hcju/isabelleps huggingface dataset:
connected_space_subconnected: fixes X :: "'c topology" shows "connected_space X = (\<forall>x\<in>topspace X. \<forall>y\<in>topspace X. \<exists>S. connectedin X S \<and> x \<in> S \<and> y \<in> S)"
weak_complete_lattice_interval: fixes L :: "('a, 'b) gorder_scheme"   and a :: "'a"   and b :: "'a" assumes "weak_complete_lattice L"   and "a \<in> carrier L"   and "b \<in> carrier L"   and "a \<sqsubseteq>\<^bsub>L\<^esub> b" shows "weak_complete_lattice (L\<lparr>carrier := \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>\<rparr>)"
1: shows "apps (Hd (head t)) (butlast (args t)) \<rightarrow>\<^sub>e\<^sub>m\<^sub>b chop (apps (Hd (head t)) (butlast (args t)))"
induct: fixes P :: "'c list \<Rightarrow> bool"   and a0 :: "'c list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. \<lbrakk>x = y \<Longrightarrow> P (x # xs); x \<noteq> y \<Longrightarrow> P (y # xs)\<rbrakk> \<Longrightarrow> P (x # y # xs)" shows "P a0"
critical_set_0: fixes A :: "ordinal set" shows "critical_set A 0 = A"
embed_split: fixes f :: "'a \<Rightarrow> real"   and s :: "'a"   and P :: "'a \<Rightarrow> bool" shows "f s = \<guillemotleft> P \<guillemotright> s * f s + \<guillemotleft> \<N> P \<guillemotright> s * f s"
awalk_verts_arc2: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and e :: "'a \<times> 'a" assumes "awalk u p v"   and "e \<in> set p" shows "snd e \<in> set (pawalk_verts u p)"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "T_isort (x # xs) = T_isort xs + T_insort x (isort xs) + 1"
gbg_to_idx_ext_impl_def: fixes eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and bhc :: "nat \<Rightarrow> 'd \<Rightarrow> nat"   and def_size :: "nat" shows "gbg_to_idx_ext_impl eq bhc def_size \<equiv> \<lambda>x xa. RETURN (id (gbgi_F xa)) \<bind> (\<lambda>xb. let y' = length xb in (let y'a = Impl_Array_Hash_Map.ahm_empty def_size in nfoldli xb (\<lambda>xc. True) (\<lambda>xc (xd, xe). LIST_FOREACH' (RETURN (id xc)) (\<lambda>xf. True) (\<lambda>xf xg. RETURN (Impl_Array_Hash_Map.ahm_update eq bhc xf (bs_insert xd (the_default (bs_empty ()) (Impl_Array_Hash_Map.ahm_lookup eq bhc xf xg))) xg)) xe \<bind> (\<lambda>xf. RETURN (Suc xd, xf))) (0, y'a) \<bind> (\<lambda>(xc, xd). RETURN (\<lambda>xe. the_default (bs_empty ()) (Impl_Array_Hash_Map.ahm_lookup eq bhc xe xd)))) \<bind> (\<lambda>xc. RETURN (y', xc))) \<bind> (\<lambda>(xb, xc). RETURN \<lparr>gi_V = gi_V xa, gi_E = gi_E xa, gi_V0 = gi_V0 xa, igbgi_num_acc = xb, igbgi_acc = xc, \<dots> = x xa\<rparr>)"
psiCasesInject: fixes CsP :: "('c \<times> ('a, 'b, 'c) psi) list"   and CsQ :: "('c \<times> ('a, 'b, 'c) psi) list" shows "(psiCases CsP = psiCases CsQ) = (CsP = CsQ)"
measurable_compose_countable: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'b \<Rightarrow> 'a" assumes "\<And>i. f i \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "(\<lambda>x. f (g x) x) \<in> M \<rightarrow>\<^sub>M N"
simps_6: fixes \<C> :: "t_context"   and t :: "t"   and uy :: "testop"   and ts :: "checker_type" shows "check_single \<C> (Testop t uy) ts = (if is_int_t t then type_update ts [TSome t] (Type [T_i32]) else Bot)"
unital_quantale_homset_iff: fixes f :: "'c \<Rightarrow> 'd" shows "(f \<in> unital_quantale_homset) = (comp_pres f \<and> Sup_pres f \<and> un_pres f)"
singular_boundary_singleton: fixes X :: "'c topology"   and a :: "'c"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "topspace X = {a}" shows "singular_relboundary p X {} c = (singular_chain p X c \<and> (odd p \<or> c = 0))"
