hcju/isabelleps huggingface dataset:
minimum_elem_sub: fixes D :: "('c, 'd) Order_scheme"   and T :: "'c set"   and X :: "'c set"   and a :: "'c" assumes "Order D"   and "T \<subseteq> carrier D"   and "X \<subseteq> T" shows "minimum_elem D X a = minimum_elem (Iod D T) X a"
assms_4: shows "\<forall>\<^sub>F x in at_top. l x \<le> f x"
rotate_map: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "rotate n (map f xs) = map f (rotate n xs)"
elims_2: fixes x :: "'c \<Rightarrow> 'd list"   and xa :: "'d \<Rightarrow> 'c"   and xb :: "nat \<times> nat list \<times> 'd list" assumes "stab_inv x xa xb" obtains f :: "'c \<Rightarrow> 'd list" and key :: "'d \<Rightarrow> 'c" and u :: "nat" and ns :: "nat list" and xs :: "'d list" where "x = f" and "xa = key" and "xb = (u, ns, xs)" and "\<forall>k. filter (\<lambda>x. key x = k) xs = f k"
subgroup_of_additive_group_of_ring_axioms: shows "subgroup_of_additive_group_of_ring I R (+) (\<cdot>) \<zero> \<one>"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
MI_pred_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and K :: "('c \<Rightarrow> univ \<Rightarrow> bool) \<times> ('a \<Rightarrow> univ list \<Rightarrow> univ) \<times> ('b \<Rightarrow> univ list \<Rightarrow> bool)" assumes "MonotProblem wtFsym wtPsym arOf resOf parOf \<Phi>" shows "MonotProblem.MI_pred wtFsym wtPsym arOf resOf parOf \<Phi> K \<equiv> InfModel wtFsym wtPsym arOf resOf parOf \<Phi> (fst3 K) (snd3 K) (trd3 K)"
impl_intro: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "P x \<Longrightarrow> Q x" shows "(P \<longrightarrow> Q) x"
base_3: shows "\<forall>i\<in>S. A i \<in> measurable_sets M"
appendPath_last: fixes Cs :: "char list list"   and Cs' :: "char list list" assumes "Cs \<noteq> []" shows "last Cs = last (Cs' @\<^sub>p Cs)"
en_eq_10: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<^bold>\<A>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<lbrace>x,F\<rbrace> in v]"
R: shows "is_mset_set X \<and> x \<notin># X"
TRUE_LabelFunSem: fixes ST :: "('d, 'e, 'f) status" shows "TRUE \<in> the (LabelFunSem (HA ST) ST)"
fMin_Min: fixes l :: "'b list" shows "fMin (fset_of_list l) = Min (set l)"
f_expand_one: fixes k :: "nat"   and a :: "'b message_af" assumes "0 < k" shows "[a] \<odot> k = a # NoMsg\<^bsup>k - Suc 0\<^esup>"
ranked: fixes q :: "'a"   and f :: "'b"   and qs :: "'a list" assumes "q \<rightarrow> f qs \<in> ta'.\<delta>" shows "A f = Some (length qs)"
integrable_spike_interior: fixes a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x"   and "f integrable_on cbox a b" shows "g integrable_on cbox a b"
minus_of_int: fixes x :: "int"   and n :: "int" shows "gauss_sum_int (x - of_int n * int n) = gauss_sum_int x"
subst_cls_lists_def: fixes Cs :: "('a, 'b) Term.term literal multiset list"   and \<sigma>s :: "('b \<Rightarrow> ('a, 'b) Term.term) list" shows "subst_cls_lists Cs \<sigma>s = Map2.map2 subst_cls Cs \<sigma>s"
epm_2: fixes w :: "'b"   and v :: "'b"   and e :: "'b" assumes "regular (stone_algebra_class.comp_inf.prim_EP w v e)"   and "bounded_distrib_lattice_class.comp_inf.vector v" shows "w \<sqinter> - stone_algebra_class.comp_inf.prim_EP w v e \<squnion> stone_algebra_class.comp_inf.prim_P w v e \<squnion> stone_algebra_class.comp_inf.prim_E w v e = w"
step: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "class.ccpo Sup less_eq less"   and "x \<in> ccpo.iterates Sup less_eq f" shows "f x \<in> ccpo.iterates Sup less_eq f"
coeff_in_coeffs: fixes p :: "'b poly"   and n :: "nat" assumes "p \<noteq> 0"   and "n \<le> degree p" shows "coeff p n \<in> set (coeffs p)"
