hcju/isabelleps huggingface dataset:
proj_exec_proj_eq_exec_proj: fixes vs :: "'d set"   and s :: "('d, 'e) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" shows "fmrestrict_set vs (exec_plan (fmrestrict_set vs s) (as_proj as vs)) = exec_plan (fmrestrict_set vs s) (as_proj as vs)"
set_tree_insert: fixes t :: "('a \<times> 'b) tree"   and x :: "'a" assumes "Tree2.bst t" shows "Tree2.set_tree (local.insert x t) = {x} \<union> Tree2.set_tree t"
eval_coprime: fixes b :: "'c"   and a :: "'c" assumes "b \<noteq> (0::'c)" shows "coprime a b = coprime b (a mod b)"
word_sdiv_def: fixes x :: "'b word"   and y :: "'b word" shows "x sdiv y = (let x' = sint x; y' = sint y; negative = (x' < 0) \<noteq> (y' < 0); result = \<bar>x'\<bar> div \<bar>y'\<bar> in word_of_int (if negative then - result else result))"
snd_eqD: fixes x :: "'d"   and y :: "'c"   and a :: "'c" assumes "snd (x, y) = a" shows "y = a"
norm_char: fixes a :: "nat" shows "cmod (dc2c \<chi> a) = (if a \<in> carrier G then 1 else 0)"
eval_body'_eval_body_final_eq: shows "eval_body_final oo unwrapC oo wrapC = eval_body'"
closed_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "path g" shows "closed (path_image g)"
distinct_35: shows "CharactersOp \<noteq> ToRealOp"
min_enat1_conv_enat: fixes a :: "nat"   and b :: "enat" shows "min (enat a) b = enat (case b of enat b' \<Rightarrow> min a b' | \<infinity> \<Rightarrow> a)"
sterm_continuous: fixes I :: "('a, 'b, 'c) interp"   and \<theta> :: "('a, 'c) trm" assumes "is_interp I"   and "dfree \<theta>" shows "continuous_on UNIV (sterm_sem I \<theta>)"
cph: shows "causal_program_history is\<^sub>s\<^sub>b sb"
verticesFrom_ram1: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list" assumes "pre_split_face f ram1 ram2 vs" shows "verticesFrom f ram1 = ram1 # between (vertices f) ram1 ram2 @ ram2 # between (vertices f) ram2 ram1"
fractrel_normalize_quot_left: fixes x :: "'b \<times> 'b"   and y :: "'b \<times> 'b" assumes "snd x \<noteq> (0::'b)" shows "fractrel (normalize_quot x) y = fractrel x y"
open_existence_ivl: fixes x0 :: "'a" shows "open (existence_ivl0 x0)"
inf_le1: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) x"
sumRes: fixes x :: "name"   and P :: "pi"   and Q :: "pi" shows "<\<nu>x>(P \<oplus> Q) \<sim>\<^sup>s <\<nu>x>P \<oplus> <\<nu>x>Q"
range2s: fixes x :: "int" assumes "x \<le> 2 ^ 31 - 1" shows "x + - 2147483647 < 2147483647"
is_span: shows "C.span HH\<mu>\<nu>\<pi>.leg0 HH\<mu>\<nu>\<pi>.leg1"
Inf_aform_def: fixes X :: "'b \<times> 'b pdevs" shows "Inf_aform X = fst X - tdev (snd X)"
ordLess_transitive: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r <o r'"   and "r' <o r''" shows "r <o r''"
simps: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" shows "shift1_inv (qs, B) = (B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)} \<and> shift2_inv qs)"
my_taut_test_ifex_def: shows "my_taut_test_ifex = taut_test Boolean_Expression_Example.ifex_of"
pmu_isotone: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "has_least_prefixpoint f"   and "has_least_prefixpoint g"   and "f \<le>\<le> g" shows "p\<mu> f \<le> p\<mu> g"
set_mode_get_child_nodes: fixes ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>Shadow_DOM.get_child_nodes_locs ptr'. r h h')"
collapse: fixes prod :: "'a \<times> 'd" shows "(fst prod, snd prod) = prod"
