hcju/isabelleps huggingface dataset:
subsumes_def: fixes t2 :: "transition"   and r :: "nat \<Rightarrow>f value option"   and t1 :: "transition" shows "subsumes t2 r t1 = (Label t1 = Label t2 \<and> Arity t1 = Arity t2 \<and> (\<forall>i. can_take_transition t1 i r \<longrightarrow> can_take_transition t2 i r) \<and> (\<forall>i. can_take_transition t1 i r \<longrightarrow> evaluate_outputs t1 i r = evaluate_outputs t2 i r) \<and> (\<forall>p1 p2 i. posterior_separate (Arity t1) (Guards t1) (Updates t2) i r = Some p2 \<longrightarrow> posterior_separate (Arity t1) (Guards t1) (Updates t1) i r = Some p1 \<longrightarrow> (\<forall>P r'. p1 $ r' = None \<or> (P (p2 $ r') \<longrightarrow> P (p1 $ r')))))"
elims_3: fixes x :: "heap \<times> nat set" assumes "\<not> in_range x" obtains h :: "heap" and as :: "nat set" where "x = (h, as)" and "\<not> (\<forall>a\<in>as. a < lim h)"
eigenvector_char_matrix: fixes A :: "'b mat"   and n :: "nat"   and v :: "'b Matrix.vec"   and e :: "'b" assumes "A \<in> carrier_mat n n" shows "eigenvector A v e = (v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> char_matrix A e *\<^sub>v v = 0\<^sub>v n)"
simps_17: fixes x21 :: "'c"   and x22 :: "'c"   and x23 :: "'b" shows "set2_arr (MkArr x21 x22 x23) = {x23}"
wt2_nneg: fixes I :: "nat set"   and i :: "nat" assumes "I \<in> P"   and "i \<in> F I" shows "0 \<le> wt (fst cf2) (snd cf2) i"
isPath_edgeset: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and e :: "nat \<times> nat" assumes "cf.isPath u p v"   and "e \<in> set p" shows "e \<in> cf.E"
distinguishing_formula_eqvt: fixes P :: "'state"   and Q :: "'state"   and p :: "perm" assumes "\<not> P =\<cdot> Q" shows "p \<bullet> distinguishing_formula P Q = distinguishing_formula (p \<bullet> P) (p \<bullet> Q)"
iso_is_section: fixes f :: "'a" assumes "B.iso f" shows "B.section f"
d_IN_Edge: fixes f :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" assumes "flow \<Delta>'' f" shows "d_IN f (Edge x y) = f (Vertex x, Edge x y)"
S_rec: shows "S = (\<lambda>s. Mprefix (Tr s) (\<lambda>e. S (Up s e)))"
satA_snvA_transE: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and at :: "('a, 'b) atm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "ModelIkMcalc wtFsym wtPsym arOf resOf parOf \<Phi> infTp intT intF intP"   and "Signature.wtA wtFsym wtPsym arOf resOf parOf at"   and "Struct.wtE (ModelIkMcalc.I_intT infTp intT) \<xi>"   and "\<And>\<sigma>. infTp \<sigma> \<or> (\<forall>x\<in>ProblemIk.nvA at. Signature.tpOfV x \<noteq> \<sigma>)" shows "Struct.satA intF intP (ModelIkMcalc.transE intT \<xi>) at = Struct.satA (ModelIkMcalc.I_intF arOf intT intF) (ModelIkMcalc.I_intP parOf intT intP) \<xi> at"
hd_remdups_adj: fixes xs :: "'g list" shows "hd (remdups_adj xs) = hd xs"
Acc_property: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>" shows "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi> (G \<psi>)) w = accepting_pair\<^sub>R (semi_mojmir_def.step \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>)) (semi_mojmir_def.initial (q\<^sub>0\<^sub>M \<psi>)) (mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q} \<union> mojmir_to_rabin_def.merge\<^sub>R \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q} (the (\<pi> (G \<psi>))), mojmir_to_rabin_def.succeed\<^sub>R \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q} (the (\<pi> (G \<psi>)))) w"
cond_assoc: fixes c :: "'b upred"   and b :: "'b upred"   and P :: "('a, 'b) uexpr"   and Q :: "('a, 'b) uexpr"   and R :: "('a, 'b) uexpr" shows "P \<triangleleft> b \<triangleright> Q \<triangleleft> c \<triangleright> R = P \<triangleleft> b \<and> c \<triangleright> (Q \<triangleleft> c \<triangleright> R)"
