hcju/isabelleps huggingface dataset:
weakSimIChainFresh: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and yvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and C :: "'e" assumes "eqvt Rel"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* Q"   and "\<And>\<Psi>'' \<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; \<alpha> \<noteq> \<tau>; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; yvec \<sharp>* \<alpha>; yvec \<sharp>* Q'; yvec \<sharp>* \<Psi>''\<rbrakk> \<Longrightarrow> \<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>'' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>'', P', Q') \<in> Rel)"   and "\<And>Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'; yvec \<sharp>* Q'\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
that: fixes a :: "'c set"   and b :: "'c set" assumes "(a, b) \<in> TI"   and "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) t s" shows "thesis"
r_one: fixes x :: "'a" assumes "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x"
mult_monom: fixes a :: "'a"   and m :: "nat"   and b :: "'a"   and n :: "nat" shows "Polynomial.monom a m * Polynomial.monom b n = Polynomial.monom (a * b) (m + n)"
relto_fun_refl: fixes A :: "('a \<times> 'a) set"   and B :: "('a \<times> 'a) set"   and a :: "'a" shows "\<exists>as sel. relto_fun A B 0 as sel 0 (a, a)"
dual_inf: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual (inf x y) = sup (dual x) (dual y)"
dim_span_insert: fixes W :: "'a vec set"   and v :: "'a vec" assumes "W \<subseteq> carrier_vec n"   and "v \<in> carrier_vec n"   and "v \<notin> local.span W" shows "dim_span (insert v W) = Suc (dim_span W)"
success_sensitive_iff_source_target_rel_weakly_respects_success: fixes success :: "'barbs" shows "((\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x))) \<and> (\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x)))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_weakly_respects_barb_set Rel (STCalWB SWB TWB) {success})"
PO_a0i_obs_inv1_iagree: shows "oreach a0i \<subseteq> a0i_inv1_iagree"
intros_1: fixes d :: "nat"   and c :: "nat" assumes "0 \<le> d"   and "d \<le> c" shows "Regions.valid_intv c (Regions.intv.Const d)"
closed_Compl: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open"   and "open S" shows "topological_space.closed open (- S)"
bc: shows "ideal_generated {b} \<subseteq> ideal_generated {c}"
encode_problem_with_operator_interference_exclusion_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat" shows "\<Phi>\<^sub>\<forall> \<Pi> t \<equiv> \<Phi>\<^sub>I \<Pi> \<^bold>\<and> (encode_operators \<Pi> t \<^bold>\<and> (encode_all_frame_axioms \<Pi> t \<^bold>\<and> (encode_interfering_operator_exclusion \<Pi> t \<^bold>\<and> (\<Phi>\<^sub>G \<Pi>) t)))"
numadd: fixes bs :: "real list"   and t :: "MIR.num"   and s :: "MIR.num" shows "Inum bs (numadd (t, s)) = Inum bs (Add t s)"
scaleR_cancel_left: fixes a :: "real"   and x :: "'a"   and y :: "'a" shows "(a *\<^sub>R x = a *\<^sub>R y) = (x = y \<or> a = 0)"
CompDiag_Diag_Dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.CompDiag C t (monoidal_language.Dom C t) = t"