wf_prog_wf_syscls: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "wf_syscls P"
gcd_mult_left_right_cancel: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one b c" shows "gcd (times a c) b = gcd a b"
stalk_axioms: shows "stalk X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X {U. is_open\<^sub>X U \<and> x \<in> U} x"
bnd: fixes l1 :: "rat"   and l2 :: "rat"   and r1 :: "rat"   and r2 :: "rat"   and sr1 :: "rat"   and sr2 :: "rat"   and l :: "rat"   and r :: "rat" assumes "I l1"   and "I l2"   and "root_cond (p1, l1, r1) x"   and "root_cond (p2, l2, r2) y"   and "bnd ((l1, r1, sr1), l2, r2, sr2) = (l, r)" shows "real_of_rat l \<le> f x y \<and> f x y \<le> real_of_rat r"
Deriv_derivable: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "(G, c, A) \<in> Deriv" shows "G \<rhd> c : A"
eval_wp_Seq: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" shows "wp (a ;; b) P = (wp a \<circ> wp b) P"
f2: fixes w :: "'a"   and z :: "'a" shows "ad (w \<cdot> z) \<cdot> (w \<cdot> ad z) = ad (w \<cdot> z) \<cdot> w"
fmdom'_notI: fixes m :: "('d, 'c) fmap"   and x :: "'d" assumes "fmlookup m x = None" shows "x \<notin> fmdom' m"
strong_inducts: fixes P :: "'b \<Rightarrow> freeRes \<Rightarrow> bool"   and z :: "'b"   and freeRes :: "freeRes" assumes "\<And>name1 name2 z. P z (name1<name2>)"   and "\<And>name1 name2 z. P z (name1[name2])"   and "\<And>z. P z (\<tau>)" shows "P z freeRes"
step_Abrupt_prop: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and c' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and x :: "'b" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Abrupt x" shows "s' = Abrupt x"
measurable_INF: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "(\<lambda>x. INF i\<in>I. F i x) \<in> M \<rightarrow>\<^sub>M count_space UNIV"
collapse_2: fixes behaviour :: "'a behaviour" assumes "is_wrong behaviour" shows "Goes_wrong (un_Goes_wrong behaviour) = behaviour"
take_bit_not_mask_eq_0: 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" 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"   and "m \<le> n" shows "take_bit m (not (mask n)) = zero"
group_subgroup_generated: fixes G :: "('b, 'c) monoid_scheme"   and S :: "'b set" assumes "Group.group G" shows "Group.group (subgroup_generated G S)"
ubx_imp_typeof_opubx: fixes opinl :: "opinl"   and ts :: "Unboxed.type option list"   and opubx :: "opubx" assumes "Op_example.ubx opinl ts = Some opubx" shows "fst (typeof_opubx opubx) = ts"
stake_as_fct: fixes n :: "nat"   and M :: "bool stream measure"   and p :: "real"   and x :: "bool stream" assumes "Suc 0 \<le> n"   and "M = bernoulli_stream p" shows "to_stream -` {w \<in> space M. stake n w = stake n x} = {X. \<forall>i. 0 \<le> i \<and> i \<le> n - 1 \<longrightarrow> X i = x !! i}"
rel_funI: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "(A ===> B) f g"
Un_insert_right: fixes A :: "'f set"   and a :: "'f"   and B :: "'f set" shows "A \<union> insert a B = insert a (A \<union> B)"
mod_poly_def: fixes f :: "'b poly"   and g :: "'b poly" shows "f mod g = (if g = 0 then f else pseudo_mod (smult (((1::'b) / lead_coeff g) ^ (Suc (degree f) - degree g)) f) g)"
Numb_def: shows "Numb \<equiv> Atom \<circ> Inr"
assms_6: shows "\<forall>w\<in>space M. 0 \<le> X w"
