hcju/isabelleps huggingface dataset:
not_postdominate_source_not_postdominate_target: fixes n :: "'node"   and a :: "'edge" assumes "\<not> n postdominates sourcenode a"   and "valid_node n"   and "valid_edge a" obtains ax :: "'edge" where "sourcenode a = sourcenode ax" and "valid_edge ax" and "\<not> n postdominates targetnode ax"
sign_simps_35: fixes am :: "'a"   and bm :: "'a"   and ck :: "'a" shows "(am + bm) * ck = am * ck + bm * ck"
lesspoll_imp_lepoll: fixes A :: "'a set"   and B :: "'b set" assumes "A \<prec> B" shows "A \<lesssim> B"
pow2: fixes k :: "nat" shows "multiplicity 2 (x ^ 2 ^ k - y ^ 2 ^ k) = multiplicity 2 (x - y) + k"
algebraic_int_root: fixes y :: "'b"   and p :: "'b poly"   and x :: "'b" assumes "algebraic_int y"   and "poly p x = y"   and "\<forall>i. coeff p i \<in> \<int>"   and "lead_coeff p = (1::'b)"   and "0 < degree p" shows "algebraic_int x"
assms_2: shows "outs_\<I> \<I> \<subseteq> outs_\<I> \<I>'"
CFGExit_axioms: shows "CFGExit sourcenode targetnode kind valid_edge (_Entry_) (_Exit_)"
C_Rep: fixes b :: "'cl"   and b' :: "'cr" shows "C b b' = (b = Rep_c b')"
elims: fixes x :: "'c list"   and y :: "'c list" assumes "remdups_adj x = y" obtains "x = []" and "ya__ = []"   | xa :: "'c" where "x = [xa__]" and "ya__ = [xa__]"   | xa :: "'c" and y :: "'c" and xs :: "'c list" where "x = xa # y # xs" and "ya__ = (if xa = y then remdups_adj (xa # xs) else xa # remdups_adj (y # xs))"
mem_0_Ord: fixes k :: "V" assumes "Ord k"   and "k \<noteq> []\<^sub>\<circ>" shows "[]\<^sub>\<circ> \<in>\<^sub>\<circ> k"
enn2ereal_Iio: fixes a :: "ereal" shows "enn2ereal -` {..<a} = (if 0 \<le> a then {..<e2ennreal a} else {})"
red_diff: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red F (p - q) r" obtains p' :: "'t \<Rightarrow>\<^sub>0 'c" and q' :: "'t \<Rightarrow>\<^sub>0 'c" where "(red F)\<^sup>*\<^sup>* p p'" and "(red F)\<^sup>*\<^sup>* q q'" and "r = p' - q'"
sum_less: fixes a :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "additive_principal a"   and "x < a"   and "y < a" shows "x + y < a"
bp: shows "bprv (PP \<langle>\<phi>\<rangle>)"
gg: fixes g :: "nat \<Rightarrow> 'a" shows "sum g {..p} = sum g {..<k} + sum g {k..p}"
iarray_of_fun_sub: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "i < n" shows "IArray.of_fun f n !! i = f i"
adjoint_unique: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "\<forall>x y. f x \<bullet> y = x \<bullet> g y" shows "adjoint f = g"
ipoly_min_int_poly: fixes x :: "'a" assumes "algebraic x" shows "ipoly (min_int_poly x) x = (0::'a)"
cong_cong_half_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a"   and B :: "'a"   and M' :: "'a"   and A' :: "'a"   and B' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A B"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M' A' B'"   and "Cong A B A' B'" shows "Cong B M B' M'"
rel_resumption_OO: fixes A :: "'f \<Rightarrow> 'k \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'l \<Rightarrow> bool"   and C :: "'k \<Rightarrow> 'i \<Rightarrow> bool"   and D :: "'l \<Rightarrow> 'j \<Rightarrow> bool" shows "Resumption.resumption.rel_resumption A B OO Resumption.resumption.rel_resumption C D = Resumption.resumption.rel_resumption (A OO C) (B OO D)"
inj_map: fixes f :: "'c \<Rightarrow> 'd" assumes "inj f" shows "inj (map_option f)"