phiUsesI: fixes n' :: "'node"   and g :: "'g"   and v' :: "'val"   and vs :: "'val list"   and n :: "'node"   and v :: "'val" assumes "n' \<in> set (\<alpha>n g)"   and "phis g (n', v') = Some vs"   and "(n, v) \<in> set (zip (predecessors g n') vs)" shows "v \<in> phiUses g n"
connectedin_Union: fixes \<U> :: "'b set set"   and X :: "'b topology" assumes "\<And>S. S \<in> \<U> \<Longrightarrow> connectedin X S"   and "\<Inter> \<U> \<noteq> {}" shows "connectedin X (\<Union> \<U>)"
\<C>_vars_\<C>: fixes x :: "'Var" assumes "x \<in> \<C>" shows "\<C>_vars x = {}"
eventually_def: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" shows "eventually P F = Rep_filter F P"
proper_interval_Compl_set_aux: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "proper_intrvl.proper_interval_Compl_set_aux less proper_interval None xs ys = (\<exists>A. ord.set_less_aux less_eq (- set xs) A \<and> ord.set_less_aux less_eq A (set ys))"
inject_4: fixes x5 :: "'b val"   and y5 :: "'b val" shows "(val.Left x5 = val.Left y5) = (x5 = y5)"
set_shadow_root_get_shadow_root_is_l_set_shadow_root_get_shadow_root: shows "l_set_shadow_root_get_shadow_root ShadowRootClass.type_wf set_shadow_root set_shadow_root_locs get_shadow_root get_shadow_root_locs"
float_of_cases: fixes x :: "float" obtains y :: "real" where "x = float_of y" and "y \<in> float"
inf_sup_ord_1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) x"
oGS_rel_def: shows "oGS_rel \<equiv> br oGS_\<alpha> oGS_invar"
quot: shows "quotient_of (fs.gs.gso i $ j) = (num, denom)"
deg_pp_plus: fixes s :: "('b, 'a) pp"   and t :: "('b, 'a) pp" shows "deg_pp (s + t) = deg_pp s + deg_pp t"
prj_simps_3: shows "C.cod \<nu>\<pi>_\<rho>.prj\<^sub>1 = \<nu>\<pi>.apex"
normFace_in_cong: fixes f :: "face"   and g :: "graph" assumes "vertices f \<noteq> []"   and "minGraphProps g"   and "normFace f \<in> set (normFaces (faces g))" shows "\<exists>f'\<in>\<F> g. f \<cong> f'"
subprob_space_distr: fixes f :: "'a cfg stream \<Rightarrow> 'b"   and s :: "'a cfg"   and M' :: "'b measure" assumes "MDP.MC.T.random_variable s M' f"   and "space M' \<noteq> {}" shows "subprob_space (distr (MDP.MC.T s) M' f)"
crel_vs_elim: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and v :: "'d"   and v\<^sub>T :: "('b, 'e) state"   and M :: "'b" assumes "crel_vs R v v\<^sub>T"   and "cmem M"   and "P M" obtains v' :: "'e" and M' :: "'b" where "run_state v\<^sub>T M = (v', M')" and "R v v'" and "cmem M'" and "P M'"
proj2_cart2: fixes p :: "proj2" assumes "z_non_zero p" shows "proj2_pt (cart2_pt p) = p"
local_policies_def: fixes G :: "igraph" shows "local_policies G \<equiv> \<lambda>x. if x = home then {(\<lambda>y. True, {put, get, move, eval})} else if x = sphone then {(\<lambda>y. has G (y, ''PIN''), {put, get, move, eval})} else if x = cloud then {(\<lambda>y. True, {put, get, move, eval})} else if x = hospital then {(\<lambda>y. \<exists>n. n @\<^bsub>G\<^esub> hospital \<and> Actor n = y \<and> has G (y, ''skey''), {put, get, move, eval})} else {}"
single_uminus: fixes k :: "'b"   and a :: "'c" shows "monomial (- a) k = - monomial a k"
ccw_convex_def: fixes ccw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and oriented :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ccw_convex ccw S oriented \<equiv> ccw_system ccw S \<and> ccw_convex_axioms ccw oriented"
