hcju/isabelleps huggingface dataset:
strict_sorted_imp_sorted: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.linorder less_eq less"   and "linorder.strict_sorted less xs" shows "linorder.sorted less_eq xs"
e_nth_snoc_small: fixes n :: "nat"   and b :: "nat"   and z :: "nat" assumes "n < e_length b" shows "e_nth (e_snoc b z) n = e_nth b n"
alternating_list_split: fixes m :: "nat"   and n :: "nat"   and s :: "'b"   and t :: "'b" shows "alternating_list (m + n) s t = alternating_list m s t @ (if gcd_nat.greater_eq m 2 then alternating_list n s t else alternating_list n t s)"
trace_comm: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "trace (A * B) = trace (B * A)"
left_commute: fixes b :: "'g"   and a :: "'g"   and c :: "'g" shows "b + (a + c) = a + (b + c)"
l_minus: fixes x :: "'a"   and y :: "'a" shows "- x * y = - (x * y)"
f'_nonneg: fixes e :: "'v \<times> 'v" shows "0 \<le> f' e"
clausal_saturated_model: fixes N :: "'a literal multiset set" assumes "saturated N"   and "{#} \<notin> N" shows "J_of N \<TTurnstile>s N"
RSpan_contains_spanset: fixes ms :: "'b list" assumes "set ms \<subseteq> V" shows "set ms \<subseteq> RSpan ms"
Var_eq_apps_conv: fixes m :: "nat"   and s :: "dB"   and ss :: "dB list" shows "(dB.Var m = s \<degree>\<degree> ss) = (dB.Var m = s \<and> ss = [])"
hyps: fixes P :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P" shows "\<forall>\<rho>\<in>density_states. trace (wlp S P * \<rho>) = trace (P * denote S \<rho>) + trace \<rho> - trace (denote S \<rho>)"
simps_11: fixes x3 :: "'b"   and x2 :: "'a" shows "Fault x3 \<noteq> Abrupt x2"
steps: shows "\<forall>i. (f i, f (Suc i)) \<in> {(A, B). (A, B) \<in> carrier_mat n n \<times> carrier_mat n n \<and> mat_arc_pos B \<and> mat_gt_arc A B}"
that: fixes x :: "'b" assumes "fmlookup m x = Some y"   and "x |\<in>| A" shows "thesis"
eventually_happens: fixes P :: "'c \<Rightarrow> bool"   and net :: "'c filter" assumes "eventually P net" shows "net = bot \<or> (\<exists>x. P x)"
control_withinI: fixes \<sigma> :: "('e \<times> ('a, 'b, 'c, 'd) seqp) set"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp" assumes "\<And>p. p \<in> Range \<sigma> \<Longrightarrow> \<exists>pn. p \<in> subterms (\<Gamma> pn)" shows "control_within \<Gamma> \<sigma>"
node_state_rel_def: shows "node_state_rel \<equiv> {(- 1, DONE)} \<union> {(int k, STACK k) |k. True}"
add_right_mono: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b" shows "a + c \<subseteq># b + c"
fst_conv: fixes x1 :: "'b"   and x2 :: "'c" shows "fst (x1, x2) = x1"
sup_inf_distrib2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a" assumes "class.distrib_lattice inf less_eq less sup" shows "sup (inf y z) x = inf (sup y x) (sup z x)"
m_closed: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and y :: "'b" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"
redT_updLs_def: fixes ls :: "('d, 'e) locks"   and t :: "'e"   and las :: "'d lock_actions" shows "redT_updLs ls t las \<equiv> (\<lambda>(l, la). upd_locks l t la) \<circ>$ ($ls, las$)"
assms_3: fixes X :: "'a set" assumes "\<And>Y. \<lbrakk>X \<subset> Y; Y \<subseteq> A\<rbrakk> \<Longrightarrow> P Y" shows "P X"
assms_7: shows "F\<epsilon> = (\<lambda>r. \<forall>rel\<in>S. \<forall>(a, b, c)\<in>X rel. \<exists>y>r. \<forall>x\<in>{r<..y}. rel (a * x\<^sup>2 + b * x + c) 0)"
sum_list_map_upto_coeff_limit: fixes L :: "rat list"   and i :: "nat" assumes "length L \<le> i" shows "Abstract_Linear_Poly.coeff (list_to_lpoly L) i = 0"
