hcju/isabelleps huggingface dataset:
tendsto_ine\<^sub>N_def: fixes N :: "'a quasinorm"   and u :: "nat \<Rightarrow> 'a"   and x :: "'a" shows "tendsto_ine\<^sub>N N u x = (\<lambda>n. eNorm N (u n - x)) \<longlonglongrightarrow> 0"
maxt34: shows "wmin (wtimes l\<^sub>1 u\<^sub>2) (wtimes u\<^sub>1 u\<^sub>2) \<equiv>\<^sub>E min (rl1 * ru2) (ru1 * ru2)"
c_merge_pairs: fixes key :: "'b \<Rightarrow> 'c"   and xss :: "'b list list" shows "Mergesort_Complexity.c_merge_pairs key xss \<le> length (concat xss)"
DirProds_is_group: fixes I :: "'d set"   and G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> Group.group (G i)" shows "Group.group (DirProds G I)"
seq_compactE: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a" assumes "seq_compact S"   and "\<forall>n. f n \<in> S" obtains l :: "'a" and r :: "nat \<Rightarrow> nat" where "l \<in> S" and "strict_mono r" and "(f \<circ> r) \<longlonglongrightarrow> l"
fusion_eq: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>x. F x" shows "(\<sigma>x. F x) = (THE x. \<forall>y. O y x = (\<exists>z. F z \<and> O y z))"
holds_imp_uses: fixes s :: "thread"   and k :: "nat" assumes "locks_ticket s k" shows "has_ticket s k"
crdt_ops_reorder: fixes xs :: "('c \<times> 'd) list"   and oid :: "'c"   and oper :: "'d"   and ys :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" assumes "crdt_ops (xs @ [(oid, oper)] @ ys) deps"   and "\<And>op2 r. \<lbrakk>op2 \<in> snd ` set ys; r \<in> deps op2\<rbrakk> \<Longrightarrow> r \<noteq> oid" shows "crdt_ops (xs @ ys @ [(oid, oper)]) deps"
OPT: shows "T\<^sub>p [x, y] qs (OPT2 qs [x, y]) = 0"
set_disconnected_nodes_typess_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_disconnected_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
poly_mapping_adds_iff: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "(s adds t) = (lookup s adds lookup t)"
is_glbD2: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c"   and u :: "'c" assumes "class.po below"   and "po.is_glb below S x"   and "po.is_lb below S u" shows "below u x"
qfree_lift_nnf_qe: fixes qe :: "atom fm \<Rightarrow> atom fm"   and \<phi> :: "atom fm" assumes "\<And>\<phi>. nqfree \<phi> \<Longrightarrow> qfree (qe \<phi>)" shows "qfree (Z.lift_nnf_qe qe \<phi>)"
part_UNlift01_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "properL dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n)" shows "part (BrnNFT dl cl) (UNlift01 cl dl P F ` UNpart1 cl dl P)"
delete_first_unroll: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "delete_first P (x # xs) = (if P x then Some xs else map_option ((#) x) (delete_first P xs))"
mult_irr_def: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly" shows "mult_irr x y = x * y mod f"
shiftr1_code: fixes w :: "'b word" shows "shiftr1 w = drop_bit 1 w"
Q_def: shows "Q \<equiv> \<lambda>f. \<forall>i. f i = b"
geseqE: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "(f, g) \<in> geseq" obtains "\<forall>i. f i \<in> A" and "\<forall>i. g i \<in> A" and "f = g"   | i :: "nat" where "\<forall>i. f i \<in> A" and "\<forall>i. g i \<in> A" and "size (g i) < size (f i)" and "\<forall>j<i. f j = g j"
convert_closed: fixes i :: "nat"   and fml :: "'a fml" assumes "i \<in> set_fml (fst (convert fml))" shows "i < length (snd (convert fml))"
inj_map_strong: fixes x :: "'c dual"   and xa :: "'c dual"   and f :: "'c \<Rightarrow> 'd"   and fa :: "'c \<Rightarrow> 'd" assumes "\<And>z za. \<lbrakk>z \<in> set_dual x; za \<in> set_dual xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_dual f x = map_dual fa xa" shows "x = xa"
d: shows "prime (9 * 2 ^ (2 * 2 - 1) - 1)"
signed_take_bit_take_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m (take_bit n a) = (if n \<le> m then take_bit n else ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m) a"
