hcju/isabelleps huggingface dataset:
is_mgu_mgu\<^sub>t\<^sub>s: fixes ts :: "fterm set"   and \<theta> :: "char list \<Rightarrow> (char list, char list) Term.term" assumes "finite ts"   and "is_imgu \<theta> (fterm_to_iterm ` ts \<times> fterm_to_iterm ` ts)" shows "mgu\<^sub>t\<^sub>s (\<lambda>x. iterm_to_fterm (\<theta> x)) ts"
subintegral: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme"   and h1 :: "'c"   and h2 :: "'c" assumes "subdomain H R"   and "h1 \<in> H"   and "h2 \<in> H"   and "h1 \<otimes>\<^bsub>R\<^esub> h2 = \<zero>\<^bsub>R\<^esub>" shows "h1 = \<zero>\<^bsub>R\<^esub> \<or> h2 = \<zero>\<^bsub>R\<^esub>"
induced_homomorphism: fixes H :: "'a set" assumes "subgroup H G" shows "\<phi> \<in> hom (G\<lparr>carrier := H\<rparr>) (BijGroup E)"
not_gr_zero: fixes n :: "'d multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
all_same_arities_def: fixes F1 :: "'i \<Rightarrow> 'j fundef option"   and F2 :: "'i \<Rightarrow> 'k fundef option" shows "all_same_arities F1 F2 \<equiv> \<forall>f. rel_option (\<lambda>fd gd. arity fd = arity gd) (F1 f) (F2 f)"
elims_2: fixes x :: "('d, 'e, 'f) FSM" assumes "finite_FSM x" obtains M :: "('d, 'e, 'f) FSM" where "x = M" and "finite (nodes M) \<and> finite (inputs M) \<and> finite (outputs M)"
Fail_HInv3_q: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and p :: "Proc"   and d :: "Disk" assumes "Fail s s' q"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
agg_sum_commute: fixes f :: "'d \<times> 'd \<Rightarrow> 'c" shows "aggregation.sum_0 (\<lambda>k. aggregation.sum_0 (\<lambda>l. f (k, l)) {l. True}) {k. True} = aggregation.sum_0 (\<lambda>l. aggregation.sum_0 (\<lambda>k. f (k, l)) {k. True}) {l. True}"
finite_lepoll_infinite: fixes A :: "'b set"   and B :: "'c set" assumes "infinite A"   and "finite B" shows "B \<lesssim> A"
amin_le_plus: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "y \<in> carrier K" shows "amin (v x) (v y) \<le> v (x \<plusminus> y)"
bv_msb_norm: fixes w :: "bit list" shows "bv_msb (norm_signed w) = bv_msb w"
worker_wrapper_fusion_new: fixes wrap :: "'b \<rightarrow> 'a"   and unwrap :: "'a \<rightarrow> 'b"   and body :: "'a \<rightarrow> 'a"   and body' :: "'b \<rightarrow> 'b" assumes "wrap oo unwrap = ID"   and "unwrap\<cdot>\<bottom> = \<bottom>"   and "unwrap oo body oo wrap = body' oo unwrap oo wrap" shows "fix\<cdot>body = wrap\<cdot>(fix\<cdot>body')"
set_plus_mono3: fixes a :: "'a"   and C :: "'a set"   and D :: "'a set" assumes "a \<in> C" shows "a +o D \<subseteq> C + D"
continuous_on_closed_invariant: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on s f = (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
exhaust: fixes y :: "dir" obtains "y = dir.Left"   | "y = dir.Right"
filterlim_ident: fixes F :: "'a filter" shows "LIM x F. x :> F"
Sym: fixes H :: "fm set"   and t :: "tm"   and u :: "tm" assumes "H \<turnstile> t EQ u" shows "H \<turnstile> u EQ t"
Obs_in_L: fixes u :: "'a"   and ou :: "'c"   and out :: "'d \<Rightarrow> 'a \<Rightarrow> 'c"   and s :: "('d, 'a, 'b) state" shows "(Obs u ou \<in> GM_sec_model.L out s) = (ou = out (local.getGMState s) u)"