children_h': shows "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r remove1 child children_h"
nat_induct2: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>k. \<lbrakk>P k; P (Suc k)\<rbrakk> \<Longrightarrow> P (Suc (Suc k))" shows "P n"
prefixIsSubset: fixes p :: "'b list"   and l :: "'b list" assumes "isPrefix p l" shows "set p \<subseteq> set l"
hp: assumes "w1 \<noteq> POS_INF \<and> 0 <s w1 \<and> w2 = POS_INF" shows "P w1 w2"
IH: fixes y :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "(y, s) \<in> R"   and "I y" shows "wp \<pi> (WHILE b DO c) Q y"
coeff_add_left_notin: fixes x :: "nat"   and p :: "int mpoly" assumes "x \<notin> vars p" shows "MPoly_Type.coeff (p - Var x) (monomial (Suc 0) x) = - 1"
assms_1: shows "(r \<restriction>\<^sup>r\<^sub>\<circ> A)\<lparr>a\<rparr> = b"
tree_ordering_left: fixes X :: "(nat \<times> nat) \<times> nat \<times> nat"   and Y :: "(nat \<times> nat) \<times> nat \<times> nat" assumes "Det X = 1"   and "Det Y = 1"   and "0 < snd (mediant X)" shows "rat_of (mediant (X \<otimes> LL \<otimes> Y)) < rat_of (mediant X)"
simps_2: fixes disc :: "'b \<Rightarrow> bool"   and neg :: "bool"   and a :: "'b" shows "has_disc_negated disc neg (Match a) = (if disc a then neg else False)"
ennreal_mult_eq_top_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(a * b = \<top>) = (a = \<top> \<and> b \<noteq> 0 \<or> b = \<top> \<and> a \<noteq> 0)"
intro: 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 t :: "'d \<Rightarrow> 'e" assumes "category A"   and "category B"   and "functor A B F"   and "functor A B G"   and "transformation_by_components_axioms A B F G t" shows "transformation_by_components A B F G t"
eq_empty_set_tree: fixes t :: "'a tree" shows "({} = set_tree t) = (t = \<langle>\<rangle>)"
Restrict_triv: fixes r :: "('d \<times> 'e) set"   and A :: "'d set" assumes "Domain r \<subseteq> A" shows "Restrict A r = r"
llist_all2_LCons_LNil: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y21 :: "'c"   and y22 :: "'c llist" shows "\<not> llist_all2 R (LCons y21 y22) LNil"
sup_least: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d"   and z :: "'d" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x"   and "less_eq z x" shows "less_eq (sup y z) x"
subt_eq: shows "subtopology (nsphere n) {x. x n = 0} = nsphere (n - Suc 0)"
ConE_1: fixes p :: "'b pfm"   and q :: "'b pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<and>\<^sub>! q)" shows "\<turnstile>\<^sub>! p"
not_coin: shows "\<not> (a = STR ''coin'' \<and> length b = 1)"
ltl_Until_Release_con_2: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> U\<^sub>c \<psi> = (\<not> \<xi> \<Turnstile>\<^sub>c not\<^sub>c \<phi> R\<^sub>c not\<^sub>c \<psi>)"
representation_zero: fixes basis :: "'d set" shows "representation basis (0::'d) = (\<lambda>b. 0)"
enumAppend_length2: fixes ls :: "nat list"   and n :: "nat"   and nmax :: "nat"   and lss :: "nat list list"   and k :: "nat"   and K :: "nat" assumes "ls \<in> set ((enumAppend nmax ^^ n) lss)"   and "\<And>l. l \<in> set lss \<Longrightarrow> |l| = k"   and "K = k + n" shows "|ls| = K"
bind_runit: fixes x :: "'m option" shows "Option.bind x Some = x"
comp_reduce: fixes f :: "'a1"   and g :: "'a1"   and k :: "'a1"   and h :: "'a1" assumes "f \<cdot>\<^sub>1 g = k"   and "C1.seq f g"   and "C1.seq g h" shows "f \<cdot>\<^sub>1 g \<cdot>\<^sub>1 h = k \<cdot>\<^sub>1 h"
