hcju/isabelleps huggingface dataset:
step: fixes x :: "'b"   and xs :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and z :: "'b" assumes "x \<notin> set xs"   and "r x y"   and "rtrancl_tab r (x # xs) y z" shows "rtrancl_tab r xs x z"
hcomp_simps\<^sub>W\<^sub>C_3: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<nu> :: "'c"   and \<mu> :: "'c" assumes "weak_composition V H"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "partial_magma.cod V (H \<nu> \<mu>) = H (partial_magma.cod V \<nu>) (partial_magma.cod V \<mu>)"
not_nonzero_Qp: fixes x :: "'a"   and Q\<^sub>p :: "('a, 'b) ring_scheme" assumes "x \<in> carrier Q\<^sub>p"   and "x \<notin> nonzero Q\<^sub>p" shows "x = \<zero>\<^bsub>Q\<^sub>p\<^esub>"
landau_product_preprocess_15: fixes m :: "num"   and n :: "num" shows "num.Bit0 m * num.Bit0 n = num.Bit0 (num.Bit0 (m * n))"
C5_b: fixes x :: "'a"   and y :: "'a" shows "-l x r\<rightarrow> -l y = y l\<rightarrow> x"
SentP_I: fixes A :: "fm" assumes "A \<in> boolean_axioms" shows "{} \<turnstile> SentP \<guillemotleft>A\<guillemotright>"
a_translate_functions: fixes R :: "('a, 'b) ring_scheme"   and c :: "'a" assumes "U_function_ring R"   and "c \<in> carrier R" shows "U_function_ring.a_translate R c \<in> carrier (function_ring (carrier R) R)"
prob_space_axioms: shows "prob_space (Pi\<^sub>M I M)"
gtrE: fixes c :: "'c multiset"   and e :: "'d"   and c' :: "'e multiset"   and T :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'e \<times> 'e multiset) set" assumes "(c, e, c') \<in> gtr T" obtains s :: "'c" and ce :: "'c multiset" and s' :: "'e" and ce' :: "'e multiset" where "c = add_mset s ce" and "c' = add_mset s' ce'" and "((s, ce), e, s', ce') \<in> T"
Units_r_inv_ex: fixes x :: "'a" assumes "x \<in> Units (units_group K)" shows "\<exists>y\<in>carrier (units_group K). x \<otimes>\<^bsub>units_group K\<^esub> y = \<one>\<^bsub>units_group K\<^esub>"
power_mult_numeral: fixes a :: "'a"   and m :: "num"   and n :: "num" shows "(a ^ numeral m) ^ numeral n = a ^ numeral (m \<cdot> n)"
some_trg_in_targets: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_trg \<mu> \<in> targets \<mu>"
isos_compose: fixes f :: "'a"   and f' :: "'a" assumes "C.iso f"   and "C.iso f'"   and "C.seq f' f" shows "C.iso (f' \<cdot> f)"
nat_abs_summable_on_comparison_test: fixes g :: "nat \<Rightarrow> real"   and I :: "nat set"   and N :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "g abs_summable_on I"   and "\<And>n. \<lbrakk>N \<le> n; n \<in> I\<rbrakk> \<Longrightarrow> norm (f n) \<le> g n" shows "f abs_summable_on I"
IH_1: assumes "is_cupcake_all_env env"   and "is_cupcake_exp e" shows "s1 = fst (s2, Rval v1)"
reachable_reaches: fixes a :: "'a"   and b :: "'a" assumes "reachable a"   and "a \<rightarrow>* b" shows "reachable b"
connectedin_iff_connected: fixes S :: "'b set" shows "connectedin euclidean S = connected S"
ttree_restr_nxt_subset: fixes x :: "'b"   and S :: "'b set"   and t :: "'b ttree" assumes "x \<in> S" shows "paths (ttree_restr S (nxt t x)) \<subseteq> paths (nxt (ttree_restr S t) x)"
Collect_empty_eq: fixes P :: "'a \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"
rtNode_1: fixes p :: "ref"   and no :: "ref"   and not :: "dag" assumes "Dag p low high lt"   and "Dag no low high not"   and "no \<in> set_of lt" shows "not \<le> lt"
SubgroupD1: fixes H :: "'g set" assumes "Subgroup H" shows "Group H"
cinner_simps_6: fixes x :: "'c"   and r :: "complex"   and y :: "'c" shows "\<langle>x, r *\<^sub>C y\<rangle> = r * \<langle>x, y\<rangle>"