Eats_EQ_Zero_E2: fixes t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (Zero EQ Eats t u) H \<turnstile> A"
cnj_scaleR: fixes r :: "real"   and x :: "quat" shows "Quaternions.cnj (r *\<^sub>R x) = r *\<^sub>R Quaternions.cnj x"
has_codomain_iff_arr: fixes f :: "'s" shows "(S.codomains f \<noteq> {}) = S.arr f"
k'_2: shows "k' cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
triangle_paths_neq: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and ab :: "'b set"   and ac :: "'b set" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c"   and "ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b"   and "ac \<in> \<P> \<and> a \<in> ac \<and> c \<in> ac \<and> a \<noteq> c" shows "ab \<noteq> ac"
weakBisimCoinductAux: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<(X \<union> weakBisim)> Q \<and> ((Q, P) \<in> X \<or> Q \<approx> P)" shows "P \<approx> Q"
\<sigma>\<tau>: shows "arr \<sigma> \<and> arr \<tau> \<and> some_src \<tau> = some_trg \<sigma>"
Diag_implies_Arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.Arr C t"
adjust_idx_rev2: fixes j :: "nat"   and i :: "nat" assumes "j \<noteq> i" shows "adjust_idx i (adjust_idx_rev i j) = j"
Polys_closed_sum: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'e"   and X :: "'d set" assumes "\<And>a. a \<in> A \<Longrightarrow> f a \<in> P[X]" shows "sum f A \<in> P[X]"
simps_1: fixes f :: "'b"   and x :: "'b"   and xs :: "'b list" shows "pderiv_coeffs_code f (x # xs) = cCons (f * x) (pderiv_coeffs_code (f + (1::'b)) xs)"
termMOD_igSubstIGAbs: shows "igSubstIGAbs termMOD"
neg_assert_def: fixes x :: "'a" shows "neg_assert x = x ^ o * \<bottom> \<sqinter> (1::'a)"
cross_ratio_1_equal: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2" assumes "cross_ratio_correct p q r s"   and "cross_ratio p q r s = 1" shows "r = s"
swapEnv_preserves_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and x :: "'var" assumes "goodEnv rho" shows "freshEnv xs (x @xs[y1 \<and> y2]_ys) (rho &[y1 \<and> y2]_ys) = freshEnv xs x rho"
a_owner_document_valid_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap" shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_owner_document_valid get_child_nodes get_disconnected_nodes h = (\<forall>node_ptr\<in>fset (node_ptr_kinds h). (\<exists>document_ptr. document_ptr |\<in>| document_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r) \<or> (\<exists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r))"
fcomp_apply: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'c"   and x :: "'d" shows "fcomp f g x = g (f x)"
xss: shows "xss = lappend (llist_of xss') (LCons (LCons x xs') xss'')"
order_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
msubstneq_nb: fixes a :: "poly"   and r :: "tm"   and t :: "tm"   and s :: "tm"   and c :: "poly"   and d :: "poly" assumes "islin (NEq (CNP 0 a r))"   and "tmbound0 t"   and "tmbound0 s" shows "bound0 (msubstneq c t d s a r)"
preserves_hseq: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.arr \<mu> \<and> B.arr \<nu> \<and> src\<^sub>B \<mu> = trg\<^sub>B \<nu>" shows "arr (UP \<mu>) \<and> arr (UP \<nu>) \<and> src (UP \<mu>) = trg (UP \<nu>)"
Exit_source: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and a :: "'g" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "valid_edge a"   and "sourcenode a = Exit" shows "False"
cp_iff_reads_contradict: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat" shows "(((\<sigma>, k), \<sigma>', k') \<in> cp) = (\<sigma> =\<^sub>L \<sigma>' \<and> cs\<^bsup>path \<sigma>\<^esup> k = cs\<^bsup>path \<sigma>'\<^esup> k' \<and> (\<exists>v\<in>local.reads (path \<sigma> k). (\<sigma>\<^bsup>k\<^esup>) v \<noteq> (\<sigma>'\<^bsup>k'\<^esup>) v))"