interval_bounds'_1: fixes a :: "'a"   and b :: "'a" assumes "cbox a b \<noteq> {}" shows "interval_upperbound (cbox a b) = b"
term_powerprod_axioms: shows "term_powerprod to_pair_unit fst"
intros_7: shows "IK_nf (K \<circ> I \<circ> K)"
rep_repa_nc: shows "\<forall>no. no \<notin> set prx \<longrightarrow> rep no = repa no"
dom_if_Some: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c option" shows "dom (\<lambda>x. if P x then Some (f x) else g x) = {x. P x} \<union> dom g"
denom_cf2_pos: fixes x :: "real" shows "0 < x\<^sup>2 - 6 * x + 12"
assms: shows "((\<lambda>x. 0::'b) has_derivative F) (at x)"
double_compl: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
is_reduced_GB_unique: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_reduced_GB A"   and "is_reduced_GB B"   and "pmdl A = pmdl B" shows "A = B"
map_iterator_genord_dom_filter_correct: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and R :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'f \<Rightarrow> bool"   and P :: "'e \<times> 'f \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "map_iterator_genord it m R"   and "\<And>k1 v1 k2 v2. \<lbrakk>m k1 = Some v1; P (k1, v1); m k2 = Some v2; P (k2, v2); R (k1, v1) (k2, v2)\<rbrakk> \<Longrightarrow> R' k1 k2" shows "set_iterator_genord (map_iterator_dom_filter P it) {k. \<exists>v. m k = Some v \<and> P (k, v)} R'"
maxsimp_singleton: fixes w :: "'w" assumes "w \<in> W" shows "SimplicialComplex.maxsimp \<Sigma> (dual_order.poset_simplex_map \<P> {w})"
elims_1: fixes x :: "'b option"   and xa :: "'b option"   and y :: "bool" assumes "(x \<sqsubseteq> xa) = y" obtains xb :: "'b" and ya :: "'b" where "x = Some xb" and "xa = Some ya__" and "y = (xb \<sqsubseteq> ya__)"   | ya :: "'b option" where "x = None" and "xa = ya" and "y"   | uu_ :: "'b" where "x = Some uu_" and "xa = None" and "\<not> y"
cancel_separate_p_fun_converse: fixes x :: "'a" assumes "is_p_fun x" shows "x\<^sup>T\<^sup>\<star> ; x\<^sup>\<star> = x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>"
auxlist'_eq: shows "auxlist' = (case auxlist0 of [] \<Rightarrow> [(\<tau> \<sigma> ne, rel2)] | x # auxlist' \<Rightarrow> if fst x = \<tau> \<sigma> ne then (fst x, snd x \<union> rel2) # auxlist' else (\<tau> \<sigma> ne, rel2) # x # auxlist')"
eq1: shows "lookup ((monomial c (pp_of_term v) when component_of_term v = component_of_term u) * proj_poly (component_of_term u) p) (pp_of_term u) = (lookup (monomial c (pp_of_term v) * proj_poly (component_of_term u) p) (pp_of_term u) when component_of_term v = component_of_term u)"
dvd_div_mult2_eq: 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 b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times (times b c) a" shows "divide a (times b c) = divide (divide a b) c"
sgn_sq_mtx_def: fixes A :: "'a sq_mtx" shows "sgn A = inverse (\<parallel>A\<parallel>) *\<^sub>R A"
thetaIfZO_ZObis: shows "thetaIfZO \<subseteq> ZObis"
