hcju/isabelleps huggingface dataset:
elims_2: fixes x :: "('b \<times> color) tree" assumes "invh x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> color) tree" and xa :: "'b" and c :: "color" and r :: "('b \<times> color) tree" where "x = \<langle>l, (xa, c), r\<rangle>" and "bheight l = bheight r \<and> invh l \<and> invh r"
l6: shows "prod Prod local.S2 = \<Prod>local.S1"
bij_betw_iff_card: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "(\<exists>f. bij_betw f A B) = (card A = card B)"
xy: shows "poincare_distance x y = poincare_distance x' y'"
m: shows "max_Var_floatarith fa \<le> DIM('a) + length params"
seqllD: fixes i :: "'e"   and P :: "('f \<times> 'g) \<times> 'h \<times> 'f \<times> 'g \<Rightarrow> bool"   and \<sigma> :: "'e \<Rightarrow> 'f"   and p :: "'g"   and a :: "'h"   and \<sigma>' :: "'e \<Rightarrow> 'f"   and p' :: "'g" assumes "seqll i P ((\<sigma>, p), a, \<sigma>', p')" shows "P ((\<sigma> i, p), a, \<sigma>' i, p')"
chambers_share_facet: fixes C :: "'a set"   and v :: "'a"   and z :: "'a set" assumes "folding_g.chamber C"   and "folding_g.chamber (insert v z)"   and "z \<lhd> C" shows "z \<lhd> insert v z"
changes_filter_empty_1: fixes xs :: "'b list" assumes "filter (\<lambda>x. x \<noteq> (0::'b)) xs = []" shows "changes xs = 0"
pslt: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b" assumes "CRR_hyps p M geom_proc u d init Mkt r risk_free_asset stk" shows "p < 1"
Cons_3: fixes i :: "nat" assumes "i < length (G # Gs)" shows "gs ! i \<in> carrier ((G # Gs) ! i)"
chine_simps_3: shows "C.cod src_f.chine = src_f.cod.apex"
ffb_Sup_dual: shows "Sup_dual fb\<^sub>\<F>"
induced_automorphism_eq_foldings_on_chambers_gf: fixes C :: "'a set" assumes "C \<in> g \<turnstile> folding_g.\<C>" shows "fun_eq_on \
 f C"
is_trail_split': fixes G :: "('a, 'b) graph"   and v :: "'a"   and p1 :: "('a \<times> 'b \<times> 'a) list"   and u :: "'a"   and w :: "'b"   and u' :: "'a"   and p2 :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.is_trail G v (p1 @ (u, w, u') # p2) v'" shows "valid_unMultigraph.is_trail G v p1 u \<and> (u, w, u') \<in> edges G \<and> valid_unMultigraph.is_trail G u' p2 v'"
skip_mem: fixes l :: "nat"   and n :: "nat"   and i :: "nat" assumes "l \<in> {i. i \<le> n}" shows "skip i l \<in> {i. i \<le> Suc n}"
diag_block_mat_last: fixes As :: "'b mat list"   and B :: "'b mat" shows "diag_block_mat (As @ [B]) = (let A = diag_block_mat As in four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B)"
is_red_addsI: fixes f :: "'t \<Rightarrow>\<^sub>0 'd"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'd" assumes "f \<in> F"   and "f \<noteq> 0"   and "v \<in> keys p"   and "lt f adds\<^sub>t v" shows "is_red F p"
Urysohn_both_ne: fixes U :: "'c set"   and S :: "'c set"   and T :: "'c set"   and a :: "'d"   and b :: "'d" assumes "closedin (top_of_set U) S"   and "closedin (top_of_set U) T"   and "S \<inter> T = {}"   and "S \<noteq> {}"   and "T \<noteq> {}"   and "a \<noteq> b" obtains f :: "'c \<Rightarrow> 'd" where "continuous_on U f" and "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b" and "\<And>x. x \<in> U \<Longrightarrow> (f x = a) = (x \<in> S)" and "\<And>x. x \<in> U \<Longrightarrow> (f x = b) = (x \<in> T)"
comp_char: fixes g :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" shows "g \<odot> f = (if Maps.seq g f then Maps.MkArr (Maps.Dom f) (Maps.Cod g) (Maps.Comp (Maps.Map g) (Maps.Map f)) else Maps.null)"
total: shows "\<Turnstile>\<^sub>t {P} S {Q}"
lfp_def: fixes f :: "'a \<Rightarrow> 'a"   and i :: "'i" shows "local.lfp f i = Glb i {a \<in> L i. f a \<le> a}"
MPair_used_D: fixes X :: "msg"   and Y :: "msg"   and H :: "event list" assumes "\<lbrace>X, Y\<rbrace> \<in> used H" shows "X \<in> used H \<and> Y \<in> used H"
sublist_altdef: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le>f ys = (\<exists>ys'. ys' \<le>p ys \<and> xs \<le>s ys')"