cod_comp: fixes g :: "'b"   and f :: "'b" assumes "J.seq g f" shows "J.cod (J.comp g f) = J.cod g"
simps_18: shows "OUTfromCh data18 = {data16}"
stepsST_refl: fixes P :: "('a, 'b) Proc"   and Source :: "'a processCalculus"   and Target :: "'b processCalculus" shows "(\<exists>S S'. P = SourceTerm S \<and> P = SourceTerm S' \<and> S \<longmapsto>Source* S') \<or> (\<exists>T T'. P = TargetTerm T \<and> P = TargetTerm T' \<and> T \<longmapsto>Target* T')"
\<Sigma>_protocol_def: shows "\<Sigma>1.\<Sigma>_protocol \<equiv> \<Sigma>1.completeness \<and> \<Sigma>1.special_soundness \<and> \<Sigma>1.HVZK"
zip_fun_simps_6: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a" shows "snd ` range (f \<parallel> g) = range g"
geq: shows "\<forall>x\<in>\<Union> \<F> \<inter> K. g x = h x"
exec_gpv_invariant: fixes x :: "'a"   and s' :: "('b, 'c) resource"   and gpv :: "('a, 'b, 'c) gpv"   and s :: "('b, 'c) resource"   and \<I> :: "('b, 'c) \<I>" assumes "(x, s') \<in> set_spmf (exec_gpv run_resource gpv s)"   and "\<I> \<turnstile>res s \<surd>"   and "\<I> \<turnstile>g gpv \<surd>" shows "\<I> \<turnstile>res s' \<surd>"
size_height_if_incomplete: fixes kdt :: "'b kdt" assumes "\<not> KD_Tree.complete kdt" shows "KD_Tree.size_kdt kdt < 2 ^ height kdt"
inv_equality: fixes y :: "cltn2"   and x :: "cltn2" assumes "y \<otimes>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = \<one>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"   and "x \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "y \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "inv\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = y"
hyps: fixes s :: "'a"   and t :: "('a, 'b) xstate" assumes "\<Gamma>\<turnstile> \<langle>sequence Seq (xs @ ys),Normal s\<rangle> =n\<Rightarrow> t" shows "\<exists>s'. \<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> =n\<Rightarrow> s' \<and> \<Gamma>\<turnstile> \<langle>sequence Seq ys,s'\<rangle> =n\<Rightarrow> t"
simps_2: fixes h :: "'b \<times> 'a"   and t :: "('b \<times> 'a) list"   and l :: "'b" shows "lookup (h # t) l = (if fst h = l then Some (snd h) else lookup t l)"
path_empty_conv: fixes E :: "('a \<times> 'a) set"   and u :: "'a"   and v :: "'a" shows "path E u [] v = (u = v)"
as_monic: fixes as :: "('c poly \<times> nat) set"   and p :: "'c poly"   and a :: "'c poly"   and i :: "nat" assumes "monic_factorization as p"   and "(a, i) \<in> as" shows "monic a"
bind_fv_changed: fixes v :: "variable \<times> typ"   and tm :: "Core.term" assumes "bind_fv v tm \<noteq> tm" shows "v \<in> fv tm"
distinct_70: fixes x9 :: "('a, 'b) form"   and x7 :: "('a, 'b) form" shows "Exists x9 \<noteq> Neg x7"
length_append: fixes xs :: "'a list"   and ys :: "'a list" shows "|xs @ ys| = |xs| + |ys|"
carrier_subgroup_generated_subgroup: shows "carrier (subgroup_generated (product_group I G) {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}) = {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}"
pred_eq_iff: fixes P :: "'a pred"   and Q :: "'a pred"   and w :: "'a" assumes "P = Q" shows "eval P w = eval Q w"
upset_to_downset3: fixes x :: "'a" shows "\<up> x = \<partial> ` \<down> (\<partial> x)"
cd_2_var: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> 1\<^sub>\<pi>) \<cdot> y = x \<cdot> 1\<^sub>\<pi> \<parallel> y"
Diff_iff: fixes c :: "'b::type"   and A :: "'b::type set"   and B :: "'b::type set" shows "((c::'b::type) \<in> (A::'b::type set) - (B::'b::type set)) = (c \<in> A \<and> c \<notin> B)"
