hcju/isabelleps huggingface dataset:
assms_2: shows "\<forall>m. P m \<longrightarrow> (\<forall>m'\<in>set (f m). Q m')"
image_mono: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "order.greater_eq B A" shows "order.greater_eq (f ` B) (f ` A)"
flatten_append: fixes xs :: "'a list list"   and ys :: "'a list list" shows "flatten (xs @ ys) = flatten xs @ flatten ys"
dom_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "local.dom (src \<mu>) = src \<mu>"
compat: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and default :: "'b"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "order_pair gt default"   and "y \<le> x"   and "gt y z" shows "gt x z"
uminus_set_def: fixes A :: "'b set" shows "- A = p2s (- (\<lambda>x. x \<in> A))"
shift1: fixes bs :: "'a list"   and p :: "poly" shows "\<lparr>shift1 p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>Mul (Bound 0) p\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
B: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "onorm (f' x) \<le> B"
pinv: shows "T \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg I), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). Q \<sigma> \<sigma>')"
assms_2: shows "h \<turnstile> adopt_node document_ptr child \<rightarrow>\<^sub>h h'"
hom_graph_insert_elim: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'a"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b" assumes "hom_graph f (insert i S) x a"   and "i \<notin> S" obtains y :: "'a formula" and z :: "'a formula" and b :: "'b" and c :: "'b" where "x = ifte (var i) y z" and "a = ifte (f i) b c" and "hom_graph f S y b" and "hom_graph f S z c"
iso_inv_iso: fixes f :: "'b" assumes "\<phi>.B.iso f" shows "\<phi>.B.iso (\<phi>.B.inv f)"
simps_1: fixes f :: "atom \<Rightarrow> nat \<Rightarrow> atom"   and n :: "nat" shows "map_fm_binders f TrueF n = TrueF"
int_mod_ge: fixes a :: "int"   and n :: "int" assumes "a < n"   and "0 < n" shows "a \<le> a mod n"
\<S>: fixes S' :: "('fun, 'var) strand_step list" assumes "S' \<in> update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)" shows "S' = S \<or> S' \<in> \<S>"
borel_measurable_less: 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 "{w \<in> space M. f w < g w} \<in> sets M"
simps_5: fixes f1 :: "'a"   and f2 :: "8 word \<Rightarrow> 'a"   and x2 :: "8 word" shows "(case Proto x2 of ProtoAny \<Rightarrow> f1 | Proto x \<Rightarrow> f2 x) = f2 x2"
transfer_operations_rl_3: shows "(msum_appr ops, \<lambda>xs ys. {map2 (+) x y |x y. x \<in> xs \<and> y \<in> ys}) \<in> appr_rell \<rightarrow> appr_rell \<rightarrow> appr_rell"
assms_1: fixes u :: "'t" assumes "u \<in> keys p" shows "v \<prec>\<^sub>t u"
f_def: shows "f = (\<lambda>a. case a of (al, fl) \<Rightarrow> (\<forall>a\<in>set al. aEval a xs) \<and> (\<forall>f\<in>set fl. eval f xs))"
insertions_subset: fixes A :: "('c \<times> ('c, 'd) list_op) list"   and B :: "('c \<times> ('c, 'd) list_op) list" assumes "list_ops A"   and "list_ops B"   and "set A \<subseteq> set B" shows "set (insertions A) \<subseteq> set (insertions B)"
zmset_of_plus: fixes M :: "'b multiset"   and N :: "'b multiset" shows "zmset_of (M + N) = zmset_of M + zmset_of N"
image_vimage_adjoints: fixes ob :: "'f \<Rightarrow> 'e"   and P :: "'f set"   and Q :: "'e set" shows "(ob ` P \<subseteq> Q) = (P \<subseteq> ob -` Q)"
lookup_pair_single: fixes k :: "'a"   and v :: "'c"   and k0 :: "'a" shows "lookup_pair [(k, v)] k0 = (if k = k0 then v else (0::'c))"
