hcju/isabelleps huggingface dataset:
l_remove_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set" shows "l_remove_child_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel \<equiv> (l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs \<and> l_get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs heap_is_wellformed parent_child_rel get_disconnected_nodes) \<and> l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes \<and> l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"
step_step': fixes S :: "'av st option"   and C :: "'av st option acom" shows "step (\<gamma>\<^sub>o S) (\<gamma>\<^sub>c C) \<le> \<gamma>\<^sub>c (step' S C)"
simps_3: fixes va :: "nat" shows "ld (Suc (Suc va)) = Suc (ld (Suc (Suc va) div 2))"
landau_flip_1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "(f \<in> O[F](g)) = (g \<in> \<Omega>[F](f))"
imirror_bounds_def: fixes I :: "nat set"   and l :: "nat"   and r :: "nat" shows "imirror_bounds I l r \<equiv> (\<lambda>x. nat_mirror x l r) ` I"
dt: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "F \<triangleright> \<Gamma> \<union> AX10 \<turnstile>\<^sub>H G" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G"
sum_list_rev: fixes xs :: "'a list" shows "sum_list (rev xs) = sum_list xs"
distinct_1: fixes x1 :: "'d"   and x2 :: "'d" shows "Le x1 \<noteq> Lt x2"
repl_invariance: fixes v :: "'b"   and t :: "'b trm"   and u :: "'b trm"   and s :: "('b \<times> 'b trm) list" assumes "v \<notin> vars_of t" shows "t \<lhd> (v, u) # s = t \<lhd> s"
assms: fixes g :: "'a" assumes "g \<in> G" shows "g \<cdot>| H = H |\<cdot> g"
mult_nonpos_nonpos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq a zero"   and "less_eq b zero" shows "less_eq zero (times a b)"
powr_bigtheta_iff: fixes g :: "'a \<Rightarrow> real"   and F :: "'a filter"   and p :: "real"   and q :: "real" assumes "filterlim g at_top F"   and "F \<noteq> bot" shows "((\<lambda>x. g x powr p) \<in> \<Theta>[F](\<lambda>x. g x powr q)) = (p = q)"
least_fixpoint: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_fixpoint f" shows "is_least_fixpoint f (\<mu> f)"
agree_comm: fixes x :: "('d, 'e) fmap"   and x' :: "('d, 'e) fmap" shows "agree x x' = agree x' x"
simps_26: fixes f :: "'e \<Rightarrow> 'd"   and x21 :: "'e rexp"   and x22 :: "'e rexp" shows "map_rexp f (Alt x21 x22) = Alt (map_rexp f x21) (map_rexp f x22)"
authKeys_are_not_AKcryptSK: fixes K :: "nat"   and evs :: "event list" assumes "K \<in> authKeys evs \<union> range shrK"   and "evs \<in> kerbV" shows "\<forall>SK. \<not> AKcryptSK SK K evs \<and> K \<in> symKeys"
ac_fProd_Prod: fixes n :: "nat"   and B :: "nat \<Rightarrow> ('d, 'e) Ring_scheme" assumes "\<forall>k\<le>n. Ring (B k)" shows "ac_fProd_Rg n B = carr_prodag {j. j \<le> n} B"
distrib_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
weighted_RP_imp_RP: fixes St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat" assumes "St \<leadsto>\<^sub>w St'" shows "state_of_wstate St \<leadsto> state_of_wstate St'"
simps_1: fixes K :: "'c \<Rightarrow> 'c pmf"   and x :: "'c"   and \<omega> :: "'c stream" shows "shd (MC_syntax.force_enabled K x \<omega>) = (if shd \<omega> \<in> set_pmf (K x) then shd \<omega> else SOME y. y \<in> set_pmf (K x))"
kas: shows "[x, y] = [x0, y0] \<or> [x, y] = [y0, x0]"
3: shows "gauge (\<lambda>x. if \<exists>xK\<in>p. f xK = x then (\<gamma> \<circ> fst \<circ> the_inv_into p f) x else \<gamma> x)"
condensation_condense1: fixes n :: "nat" shows "(\<Sum>k = 1..<2 ^ n. f (2 ^ Discrete.log k)) = (\<Sum>k<n. 2 ^ k * f (2 ^ k))"