reachable_V: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'c"   and m :: "'b" shows "(g_E (b_graph.to_gbg_ext (degeneralize_ext ma) m))\<^sup>* `` g_V0 (b_graph.to_gbg_ext (degeneralize_ext ma) m) \<subseteq> g_V (b_graph.to_gbg_ext (degeneralize_ext ma) m)"
adm_wf_def: fixes R :: "('c \<times> 'c) set"   and F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" shows "adm_wf R F = (\<forall>f g x. (\<forall>z. (z, x) \<in> R \<longrightarrow> f z = g z) \<longrightarrow> F f x = F g x)"
admissible_targets_2: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "(l, g, \<mu>) \<in> PTA.trans_of A"   and "(X, l') \<in> set_pmf \<mu>" shows "X \<subseteq> clocks A"
V_mono: fixes l :: "nat"   and m :: "nat" assumes "l \<le> m" shows "V l \<subseteq> V m"
mat_times_vec_leqD_2: fixes A :: "'a mat"   and x :: "'a vec"   and b :: "'a vec" assumes "[A*\<^sub>vx]\<le>b" shows "dim_col A = dim_vec x"
zero_neq_one_ow: shows "class.zero_neq_one = zero_neq_one_ow UNIV"
mod_mult_self4: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (times b c) a) b = modulo a b"
prems_4: shows "length lm2 = aa - ba - Suc n"
cmp_def: fixes g :: "'j \<Rightarrow> 'i"   and f :: "'h \<Rightarrow> 'j" shows "cmp g f = (\<lambda>x. g (f x))"
ins_eq: fixes x :: "'a"   and y :: "'a"   and S :: "'b" assumes "isNode x"   and "isNode y"   and "invariant S"   and "\<not> memb y S" shows "memb x (ins y S) = (nodeAbs x = nodeAbs y \<or> memb x S)"
lang_pow_empty: fixes n :: "nat" shows "{} ^^ n = (if n = 0 then {[]} else {})"
simps_1: fixes t :: "([:'r:] \<times> [:'r:]) tree" shows "next\<cdot>[::]\<cdot>t = t"
adm_not_mem: fixes f :: "'a \<Rightarrow> 'b set"   and y :: "'b" assumes "cont f" shows "adm (\<lambda>x. y \<notin> f x)"
Pi_pmf_subset': fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c pmf"   and dflt :: "'c" assumes "finite A"   and "B \<subseteq> A"   and "\<And>x. x \<in> A - B \<Longrightarrow> f x = return_pmf dflt" shows "Pi_pmf A dflt f = Pi_pmf B dflt f"
subdiv_ate_3: shows "arc_to_ends H uw = (u, w)"
algebraic_imp_represents_unique: fixes x :: "'b" assumes "algebraic x" shows "\<exists>!p. p represents x \<and> irreducible p \<and> 0 < Polynomial.lead_coeff p"
deadends_def: fixes p :: "Player" shows "G'.deadends p \<equiv> {v \<in> G'.VV p. G'.deadend v}"
finsum_dim: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a vec" assumes "finite A"   and "f \<in> A \<rightarrow> carrier_vec n" shows "dim_vec (finsum V f A) = n"
not_less_iff_gr_or_eq: fixes x :: "'a list"   and y :: "'a list" shows "(\<not> x <lex y) = (y <lex x \<or> x = y)"
succ_rel_rem_head: fixes x :: "'b"   and xs :: "'b list" assumes "distinct (x # xs)" shows "{(p, n). (p, n) \<in> succ_rel (x # xs) \<and> p \<noteq> x} = succ_rel xs"
CAND_AndL2_elim: fixes x :: "name"   and y :: "name"   and M :: "trm"   and B :: "ty" assumes "x:AndL2 y.M x \<in> \<parallel>(B)\<parallel>"   and "x:AndL2 y.M x \<notin> BINDINGn B (\<parallel><B>\<parallel>)" shows "\<exists>B1 B2. B = B1 AND B2 \<and> y:M \<in> \<parallel>(B2)\<parallel>"
elims: fixes x :: "'a ifex"   and y :: "'a option" assumes "ifex_top_var x = y" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e" and "y = Some v"   | "x = Trueif" and "y = None"   | "x = Falseif" and "y = None"
atLeast_def: fixes l :: "'a" shows "{l..} \<equiv> {x. l \<le> x}"
checked_s_def: shows "checked_s = Incremental_State_Ops_Simplex.checked_s"
vmc_path_lnull_ltl_no_deadend: assumes "\<not> lnull (ltl P')" shows "vmc_path_no_deadend G P' v0 p \<sigma>"