elims_2: fixes x :: "'v"   and xa :: "('v \<times> 'v) list"   and xb :: "'v" assumes "is_path x xa xb" obtains v :: "'v" and v' :: "'v" where "x = vb__" and "xa = []" and "xb = v'a__" and "vb__ = v'a__ \<and> v'a__ \<in> V"   | v :: "'v" and v1 :: "'v" and v2 :: "'v" and p :: "('v \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, v2) # p" and "xb = v'" and "v = v1 \<and> (v1, v2) \<in> E \<and> is_path v2 p v'"
fls_of_fpxs_eq_0_iff: fixes x :: "'b fpxs" shows "(fls_of_fpxs x = 0) = (x = 0)"
intros_31: fixes E :: "char list \<Rightarrow> ty option"   and Copt :: "char list option"   and M :: "char list"   and vs :: "val list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>Call null Copt M (map Val vs),s\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<rangle>"
\<a>_def: fixes \<tau> :: "('a E.term, 'a) arr"   and \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr" shows "\<a> \<tau> \<mu> \<nu> \<equiv> if VVV.arr (\<tau>, \<mu>, \<nu>) then \<tau> \<star> \<mu> \<star> \<nu> else null"
discount_factor_borel_measurable: fixes r :: "real"   and n :: "nat"   and M :: "'b measure" shows "discount_factor r n \<in> borel_measurable M"
repeat_mset_distrib: fixes m :: "nat"   and n :: "nat"   and A :: "'k multiset" shows "repeat_mset (m + n) A = repeat_mset m A + repeat_mset n A"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat list \<Rightarrow> bexp list \<Rightarrow> 'a"   and f2 :: "bexp \<Rightarrow> int \<Rightarrow> 'a"   and f3 :: "(bexp \<times> int) list \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and instr :: "instr" shows "P (case instr of AssI x xa \<Rightarrow> f1 x xa | TestI x xa \<Rightarrow> f2 x xa | ChoiceI x \<Rightarrow> f3 x | GotoI x \<Rightarrow> f4 x) = (\<not> ((\<exists>x11 x12. instr = AssI x11 x12 \<and> \<not> P (f1 x11 x12)) \<or> (\<exists>x21 x22. instr = TestI x21 x22 \<and> \<not> P (f2 x21 x22)) \<or> (\<exists>x3. instr = ChoiceI x3 \<and> \<not> P (f3 x3)) \<or> (\<exists>x4. instr = GotoI x4 \<and> \<not> P (f4 x4))))"
S_intersection1_def: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection1 p q \<equiv> proj2_abs (S_intersection1_rep p q)"
0: fixes ps :: "(int \<times> int) list" shows "real (time (closest_pair_rec_tm ps)) \<le> (closest_pair_recurrence \<circ> length) ps"
fps_minus_def: shows "(-) = (\<lambda>f g. Abs_fps (\<lambda>n. f $ n - g $ n))"
backtrack_simplex': fixes c :: "(nat, 'a \<times> QDelta) mapping \<times> (nat, 'a \<times> QDelta) mapping"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "backtrack_simplex (Simplex_Checkpoint c) (Simplex_State s) = Simplex_State (backtrack_simplex' c s)"
continuous_map_snd: fixes X :: "'d topology"   and Y :: "'e topology" shows "continuous_map (prod_topology X Y) Y snd"
prod_mset_Un: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one (A + B) = times (comm_monoid_mult.prod_mset times one A) (comm_monoid_mult.prod_mset times one B)"
sas_plus_problem_has_serial_solution_iff_ii: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list"   and h :: "nat" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi>"   and "h = length \<psi>" shows "\<exists>\<A>. \<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<phi> \<Psi>  h"
P: shows "\<forall>x. b \<le> norm x \<longrightarrow> P x"
make_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "Array.make n f = Heap_Monad.heap (Array.alloc (map f [0..<n]))"
l7: fixes n :: "nat" shows "fact n = \<Prod>{0<..int n}"
le_mult_floor: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a"   and "(0::'b) \<le> b" shows "\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor> \<le> \<lfloor>a * b\<rfloor>"
assms_5: shows "mod_arr = arr'(m - 1 := Min_mset (arr_mset arr l m))"
