hcju/isabelleps huggingface dataset:
heap_is_wellformed_one_disc_parent: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and document_ptr' :: "(_) document_ptr"   and disc_nodes' :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "h \<turnstile> get_disconnected_nodes document_ptr' \<rightarrow>\<^sub>r disc_nodes'"   and "set disc_nodes \<inter> set disc_nodes' \<noteq> {}" shows "document_ptr = document_ptr'"
ngr_res: shows "ord_resolve_rename S CAs0 DA0 AAs0 As0 \<tau> E0"
map_of_is_SomeI: fixes xys :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "map_of xys x = Some y"
prod_relI: fixes a :: "'e"   and a' :: "'f"   and R1 :: "('e \<times> 'f) set"   and b :: "'g"   and b' :: "'h"   and R2 :: "('g \<times> 'h) set" assumes "(a, a') \<in> R1"   and "(b, b') \<in> R2" shows "((a, b), a', b') \<in> R1 \<times>\<^sub>r R2"
atoms_formula_resolvents: fixes S :: "'at Literal set set"   and A :: "'at" shows "atoms_formula (all_resolvents_upon S A) \<subseteq> atoms_formula S"
a2: fixes u :: "'t" assumes "u \<in> keys p" shows "u \<preceq>\<^sub>t v"
bij_betw_chambers: fixes C :: "'a set" assumes "domain.chamber C" shows "bij_betw f C (f ` C)"
stream_rel_snth: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and u :: "'d stream"   and v :: "'e stream" shows "stream_all2 P u v = (\<forall>i. P (u !! i) (v !! i))"
nn: assumes "w1 = NEG_INF \<and> w2 = NEG_INF" shows "P w1 w2"
val_le_refl: fixes v :: "val" shows "v \<sqsubseteq> v"
emeasure_eq_measure: fixes xy :: "real \<times> 'a"   and A :: "(real \<times> 'a) stream set" shows "emeasure (K.lim_stream xy) A = ennreal (L.prob xy A)"
paste_Range: fixes P :: "('b \<times> 'a) set"   and Q :: "('b \<times> 'a) set" shows "Range (P +< Q) \<subseteq> Range P \<union> Range Q"
1: shows "sources (local.inv \<nu>) \<inter> targets (local.inv \<mu>) \<noteq> {}"
3: shows "length As # ds @ dims B = length (map (\<lambda>A. tensor_from_vec (dims A @ dims B) (concat (map (\<lambda>a. vec_smult a (vec B)) (vec A)))) As) # ds @ dims B"
sup_right_isotone: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x" shows "z \<sqinter> y \<le> z \<sqinter> x"
RDirProd_ring: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme" assumes "ring R"   and "ring S" shows "ring (RDirProd R S)"
us_IInf_inv: shows "us \<circ> IInf = id"
normalize_replicate_zero: fixes n :: "nat"   and p :: "'a list" shows "local.normalize (replicate n \<zero> @ p) = local.normalize p"
pseudofunctor_axioms: shows "pseudofunctor S.comp (\<star>) S.\<a> \<i> S.src S.trg (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B local.map cmp"
valid_arbitrary_strategy: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player" assumes "ParityGame G" shows "ParityGame.strategy G p (ParityGame.\<sigma>_arbitrary G)"
char_of_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "'a" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "unique_euclidean_semiring_with_bit_shifts.char_of plus times one bit n = Char (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) n) (bit n 1) (bit n 2) (bit n 3) (bit n 4) (bit n 5) (bit n 6) (bit n 7)"
rec_inpt_def: fixes vl :: "nat" shows "rec_inpt vl = Cn vl rec_trpl [Cn vl (constn 0) [recf.id vl 0], Cn vl (constn 1) [recf.id vl 0], Cn vl (rec_strt (vl - 1)) (map (recf.id vl) [1..<vl])]"