fst_sum: fixes f :: "'f \<Rightarrow> 'd \<times> 'e"   and A :: "'f set" shows "fst (sum f A) = (\<Sum>x\<in>A. fst (f x))"
filter_min_aux_minimal: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and y :: "'a" assumes "transp rel"   and "x \<in> set (local.filter_min_aux xs ys)"   and "y \<in> set (local.filter_min_aux xs ys)"   and "rel x y"   and "\<And>a b. \<lbrakk>a \<in> set xs \<union> set ys; b \<in> set ys; rel a b\<rbrakk> \<Longrightarrow> a = b" shows "x = y"
lookup_smult: fixes "is" :: "nat list"   and A :: "'a tensor"   and \<alpha> :: "'a" assumes "is \<lhd> dims A" shows "lookup (\<alpha> \<cdot> A) is = \<alpha> * lookup A is"
unit_simps_5: fixes a :: "'a" assumes "B.obj a" shows "cod (UP.unit a) = UP a"
sq_mtx_vec_mult_eq: fixes m :: "'a sq_mtx"   and x :: "(real, 'a) vec" shows "m *\<^sub>V x = (\<chi>i. \<Sum>j\<in>UNIV. m $$ i $ j * x $ j)"
advantage_def: fixes \<G> :: "'a cyclic_group"   and \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "Diffie_Hellman.ddh.advantage \<G> \<A> = \<bar>spmf (Diffie_Hellman.ddh.ddh_0 \<G> \<A>) True - spmf (Diffie_Hellman.ddh.ddh_1 \<G> \<A>) True\<bar>"
add_ac_2: fixes a :: "'b"   and b :: "'b" shows "a + b = b + a"
negGorbit_list_Cons0: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'d \<Rightarrow> 'c"   and as :: "'d list"   and gs :: "'a list" assumes "FGModule G smult V"   and "T ` set as \<subseteq> V" shows "aezfun_scalar_mult.negGorbit_list smult ((0::'a) # gs) T as = map T as # aezfun_scalar_mult.negGorbit_list smult gs T as"
has_codomain_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "(partial_magma.codomains (product_category.comp C1 C2) f \<noteq> {}) = (partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f))"
simps_6: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "step ((Pr n f g, xs, ls) # fs, rv) = (if ls = [] then if rv \<up> then ((f, tl xs, []) # (Pr n f g, xs, ls) # fs, None) else ((Pr n f g, xs, [the rv]) # fs, None) else if length ls = Suc (hd xs) then (fs, Some (hd ls)) else if rv \<up> then ((g, (length ls - 1) # hd ls # tl xs, []) # (Pr n f g, xs, ls) # fs, None) else ((Pr n f g, xs, the rv # ls) # fs, None))"
majorities_intersect: fixes Q :: "process set"   and Q' :: "process set" assumes "N < card Q + card Q'" shows "Q \<inter> Q' \<noteq> {}"
external_invariant_preservation: fixes Sa :: "('b, 'a) spec"   and Q :: "'a set"   and pi :: "'c \<Rightarrow> 'a"   and Sc :: "('d, 'c) spec" assumes "oreach Sa \<subseteq> Q"   and "implements pi Sa Sc" shows "pi ` oreach Sc \<subseteq> Q"
scaleC_scaleC: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "complex"   and b :: "complex"   and x :: "'c" assumes "class.complex_vector scaleR scaleC plus zero minus uminus" shows "scaleC a (scaleC b x) = scaleC (a * b) x"
wf_tuple_Suc: fixes m :: "nat"   and A :: "nat set"   and a :: "'b option list" shows "wf_tuple (Suc m) A a = (a \<noteq> [] \<and> wf_tuple m ((\<lambda>x. x - 1) ` (A - {0})) (tl a) \<and> (0 \<in> A) = (hd a \<noteq> None))"