finite_measureI: fixes M :: "'c measure" assumes "emeasure M (space M) \<noteq> \<infinity>" shows "finite_measure M"
name_swap: fixes a :: "name"   and b :: "name"   and x :: "'b" shows "[(a, b)] \<bullet> x = [(b, a)] \<bullet> x"
rev_filter: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "rev (filter P xs) = filter P (rev xs)"
ereal_plus_1_1: fixes r :: "real" shows "1 + ereal r = ereal (r + 1)"
unat_div: fixes x :: "'a word"   and y :: "'a word" shows "unat (x div y) = unat x div unat y"
disjnt_Times1_iff: fixes C :: "'c set"   and A :: "'d set"   and B :: "'d set" shows "disjnt (C \<times> A) (C \<times> B) = (C = {} \<or> disjnt A B)"
b: shows "divisor_set 6 = {1, 2, 3, 6}"
assms_2: shows "\<forall>x\<in>P. finite (cl_ecl x)"
length_rev: fixes xs :: "'c list" shows "length (rev xs) = length xs"
inject_3: fixes x2 :: "ccs"   and x1 :: "ccs"   and y2 :: "ccs"   and y1 :: "ccs" shows "(x2 \<parallel> x1 = y2 \<parallel> y1) = (x2 = y2 \<and> x1 = y1)"
span_image_scale_eq_image_scale: fixes q :: "'a"   and F :: "'a set" shows "ideal ((*) q ` F) = (*) q ` ideal F"
equal: fixes equal :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
rep_eq: fixes x :: "('g, 'e) generator"   and xa :: "('g, 'f) generator" shows "generator (splice_trans x xa) = splice_raw (generator x) (generator xa)"
bijec_def: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" shows "bijec\<^bsub>F,G\<^esub> f = (injec\<^bsub>F,G\<^esub> f \<and> surjec\<^bsub>F,G\<^esub> f)"
not_outside_2: fixes x :: "'b pmf"   and y :: "'b pmf" assumes "Pareto (SD \<circ> R) x y" shows "y \<in> lotteries_on alts"
simps_4: fixes P :: "'a prog"   and D :: "char list"   and C :: "char list"   and M :: "char list"   and es :: "'b exp list" shows "icheck P D (C\<bullet>\<^sub>sM(es)) = ((\<exists>vs. es = map Val vs) \<and> (\<exists>Ts T m. P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D))"
juncts_eq_set_juncts_list_1: fixes \<phi> :: "('c, 'd) aformula" shows "disjuncts \<phi> = set (disjuncts_list \<phi>)"
HStartBallot_HInv5_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "HInv5_inner s p" shows "HInv5_inner s' p"
eq_numeral_iff_iszero_10: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and y :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "(zero = numeral.numeral one plus y) = ring_1.iszero zero (numeral.numeral one plus y)"
integral_scaleR_left: fixes c :: "'d"   and M :: "'e measure"   and f :: "'e \<Rightarrow> real" assumes "c \<noteq> (0::'d) \<Longrightarrow> integrable M f" shows "LINT x|M. f x *\<^sub>R c = integral\<^sup>L M f *\<^sub>R c"
assms_1: fixes i :: "nat" shows "sigma_algebra (space M) (A i)"
2: shows "absCase xs s f (Abs xs x X) = f x' X'"
filter_cong: fixes xs :: "'a list"   and ys :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
static_empty_order_equiv_static: shows "statically_complete_calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> local.Red_F_\<G> = statically_complete_calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> empty_ord.Red_F_\<G>"
fup_Aexp_eq: fixes e :: "exp"   and a :: "Arity\<^sub>\<bottom>" shows "Aexp_bot_syn' a e = fst (fup\<cdot>(cCCexp e)\<cdot>a)"
possible_steps_alt3: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and d :: "nat"   and t :: "transition" shows "(possible_steps e s r l i = {|(d, t)|}) = (ffilter (\<lambda>((origin, dest), t). origin = s \<and> Label t = l \<and> can_take_transition t i r) e = {|((s, d), t)|})"