prim_invar1_def: fixes A :: "('v \<times> 'v) set" shows "prim_invar1 A \<equiv> is_subset_MST w rg A \<and> (\<forall>(u, v)\<in>A. (v, r) \<in> A\<^sup>*)"
1: shows "Rep_nat_int (i \<squnion> j) = {minimum k..maximum k}"
strong_supplementation: fixes x :: "i"   and y :: "i" assumes "GEM1"   and "\<not> P x y" shows "\<exists>z. P z x \<and> \<not> O z y"
valuation_iff_close_world: fixes M :: "object atom formula set"   and \<phi> :: "object atom formula" assumes "wm_basic M" shows "valuation M \<Turnstile> \<phi> = M \<^sup>c\<TTurnstile>\<^sub>= \<phi>"
lead_coeff_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "lead_coeff (p * q) = lead_coeff p * lead_coeff q"
card_eq: shows "card B = card (A \<inter> B) + card (B - A)"
\<theta>: shows "[\<lparr>A!,a\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>a\<^sup>P,F\<rbrace> \<^bold>\<equiv> (\<^bold>\<exists>y. \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& F \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,y\<^sup>P\<rparr>) \<^bold>& \<^bold>\<not>\<lbrace>y\<^sup>P,F\<rbrace>)) in v]"
simps_1: shows "X4d X4.Xd1 = D1"
alphaAbs_ex_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "alphaAbs_ex_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
roifex_Trueif_unique: fixes b :: "'a ifex" assumes "ro_ifex b"   and "\<forall>ass. val_ifex b ass" shows "b = Trueif"
ThinishChamberComplexFolding_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" shows "ThinishChamberComplexFolding X f \<equiv> ThinishChamberComplex X \<and> ChamberComplexFolding X f"
extra_wt_subseteq: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm"   and \<rho> :: "'a \<Rightarrow> ('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "vars_mset s \<subseteq># vars_mset t" shows "kbo_std.extra_wt ground_heads_var \<delta> arity_sym wt_sym \<rho> s \<le> kbo_std.extra_wt ground_heads_var \<delta> arity_sym wt_sym \<rho> t"
lmember_code_1: fixes x :: "'a" shows "lmember x LNil = False"
fmlookup_restrict_set: fixes A :: "'d set"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmrestrict_set A m $$ x = (if x \<in> A then m $$ x else None)"
reachable_full: fixes y :: "'v" assumes "reachable y" shows "j (None, \<langle>y\<rangle>) = u y"
assms_1: shows "x\<^sub>1 \<le> x"
assms_2: shows "(\<box>(\<lambda>s. P' s \<longrightarrow> (Q' \<U> R) s)) \<sigma>"
fr_nGet1: fixes n :: "'a"   and nReturn :: "'a \<Rightarrow> nat \<Rightarrow> 'b Frame list"   and recv :: "nat \<Rightarrow> 'b Frame list"   and nC :: "'a \<Rightarrow> Config"   and nStore :: "'a \<Rightarrow> nat \<Rightarrow> 'b Frame list"   and nSend :: "'a \<Rightarrow> nat \<Rightarrow> 'b Frame list"   and nGet :: "'a \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "'a" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n" shows "nGet k t = [t mod cycleLength (nC k)]"
g_bnd_def: fixes n :: "nat"   and m :: "nat"   and B :: "rat"   and fs :: "int vec list" shows "LLL.g_bnd n m B fs = (\<forall>i<m. \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 \<le> B)"
mergesort_remdups_def: fixes xs :: "'a list" shows "mergesort_remdups xs = merge_list [] (map (\<lambda>x. [x]) xs)"
distinct_3: shows "cospan_shape.Null \<noteq> cospan_shape.BB"
divmod_int_def: fixes m :: "num"   and n :: "num" shows "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
disc_2: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "\<not> lnull xs \<or> \<not> lnull ys" shows "\<not> lnull (lappend xs ys)"
nrows_singleton: fixes j :: "nat"   and i :: "nat"   and e :: "'b" shows "nrows (singleton_matrix j i e) = (if e = (0::'b) then 0 else Suc j)"
prems_1: shows "tgtOf trn = srcOf (hd (tr_ ## trn_))"