edges_join: fixes g\<^sub>1 :: "'a ugraph"   and g\<^sub>2 :: "'a ugraph" shows "edges (graph_join g\<^sub>1 g\<^sub>2) = edges g\<^sub>1 \<union> edges g\<^sub>2"
assms_2: shows "B \<subseteq>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>"
Span_is_add_subgroup: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "subgroup (R.Span K Us) (add_monoid R)"
empty_ivl_def: shows "empty_ivl \<equiv> abs_ivl empty_rep"
updatable_def: fixes c :: "('c, 'd) conf" shows "updatable c \<equiv> \<forall>v. \<exists>sv. fst sv = v \<and> Conf.fresh_symvar sv c"
CONS: fixes s :: "'b" assumes "I s"   and "\<not> b s" shows "\<Phi> s"
reachable_def: fixes r :: "('b \<times> 'b) set"   and P :: "'b ref set" shows "reachable r P = r\<^sup>* `` addrs P"
Init_eq: fixes init :: "'c set"   and acts :: "('c \<times> 'c) set set"   and allowed :: "('c \<times> 'c) set set" shows "Init (mk_program (init, acts, allowed)) = init"
parallel_fix_ind: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and F :: "'d \<rightarrow> 'd"   and G :: "'e \<rightarrow> 'e" assumes "adm (\<lambda>x. P (fst x) (snd x))"   and "P \<bottom> \<bottom>"   and "\<And>x y. P x y \<Longrightarrow> P (F\<cdot>x) (G\<cdot>y)" shows "P (fix\<cdot>F) (fix\<cdot>G)"
power_int_eq_0_iff: fixes x :: "'a"   and n :: "int" shows "(x powi n = (0::'a)) = (x = (0::'a) \<and> n \<noteq> 0)"
spmf_conv_measure_spmf: fixes p :: "'b spmf"   and x :: "'b" shows "spmf p x = Sigma_Algebra.measure (measure_spmf p) {x}"
eq_inf_symp: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream" assumes "w\<^sub>1 =\<^sub>I w\<^sub>2" shows "w\<^sub>2 =\<^sub>I w\<^sub>1"
conc_empty_1: fixes A :: "'b list set" shows "{} @@ A = {}"
snth_smap: fixes f :: "'d \<Rightarrow> 'b"   and s :: "'d stream"   and n :: "nat" shows "smap f s !! n = f (s !! n)"
existsA: fixes x :: "nat \<times> nat" assumes "x \<in> A" shows "\<exists>y. P x y"
lm084: fixes x :: "'c"   and X :: "'c set"   and g :: "('c \<times> 'd) set"   and Y :: "'d set" assumes "x \<notin> X"   and "g \<in> injections ({x} \<union> X) Y" shows "\<exists>f\<in>injections X Y. g \<in> {f \<union> {(x, y)} |y. y \<in> Y - Range f}"
that: fixes c :: "'a" assumes "c \<in> set cs"   and "P c w" shows "thesis"
homologous_rel_set_eq: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "(homologous_rel_set p X S a = homologous_rel_set p X S b) = homologous_rel p X S a b"
ideal_eq_UNIV_iff_reduced_GB_eq_one_finite: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite F" shows "(ideal F = UNIV) = (punit.reduced_GB F = {1})"
F2: shows "R2 \<Longrightarrow>\<^sub>A P y p * F2"
insertion_mpoly_of_poly: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat"   and p :: "'a poly" shows "insertion f (mpoly_of_poly i p) = poly p (f i)"
complete_def: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.metric_space dist uniformity open" shows "uniform_space.complete open uniformity S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> uniform_space.Cauchy uniformity f \<longrightarrow> (\<exists>l\<in>S. filterlim f (topological_space.nhds open l) sequentially))"
resolvingBidBridgingLemmaVariant2: fixes N :: "integer set"   and \<Omega> :: "'c list"   and a :: "(integer \<times> 'c set) set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "finite N"   and "distinct \<Omega>"   and "a \<in> maximalStrictAllocations N (set \<Omega>) b" shows "sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>)) a = sum (randomBidsAlg N \<Omega> b r) a"