fmdom'I: fixes m :: "('d, 'c) fmap"   and x :: "'d"   and y :: "'c" assumes "fmlookup m x = Some y" shows "x \<in> fmdom' m"
kbo_as_wpo: fixes t :: "('f, 'a) Term.term"   and s :: "('f, 'a) Term.term" assumes "t \<in> {t, t \<cdot> \<sigma>}" shows "kbo s t = local.wpo.wpo s t"
finite_ordLess_infinite: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set" assumes "Well_order r"   and "Well_order r'"   and "finite (Field r)"   and "infinite (Field r')" shows "r <o r'"
max_power_distrib_right: fixes a :: "'a"   and b :: "nat"   and c :: "nat" assumes "(1::'a) < a" shows "max (a ^ b) (a ^ c) = a ^ max b c"
consume_weaken_type: fixes tn :: "t list"   and cons :: "ct list"   and t_int :: "t list"   and ts :: "t list" assumes "consume (Type tn) cons = Type t_int" shows "consume (Type (ts @ tn)) cons = Type (ts @ t_int)"
BotE: fixes \<Gamma> :: "'a formula set"   and F :: "'a formula" assumes "\<Gamma> \<turnstile> \<bottom>" shows "\<Gamma> \<turnstile> F"
substitute''_Nil: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set" shows "substitute'' f T [] []"
branch_prob_cexpr_def: shows "branch_prob_cexpr \<equiv> \<lambda>(vs, vs', \<Gamma>, \<delta>). integrate_vars \<Gamma> vs \<delta>"
permutation_inverse_works_2: fixes p :: "'a \<Rightarrow> 'a" assumes "permutation p" shows "p \<circ> inv p = id"
pn_atoms_atoms: fixes F :: "'b formula" shows "atoms F = fst (pn_atoms F) \<union> snd (pn_atoms F)"
assms: fixes x :: "'a" shows "f (x, source \<Delta>) = 0"
par_col2_par_bis: fixes E :: "'p"   and F :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "E \<noteq> F"   and "A B Par C D"   and "Col E F C"   and "Col E F D" shows "A B Par E F"
filter_pdevs_def: shows "filter_pdevs \<equiv> map_fun id (map_fun pdevs_apply Abs_pdev) filter_pdevs_raw"
quicksort_post_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'c"   and lo :: "nat"   and hi :: "nat"   and xs :: "'d list"   and xs' :: "'d list" shows "quicksort_post R h lo hi xs xs' \<equiv> mset xs' = mset xs \<and> sorted_sublist_map R h xs' lo hi \<and> (\<forall>i<lo. xs' ! i = xs ! i) \<and> (\<forall>j. hi < j \<and> j < length xs \<longrightarrow> xs' ! j = xs ! j)"
simps_4: fixes e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" shows "coste (Add e\<^sub>1 e\<^sub>2) = 1 + coste e\<^sub>1 + coste e\<^sub>2"
conga_trans: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and A'' :: "'a"   and B'' :: "'a"   and C'' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Tarski_neutral_dimensionless.CongA Bet Cong A' B' C' A'' B'' C''" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A'' B'' C''"
invariance_of_dimension_affine_sets: fixes U :: "'a set"   and S :: "'a set"   and V :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "openin (top_of_set U) S"   and "affine U"   and "affine V"   and "continuous_on S f"   and "f ` S \<subseteq> V"   and "inj_on f S"   and "S \<noteq> {}" shows "aff_dim U \<le> aff_dim V"
of_int_0_less_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (ring_1.of_int one plus zero minus z) = (0 < z)"
last: shows "(ts'', m'', \<S>'') \<Rightarrow>\<^sub>d\<^sup>* (ts', m', \<S>')"
line_extension_consistent: fixes K :: "'a set" assumes "subring K R" shows "ring.line_extension (R\<lparr>carrier := K\<rparr>) = line_extension"
R_def: shows "R = cbox (case (real_of_int l - \<delta>, snd p\<^sub>0) of (x, y) \<Rightarrow> (x, real_of_int y)) (real_of_int l + \<delta>, real_of_int (snd p\<^sub>0) + \<delta>)"
