hcju/isabelleps huggingface dataset:
pdevs_val_selsort_ccw: fixes xs :: "(real \<times> real) list"   and e :: "nat \<Rightarrow> real"   and I :: "real set" assumes "distinct xs"   and "e \<in> UNIV \<rightarrow> I" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> UNIV \<rightarrow> I" and "pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list (ccw.selsort 0 xs))"
supp_def: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.pt permute" shows "pt.supp permute x = {a. infinite {b. permute (a \<rightleftharpoons> b) x \<noteq> x}}"
ev_mono: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream"   and \<psi> :: "'b stream \<Rightarrow> bool" assumes "ev \<phi> xs"   and "\<And>xs. \<phi> xs \<Longrightarrow> \<psi> xs" shows "ev \<psi> xs"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "inj_idom_divide_hom hom" shows "map_poly_inj_idom_divide_hom hom"
1: fixes A :: "'a set" assumes "i \<in> A" shows "(A \<in> Rep_formula x) = (A \<in> Rep_formula x')"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) lit" assumes "\<not> I.satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>'__" and "xa = Pos a__" and "\<not> I.satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> I.satA \<xi> a"
bind_distr: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and X :: "'d measure"   and N :: "'d \<Rightarrow> 'e measure"   and K :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M X"   and "N \<in> X \<rightarrow>\<^sub>M subprob_algebra K"   and "space M \<noteq> {}" shows "distr M X f \<bind> N = M \<bind> (\<lambda>x. N (f x))"
np: fixes \<eta> :: "nat" shows "np (\<G> \<eta>)"
that: fixes k :: "nat" assumes "k \<le> n"   and "\<pi> k = ipd (\<pi> 0)" shows "thesis"
rreq_rrep_sn_def: fixes m :: "msg" shows "rreq_rrep_sn m \<equiv> case m of Rreq x xa xb xc xd osnc xe \<Rightarrow> 1 \<le> osnc | Rrep x xa dsnc xb xc \<Rightarrow> 1 \<le> dsnc | _ \<Rightarrow> True"
sup_commute: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<union># y = y \<union># x"
dyn_equiv_stat: shows "dynamically_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) empty_ord.Red_I Red_F_\<G>_empty = statically_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) empty_ord.Red_I Red_F_\<G>_empty"
pickQFresh_def: fixes xs :: "'e"   and V :: "'d set"   and XS :: "('f, 'g, 'e, 'd, 'h) qTerm set" shows "pickQFresh xs V XS \<equiv> SOME x. x \<notin> V \<and> (\<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
TA: fixes p :: "('b, 'a) fm" assumes "tautology p" shows "\<turnstile> p"
And\<^sub>n_prop_semantics: fixes \<Phi> :: "'b ltln set"   and \<A> :: "'b ltln set" assumes "finite \<Phi>" shows "\<A> \<Turnstile>\<^sub>P And\<^sub>n \<Phi> = (\<forall>\<phi>\<in>\<Phi>. \<A> \<Turnstile>\<^sub>P \<phi>)"
exists_worst_and_degenerate: shows "local.degenerate_lotteries \<inter> local.worst \<noteq> {}"
cases: fixes x :: "('c \<times> nat) tree" obtains "x = \<langle>\<rangle>"   | l :: "('c \<times> nat) tree" and a :: "'c" and h :: "nat" and r :: "('c \<times> nat) tree" where "x = \<langle>l, (a, h), r\<rangle>"
region_not_empty: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set" assumes "finite X"   and "Regions.valid_region X k I r" shows "\<exists>u. u \<in> Regions.region X I r"
cases: fixes f :: "nat \<Rightarrow> real" assumes "polynomial f" obtains n :: "real" where "f \<in> O(\<lambda>x. real x powr n)"
full_SetCompr_eq: fixes f :: "'c \<Rightarrow> 'b" shows "{u. \<exists>x. u = f x} = range f"
a_set_val_locs_def: fixes character_data_ptr :: "(_) character_data_ptr" shows "local.a_set_val_locs character_data_ptr \<equiv> all_args (put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr val_update)"
diffeomorphism_axioms_def: fixes charts1 :: "('c, 'd) chart set"   and charts2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and f' :: "'e \<Rightarrow> 'c" shows "diffeomorphism_axioms charts1 charts2 f f' \<equiv> (\<forall>x. x \<in> manifold.carrier charts1 \<longrightarrow> f' (f x) = x) \<and> (\<forall>y. y \<in> manifold.carrier charts2 \<longrightarrow> f (f' y) = y)"