exhaust_sel: fixes pheap :: "pheap" obtains "pheap = pHeap (heapOf pheap) (addrOf pheap)"
stable_no_blocking_on_def: fixes ds :: "D2 set"   and X :: "X4 set" shows "StableNoDecomp.stable_no_blocking_on ds X = (\<forall>h X'. \<not> StableNoDecomp.blocking_on ds X h X')"
cSUP_singleton: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "\<Squnion> (f ` {x}) = f x"
init_inv_sys: shows "\<forall>s. initial_state \<lparr>PGMs = gc_coms, INIT = gc_system_init, FAIR = \<lambda>s. True\<rparr> s \<longrightarrow> invs \<lparr>GST = s, HST = []\<rparr>\<down>"
geom_rand_walk_borel_adapted: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real" assumes "prob_grw p M geom_proc u d init" shows "borel_adapt_stoch_proc (infinite_coin_toss_space.nat_filtration M) geom_proc"
distinct_131: fixes x6 :: "char list"   and x121 :: "expr"   and x122 :: "expr" shows "Var x6 \<noteq> x121;; x122"
hyps: shows "sum_list xs / n + sum_list xs = ((1::'a) + (1::'a) / n) * sum_list xs"
I_cyclic: fixes a :: "atom"   and i :: "int"   and j :: "int"   and e :: "int list" assumes "is_dvd a"   and "hd_coeff a = 1"   and "i mod divisor a = j mod divisor a" shows "I\<^sub>Z a (i # e) = I\<^sub>Z a (j # e)"
partition_between_ref_partition_between: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and lo :: "nat"   and hi :: "nat"   and xs :: "'c list" shows "partition_between_ref R h lo hi xs \<le> partition_between R h lo hi xs"
c_manifold_def: fixes charts :: "('c, 'd) chart set"   and k :: "enat" shows "c_manifold charts k \<equiv> \<forall>c1 c2. c1 \<in> charts \<longrightarrow> c2 \<in> charts \<longrightarrow> k-smooth_compat c1 c2"
closed_csubspace_inter: fixes A :: "'d set"   and B :: "'d set" assumes "closed_csubspace A"   and "closed_csubspace B" shows "closed_csubspace (A \<inter> B)"
assms_6: shows "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-Class statC"
Gamma_series_Weierstrass_def: fixes z :: "'a"   and n :: "nat" shows "Gamma_series_Weierstrass z n = exp (- euler_mascheroni * z) / z * (\<Prod>k = 1..n. exp (z / of_nat k) / ((1::'a) + z / of_nat k))"
rep_eq: fixes x :: "'d set!['e]" shows "set_bset (bset_of_nebset x) = set_nebset x"
bind_eq_Some_conv: fixes f :: "\<langle>'e\<rangle>\<^sub>\<bottom>"   and g :: "'e \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and x :: "'d" shows "(Option.bind f g = \<lfloor>x\<rfloor>) = (\<exists>y. f = \<lfloor>y\<rfloor> \<and> g y = \<lfloor>x\<rfloor>)"
intros_7: shows "(\<lambda>x y z. y) \<in> PrimRec3"
commute: fixes a :: "'b"   and b :: "'b" shows "lcm a b = lcm b a"
simps: fixes n :: "nat"   and s :: "nat"   and tp :: "cell list \<times> cell list" shows "inv_begin n (s, tp) = (if s = 0 then inv_begin0 n tp else if s = 1 then inv_begin1 n tp else if s = 2 then inv_begin2 n tp else if s = 3 then inv_begin3 n tp else if s = 4 then inv_begin4 n tp else False)"
ide_char: fixes a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" shows "S.VV.ide a = (S.VV.arr a \<and> S.VxV.ide a)"
top_apply: fixes x :: "'d" shows "\<top> x = \<top>"
has_root: fixes r :: "'a"   and x :: "'a" assumes "connected_root r x"   and "point r"   and "x \<noteq> 0" shows "x\<^sup>T ; r \<noteq> 0"