rbt_comp_lookup_map_const: fixes c :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and f :: "'c \<Rightarrow> 'b"   and t :: "('a, 'c) rbt" shows "rbt_comp_lookup c (RBT_Impl.map (\<lambda>_. f) t) = map_option f \<circ> rbt_comp_lookup c t"
N_def: shows "N = Missing_Lemmas.max_list (map (nat \<circ> sq_norm) fs_init)"
Y_l: shows "\<forall>(t, n)\<in>set (remdups (map simp_num_pair (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs (\<Upsilon> (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))))))). numbound0 t \<and> 0 < n"
that: fixes p :: "nat" assumes "p \<in> {m<..n}"   and "prime p" shows "thesis"
case_1: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c" shows "(case None of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f1"
fin_PQ: shows "finite {l + n |l n. l \<in> Poly_Mapping.keys p \<and> n \<in> Poly_Mapping.keys q}"
hyperplane_eq_Ex: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a)" obtains x :: "'a" where "a \<bullet> x = b"
hyps: shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.Q \<longmapsto> \<tau> \<prec> Q'"
simps_10: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case flat_complete_lattice.Top of flat_complete_lattice.Value x \<Rightarrow> f1 x | flat_complete_lattice.Bot \<Rightarrow> f2 | flat_complete_lattice.Top \<Rightarrow> f3) = f3"
fls_base_factor_to_fps_const: fixes c :: "'a" shows "fls_base_factor_to_fps (fls_const c) = fps_const c"
accontr: shows "\<Union> (get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = stateful \<union> X\<rparr>)) \<subseteq> backflows (filternew_flows_state \<lparr>hosts = V, flows_fix = E, flows_state = stateful \<union> X\<rparr>)"
independece_dest_alt: fixes C :: "'a pmf set"   and P :: "('a pmf \<times> 'a pmf) set" assumes "independent_vnm C P" shows "\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r"
fix_def2: fixes F :: "'b \<rightarrow> 'b" shows "fix\<cdot>F = (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
thetaSeqZOD_ZOretr: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaSeqZOD tval aval indis \<subseteq> PL_Indis.ZOretr tval aval indis (PL_Indis.thetaSeqZOD tval aval indis \<union> PL_Indis.ZObis tval aval indis)"
basic_subtype_irrefl: fixes \<tau> :: "'c basic_type" assumes "\<tau> < \<tau>" shows "False"
postdominate_path_branch: fixes n :: "'node"   and as :: "'edge list"   and n'' :: "'node"   and n' :: "'node" assumes "n -as\<rightarrow>* n''"   and "n' postdominates n''"   and "\<not> n' postdominates n" obtains a :: "'edge" and as' :: "'edge list" and as'' :: "'edge list" where "as = as' @ a # as''" and "valid_edge a" and "\<not> n' postdominates sourcenode a" and "n' postdominates targetnode a"
a1: shows "\<beta> = \<phi> $$ (1, 0)"
ket_zero_k_decode: fixes k :: "nat" assumes "k < n" shows "ket_zero_k k = Matrix.vec (2 ^ Suc k) (\<lambda>k. if k = 0 then 1 else 0)"
le: fixes x__ :: "'a"   and x'__ :: "'b"   and x :: "'a"   and x' :: "'b" assumes "(x__, x'__) \<in> A"   and "(x, x') \<in> A" shows "(le x__ x, x'__ \<le> x') \<in> bool_rel"
enat_add_left_cancel_le: fixes a :: "enat"   and b :: "enat"   and c :: "enat" shows "(a + b \<le> a + c) = (a = \<infinity> \<or> b \<le> c)"
lprefix_LCons_conv: fixes xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" shows "lprefix xs (LCons y ys) = (xs = LNil \<or> (\<exists>xs'. xs = LCons y xs' \<and> lprefix xs' ys))"
set_ConsD: fixes y :: "'a"   and x :: "'a"   and xs :: "'a list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
simps_1: fixes x1 :: "nat"   and x2 :: "'b"   and x3 :: "'b tree list"   and y1 :: "nat"   and y2 :: "'b"   and y3 :: "'b tree list" shows "(Node x1 x2 x3 = Node y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