trap: fixes es :: "e list"   and lholed :: "Lholed" assumes "es \<noteq> [Trap]"   and "Lfilled 0 lholed [Trap] es" shows "\<lparr>es\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
lossless_iff_set_pmf_None: fixes p :: "'a spmf" shows "lossless_spmf p = (None \<notin> set_pmf p)"
global_in_ass: shows "global_assms TYPE('var) |\<subseteq>| ass_forms"
parts_UN_subset1: fixes H :: "'a \<Rightarrow> msg set"   and A :: "'a set" shows "(\<Union>x\<in>A. parts (H x)) \<subseteq> parts (\<Union> (H ` A))"
length_map2: fixes ys :: "'a list"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "length ys = length xs" shows "length (map2 f xs ys) = length xs"
is_word_terminals_drop: fixes p :: "(symbol \<times> character list) list"   and n :: "nat" assumes "wellformed_tokens p" shows "is_word (terminals (drop n p))"
key: shows "{\<beta>} \<subseteq> (ars\<^sup>+)\<inverse> \<down>s {\<alpha>}"
lset_subset: fixes xs :: "'a llist"   and A :: "'a set" assumes "\<not> lset xs \<subseteq> A" shows "\<exists>n. enat n < llength xs \<and> lnth xs n \<notin> A"
less_mod_ring_def: fixes x :: "'b mod_ring"   and y :: "'b mod_ring" shows "(x < y) = (to_int_mod_ring x < to_int_mod_ring y)"
cases: fixes x :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" obtains "x = \<langle>\<rangle>"   | l :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" and xa :: "'c \<times> 'd" and uu_ :: "'e" and mkp :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> 'e \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (xa, uu_, mkp), r\<rangle>"
inner_add_right: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "inner x (y + z) = inner x y + inner x z"
rotate_topneg_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "rotate_topneg w1 w2" shows "kauff_mat w1 = kauff_mat w2"
directedI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. r x z \<and> r y z" shows "directed A r"
simps_1: shows "SkewBinomialHeapStruc.height_queue [] = 0"
B_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "B j \<le> B i"
test_conv: fixes x :: "'a" assumes "is_test x" shows "is_test (x\<^sup>\<smile>)"
arcs_G: shows "arcs G = insert a (insert a' (arcs H))"
replicate_spec: fixes n :: "nat"   and xs :: "'d list"   and x :: "'d" assumes "\<forall>i<n. xs ! i = x"   and "n = length xs" shows "replicate (length xs) x = xs"
at_eq_sup_left_right: fixes x :: "'a" shows "at x = sup (at_left x) (at_right x)"
elims: fixes x :: "'g list"   and xa :: "'g list"   and y :: "'g list" assumes "splice x xa = y" obtains ys :: "'g list" where "x = []" and "xa = ys__" and "y = ys__"   | xb :: "'g" and xs :: "'g list" and ys :: "'g list" where "x = xb # xs" and "xa = ys" and "y = xb # splice ys xs"
cancels_to_1_at_def: fixes i :: "nat"   and l1 :: "(bool \<times> 'e) list"   and l2 :: "(bool \<times> 'e) list" shows "cancels_to_1_at i l1 l2 = (0 \<le> i \<and> 1 + i < length l1 \<and> canceling (l1 ! i) (l1 ! (1 + i)) \<and> l2 = cancel_at i l1)"
borel_measurable_implies_simple_function_sequence_real: fixes u :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "u \<in> borel_measurable M"   and "\<And>x. 0 \<le> u x" shows "\<exists>f. incseq f \<and> (\<forall>i. simple_function M (f i)) \<and> (\<forall>x. bdd_above (range (\<lambda>i. f i x))) \<and> (\<forall>i x. 0 \<le> f i x) \<and> u = Sup (range f)"
SN: shows "SN_on R (R\<^sup>* `` X)"