keys_replace: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "keys (replace k v m) = keys m"
jozsa_transform_coeff_3: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "jozsa f n"   and "i < dim_row (jozsa.jozsa_transform f n) \<and> j < dim_col (jozsa.jozsa_transform f n)" shows "1 \<le> j \<and> i = j - 1 \<and> even i \<longrightarrow> jozsa.jozsa_transform f n $$ (i, j) = of_nat (f (i div 2))"
C_eq_sortedQ: fixes p :: "((int \<times> int) set set, int) Combinators list"   and l :: "(int \<times> int) set set set list" assumes "distinct p"   and "all_in_list p l"   and "singleCombinators p"   and "wellformed_policy1_strong p"   and "wellformed_policy3 p"   and "allNetsDistinct p" shows "C (list2FWpolicy (qsort p l)) = C (list2FWpolicy p)"
mono_F: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "F x \<le> F y"
hj_def: shows "hj = H_rcoset_reps ! j"
cscc_invar_loc_axioms: shows "cscc_invar_loc G v0 D0 l p D pE"
nonzero_abs_divide: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "b \<noteq> zero" shows "abs (divide a b) = divide (abs a) (abs b)"
assms_3: assumes "\<not> b"   and "\<not> b'" shows "RETURN S2 \<le> \<Down> R S2'"
facs_1: shows "u \<le>f r \<^sup>@ k"
DiffI: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A - B"
infinite_iff_natLeq_ordLeq: fixes A :: "'a set" shows "finite A \<noteq> (natLeq \<le>o |A|)"
R_def: shows "R \<equiv> {(h, w). h = \<^bold>g [^] fst w \<otimes> g' [^] snd w}"
hseq_char': fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "horizontal_composition V H src trg" shows "partial_magma.arr V (H \<nu> \<mu>) = (H \<nu> \<mu> \<noteq> partial_magma.null V)"
assms_6: fixes x :: "'b" assumes "x \<in> S" shows "x \<in> P"
L2_def: fixes x :: "nat"   and y :: "nat" shows "L2 x y = {(x, y), (x + 1, y), (x, y + 1)}"
fold_eq_nfoldli_assert: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and l :: "'d list"   and s :: "'c" shows "RETURN (fold f l s) = nfoldli l (\<lambda>_. True) (\<lambda>x s. ASSERT (x \<in> set l) \<bind> (\<lambda>_. RETURN (f x s))) s"
coalesce_eq_Err_D: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and xs :: "'b list"   and n :: "nat"   and ys :: "'b list" assumes "semilat (err A, Err.le r, lift2 f)"   and "xs \<in> list n A"   and "ys \<in> list n A"   and "coalesce (xs [+f] ys) = Err" shows "\<not> (\<exists>zs\<in>list n A. xs [\<sqsubseteq>\<^bsub>r\<^esub>] zs \<and> ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs)"
simps_15: fixes x21 :: "'a"   and x22 :: "'a"   and x23 :: "'b" shows "set1_arr (MkArr x21 x22 x23) = {x21, x22}"
edom_iff2: fixes m :: "'b \<Rightarrow> 'a"   and a :: "'b" shows "(m a = \<bottom>) = (a \<notin> edom m)"
max_prob_init: fixes \<O>\<^sub>s :: "'a set"   and \<K> :: "'b \<Rightarrow> 'b pmf"   and \<O> :: "'b \<Rightarrow> 'a pmf"   and \<S> :: "'b set"   and s :: "'b"   and os :: "'a list"   and o :: "'a" assumes "HMM2 \<O>\<^sub>s \<K> \<O> \<S>" shows "HMM2.max_prob \<O>\<^sub>s \<K> \<O> \<S> s os = Max {emeasure (MC_syntax.T (HMM_defs.K \<K> \<O>) (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'} |as. length as = length os \<and> set as \<subseteq> \<S>}"
map_pair_cong: fixes xs :: "('e \<times> 'f) list"   and f :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'g"   and g :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'g" assumes "\<And>kv. kv \<in> set xs \<Longrightarrow> f kv = g kv" shows "map_pair f xs = map_pair g xs"
id: shows "map fst (map (\<lambda>(A, jbs). (A, jordan_matrix jbs)) Ms) = map fst Ms"
that_2: fixes ta' :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "t \<turnstile> \<langle>x, shr s\<rangle> -ta'\<rightarrow> \<langle>x', m'\<rangle>"   and "actions_ok s t ta'" shows "thesis"
assms: shows "(\<sigma>, P) \<rightarrow>\<^sub>u (\<rho>, Q)"
update_f: fixes F :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'k \<Rightarrow> 'a option"   and x :: "'k"   and g :: "'a \<Rightarrow> 'm" shows "update F (bind (f x) g) = bind (f x) (\<lambda>a. update F (g a))"