M0_def: shows "M0 = mat K K (\<lambda>(i, j). if i = j \<and> R \<le> i then 1 else 0)"
predicate1I: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "\<And>x. P x \<Longrightarrow> Q x" shows "P \<le> Q"
hyps_2: fixes s :: "'a llist" assumes "l @@ s \<in> A\<^sup>\<omega>" shows "s \<in> A\<^sup>\<omega>"
CofinitaryGroup_def: fixes dom :: "(nat \<Rightarrow> nat) set" shows "CofinitaryGroup dom \<equiv> (dom \<subseteq> S_inf \<and> id \<in> dom) \<and> (\<forall>f g. f \<in> dom \<and> g \<in> dom \<longrightarrow> f \<circ> g \<in> dom) \<and> (\<forall>f. f \<in> dom \<longrightarrow> inv f \<in> dom) \<and> (\<forall>f. f \<in> dom \<and> f \<noteq> id \<longrightarrow> finite (Fix f))"
meets_wd: fixes p :: "'a"   and q :: "'a" assumes "p \<parallel> q" shows "\<I> p \<and> \<I> q"
def_prg_Allowed: fixes F :: "'c program"   and init :: "'c set"   and acts :: "('c \<times> 'c) set set"   and X :: "'c program set" assumes "F \<equiv> mk_program (init, acts, \<Union> (Acts ` X))"   and "safety_prop X" shows "Allowed F = X"
unreach_connected: fixes Q :: "'a set"   and b :: "'a"   and Q\<^sub>x :: "'a"   and Q\<^sub>z :: "'a"   and Q\<^sub>y :: "'a" assumes "Q \<in> \<P>"   and "b \<notin> Q"   and "b \<in> \<E>"   and "Q\<^sub>x \<in> \<emptyset> Q b"   and "Q\<^sub>z \<in> \<emptyset> Q b"   and "Q\<^sub>x \<noteq> Q\<^sub>z"   and "[[Q\<^sub>x Q\<^sub>y Q\<^sub>z]]" shows "Q\<^sub>y \<in> \<emptyset> Q b"
step_16: shows "InvariantVarsF (getF state') F0 Vbl"
cauchy_in\<^sub>N_I: fixes N :: "'c quasinorm"   and u :: "nat \<Rightarrow> 'c" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. Norm N (u n - u m) < e" shows "cauchy_in\<^sub>N N u"
dsj_mono: fixes A :: "fmla"   and H :: "fmla set"   and B :: "fmla"   and C :: "fmla"   and D :: "fmla" assumes "insert A H \<turnstile> B"   and "insert C H \<turnstile> D" shows "insert (A OR C) H \<turnstile> B OR D"
simps: fixes x :: "nat"   and n :: "nat" shows "\<MM>.token_run x n = run \<up>af\<^sub>G (Abs \<phi>) (suffix x w) (n - x)"
Abs_fnat_hom_1: shows "1 = word_of_nat (Suc 0)"
stirling_sum'_complex_of_real: fixes j :: "nat"   and m :: "nat"   and x :: "real" shows "stirling_sum' j m (complex_of_real x) = complex_of_real (stirling_sum' j m x)"
simps_1: fixes s :: "'c \<Rightarrow> bool option" shows "execute_parallel_plan s [] = s"
refl_symm_closure_is_symm_refl_closure: fixes Rel :: "('c \<times> 'c) set" shows "symcl (Rel\<^sup>=) = (symcl Rel)\<^sup>="
fun_add_zeroL: fixes a :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier F" shows "\<zero>\<^bsub>F\<^esub> \<oplus>\<^bsub>F\<^esub> a = a"
big_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Omega>[F](g)"   and "g \<in> \<Omega>[F](h)" shows "f \<in> \<Omega>[F](h)"
funionCI: fixes c :: "'a"   and B :: "'a fset"   and A :: "'a fset" assumes "c |\<notin>| B \<Longrightarrow> c |\<in>| A" shows "c |\<in>| A |\<union>| B"
RDirProd_def: fixes R :: "('d, 'f) ring_scheme"   and S :: "('e, 'g) ring_scheme" shows "RDirProd R S = monoid.extend (R \<times>\<times> S) \<lparr>zero = \<one>\<^bsub>add_monoid R \<times>\<times> add_monoid S\<^esub>, add = (\<otimes>\<^bsub>add_monoid R \<times>\<times> add_monoid S\<^esub>)\<rparr>"
jkbpC_tLength_inv: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and n :: "nat" assumes "t \<in> SPR.jkbpC"   and "tLength t = n" shows "t \<in> SPR.jkbpCn n"