cong_commutativity: 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"   and "Cong A B C D" shows "Cong B A D C"
cSup_singleton: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Sup {x} = x"
scale_right_diff_distrib: fixes a :: "complex"   and x :: "'b"   and y :: "'b" shows "a *\<^sub>C (x - y) = a *\<^sub>C x - a *\<^sub>C y"
Inl_def_alt: shows "Inl \<equiv> \<lambda>a. Basic_BNF_LFPs.xtor (BNF_Composition.id_bnf (Inl a))"
append_in_starI: fixes u :: "'a list"   and A :: "'a list set"   and v :: "'a list" assumes "u \<in> star A"   and "v \<in> star A" shows "u @ v \<in> star A"
simps_85: fixes f1 :: "char list \<Rightarrow> 'c trm list \<Rightarrow> 'b"   and f2 :: "'c trm \<Rightarrow> 'c trm \<Rightarrow> 'b"   and f3 :: "'c formula \<Rightarrow> 'b"   and f4 :: "'c formula \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f5 :: "'c formula \<Rightarrow> 'b"   and f6 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f7 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f8 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f9 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and x41 :: "'c formula"   and x42 :: "'c formula" shows "(case Or x41 x42 of Pred x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa | Neg x \<Rightarrow> f3 x | Or x xa \<Rightarrow> f4 x xa | Exists x \<Rightarrow> f5 x | Prev x xa \<Rightarrow> f6 x xa | Next x xa \<Rightarrow> f7 x xa | Since x xa xb \<Rightarrow> f8 x xa xb | Until x xa xb \<Rightarrow> f9 x xa xb) = f4 x41 x42"
hereditarily_fs_alpha_renaming: fixes \<alpha> :: "'d"   and t\<^sub>\<alpha> :: "('b, 'c, 'd) Tree\<^sub>\<alpha>"   and \<alpha>' :: "'d"   and t\<^sub>\<alpha>' :: "('b, 'c, 'd) Tree\<^sub>\<alpha>" assumes "Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> \<alpha>' t\<^sub>\<alpha>'" shows "hereditarily_fs t\<^sub>\<alpha> = hereditarily_fs t\<^sub>\<alpha>'"
fun_upd_idem: fixes f :: "'g \<Rightarrow> 'f"   and x :: "'g"   and y :: "'f" assumes "f x = y" shows "f(x := y) = f"
intro: fixes T :: "real set" assumes "is_interval T" shows "interval T"
protocol_def: fixes M :: "'grp \<times> 'grp"   and v :: "bool" shows "protocol M v = (let (m0, m1) = M in sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. let c\<^sub>v = a * b mod order \<G> in sample_uniform (order \<G>) \<bind> (\<lambda>c\<^sub>v'. sample_uniform_units (order \<G>) \<bind> (\<lambda>r0. sample_uniform_units (order \<G>) \<bind> (\<lambda>s0. let w0 = (\<^bold>g [^] a) [^] s0 \<otimes> \<^bold>g [^] r0; z0' = (\<^bold>g [^] (if v then c\<^sub>v' else c\<^sub>v)) [^] s0 \<otimes> (\<^bold>g [^] b) [^] r0 in sample_uniform_units (order \<G>) \<bind> (\<lambda>r1. sample_uniform_units (order \<G>) \<bind> (\<lambda>s1. let w1 = (\<^bold>g [^] a) [^] s1 \<otimes> \<^bold>g [^] r1; z1' = (\<^bold>g [^] (if v then c\<^sub>v else c\<^sub>v')) [^] s1 \<otimes> (\<^bold>g [^] b) [^] r1; enc_m0 = z0' \<otimes> m0; enc_m1 = z1' \<otimes> m1; out_2 = if v then enc_m1 \<otimes> inv (w1 [^] b) else enc_m0 \<otimes> inv (w0 [^] b) in return_spmf ((), out_2)))))))))"
div_mult2_eq': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide a (times (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)) = divide (divide a (semiring_1.of_nat one plus zero m)) (semiring_1.of_nat one plus zero n)"