col_transitivity_2: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p" assumes "P \<noteq> Q"   and "Col P Q A"   and "Col P Q B" shows "Col Q A B"
real_of_nat_indicator: fixes A :: "'a set"   and x :: "'a" shows "real (indicator A x) = indicat_real A x"
Pyz: shows "distributed M (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x)) (\<lambda>x. ennreal (Pyz x))"
rel_arg_cong: fixes x :: "'c"   and y :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "(x, y) \<in> Id" shows "(f x, f y) \<in> Id"
principal_g3up: fixes r :: "(C, BD) sequent list \<times> (C, BD) sequent"   and A :: "(C, BD) form"   and R :: "((C, BD) sequent list \<times> (C, BD) sequent) set" assumes "r \<in> g3up"   and "rightPrincipal r (\<box> A) R" shows "( \<Empt> \<Rightarrow>* \<LM> A  \<RM>) \<in> set (fst r)"
walk_edges_subset: fixes xs :: "'a list"   and ys :: "'a list" assumes "edges_of_walk xs \<subseteq> edges_of_walk ys"   and "tl xs \<noteq> []" shows "set xs \<subseteq> set ys"
Cond_True_or_False: fixes b :: "expr"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and s :: "char list \<Rightarrow> val option"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>if (b) c\<^sub>1 else c\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>" shows "\<lbrakk>b\<rbrakk>s = Some true \<or> \<lbrakk>b\<rbrakk>s = Some false"
a: shows "[\<^bold>\<forall>x. \<phi> x \<^bold>\<rightarrow> \<^bold>\<box>\<phi> x in v]"
lcl_conv: fixes h :: "'f"   and l :: "'e" shows "lcl (h, l) = l"
\<mu>: shows "\<guillemotleft>\<mu> : g' \<Rightarrow>\<^sub>D F g\<guillemotright> \<and> D.iso \<mu>"
group_incl_imp_subgroup: fixes H :: "'a set" assumes "H \<subseteq> carrier G"   and "Group.group (G\<lparr>carrier := H\<rparr>)" shows "subgroup H G"
fst_netgmap_subnet: fixes sr :: "'f \<Rightarrow> 'd \<times> 'e"   and s2 :: "'f net_state"   and s1 :: "'f net_state" shows "fst (case netgmap sr s1 of (\<sigma>\<^sub>1, ss) \<Rightarrow> case netgmap sr s2 of (\<sigma>\<^sub>2, tt) \<Rightarrow> (\<sigma>\<^sub>1 ++ \<sigma>\<^sub>2, SubnetS ss tt)) = fst (netgmap sr s1) ++ fst (netgmap sr s2)"
frameResChainPres: fixes F :: "'b frame"   and G :: "'b frame"   and xvec :: "name list" assumes "F \<simeq>\<^sub>F G" shows "\<lparr>\<nu>*xvec\<rparr>F \<simeq>\<^sub>F \<lparr>\<nu>*xvec\<rparr>G"
inv_unique_comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "f \<circ> g = id"   and "g \<circ> f = id" shows "inv f = g"
Cauchy_Re: fixes X :: "nat \<Rightarrow> complex" assumes "Cauchy X" shows "Cauchy (\<lambda>n. Re (X n))"
1: shows "\<p>\<^sub>1[g.leg0, g.dsrc] \<cdot> chine_hcomp g \<epsilon> \<cdot> gfg.chine_assoc \<cdot> chine_hcomp \<eta> g \<cdot> \<langle>g.leg1 \<lbrakk>g.dtrg, g.leg1\<rbrakk> g.chine\<rangle> = gfg.Prj\<^sub>1 \<cdot> gfg.chine_assoc \<cdot> chine_hcomp \<eta> g \<cdot> \<langle>g.leg1 \<lbrakk>g.dtrg, g.leg1\<rbrakk> g.chine\<rangle>"
correctCompositionDiffLevelsA72: shows "correctCompositionDiffLevels sA72"
gen_ramanujan_sum_periodic: fixes f :: "nat \<Rightarrow> complex"   and g :: "nat \<Rightarrow> complex"   and k :: "nat" shows "periodic_arithmetic (s f g k) k"
connectedin_connected_component_of: fixes X :: "'b topology"   and x :: "'b" shows "connectedin X (connected_component_of_set X x)"
Cons_1: assumes "\<And>x. x \<in> set xs - set ys \<Longrightarrow> \<exists>y. y \<in> set ys \<and> x \<le> y" shows "max_list xs \<le> max_list ys"