lemma3_5_1: fixes r :: "('b \<times> 'b) set"   and \<tau> :: "'b list"   and \<sigma> :: "'b list"   and \<sigma>' :: "'b list"   and \<tau>' :: "'b list"   and \<upsilon> :: "'b list"   and \<sigma>'' :: "'b list"   and \<upsilon>' :: "'b list" assumes "trans r"   and "irrefl r"   and "D r \<tau> \<sigma> \<sigma>' \<tau>'"   and "D r \<upsilon> \<sigma>' \<sigma>'' \<upsilon>'" shows "(r|\<tau> @ \<upsilon> @ \<sigma>''|, r|\<tau> @ \<upsilon>| + r|\<sigma>|) \<in> mul_eq r"
f_in_Arr: shows "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"
differentiableI_vector: fixes f :: "real \<Rightarrow> 'a"   and y :: "'a"   and F :: "real filter" assumes "(f has_vector_derivative y) F" shows "f differentiable F"
eq_equiv_class: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and A :: "'a set" assumes "r `` {a} = r `` {b}"   and "equiv A r"   and "b \<in> A" shows "(a, b) \<in> r"
2: shows "fmrestrict_set (fmdom' s - vs) s = fmrestrict_set (fmdom' s - vs) (state_succ s a)"
add_block_fin: fixes b :: "'a set" assumes "finite b" shows "finite_incidence_system (\<V> \<union> b) (add_block b)"
cprank_max1: fixes A :: "'b tensor" assumes "cprank_max1 A" shows "cprank_max 1 A"
HInv3_inner_def: fixes s :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" shows "HInv3_inner s p q d = (HInv3_L s p q d \<longrightarrow> HInv3_R s p q d)"
X: fixes i :: "nat" shows "X i \<in> sets (Pi\<^sub>M (J i) M)"
nfa_steps_is_node: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and q :: "bool list"   and w :: "bool list list" assumes "wf_nfa A n"   and "nfa_is_node A q"   and "list_all (is_alph n) w" shows "nfa_is_node A (nfa_steps A q w)"
Proc_CFG_edge_det: fixes prog :: "cmd"   and n :: "label"   and et :: "p_edge_kind"   and n' :: "label"   and et' :: "p_edge_kind" assumes "prog \<turnstile> n -et\<rightarrow>\<^sub>p n'"   and "prog \<turnstile> n -et'\<rightarrow>\<^sub>p n'" shows "et = et'"
numeral_tree_simps_3: fixes n :: "num" shows "right (numeral n) = numeral n"
satisfies_sum_le_constraints: fixes v :: "nat \<Rightarrow> 'a"   and c :: "'a le_constraint"   and d :: "'a le_constraint" assumes "v \<Turnstile>\<^sub>l\<^sub>e c"   and "v \<Turnstile>\<^sub>l\<^sub>e d" shows "v \<Turnstile>\<^sub>l\<^sub>e (c + d)"
Standard_simps_4: fixes x :: "'b star"   and y :: "'b star" assumes "x \<in> Standard"   and "y \<in> Standard" shows "x - y \<in> Standard"
ntrs_stack_comp_s: fixes s :: "'a list"   and c :: "'a list multiset"   and ee :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and r :: "'a list" assumes "((s, c), ee, s', c') \<in> ntrs fg" shows "((s @ r, c), ee, s' @ r, c') \<in> ntrs fg"
append_def: fixes xq :: "'b lazy_sequence"   and yq :: "'b lazy_sequence" shows "Lazy_Sequence.append xq yq = lazy_sequence_of_list (list_of_lazy_sequence xq @ list_of_lazy_sequence yq)"
lifting_inv_axioms: shows "lifting_inv final r (\<lambda>_. P)"
memory_write_asi_privilege: fixes r :: "'b sparc_state option"   and asi :: "word8"   and addr :: "word32"   and byte_mask :: "word4"   and data :: "word32"   and s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "r = memory_write_asi asi addr byte_mask data s \<and> r = Some s' \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
finite_Diff: fixes A :: "'b set"   and B :: "'b set" assumes "finite A" shows "finite (A - B)"