simps_1: fixes \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option" shows "ptrm_infer_type \<Gamma> PUnit = Some TUnit"
A: shows "ipurge_tr_aux I D {u} xs = ipurge_tr I D u xs"
transitive_asymmetric_irreflexive: fixes x :: "'a" assumes "x \<sqinter> x \<le> x" shows "(x \<sqinter> id x = bot) = (x \<le> - top)"
subseteq_qle_def: fixes M :: "'a set"   and N :: "'a set" shows "M \<subseteq>\<^sub>\<preceq> N \<equiv> \<forall>x\<in>M. x \<in>\<^sub>\<preceq> N"
prems_8: shows "\<forall>V\<in>rv S (CFG_node m). state_val s V = state_val s' V"
top_ennreal_def: shows "top \<equiv> e2ennreal top"
r_leap_total: fixes t :: "nat"   and i :: "nat"   and x :: "nat" shows "eval r_leap [t, i, x] \<down>"
le_imp_bigo_real: fixes c :: "real"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "0 \<le> c"   and "\<forall>\<^sub>F x in F. f x \<le> c * g x"   and "\<forall>\<^sub>F x in F. 0 \<le> f x" shows "f \<in> O[F](g)"
map_add_assoc: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option"   and m3 :: "'b \<Rightarrow> 'c option" shows "m3 \<Oplus> m2 \<Oplus> m1 = m3 \<Oplus> (m2 \<Oplus> m1)"
map_reindex: fixes l :: "'a list"   and g :: "'a \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b" assumes "\<forall>i<length l. g (l ! i) = f i" shows "map f [0..<length l] = map g l"
order_gt_0_iff_finite: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G" shows "(0 < order G) = finite (carrier G)"
dvd_times_left_cancel_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> (0::'a)" shows "(a * b dvd a * c) = (b dvd c)"
2: shows "Dom (g \<cdot> f) \<subseteq> Cod (g \<cdot> f)"
assign_annos_eval: fixes x :: "'Var"   and e :: "'AExp"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>(x \<leftarrow> e) \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>Stop, mds \<oplus> annos, mem(x := ev\<^sub>A mem e)\<rangle>"
hull_drop_one: fixes G :: "'b list set" shows "\<langle>G\<rangle> = \<langle>G\<^sub>+\<rangle>"
\<theta>2: shows "unify (M' \<lhd> \<theta>1) (N' \<lhd> \<theta>1) = Some \<theta>2"
gcsrp_tassoc: fixes N :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "Group G"   and "G \<triangleright> N"   and "a \<in> gcsrp_map G N ` set_rcs G N"   and "b \<in> gcsrp_map G N ` set_rcs G N"   and "c \<in> gcsrp_map G N ` set_rcs G N" shows "gcsrp_top G N (gcsrp_top G N a b) c = gcsrp_top G N a (gcsrp_top G N b c)"
tanh_minus: fixes x :: "'a" shows "tanh (- x) = - tanh x"
cINF_lower: fixes f :: "'b \<Rightarrow> 'a multiset"   and A :: "'b set"   and x :: "'b" assumes "subset_mset.bdd_below (f ` A)"   and "x \<in> A" shows "Inf (f ` A) \<subseteq># f x"
arthan_wlog: fixes xs :: "'b list"   and x :: "'b"   and y :: "'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> bool" assumes "xs \<noteq> []"   and "last xs \<noteq> (0::'b)"   and "x + y + sum_list xs = (0::'b)"   and "\<And>x y xs. \<lbrakk>xs \<noteq> []; last xs \<noteq> (0::'b); x + y + sum_list xs = (0::'b); (0::'b) \<le> x\<rbrakk> \<Longrightarrow> P x y xs"   and "\<And>x y xs. \<lbrakk>xs \<noteq> []; P x y xs\<rbrakk> \<Longrightarrow> P (- x) (- y) (map uminus xs)" shows "P x y xs"
expand: fixes llist :: "'c llist"   and llist' :: "'c llist" assumes "lnull llist = lnull llist'"   and "\<lbrakk>\<not> lnull llist; \<not> lnull llist'\<rbrakk> \<Longrightarrow> lhd llist = lhd llist' \<and> ltl llist = ltl llist'" shows "llist = llist'"