MClkISpec_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState" shows "MClkISpec send rcv cst = (\<forall>p. MClkIPSpec send rcv cst p)"
natural_transformation_axioms: shows "natural_transformation A B F H \<sigma>\<tau>.map"
S3: shows "nat_to_sch (Suc (Suc 3)) = (let u = mod7 (c_fst (Suc (Suc 3))); v = c_snd (Suc (Suc 3)); v1 = c_fst v; v2 = c_snd v; sch1 = nat_to_sch v1 in Let (nat_to_sch v2) (loc_f u sch1))"
common_sudiv_exists_def: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "common_sudiv_exists one_chain1 one_chain2 \<equiv> \<exists>subdiv ps1 ps2. chain_subdiv_chain (one_chain1 - ps1) subdiv \<and> chain_subdiv_chain (one_chain2 - ps2) subdiv \<and> (\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>) \<and> boundary_chain subdiv \<and> (\<forall>(k, \<gamma>)\<in>ps1. point_path \<gamma>) \<and> (\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>)"
concrete_category_axioms: shows "concrete_category IDE HOM EVAL (\<lambda>_ _ _. (\<cdot>\<^sub>B))"
leftderives_trans_step: fixes a :: "symbol list"   and u :: "symbol list"   and b :: "symbol list"   and v :: "symbol list"   and c :: "symbol list" assumes "leftderives a (u @ b @ v)"   and "leftderives b c"   and "is_word u"   and "is_sentence v" shows "leftderives a (u @ c @ v)"
ik\<^sub>s\<^sub>s\<^sub>t_def: fixes A :: "('c, 'd) stateful_strand_step list" shows "ik\<^sub>s\<^sub>s\<^sub>t A \<equiv> {t. receive\<langle>t\<rangle> \<in> set A}"
insert_lem: fixes x :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "insert x A = B" shows "x \<in> B"
is_cnf_foldr_and_if: fixes fs :: "'b formula list" assumes "\<forall>f\<in>set fs. is_cnf f" shows "is_cnf (foldr (\<^bold>\<and>) fs (\<^bold>\<not> \<bottom>))"
meet_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (division_rel G)"   and "y \<in> carrier (division_rel G)" shows "x \<sqinter>\<^bsub>division_rel G\<^esub> y \<in> carrier (division_rel G)"
star_of_abs: fixes x :: "'a" shows "star_of \<bar>x\<bar> = \<bar>star_of x\<bar>"
diff_divide_eq_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and z :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "minus x (divide y z) = divide (minus (times x z) y) z"
zero: fixes x :: "'v"   and y :: "'v" assumes "x \<noteq> y" shows "f (Inl x, Inr y) = 0"
rel_language_alt: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and L :: "'b language"   and K :: "'c language" shows "rel_language R L K = rel_fun (list_all2 R) (=) (in_language L) (in_language K)"
botE: fixes x :: "'a"   and P :: "bool" assumes "eval \<bottom> x" shows "P"
degeneralize_language: fixes A :: "('d, 'e) dgbta" shows "DBTA.language (degeneralize A) = DGBTA.language A"
ProofStepInduct_NonTrivial_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat" shows "ProofStepInduct_NonTrivial G v0 v1 paths P_new sep_size \<equiv> ProofStepInduct G v0 v1 paths P_new sep_size \<and> ProofStepInduct_NonTrivial_axioms v1 P_new"
outside_exists_line_not_intersect_S: fixes p :: "proj2" assumes "conic_sgn p = 1" shows "\<exists>l. proj2_incident p l \<and> (\<forall>q. proj2_incident q l \<longrightarrow> q \<notin> S)"
HF1_def: shows "HF1 = Var X0 EQ Zero IFF SyntaxN.All X1 (Neg (Var X1 IN Var X0))"
monoid_homomorphism_def: fixes \<eta> :: "'d \<Rightarrow> 'e"   and M :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d"   and M' :: "'e set"   and composition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and unit' :: "'e" shows "monoid_homomorphism \<eta> M composition unit M' composition' unit' \<equiv> (Set_Theory.map \<eta> M M' \<and> Group_Theory.monoid M composition unit) \<and> Group_Theory.monoid M' composition' unit' \<and> monoid_homomorphism_axioms \<eta> M composition unit composition' unit'"
not_minus_numeral_inc_eq: fixes n :: "num" shows "NOT (- numeral (Num.inc n)) = numeral n"