pmod_zero_iff_pdivides: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "(UP.pmod p q = []) = q pdivides\<^bsub>K [X]\<^esub> p"
lincomb_scalar_prod_right: fixes W :: "'b vec set"   and n :: "nat"   and v :: "'b vec"   and a :: "'b vec \<Rightarrow> 'b" assumes "W \<subseteq> carrier_vec n"   and "v \<in> carrier_vec n" shows "v \<bullet> module.lincomb (module_vec TYPE('b) n) a W = (\<Sum>w\<in>W. a w * (v \<bullet> w))"
lockstep_simulation: fixes s1 :: "'state1"   and s2 :: "'state2"   and s2' :: "'state2" assumes "match s1 s2"   and "step2 s2 s2'" shows "\<exists>s1'. step1 s1 s1' \<and> match s1' s2'"
LFOc_refine: fixes li :: "'a list"   and l :: "'b set"   and A :: "('a \<times> 'b) set"   and R :: "('d \<times> 'c) set"   and ci :: "'d \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool"   and fi :: "'a \<Rightarrow> 'd \<Rightarrow> 'd nres"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s0i :: "'d"   and s0 :: "'c" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "\<And>s si. (si, s) \<in> R \<Longrightarrow> ci si = c s"   and "\<And>x xi s si. \<lbrakk>(xi, x) \<in> A; (si, s) \<in> R\<rbrakk> \<Longrightarrow> fi xi si \<le> \<Down> R (f x s)"   and "(s0i, s0) \<in> R" shows "nfoldli li ci fi s0i \<le> \<Down> R (FOREACH\<^sub>C l c f s0)"
polyn_expr_def: fixes R :: "('d, 'e) Ring_scheme"   and X :: "'d"   and k :: "nat"   and c :: "nat \<times> (nat \<Rightarrow> 'd)" shows "polyn_expr R X k c \<equiv> \<Sigma>\<^sub>e R (\<lambda>j. snd c j \<cdot>\<^sub>r\<^bsub>R\<^esub> X^\<^bsup>R j\<^esup>) k"
dq_def: shows "dq = last (prio_emb_pos prio t s)"
emeasure_lim_emb: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "J \<subseteq> I"   and "finite J"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>i. borel))" shows "emeasure local.lim (emb I J X) = emeasure (P J) X"
sep_disj_addD1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x (plus y z)"   and "sep_disj y z" shows "sep_disj x y"
max_normal_subgroup_def: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" shows "max_normal_subgroup H G \<equiv> H \<lhd> G \<and> max_normal_subgroup_axioms H G"
assms: shows "\<rho> \<in> \<lbrakk> \<S>\<^sub>1 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
sup_measure_F_mono: fixes I :: "'a measure set"   and J :: "'a measure set" assumes "finite I"   and "J \<subseteq> I" shows "sup_measure.F id J \<le> sup_measure.F id I"
span_clauses_1: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and S :: "'e set" assumes "module scale"   and "a \<in> S" shows "a \<in> module.span scale S"
sep_impl_sep_true_false: shows "((\<lambda>s. True) \<longrightarrow>* (\<lambda>s. False)) = (\<lambda>s. False)"
simps_2: shows "nat_to_sch (Suc 0) = Base_zero"
Quotient_abs_rep: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "Quotient R Abs Rep T" shows "Abs (Rep a) = a"
out_arcs_int_arcs: fixes G :: "('d, 'c) pre_digraph"   and u :: "'d" shows "out_arcs G u \<inter> arcs G = out_arcs G u"
tendsto_Min: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b \<Rightarrow> 'a"   and F :: "'c filter" assumes "class.linorder_topology less_eq less open"   and "I \<noteq> {}"   and "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> filterlim (f i) (topological_space.nhds open (a i)) F" shows "LIM x F. linorder.Min less_eq ((\<lambda>i. f i x) ` I) :> topological_space.nhds open (linorder.Min less_eq (a ` I))"
simps_1: fixes uu :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and uv :: "'f"   and t :: "'f" shows "cond_action_ok' uu uv (Join t) = True"