add_commute: fixes a :: "'a hrel"   and b :: "'a hrel" shows "a \<sqinter> b = b \<sqinter> a"
ridmap_def: fixes R :: "('d, 'e) Ring_scheme" shows "ridmap R = (\<lambda>x\<in>carrier R. x)"
simps_1: fixes x :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators" shows "first_srcNet (x \<oplus> y) = first_srcNet x"
fixp_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "monotone orda orda f" shows "a.fixp f = f (a.fixp f)"
trcl_uncons_cases: fixes c :: "'c"   and e :: "'d"   and w :: "'d list"   and c' :: "'c"   and S :: "('c \<times> 'd \<times> 'c) set" assumes "(c, e # w, c') \<in> trcl S" obtains ch :: "'c" where "(c, e, ch) \<in> S" and "(ch, w, c') \<in> trcl S"
state_in_range_step: fixes a :: "nat"   and A :: "(action \<times> nat) list"   and b :: "cell list"   and c :: "cell list"   and st :: "nat"   and l :: "cell list"   and r :: "cell list" assumes "a \<le> length A div 2"   and "step0 (a, b, c) A = (st, l, r)"   and "tm_wf0 A" shows "st \<le> length A div 2"
restrict_group_imp_subgroup: fixes H :: "'a set set" assumes "H \<subseteq> carrier (G Mod N)"   and "Group.group ((G Mod N)\<lparr>carrier := H\<rparr>)" shows "subgroup H (G Mod N)"
borel_measurable_divide: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x / g x) \<in> borel_measurable M"
projection_rank1: fixes t :: "'b"   and x :: "'b" shows "projection (cspan {t}) x = (\<langle>t, x\<rangle> / \<langle>t, t\<rangle>) *\<^sub>C t"
Cauchy_Binet1: shows "Determinant.det (A * B) = (\<Sum>f\<in>local.F_strict. Determinant.det (submatrix A UNIV (f ` {0..<n})) * Determinant.det (submatrix B (f ` {0..<n}) UNIV))"
fmap_rel_in_dom_iff: assumes "(aa, a'a) \<in> \<langle>K, V\<rangle>fmap_rel"   and "(a, a') \<in> K" shows "(a' \<in># dom_m a'a) = (a \<in># dom_m aa)"
fPow_top: fixes A :: "'a fset" shows "A |\<in>| fPow A"
posX: shows "0 \<le> PC l (te p (i + 1)) - PC l (te q (i + 1))"
to_mtx_ith_1: fixes A :: "((real, 'a) vec, 'a) vec"   and i1 :: "'a" shows "to_mtx A $$ i1 = A $ i1"
igFreshInp_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme"   and ys :: "'c"   and y :: "'f"   and inp :: "'j \<Rightarrow> 'g option" shows "igFreshInp MOD ys y inp \<equiv> liftAll (igFresh MOD ys y) inp"
SF1: fixes P :: "state \<Rightarrow> bool"   and N :: "state \<times> state \<Rightarrow> bool"   and Q :: "state \<Rightarrow> bool"   and A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a"   and F :: "'b \<Rightarrow> bool" assumes "\<turnstile> $P \<and> N \<longrightarrow> P$ \<or> Q$"   and "\<turnstile> ($P \<and> N) \<and> <A>_v \<longrightarrow> Q$"   and "\<turnstile> \<box>P \<and> \<box>N \<and> \<box>F \<longrightarrow> \<diamond>Enabled (<A>_v)" shows "\<turnstile> \<box>N \<and> SF(A)_v \<and> \<box>F \<longrightarrow> (P \<leadsto> Q)"
prems_1: shows "obsp \<sigma> i \<noteq> obsp \<sigma>' i' \<or> cs\<^bsup>\<pi>\<^esup> i \<noteq> cs\<^bsup>\<pi>'\<^esup> i'"
interval_def: fixes l :: "nat"   and r :: "enat" shows "interval l r = (if enat l \<le> r then Abs_\<I> (l, r) else undefined)"
wordinterval_CIDR_split1_preserve: fixes r :: "'b wordinterval"   and s :: "'b prefix_match"   and u :: "'b wordinterval" assumes "wordinterval_CIDR_split1 r = (Some s, u)" shows "wordinterval_eq (wordinterval_union (prefix_to_wordinterval s) u) r"
lat_filters: fixes X :: "'a set" shows "filterp X = (X \<noteq> {} \<and> upset_setp X \<and> (\<forall>x\<in>X. \<forall>y\<in>X. x \<sqinter> y \<in> X))"
assms: shows "total1 (\<phi> i) \<or> total1 (\<phi> j)"