sminus_def: fixes v :: "'t"   and t :: "'a" shows "v \<ominus> t = term_of_pair (pp_of_term v - t, component_of_term v)"
rtranclp_tranclp_absorb: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "R\<^sup>*\<^sup>*\<^sup>+\<^sup>+ = R\<^sup>*\<^sup>*"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and R :: "('procS, 'procT) Proc" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q"   and "Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> R" shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> R"
card_insert_ge: fixes A :: "'a set"   and x :: "'a" shows "card A \<le> card (insert x A)"
get_scdom_component_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "ptr' |\<in>| object_ptr_kinds h"
correctCompositionDiffLevelsS4opt: shows "correctCompositionDiffLevels sS4opt"
lbounds_def: fixes as :: "atom list" shows "lbounds as = concat (map (\<lambda>x. case x of Less 0 nat2 \<Rightarrow> [] | Less (Suc i) 0 \<Rightarrow> [i] | Less (Suc i) (Suc nata) \<Rightarrow> [] | Eq nat1 nat2 \<Rightarrow> []) as)"
map_map_filter: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b option"   and xs :: "'c list" shows "map f (List.map_filter g xs) = List.map_filter (map_option f \<circ> g) xs"
extremum_uniqueI: fixes a :: "'b" assumes "top \<le> a" shows "a = top"
bind_refine': fixes M :: "'c nres"   and R' :: "('c \<times> 'd) set"   and M' :: "'d nres"   and f :: "'c \<Rightarrow> 'e nres"   and R :: "('e \<times> 'f) set"   and f' :: "'d \<Rightarrow> 'f nres" assumes "M \<le> \<Down> R' M'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; inres M x; inres M' x'; nofail M; nofail M'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "M \<bind> f \<le> \<Down> R (M' \<bind> f')"
map_ident: fixes t :: "'b option" shows "map_option (\<lambda>x. x) t = t"
self_append_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs = xs @ ys) = (ys = [])"
divide_nonpos_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less zero y" shows "less_eq (divide x y) zero"
f_Exec_Stream_Init_prefix: fixes xs :: "'d list"   and ys :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "prefix xs ys" shows "prefix (f_Exec_Comp_Stream_Init trans_fun xs c) (f_Exec_Comp_Stream_Init trans_fun ys c)"
W1_nneg: fixes I :: "nat set" assumes "I \<in> P" shows "0 \<le> W cf1 I"
map_option_bind: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'e option"   and g :: "'e \<Rightarrow> 'd option" shows "map_option f (x \<bind> g) = x \<bind> map_option f \<circ> g"
assms_6: shows "decisionVars \<subseteq> vars (elements (getM state))"
map_of_filter_in: fixes xs :: "('d \<times> 'c) list"   and k :: "'d"   and z :: "'c"   and P :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "map_of xs k = Some z"   and "P k z" shows "map_of (filter (\<lambda>(x, y). P x y) xs) k = Some z"
simps_3: fixes c1 :: "('a, 'b, 'c) cmd"   and c2 :: "('a, 'b, 'c) cmd" shows "proper (c1 ;; c2) = (proper c1 \<and> proper c2)"
rpre_def: shows "rpre \<equiv> \<lambda>S (V, v). v \<in> local.reachable - V \<and> V \<subseteq> local.reachable \<and> S \<subseteq> V \<and> tgt \<notin> V \<and> E `` (V - S) \<subseteq> V"
less: fixes t :: "real" assumes "t0 < t"   and "t < t0 + e" shows "v t < w t"
prems: shows "is_tangle_diagram (w1 * ws1) \<and> is_tangle_diagram ws2 \<and> domain_wall ws2 = codomain_wall (w1 * ws1)"