homeomorphic_eq_everything_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "homeomorphic_map X Y f = (continuous_map X Y f \<and> open_map X Y f \<and> closed_map X Y f \<and> f ` topspace X = topspace Y \<and> inj_on f (topspace X))"
swap_product: fixes A :: "'c set"   and B :: "'b set" shows "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
assms: shows "\<tau>.P \<longmapsto> a[b] \<prec> P'"
times_divide_eq_left: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "b / c * a = b * a / c"
da_Init: fixes A :: "assigned"   and B :: "lname set"   and Env :: "env"   and C :: "qtname" assumes "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>" shows "Env\<turnstile> B \<guillemotright>\<langle>Init C\<rangle>\<guillemotright> A"
hyps_1: assumes "fv exp1 \<subseteq> set Vs" shows "fv (compE1 Vs exp1) = index Vs ` fv exp1"
BIGTHETA_CONST'_tag: fixes c :: "'b" shows "\<Theta>(\<lambda>x. c) = BIGTHETA_CONST' c"
dim_a_1: shows "vec_normalize v \<in> carrier_vec n"
fold2: fixes s1 :: "('e, 'f, 'd) F1 \<Rightarrow> 'f"   and s2 :: "('e, 'f, 'd) F2 \<Rightarrow> 'd"   and x :: "('e, 'e IF1, 'e IF2) F2" shows "fold2 s1 s2 (ctor2 x) = s2 (F2map id (fold1 s1 s2) (fold2 s1 s2) x)"
set_list_bind: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'a list" shows "set (List.bind xs f) = (\<Union>x\<in>set xs. set (f x))"
dep_vars_Cons_keep: fixes V :: "'var"   and as' :: "'edge list"   and as :: "'edge list"   and n' :: "'node"   and a :: "'edge" assumes "(V, as', as) \<in> dependent_live_vars n'"   and "n' = last (targetnodes (a # as))"   and "\<not> sourcenode a -{V}a # as'\<rightarrow>\<^sub>d\<^sub>d last (targetnodes (a # as'))" shows "(V, a # as', a # as) \<in> dependent_live_vars n'"
wt_step_wtl_lemma: fixes pc :: "nat"   and ls :: "'b list"   and s :: "'a" assumes "wt_step r \<top> step \<tau>s"   and "pc + length ls = length \<tau>s"   and "s \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! pc"   and "s \<in> A"   and "s \<noteq> \<top>" shows "wtl ls c pc s \<noteq> \<top>"
seq_char: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" shows "A1xA2.seq g f = (A1.seq (fst g) (fst f) \<and> A2.seq (snd g) (snd f))"
max_test_bit: fixes n :: "nat" shows "(- 1) !! n = (n < LENGTH('b))"
codomain_block_nonnegative: fixes x :: "brick list" shows "0 \<le> codomain_block x"
simps: fixes x :: "'a"   and y :: "'b Filters.filter" shows "pairs_uminus (x, y) = (- x, phi x)"
mat_multcol_def: fixes k :: "nat"   and a :: "'a"   and A :: "'a mat" shows "multcol k a A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = j then a * A $$ (i, j) else A $$ (i, j))"
plus_nmultiset_def: fixes X :: "unit nmultiset"   and Y :: "unit nmultiset" shows "plus_nmultiset X Y = Hereditary_Multiset.Rep_hmultiset (Hereditary_Multiset.Abs_hmultiset X + Hereditary_Multiset.Abs_hmultiset Y)"
diagvector_eq_mat: fixes x :: "'b \<Rightarrow> 'a"   and a :: "'a" shows "(diagvector x = mat a) = (x = (\<lambda>x. a))"
h: shows "steps0 (1, l, r) B (Suc stp) = (s', l', r')"
rep_eq: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'b" assumes "bounded_bilinear prod" shows "blinfun_apply (bounded_bilinear.prod_right prod x) = prod x"
10: fixes x :: "'a" shows "x \<squnion> bot = x"
h: fixes t2 :: "lam" assumes " t \<longrightarrow>\<^sub>1 t2" shows "\<exists>t3.  s1 \<longrightarrow>\<^sub>1 t3 \<and>  t2 \<longrightarrow>\<^sub>1\<^sup>* t3"