fbrelation_frrestriction: fixes r :: "V"   and A :: "V" shows "fbrelation (r \<restriction>\<^sup>r\<^sub>\<bullet> A)"
crel_vs_elim: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and v :: "'c"   and v\<^sub>T :: "(heap, 'd) state"   and M :: "heap" assumes "state_dp_consistency.crel_vs R v v\<^sub>T"   and "state_dp_consistency.cmem M"   and "P M" obtains v' :: "'d" and M' :: "heap" where "run_state v\<^sub>T M = (v', M')" and "R v v'" and "state_dp_consistency.cmem M'" and "P M'"
repeatable_both_both2: fixes t :: "'b ttree"   and t' :: "'b ttree" assumes "repeatable t" shows "t' \<otimes>\<otimes> t \<otimes>\<otimes> t = t' \<otimes>\<otimes> t"
hom_eq: fixes f :: "('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme"   and f' :: "('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c" assumes "f \<in> hom (free_Abelian_group S) H"   and "\<And>x. x \<in> carrier (free_Abelian_group S) \<Longrightarrow> f' x = f x" shows "f' \<in> hom (free_Abelian_group S) H"
simps_8: fixes f :: "'m \<Rightarrow> 'l" shows "map f [] = []"
inf_mono: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and c :: "'d"   and b :: "'d"   and d :: "'d" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (inf a b) (inf c d)"
simps_2: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and xs :: "'e list"   and x :: "'e" shows "ipurge_tr_aux I D U (xs @ [x]) = (if \<exists>v\<in>sinks_aux I D U xs. (v, D x) \<in> I then ipurge_tr_aux I D U xs else ipurge_tr_aux I D U xs @ [x])"
chi2: fixes val :: "('a \<Rightarrow> bool) set"   and \<tau> :: "nat \<Rightarrow> 'a" assumes "val \<subseteq> A"   and "\<tau> \<Turnstile>\<^sub>p chi val" shows "val = stval (\<tau> 0)"
winding_number_constI: fixes c :: "complex"   and z :: "complex"   and g :: "real \<Rightarrow> complex" assumes "c \<noteq> z"   and "\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> g t = c" shows "winding_number g z = 0"
special_nsg_e: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "\<natural>H \<triangleright> {\<one>}"
collapse_1: fixes llist :: "'b llist" assumes "lnull llist" shows "llist = <>"
sigma_finite: fixes i :: "'i" obtains A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets (M i)" and "\<Union> (range A) = space (M i)" and "\<And>ia. emeasure (M i) (A ia) \<noteq> \<infinity>"
H_eq_2: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "(\<not> MC_syntax.recurrent K s) = (MC_syntax.H K s s = 0)"
rec_2: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
simps_2: fixes xs :: "'a list" shows "proper_interval None (Some xs) = (xs \<noteq> [])"
sg_subset: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "H \<subseteq> carrier G"
distinct_1: fixes x2 :: "int" shows "MinInfty \<noteq> Num_arc x2"
comp_unitr: fixes x :: "'a" shows "x ; 1' = x"
mkAutoSim_simps_3: fixes aOps :: "('e, 'b, 'c list) MapOps"   and tOps :: "('f, 'b \<times> 'd, 'b) MapOps"   and simObs :: "'g \<Rightarrow> 'b \<Rightarrow> 'd"   and simInit :: "'g \<Rightarrow> 'd \<Rightarrow> 'b"   and simTrans :: "'g \<Rightarrow> 'b \<Rightarrow> 'b list"   and simAction :: "'g \<Rightarrow> 'b \<Rightarrow> 'c list"   and frontier :: "'g \<Rightarrow> 'b list"   and a :: "'g" shows "pAct (mkAlgAuto aOps tOps simObs simInit simTrans simAction frontier a) = (\<lambda>ec. the (lookup aOps (aActs (alg_dfs aOps tOps (simObs a) (simTrans a) (simAction a) (frontier a))) ec))"
integrable_point_measure_finite: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> 'b" assumes "finite A" shows "integrable (point_measure A (\<lambda>x. ennreal (f x))) g"