T_vrestr_composed_3: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and m :: "nat" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.vimage_restr M (T ^^ m) (qmpt.vimage_restr M T A) = qmpt.vimage_restr M (T ^^ (m + 1)) A"
simple_match: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw"   and "StableMarriage.simple Pm Pw \<mu>" shows "StableMarriage.match Pm Pw \<mu>"
PPsD: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and X :: "'a set" assumes "t \<in> .[X]" shows "keys t \<subseteq> X"
dual_space_def: fixes S :: "'c set" shows "dual_space S = {E. linear_fun_on S E \<and> extensional0 S E}"
discovered_not_stack_imp_finished: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and x :: "'c" assumes "DFS_invar G param s"   and "x \<in> dom (discovered s)"   and "x \<notin> set (stack s)" shows "x \<in> dom (finished s)"
k_frontier_def: fixes a :: "'a" shows "SPRsingle.k_frontier a \<equiv> map (local.spr_simInit a \<circ> envObs a) envInit"
DERIV_pos_inc_right: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real" assumes "(f has_real_derivative l) (at x)"   and "0 < l" shows "\<exists>d>0. \<forall>h>0. h < d \<longrightarrow> f x < f (x + h)"
get_owner_document_disconnected_nodes: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "known_ptrs h"   and "type_wf h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node_ptr \<in> set disc_nodes" shows "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr) \<rightarrow>\<^sub>r document_ptr"
wrap_iso_arcs_iso_arcs: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "x \<in> arcs G" shows "wrap_iso_arcs hom f (iso_arcs hom x) = iso_arcs hom (f x)"
x_prop: shows "\<forall>y. a * y\<^sup>2 + b * y + c = 0 \<longrightarrow> x < y"
vfsequence_vrange_vproduct: fixes A :: "V \<Rightarrow> V" assumes "\<And>i. i \<in>\<^sub>\<circ> vcard xs \<Longrightarrow> xs\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i" shows "xs \<in>\<^sub>\<circ> vproduct (vcard xs) A"
inf_commute: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
ev_signal_precondition_def: fixes tid :: "thread_id_t"   and partner :: "thread_id_t"   and s :: "state_t" shows "ev_signal_precondition tid partner s \<equiv> sp_impl_subj_subj s (Step.partition tid) (Step.partition partner)"
assms_3: shows "p \<cdot> s \<cdot> x \<le> x \<cdot> p"
complement_blocks_def: shows "\<B>\<^sup>C \<equiv> image_mset fin.block_complement \<B>"
funk: fixes s :: "char list \<Rightarrow> int" assumes "\<up> (P s) + enat n < \<infinity>" shows "\<exists>s' m. (c, s) \<Rightarrow> m \<Down> s' \<and> enat m + enat k * \<up> (Q s') \<le> enat k * (\<up> (P s) + enat n)"
polynomial_function_minus: fixes f :: "'c \<Rightarrow> 'd" assumes "polynomial_function f" shows "polynomial_function (\<lambda>x. - f x)"
intro\<^sub>1: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'" shows "\<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>"
S'_def: fixes i :: "'a" shows "S' i = {s \<in> S i. i = index s}"
set_partition_on_insert_with_fixed_card_eq: fixes A :: "'b set"   and a :: "'b"   and k :: "nat" assumes "finite A"   and "a \<notin> A" shows "{P. partition_on (insert a A) P \<and> card P = Suc k} = ({P. partition_on A P \<and> card P = Suc k} \<bind> (\<lambda>P. P \<bind> (\<lambda>p. {insert (insert a p) (P - {p})}))) \<union> ({P. partition_on A P \<and> card P = k} \<bind> (\<lambda>P. {insert {a} P}))"