eqcard: fixes xa :: "'a set" assumes "xa \<subseteq> b1"   and "xa \<subseteq> b2" shows "card (\<pi> ` xa) = card xa"
hyps_3: shows "\<Psi> \<rhd> P' \<longmapsto> \<tau> \<prec> P''"
Collect_neg_eq: fixes P :: "'c \<Rightarrow> bool" shows "{x. \<not> P x} = - {x. P x}"
\<tau>move2AAssRed: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and a :: "'addr expr1"   and i :: "'addr expr1"   and e :: "'addr expr1" shows "\<tau>move2 P h stk (a\<lfloor>i\<rceil> := e) (Suc (length (compE2 a) + length (compE2 i) + length (compE2 e))) None"
homotopic_with_mono: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and Q :: "('b \<Rightarrow> 'c) \<Rightarrow> bool" assumes "homotopic_with P X Y f g"   and "\<And>h. \<lbrakk>continuous_map X Y h; P h\<rbrakk> \<Longrightarrow> Q h" shows "homotopic_with Q X Y f g"
augment_edge_zero: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and e :: "nat \<times> nat" shows "augment_edge f e (0::'capacity) = f"
poly_poly_as_poly2: fixes p :: "'a poly poly"   and x :: "'a"   and q :: "'a poly" shows "poly2 p x (poly q x) = poly (poly p q) x"
wf: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)"
proj2_rep_independent: fixes p :: "proj2"   and q :: "proj2" assumes "p \<noteq> q" shows "independent {proj2_rep p, proj2_rep q}"
rep_eq: fixes x :: "'a ltln\<^sub>L"   and xa :: "'a ltln\<^sub>L" shows "local.ltln\<^sub>L_eq_test x xa = rep_ltln\<^sub>L x \<sim>\<^sub>L rep_ltln\<^sub>L xa"
left_total_OO: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "left_total R"   and "left_total S" shows "left_total (R OO S)"
min_product_set_insert: fixes X :: "'b fset set set"   and x :: "'b fset set" assumes "finite X" shows "\<Otimes>\<^sub>m (insert x X) = x \<otimes>\<^sub>m \<Otimes>\<^sub>m X"
it_paths_strict_prefix: fixes "is" :: "nat list"   and t :: "'b rose_tree"   and is' :: "nat list" assumes "is \<in> it_paths t"   and "strict_prefix is' is" shows "is' \<in> it_paths t"
sups_def: fixes x :: "'a"   and y :: "'a" shows "Sup_lattice.sups Inf x y = \<Sqinter> {x, y}"
constructive_security_aux_obsf: fixes \<eta> :: "nat" shows "constructive_security_aux_obsf (real_resource \<eta>) (ideal_resource \<eta>) (sim \<eta>) (\<I>_real \<eta>) (\<I>_ideal \<eta>) (\<I>_common \<eta>) 0"
Suc_2: fixes x :: "'a" assumes "x \<in> C"   and "enat (Suc n) \<le> esize x" shows "thesis"
cindependent_cfinite_dim_finite: fixes S :: "'d set" assumes "cindependent S" shows "finite S"
Hrew': shows "drop (Suc n) xs ! (p - n) = xs ! Suc p"
gtrace_infinite: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and r :: "(nat \<times> 'b) stream"   and v :: "nat \<times> 'b" assumes "grun A w V r v" shows "infinite (sset (gtrace r v))"
unit_disc_fix_discI: fixes M :: "moebius"   and u :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc" shows "moebius_pt M u \<in> unit_disc"
span_finite: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set" assumes "module scale"   and "finite S" shows "module.span scale S = range (\<lambda>u. \<Sum>v\<in>S. scale (u v) v)"
disc_4: fixes x2 :: "'l" shows "is_ok (Ok x2)"
assms: fixes n :: "nat" shows "n < length cl \<and> \<not> finished (cl ! n) \<and> n < length dl \<and> \<not> finished (dl ! n)"
one_extra_update_directly_subsumes: fixes t1 :: "transition"   and t2 :: "transition"   and r :: "nat"   and u :: "vname aexp"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat" assumes "Label t1 = Label t2"   and "Arity t1 = Arity t2"   and "set (Guards t1) \<subseteq> set (Guards t2)"   and "Outputs t1 = Outputs t2"   and "Updates t1 = (r, u) # Updates t2"   and "not_updated r t2"   and "initially_undefined_context_check e2 r s2" shows "directly_subsumes e1 e2 s1 s2 t1 t2"