simple_distributed_measure: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> real"   and a :: "'d" assumes "simple_distributed M X P"   and "a \<in> X ` space M" shows "P a = Sigma_Algebra.measure M (X -` {a} \<inter> space M)"
universal: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "\<exists>!l. prj1 f g \<cdot> l = h \<and> prj0 f g \<cdot> l = k"
phiDefNodes_aux_in_unvisited: fixes g :: "'g"   and v :: "'var"   and un :: "'node list"   and n :: "'node" shows "phiDefNodes_aux g v un n \<subseteq> set un"
pdouble_def: fixes a :: "'a"   and p :: "'a \<times> 'a \<times> 'a" shows "pdouble a p = (let (x, y, z) = p in if z = \<zero> then p else let l = \<guillemotleft>2\<guillemotright> \<otimes> y \<otimes> z; m = \<guillemotleft>3\<guillemotright> \<otimes> x [^] 2 \<oplus> a \<otimes> z [^] 2 in (l \<otimes> (m [^] 2 \<ominus> \<guillemotleft>4\<guillemotright> \<otimes> x \<otimes> y \<otimes> l), m \<otimes> (\<guillemotleft>6\<guillemotright> \<otimes> x \<otimes> y \<otimes> l \<ominus> m [^] 2) \<ominus> \<guillemotleft>2\<guillemotright> \<otimes> y [^] 2 \<otimes> l [^] 2, l [^] 3))"
interval_split_1: fixes k :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "real" assumes "k \<in> Basis" shows "cbox a b \<inter> {x. x \<bullet> k \<le> c} = cbox a (\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i)"
zip_Cons: fixes xs :: "'g list"   and y :: "'h"   and ys :: "'h list" shows "zip xs (y # ys) = (case xs of [] \<Rightarrow> [] | z # zs \<Rightarrow> (z, y) # zip zs ys)"
pf_3: assumes "n = 0" shows "prime_factorization_nat n = []"
check_instr_imp_check_instr': fixes ins :: "'addr 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 h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'addr frame list" assumes "check_instr ins P h stk loc C M pc frs" shows "check_instr' ins P h stk loc C M pc frs"
A: fixes n :: "nat"   and k :: "nat" assumes "n \<in> {1..2 ^ k}" shows "dist x ((f ^^ n) x) - real n * additive_strength f xi \<le> dist x (f x) + real k * 16 * deltaG TYPE('a)"
least_def: fixes P :: "'b \<Rightarrow> bool"   and po :: "'b potype" shows "least P po = (SOME x. x \<in> pset po \<and> P x \<and> (\<forall>y\<in>pset po. P y \<longrightarrow> (x, y) \<in> order po))"
rule_charn2: fixes p :: "((int \<times> int) set set, int) Combinators list"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set"   and po :: "int"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "allNetsDistinct p"   and "wellformed_policy1 p"   and "singleCombinators p"   and "wellformed_policy3 p"   and "AllowPortFromTo c d po \<in> set p"   and "x \<in> dom (C (AllowPortFromTo c d po))" shows "applied_rule_rev C x p = \<lfloor>AllowPortFromTo c d po\<rfloor>"
cut_def: fixes A :: "rat set" shows "Dedekind_Real.cut A \<equiv> {} \<subset> A \<and> A \<subset> {0<..} \<and> (\<forall>y\<in>A. (\<forall>z. 0 < z \<and> z < y \<longrightarrow> z \<in> A) \<and> (\<exists>u\<in>A. y < u))"
subseq_of_lengthD_1: fixes n :: "nat"   and xs :: "'c list"   and ys :: "'c list" assumes "subseq_of_length n xs ys" shows "ys \<in> set (subseqs xs)"
filter_cong: fixes xs :: "'d list"   and ys :: "'d list"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
at_within_interior: fixes S :: "'c set"   and x :: "'c" assumes "NO_MATCH UNIV S"   and "x \<in> interior S" shows "at x within S = at x"
assms_2: shows "1 + c / 2 + (4 * ln (real m + 1) + 3) / (ln 2 * (real m + 1)) \<le> c'"