set_intros_2: fixes x22 :: "'a"   and x21 :: "nat" shows "x22 \<in> set_atom (Geq x21 x22)"
progress_L0_trap: fixes cs :: "e list"   and es :: "e list"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "const_list cs"   and "cs \<noteq> [] \<or> es \<noteq> []" shows "\<exists>a. \<lparr>s;vs;cs @ [Trap] @ es\<rparr> \<leadsto>_ i \<lparr>s;vs;[Trap]\<rparr>"
bounded_check_eqv_soundness: fixes \<AA> :: "interp"   and idx :: "idx"   and \<phi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and \<psi> :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" assumes "#\<^sub>V \<AA> = idx"   and "WS1S.bounded.check_eqv idx \<phi> \<psi>" shows "WS1S.sat\<^sub>b \<AA> \<phi> = WS1S.sat\<^sub>b \<AA> \<psi>"
lookup_edom: fixes m :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "m x \<noteq> \<bottom>" shows "x \<in> edom m"
fm_def: shows "fm = (\<lambda>x. ennreal (- f x))"
assms_3: fixes s :: "'s"   and a :: "'a" shows "0 \<le> P s a"
reduce_column_aux: fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P_init :: "'a mat"   and K_init :: "'a mat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "nat list" assumes "H \<in> carrier_mat m n"   and "P_init \<in> carrier_mat m m"   and "K_init \<in> carrier_mat m n"   and "P_init * H = K_init"   and "(P, K) = local.reduce_column_aux div_op xs H (P_init, K_init)"   and "0 < m"   and "invertible_mat P_init"   and "0 \<notin> set xs" shows "P \<in> carrier_mat m m \<and> K \<in> carrier_mat m n \<and> P * H = K \<and> invertible_mat P"
is_natural: fixes f :: "'a" assumes "A.arr f" shows "t (A.cod f) \<cdot>\<^sub>B F f = G f \<cdot>\<^sub>B t (A.dom f)"
iMinI_ex2: fixes I :: "'a set" assumes "I \<noteq> {}" shows "iMin I \<in> I"
cases: fixes a1 :: "trm"   and a2 :: "trm" assumes "a1 \<mapsto> a2" obtains s :: "trm" and s' :: "trm" and t :: "trm" where "a1 = App s__ ta__" and "a2 = App s'__ ta__" and "s__ \<mapsto> s'__"   | t :: "trm" and t' :: "trm" and s :: "trm" where "a1 = App sa__ tb__" and "a2 = App sa__ t'__" and "tb__ \<mapsto> t'__"   | x :: "name" and s :: "trm" and t :: "trm" where "a1 = App (\<Lambda> x__ . tc__) sb__" and "a2 = tc__[x__::=sb__]" and "x__ \<sharp> sb__"   | t :: "trm" and t' :: "trm" and x :: "name" where "a1 = \<Lambda> xa__ . td__" and "a2 = \<Lambda> xa__ . t'a__" and "td__ \<mapsto> t'a__"   | x :: "name" and t :: "trm" where "a1 = \<Lambda> xb__ . App te__ (Var xb__)" and "a2 = te__" and "xb__ \<sharp> te__"   | x :: "name" and s :: "trm" and s' :: "trm" and t :: "trm" where "a1 = sc__ to xc__ in tf__" and "a2 = s'a__ to xc__ in tf__" and "xc__ \<sharp> (sc__, s'a__)" and "sc__ \<mapsto> s'a__"   | x :: "name" and s :: "trm" and t :: "trm" and t' :: "trm" where "a1 = sd__ to xd__ in tg__" and "a2 = sd__ to xd__ in t'" and "xd__ \<sharp> sd__" and "tg__ \<mapsto> t'"   | x :: "name" and s :: "trm" and t :: "trm" where "a1 = [se__] to xe__ in th__" and "a2 = th__[xe__::=se__]" and "xe__ \<sharp> se__"   | x :: "name" and s :: "trm" where "a1 = sf__ to xf__ in [Var xf__]" and "a2 = sf__" and "xf__ \<sharp> sf__"   | x :: "name" and y :: "name" and s :: "trm" and u :: "trm" and t :: "trm" where "a1 = (sg__ to x in t) to y in u" and "a2 = sg__ to x in t to y in u" and "x \<sharp> (y, sg__, u)" and "y \<sharp> (sg__, t)"   | s :: "trm" and s' :: "trm" where "a1 = [s]" and "a2 = [s']" and "s \<mapsto> s'"