tendsto_snd: fixes f :: "'a \<Rightarrow> 'b \<times> 'c"   and a :: "'b \<times> 'c"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. snd (f x)) \<longlongrightarrow> snd a) F"
outer_invar_loc_axioms: shows "outer_invar_loc G it D"
additive: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and g :: "'c"   and g' :: "'c" assumes "FGModuleHom G smult V smult' T"   and "g \<in> V"   and "g' \<in> V" shows "T (g + g') = T g + T g'"
bdd_above_uminus: fixes X :: "'a set" shows "bdd_above (uminus ` X) = bdd_below X"
rewrite_X_def: fixes \<phi> :: "'b ltln" shows "rewrite_X \<phi> = to_ltln (rewrite_X_enat \<phi>)"
MM: shows "Ik.MModel (ProblemIkTpartG_GEModel.intT prot protFw eintT eintP) (ProblemIkTpartG_GEModel.intF eintF) (ProblemIkTpartG_GEModel.intP eintP)"
in_in_arcs_conv: fixes e :: "'c"   and G :: "('d, 'c) pre_digraph"   and v :: "'d" shows "(e \<in> in_arcs G v) = (e \<in> arcs G \<and> head G e = v)"
inf_absorb2: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and x :: "'e" assumes "class.semilattice_inf inf less_eq less"   and "less_eq y x" shows "inf x y = y"
Ints_cases: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and q :: "'a" assumes "class.ring_1 one times plus zero minus uminus"   and "q \<in> ring_1.Ints one plus zero minus" obtains z :: "int" where "q = ring_1.of_int one plus zero minus z"
P_def: shows "P = (\<lambda>D A. (\<forall>(t, t')\<in>set D. (fv t \<union> fv t') \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}) \<and> fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {})"
x: shows "x = (\<Sum>a\<in>A. q a *s a)"
tendsto_set_lebesgue_integral_at_top: fixes a :: "real"   and M :: "real measure"   and f :: "real \<Rightarrow> 'c" assumes "\<And>b. a \<le> b \<Longrightarrow> {a..b} \<in> sets M"   and "set_integrable M {a..} f" shows "((\<lambda>b. set_lebesgue_integral M {a..b} f) \<longlongrightarrow> set_lebesgue_integral M {a..} f) at_top"
ipd_icd_greatest_cd_not_ipd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and k :: "nat"   and j :: "nat" assumes "\<pi> m = ipd (\<pi> k)"   and "\<forall>n\<in>{k..<m}. \<pi> n \<noteq> ipd (\<pi> k)"   and "k < m"   and "m icd\<^bsup>\<pi>\<^esup>\<rightarrow> j" shows "j = (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> \<pi> m)"
idempotent: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "Idem\<^bsub>L\<^esub> f"   and "x \<in> carrier L" shows "f (f x) .=\<^bsub>L\<^esub> f x"
func: shows "f \<in> carrier \<rightarrow> carrier\<^bsub>S\<^esub>"
seqI': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "category C"   and "partial_magma.in_hom C f a b"   and "partial_magma.in_hom C g b c" shows "partial_magma.arr C (C g f)"
sWlsDisj_def: fixes SEM :: "('b, 'c, 'd, 'e, 'f, 'g, 'h) semDom_scheme" shows "sWlsDisj SEM \<equiv> \<forall>s s' sX. sWls SEM s sX \<and> sWls SEM s' sX \<longrightarrow> s = s'"
indexed_weak_nominal_ts_axioms: shows "indexed_weak_nominal_ts TYPE('idx) \<tau> (\<turnstile>) (\<rightarrow>)"
conforms_NormI: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv" assumes "(x, s)\<Colon>\<preceq>(G, L)" shows "Norm s\<Colon>\<preceq>(G, L)"