Let_def: fixes s :: "'c"   and f :: "'c \<Rightarrow> 'b" shows "Let (s::'c) (f::'c \<Rightarrow> 'b) \<equiv> f s"
abelian_subgroup_def: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" shows "abelian_subgroup H G \<equiv> additive_subgroup H G \<and> abelian_group G \<and> abelian_subgroup_axioms H G"
all_wf: fixes g :: "'g" shows "minimal_element (Prec_F_g g) UNIV"
BfunE: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "Bfun f F" obtains B :: "real" where "0 < B" and "\<forall>\<^sub>F x in F. norm (f x) \<le> B"
notnot_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(not_PL (not_PL p) \<in> Cn A) = (p \<in> Cn A)"
PiM_def: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" shows "Pi\<^sub>M I M = extend_measure (\<Pi>\<^sub>E i\<in>I. space (M i)) {(J, X). (J \<noteq> {} \<or> I = {}) \<and> finite J \<and> J \<subseteq> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))} (\<lambda>(J, X). prod_emb I M J (Pi\<^sub>E J X)) (\<lambda>(J, X). \<Prod>j\<in>J \<union> {i \<in> I. emeasure (M i) (space (M i)) \<noteq> 1}. if j \<in> J then emeasure (M j) (X j) else emeasure (M j) (space (M j)))"
less_nat_zero_code: fixes n :: "nat" shows "(n \<sqsubset> 0) = False"
gfinprod_def: fixes G :: "('b, 'c) monoid_scheme"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set" shows "gfinprod G f A = (if finite {x \<in> A. f x \<noteq> \<one>\<^bsub>G\<^esub>} then finprod G f {x \<in> A. f x \<noteq> \<one>\<^bsub>G\<^esub>} else \<one>\<^bsub>G\<^esub>)"
less_list_length: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs < ys" shows "length xs < length ys"
rbt_map_entry_color_of: fixes k :: "'a"   and f :: "'c \<Rightarrow> 'c"   and t :: "('a, 'c) rbt" shows "color_of (rbt_map_entry k f t) = color_of t"
prems_3: shows "\<forall>c\<in>set (c' # cs). v c \<le> n"
list_all2_elimL: fixes xs :: "'a list"   and x :: "'a"   and xs' :: "'a list"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" assumes "xs = x # xs'"   and "list_all2 R xs ys" obtains y :: "'b" and ys' :: "'b list" where "ys = y # ys'" and "R x y" and "list_all2 R xs' ys'"
order_eq_refl: fixes x :: "'b"   and y :: "'b" assumes "x = y" shows "x \<le> y"
length_isort: fixes xs :: "'a list" shows "length (isort xs) = length xs"
mergesort_remdups_correct: fixes l :: "'a list" shows "distinct (mergesort_remdups l) \<and> sorted (mergesort_remdups l) \<and> set (mergesort_remdups l) = set l"
induct: fixes P :: "f \<Rightarrow> bool"   and f :: "f" assumes "P val"   and "P Aodv_Basic.inv" shows "P f"
connected_awalkE: fixes u :: "'a"   and v :: "'a" assumes "Digraph_Component.connected G"   and "u \<in> verts G"   and "v \<in> verts G" obtains p :: "('a \<times> 'a) list" where "pre_digraph.awalk (with_proj (mk_symmetric G)) u p v"
filterlim_tendsto_pos_mult_at_bot_iff: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "0 < c" shows "(LIM x F. f x * g x :> at_bot) = filterlim g at_bot F"