split: fixes P :: "'e \<Rightarrow> bool"   and f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and option :: "'f option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
next_preserves_inv_implications_nonneg: fixes s :: "('loc, 't) configuration stream" assumes "next s"   and "holds inv_implications_nonneg s"   and "holds inv_imp_plus_work_nonneg s" shows "nxt (holds inv_implications_nonneg) s"
fps_inverse_idempotent: fixes f :: "'a fps" assumes "fps_nth f 0 \<noteq> (0::'a)" shows "inverse (inverse f) = f"
induct: fixes P :: "'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b up\<^sub>d \<Rightarrow> bool"   and a0 :: "'b tree234"   and a1 :: "'b"   and a2 :: "'b tree234"   and a3 :: "'b"   and a4 :: "'b up\<^sub>d" assumes "\<And>l a m b r. P l a m b (T\<^sub>d r)"   and "\<And>t1 a t2 b t3 c t4. P t1 a \<langle>t2, b, t3\<rangle> c (Up\<^sub>d t4)"   and "\<And>t1 a t2 b t3 c t4 d t5. P t1 a \<langle>t2, b, t3, c, t4\<rangle> d (Up\<^sub>d t5)"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6. P t1 a \<langle>t2, b, t3, c, t4, d, t5\<rangle> e (Up\<^sub>d t6)"   and "\<And>a b d v. P a b \<langle>\<rangle> d (Up\<^sub>d v)" shows "P a0 a1 a2 a3 a4"
res: shows "euclid_ext_poly_mod_main m ff_ops f g = (a, b)"
mon_w_def: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and w :: "('h, 'k) label list" shows "mon_w fg w \<equiv> \<Union> {mon_e fg e |e. e \<in> set w}"
fmdom'_notD: fixes x :: "'a"   and m :: "('a, 'b) fmap" assumes "x \<notin> fmdom' m" shows "m $$ x = None"
word_ao_absorbs_8: fixes x :: "'b word"   and y :: "'b word" shows "x AND y OR x = x"
DynCom: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and c :: "'f \<Rightarrow> ('f, 'e, 'g) com"   and s :: "'f"   and t :: "('f, 'g) xstate" assumes "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> \<Rightarrow> t"
hpsafe_Pvar_simps: fixes a :: "'c" shows "hpsafe ($\<alpha> a) = True"
rev_induct: fixes P :: "'g list \<Rightarrow> bool"   and xs :: "'g list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (xs @ [x])" shows "P xs"
omega_continuousDdom: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "omega_continuous A less_eq f" shows "f ` A \<subseteq> A"
mset_subset_size: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "A \<subset># B" shows "size A < size B"
OrdP_Zero: fixes H :: "fm set" shows "H \<turnstile> OrdP Zero"
i_Exec_Stream_Init_nth_Suc_eq_i_Exec_Stream_nth: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d"   and n :: "nat" shows "i_Exec_Comp_Stream_Init trans_fun input c (Suc n) = i_Exec_Comp_Stream trans_fun input c n"
terminal_lappendt_lfinite: fixes xs :: "'a llist"   and ys :: "('a, 'b) tllist" assumes "lfinite xs" shows "terminal (lappendt xs ys) = terminal ys"
ltlr_rel_cases: fixes a1 :: "'a ltlr"   and a2 :: "'b ltlr"   and R :: "('a \<times> 'b) set" assumes "(a1, a2) \<in> \<langle>R\<rangle>ltlr_rel" obtains "a1 = true\<^sub>r" and "a2 = true\<^sub>r"   | "a1 = false\<^sub>r" and "a2 = false\<^sub>r"   | a :: "'a" and a' :: "'b" where "a1 = prop\<^sub>r(a__)" and "a2 = prop\<^sub>r(a'__)" and "(a__, a'__) \<in> R"   | a :: "'a" and a' :: "'b" where "a1 = nprop\<^sub>r(aa__)" and "a2 = nprop\<^sub>r(a'a__)" and "(aa__, a'a__) \<in> R"   | a :: "'a ltlr" and a' :: "'b ltlr" and b :: "'a ltlr" and b' :: "'b ltlr" where "a1 = ab__ and\<^sub>r b__" and "a2 = a'b__ and\<^sub>r b'__" and "(ab__, a'b__) \<in> \<langle>R\<rangle>ltlr_rel" and "(b__, b'__) \<in> \<langle>R\<rangle>ltlr_rel"   | a :: "'a ltlr" and a' :: "'b ltlr" and