finite_UnionD: fixes A :: "'b set set" assumes "finite (\<Union> A)" shows "finite A"
fresh_atom_at_base: fixes a :: "atom"   and b :: "'c" shows "a \<sharp> atom b = a \<sharp> b"
Chains_relation_of: fixes C :: "'c set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "C \<in> Chains (relation_of P A)" shows "C \<subseteq> A"
a1: shows "get_S (cpu_reg_val PSR s1) = 0 \<and> good_context s1 \<and> get_delayed_pool s1 = [] \<and> get_trap_set s1 = {} \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> get_delayed_pool s2 = [] \<and> get_trap_set s2 = {} \<and> good_context s2 \<and> user_seq_exe n s1 \<and> user_seq_exe n s2 \<and> low_equal s1 s2"
UN_simps_10: fixes B :: "'g \<Rightarrow> 'f set"   and f :: "'h \<Rightarrow> 'g"   and A :: "'h set" shows "\<Union> (B ` f ` A) = (\<Union>a\<in>A. B (f a))"
power_eq_0_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.semiring_1_no_zero_divisors one times plus zero" shows "(power.power one times a n = zero) = (a = zero \<and> 0 < n)"
atlas_is_atlas: fixes a1 :: "('b, 'f) chart"   and a2 :: "('b, 'f) chart" assumes "a1 \<in> dest.atlas"   and "a2 \<in> dest.atlas" shows "k-smooth_compat a1 a2"
fundfold_comp_endomorphism: fixes ss :: "'a permutation list" assumes "order.greater_eq (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) (set ss)" shows "ChamberComplexEndomorphism X (fold fst (map Spair ss))"
absolutely_integrable_restrict_UNIV: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "((\<lambda>x. if x \<in> S then f x else (0::'d)) absolutely_integrable_on UNIV) = (f absolutely_integrable_on S)"
IH: fixes w :: "'a" shows "(deriv ^^ j) (\<lambda>w. (w - z) ^ n) w = pochhammer (of_nat (Suc n - j)) j * (w - z) ^ (n - j)"
comp_inv_arr': fixes f :: "'c" assumes "C.iso f" shows "C.inv f \<cdot>\<^sub>C f = C.dom f"
atomic_step_ipc_def: fixes tid :: "thread_id_t"   and dir :: "ipc_direction_t"   and stage :: "ipc_stage_t"   and partner :: "thread_id_t"   and page :: "page_t"   and s :: "state_t" shows "atomic_step_ipc tid dir stage partner page s \<equiv> case stage of BUF page' \<Rightarrow> case dir of SEND \<Rightarrow> set_object_value (PAGE page') (obj s (PAGE page)) s | RECV \<Rightarrow> s | _ \<Rightarrow> s"
ins_list_idem_eq_isin: fixes xs :: "'c list"   and x :: "'c" assumes "sorted_less xs" shows "(x \<in> set xs) = (ins_list x xs = xs)"
Inter_FullL: fixes r :: "'a language" shows "Inter \<Sigma>\<^sup>* r = r"
nn_set_integral_space: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" shows "set_nn_integral M (space M) f = integral\<^sup>N M f"
inext_neq_iMin_not_singleton: fixes I :: "nat set"   and n :: "nat" assumes "I \<noteq> {}"   and "\<nexists>a. I = {a}" shows "inext n I \<noteq> iMin I"
nlistToTree_list: fixes nl :: "('a, 'b) Node list" shows "FingerTreeStruc.toList (FingerTreeStruc.nlistToTree nl) = FingerTreeStruc.nlistToList nl"
subst_subst1: fixes a :: "coname"   and Q :: "trm"   and b :: "coname"   and x :: "name"   and y :: "name"   and P :: "trm"   and M :: "trm" assumes "a \<sharp> (Q, b)"   and "x \<sharp> (y, P, Q)"   and "b \<sharp> Q"   and "y \<sharp> P" shows "M{x:=<a>.P}{b:=y.Q} = M{b:=y.Q}{x:=<a>.P{b:=y.Q}}"