case_1: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b pred \<Rightarrow> 'a"   and f3 :: "'b pred \<Rightarrow> 'b seq \<Rightarrow> 'a" shows "(case Empty of Empty \<Rightarrow> f1 | Insert x xa \<Rightarrow> f2 x xa | Join x xa \<Rightarrow> f3 x xa) = f1"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "comm_semiring_hom hom"   and "inj_semiring_hom hom" shows "inj_comm_semiring_hom hom"
iso_tuple_update_accessor_cong_assist_id: fixes upd :: "('g \<Rightarrow> 'g) \<Rightarrow> 'h \<Rightarrow> 'h"   and ac :: "'h \<Rightarrow> 'g" assumes "Record.iso_tuple_update_accessor_cong_assist upd ac" shows "upd id = id"
countable_null_set: fixes A :: "'a set" assumes "countable A"   and "A \<subseteq> space M" shows "A \<in> null_sets M"
sets_PiM: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" shows "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
degree0_polybound0: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p n"   and "Reflected_Multivariate_Polynomial.degree p = 0" shows "polybound0 p"
zeroTrans: fixes Rs :: "residual" assumes "\<zero> \<longmapsto> Rs" shows "False"
perzeta_one_half_left': fixes s :: "complex" shows "perzeta (1 / 2) s = (if s = 1 then - Ln 2 else (2 powr (1 - s) - 1) / (s - 1)) * ((s - 1) * pre_zeta 1 s + 1)"
Mp_lift_modulus: fixes f :: "int poly"   and g :: "int poly"   and k :: "int" assumes "q.eq_m f g" shows "poly_mod.Mp (q * k) (Polynomial.smult k f) = poly_mod.Mp (q * k) (Polynomial.smult k g)"
pred_mono_strong: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a list"   and Pa :: "'a \<Rightarrow> bool" assumes "list_all P x"   and "\<And>z. \<lbrakk>z \<in> set x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "list_all Pa x"
bij_betw_vertices: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexAutomorphism X f" shows "bij_betw f (\<Union> X) (\<Union> X)"
Glb_greatest: fixes b :: "'a"   and A :: "'a set" assumes "b \<in> L"   and "\<forall>a\<in>A. b \<le> a" shows "b \<le> Glb A"
2: shows "w\<^sub>3 = w\<^sub>2 @ v\<^sub>2"
subspace_span: fixes Sa :: "'a set" assumes "Sa \<subseteq> S" shows "subspace (span Sa)"
eval\<Pi>\<^sub>2_inject: fixes x :: "\<Pi>\<^sub>2"   and y :: "\<Pi>\<^sub>2" shows "(eval\<Pi>\<^sub>2 x = eval\<Pi>\<^sub>2 y) = (x = y)"
of_int_neg_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (- numeral k) = uminus (numeral.numeral one plus k)"
WTrtCallNT: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and es :: "'addr expr list"   and Ts :: "ty list"   and M :: "String.literal"   and T :: "ty" assumes "P,E,h \<turnstile> e : NT"   and "P,E,h \<turnstile> es [:] Ts" shows "P,E,h \<turnstile> e\<bullet>M(es) : T"
dg_Rel_is_arrD_2: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Rel \<alpha>\<^esub> B" shows "S\<lparr>ArrDom\<rparr> = A"
set_locals_def: fixes l :: "lname \<Rightarrow> val option" shows "set_locals l = case_st (\<lambda>g l'. st g l)"
concat_vpath_is_vpath: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a"   and q :: "'a list"   and w :: "'a" assumes "vwalk p G"   and "hd p = u"   and "last p = v"   and "vwalk q G"   and "hd q = v"   and "last q = w" shows "vpath (concat_vpath p q) G \<and> hd (concat_vpath p q) = u \<and> last (concat_vpath p q) = w"
lnth_iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and n :: "nat" shows "iterates f x $ n = (f ^^ n) x"
rev_iexI: fixes t :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "t \<in> I"   and "P t" shows "\<diamond> t I. P t"