greatest_LowerI: fixes A :: "'c set"   and L :: "('c, 'd) gorder_scheme"   and i :: "'c" assumes "\<And>x. x \<in> A \<Longrightarrow> i \<sqsubseteq>\<^bsub>L\<^esub> x"   and "\<And>y. y \<in> Lower L A \<Longrightarrow> y \<sqsubseteq>\<^bsub>L\<^esub> i"   and "A \<subseteq> carrier L"   and "i \<in> carrier L" shows "is_glb L i A"
hom_nat_pow: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and x :: "'a"   and n :: "nat" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h (x [^]\<^bsub>G\<^esub> n) = h x [^]\<^bsub>H\<^esub> n"
maddux_3_21_pp: fixes 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 bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.stone_algebra inf less_eq less sup bot top uminus" shows "sup (uminus (uminus x)) (inf (uminus x) y) = sup (uminus (uminus x)) y"
rep_eq: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "'c fset"   and xb :: "'d fset" shows "rel_fset x xa xb = rel_set x (fset xa) (fset xb)"
a_2: shows "\<forall>b\<in>S. \<not> (a \<noteq> b \<and> (a, b) \<in> r)"
WT_unique: fixes is_lub :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'b expr"   and T :: "ty"   and T' :: "ty" assumes "\<And>T1 T2 T3 T4. \<lbrakk>is_lub T1 T2 T3; is_lub T1 T2 T4\<rbrakk> \<Longrightarrow> T3 = T4"   and "is_lub,P,E \<turnstile> e :: T"   and "is_lub,P,E \<turnstile> e :: T'" shows "T = T'"
div_diff_self2: fixes n :: "nat"   and m :: "nat" shows "(n - m) div m = n div m - Suc 0"
induct: fixes P :: "'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b up\<^sub>d \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> bool"   and a0 :: "'b tree234"   and a1 :: "'b"   and a2 :: "'b up\<^sub>d"   and a3 :: "'b"   and a4 :: "'b tree234" assumes "\<And>t1 a t2 b t3. P t1 a (T\<^sub>d t2) b t3"   and "\<And>t1 a t2 b t3 c t4. P t1 a (Up\<^sub>d t2) b \<langle>t3, c, t4\<rangle>"   and "\<And>t1 a t2 b t3 c t4 d t5. P t1 a (Up\<^sub>d t2) b \<langle>t3, c, t4, d, t5\<rangle>"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6. P t1 a (Up\<^sub>d t2) b \<langle>t3, c, t4, d, t5, e, t6\<rangle>"   and "\<And>a b v d. P a b (Up\<^sub>d v) d \<langle>\<rangle>" shows "P a0 a1 a2 a3 a4"
Compr_image_eq: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and P :: "'d \<Rightarrow> bool" shows "{x \<in> f ` A. P x} = f ` {x \<in> A. P (f x)}"
must_sync_def2: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and t :: "'b"   and x :: "'a"   and m :: "'c" shows "multithreaded_base.must_sync final r t x m = (\<exists>ta x' m' s. r t (x, m) ta (x', m') \<and> final_thread.actions_ok final s t ta)"
Retr_defs_2: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set" assumes "PL_Indis cval indis" shows "PL_Indis.ZOretr aval tval cval indis theta \<equiv> {(c, d). PL_Indis.matchC_LC aval tval cval indis theta c d}"
elims_2: fixes x :: "'b tree"   and xa :: "'b list" assumes "braun_list x xa" obtains xs :: "'b list" where "x = \<langle>\<rangle>" and "xa = xs__" and "xs__ = []"   | l :: "'b tree" and xb :: "'b" and r :: "'b tree" and xs :: "'b list" where "x = \<langle>l, xb, r\<rangle>" and "xa = xs" and "xs \<noteq> [] \<and> xb = hd xs \<and> braun_list l (take_nths 1 1 xs) \<and> braun_list r (take_nths 2 1 xs)"
Abs: fixes env :: "nat \<Rightarrow> LambdaType.type"   and T :: "LambdaType.type"   and t :: "dB"   and U :: "LambdaType.type" assumes "env\<langle>0:T\<rangle> \<turnstile> t : U" shows "env \<turnstile> Abs t : T \<Rightarrow> U"