nth_prime_lower_bound: fixes n :: "nat" shows "Suc (Suc n) \<le> nth_prime n"
simps_4: fixes v :: "(int \<times> int \<times> protocol) set set"   and va :: "(int \<times> int \<times> protocol) set set"   and vb :: "protocol \<times> int"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "wellformed_policy2Pr (AllowPortFromTo v va vb # xs) = ((\<forall>c a b. c = DenyAllFromTo a b \<and> c \<in> set xs \<longrightarrow> dom (Cp (AllowPortFromTo v va vb)) \<inter> dom (Cp c) = {}) \<and> wellformed_policy2Pr xs)"
P2': fixes x :: "'c"   and x' :: "'d" shows "P2 x x' = \<up> (P2' x x')"
set_tag_name_is_l_set_tag_name: shows "l_set_tag_name DocumentClass.type_wf set_tag_name set_tag_name_locs"
scale: fixes b :: "'b"   and r :: "'a" assumes "b \<in> U\<^sub>M\<^sub>_\<^sub>1" shows "f (r *s\<^sub>1 b) = r *s\<^sub>2 f b"
ahm_iteratei_aux_def: shows "ahm_iteratei_aux \<equiv> rec_array (\<lambda>xs. foldli (concat xs))"
pathsConverge_def: fixes g :: "'g"   and x :: "'node"   and xs :: "'node list"   and y :: "'node"   and ys :: "'node list"   and z :: "'node" shows "pathsConverge g x xs y ys z \<equiv> g \<turnstile> x-xs\<rightarrow>z \<and> g \<turnstile> y-ys\<rightarrow>z \<and> 1 < length xs \<and> 1 < length ys \<and> x \<noteq> y \<and> (\<forall>j\<in>{0..<length xs}. \<forall>k\<in>{0..<length ys}. xs ! j = ys ! k \<longrightarrow> j = length xs - 1 \<or> k = length ys - 1)"
comp_galcon_closed: fixes G :: "('d, 'e, unit, unit) galcon"   and F :: "('f, 'd, unit, unit) galcon" assumes "galois_connection G"   and "galois_connection F"   and "\<Y>\<^bsub>F\<^esub> = \<X>\<^bsub>G\<^esub>" shows "galois_connection (G \<circ>\<^sub>g F)"
continuous_imp_integrable_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "continuous_on {a..b} f" shows "integrable (lebesgue_on {a..b}) f"
negation_implies: fixes x :: "'a"   and y :: "'a" shows "- (x \<leadsto> y) = - - x \<sqinter> - y"
step_elim_cases_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> ([], css, Abrupt s) \<rightarrow> u" obtains nrms :: "('e, 'd, 'f) com list" and abrs :: "('e, 'd, 'f) com list" and cssa :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" where "css = (nrms, abrs) # cssa" and "u = (abrs, cssa, Normal s)"
B: fixes x :: "'a"   and y :: "'a" shows "(x, y) \<in> r \<or> (y, x) \<in> r"
set_quotient: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c" assumes "Quotient3 R Abs Rep" shows "Quotient3 (rel_vset R) ((-`) Rep) ((-`) Abs)"
subspace_def: fixes S :: "'b set" shows "local.subspace S = ((0::'b) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *s x \<in> S))"
af_congruent_axioms: shows "af_congruent (\<sim>)"
e2rv_rv: fixes ss :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "e2rv (encode_config (ss, rv)) = encode_option rv"
arithmetic_simps_2: fixes n :: "num" shows "num.One + num.Bit0 n = num.Bit1 n"
VcompNml_Can_Inv: fixes t :: "'a local.term" assumes "Can t"   and "Nml t" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Inv t = Cod t"
distinct_29: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x31@[x32] \<noteq> Stop"
pure_def: fixes f :: "('ap, 'aq, 'ar) prog"   and h :: "'ap" shows "pure f h = (h \<turnstile> ok f \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h)"
join_commute: fixes x :: "'c"   and y :: "'c" shows "x \<squnion> y = y \<squnion> x"
distinct_1: fixes x1 :: "nat"   and x21 :: "nat"   and x22 :: "exp"   and x23 :: "(nat \<times> bval) list" shows "BNat x1 \<noteq> BClos x21 x22 x23"