glb_bin: fixes x :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> y" shows "\<Sqinter>{x, y} = x"
get_element_by_id_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and idd :: "char list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_element_by_id ptr' idd \<rightarrow>\<^sub>r Some result" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set c) = (ptr' \<in> set c)"
nths_reencode_eq3: fixes C :: "nat set"   and B :: "nat set"   and A :: "nat set"   and xs :: "'b list" assumes "C \<subseteq> B"   and "B \<subseteq> A" shows "nths (nths xs (ind_in_set A ` B)) (ind_in_set B ` C) = nths xs (ind_in_set A ` C)"
unit_vecs_length: fixes n :: "nat" shows "length (unit_vecs n) = n"
replicate_mset_minus_replicate_mset_same: fixes m :: "nat"   and x :: "'a"   and n :: "nat" shows "replicate_mset m x - replicate_mset n x = replicate_mset (m - n) x"
evaln_elim_cases_29: fixes G :: "prog"   and s :: "st"   and e1 :: "expr"   and e2 :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In2 (e1.[e2])\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains a :: "val" and s1 :: "state" and i :: "val" and s2 :: "state" and va :: "vvar" where "v = In2 va" and "G\<turnstile>Norm s \<midarrow>e1-\<succ>a\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>e2-\<succ>i\<midarrow>n\<rightarrow> s2" and "(va, s') = avar G i a s2"
over_def: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d" shows "f over a = f a"
insert_if: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "finite A" shows "aggregation.sum_0 g (insert x A) = (if x \<in> A then aggregation.sum_0 g A else g x + aggregation.sum_0 g A)"
fbox_star_induct_var: fixes y :: "'a"   and x :: "'a" assumes "d y \<le>  |x] y" shows "d y \<le>  |x\<^sup>\<star>] y"
atLeastAtMost_diff_ends: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b - {a, b} = ord.greaterThanLessThan less a b"
elims_3: fixes x :: "atom set \<times> 'e"   and xa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xb :: "'e \<Rightarrow> atom set"   and xc :: "perm"   and xd :: "atom set \<times> 'e" assumes "\<not> x \<approx>set xa xb xc xd" obtains bs :: "atom set" and x :: "'e" and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" and f :: "'e \<Rightarrow> atom set" and p :: "perm" and cs :: "atom set" and y :: "'e" where "xb__ = (bs, x)" and "xa = R" and "xb = f" and "xc = p" and "xd = (cs, y)" and "\<not> (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
fst_safe_distance_def: shows "fst_safe_distance = v\<^sub>o\<^sup>2 / (2 * a\<^sub>o) - v\<^sub>e\<^sup>2 / (2 * a\<^sub>e)"
front_tickFree_dw_closed: fixes s :: "'b event list"   and t :: "'b event list" assumes "front_tickFree (s @ t)" shows "front_tickFree s"
obj_self_composable_2: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.isomorphic (a \<circ> a) a"
seqE_cons: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r # rs, s\<rangle> \<Rightarrow> t" obtains ti :: "state" where "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[r], s\<rangle> \<Rightarrow> ti" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, ti\<rangle> \<Rightarrow> t"
l_set_disconnected_nodes_get_tag_name_axioms: shows "l_set_disconnected_nodes_get_tag_name type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs"
RSpan_zero_closed: fixes ms :: "'m list" assumes "set ms \<subseteq> M" shows "(0::'m) \<in> RSpan ms"
depth_sibling: fixes t :: "'a tree"   and a :: "'a" assumes "consistent t" shows "depth t (sibling t a) = depth t a"