combine_infinity_invariant_4: fixes i :: "enat"   and x :: "'a ltln"   and j :: "enat"   and y :: "'a ltln"   and z :: "'a ltln"   and k :: "enat" assumes "(i = \<infinity>) = is_constant x"   and "(j = \<infinity>) = is_constant y"   and "combine mk_release (x, i) (y, j) = (z, k)" shows "(k = \<infinity>) = is_constant z"
monomial_sum: fixes a :: "'a"   and f :: "'c \<Rightarrow> 'b"   and C :: "'c set" shows "monomial (sum f C) a = (\<Sum>c\<in>C. monomial (f c) a)"
Throw: fixes e :: "expr"   and xs :: "expr list" shows "map_val2 (throw e # xs) [] (throw e # xs)"
sum_le_suminf: fixes I :: "nat set" assumes "summable f"   and "finite I"   and "\<And>n. n \<in> - I \<Longrightarrow> (0::'a) \<le> f n" shows "sum f I \<le> suminf f"
2: shows "(e\<^sup>T * top)\<^sup>T = top * e"
take_rews_1: fixes n :: "nat"   and cfun :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" shows "\<psi>\<^sup>E\<^bsub>Suc n\<^esub>\<cdot>(CFn\<cdot>cfun) = CFn\<cdot>(cfun_map\<cdot>\<psi>\<^sup>A\<^bsub>n\<^esub>\<cdot>\<psi>\<^sup>A\<^bsub>n\<^esub>\<cdot>cfun)"
lappend_eq_lappend_conv: fixes xs :: "'c llist"   and us :: "'c llist"   and ys :: "'c llist"   and vs :: "'c llist" assumes "llength xs = llength us" shows "(lappend xs ys = lappend us vs) = (xs = us \<and> (lfinite xs \<longrightarrow> ys = vs))"
insert_Diff_single: fixes a :: "'b"   and A :: "'b set" shows "insert a (A - {a}) = insert a A"
ffact_Suc_rev: fixes n :: "nat"   and m :: "'b" shows "ffact (Suc n) m = (m - of_nat n) * ffact n m"
oneBIT_step1x: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a" assumes "x \<noteq> y"   and "x \<in> {x0, y0}"   and "y \<in> {x0, y0}" shows "type1 [x0, y0] x y \<bind> (\<lambda>s. BIT_step s x \<bind> (\<lambda>(a, is'). return_pmf (step (fst s) x a, is'))) = type0 [x0, y0] x y"
h_IN: fixes n :: "nat"   and y :: "'v" shows "d_IN (\<lambda>x. ennreal (h n x)) y = ennreal (if y \<in> Y n then enn2real (weight \<Gamma> y - h' n None y) else 0)"
S4: shows "2 ^ log2 u div 2 ^ log2 u = 1"
fixf_le_lubH: fixes H :: "'a set" assumes "H = {x. (x, f x) \<in> r \<and> x \<in> A}" shows "\<forall>x\<in>fix f A. (x, lub H cl) \<in> r"
relabel_increase_u: fixes u :: "nat" assumes "relabel_precond f l u" shows "l u < relabel_effect f l u u"
adm_subst2: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "cont f"   and "cont g" shows "adm (\<lambda>x. f (fst x) = g (snd x))"
sup_ccINF: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d set"   and a :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_distrib_lattice Inf Sup inf less_eq less sup bot top"   and "countable B" shows "sup a (Inf (f ` B)) = Inf ((\<lambda>b. sup a (f b)) ` B)"
ang_circ_c_def: fixes E :: "complex"   and \<mu>1 :: "complex"   and \<mu>2 :: "complex"   and p1 :: "bool"   and p2 :: "bool" shows "ang_circ_c E \<mu>1 \<mu>2 p1 p2 = \<angle>c (circ_tang_vec \<mu>1 E p1) (circ_tang_vec \<mu>2 E p2)"
bet_out__bet: fixes A :: "'p"   and P :: "'p"   and C :: "'p"   and B :: "'p" assumes "Bet A P C"   and "P Out A B" shows "Bet B P C"
AllPInProclist: shows "\<forall>p. \<exists>i<n. procList ! i = p"
iffD1: fixes Q :: "bool"   and P :: "bool" assumes "Q = P"   and "Q" shows "P"
prems_3: fixes g :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "g \<in> set gs"   and "p = monom_mult (1::'b) t g" shows "thesis"
prems_1: shows "upd_rev_cs cs (as @ [a]) = []"