nodemap_induct: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and ii :: "nat"   and \<zeta> :: "'e"   and R :: "nat set"   and T :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> nat set \<Rightarrow> bool" assumes "(\<sigma>, NodeS ii \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U"   and "\<And>\<sigma> \<zeta>. (\<sigma>, NodeS ii \<zeta> R\<^sub>i) \<in> init (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) \<Longrightarrow> P \<sigma> \<zeta> R\<^sub>i"   and "\<And>\<sigma> \<zeta> R \<sigma>' a. \<lbrakk>(\<sigma>, NodeS ii \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U; U \<sigma> \<sigma>'; P \<sigma> \<zeta> R\<rbrakk> \<Longrightarrow> P \<sigma>' \<zeta> R"   and "\<And>\<sigma> \<zeta> R \<sigma>' \<zeta>' R' a. \<lbrakk>(\<sigma>, NodeS ii \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U; ((\<sigma>, NodeS ii \<zeta> R), a, \<sigma>', NodeS ii \<zeta>' R') \<in> automaton.trans (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o); S \<sigma> \<sigma>' a; P \<sigma> \<zeta> R\<rbrakk> \<Longrightarrow> P \<sigma>' \<zeta>' R'" shows "P \<sigma> \<zeta> R"
subnet_reachable_2: fixes s1 :: "'c net_state"   and s2 :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree" assumes "SubnetS s1 s2 \<in> reachable (pnet np (p1 \<parallel> p2)) TT" shows "s2 \<in> reachable (pnet np p2) TT"
is_sig_GB_uptD1: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t" assumes "is_sig_GB_upt d G u" shows "G \<subseteq> dgrad_sig_set' (length fs) d"
card_eq_UNIV_imp_eq_UNIV: fixes A :: "'a set" assumes "finite UNIV"   and "card A = card UNIV" shows "A = UNIV"
nxtAct_def: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub> \<equiv> THE n'. n \<le> n' \<and> \<parallel>c\<parallel>\<^bsub>t n'\<^esub> \<and> \<not> (\<exists>k\<ge>n. latestAct_cond c t n' k)"
list_induct_CCons: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P xs"
run_shift: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and r :: "(nat \<times> 'd) list"   and p :: "nat \<times> 'd"   and s :: "(nat \<times> 'd) stream" assumes "gpath A w V r p"   and "grun A w V s (gtarget r p)" shows "grun A w V (r @- s) p"
length_lang_pow_lb: fixes X :: "'a list set"   and k :: "nat"   and x :: "'a list"   and n :: "nat" assumes "\<forall>x\<in>X. k \<le> |x|"   and "x \<in> X\<^bsup>n\<^esup>" shows "k \<cdot> n \<le> |x|"
nasc_seqn1: fixes a :: "nat"   and A :: "nat set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)" shows "Suc (nasc_seq A a n) \<le> nasc_seq A a (Suc n)"
DERIV_pow: fixes n :: "nat"   and x :: "real"   and s :: "real set" shows "((\<lambda>x. x ^ n) has_real_derivative real n * x ^ (n - Suc 0)) (at x within s)"
sub_q65_array: fixes i :: "nat" assumes "i \<in> {..<65}" shows "q65_array !! i = (i \<in> q65)"
viewR_correct_2: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t"   and "t \<noteq> Empty" shows "\<exists>a s. FingerTreeStruc.viewR t = Some (a, s) \<and> FingerTreeStruc.ft_invar s \<and> FingerTreeStruc.toList t = FingerTreeStruc.toList s @ [a]"
j_le_k: shows "mod_type_class.to_nat j < k"
is_span: shows "C.span HfHgh.dom.leg0 HfHgh.dom.leg1"
overlap_intro: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "P z x"   and "P z y" shows "O x y"