hcju/isabelleps huggingface dataset:
cball_eq_atLeastAtMost: fixes a :: "real"   and b :: "real" shows "cball a b = {a - b..a + b}"
pickT_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and II :: "nat set" shows "pickT cl P II \<equiv> SOME n. pickT_pred cl P II n"
simps_1: fixes OPERATION :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and N :: "'c" shows "rexp_of_list OPERATION N [] = N"
eval: shows "Hom.R.eval p x = \<zero>\<^bsub>R\<^esub>"
cases: fixes G :: "('c, 'd) Graph_scheme"   and a :: "'c list" assumes "Graph G"   and "Graph.walk G a" obtains "a = []"   | v :: "'c" where "a = [va__]" and "va__ \<in> V\<^bsub>G\<^esub>"   | v :: "'c" and w :: "'c" and vs :: "'c list" where "a = v # w # vs" and "v \<rightarrow>\<^bsub>G\<^esub> w" and "Graph.walk G (w # vs)"
closure_of_eq: fixes X :: "'a topology"   and S :: "'a set" shows "(X closure_of S = S) = closedin X S"
ComplD: fixes c :: "'c"   and A :: "'c set" assumes "c \<in> - A" shows "c \<notin> A"
bdd_below_image_dist: fixes x :: "'b"   and A :: "'b set" shows "bdd_below (dist x ` A)"
state_in_range: fixes tp :: "cell list \<times> cell list"   and A :: "(action \<times> nat) list"   and stp :: "nat"   and st :: "nat"   and l :: "cell list"   and r :: "cell list" assumes "steps0 (Suc 0, tp) A stp = (st, l, r)"   and "tm_wf0 A" shows "st \<le> length A div 2"
distinct_1: fixes x21 :: "'g"   and x22 :: "'g list" shows "[] \<noteq> x21 # x22"
eq_\<I>_gpv_Done: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and \<I> :: "('c, 'd) \<I>" assumes "A x y" shows "eq_\<I>_gpv A \<I> (Generative_Probabilistic_Value.Done x) (Generative_Probabilistic_Value.Done y)"
inject_6: fixes x61 :: "nat"   and x62 :: "msg"   and y61 :: "nat"   and y62 :: "msg" shows "(\<langle>x61, x62\<rangle> = \<langle>y61, y62\<rangle>) = (x61 = y61 \<and> x62 = y62)"
diag_Sup: fixes le_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Y :: "'d set"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'a" assumes "Complete_Partial_Order.chain le_b Y"   and "\<And>y. y \<in> Y \<Longrightarrow> monotone le_b (\<le>) (\<lambda>x. f x y)"   and "\<And>x a b. \<lbrakk>x \<in> Y; le_b a b; a \<in> Y; b \<in> Y\<rbrakk> \<Longrightarrow> f x a \<le> f x b" shows "(\<Squnion>x\<in>Y. \<Squnion> (f x ` Y)) = (\<Squnion>x\<in>Y. f x x)"
add_independentS_doubleI: fixes B :: "'a set"   and A :: "'a set" assumes "\<And>b a. \<lbrakk>b \<in> B; a \<in> A; a + b = (0::'a)\<rbrakk> \<Longrightarrow> a = (0::'a)" shows "add_independentS [A, B]"
orecvmsg_simps_7: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and m :: "'b" shows "orecvmsg P \<sigma> (receive m) = P \<sigma> m"
is_const_simps_3: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "\<not> pre_term.is_const unconst (free name)"
monotone_onD: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "monotone_on X r s f"   and "r x y"   and "x \<in> X"   and "y \<in> X" shows "s (f x) (f y)"
ts_c': fixes x :: "nat"   and y :: "nat"   and v :: "nat list"   and h :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "v \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom x])"   and "(\<exists>hs. h = [x, x] @ hs) \<or> h = [x] \<or> h = []" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) v = length v - 2 \<and> Partial_Cost_Model.config' (rTS h0) ([x, y], h) v = ([x, y], rev v @ h) \<and> (\<exists>hs. rev v @ h = [x, x] @ hs)"
real_root_pos_unique: fixes n :: "nat"   and y :: "real"   and x :: "real" assumes "0 < n"   and "0 \<le> y"   and "y ^ n = x" shows "root n x = y"
simps_6: fixes uz :: "tag" shows "catomic (cTag uz) = True"
up_to_I0o: fixes n :: "nat" shows "up_to {0..<n} = n"
sequential_def: fixes P :: "'a option process" shows "sequential P \<equiv> (\<forall>xs\<in>traces P. None \<notin> set (butlast xs)) \<and> (\<forall>xs\<in>sentences P. next_events P xs = {None})"