set_l_is: shows "set (map (map_of \<circ> effect_to_assignments) ops) = (map_of \<circ> effect_to_assignments) ` set ops"
intro: fixes x :: "'a \<Longrightarrow> 'b" assumes "weak_lens x"   and "mwb_lens_axioms x" shows "mwb_lens x"
assms_2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid left_forkee)]), left_forkee \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e))"
WT_converter_id: fixes \<I> :: "('m, 'n) \<I>" shows "\<I>, \<I> \<turnstile>\<^sub>C 1\<^sub>C \<surd>"
scaleR_left: fixes r :: "real"   and a :: "'e \<Rightarrow>\<^sub>L 'd"   and b :: "'e" shows "blinfun_apply (r *\<^sub>R a) b = r *\<^sub>R blinfun_apply a b"
bounded_plus_image: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "bounded (f ` s)"   and "bounded (g ` s)" shows "bounded ((\<lambda>x. f x + g x) ` s)"
not_INFM: fixes P :: "'a \<Rightarrow> bool" shows "(\<not> (\<exists>\<^sub>\<infinity>x. P x)) = (\<forall>\<^sub>\<infinity>x. \<not> P x)"
splus_left_canc: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a"   and u :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "(term_powerprod.splus pair_of_term term_of_pair t v = term_powerprod.splus pair_of_term term_of_pair t u) = (v = u)"
surjective: fixes r :: "'a transition_scheme" shows "r = \<lparr>Label = Label r, Arity = Arity r, Guards = Guards r, Outputs = Outputs r, Updates = Updates r, \<dots> = more r\<rparr>"
system_iff: fixes Q :: "'b mat"   and v :: "'b vec" assumes "dim_row Q = dim_col Q"   and "dim_row Q = dim_vec v" shows "(Q\<^sup>T *\<^sub>v v = v) = ((Q\<^sup>T - 1\<^sub>m (dim_row Q)) *\<^sub>v v = 0\<^sub>v (dim_vec v))"
u_def: shows "u = Max (insert 0 {x. poly p x = 0}) + 1"
convex_scaleR_image_iff: fixes r :: "real"   and f :: "'c \<Rightarrow> 'b"   and R :: "'c set" shows "convex ((\<lambda>x. r *\<^sub>R f x) ` R) = (r = 0 \<or> convex (f ` R))"
WHILEI_weaken: fixes I :: "'c \<Rightarrow> bool"   and I' :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and x :: "'c" assumes "\<And>x. I x \<Longrightarrow> I' x" shows "WHILE\<^bsup>I'\<^esup> b f x \<le> WHILE\<^bsup>I\<^esup> b f x"
r_def: shows "r = nat \<lceil>log 2 (1 / c)\<rceil>"
holdffm: fixes m :: "nat" shows "(\<lambda>z. deriv f ((f ^^ m) z)) holomorphic_on s"
inducts: fixes x :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite x"   and "P {}"   and "\<And>A a. \<lbrakk>finite A; P A\<rbrakk> \<Longrightarrow> P (insert a A)" shows "P x"
conv: shows "(\<lambda>N. emeasure M (K N)) \<longlonglongrightarrow> emeasure M A"
cases: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and a1 :: "'heap"   and a2 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and a3 :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and a4 :: "'heap"   and a5 :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_meth ci P ins xt t a1 a2 a3 a4 a5" obtains ta :: "('addr, 'thread_id, 'heap) jvm_thread_action" and xcp :: "'addr option" and h' :: "'heap" and stk' :: "'addr val list" and loc' :: "'addr val list" and pc' :: "nat" and pc :: "nat" and h :: "'heap" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = ha__" and "a2 = (stka__, loca__, pca__, None)" and "a3 = ta" and "a4 = h'" and "a5 = (stk', loc', pc'a__, xcpa__)" and "(ta, xcpa__, h', [(stk', loc', undefined, undefined, pc'a__)]) \<in> exec_instr (ins ! pca__) P t ha__ stka__ loca__ undefined undefined pca__ []" and "pca__ < length ins" and "ci_app ci (ins ! pca__) P ha__ stka__ loca__ undefined undefined pca__ []"   | h :: "'heap" and xcp :: "'addr" and pc :: "nat" and pc' :: "nat" and d :: "nat" and stk :: "'addr val list" and loc :: "'addr val list" where "a1 = h" and "a2 = (stk, loc, pc, \<lfloor>xcp\<rfloor>)" and "a3 = \<lbrace>\<rbrace>" and "a4 = h" and "a5 = (Addr xcp # drop (length stk - d) stk, loc, pc', None)" and "match_ex_table P (cname_of h xcp) pc xt = \<lfloor>(pc', d)\<rfloor>" and "d \<le> length stk"