induct: fixes x1 :: "ExVar \<Rightarrow> Val option"   and x2 :: "Ex"   and x3 :: "Val"   and P :: "(ExVar \<Rightarrow> Val option) \<Rightarrow> Ex \<Rightarrow> Val \<Rightarrow> bool" assumes "x1 \<turnstile> x2 ---> x3"   and "\<And>ve c. P ve (e_const c) (v_const c)"   and "\<And>ev ve. ev \<in> dom ve \<Longrightarrow> P ve (e_var ev) (the (ve ev))"   and "\<And>ve ev e. P ve fn ev => e (v_clos \<langle>ev , e , ve\<rangle>)"   and "\<And>cl ev1 e ve ev2. cl = \<langle>ev1 , e , ve(ev2 \<mapsto> v_clos cl)\<rangle> \<Longrightarrow> P ve fix ev2  ev1  = e (v_clos cl)"   and "\<And>ve e1 c1 e2 c2. \<lbrakk>ve \<turnstile> e1 ---> v_const c1; P ve e1 (v_const c1); ve \<turnstile> e2 ---> v_const c2; P ve e2 (v_const c2)\<rbrakk> \<Longrightarrow> P ve e1 @@ e2 (v_const (c_app c1 c2))"   and "\<And>ve e1 xm em vem e2 v2 v. \<lbrakk>ve \<turnstile> e1 ---> v_clos \<langle>xm , em , vem\<rangle>; P ve e1 (v_clos \<langle>xm , em , vem\<rangle>); ve \<turnstile> e2 ---> v2; P ve e2 v2; vem(xm \<mapsto> v2) \<turnstile> em ---> v; P (vem(xm \<mapsto> v2)) em v\<rbrakk> \<Longrightarrow> P ve e1 @@ e2 v" shows "P x1 x2 x3"
below_antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
carrier_def: fixes charts :: "('b, 'c) chart set" shows "manifold.carrier charts = \<Union> (domain ` charts)"
cong: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" shows "comm_monoid_add.sum plus zero \<equiv> comm_monoid_add.sum plus zero"
prems: shows "\<forall>i. var \<notin> vars (f i)"
IH: fixes v1 :: "'a \<times> nat" assumes "SubRel.subpath g v1 es v2 subs"   and "e \<in> set es" shows "\<exists>es'. es = es' @ [e] \<and> e \<notin> set es'"
is_zero_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" shows "is_zero p = (p = 0)"
weak_lens_def: fixes x :: "'a \<Longrightarrow> 'b" shows "weak_lens x \<equiv> \<forall>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
of_real_harm: fixes n :: "nat" shows "of_real (harm n) = harm n"
PSQ: shows "(P, S, Q) = diagonal_to_Smith_PQ_JNF A bezout"
AnalBinds_delete_bot: fixes ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and x :: "Vars.var"   and exp :: "exp \<Rightarrow> 'a \<rightarrow> 'b"   and \<Gamma> :: "(Vars.var \<times> exp) list" assumes "ae x = \<bottom>" shows "ExpAnalysis.AnalBinds exp (delete x \<Gamma>)\<cdot>ae = ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and f :: "('fsym, 'tp) efsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "int \<xi> (Fn f Tl) = eintF f (map (int \<xi>) Tl)"
induct: fixes x :: "'c llist"   and P :: "'c llist \<Rightarrow> bool" assumes "lfinite x"   and "P LNil"   and "\<And>xs x. \<lbrakk>lfinite xs; P xs\<rbrakk> \<Longrightarrow> P (LCons x xs)" shows "P x"
vproduct_vrange: fixes f :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct I A" shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda I A `\<^sub>\<circ> I)"
"1_2": shows "\<forall>(t, t')\<in>set (map snd (dbproj i D)). (fv t \<union> fv t') \<inter> set X = {}"
not_neutral_contains_not_neutral: fixes g :: "'d \<Rightarrow> 'a"   and A :: "'d set" assumes "F.F g A \<noteq> \<^bold>1" obtains a :: "'d" where "a \<in> A" and "g a \<noteq> \<^bold>1"
assms_2: shows "\<forall>m. P m \<longrightarrow> (\<forall>m'\<in>set (f m). P m')"
resultant_0_2: fixes p :: "'a poly" assumes "0 < degree p" shows "resultant p 0 = (0::'a)"
proper_Par_properL: fixes cl :: "('d, 'e, 'f) cmd list" shows "proper (Par cl) = properL cl"
sams_lea_lta456_suma2__lta123: fixes D' :: "'p"   and E' :: "'p"   and F' :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p"   and G' :: "'p"   and H' :: "'p"   and I' :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "D' E' F' LtA D E F"   and "G H I LeA G' H' I'"   and "SAMS A B C D E F"   and "A B C D E F SumA G H I"   and "A' B' C' D' E' F' SumA G' H' I'" shows "A B C LtA A' B' C'"
sig_poly_rel_vec_of: fixes sp :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "spp_inv sp" shows "spp_rel sp (vec_of sp)"