pos_divideC_less_eq: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 c :: "complex"   and b :: "'b"   and a :: "'b" assumes "class.ordered_complex_vector scaleR scaleC plus zero minus uminus less_eq less"   and "0 < c" shows "less (scaleC (inverse c) b) a = less b (scaleC c a)"
exhaust_sel: fixes prod :: "'a \<times> 'b" obtains "prod = (fst prod, snd prod)"
is_extensional: fixes f :: "'d" assumes "\<not> D.arr f" shows "\<epsilon>'F_F\<eta>.map f = C.null"
intersect_num_commute: fixes b1 :: "'a set"   and b2 :: "'a set" shows "b1 |\<inter>| b2 = b2 |\<inter>| b1"
closed_sequentially: fixes s :: "'d set"   and f :: "nat \<Rightarrow> 'd"   and l :: "'d" assumes "closed s"   and "\<forall>n. f n \<in> s"   and "f \<longlonglongrightarrow> l" shows "l \<in> s"
retraction_is_epi: fixes g :: "'a \<times> 'a" assumes "CC.retraction g" shows "CC.epi g"
eval_conjoin: fixes e :: "nat \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f list \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f list \<Rightarrow> bool"   and G :: "('d, 'e) form list" shows "list_all (eval e f g) G = eval e f g (conjoin G)"
pred_sum_inject_1: fixes P1 :: "'b \<Rightarrow> bool"   and P2 :: "'c \<Rightarrow> bool"   and a :: "'b" shows "pred_sum P1 P2 (Inl a) = P1 a"
fin_dim_def: shows "V.fin_dim = (\<exists>A. finite A \<and> A \<subseteq> carrier local.V \<and> V.gen_set A)"
path_connected_complement_homeomorphic_interval: fixes S :: "'c set"   and a :: "'d"   and b :: "'d" assumes "S homeomorphic cbox a b"   and "2 \<le> DIM('c)" shows "path_connected (- S)"
nths_reencode_eq_comp: fixes B :: "nat set"   and A :: "nat set"   and xs :: "'a list" assumes "B \<subseteq> A" shows "nths (nths xs A) (- ind_in_set A ` B) = nths xs (A - B)"
a_group: shows "Group.group \<lparr>carrier = UNIV, monoid.mult = (+), one = 0, \<dots> = undefined\<rparr>"
assms: shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> B"
retraction_def: fixes g :: "'c" shows "C.retraction g \<equiv> \<exists>f. C.ide (g \<cdot>\<^sub>C f)"
simps: fixes i :: "nat"   and j :: "nat" shows "gs.\<mu> i j = (if j < i then map of_int_hom.vec_hom fs_init ! i \<bullet> gs.gso j / \<parallel>gs.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> xs" shows "xs\<lparr>a\<rparr> = b"
a: shows "<a>:M \<in> ANDRIGHT (B AND C) (\<parallel><B>\<parallel>) (\<parallel><C>\<parallel>)"
app_def: fixes s :: "('a, 'b) FingerTree"   and t :: "('a, 'b) FingerTree" shows "FingerTree.app s t = Abs_FingerTree (FingerTreeStruc.app (Rep_FingerTree s) (Rep_FingerTree t))"
join_ti_1nempty: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'a list"   and x :: "nat" assumes "\<forall>i. 0 < i \<and> i < Suc n \<longrightarrow> s (x + i) = []" shows "join_ti s x n = s x"
exprel_imp_eq_freevars: fixes U :: "freeExp"   and V :: "freeExp" assumes "U \<sim> V" shows "freevars U = freevars V"
p'_zero: shows "delayed_safe_distance.other.p' 0 = other.s' \<delta>"
b'_insertion: shows "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0)) = b'"
subspace_add: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "m1.subspace S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
ennreal_inverse_eq_0_iff: fixes a :: "ennreal" shows "(inverse a = 0) = (a = \<top>)"
insert_subsetI: fixes x :: "'b"   and A :: "'b set"   and X :: "'b set" assumes "x \<in> A"   and "X \<subseteq> A" shows "x \<triangleright> X \<subseteq> A"
set_trie_def: fixes t :: "trie" shows "set_trie t = {xs. isin t xs}"