mini_formula_atoms: fixes F :: "'b formula" shows "atoms (to_mini_formula F) = atoms F"
pmdl_Macaulay_list: fixes ps :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "local.punit.pmdl (set (punit.Macaulay_list ps)) = local.punit.pmdl (set ps)"
finite_code: fixes A :: "'b set" assumes "class.finite TYPE('b)" shows "finite A = True"
A1: shows "c_f_list f \<in> PrimRec2"
offs_append: fixes ms :: "nat list"   and ns :: "nat list"   and k :: "nat" shows "offs (ms @ ns) k = offs ms k @ offs ns (foldl (+) k ms)"
distinct_deleteAround: fixes ps :: "(nat \<times> nat) list"   and g :: "graph"   and a :: "nat"   and b :: "'a" assumes "distinct (map fst ps)" shows "distinct (map fst (deleteAround g (fst (a, b)) ps))"
simps_2: fixes v :: "nat"   and va :: "nat" shows "is_Less (Eq v va) = False"
supp_rev: fixes xs :: "'a list" shows "supp (rev xs) = supp xs"
incseq_const: fixes k :: "'a" shows "incseq (\<lambda>x. k)"
ctr_transfer_1: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_Expression R1 R2 Vx Vx"
tauChainInduct: fixes P :: "pi"   and P' :: "pi"   and F :: "pi \<Rightarrow> bool" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "F P"   and "\<And>P'' P'''. \<lbrakk>P \<Longrightarrow>\<^sub>\<tau> P''; P'' \<longmapsto> \<tau> \<prec> P'''; F P''\<rbrakk> \<Longrightarrow> F P'''" shows "F P'"
hom_zero: fixes d :: "nat" shows "monom (0::'c) d = 0"
functor_implies_equals_dom: assumes "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) A.map" shows "A.map = A.map"
bound_of_least: fixes b :: "real"   and P :: "'c \<Rightarrow> real" assumes "bounded_by b P" shows "bound_of P \<le> b"
simps: fixes X :: "'c set" shows "the_RES (RES X) = X"
card_of_Plus_infinite: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "|B| \<le>o |A|" shows "|A <+> B| =o |A| \<and> |B <+> A| =o |A|"
circle_y_bounds_1: fixes x :: "real" assumes "- 1 / 2 \<le> x \<and> x \<le> 1 / 2" shows "0 \<le> circle_y x"
atLeastOneRes: fixes ts :: "traffic"   and c :: "cars" shows "1 \<le> |res ts c|"
at_in_mapi_branch: fixes a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and f :: "nat \<Rightarrow> nat \<Rightarrow> ('b, 'c) fm \<Rightarrow> ('b, 'c) fm" assumes "p at a in' set branch"   and "p \<noteq> Nom a" shows "\<exists>v v'. f v v' p at a in' set (mapi_branch f branch)"
assms_2: shows "\<exists>c. \<forall>x\<in>space M. f x = c"
remdups_fwd_split: fixes Acc :: "'b set"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "remdups_fwd_acc Acc xs = ys @ zs" shows "\<exists>us vs. xs = us @ vs \<and> remdups_fwd_acc Acc us = ys \<and> remdups_fwd_acc (Acc \<union> set ys) vs = zs"
q'_irred: shows "\<not> is_sig_red (\<prec>\<^sub>t) (\<preceq>) G q'"
inject_2: fixes x21 :: "'e"   and x22 :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and x23 :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option"   and y21 :: "'e"   and y22 :: "'a \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm option"   and y23 :: "'b \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs option" shows "(qOp x21 x22 x23 = qOp y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
assms_4: shows "(i, Store e1 mo i) \<in> set (node_deliver_messages xs)"
Null_notin_Dag: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" assumes "Dag p l r t" shows "Null \<notin> set_of t"
avg_aux_correct: fixes s :: "(nat, unit) HashMap.hashmap" assumes "my_invar s" shows "hs_ctx.avg_aux s = (card (my_\<alpha> s), \<Sum> (my_\<alpha> s))"
proj_filter: fixes i :: "'a"   and Di :: "('a strand_label \<times> 'b) list"   and D :: "('a strand_label \<times> 'b) list" shows "proj i (filter (\<lambda>d. d \<notin> set Di) D) = filter (\<lambda>d. d \<notin> set Di) (proj i D)"