subspace_def: fixes S :: "'c set" shows "m2.subspace S = ((0::'c) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *b x \<in> S))"
wadjust_goon_left_moving_B_no_Oc: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" shows "wadjust_goon_left_moving_B m rs (c, Oc # list) = False"
msg_after_marker_and_nonempty_implies_postrecording_event: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "Marker \<in> set (msgs (S t i) cid)"   and "Marker \<noteq> last (msgs (S t i) cid)"   and "i \<le> length t"   and "channel cid = Some (p, q)" shows "\<exists>j<i. postrecording_event t j"
nonconstant_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b list" assumes "class.semiring_0 plus zero times" shows "semiring_0.nonconstant zero p = (semiring_0.pnormal zero p \<and> (\<forall>x. p \<noteq> [x]))"
cases: fixes x :: "('b, 'c) ioa \<times> 'b \<times> ('c \<times> 'b) list" obtains A :: "('b, 'c) ioa" and s :: "'b" and ps :: "('c \<times> 'b) list" and p' :: "'c \<times> 'b" and p :: "'c \<times> 'b" where "x = (Aa__, s__, Cons p__ (Cons p' ps))"   | A :: "('b, 'c) ioa" and s :: "'b" and p :: "'c \<times> 'b" where "x = (Ab__, sa__, [p])"   | A :: "('b, 'c) ioa" and s :: "'b" where "x = (A, s, [])"
chain_homotopic_iterated_singular_subdivision: fixes n :: "nat" obtains h :: "nat \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" where "\<And>p. h p 0 = 0" and "\<And>p c1 c2. h p (c1 - c2) = h p c1 - h p c2" and "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (h p c)" and "\<And>p X c. singular_chain p X c \<Longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c) = (singular_subdivision p ^^ n) c - c"
fin_mono: fixes A :: "'b fset"   and B :: "'b fset"   and x :: "'b" assumes "A |\<subseteq>| B" shows "x |\<in>| A \<longrightarrow> x |\<in>| B"
zero_less_mult_iff: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less zero (times a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"
B_axiom: shows "universal (\<lambda>x y. x r y \<longrightarrow> y r x)"
emp_act: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and p :: "'d" assumes "unital_quantale_module act" shows "act (\<Squnion>{}) p = \<bottom>"
converse_lehmer: fixes p :: "nat" assumes "prime p" shows "\<exists>a. [a ^ (p - 1) = 1] (mod p) \<and> (\<forall>q. q \<in># prime_factorization (p - 1) \<longrightarrow> [a ^ ((p - 1) div q) \<noteq> 1] (mod p)) \<and> 0 < a \<and> a < p"
x_lin_x: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "x \<in> lin x"
topl: fixes x :: "'a" shows "\<top> \<cdot> - (\<top> \<cdot> x) = - (\<top> \<cdot> x)"
idx_valid: fixes bound :: "'b bound"   and x :: "'b" assumes "in_bound less_eq1 less1 bound x" shows "idx1 bound x < size1 bound"
inf_Sup_absorb: fixes 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 A :: "'b set"   and a :: "'b" assumes "class.lattice inf less_eq less sup"   and "finite A"   and "a \<in> A" shows "inf a (semilattice_sup.Sup_fin sup A) = a"
base: fixes C :: "'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "([], C) \<in> R" shows "C \<in> derivable' R"
binop_axioms: shows "binop \<rat>\<^sub>\<circ> vrat_plus"
substitution_code_def: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and "next" :: "'node \<times> 'val" shows "uninst.substitution_code p g next = the (the_trivial (snd next) (the (p g next)))"
p_def: shows "p = (\<lambda>f. cbox (\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)) ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
obsf_resource_sel: fixes res :: "('b, 'a) resource"   and x :: "'b" shows "run_resource (obsf_resource res) x = map_spmf (map_prod id (\<lambda>resF. obsf_resource (case resF of Fault \<Rightarrow> fail_resource | OK res' \<Rightarrow> res'))) (TRY map_spmf (map_prod OK OK) (run_resource res x) ELSE return_spmf (Fault, Fault))"
assms_1: fixes m :: "nat" shows "f m \<in> carrier (residue_ring (p ^ m))"