assms: fixes \<pi> :: "('state \<times> 'aprop set) stream"   and \<pi>' :: "('state \<times> 'aprop set) stream" assumes "wfp AP' \<pi>"   and "wfp AP' \<pi>'"   and "\<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l)"   and "\<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0"   and "stateOf \<pi>' = stateOf \<pi>" shows "F \<pi> \<pi>' \<pi>l"
borel_measurable_add: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
equal: fixes equal :: "'m \<Rightarrow> 'm \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
to_fract_0: shows "to_fract (0::'b) = 0"
vp_split_2: fixes n :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as @ a # as'\<rightarrow>\<^sub>\<surd>* n'" shows "valid_edge a"
axioms_3: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'" shows "ring R' addition' multiplication' zero' unit'"
S_relator_list: fixes s :: "'w" assumes "s \<in> S" shows "pair_relator_list s s = [s, s]"
part_UN: fixes N :: "'a set"   and I :: "'a \<Rightarrow> 'b set"   and P :: "'a \<Rightarrow> 'b set set" assumes "\<And>n. n \<in> N \<Longrightarrow> part (I n) (P n)"   and "\<And>n1 n2. {n1, n2} \<subseteq> N \<and> n1 \<noteq> n2 \<Longrightarrow> I n1 \<inter> I n2 = {}" shows "part (\<Union> (I ` N)) (\<Union> (P ` N))"
A: fixes u :: "'a"   and w :: "'a" shows "u \<in> sinks_aux I D (U \<union> V) xs \<longrightarrow> (u, w) \<in> I \<longrightarrow> \<not> (\<exists>v\<in>sinks_aux I D V xs. (v, w) \<in> I) \<longrightarrow> u \<in> sinks_aux I D U (ipurge_tr_aux I D V xs)"
l4_2: 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 D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.IFSC Bet Cong A B C D A' B' C' D'" shows "Cong B D B' D'"
rbt_comp_insert_with_key: shows "local.rbt_comp_insert_with_key = ord.rbt_insert_with_key (lt_of_comp c)"
dom_fmlookup: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "dom (fmlookup m) = fmdom' m"
hyps_2: shows "map_poly f (p * q) = map_poly f p * map_poly f q"
exec_assign_global: fixes \<sigma> :: "'a control_state_scheme"   and upd :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and rhs :: "'a control_state_scheme \<Rightarrow> 'b"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> _bind_SE _ (assign_global upd rhs) M) = (\<sigma> \<Turnstile> M)"
pp_eqI: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" assumes "\<And>u. lookup_pp s u = lookup_pp t u" shows "s = t"
snd_sum: fixes f :: "'c \<Rightarrow> 'b \<times> 'a"   and A :: "'c set" shows "snd (sum f A) = (\<Sum>x\<in>A. snd (f x))"
Domain_converse: fixes r :: "('c \<times> 'b) set" shows "Domain (r\<inverse>) = Range r"
bin_exhaust: fixes bin :: "int" obtains x :: "int" and b :: "bool" where "bin = x BIT b"
inf_ordering_inj': fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "infinite X"   and "TernaryOrdering.ordering f ord X"   and "\<forall>a b c. ord a b c \<longrightarrow> a \<noteq> b \<and> a \<noteq> c \<and> b \<noteq> c"   and "f m = f n" shows "m = n"
assms_2: shows "\<delta>\<^sub>R' = dist_code p\<^sub>0\<^sub>R p\<^sub>1\<^sub>R"
seq_stop_eval\<^sub>w: fixes c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat" shows "eval_abv\<^sub>C \<langle>Stop ;; c', mds, mem\<rangle>\<^sub>C \<langle>c', mds, mem\<rangle>\<^sub>C"
simps_6: fixes v :: "'b" shows "sup dSUCCEEDi (dRETURN v) = dRETURN v"
bij_betw_combine: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and C :: "'c set"   and D :: "'d set" assumes "bij_betw f A B"   and "bij_betw f C D"   and "B \<inter> D = {}" shows "bij_betw f (A \<union> C) (B \<union> D)"