one_le_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one (numeral.numeral one plus n)"
ccHeap_Extra_Edges: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e' :: "exp"   and e :: "exp"   and a :: "Arity"   and a' :: "Arity" assumes "map_of \<Delta> x = Some e'"   and "(Aheap \<Delta> e\<cdot>a) x = up\<cdot>a'" shows "fv e' G\<times> (ccNeighbors x (ccHeap \<Delta> e\<cdot>a) - {x} \<inter> thunks \<Delta>) \<sqsubseteq> ccHeap \<Delta> e\<cdot>a"
total_on_diff_Id: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "total_on A (r - Id) = total_on A r"
unsigned_and_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and v :: "'c word"   and w :: "'c word" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "semiring_1.unsigned one plus zero (v AND w) = and (semiring_1.unsigned one plus zero v) (semiring_1.unsigned one plus zero w)"
fdom_res_swap: fixes A :: "'a set"   and f :: "('a, 'b) ffun"   and B :: "'b set" shows "A \<lhd>\<^sub>f f \<rhd>\<^sub>f B = A \<lhd>\<^sub>f (f \<rhd>\<^sub>f B)"
remdups'_map: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and xs :: "'e list" shows "remdups' f (map g xs) = map g (remdups' (f \<circ> g) xs)"
EphK_Exp: fixes X :: "ephkey"   and X' :: "msg"   and Y' :: "msg" shows "EphK X \<noteq> Exp X' Y'"
T4_def: shows "T4 \<equiv> unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_updates T))"
ladder_n_pairwise_bound: fixes u :: "nat"   and v :: "nat" assumes "u < v"   and "v < length L'" shows "ladder_n L' u < ladder_n L' v"
hyps: shows "Stirling (Suc (Suc n)) (Suc (Suc 0)) = 2 ^ Suc n - 1"
vertical_composite_axioms: shows "vertical_composite A B H G F \<tau>' \<sigma>'"
sets_Collect_neg: fixes P :: "'a \<Rightarrow> bool" assumes "{x \<in> \<Omega>. P x} \<in> M" shows "{x \<in> \<Omega>. \<not> P x} \<in> M"
inv_del: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "invh t"   and "invc t" shows "invh (RBT_Set.del x t) \<and> (color t = Red \<longrightarrow> bheight (RBT_Set.del x t) = bheight t \<and> invc (RBT_Set.del x t)) \<and> (color t = Black \<longrightarrow> bheight (RBT_Set.del x t) = bheight t - 1 \<and> invc2 (RBT_Set.del x t))"
rep_eq: fixes x :: "('c, 'd) mapping"   and xa :: "('c, 'd) mapping" shows "mapping.rep (mapping_add x xa) = mapping.rep x ++ mapping.rep xa"
prefix_match_dtor_def: fixes m :: "'b prefix_match" shows "prefix_match_dtor m \<equiv> case m of PrefixMatch p l \<Rightarrow> (p, l)"
induct: fixes P :: "char list \<Rightarrow> ('a \<Rightarrow> char list) \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "char list"   and a1 :: "'a \<Rightarrow> char list"   and a2 :: "'a list" assumes "\<And>sep toStr. P sep toStr []"   and "\<And>sep toStr l. P sep toStr [l]"   and "\<And>sep toStr l v va. P sep toStr (v # va) \<Longrightarrow> P sep toStr (l # v # va)" shows "P a0 a1 a2"
in_keys_monom_mult_ge: fixes v :: "'t"   and c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (monom_mult c t p)" shows "t \<oplus> tt p \<preceq>\<^sub>t v"
conga__lea456123: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C CongA D E F" shows "D E F LeA A B C"
convex_Int: fixes s :: "'a set"   and t :: "'a set" assumes "convex s"   and "convex t" shows "convex (s \<inter> t)"
isPath_rtc: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "cf.isPath u p v" shows "(u, v) \<in> cf.E\<^sup>*"
prems_1: fixes env' :: "(name, 'a) fmap" assumes "match (v\<^sub>1 $ v\<^sub>2) t = Some env'"   and "P_env env' (env\<^sub>1 ++\<^sub>f env\<^sub>2)" shows "thesis"
set_subset_Cons: fixes xs :: "'x list"   and x :: "'x" shows "set xs \<subseteq> set (x # xs)"