cyclic_group_def: fixes G :: "('a, 'b) cyclic_group_scheme" shows "Cyclic_Group.cyclic_group G \<equiv> Group.group G \<and> cyclic_group_axioms G"
disjoint_projection: fixes X :: "'b set"   and Y :: "'b set"   and l :: "'b list" assumes "X \<inter> Y = {}" shows "l \<upharpoonleft> X \<upharpoonleft> Y = []"
ball_max_Un: fixes a :: "'a"   and r :: "real"   and s :: "real" shows "ball a (max r s) = ball a r \<union> ball a s"
f_range_trans: shows "f ` range to_nat = range to_nat"
all_opaque: shows "set (opaque x) \<subseteq> set vs"
intro: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>less_eq x y; less_eq y x\<rbrakk> \<Longrightarrow> x = y" shows "class.order_axioms less_eq"
terminates_on_maps_raw_Some: fixes s :: "'e"   and f :: "'h \<Rightarrow> ('f, 'g) generator \<times> 'g"   and g :: "'e \<Rightarrow> ('h, 'e) step"   and g'' :: "('f, 'g) generator"   and s'' :: "'g" assumes "(s, None) \<in> terminates_on (maps_raw f g)" shows "(s, Some (g'', s'')) \<in> terminates_on (maps_raw f g)"
times_fun_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(f * g) x = f x * g x"
octo_of_real_eq_Octo: fixes a :: "real" shows "octo_of_real a = Octo a 0 0 0 0 0 0 0"
X_subseteq_XD_smallest: shows "X \<subseteq> XD_smallest"
def_scheme: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b" shows "f X Y = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> \<and> (\<delta> Y) \<tau> = true \<tau> then g (X \<tau>) (Y \<tau>) else invalid \<tau>)"
asm: shows "v = Tensor.mat_of_cols_list 8 [[\<alpha> / complex_of_real (sqrt 2), \<beta> / complex_of_real (sqrt 2), \<beta> / complex_of_real (sqrt 2), \<alpha> / complex_of_real (sqrt 2), 0, 0, 0, 0]]"
csol_mem_csols: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "(csol t0 x0 t, t) \<in> csols t0 x0"
assms_2: shows "M' \<longrightarrow>\<^sub>l M''"
frees_strip_comb: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "frees t = frees (fst (term.strip_comb unapp t)) |\<union>| pre_term.freess frees (snd (term.strip_comb unapp t))"
leadsTo_UNIV: fixes F :: "'c program"   and A :: "'c set" shows "F \<in> A \<longmapsto> UNIV"
bin_sign_simps_1: shows "bin_sign 0 = 0"
univ_commute: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "equiv A r"   and "f respects r"   and "x \<in> A" shows "univ f (Equiv_Relations.proj r x) = f x"
bindK_def: shows "bindK \<equiv> \<Lambda> m k c. m\<cdot>(\<Lambda> a. k\<cdot>a\<cdot>c)"
sin_coeff_def: shows "sin_coeff = (\<lambda>n. if even n then 0 else (- 1) ^ ((n - Suc 0) div 2) / fact n)"
elims: fixes x :: "nat list"   and xa :: "'var \<Rightarrow> 'var"   and xb :: "('form, 'rule, 'subst, 'var) itnode"   and y :: "('form, 'rule, 'subst, 'var) itnode" assumes "globalize_node x xa xb = y" obtains "is" :: "nat list" and f :: "'var \<Rightarrow> 'var" and n :: "('form, 'rule) graph_node" and p :: "'form" and i :: "nat" and s :: "'subst" where "x = isa__" and "xa = fa__" and "xb = I n p i__ s__" and "y = I n p (isidx isa__) (subst_renameLCs fa__ s__)"   | "is" :: "nat list" and f :: "'var \<Rightarrow> 'var" and i :: "nat" and s :: "'subst" where "x = is" and "xa = f" and "xb = H i s" and "y = H (isidx is) (subst_renameLCs f s)"