mod_minus_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo a (uminus b) = uminus (modulo (uminus a) b)"
UQ: shows "P = (\<lambda>x. Elem x U \<longrightarrow> Q x)"
lim_imp_Liminf: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and f0 :: "'b" assumes "F \<noteq> \<bottom>"   and "(f \<longlongrightarrow> f0) F" shows "Liminf F f = f0"
part_equiv_trans: fixes R :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "part_equiv R"   and "(x, y) \<in> R"   and "(y, z) \<in> R" shows "(x, z) \<in> R"
direct_decomp_split_map: fixes A :: "'a set"   and f :: "'b \<Rightarrow> 'a set"   and ss :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "direct_decomp A (map f ss)" shows "direct_decomp A (map f (filter P ss) @ map f (filter (- P) ss))"
qfN_oddprime_cube: fixes p :: "int"   and N :: "int"   and q :: "int" assumes "prime (p\<^sup>2 + N * q\<^sup>2)"   and "odd (p\<^sup>2 + N * q\<^sup>2)"   and "p \<noteq> 0"   and "1 \<le> N" shows "\<exists>a b. (p\<^sup>2 + N * q\<^sup>2) ^ 3 = a\<^sup>2 + N * b\<^sup>2 \<and> coprime a (N * b)"
minus_closed: fixes G :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<ominus>\<^bsub>G\<^esub> y \<in> carrier G"
card_of_Un_infinite: fixes A :: "'a set"   and B :: "'a set" assumes "infinite A"   and "|B| \<le>o |A|" shows "|A \<union> B| =o |A| \<and> |B \<union> A| =o |A|"
2: shows "dom subs' \<subseteq> tvs f \<union> tvs u \<union> dom subs"
le: shows "(\<Sum>i | i dvd order (mult_of R). card {a \<in> carrier (mult_of R). G.ord a = i}) \<le> (\<Sum>i | i dvd order (mult_of R). phi' i)"
intFF_def: shows "M.M.intFF \<equiv> local.M.intFF"
return_writer_def: fixes return :: "'e \<times> 'd list \<Rightarrow> 'f"   and x :: "'e" shows "return_writer return x = WriterT (return (x, []))"
prj_simps_1: shows "C.seq \<mu>\<nu>.prj\<^sub>1 \<p>\<^sub>1[\<nu>.leg0 \<cdot> \<mu>\<nu>.prj\<^sub>0, \<pi>\<rho>.leg1]"
uniformity_QuantT_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
mult_Suc_add_less: fixes \<alpha> :: "V"   and \<gamma> :: "V"   and \<beta> :: "V"   and m :: "nat" assumes "\<alpha> < \<gamma>"   and "\<beta> < \<gamma>"   and "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "\<gamma> * ord_of_nat m + \<alpha> < \<gamma> * ord_of_nat (Suc m) + \<beta>"
B: shows "B \<equiv> fst (echelon_form_of_column_k bezout (A, i) k)"
homotopic_paths_refl: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a" shows "homotopic_paths s p p = (path p \<and> path_image p \<subseteq> s)"
wf_TVarB: fixes \<Gamma> :: "binding list"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f" shows "TVarB T \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f"
update_def: fixes f :: "('c \<Rightarrow> 'd list) \<times> (('d list \<Rightarrow> 'd list) \<Rightarrow> 'c \<Rightarrow> 'c)"   and upd :: "'d \<Rightarrow> 'd" shows "update f upd \<equiv> snd f (\<lambda>l. upd (hd l) # tl l)"
finite_fv: shows "finite (fv\<^sub>s\<^sub>t (S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S'))"
Nil3: fixes xs :: "'c list"   and ys :: "'a list" assumes "as = xs"   and "bs = ys"   and "cs = []" shows "P"
sphere_rotate: fixes c :: "complex"   and x :: "real"   and r :: "real" shows "(\<lambda>z. c + cis x * (z - c)) ` sphere c r = sphere c r"
subindex_permutation2: fixes ss :: "nat list"   and ls :: "'b list" assumes "distinct ss"   and "set ss \<subseteq> {..<length ls}" obtains f :: "nat \<Rightarrow> nat" where "f permutes {..<length ls}" and "ls = permute_list f (map ((!) ls) (filter (\<lambda>i. i \<notin> set ss) [0..<length ls]) @ map ((!) ls) ss)"