le_iff_add: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subseteq># b) = (\<exists>c. b = a + c)"
PDG_path_CFG_path: fixes n :: "'node"   and n' :: "'node" assumes "PDG.PDG_path sourcenode targetnode valid_edge Def Use standard_control_dependence n n'" obtains as :: "'edge list" where "n -as\<rightarrow>* n'"
realfun_mon_conv_iff: fixes u :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and f :: "'b \<Rightarrow> real" shows "u\<up>f = (\<forall>w. (\<lambda>n. u n w)\<up>f w)"
allE: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "\<forall>x. P x" obtains "P x"
Max_enat_minus1: fixes A :: "enat set"   and m :: "enat" assumes "finite A"   and "A \<noteq> {}" shows "Max A - m = (MAX n\<in>A. n - m)"
epm_13: fixes v :: "'b"   and w :: "'b"   and e :: "'b" assumes "bounded_distrib_lattice_class.comp_inf.vector v" shows "stone_algebra_class.comp_inf.prim_P w v e \<sqinter> stone_algebra_class.comp_inf.prim_E w v e = bot"
funpow_0: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "f^0 x = x"
min_union_finite: fixes A :: "'b fset set"   and B :: "'b fset set" assumes "finite A"   and "finite B" shows "finite (A \<union>\<^sub>m B)"
Zr_gen_Zleast: fixes I :: "int set" assumes "ideal Zr I"   and "I \<noteq> {0}" shows "Zr \<diamondsuit>\<^sub>p lev I = I"
lt_def: fixes c :: "int"   and t :: "MIR.num" shows "lt c t = (if c = 0 then Lt t else if 0 < c then Lt (CN 0 c t) else Gt (CN 0 (- c) (Neg t)))"
simps_9: fixes vk :: "'a \<Rightarrow> 'b fundef option"   and op :: "'op"   and \<Sigma> :: "Unboxed.type option list" shows "Subx.sp_instr vk (Ubx.instr.IOp op) \<Sigma> = Subx.sp_gen_pop_push (replicate (\<AA>\<rr>\<ii>\<tt>\<yy> op) None, None) \<Sigma>"
that: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ennreal" assumes "\<And>i. simple_function M (f i)"   and "incseq f"   and "\<And>i x. f i x < top"   and "\<And>x. (SUP i. f i x) = u x" shows "thesis"
ground_superposition_is_sound: fixes P1 :: "'a eclause"   and P2 :: "'a eclause"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set"   and \<theta> :: "('a \<times> 'a trm) list" assumes "finite (cl_ecl P1)"   and "finite (cl_ecl P2)"   and "superposition P1 P2 C \<sigma> k C'"   and "ground_clause (subst_cl (cl_ecl C) \<theta>)" shows "set_entails_clause {subst_cl (subst_cl (cl_ecl P1) \<sigma>) \<theta>, subst_cl (subst_cl (cl_ecl P2) \<sigma>) \<theta>} (subst_cl (cl_ecl C) \<theta>)"
strict_implies_order: fixes b :: "'b list"   and a :: "'b list" assumes "strict_prefix b a" shows "??.Sublist.prefix b a"
simps_1: fixes s :: "('c, 'd) fmap" shows "sat_precond_as s [] = True"
V_disjoint_iff: fixes x :: "V"   and y :: "V" shows "(x \<inter>\<^sub>\<circ> y = []\<^sub>\<circ>) = (elts x \<inter> elts y = {})"
inv_P: shows "invertible (row_add (mat (1::'b)) (mod_type_class.from_nat n) i ((res (A $ i $ j) (A $ mod_type_class.from_nat n $ j) - A $ mod_type_class.from_nat n $ j) div A $ i $ j))"
emeasure_HLD_nxt: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s"   and X :: "'s set" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "emeasure (T s) {\<omega> \<in> space (T s). HLD X \<omega> \<and> nxt P \<omega>} = \<integral>\<^sup>+x\<in>X. emeasure (T x) {\<omega> \<in> space (T x). P \<omega>}\<partial>measure_pmf (K s)"
take_bit_of_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n zero = zero"
snapshot_bound_leq_S: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and vs :: "'a set"   and s :: "('a, 'b) fmap"   and lss :: "('a, 'b) fmap list" shows "problem_plan_bound (snapshot PROB (fmrestrict_set vs s)) \<le> S vs lss PROB (fmrestrict_set vs s)"