generator_inverse: fixes x :: "('e, 'f) generator" shows "Generator (generator x) = x"
simps_3: fixes x :: "'r"   and xs :: "[:'r:]"   and us :: "[:'r:]"   and l :: "([:'r:] \<times> [:'r:]) tree"   and r :: "([:'r:] \<times> [:'r:]) tree" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "next\<cdot>(x :# xs)\<cdot>(Node\<cdot>(us, [::])\<cdot>l\<cdot>r) = Node\<cdot>(us, [::])\<cdot>l\<cdot>r"
up_filter_antitone: fixes x :: "'c"   and y :: "'c" shows "(x \<le> y) = (up_filter y \<le> up_filter x)"
simps_1: fixes r :: "'c rexp"   and s :: "'c rexp" shows "toplevel_summands (Plus r s) = toplevel_summands r \<union> toplevel_summands s"
times_real_alg_def: shows "(*) \<equiv> (rep_real_alg ---> rep_real_alg ---> Real_Alg_Quotient) mult_3"
trg_hcomp: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>"   and "S.arr \<nu>"   and "S.src \<mu> = S.trg \<nu>" shows "S.trg (\<mu> \<star>\<^sub>S \<nu>) = S.trg \<mu>"
nth_in_AffineI: fixes xs :: "'a list"   and XS :: "('a \<times> 'a pdevs) list"   and i :: "nat" assumes "xs \<in> Joints XS"   and "i < length XS" shows "xs ! i \<in> Affine (XS ! i)"
sint_ge: fixes x :: "'a word" shows "- (2 ^ (LENGTH('a) - 1)) \<le> sint x"
\<theta>\<delta>_disj_2: shows "subst_domain \<theta> \<inter> range_vars \<delta> = {}"
ordertype_eq_Ord: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "ordertype (elts \<alpha>) VWF = \<alpha>"
func_to_img: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B" shows "f \<in> A \<rightarrow> f ` A"
extremum_uniqueI: fixes a :: "'b multiset" assumes "a \<subseteq># {#}" shows "a = {#}"
Cons_mem_clw_rel_iff: fixes A :: "('c \<times> 'd set) set"   and x :: "'c"   and xs :: "'c list"   and X :: "'d set" assumes "single_valued A" shows "((x # xs, X) \<in> clw_rel A) = (\<exists>Y YS. (x, Y) \<in> A \<and> (set xs, YS) \<in> \<langle>A\<rangle>set_rel \<and> X = Y \<union> \<Union> YS)"
not_eq_icard_imp_inj_on: shows "\<exists>f A. icard (f ` A) = icard A \<and> \<not> inj_on f A"
fps_integral0_fps_const_mult_left: fixes c :: "'a"   and a :: "'a fps" shows "fps_integral0 (fps_const c * a) = fps_const c * fps_integral0 a"
hyps_21: shows "\<forall>h'\<in>{0..h}. flow0 x0 h' \<in> CX"
oreturn_bind: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b option" shows "oreturn x |>> f = f x"
simps_13: fixes x1 :: "'b"   and x31 :: "'b local.term"   and x32 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle>\<^sub>0 \<noteq> x31 \<^bold>\<star> x32"
domA_restrictA: fixes S :: "'b set"   and \<Gamma> :: "('b \<times> 'c) list" shows "domA (restrictA S \<Gamma>) = domA \<Gamma> \<inter> S"
dim_def: fixes V :: "('b \<Rightarrow> real) set" shows "a.dest.diff_fun_space.m1.dim V = (if \<exists>b\<subseteq>a.dest.diff_fun_space. \<not> a.dest.diff_fun_space.m1.dependent b \<and> a.dest.diff_fun_space.m1.span b = a.dest.diff_fun_space.m1.span V then card (SOME b. b \<subseteq> a.dest.diff_fun_space \<and> \<not> a.dest.diff_fun_space.m1.dependent b \<and> a.dest.diff_fun_space.m1.span b = a.dest.diff_fun_space.m1.span V) else 0)"
assm: shows "\<forall>x. x \<rhd> (1::'a) \<rhd> (1::'a) = x"
L1: shows "\<forall>x. (x \<in> A) = (f x \<in> B)"
thm_relation_negation_7: fixes v :: "i"   and p :: "\<o>" shows "[p\<^sup>- \<^bold>= \<^bold>\<not>p in v]"
cnt_B': fixes x :: "'a" assumes "x \<in> A" shows "countable (B' x)"