line_integral_sum_gen_2: fixes basis :: "'a set"   and F :: "'a \<Rightarrow> 'a"   and basis1 :: "'a set"   and \<gamma> :: "real \<Rightarrow> 'a"   and basis2 :: "'a set" assumes "finite basis"   and "line_integral_exists F basis1 \<gamma>"   and "line_integral_exists F basis2 \<gamma>"   and "basis1 \<union> basis2 = basis"   and "basis1 \<inter> basis2 = {}" shows "line_integral_exists F basis \<gamma>"
H: fixes c :: "real"   and f :: "'a \<Rightarrow> real" assumes "c \<noteq> 0" shows "e2ennreal (esssup M (\<lambda>x. ereal \<bar>(c *\<^sub>R f) x\<bar>)) \<le> ennreal \<bar>c\<bar> * e2ennreal (esssup M (\<lambda>x. ereal \<bar>f x\<bar>))"
asubg_nsubg: fixes H :: "'a set" assumes "A +> H" shows "b_ag A \<triangleright> H"
\<alpha>': shows "\<alpha>' = take i \<alpha> @ snd r @ drop (i + 1) \<alpha>"
simps_1: fixes Y :: "tyvrs"   and T :: "ty" shows "[][Y \<mapsto> T]\<^sub>e = []"
M: fixes i :: "'n" shows "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < r / real CARD('n)"
hyps_2: shows "P \<parallel> !P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'"
assms_4: shows "prv (imp \<phi> \<psi>)"
clo: fixes C :: "'a set" assumes "C \<in> \<S>" shows "closed C"
perp_inter_perp_in: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "\<exists>P. Col A B P \<and> Col C D P \<and> P PerpAt A B C D "
Suc_0_or_eq: fixes n :: "nat" shows "Suc 0 OR n = n + of_bool (even n)"
le_fininf_eq_inf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item stream"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>2 =\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
\<Phi>_ide: fixes a :: "'s" assumes "S.ide a" shows "\<Phi> a = \<Phi>o a"
filter_False: fixes xs :: "'e list"   and P :: "'e \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> P x" shows "filter P xs = []"
of_int_round_abs_le: fixes x :: "'b" shows "\<bar>of_int (round x) - x\<bar> \<le> (1::'b) / (2::'b)"
Zorns_po_lemma: fixes r :: "('b \<times> 'b) set" assumes "Partial_order r"   and "\<And>C. C \<in> Chains r \<Longrightarrow> \<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r" shows "\<exists>m\<in>Field r. \<forall>a\<in>Field r. (m, a) \<in> r \<longrightarrow> a = m"
finite_measure_Diff: fixes A :: "real set"   and B :: "real set" assumes "A \<in> sets M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M B"
run_bind_state: fixes bind :: "'d \<Rightarrow> ('f \<times> 'e \<Rightarrow> 'd) \<Rightarrow> 'd"   and x :: "('e, 'd) stateT"   and f :: "'f \<Rightarrow> ('e, 'd) stateT"   and s :: "'e" shows "run_state (bind_state bind x f) s = bind (run_state x s) (\<lambda>(a, s'). run_state (f a) s')"
unat_add_lem: fixes x :: "'b word"   and y :: "'b word" shows "(unat x + unat y < 2 ^ LENGTH('b)) = (unat (x + y) = unat x + unat y)"
C: shows "C = ''y'' ::= N 0;; ''z'' ::= N 0;; {(I, S, E)} WHILE Less (V ''z'') (V ''a'') DO (''y'' ::= Plus (V ''y'') (V ''b'');; ''z'' ::= Plus (V ''z'') (N 1))"
assms_1: shows "A B C LtA A1 B1 C1"
nsClose: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows " n \<times>\<^bsub>R\<^esub> x \<in> carrier R"
create_character_data_is_weakly_dom_component_safe_step: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'a" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^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 document_ptr)|\<^sub>r"   and "ptr \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^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 |h \<turnstile> create_character_data document_ptr text|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