circ_sup_sub_sup_one_2: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<circ> * (x \<squnion> y) \<le> x\<^sup>\<circ> * ((1::'a) \<squnion> y)"
stable_def: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and p :: "('d, 'e, 'f) configuration \<Rightarrow> bool" assumes "computation channel trans send recv init final" shows "computation.stable channel trans send recv p \<equiv> \<forall>c. p c \<longrightarrow> (\<forall>t c'. distributed_system.trace channel trans send recv c t c' \<longrightarrow> p c')"
redT_normal: fixes t :: "'t"   and x :: "'x"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "actions_ok s t ta"   and "redT_upd s t ta x' m' s'" shows "s -t\<triangleright>ta\<rightarrow> s'"
Call: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and s :: "'d"   and t :: "('d, 'f) xstate" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow> t"
vfield_vsubset_VUnion2: fixes r :: "V" shows "\<F>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> r)"
multiplicity_prime_power: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p" shows "factorial_semiring.multiplicity times one p (power.power one times p n) = n"
swap_bij: fixes x :: "'a"   and x' :: "'a" shows "bij (id(x := x', x' := x))"
G'_gba_fields: shows "G'.L = (\<lambda>q l. if q \<in> f ` V then L (fi q) l else False)"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "bo x y"   and "bo y z"   and "x \<in> A" shows "bo x z"
comm_monoid_axioms: shows "Group.comm_monoid (G\<lparr>carrier := adjoin G H a\<rparr>)"
that: fixes l :: "complex" assumes "f \<midarrow>z\<rightarrow> l \<or> (inverse \<circ> f) \<midarrow>z\<rightarrow> l" shows "thesis"
create_gba_nds_invars: fixes \<phi> :: "'b ltlr" shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. nds_invars (g_V \<A>))"
map_poly_of_real_Re: fixes p :: "complex poly" assumes "set (coeffs p) \<subseteq> \<real>" shows "map_poly complex_of_real (map_poly Re p) = p"
valid_reqs_ACS_D: fixes M :: "'b SecurityInvariant_configured list" assumes "valid_reqs M" shows "valid_reqs (get_ACS M)"
borel_open: fixes A :: "'b set" assumes "open A" shows "A \<in> sets borel"
raw_match_term_extend_map_preserve: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs'' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and p :: "variable \<times> String.literal set" assumes "raw_match_term t u subs = Some subs'"   and "subs' \<subseteq>\<^sub>m subs''"   and "p \<in> tvs t" shows "subs'' p = subs' p"
minus_finite_4_def: fixes x :: "finite_4"   and y :: "finite_4" shows "x - y = x \<sqinter> - y"
Limit_oexp: fixes \<gamma> :: "V"   and \<alpha> :: "V" assumes "Limit \<gamma>"   and "Ord \<alpha>"   and "1 < \<alpha>" shows "Limit (\<alpha> \<up> \<gamma>)"
simps_5: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "pre ({P} WHILE b DO C) Q = (\<lambda>s. P s + 1)"
\<delta>_prodE: fixes r :: "('a \<times> 'b, 'c) ta_rule"   and \<delta>1 :: "('a, 'c) ta_rule set"   and \<delta>2 :: "('b, 'c) ta_rule set" assumes "r \<in> \<delta>_prod \<delta>1 \<delta>2" obtains q1 :: "'a" and q2 :: "'b" and l :: "'c" and qs1 :: "'a list" and qs2 :: "'b list" where "length qs1 = length qs2" and "q1 \<rightarrow> l qs1 \<in> \<delta>1" and "q2 \<rightarrow> l qs2 \<in> \<delta>2" and "r = (q1, q2) \<rightarrow> l zip qs1 qs2"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>ip = ip r, sn = sn r, rt = rt r, rreqs = rreqs r, store = store r, msg = msg r, data = data r, dests = dests r, pre = pre r, rreqid = rreqid r, dip = dip r, oip = oip r, hops = hops r, dsn = dsn r, dsk = dsk r, osn = osn r, sip = sip r, \<dots> = state.more r\<rparr>"