distinct_1: fixes x1 :: "'a"   and x2 :: "'a \<times> nat" shows "CFG_node x1 \<noteq> Formal_in x2"
top_times_top: shows "times_prod' top_prod top_prod = top_prod"
bij_to_nat: shows "bij_betw Bij_Nat.to_nat UNIV {0..<CARD('b)}"
agree_UNIV_fst: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<omega> :: "(real, 'd) vec \<times> (real, 'd) vec" assumes "Vagree \<nu> \<omega> (range Inl)" shows "fst \<nu> = fst \<omega>"
max_tstp_intro': fixes tstp :: "nat + nat"   and tstp' :: "nat + nat"   and ts' :: "nat"   and tp' :: "nat" assumes "isl tstp = isl tstp'"   and "ts_tp_le ts' tp' tstp" shows "ts_tp_le ts' tp' (max_tstp tstp tstp')"
characterize_consistent_signs_at_roots_rw: fixes p :: "real poly"   and fs :: "real poly list" assumes "p \<noteq> 0"   and "\<And>q. q \<in> set fs \<Longrightarrow> comm_monoid_mult_class.coprime p q" shows "set (characterize_consistent_signs_at_roots p fs) = consistent_sign_vectors_r fs {x. poly p x = 0}"
rev_bexI: fixes x :: "'e"   and A :: "'e set"   and P :: "'e \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>x\<in>A. P x"
x_1: shows "[A*\<^sub>vx]\<le>b"
mono_ndet_F_right: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>F Q" shows "(S |-| P) \<sqsubseteq>\<^sub>F Q"
that: assumes "r a b"   and "a \<in> A"   and "b \<in> A" shows "thesis"
root: shows "of_int_poly p = Polynomial.smult (Polynomial.lead_coeff (of_int_poly p)) (\<Prod>i<n. [:- root i, 1:])"
execution_def: fixes e :: "('r, 'l, 'v) expr"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" shows "execution e s s' \<equiv> (\<exists>r. s = \<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)) \<and> program_expr e) \<and> s \<leadsto>\<^sup>* s'"
case_prodI2': fixes p :: "'b \<times> 'c"   and c :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "\<And>a b. (a, b) = p \<Longrightarrow> c a b x" shows "(case p of (a, b) \<Rightarrow> c a b) x"
one_strongly_connected_iff_6_eq: fixes x :: "'a" assumes "is_p_fun x"   and "is_inj x" shows "one_strongly_connected x = (x ; 1 ; x = x ; x\<^sup>+)"
lconjseq_alternating_order2_repeats: fixes s :: "'a"   and t :: "'a"   and k :: "nat" assumes "s + s = (0::'a)"   and "t + t = (0::'a)"   and "(s + t) +^ k = (0::'a)" shows "lconjseq (alternating_list (2 * k) s t) = lconjseq (alternating_list k s t) @ lconjseq (alternating_list k s t)"
try_SE_def: fixes ioprog :: "'d \<Rightarrow> ('e \<times> 'd) option" shows "try\<^sub>S\<^sub>E ioprog = (\<lambda>\<sigma>. case ioprog \<sigma> of None \<Rightarrow> Some (None, \<sigma>) | Some (outs, \<sigma>') \<Rightarrow> Some (Some outs, \<sigma>'))"
calc_ideal_point1_on_circline: fixes z1 :: "complex"   and z2 :: "complex"   and A :: "complex"   and B :: "complex" assumes "(z1, z2) = calc_ideal_point1_cvec A B"   and "is_real A"   and "(cmod A)\<^sup>2 < (cmod B)\<^sup>2" shows "A * z1 * cnj z1 + B * cnj z1 * z2 + cnj B * z1 * cnj z2 + A * z2 * cnj z2 = 0"
result_map_generat: fixes x :: "('g, 'h, 'i) generat"   and f :: "'g \<Rightarrow> 'j"   and g :: "'h \<Rightarrow> 'k"   and h :: "'i \<Rightarrow> 'l" assumes "is_Pure x" shows "generat.result (map_generat f g h x) = f (generat.result x)"
H_on_ket_one_is_state: shows "state 1 (H * |Deutsch.one\<rangle>)"
assms: shows "prob0_fst_qubits n jozsa_algo = 1"
left_mult_one_mat': fixes A :: "'b mat"   and n :: "nat" assumes "dim_row A = n" shows "1\<^sub>m n * A = A"
fresh_var_fresh: fixes Vs :: "String.literal list" shows "fresh_var Vs \<notin> set Vs"