgroup H (add_monoid G)"   and "R \<in> rcosets\<^bsub>add_monoid G\<^esub> H"   and "r1 \<in> R"   and "r2 \<in> R" shows "r1 \<oplus> \<ominus> r2 \<in> H"
mk_coeff_def: fixes vs :: "'f list"   and c :: "nat \<Rightarrow> 'a"   and v :: "'f" shows "mk_coeff vs c v \<equiv> R.sumlist (map c (find_indices v vs))"
simps_2: fixes C :: "char list"   and e :: "expr"   and A :: "char list set option" shows "\<D> (Cast C e) A = \<D> e A"
PC_fixed: fixes P :: "'b \<Rightarrow> real"   and a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and ab :: "bool"   and R :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and p :: "real" assumes "P \<tturnstile> a ab R"   and "Q \<tturnstile> b ab R"   and "0 \<le> p"   and "p \<le> 1" shows "\<lambda>s. p * P s + (1 - p) * Q s \<tturnstile> (a \<^bsub>(\<lambda>s. p)\<^esub>\<oplus> b) ab R"
axioms_3: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplexMorphism_axioms X Y f"
stable_def: fixes \<mu> :: "(M \<times> W) set" shows "Non_Strict.stable \<mu> = (Non_Strict.match \<mu> \<and> (\<forall>m w. \<not> Non_Strict.blocks m w \<mu>))"
v_in_prime_v: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "v \<in> P\<^bsub> K v\<^esub>"
surj_imp_ordLeq: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "B \<subseteq> f ` A" shows "|B| \<le>o |A|"
prefix_append: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "prefix xs (ys @ zs) = (prefix xs ys \<or> (\<exists>us. xs = ys @ us \<and> prefix us zs))"
id_union_univ_1: shows "\<Union> (range \<Sigma>n) - \<C> \<union> \<C> = UNIV"
some_geodesic_is_geodesic_segment_2: fixes x :: "'a"   and y :: "'a" shows "geodesic_segment {x--y}"
match_list_True: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and ms :: "'b match_expr list"   and a :: "action"   and p :: "'c" assumes "match_list \<gamma> ms a p" shows "approximating_bigstep_fun \<gamma> p (map (\<lambda>m. Rule m a) ms) Undecided = (case a of Accept \<Rightarrow> Decision FinalAllow | Drop \<Rightarrow> Decision FinalDeny | Log \<Rightarrow> Undecided | Reject \<Rightarrow> Decision FinalDeny | Empty \<Rightarrow> Undecided)"
Chi_range_disjoint: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and U :: "'a" assumes "S \<noteq> T"   and "T \<in> HAStates A"   and "S \<in> HAStates A"   and "U \<in> Chi A S" shows "U \<notin> Chi A T"
reflclp_ACI: shows "(~)\<^sup>=\<^sup>= = (~)"
verts_slim_in_G: shows "pverts slim.slim \<subseteq> pverts slim"
simps_5: fixes f :: "form" shows "fv (FAll f) = preSuc (fv f)"
cis_mult: fixes a :: "real"   and b :: "real" shows "cis a * cis b = cis (a + b)"
supt_size: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "s \<rhd> t" shows "size t < size s"
P_D2_H2_H1_def: fixes A :: "(D2 \<times> H2) set" shows "P_D2_H2_H1 A = (if (D1, H1) \<in> A then {(D1, H1)} else if (D2, H1) \<in> A then {(D2, H1)} else {})"
T_vrestr_same_emeasure_1: fixes A :: "'a set" assumes "A \<in> sets M" shows "emeasure M (T --` A) = emeasure M A"
pat_elem_less_size: fixes pat :: "'b"   and e :: "exp0"   and pes :: "('b \<times> exp0) list" assumes "(pat, e) \<in> set pes" shows "size_exp' e < size_list (size_prod size size_exp') pes"
YM4_Key_parts_knows_Spy: fixes A :: "agent"   and B :: "msg"   and K :: "msg"   and NA :: "msg"   and NB :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>B, K, NA, NB\<rbrace>, X\<rbrace> \<in> set evs" shows "K \<in> parts (knows Spy evs)"
assms: shows "(u, v) \<in> (edges g - {(u, v), (v, u)})\<^sup>*"
assms: fixes T :: "('a \<times> real) set"   and f :: "'a \<times> real \<Rightarrow> 'a"   and U :: "('a \<times> real) set" assumes "continuous_on T f"   and "f ` T \<subseteq> S"   and "closedin (top_of_set U) T" shows "\<exists>g. continuous_on U g \<and> g ` U \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)"
ceil_elim: fixes r :: "'a" shows "r \<le> of_nat (nat \<lceil>r\<rceil>)"
assms: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ [Rule m Empty] @ rs\<^sub>2, s\<rangle> \<Rightarrow> t"
stable_translation_length_as_pointwise_limit: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "1-lipschitz_on UNIV f" shows "(\<lambda>n. dist x ((f ^^ n) x) / real n) \<longlonglongrightarrow> stable_translation_length f"
cases: fixes a :: "'b formula" assumes "a \<in> AX0" obtains F :: "'b formula" and G :: "'b formula" where "a = F__ \<^bold>\<rightarrow> G__ \<^bold>\<rightarrow> F__"   | F :: "'b formula" and G :: "'b formula" and H :: "'b formula" where "a = (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H"