cooper_def: fixes p :: "fm" shows "cooper p = (let (q, B, d) = unit p; js = [1..d]; mq = simpfm (minusinf q); md = evaldjf (\<lambda>j. simpfm (subst0 (C j) mq)) js in if md = T then T else let qd = evaldjf (\<lambda>(b, j). simpfm (subst0 (Add b (C j)) q)) (concat (map (\<lambda>b. map (Pair b) js) B)) in decr (Cooper.disj md qd))"
minus_flip: fixes a :: "'a"   and b :: "'a" shows "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
S2P_Collect: fixes \<phi> :: "'b \<Rightarrow> bool" shows "S2P (Collect \<phi>) = \<phi>"
antimonoI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f y \<le> f x" shows "order.antimono less_eq f"
step: fixes m0 :: "'a nres"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres"   and x :: "'a"   and Y :: "'a set"   and y :: "'a" assumes "rwof m0 cond step x"   and "cond x"   and "step x = RES Y"   and "y \<in> Y" shows "rwof m0 cond step y"
is_Inf_Meet: fixes A :: "'b set" shows "is_Inf A (\<Sqinter>A)"
map_simp: fixes Fg :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2" assumes "A2_BxA2.arr Fg" shows "E.map Fg = A2_B.Map (fst Fg) (snd Fg)"
get_child_nodes_new_element: fixes ptr' :: "(_) object_ptr"   and new_element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^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 new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "r \<in> get_child_nodes_locs ptr'" shows "r h h'"
I': fixes t :: "real"   and x :: "real" assumes "t \<in> {0..x} - D" shows "(I has_vector_derivative f t) (at t within {0..x} - D)"
simps_1: fixes f :: "'c \<Rightarrow> 'b option" shows "traverse f [] = Some []"
univ_is_ce: shows "univ_ce \<in> ce_sets"
chain: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> \<F> \<and> T \<in> \<F>" shows "S \<subseteq> T \<or> T \<subseteq> S"
IH: assumes "i < card S"   and "wellorder_class.enumerate S i < x" shows "wellorder_class.enumerate (S - {x}) i = wellorder_class.enumerate S i"
assms_4: fixes U :: "'a set" assumes "openin (top_of_set S) U" shows "openin (top_of_set T) (f ` U)"
flow_solves_ode: fixes x0 :: "'a" assumes "t\<^sub>0 \<in> T"   and "x0 \<in> S" shows "(flow t\<^sub>0 x0 solves_ode f) (ex_ivl x0) S"
hequiv_refl': fixes ps :: "('b, 'c) fm list"   and i :: "'c"   and H :: "(('b, 'c) fm list \<times> 'c) set" assumes "(ps, i) \<in> H" shows "hequiv H i i"
Apps_append: fixes f :: "expr"   and xs :: "expr list"   and ys :: "expr list" shows "f \<bullet> (xs @ ys) = f \<bullet> xs \<bullet> ys"
cr_Bonk_Schramm_extension_def: shows "cr_Bonk_Schramm_extension \<equiv> \<lambda>x y. (x \<in> extended_distance_set \<and> x \<in> extended_distance_set \<and> extended_distance x x = 0) \<and> abs_Bonk_Schramm_extension x = y"
well_def: shows "\<forall>q. List.member index_list q \<longrightarrow> q < length arb_list"
relpow_imp_rtrancl: fixes p :: "'b \<times> 'b"   and n :: "nat"   and R :: "('b \<times> 'b) set" assumes "p \<in> R ^^ n" shows "p \<in> R\<^sup>*"
that: fixes y :: "'a" assumes "y \<in> C"   and "eSuc (esize x) \<le> esize y" shows "thesis"
ip_cidr_intersect: fixes b2 :: "'b word"   and m2 :: "nat"   and b1 :: "'b word"   and m1 :: "nat" assumes "\<not> ipset_from_cidr b2 m2 \<subseteq> ipset_from_cidr b1 m1"   and "\<not> ipset_from_cidr b1 m1 \<subseteq> ipset_from_cidr b2 m2" shows "ipset_from_cidr b1 m1 \<inter> ipset_from_cidr b2 m2 = {}"