assms_1: shows "f \<equiv> \<lambda>M. {t \<cdot> \<delta> |t \<delta>. t \<in> M \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> fv (t \<cdot> \<delta>) = {}}"
V_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b" shows "Graph.V c \<equiv> {u. \<exists>v. (u, v) \<in> Graph.E c \<or> (v, u) \<in> Graph.E c}"
inner2_def: fixes x :: "'a"   and n :: "'a"   and l :: "'a" shows "inner2 x n l = (x \<bullet> n, x \<bullet> l)"
beta: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm"   and x :: "'a"   and T :: "PreSimplyTyped.type" assumes "A >> A'"   and "B >> B'" shows "App (Fn x T A) B >> A'[x ::= B']"
disc_2: fixes x2 :: "bool" shows "is_BoolVal (BoolVal x2)"
ex_un_null: shows "\<exists>!n. \<forall>f. R.comp n f = n \<and> R.comp f n = n"
take_numeral_Cons_1: fixes x :: "'b"   and xs :: "['b]" shows "take\<cdot>1\<cdot>(x : xs) = [x]"
bdd_aboveI2: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset"   and M :: "'b multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "subset_mset.bdd_above (f ` A)"
igFreshIGAbs1_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igFreshIGAbs1 MOD \<equiv> \<forall>ys y s X. isInBar (ys, s) \<and> igWls MOD s X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD ys y X)"
minus_monom: fixes a :: "'b"   and n :: "nat" shows "- monom a n = monom (- a) n"
flist_arb_def: fixes A :: "'b fset" shows "flist_arb A = (SOME xs. xs \<in> flists A)"
CollectD: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "s2p {x. P x} a" shows "P a"
key_map_is_ring_morphism: fixes U :: "'a set set" assumes "\<pp> \<in> U"   and "is_zariski_open U" shows "ring_homomorphism (local.key_map U) (\<O> U) (add_sheaf_spec U) (mult_sheaf_spec U) (zero_sheaf_spec U) (one_sheaf_spec U) pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at"
assoc'_simps_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "src \<a>\<^sup>-\<^sup>1[f, g, h] = src h"
pred_map: fixes Q :: "'e \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'e"   and x :: "'g set['f]" shows "pred_bset Q (map_bset f x) = pred_bset (Q \<circ> f) x"
simps_3: fixes x21 :: "'c"   and x22 :: "'c" shows "Point x21 x22 \<noteq> Infinity"
q_in: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G" shows "q g \<in> punit.dgrad_p_set d m'"
Gp_smult_nsg1: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "\<natural>\<^bsub>G\<^esub>N \<diamondop>\<^bsub>G\<^esub> H \<triangleright> N"
main_connect: assumes "length (map of_int_hom.vec_hom fs) \<le> n" shows "gram_schmidt n (map of_int_hom.vec_hom fs) = map gs1.gso [0..<length (map of_int_hom.vec_hom fs)]"
inf_sup_distrib1: fixes x :: "'e multiset"   and y :: "'e multiset"   and z :: "'e multiset" shows "x \<inter># (y \<union># z) = x \<inter># y \<union># (x \<inter># z)"
S: fixes a :: "'a" assumes "a \<in> S" shows "a \<notin> convex hull (S - {a})"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<R>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> TargetTerm T2"
enabled_def: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and r :: "'a"   and s :: "'b" shows "RuleSystem_Defs.enabled eff r s \<equiv> \<exists>sl. eff r s sl"
MergeGuardsD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P merge_guards c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
IH: shows "\<forall>A. A \<subset># A \<longrightarrow> P A"
CSS_def: shows "CSS \<equiv> \<lambda>c. fst (snd c)"
uniformly_continuous_on: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "bounded_linear f"   and "uniformly_continuous_on s g" shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