gb_sig_z_pprod_def: fixes to :: "(('a, 'b) pp \<times> nat) nat_term_order"   and rword_strict :: "(('a, 'b) pp \<times> nat) nat_term_order \<Rightarrow> (('a, 'b) pp \<times> nat) \<times> (('a, 'b) pp \<Rightarrow>\<^sub>0 'c) \<Rightarrow> (('a, 'b) pp \<times> nat) \<times> (('a, 'b) pp \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and fs :: "(('a, 'b) pp \<Rightarrow>\<^sub>0 'c) list" shows "gb_sig_z_pprod to rword_strict fs \<equiv> let res = gb_sig_z_pprod' to (rword_strict to) (map (change_ord (proj_ord to)) fs) in (length (fst res), snd res)"
diff: fixes x :: "'b"   and y :: "'b" shows "g (x - y) = g x - g y"
rupture_surj_hom_1: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "UP.rupture_surj K p \<in> ring_hom (K [X]\<^bsub>K [X]\<^esub>) (Rupt\<^bsub>K [X]\<^esub> K p)"
ket_pre_def: shows "ket_pre = Matrix.vec N (\<lambda>k. if k = 0 then 1 else 0)"
sel_1: fixes x1 :: "('e \<times> 'f \<times> 'f) set"   and x2 :: "'f set" shows "edges (LG x1 x2) = x1"
locally_path_connected_space_quotient_map_image: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "quotient_map X Y f"   and "locally_path_connected_space X" shows "locally_path_connected_space Y"
det_swap_final_rows: fixes A :: "'b mat"   and m :: "nat"   and l :: "nat"   and k :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m m"   and "m = l + k + n" shows "det A = (- (1::'b)) ^ (k * n) * det (mat m m (\<lambda>(i, j). A $$ (if i < l then i else if i < l + n then i + k else i - n, j)))"
nat_term_compare_conv: fixes to :: "'a nat_term_order" shows "nat_term_compare to = key_compare (key_order_of_nat_term_order to)"
dims0_def: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and dims :: "nat list" assumes "partial_state2 vars1 vars2" shows "partial_state2.dims0 dims vars1 vars2 = nths dims (partial_state2.vars0 vars1 vars2)"
vpairsD_2: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> vpairs r" shows "\<exists>a b. x = \<langle>a, b\<rangle>"
CondTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and t :: "('d, 'f) xstate"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Cond b c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
t_def: shows "compute_tm_by_comp prec ord I a f (Some tg) c = Some t"
simps_3: fixes x :: "num" shows "Num.inc (num.Bit1 x) = num.Bit0 (Num.inc x)"
nrename_NotL: fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and N :: "trm"   and z :: "name" assumes "R[x\<turnstile>n>y] = NotL <c>.N z"   and "c \<sharp> R"   and "z \<sharp> (x, y)" shows "\<exists>N'. R = NotL <c>.N' z \<and> N'[x\<turnstile>n>y] = N"
word_less_1: fixes x :: "'b word" shows "(x < 1) = (x = 0)"
v0_v1_Digraph_axioms_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c" shows "v0_v1_Digraph_axioms G v0 v1 \<equiv> (v0 \<in> V\<^bsub>G\<^esub> \<and> v1 \<in> V\<^bsub>G\<^esub>) \<and> (v0, v1) \<notin> E\<^bsub>G\<^esub> \<and> v0 \<noteq> v1"
strand_sem_mono_ik_2: fixes arity :: "'c \<Rightarrow> nat"   and Ana :: "('c, 'd) Term.term \<Rightarrow> ('c, 'd) Term.term list \<times> ('c, 'd) Term.term list"   and M :: "('c, 'd) Term.term set"   and M' :: "('c, 'd) Term.term set"   and public :: "'c \<Rightarrow> bool"   and S :: "('c, 'd) strand_step list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "intruder_model arity Ana"   and "M \<subseteq> M'"   and "intruder_model.strand_sem_d arity public Ana M S \<theta>" shows "intruder_model.strand_sem_d arity public Ana M' S \<theta>"
intro: fixes get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" assumes "\<And>ptr h. pure (get_root_node ptr) h"   and "\<And>h node_ptr. h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None \<Longrightarrow> h \<turnstile> get_root_node (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) \<rightarrow>\<^sub>r 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" shows "l_get_root_node get_root_node get_parent"
Pi_cancel_fupd: fixes i :: "'a"   and I :: "'a set"   and x :: "'a \<Rightarrow> 'b"   and a :: "'b"   and B :: "'a \<Rightarrow> 'b set" assumes "i \<notin> I" shows "(x(i := a) \<in> Pi I B) = (x \<in> Pi I B)"
inj: shows "inj_on (\<lambda>b. x * b mod q) {..<q}"