grid_single_dimensional_specification: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and i :: "int"   and l' :: "nat"   and l :: "nat" assumes "d < length b"   and "odd i"   and "lv b d + l' = l"   and "i < (ix b d + 1) * 2 ^ l'"   and "(ix b d - 1) * 2 ^ l' < i" shows "b[d := (l, i)] \<in> grid b {d}"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Language.normalize c Q,A"
literal_to_term_sub: fixes l :: "fterm literal"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "literal_to_term (l \<cdot>\<^sub>l \<sigma>) = literal_to_term l \<cdot>\<^sub>t \<sigma>"
insertI1: fixes a :: "'b::type"   and B :: "'b::type set" shows "(a::'b::type) \<in> insert a (B::'b::type set)"
A_B_step: fixes a :: "'a"   and b :: "'a"   and a' :: "'b" assumes "A a b"   and "a \<sim> a'"   and "PA a"   and "PB a'" shows "\<exists>b'. B a' b' \<and> b \<sim> b'"
semi_mojmir_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c" shows "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w \<equiv> finite \<Sigma> \<and> finite (reach \<Sigma> \<delta> q\<^sub>0) \<and> range w \<subseteq> \<Sigma>"
fps_nth_of_nat: fixes c :: "nat"   and n :: "nat" shows "fps_nth (of_nat c) n = (if n = 0 then of_nat c else (0::'a))"
the_sym_eq_trivial: fixes x :: "'d" shows "(THE y. x = y) = x"
simps_5: fixes 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"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and r :: "nat \<times> char list list" shows "P \<turnstile> typeof\<^bsub>h\<^esub> (Ref r) = (case h (the_addr (Ref r)) of None \<Rightarrow> None | \<lfloor>(C, S)\<rfloor> \<Rightarrow> if Subobjs P C (the_path (Ref r)) then \<lfloor>Class (last (the_path (Ref r)))\<rfloor> else None)"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Normal x1 = Normal y1) = (x1 = y1)"
Field_square: fixes x :: "'d set" shows "Field (x \<times> x) = x"
path_connectedin: fixes X :: "'a topology"   and S :: "'a set" shows "path_connectedin X S = (S \<subseteq> topspace X \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>g. pathin X g \<and> g ` {0..1} \<subseteq> S \<and> g 0 = x \<and> g 1 = y))"
intros_2: fixes "is" :: "nat list"   and ds :: "nat list"   and i :: "nat"   and d :: "nat" assumes "is \<lhd> ds"   and "i < d" shows "i # is \<lhd> d # ds"
map_graph_fn_id_2: fixes X :: "('b, 'c) labeled_graph" shows "map_graph (Id_on (vertices X)) X = restrict X"
nextl_path_to: fixes q :: "hf" assumes "q \<in> MN.accessible" shows "MN.nextl (dfa.init Canon.DFA) (MN.path_to q) = q"
eqs: fixes i :: "'sz" shows "sterm_sem I (Tsubst (args i) \<sigma>) (fst \<nu>) = sterm_sem (local.adjoint I \<sigma> \<nu>) (args i) (fst \<nu>)"
1: shows "\<forall>v. v \<in> {a, b, c, d}"
mkarr_memb_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "t \<in> f"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "free_monoidal_category.mkarr C t = f"
metric_LIMSEQ_D: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a"   and r :: "real" assumes "X \<longlonglongrightarrow> L"   and "0 < r" shows "\<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
ipurge_tr_rev_t_subset: fixes Y :: "('c, 'd) ipurge_rec"   and X :: "('c, 'd) ipurge_rec" assumes "Y \<in> ipurge_tr_rev_t_set X" shows "ipurge_tr_rev_t_set Y \<subseteq> ipurge_tr_rev_t_set X"
simps: fixes \<Gamma> :: "('k, 'l) web_scheme" shows "hindered \<Gamma> = (\<exists>f. hindrance \<Gamma> f \<and> current \<Gamma> f \<and> wave \<Gamma> f)"
simps_1: fixes r :: "real"   and p :: "real" shows "ereal r + ereal p = ereal (r + p)"