set_tries_empty: shows "set_tries (Trie None []) = {}"
inj_extend: fixes h :: "'c \<times> 'd \<Rightarrow> 'e" assumes "Extend h" shows "inj (extend h)"
tendsto_add_ereal: fixes x :: "ereal"   and y :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "\<bar>y\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
span_induct_alt: fixes x :: "('a, 'b) vec"   and S :: "('a, 'b) vec set"   and h :: "('a, 'b) vec \<Rightarrow> bool" assumes "x \<in> vec.span S"   and "h 0"   and "\<And>c x y. \<lbrakk>x \<in> S; h y\<rbrakk> \<Longrightarrow> h (c *s x + y)" shows "h x"
norm_le_transpose: fixes A :: "'a sq_mtx" shows "\<parallel>A\<parallel> \<le> \<parallel>A\<^sup>\<dagger>\<parallel>"
fres_canc1: fixes y :: "'c \<times> 'd"   and x :: "'c \<times> 'd" shows "less_eq_prod (times_prod' (prod.fres y x) x) y"
simps_6: fixes f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" shows "rec_nat f1 f2 0 = f1"
base: fixes C :: "'a literal multiset"   and CC :: "'a literal multiset set" assumes "C \<in> CC" shows "C \<in> saturate CC"
binrelchain_sym_rev: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "\<And>x y. P x y \<Longrightarrow> P y x"   and "binrelchain P xs" shows "binrelchain P (rev xs)"
abs_sgn: fixes a :: "'k" shows "sgn \<bar>a\<bar> = of_bool (a \<noteq> (0::'k))"
greedy_approach_leads_to_minBasis: fixes F :: "'a set" assumes "indep F"   and "\<forall>e\<in>carrier - F. \<not> indep (insert e F)"   and "no_smallest_element_skipped {} F" shows "minBasis F"
circline_set_fix_iff_circline_fix: fixes H' :: "circline"   and M :: "moebius"   and H :: "circline" assumes "circline_set H' \<noteq> {}" shows "(circline_set (moebius_circline M H) = circline_set H') = (moebius_circline M H = H')"
le_onorm: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "bounded_linear f" shows "norm (f x) / norm x \<le> onorm f"
grey_protects_white_reaches: fixes g :: "'a"   and w :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" assumes "(g grey_protects_white w) s" shows "(g reaches w) s"
g_nonneg: fixes y :: "nat" shows "0 \<le> g y"
assms: shows "<\<nu>x>a{b}.P \<longmapsto> \<tau> \<prec> P'"
fcomp_assoc: fixes f :: "'b \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd"   and h :: "'d \<Rightarrow> 'c" shows "f ;; g ;; h = f ;; (g ;; h)"
Interp_def: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.Interp S N C = ground_resolution_with_selection.interp S N C \<union> ground_resolution_with_selection.production S N C"
space_vimage_algebra: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and M :: "'d measure" shows "space (vimage_algebra X f M) = X"
dual_transaction_ik_is_transaction_send: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" assumes "wellformed_transaction T" shows "ik\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_strand T)) = trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T)"
SS_def: shows "SS = {y \<in> supp f. - y + x \<in> supp g}"
direct_decomp_appendD_1: fixes A :: "'b set"   and ss1 :: "'b set list"   and ss2 :: "'b set list" assumes "direct_decomp A (ss1 @ ss2)"   and "{} \<notin> set ss2" shows "direct_decomp (sum_list ` listset ss1) ss1"
ide_def: fixes a :: "'a" shows "S.ide a \<equiv> a \<cdot>\<^sub>S a \<noteq> S.null \<and> (\<forall>f. (f \<cdot>\<^sub>S a \<noteq> S.null \<longrightarrow> f \<cdot>\<^sub>S a = f) \<and> (a \<cdot>\<^sub>S f \<noteq> S.null \<longrightarrow> a \<cdot>\<^sub>S f = f))"
hoare_test: fixes p :: "'a"   and t :: "'a"   and q :: "'a" assumes "\<iota> p \<cdot> \<iota> t \<le> \<iota> q" shows "\<lbrace>\<iota> p\<rbrace> \<iota> t \<lbrace>\<iota> q\<rbrace>"
nfa_def: fixes M :: "'b nfa" shows "nfa M \<equiv> (nfa.init M \<subseteq> nfa.states M \<and> nfa.final M \<subseteq> nfa.states M) \<and> (\<forall>q x. q \<in> nfa.states M \<longrightarrow> nfa.nxt M q x \<subseteq> nfa.states M) \<and> finite (nfa.states M)"