simps_3: fixes x31 :: "'a itrm"   and x32 :: "'a itrm"   and y31 :: "'a itrm"   and y32 :: "'a itrm" shows "(x31 \<diamondop> x32 = y31 \<diamondop> y32) = (x31 = y31 \<and> x32 = y32)"
optimize_matches_option_preserves: fixes rs :: "'b rule list"   and f :: "'b match_expr \<Rightarrow> 'b match_expr option"   and P :: "'b match_expr \<Rightarrow> bool" assumes "\<And>r m. \<lbrakk>r \<in> set rs; f (get_match r) = Some m\<rbrakk> \<Longrightarrow> P m" shows "\<forall>r\<in>set (optimize_matches_option f rs). P (get_match r)"
ChiPlus_NoCycles: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(S, T) \<in> ChiPlus A" shows "S \<noteq> T"
coset_join3: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and H :: "'c set" assumes "Group.group G"   and "x \<in> carrier G"   and "subgroup H G"   and "x \<in> H" shows "x <#\<^bsub>G\<^esub> H = H"
map_semantics_ltlc: fixes f :: "'b \<Rightarrow> 'c option"   and \<phi> :: "'b ltlc"   and \<xi> :: "nat \<Rightarrow> 'b set" assumes "inj_on f (dom f)"   and "atoms_ltlc \<phi> \<subseteq> dom f" shows "\<xi> \<Turnstile>\<^sub>c \<phi> = (map_props f \<circ> \<xi>) \<Turnstile>\<^sub>c map_ltlc (the \<circ> f) \<phi>"
ipv6addr_of_nat_def: fixes n :: "nat" shows "ipv6addr_of_nat n = word_of_nat n"
prj_simps_6: shows "C.dom fgf.Prj\<^sub>0 = Chn ((f \<star> g) \<star> f)"
the_apartment_iso_apartment_simplex_map: fixes X :: "'a set set"   and Y :: "'a set set"   and C :: "'a set"   and x :: "'a set" assumes "X \<in> \<A>"   and "Y \<in> \<A>"   and "chamber C"   and "C \<in> X \<inter> Y"   and "x \<in> X" shows "the_apartment_iso X Y ` x \<in> Y"
i: shows "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) + integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
p_recurrence'': fixes k :: "nat" assumes "3 < k" shows "p k = - (\<Sum>i = 1..3. (- 1) ^ i * e i * p (k - i))"
semiring_norm_8: fixes m :: "num" shows "num.Bit1 m + num.One = num.Bit0 (m + num.One)"
Rep_reg_lan_inverse: fixes x :: "'a reg_lan" shows "Abs_reg_lan (Rep_reg_lan x) = x"
range_composition: fixes f :: "'f \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'f" shows "range (\<lambda>x. f (g x)) = f ` range g"
P: shows "par (incl_of (img f) (cod f) \<cdot> corestr f) f"
surjD: fixes f :: "'d \<Rightarrow> 'c"   and y :: "'c" assumes "surj f" shows "\<exists>x. y = f x"
nth: fixes j :: "nat"   and v :: "nat" shows "eval nth [j, v, i, e] \<down>= e_nth e j"
REFL: shows "(l, l') \<in> \<langle>Rl\<rangle>list_rel"
less_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs < ys \<equiv> xs \<noteq> [] \<and> ys \<noteq> [] \<longrightarrow> last xs < hd ys"
intro: fixes X :: "'c \<Longrightarrow> 'a"   and Y :: "'b \<Longrightarrow> 'a" assumes "\<And>\<sigma> v u. put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v"   and "\<And>\<sigma> v. get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>"   and "\<And>\<sigma> u. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>" shows "X \<bowtie> Y"
ind_is_open_iff_open: fixes U :: "'b set" shows "ind_topology.ind_is_open X is_open X U = (is_open U \<and> U \<subseteq> X)"
elapsed_ge0: fixes x :: "'c \<Rightarrow> 'b"   and y :: "'c \<Rightarrow> 'b" shows "(0::'b) \<le> elapsed x y"
simps_27: fixes vb :: "char list"   and vc :: "aexp"   and vd :: "'a"   and v :: "'a acom"   and va :: "'a acom" shows "(vb ::= vc {vd} \<sqsubseteq> v;; va) = False"
not_accp_down: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "\<not> Wellfounded.accp R x" obtains z :: "'b" where "R z x" and "\<not> Wellfounded.accp R z"