b :: "'a ltlr" and b' :: "'b ltlr" where "a1 = ac__ or\<^sub>r ba__" and "a2 = a'c__ or\<^sub>r b'a__" and "(ac__, a'c__) \<in> \<langle>R\<rangle>ltlr_rel" and "(ba__, b'a__) \<in> \<langle>R\<rangle>ltlr_rel"   | a :: "'a ltlr" and a' :: "'b ltlr" where "a1 = X\<^sub>r ad__" and "a2 = X\<^sub>r a'd__" and "(ad__, a'd__) \<in> \<langle>R\<rangle>ltlr_rel"   | a :: "'a ltlr" and a' :: "'b ltlr" and b :: "'a ltlr" and b' :: "'b ltlr" where "a1 = ae__ U\<^sub>r bb__" and "a2 = a'e__ U\<^sub>r b'b__" and "(ae__, a'e__) \<in> \<langle>R\<rangle>ltlr_rel" and "(bb__, b'b__) \<in> \<langle>R\<rangle>ltlr_rel"   | a :: "'a ltlr" and a' :: "'b ltlr" and b :: "'a ltlr" and b' :: "'b ltlr" where "a1 = a R\<^sub>r b" and "a2 = a' R\<^sub>r b'" and "(a, a') \<in> \<langle>R\<rangle>ltlr_rel" and "(b, b') \<in> \<langle>R\<rangle>ltlr_rel"
ssubst: fixes t :: "'c"   and s :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "t = s"   and "P s" shows "P t"
ncols_iarray_def: fixes A :: "'a iarray iarray" shows "ncols_iarray A = IArray.length (A !! 0)"
orient: shows "\<forall>x y. (x, y) \<in> set xys \<longrightarrow> gt_arctic_delta \<delta> x y"
llist_of_tllist_TCons: fixes xa :: "'a"   and x :: "('a, 'b) tllist" shows "llist_of_tllist (TCons xa x) = LCons xa (llist_of_tllist x)"
Basis_pair_def: shows "Basis_pair = B1 \<times> {0::'c} \<union> {0::'b} \<times> B2"
empty_sets: fixes M :: "'a measure" shows "{} \<in> null_sets M"
conf_keep: fixes m :: "nat"   and lm :: "nat"   and stp :: "nat"   and r :: "nat"   and n :: "nat" assumes "conf m lm stp = trpl 0 0 r" shows "conf m lm (stp + n) = trpl 0 0 r"
small_vsv: fixes A :: "V"   and B :: "V" shows "small {f. vsv f \<and> \<D>\<^sub>\<circ> f = A \<and> \<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> B}"
WR_succ_iff: fixes i :: "hf"   and z :: "hf" assumes "Ord i" shows "WR (succ i) z = (\<exists>y. z = q_Eats y y \<and> WR i y)"
set_map: fixes f :: "'h \<Rightarrow> 'g"   and v :: "'h list" shows "set (map f v) = f ` set v"
FAssThrow1: fixes e :: "char list exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>throw e\<bullet>F{D} := e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
rbbl_Cons: fixes b :: "bool"   and n :: "nat"   and x :: "int" shows "b # rev (bin_to_bl n x) = rev (bin_to_bl (Suc n) (of_bool b + 2 * x))"
bounded_countable_web_axioms_def: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "bounded_countable_web_axioms \<Gamma> \<equiv> \<forall>x. x \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub> - B \<Gamma> \<longrightarrow> integral\<^sup>N (count_space (\<^bold>E\<^bsub>\<Gamma>\<^esub> `` {x})) (weight \<Gamma>) < \<top>"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_set_tag_name type_wf set_tag_name set_tag_name_locs"   and "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_disconnected_nodes set_disconnected_nodes set_tag_name create_element known_ptr" shows "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs type_wf create_element known_ptr"
lift_f_trivial_Q: fixes q :: "'a freeword" assumes "q \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>" shows "lift_f q = (0::'b)"
prv_exi_imp_gen: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and x :: "'a"   and \<phi> :: "'c"   and \<chi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<chi>"   and "prv (imp \<phi> \<chi>)" shows "prv (imp (exi x \<phi>) \<chi>)"
assms: shows "det2 a11 a12 a21 a22 \<noteq> (0::'a)"