simps: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and a1 :: "'g"   and a2 :: "'g" shows "\<tau>trsys.silent_move trsys \<tau>move a1 a2 = (\<exists>s s' tl. a1 = s \<and> a2 = s' \<and> trsys s tl s' \<and> \<tau>move s tl s')"
pderivs_snoc: fixes s :: "'b list"   and c :: "'b"   and r :: "'b rexp" shows "pderivs (s @ [c]) r = pderiv_set c (pderivs s r)"
R29_bc_2: shows "pmf (sds R29) c = 0"
af\<^sub>G_simps_1: fixes w :: "'a set list" shows "af\<^sub>G true w = true"
not_dbm_le_le_impl: fixes a :: "'c"   and b :: "'c" assumes "\<not> Le a \<prec> Le b" shows "b \<le> a"
assms_1: shows "igSwapAllIPresIGWlsAll MOD"
oiface_Some: fixes iface :: "iface" assumes "iface_conjunct oif1 oif2 = Some iface" shows "match_iface iface (p_oiface p) = (match_iface oif1 (p_oiface p) \<and> match_iface oif2 (p_oiface p))"
part_2_4: fixes v :: "nat \<Rightarrow> QDelta" assumes "v \<Turnstile>\<^sub>t t1" shows "v \<Turnstile>\<^sub>t t"
elims_2: fixes x :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list \<times> ((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set" assumes "shift1_inv x" obtains 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" where "x = (qs, B)" and "B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)} \<and> shift2_inv qs"
ln_ge_iff: fixes x :: "real"   and y :: "real" assumes "0 < x" shows "(y \<le> ln x) = (exp y \<le> x)"
Gt_lt_conv: fixes x :: "'a"   and y :: "'a" shows "(comp x y = Gt) = lt y x"
simps: fixes S :: "'b set"   and B :: "'b set set"   and a :: "'b set" shows "generated_topology S B a = (a = S \<or> (\<exists>U V. a = U \<inter> V \<and> generated_topology S B U \<and> generated_topology S B V) \<or> (\<exists>K. a = \<Union> K \<and> (\<forall>x. x \<in> K \<longrightarrow> generated_topology S B x)) \<or> (\<exists>b. a = b \<and> b \<in> B \<and> b \<subseteq> S))"
dim_col_transfer_rule: fixes A :: "'d mat"   and A' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" assumes "HMA_Connect.HMA_M A A'" shows "dim_col A = CARD('e)"
fs_deriv: fixes k :: "nat"   and x :: "real" assumes "k < N"   and "x \<in> {real_of_int a..} - Y" shows "(fs k has_vector_derivative fs (Suc k) x) (at x)"
simps_3: fixes a :: "'b"   and u :: "'b list"   and b :: "'b"   and v :: "'b list" shows "mismatch_pair (a # u) (b # v) = (if a = b then mismatch_pair u v else (a, b))"
in_minus_pairs: fixes p :: "'e \<times> 'e"   and A :: "('e \<times> 'e) set"   and B :: "('e \<times> 'e) set" shows "(p \<in> A -\<^sub>p B) = (p \<in> A \<and> p \<notin>\<^sub>p B)"
finite_subs: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "finite (subs gamma)"
Choiceo_undef: fixes \<alpha> :: "game option"   and \<beta> :: "game option" shows "(Choiceo \<alpha> \<beta> = undefg) = (\<alpha> = undefg \<or> \<beta> = undefg)"
le_approx_lemma_T: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<sqsubseteq> Q" shows "T Q \<subseteq> T P"
simps_11: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and f3 :: "('d \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'c"   and x2 :: "'e" shows "(case Load x2 of Const x \<Rightarrow> f1 x | Load x \<Rightarrow> f2 x | Apply x \<Rightarrow> f3 x) = f2 x2"
dickson_grading_dgrad_dummy: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and lcs :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "class.graded_dickson_powerprod plus minus zero lcs" shows "plus.dickson_grading plus (graded_dickson_powerprod.dgrad_dummy plus)"
unl_before: fixes v :: "state \<Rightarrow> 'a"   and s :: "state"   and t :: "state" shows "($v) (s, t) \<equiv> v s"
alluopairs_set: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "x \<in> set xs"   and "y \<in> set xs" shows "(x, y) \<in> set (alluopairs xs) \<or> (y, x) \<in> set (alluopairs xs)"