BRBangCases: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and F :: "pi \<Rightarrow> bool" assumes "(P, !Q) \<in> bangRel Rel"   and "\<And>P. (P, Q) \<in> Rel \<Longrightarrow> F (!P)" shows "F P"
bij_betw_pos_interval: fixes a :: "real"   and b :: "real" assumes "a < b" shows "bij_betw (\<lambda>x. (a + b * x) / (1 + x)) {x. 0 < x} {x. a < x \<and> x < b}"
kuhn_complete_lemma: fixes simplices :: "'a set set"   and face :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and n :: "nat"   and rl :: "'a \<Rightarrow> nat"   and bnd :: "'a set \<Rightarrow> bool" assumes "finite simplices"   and "\<And>f s. face f s = (\<exists>a\<in>s. f = s - {a})"   and "\<And>s. s \<in> simplices \<Longrightarrow> card s = n + 2"   and "\<And>s. s \<in> simplices \<Longrightarrow> rl ` s \<subseteq> {..Suc n}"   and "\<And>f s. \<lbrakk>s \<in> simplices; face f s; bnd f\<rbrakk> \<Longrightarrow> card {s \<in> simplices. face f s} = 1"   and "\<And>f s. \<lbrakk>s \<in> simplices; face f s; \<not> bnd f\<rbrakk> \<Longrightarrow> card {s \<in> simplices. face f s} = 2"   and "odd (card {f. (\<exists>s\<in>simplices. face f s) \<and> rl ` f = {..n} \<and> bnd f})" shows "odd (card {s \<in> simplices. rl ` s = {..Suc n}})"
not_lfinite_llength: fixes xs :: "'g llist" assumes "\<not> lfinite xs" shows "llength xs = \<infinity>"
ball_UNIV: fixes P :: "'b \<Rightarrow> bool" shows "Ball UNIV P = All P"
tendsto_sandwich: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and "open" :: "'e set \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e"   and net :: "'f filter"   and h :: "'f \<Rightarrow> 'e"   and c :: "'e" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in net. less_eq (f n) (g n)"   and "\<forall>\<^sub>F n in net. less_eq (g n) (h n)"   and "filterlim f (topological_space.nhds open c) net"   and "filterlim h (topological_space.nhds open c) net" shows "filterlim g (topological_space.nhds open c) net"
Group: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "RModule R smult M" shows "Group M"
AssertAllState''_index_valid: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" assumes "\<triangle> t" shows "index_valid (set ats) (assert_bound_loop ats (init t))"
rtranclp_rtrancl_eq: fixes r :: "('b \<times> 'b) set" shows "(\<lambda>x xa. (x, xa) \<in> r)\<^sup>*\<^sup>* = (\<lambda>x xa. (x, xa) \<in> r\<^sup>*)"
Suc: shows "ffact n (x + y) = (\<Sum>k = 0..n. (n choose k) * ffact k x * ffact (n - k) y)"
bounded_by_imp_bounded: fixes b :: "real"   and P :: "'b \<Rightarrow> real" assumes "bounded_by b P" shows "Expectations.bounded P"
fun_eqvtI: fixes f :: "'a \<Rightarrow> 'b"   and p :: "perm" assumes "\<And>p x. p \<bullet> f x = f (p \<bullet> x)" shows "p \<bullet> f = f"
steps_append: fixes xs :: "'a list"   and ys :: "'a list" assumes "steps xs"   and "steps ys"   and "last xs = hd ys" shows "steps (xs @ tl ys)"
nth_rotate1_Suc: fixes n :: "nat"   and ls :: "'c list" assumes "Suc n < |ls|" shows "ls ! Suc n = rotate1 ls ! n"
keysFor_insert_Crypt: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" shows "keysFor (insert (Crypt K X) H) = insert (invKey K) (keysFor H)"
meet_left: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "le (dlat.meet le x y) x"
distinct_5: fixes x2 :: "'b" shows "Lt x2 \<noteq> \<infinity>"
ps_def: shows "ps = pders (pderiv (pderiv p))"
seq_comp_prop_4: fixes P :: "'c option process"   and xs :: "'c option list"   and X :: "'c option set"   and Q :: "'c option process"   and x :: "'c option" assumes "weakly_sequential P"   and "(xs, X) \<in> seq_comp_failures P Q" shows "(xs @ [x], {}) \<in> seq_comp_failures P Q \<or> (xs, insert x X) \<in> seq_comp_failures P Q"
intros_7: fixes F :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset"   and G :: "'a formula" assumes "F, \<Gamma> \<Rightarrow> \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<or> G, \<Gamma> \<Rightarrow> \<Delta>"
a: shows "\<forall>(a, b)\<in>set pi. c \<noteq> a \<and> c \<noteq> b"