lookup_pair_map_val_pair: fixes xs :: "('a \<times> 'e) list"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f"   and k :: "'a" assumes "oalist_inv_raw xs"   and "f k (0::'e) = (0::'f)" shows "lookup_pair (map_val_pair f xs) k = f k (lookup_pair xs k)"
cases: fixes lub :: "'b set \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "ccpo.compact lub ord x" obtains "ccpo.admissible lub ord (\<lambda>y. \<not> ord x y)" and "ccpo.admissible lub ord ((\<noteq>) x)"
bij_Abs: shows "bij_betw Abs {0..<int CARD('a)} UNIV"
aux1: assumes "0 < \<theta>" shows "LBINT x. indicat_real {0<..<T} x * sin (x * \<theta>) / x = LBINT x. indicat_real {0<..<T * \<theta>} x * sin x / x"
"2_2": fixes i :: "nat" shows "target (stake i wr) (r 0) = r i"
simps_1: fixes C :: "'b set set"   and S :: "'b set" shows "pcp_seq (C::'b set set) (S::'b set) (0::nat) = S"
hoare_tripleD_4: fixes P :: "assn"   and c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set"   and \<sigma> :: "heap option"   and r :: "'d" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P"   and "run c (Some h) \<sigma> r" shows "lim h \<le> lim (the_state \<sigma>)"
steps: shows "\<forall>i<n. (S i, S (Suc i)) \<in> R"
shift1_isnpolyh: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0"   and "p \<noteq> 0\<^sub>p" shows "isnpolyh (shift1 p) 0"
image_diff_subset: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" shows "order.greater_eq (f ` (A - B)) (f ` A - f ` B)"
separated_disj_Union2: fixes g :: "graph"   and V :: "nat list"   and w :: "face \<Rightarrow> nat" assumes "Invariants.inv g"   and "final g"   and "noExceptionals g (set V)"   and "separated g (set V)"   and "distinct V"   and "set V \<subseteq> \<V> g" shows "(\<Sum>\<^bsub>v\<in>V\<^esub> ListSum (facesAt g v) w) = ListSum (filter (\<lambda>f. \<exists>v\<in>set V. f \<in> set (facesAt g v)) (faces g)) w"
not_iless0: fixes n :: "enat" shows "\<not> n < 0"
is_weakly_dom_component_safe_def: shows "is_weakly_dom_component_safe \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe get_root_node to_tree_order"
bdd_below_Icc: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.preorder less_eq less" shows "preorder.bdd_below less_eq (ord.atLeastAtMost less_eq a b)"
poly_subst_alt: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f p = (\<Sum>t. MPoly_Type_Class.punit.monom_mult (lookup p t) 0 (subst_pp f t))"
Quotient_interface: shows "Quotient (\<simeq>) abs_interface rep_interface cr_interface"
subfrmlsn_finite: fixes \<phi> :: "'b ltln" shows "finite (subfrmlsn \<phi>)"
rep_eq: fixes x :: "'b mpoly"   and xa :: "nat" shows "poly.coeff (mpoly_to_nested_poly x xa) = isolate_variable_sparse x xa"
fps_nth_Abs_fps: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "fps_nth (Abs_fps f) n = f n"
xenv_le_def: fixes X :: "nat set"   and \<rho> :: "(nat \<times> val) list"   and \<rho>' :: "(nat \<times> val) list" shows "X \<turnstile> \<rho> \<sqsubseteq> \<rho>' \<equiv> \<forall>x v. x \<in> X \<and> lookup \<rho> x = Some v \<longrightarrow> (\<exists>v'. lookup \<rho>' x = Some v' \<and> v \<sqsubseteq> v')"
upset_prop: fixes x :: "'a" shows "\<up> x = {y. x \<le> y}"
coinduct_strong: fixes R :: "('b, 'd, 'e) gpv \<Rightarrow> ('b, 'd, 'e) gpv \<Rightarrow> bool"   and gpv :: "('b, 'd, 'e) gpv"   and gpv' :: "('b, 'd, 'e) gpv" assumes "R gpv gpv'"   and "\<And>gpv gpv'. R gpv gpv' \<Longrightarrow> rel_spmf (rel_generat (=) (=) (rel_fun (=) (\<lambda>gpv gpv'. R gpv gpv' \<or> gpv = gpv'))) (the_gpv gpv) (the_gpv gpv')" shows "gpv = gpv'"