ResSet_image: fixes p :: "nat"   and a :: "int"   and m :: "int"   and f :: "int \<Rightarrow> int" assumes "GAUSS p a"   and "0 < m"   and "ResSet m (GAUSS.A p)"   and "\<forall>x\<in>GAUSS.A p. \<forall>y\<in>GAUSS.A p. [f x = f y] (mod m) \<longrightarrow> x = y" shows "ResSet m (f ` GAUSS.A p)"
assms_1: fixes n :: "nat" shows "\<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s \<and> n = f s} c {\<lambda>s. P s \<and> f s < n}"
assms_3: fixes x :: "'a" shows "(V has_derivative V' x) (at x)"
simulation_silent1: fixes s1 :: "'s1"   and s2 :: "'s2"   and s1' :: "'s1" assumes "s1 \<approx> s2"   and "s1 -\<tau>1\<rightarrow> s1'" shows "s1' \<approx> s2 \<and> \<mu>1\<^sup>+\<^sup>+ s1' s1 \<or> (\<exists>s2'. s2 -\<tau>2\<rightarrow>+ s2' \<and> s1' \<approx> s2')"
ACIcl_le_ACIEQ: fixes r :: "'d rexp"   and s :: "'d rexp" assumes "r ~~ s" shows "r \<approx> s"
elims_3: fixes x :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "'d list" assumes "\<not> sorted_wrt x xa" obtains P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" and xb :: "'d" and ys :: "'d list" where "x = P" and "xa = xb # ys" and "\<not> (Ball (set ys) (P xb) \<and> sorted_wrt P ys)"
decn: fixes s :: "'a"   and v :: "'b"   and sn :: "'a"   and sz :: "'a"   and M :: "('a \<times> ('a, 'b) Op) set"   and vs :: "'b \<Rightarrow> nat"   and n :: "nat" assumes "(s, Dec v sn sz) \<in> M"   and "vs v = Suc n" shows "((s, vs), sn, \<lambda>x. if x = v then n else vs x) \<in> step M"
ex1: shows "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> P1 x = P1'"
intro: fixes final :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>s. final s \<Longrightarrow> finished step s" shows "semantics step final"
BuildSeq_1: fixes B :: "hf \<Rightarrow> bool"   and y :: "hf"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool" assumes "B y" shows "BuildSeq B C \<lbrace>\<langle>0, y\<rangle>\<rbrace> 0 y"
preserves_dom_def: fixes G :: "('o, 'a, 'a set, 'a set_arrow) functor" shows "preserves_dom G \<equiv> \<forall>f\<in>Ar. G \<^bsub>\<o>\<^esub> (Dom f) = Dom\<^bsub>BB\<^esub> G \<^bsub>\<a>\<^esub> f"
pivot_tableau_def: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and t :: "(nat \<times> linear_poly) list" shows "pivot_tableau x\<^sub>i x\<^sub>j t \<equiv> let eqa = eq_for_lvar t x\<^sub>i; eq' = pivot_eq eqa x\<^sub>j in map (\<lambda>e. if lhs e = lhs eqa then eq' else subst_var_eq x\<^sub>j (rhs eq') e) t"
wt_sym\<^sub>h_def: fixes n :: "nat" shows "wt_sym\<^sub>h n = 1"
pcr_pmf_def: fixes T :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "pcr_pmf T \<equiv> rel_fun T (=) OO cr_pmf"
cases: fixes a1 :: "'a tag_const"   and a2 :: "'a tag_const" assumes "a1 \<le> a2" obtains n :: "'a" and m :: "'a" where "a1 = \<tau>\<^sub>c\<^sub>s\<^sub>t n" and "a2 = \<tau>\<^sub>c\<^sub>s\<^sub>t m" and "n \<le> m"
plus_fun_conv: fixes a :: "'a \<Rightarrow> 'b option"   and b :: "'a \<Rightarrow> 'b option" shows "a + b = a ++ b"