bal_inorder: fixes n :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and zs :: "'a list" assumes "n \<le> length xs"   and "bal n xs = (t, zs)" shows "xs = inorder t @ zs \<and> size t = n"
inf1D2: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "inf A B x" shows "B x"
dir2: shows "\<forall>s\<^sub>1 s\<^sub>2. S (s\<^sub>1, s\<^sub>2) \<sqsubseteq> (P s\<^sub>1|||Q s\<^sub>2)"
continuous_on_imp_continuous_within: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and t :: "'a set"   and x :: "'a" assumes "continuous_on s f"   and "t \<subseteq> s"   and "x \<in> s" shows "continuous (at x within t) f"
STCal_steps_1: fixes S :: "'procS"   and P' :: "('procS, 'procT) Proc" shows "SourceTerm S \<longmapsto>STCal Source Target* P' = (\<exists>S'. S' \<in>S P' \<and> S \<longmapsto>Source* S')"
sel_1: fixes x1 :: "'a list"   and x2 :: "'a" shows "prems_of (Infer x1 x2) = x1"
dioid_tests_axioms: shows "dioid_tests \<iota> -"
H: shows "\<forall>i<length xs. \<exists>x. fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma ! i = Some x"
E_ss: shows "E \<subseteq> {0..<N} \<times> {0..<N}"
all_nat_less: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>m\<le>n. P m) = (\<forall>m\<in>{0..n}. P m)"
min_trms_not_empty: fixes E :: "('a trm \<times> 'a trm) set" assumes "E \<noteq> {}" shows "min_trms E \<noteq> {}"
prim_conjug: fixes u :: "'b list"   and v :: "'b list" assumes "primitive u"   and "u \<sim> v" shows "primitive v"
leI: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
bind_refine: fixes M :: "'b nres"   and R' :: "('b \<times> 'c) set"   and M' :: "'c nres"   and f :: "'b \<Rightarrow> 'd nres"   and R :: "('d \<times> 'e) set"   and f' :: "'c \<Rightarrow> 'e nres" assumes "M \<le> \<Down> R' M'"   and "\<And>x x'. (x, x') \<in> R' \<Longrightarrow> f x \<le> \<Down> R (f' x')" shows "M \<bind> f \<le> \<Down> R (M' \<bind> f')"
ivl_subset_existence_ivl: fixes t :: "real"   and t0 :: "real"   and x0 :: "real" assumes "t \<in> existence_ivl t0 x0" shows "{t0..t} \<subseteq> existence_ivl t0 x0"
literal_less_def: fixes xs :: "'a literal"   and ys :: "'a literal" shows "(xs < ys) = (if atoms_of_lit xs = atoms_of_lit ys then case xs of a\<^sup>+ \<Rightarrow> False | x\<inverse> \<Rightarrow> case ys of x\<^sup>+ \<Rightarrow> True | a\<inverse> \<Rightarrow> False else atoms_of_lit xs < atoms_of_lit ys)"
has_bochner_integral_discrete_difference: fixes X :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "has_bochner_integral M f x = has_bochner_integral M g x"
sup_supF: fixes H :: "'b set"   and h :: "'b \<Rightarrow> real"   and c :: "('b \<times> real) set set"   and M :: "('b \<times> real) set set"   and E :: "'b set"   and p :: "'b \<Rightarrow> real"   and F :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "graph H h = \<Union> c"   and "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "\<exists>x. x \<in> c"   and "Subspace.subspace F E" shows "Subspace.subspace F H"
simps_2: fixes S :: "'node SDG_node set"   and cs :: "'edge list"   and a :: "'edge"   and as :: "'edge list" assumes "slice_edge S cs a" shows "slice_edges S cs (a # as) = a # slice_edges S (upd_cs cs [a]) as"
sub_composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> I"   and "b \<in> I" shows "a + b \<in> I"
k_i'_le: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "k_i' i e \<le> \<alpha>' / 2 ^ Suc i"
cases: fixes A :: "'c ring"   and B :: "'c ring" assumes "A \<lesssim> B" obtains "id \<in> ring_hom A B"
img_is_ring: shows "ring (S\<lparr>carrier := h ` carrier R\<rparr>)"
assms: shows "prefs_from_table xs = prefs_from_table ys"
ennreal_less_iff: fixes r :: "real"   and q :: "real" assumes "0 \<le> r" shows "(ennreal r < ennreal q) = (r < q)"
multiplicity_self: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "factorial_semiring.multiplicity times one p p = 1"