mkChannels_pState_inv: fixes prog :: "program"   and p :: "pState"   and g :: "'a gState_scheme"   and cs :: "channel list" assumes "pState_inv prog p"   and "cl_inv (g, p)" shows "pState_inv prog (snd (mkChannels g p cs))"
sel_2: fixes x21 :: "'a"   and x22 :: "nat"   and x23 :: "'a interval" shows "pos (DEL x21 x22 x23) = x21"
ccInf_lower2: fixes A :: "'a set"   and u :: "'a"   and v :: "'a" assumes "countable A"   and "u \<in> A"   and "u \<le> v" shows "Inf A \<le> v"
assoc_simps_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and h :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f"   and "partial_magma.ide (span_vertical_category.vcomp C) g"   and "partial_magma.ide (span_vertical_category.vcomp C) h"   and "span_vertical_category.src C f = span_vertical_category.trg C g"   and "span_vertical_category.src C g = span_vertical_category.trg C h" shows "partial_magma.arr (span_vertical_category.vcomp C) \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc C prj0 prj1 f g h, Dom = Dom (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h), Cod = Cod (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h))\<rparr>"
adj_path_V': fixes v :: "'v"   and ps :: "'v list" assumes "adj_path v ps" shows "v \<in> V"
max_coeff_bound'_1: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Minimal_Solutions" shows "\<forall>i<length x. x ! i \<le> Max (set b)"
set_s_A: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list"   and n :: "nat" assumes "MTF_Off as rs s0" shows "set (MTF_Off.s_A as rs s0 n) = set s0"
gap_pres_Labeling: fixes k :: "nat" assumes "gap_precond l' k" shows "Labeling c s t f (gap_effect l' k)"
inv_loop_steps: fixes x :: "nat"   and cf :: "nat \<times> cell list \<times> cell list"   and stp :: "nat" assumes "inv_loop x cf"   and "0 < x" shows "inv_loop x (steps0 cf tcopy_loop stp)"
insert_3: shows "pderiv (prod f as) = (\<Sum>a\<in>as. prod f (as - {a}) * pderiv (f a))"
obj_trg: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "obj (trg \<mu>)"
det2_def: fixes a11 :: "'b"   and a12 :: "'b"   and a21 :: "'b"   and a22 :: "'b" shows "det2 a11 a12 a21 a22 \<equiv> a11 * a22 - a12 * a21"
simps_5: fixes a :: "Ferrack.num"   and b :: "Ferrack.num" shows "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
simps_5: fixes g :: "char list \<Rightarrow> bool"   and h :: "'a pfm \<Rightarrow> bool"   and p :: "'a pfm"   and q :: "'a pfm" shows "peval g h (p \<^bold>\<longrightarrow>\<^sub>! q) = (peval g h p \<longrightarrow> peval g h q)"
pull_up_configuration_senior: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and q :: "'d"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n"   and "y \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n" shows "semi_mojmir_def.senior \<delta> q\<^sub>0 w x n = semi_mojmir_def.senior \<delta> q\<^sub>0 w y n"
comp_cong: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e"   and f' :: "'f \<Rightarrow> 'c"   and g' :: "'g \<Rightarrow> 'f"   and x' :: "'g" assumes "f (g x) = f' (g' x')" shows "(f \<circ> g) x = (f' \<circ> g') x'"
1: shows "invertible P'' \<and> invertible Q'' \<and> isDiagonal D \<and> D = P'' ** A ** Q''"
rt_fresher_def2: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2" shows "(rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2) = (nsqn rt1 dip < nsqn rt2 dip \<or> nsqn rt1 dip = nsqn rt2 dip \<and> the (dhops rt2 dip) \<le> the (dhops rt1 dip))"
gamma_num': fixes n :: "int" shows "n \<in> \<gamma> (num' n)"
C_def: shows "C = (\<Inter>n. (T ^^ n) --` A)"
assms: shows "V \<equiv> {root n (norm (x ^ n)) |n. 0 < n}"