bi_unique_alt_def: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
assms_1: fixes y :: "'a" assumes "y \<in> Y" shows "countable (X `` {y})"
vimage_Inr_image_Inl: fixes A :: "'j set" shows "Inr -` Inl ` A = {}"
card_le_Suc_iff: fixes n :: "nat"   and A :: "'d set" shows "(Suc n \<le> card A) = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
g_isSng_def: fixes m :: "'s" shows "g_isSng m \<equiv> g_size_abort 2 m = 1"
retraction_def: fixes g :: "'a" shows "C.retraction g \<equiv> \<exists>f. C.ide (g \<cdot>\<^sub>C f)"
g_def: fixes x :: "'a" shows "g x = (if x \<in> closure S then f x else \<infinity>)"
clinear_iff: fixes f :: "'d \<Rightarrow> 'e" shows "clinear f = ((\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>C x) = c *\<^sub>C f x))"
Close1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<longmapsto>\<^sub>e a<x> \<prec>\<^sub>e P'"   and "Q \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e <\<nu>x>(P' \<parallel> Q')"
contra1: fixes s :: "real \<times> real"   and p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "distinct4 s p q r" shows "\<not> (Counterclockwise_2D_Arbitrary.lex p q \<and> Counterclockwise_2D_Arbitrary.lex q r \<and> Counterclockwise_2D_Arbitrary.lex r s \<and> indelta s p q r)"
Lambert_W_ge: fixes x :: "real" shows "- 1 \<le> Lambert_W x"
ts_process_traces: fixes T :: "'a list set" assumes "trace_set T" shows "traces (ts_process T) = T"
w1_def: shows "w1 = (\<lambda>r. Re (winding_number (poly p \<circ> (\<lambda>x. complex_of_real (linepath (- r) (of_real r) x))) 0))"
all_set_conv_all_nth: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>set xs. P x) = (\<forall>i<length xs. P (xs ! i))"
assms_3: shows "X \<subseteq>\<^sub>L' Y"
dSUCCEED_abort_def: fixes msg :: "String.literal"   and dm :: "'b dres"   and m :: "'b dres" shows "dSUCCEED_abort msg dm m = (case m of dSUCCEEDi \<Rightarrow> Code.abort msg (\<lambda>_. dm) | _ \<Rightarrow> m)"
Y_fin: fixes n :: "nat" shows "emeasure M (Y n) \<noteq> \<infinity>"
element_exists: fixes A :: "'a set" assumes "A \<in> left.transformation.orbit.Partition" shows "\<exists>a\<in>G. a \<in> A"
of_int_Gramian_determinant: fixes k :: "nat"   and F :: "int vec list" assumes "k \<le> length F"   and "\<And>i. i < length F \<Longrightarrow> dim_vec (F ! i) = n" shows "gs.Gramian_determinant (map of_int_hom.vec_hom F) k = of_int (gs.Gramian_determinant F k)"
cls': shows "class (class_add P (C, cdec)) Object = \<lfloor>(C', fs, ms)\<rfloor>"
m_closed: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F" shows "x \<oplus>\<^bsub>F\<^esub> y \<in> carrier F"
cases: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "a1 \<approx> a2" obtains \<R> :: "((('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" and lc\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" and lc\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" where "a1 = lc\<^sub>1" and "a2 = lc\<^sub>2" and "strong_low_bisim_mm \<R>" and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>"
strict_boundedE: fixes b :: "'c"   and c :: "'c"   and a :: "'c" assumes "b \<squnion> c < a" obtains "b < a" and "c < a"
finite_set_choice: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite A"   and "\<forall>x\<in>A. \<exists>y. P x y" shows "\<exists>f. \<forall>x\<in>A. P x (f x)"
simps_1: fixes e :: "'b equation"   and s :: "('b \<times> 'b trm) list" shows "subst_lit (Pos e) s = Pos (subst_equation e s)"