nabla_unfold: fixes x :: "'a" shows "\<nabla>x \<le>  |x\<rangle> \<nabla>x"
consistent_extend: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set"   and f :: "nat \<Rightarrow> ('c, 'b) fm list \<times> 'b"   and n :: "nat" assumes "infinite UNIV"   and "finite A"   and "consistent A (extend A S f n)"   and "finite (\<Union> (block_nominals ` extend A S f n))" shows "consistent A (extend A S f (Suc n))"
Suc_6: shows "x \<in> lvars (\<T> (l ! i))"
refine: fixes mrel :: "('c \<times> 'd) set"   and node_rel :: "('e \<times> 'f) set"   and node_eq_impl :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'e \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and G_impl :: "('e \<Rightarrow> bool, 'e \<Rightarrow> 'e list, 'e list, 'e \<Rightarrow> integer, 'c) gen_igbg_impl_scheme"   and G :: "('f, 'd) igb_graph_rec_scheme" assumes "impl_lasso_loc mrel node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G" shows "RETURN (find_lasso_tr node_eq_impl node_hash_impl node_def_hash_size G_impl) \<le> find_lasso_code node_eq_impl node_hash_impl node_def_hash_size G_impl"
in_CompI: fixes F :: "'a set"   and G :: "'a set"   and f :: "'a"   and g :: "'a"   and h :: "'a" assumes "is_iso_class F"   and "is_iso_class G"   and "f \<in> F"   and "g \<in> G"   and "g \<star> f \<cong> h" shows "h \<in> Maps.Comp G F"
conf_f_def2: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and ST :: "ty list"   and LT :: "ty err list"   and "is" :: "bytecode"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and ics :: "init_call_status" shows "conf_f P h sh (ST, LT) is (stk, loc, C, M, pc, ics) \<equiv> P,h \<turnstile> stk [:\<le>] ST \<and> P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT \<and> pc < length is \<and> P,h,sh \<turnstile>\<^sub>i (C, M, pc, ics)"
gen_length_def: fixes n :: "nat"   and xs :: "'b list" shows "List.gen_length n xs = n + length xs"
assms_2: shows "a \<noteq> c \<or> b \<noteq> d"
irreducible_const_poly_iff: fixes c :: "'b" shows "irreducible [:c:] = irreducible c"
interleave_cont_rev_conc: fixes b :: "'b list"   and a :: "'b list" shows "b @ a \<in> a \<otimes> b"
NSBseq_Bseq: fixes X :: "nat \<Rightarrow> 'a" assumes "NSBseq X" shows "Bseq X"
assms_6: shows "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)|\<^sub>r"
AssertionsImplyInvariantsAux: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "G \<rhd> \<lbrace> A \<rbrace> C,m,l \<lbrace> B \<rbrace> I" shows "(\<forall>s0 s t. A s0 s \<longrightarrow> I s0 s (heap s)) \<and> (\<forall>Mspec Minv Anno. MST\<down>(C, m) = Some (Mspec, Minv, Anno) \<longrightarrow> (\<forall>Q. Anno\<down>l = Some Q \<longrightarrow> (\<forall>s0 s. A s0 s \<longrightarrow> Q s0 s)))"
adjunction_in_strict_bicategory_axioms: shows "adjunction_in_strict_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg (h \<star> f) (g \<star> k) \<eta> \<epsilon>"
cont_const: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'c" assumes "class.ccpo Sup less_eq less" shows "cont luba orda Sup less_eq (\<lambda>x. c)"
ldistinct_ldropn: fixes xs :: "'c llist"   and n :: "nat" assumes "ldistinct xs" shows "ldistinct (ldropn n xs)"
collinear_def: fixes S :: "'b set" shows "collinear S = (\<exists>u. \<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u)"
integrable_fst': fixes f :: "'a \<times> 'b \<Rightarrow> 'd" assumes "integrable (count_space A \<Otimes>\<^sub>M count_space B') f" shows "integrable (count_space A) (\<lambda>x. LINT y|count_space B'. f (x, y))"
Limsup_bounded: fixes X :: "'b \<Rightarrow> 'd"   and C :: "'d"   and F :: "'b filter" assumes "\<forall>\<^sub>F n in F. X n \<le> C" shows "Limsup F X \<le> C"
image_iff: fixes z :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
b4: shows "(id - projection M) -` {0::'a} = M"
f2: shows "\<forall>mo. Deliver (i, Expunge e2 mo i) \<notin> set (history k)"