iotam_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set"   and K :: "'e set" shows "\<iota>m\<^bsub>M H,K\<^esub> = (\<lambda>x\<in>H. x \<plusminus>\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub>)"
sas_plus_equivalent_to_strips_i_a_XI: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op' :: "('c \<times> 'd) strips_operator"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)" shows "(\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op') = \<phi>\<^sub>S \<Psi> s ++ map_of (effect_of (\<phi>\<^sub>O\<inverse> \<Psi> op'))"
starfun_exp_Infinitesimal: fixes x :: "real star" assumes "x \<in> HInfinite"   and "x \<le> 0" shows "(*f* exp) x \<in> Infinitesimal"
U_2: fixes n :: "nat" shows "ball (0::'a) (real n) \<inter> B \<subseteq> U n"
False: shows "snd (rep_nat_term u) \<noteq> snd (rep_nat_term v)"
valid: fixes g :: "'g" assumes "invar g" shows "valid_graph (\<alpha> g)"
take_tl: fixes n :: "nat"   and xs :: "'c list" shows "take n (tl xs) = tl (take (Suc n) xs)"
bij_betw_image_quotient: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "bij_betw (\<lambda>y. f -` {y}) (f ` A) (A // {(a, b). f a = f b})"
lift_step_into_qmsg_statelessassm: fixes A :: "((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and R :: "'c \<Rightarrow> bool"   and i :: "nat"   and P :: "(nat \<Rightarrow> 'a) \<times> 'c seq_action \<times> (nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "A \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala P"   and "\<And>\<sigma> \<sigma>' m. \<sigma>' i = \<sigma> i \<Longrightarrow> P (\<sigma>, receive m, \<sigma>')"   and "\<And>\<sigma> \<sigma>' m. P (\<sigma>, receive m, \<sigma>') \<Longrightarrow> P (\<sigma>, \<tau>, \<sigma>')" shows "A \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. orecvmsg (\<lambda>_. R) \<sigma>, other (\<lambda>_ _. True) {i} \<rightarrow>) globala P"
lbvs_def: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c"   and B :: "'c"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and c :: "'c list"   and ins :: "'d list" shows "lbvs A r f T B step c ins \<equiv> lbv A r f T B \<and> lbvs_axioms A T B step c ins"
pointbased_implies_continuous: fixes f :: "'a set \<Rightarrow> 'b set" assumes "pointbased f" shows "continuous f"
mset_neg_uminus: fixes A :: "'b zmultiset" shows "mset_neg (- A) = mset_pos A"
F_filter: fixes F :: "'b set set"   and T :: "'b set set" assumes "Topology.filter F T" shows "F \<in> Filters\<^bsub>T\<^esub>"
inject: fixes x21 :: "'m"   and x22 :: "'m list"   and y21 :: "'m"   and y22 :: "'m list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
sum_match_cong: fixes e :: "'a + 'c"   and e' :: "'b + 'd"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and B' :: "'c \<Rightarrow> 'd \<Rightarrow> assn" assumes "\<And>x y. \<lbrakk>e = Inl x; e' = Inl y\<rbrakk> \<Longrightarrow> hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y"   and "\<And>x y. \<lbrakk>e = Inr x; e' = Inr y\<rbrakk> \<Longrightarrow> hn_ctxt B x y \<Longrightarrow>\<^sub>t hn_ctxt B' x y" shows "hn_ctxt (A +\<^sub>a B) e e' \<Longrightarrow>\<^sub>t hn_ctxt (A' +\<^sub>a B') e e'"
SComplex_approx_iff: fixes x :: "complex star"   and y :: "complex star" assumes "x \<in> SComplex"   and "y \<in> SComplex" shows "(x \<approx> y) = (x = y)"
epo_axioms: shows "epo ground_heads_var (>\<^sub>s) arity_sym arity_var extf"
nou_equiv: shows "uninst_code.phi_equiv_mapping (\<lambda>_. s1_phis) g s1_nodes_of_uses (uninst_code.ssa.useNodes_of (\<lambda>_. s1_uses) g)"
append_is_Nil_conv: fixes xs :: "'f list"   and ys :: "'f list" shows "(xs @ ys = []) = (xs = [] \<and> ys = [])"
Guard: fixes P :: "'d set"   and g :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f" assumes "P \<subseteq> g \<inter> R"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Guard f g c Q,A"