cfnl_def: fixes p :: "nat"   and f :: "nat \<Rightarrow> real" shows "cfnl p f = (Max (reduce f PR) + Min (reduce f PR)) / 2"
i: shows "column i (fst (QR_decomposition A)) \<in> columns (fst (QR_decomposition A))"
max_less_iff_conj: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(max x y < z) = (x < z \<and> y < z)"
Suc_pred: fixes n :: "nat" assumes "0 < (n::nat)" shows "Suc ((n::nat) - Suc 0) = n"
simps_2: fixes sh :: "sheap"   and e :: "'a exp"   and es :: "'a exp list" shows "iconfs sh (e # es) = (case val_of e of None \<Rightarrow> iconf sh e \<and> \<not> sub_RIs es | \<lfloor>v\<rfloor> \<Rightarrow> iconfs sh es)"
prod_eq_iff: fixes s :: "'b \<times> 'c"   and t :: "'b \<times> 'c" shows "(s = t) = (fst s = fst t \<and> snd s = snd t)"
list_separated_toString_def: fixes sep :: "char list"   and toStr :: "'a \<Rightarrow> char list"   and ls :: "'a list" shows "list_separated_toString sep toStr ls = concat (splice (map toStr ls) (replicate (length ls - 1) sep))"
assms_2: shows "w `\<rightarrow> C0 = C0"
strict_not_outside_1: fixes x :: "'b"   and y :: "'b" assumes "x \<prec>[Pareto R] y" shows "x \<in> alts"
f25: shows "(fst x \<otimes> fst x', snd x \<otimes> snd x') \<in> carrier rel"
mult: fixes a :: "nat"   and b :: "nat" shows "\<chi>\<^sub>0 (a * b) = \<chi>\<^sub>0 a * \<chi>\<^sub>0 b"
g: shows "g = (THE g. f = Abs_finfun (map_default b g) \<and> finite (dom g) \<and> b \<notin> ran g)"
power_add_numeral: fixes a :: "'b"   and m :: "num"   and n :: "num" shows "a ^ numeral m * a ^ numeral n = a ^ numeral (m + n)"
parts_connection_http_def: shows "parts_connection_http \<equiv> \<lparr>pc_iiface = ''1'', pc_oiface = ''1'', pc_proto = TCP, pc_sport = 10000, pc_dport = 80\<rparr>"
case_cmp_if: fixes l :: "'b"   and e :: "'b"   and g :: "'b"   and c :: "cmp_val" shows "(case c of LT \<Rightarrow> l | EQ \<Rightarrow> e | GT \<Rightarrow> g) = (if c = LT then l else if c = GT then g else e)"
rep_eq: fixes x :: "'b coreflexive" shows "Rep_coreflexive (- x) = Rep_coreflexive x '"
leftInvertible: fixes R' :: "('a sequent list \<times> 'a sequent) set"   and R :: "('a sequent list \<times> 'a sequent) set"   and \<Gamma> :: "'a form multiset"   and F :: "'a"   and Fs :: "'a form list"   and \<Delta> :: "'a form multiset"   and n :: "nat"   and \<Gamma>' :: "'a form multiset"   and \<Delta>' :: "'a form multiset" assumes "R' \<subseteq> upRules \<and> R = Ax \<union> R'"   and "( \<Gamma> \<oplus> Compound F Fs \<Rightarrow>* \<Delta>, n) \<in> derivable R*"   and "\<forall>r'\<in>R. leftPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')" shows "\<exists>m\<le>n. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable R*"
cut_le_inext_nth_card_eq1: fixes n :: "nat"   and I :: "nat set" assumes "n < card I \<or> infinite I" shows "card (I \<down>\<le> (I \<rightarrow> n)) = Suc n"
sig_red_tail_rtrancl_lt_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg (\<prec>) F)\<^sup>*\<^sup>* p q" shows "punit.lt (rep_list q) = punit.lt (rep_list p)"
const_def: fixes x :: "'b" shows "pure x = (\<lambda>_. x)"
assms: shows "GEN_OP is_empty_i is_empty (A \<rightarrow> bool_rel)"