echelon_form_Hermite_of_condition2: fixes A :: "(('a, 'b) vec, 'c) vec"   and ass :: "'a \<Rightarrow> 'a"   and i :: "'c"   and ia :: "'c"   and j :: "'c"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "echelon_form A"   and "ass_function ass"   and "\<not> is_zero_row i A"   and "ia < j"   and "\<not> is_zero_row ia (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res)"   and "\<not> is_zero_row j (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res)" shows "(LEAST n. Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res $ ia $ n \<noteq> (0::'a)) < (LEAST n. Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res $ j $ n \<noteq> (0::'a))"
oc2: shows "fs ! i - (fs ! i - gso i) \<in> orthogonal_complement (local.span (gso ` {0..<i}))"
rquot_node: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and q :: "nat" shows "dfa_is_node (rquot M n) q = dfa_is_node M q"
atLeastAtMost_singleton: fixes a :: "'a Word.word" shows "signed.atLeastAtMost a a = {a}"
unique_dimension: fixes K :: "'a set"   and E :: "'a set" assumes "subfield K (L\<lparr>carrier := A\<rparr>)"   and "Hom.finite_dimension K E" shows "\<exists>!n. Hom.dimension n K E"
test_mult_comm_var: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "p \<cdot> q = q \<cdot> p"
restr_can_restrict_env_heap: fixes r :: "C" shows "(\<N>\<lbrace>\<Gamma>\<rbrace>\<rho>)|\<^sup>\<circ>\<^bsub>r\<^esub> = (\<N>\<lbrace>\<Gamma>\<rbrace>\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>)|\<^sup>\<circ>\<^bsub>r\<^esub>"
size_2: fixes x :: "'a \<Rightarrow> nat"   and x21 :: "'a"   and x22 :: "'a list" shows "size_list x (x21 # x22) = x x21 + size_list x x22 + Suc 0"
invimage_eq_vdomain_vrestriction: fixes r :: "V"   and A :: "V" shows "r -`\<^sub>\<circ> A = \<D>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> A)"
ts_tuple_rel_ext_Cons: fixes tas :: "nat \<times> 'a option list"   and nt :: "nat"   and X :: "'a option list set"   and tass :: "(nat \<times> 'a option list set) list" assumes "tas \<in> ts_tuple_rel {(nt, X)}" shows "tas \<in> ts_tuple_rel (set ((nt, X) # tass))"
compl_sets: fixes a :: "'a set" assumes "a \<in> M" shows "\<Omega> - a \<in> M"
poly_rel_mod: shows "(poly_rel ===> poly_rel ===> poly_rel) (mod_field_poly_i ops) (mod)"
insert_Diff_single: fixes a :: "'g"   and A :: "'g set" shows "insert a (A - {a}) = insert a A"
f0: shows "(\<Oplus>\<^bsub>V\<^esub>v\<in>set vs - A. (if v \<in> set vs - A then 0::'a else f v) \<cdot>\<^sub>v v) = 0\<^sub>v n"
module_hom_axioms_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "module_hom_axioms s1 s2 f \<equiv> (\<forall>b1 b2. f (b1 + b2) = f b1 + f b2) \<and> (\<forall>r b. f (s1 r b) = s2 r (f b))"
seq_if_composable: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "\<nu> \<star>\<^sub>D \<mu> \<noteq> D.null" shows "src\<^sub>D \<nu> = trg\<^sub>D \<mu>"
continuous_on_real_root: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real"   and n :: "nat" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. root n (f x))"
fundamental_theorem_of_calculus_interior_strong: fixes S :: "real set"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a" assumes "finite S"   and "a \<le> b"   and "\<And>x. x \<in> {a<..<b} - S \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "continuous_on {a..b} f" shows "(f' has_integral f b - f a) {a..b}"
Int_atMost: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.linorder less_eq less" shows "ord.atMost less_eq a \<inter> ord.atMost less_eq b = ord.atMost less_eq (ord.min less_eq a b)"
resl_superdist_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<leftarrow> y \<squnion> z \<le> x \<leftarrow> y"
infinite_super: fixes S :: "'q set"   and T :: "'q set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
DomainE: fixes a :: "'h"   and r :: "('h \<times> 'i) set" assumes "a \<in> Domain r" obtains b :: "'i" where "(a, b) \<in> r"
same_suffix_suffix: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" shows "ys \<cdot> xs \<le>s zs \<cdot> xs = ys \<le>s zs"