M_def: shows "M = 188 * lambda ^ 3 * (C + deltaG TYPE('b))"
\<pi>_le_self': fixes x :: "real" assumes "1 \<le> x" shows "\<pi> x \<le> x - 1"
nn_integral_embed_spmf_eq_1: shows "\<integral>\<^sup>+ x. ennreal (case x of None \<Rightarrow> 1 - enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV) | Some x' \<Rightarrow> max 0 (f x')) \<partial>count_space UNIV = 1"
transform_graph_aux_def: shows "transform_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>transform_aux a e x. x1 = (a, App e x) \<and> x2 = TransApp a (((\<lambda>uu_. uu_) \<circ> transform_aux) (PropApp a, e)) x \<and> p (PropApp a, e) (((\<lambda>uu_. uu_) \<circ> transform_aux) (PropApp a, e))) \<or> (\<exists>transform_aux a x e. x1 = (a, Lam [x]. e) \<and> x2 = TransLam a x (((\<lambda>uu_. uu_) \<circ> transform_aux) (PropLam a, e)) \<and> p (PropLam a, e) (((\<lambda>uu_. uu_) \<circ> transform_aux) (PropLam a, e))) \<or> (\<exists>transform_aux a x. x1 = (a, Var x) \<and> x2 = TransVar a x) \<or> (\<exists>transform_aux a \<Gamma> e. x1 = (a, Terms.Let \<Gamma> e) \<and> x2 = TransLet (AnalLet \<Gamma> e a) (map_transform (\<lambda>x0 x1. ((\<lambda>uu_. uu_) \<circ> transform_aux) (x0, x1)) (PropLetHeap (AnalLet \<Gamma> e a)) \<Gamma>) (((\<lambda>uu_. uu_) \<circ> transform_aux) (PropLetBody (AnalLet \<Gamma> e a), e)) \<and> HOL.induct_forall (\<lambda>x. HOL.induct_forall (\<lambda>ea. HOL.induct_forall (\<lambda>aa. HOL.induct_implies ((x, ea) \<in> set \<Gamma>) (p (aa, ea) (((\<lambda>uu_. uu_) \<circ> transform_aux) (aa, ea)))))) \<and> p (PropLetBody (AnalLet \<Gamma> e a), e) (((\<lambda>uu_. uu_) \<circ> transform_aux) (PropLetBody (AnalLet \<Gamma> e a), e))) \<or> (\<exists>transform_aux a b. x1 = (a, Bool b) \<and> x2 = Bool b) \<or> (\<exists>transform_aux a scrut e1 e2. x1 = (a, scrut ? e1 : e2) \<and> x2 = (((\<lambda>uu_. uu_) \<circ> transform_aux) (PropIfScrut a, scrut) ? ((\<lambda>uu_. uu_) \<circ> transform_aux) (a, e1) : ((\<lambda>uu_. uu_) \<circ> transform_aux) (a, e2)) \<and> p (PropIfScrut a, scrut) (((\<lambda>uu_. uu_) \<circ> transform_aux) (PropIfScrut a, scrut)) \<and> p (a, e1) (((\<lambda>uu_. uu_) \<circ> transform_aux) (a, e1)) \<and> p (a, e2) (((\<lambda>uu_. uu_) \<circ> transform_aux) (a, e2))))"
plus_permutation_abs_eq: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "bij f"   and "bij g" shows "Abs_permutation f + Abs_permutation g = Abs_permutation (f \<circ> g)"
rel_state_elim: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "(heap, 'g) state"   and g :: "'h Heap"   and heap :: "heap" assumes "local.rel_state R f g"   and "P heap" obtains heap' :: "heap" and v :: "'g" and v' :: "'h" where "run_state f heap = (v, heap')" and "execute g heap = Some (v', heap')" and "R v v'" and "P heap'"
chain_bounds_unique2: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and X :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and x :: "'a"   and z :: "'a" assumes "[f[a .. c]X]"   and "[g[x .. z]X]"   and "3 \<le> card X" shows "a = x \<and> c = z \<or> a = z \<and> c = x"