fgmodule_is_module: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and z :: "'d" assumes "fgmodule_condition R f i s A z" shows "R module fgmodule R A z i f s"
p2_def: shows "p2 = p \<circ>\<^sub>p [:Complex (Re ub) (Im lb), Complex 0 (Im ub - Im lb):]"
abelian_group: fixes m :: "int" assumes "residues m" shows "abelian_group (residue_ring m)"
BRParCases: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and F :: "pi \<Rightarrow> bool" assumes "(P, Q \<parallel> !Q) \<in> bangRel Rel"   and "\<And>P R. \<lbrakk>(P, Q) \<in> Rel; (R, !Q) \<in> bangRel Rel\<rbrakk> \<Longrightarrow> F (P \<parallel> R)" shows "F P"
card_Un_disjoint: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "card (A \<union> B) = card A + card B"
simps_1: fixes uu :: "'a tree" shows "mrbst_inter \<langle>\<rangle> uu = return_pmf \<langle>\<rangle>"
self_fin_delta_pf_def: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and v0 :: "real" shows "self_fin_delta_pf N der matur v0 = self_finance Mkt v0 (delta_pf N der matur) risk_free_asset"
polybound0_I: fixes a :: "'b poly"   and b :: "'b"   and bs :: "'b list"   and b' :: "'b" assumes "polybound0 a" shows "Ipoly (b # bs) a = Ipoly (b' # bs) a"
simps_1: fixes x :: "name"   and T :: "ty" shows "pat_type (PVar x T) = T"
simps_1: fixes x :: "'a"   and y :: "'a" shows "(Some x \<sqsubseteq> Some y) = (x \<sqsubseteq> y)"
learn_finI: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "environment \<psi> U s"   and "\<And>f. f \<in> U \<Longrightarrow> \<exists>i n\<^sub>0. \<psi> i = f \<and> (\<forall>n<n\<^sub>0. s (f \<triangleright> n) \<down>= 0) \<and> (\<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= Suc i)" shows "learn_fin \<psi> U s"
atLeast0_lessThan_Suc: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "prod g {0..<Suc n} = prod g {0..<n} * g n"
plus_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "f \<oplus>\<^sub>M g = (\<lambda>e. f e + g e)"
state_\<alpha>_conv_3: fixes s :: "('l, 't) locks \<times> ('m_t \<times> 'm) \<times> 'm_w \<times> 's_i" shows "shr (state_\<alpha> s) = shr s"
le_length_plus: fixes xs :: "['b]"   and n :: "Integer" assumes "length\<cdot>xs \<noteq> \<bottom>"   and "n \<noteq> \<bottom>" shows "le\<cdot>n\<cdot>(length\<cdot>xs + n) = TT"
refl_on_allrel: fixes A :: "'b set" shows "refl_on A (A \<times> A)"
intros_2: fixes ss :: "('c, 'd) Term.term list"   and ts :: "('c, 'd) Term.term list"   and f :: "'c"   and E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "length ss = length ts" shows "UNIF1 Var (add_mset (Fun f ss, Fun f ts) E) (E + mset (zip ss ts))"
absorb_iff2: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (a + b = b)"
SKIP_le: fixes S :: "'a"   and c :: "'a acom" shows "(SKIP {S} \<le> c) = (\<exists>S'. c = SKIP {S'} \<and> S \<le> S')"
span_eq_real1: fixes S :: "'c set"   and T :: "'d set" assumes "real_vector_space_pair_on S T" shows "module_on.span (*\<^sub>R) = span"
divisor_sigma_1_left: fixes n :: "nat" shows "divisor_sigma (1::'a) n = of_nat (divisor_sum n)"
assms_3: shows "A $$ (i, i) = (0::'a)"
clm_sing_not_interrupted: fixes c :: "cars" shows "\<forall>ts v. \<nexists>va u. v=va--u \<and> (0 < \<parallel>ext va\<parallel> \<and> len va ts c = ext va \<and> restrict va (clm ts) c = lan va \<and> |lan va| = 1) \<and> (\<exists>v ua. u=v--ua \<and> True \<and> 0 < \<parallel>ext ua\<parallel> \<and> len ua ts c = ext ua \<and> restrict ua (clm ts) c = lan ua \<and> |lan ua| = 1)"