hcju/isabelleps huggingface dataset:
entries_eqI: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt" assumes "rbt_sorted t1"   and "rbt_sorted t2"   and "rbt_lookup t1 = rbt_lookup t2" shows "entries t1 = entries t2"
map_inv_id: fixes xs :: "'c set" shows "map_inv (map_id_on xs) = map_id_on xs"
hom_mult: fixes x :: "int poly"   and y :: "int poly" shows "Ring_Hom_Poly.of_int_poly (x * y) = Ring_Hom_Poly.of_int_poly x * Ring_Hom_Poly.of_int_poly y"
k_def: shows "k = nat \<lfloor>1 / \<delta>\<rfloor> + 1"
foundation12: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x" shows "(\<tau> \<Turnstile> x implies y) = (\<tau> \<Turnstile> x \<longrightarrow> \<tau> \<Turnstile> y)"
test_inf: fixes p :: "'a"   and q :: "'a" shows "\<iota> (inf p q) = \<iota> p \<cdot> \<iota> q"
intro: fixes lookup :: "'c \<Rightarrow> ('d, 'e option) state"   and update :: "'c \<Rightarrow> 'e \<Rightarrow> ('d, unit) state"   and P :: "'d \<Rightarrow> bool" assumes "mem_correct lookup update P" shows "dp_consistency lookup update P"
density_distr: fixes f :: "'b \<Rightarrow> ennreal"   and N :: "'b measure"   and X :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "f \<in> borel_measurable N"   and "X \<in> M \<rightarrow>\<^sub>M N" shows "density (distr M N X) f = distr (density M (\<lambda>x. f (X x))) N X"
cases: fixes x :: "('r + 's) + 't + 'u" obtains iadv_core :: "'r" where "x = Inl (Inl iadv_core)"   | iadv_rest :: "'s" where "x = Inl (Inr iadv_rest)"   | iusr_core :: "'t" where "x = Inr (Inl iusr_core)"   | iusr_rest :: "'u" where "x = Inr (Inr iusr_rest)"
ind_f: fixes n :: "nat" shows "domain (f n) = domain g"
filterlim_at_bot: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "filterlim f at_bot F = (\<forall>Z. \<forall>\<^sub>F x in F. f x \<le> Z)"
subst_ident: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and u :: "('b, 'a) Term.term" assumes "x \<notin> fv t" shows "t \<cdot> subst x u = t"
simps_19: fixes x21 :: "'c"   and x22 :: "'c llist" shows "lset (LCons x21 x22) = insert x21 (lset x22)"
sum_atLeastAtMost_of_int_nat_transfer: fixes f :: "'b \<Rightarrow> 'a"   and a :: "nat"   and b :: "nat" shows "(\<Sum>k = int a..int b. f (of_int k)) = (\<Sum>k = a..b. f (of_nat k))"
coeff_add: fixes p :: "'c poly"   and q :: "'c poly"   and n :: "nat" shows "coeff (p + q) n = coeff p n + coeff q n"
simps: fixes q :: "'b queue" shows "front_list q = (case status q of Idle \<Rightarrow> front q | Rev ok f f' r r' \<Rightarrow> rev (take ok f') @ f @ rev r @ r' | App ok f' x \<Rightarrow> rev (take ok f') @ x | Done f \<Rightarrow> f)"
delete_block_wf: fixes bl :: "'a set"   and b :: "'a set" shows "incidence_system (\<V> \<union> bl) (add_block_sys.del_block bl b)"
\<sigma>: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> w) \<sigma> m (g \<star> n)"
list_all2_nthD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length xs" shows "P (xs ! p) (ys ! p)"
fds_nth_one: shows "fds_nth 1 = (\<lambda>n. if n = 1 then 1::'b else (0::'b))"
\<omega>\<omega>: shows "\<omega> \<up> \<omega> = tp (elts (\<omega> \<up> \<omega>))"
wqo_on_with_bot: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wqo_on P A" shows "wqo_on (option_le P) A\<^sub>\<bottom>"
starfun_def: fixes f :: "'c \<Rightarrow> 'd" shows "*f* f \<equiv> Ifun (star_of f)"
IH: assumes "v \<in> fv\<^sub>s\<^sub>t S'"   and "v \<notin> subst_domain \<delta>" shows "v \<in> fv\<^sub>s\<^sub>t (S' \<cdot>\<^sub>s\<^sub>t \<delta>)"
l2_inv2I: fixes x :: "l2_state" assumes "\<And>Ra A B gny. \<lbrakk>Auth B A \<langle>Number 0, gny, Exp Gen (NonceF (Ra $ nx))\<rangle> \<in> chan x; A \<notin> bad x \<and> B \<notin> bad x\<rbrakk> \<Longrightarrow> \<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress x Rb) {xny, xgnx, xgny, xsk} \<and> gny = Exp Gen (NonceF (Rb $ ny)) \<and> guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx)))" shows "x \<in> l2_inv2"
exhaust: fixes y :: "finite_4" obtains "y = a\<^sub>1"   | "y = a\<^sub>2"   | "y = a\<^sub>3"   | "y = a\<^sub>4"
dependent_explicit: fixes s :: "'a set" assumes "s \<subseteq> S" shows "subspace.dependent s = (\<exists>t\<subseteq>S. finite t \<and> t \<subseteq> s \<and> (\<exists>u. (\<Sum>v\<in>t. u v *\<^sub>R v) = (0::'a) \<and> (\<exists>v\<in>t. u v \<noteq> 0)))"
Minr_is_Suc: fixes Rr :: "nat list \<Rightarrow> bool"   and xs :: "nat list"   and w :: "nat" assumes "Minr Rr xs w = Suc w"   and "Rr (xs @ [Suc w])" shows "Minr Rr xs (Suc w) = Suc w"
vec_in_image_vec: fixes x :: "'a"   and S :: "'a set" shows "(vec x \<in> vec ` S) = (x \<in> S)"
intros_6: shows "IK_nf (I \<circ> K \<circ> I)"
fA_neq_locs_def: shows "fA_neq_locs = {idle_phase_init, idle_store_fM, mark_load_fM, mark_store_fA} \<union> prefixed ''idle_flip_noop'' \<union> init_locs"
hoare_triple_preI: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn" assumes "\<And>h. h \<Turnstile> P \<Longrightarrow> <P> c <Q>" shows "<P> c <Q>"
truncate_up_zero: fixes prec :: "nat" shows "truncate_up prec 0 = 0"
tna_ant: fixes z :: "int" shows "tna (ant z) = z"
phis'_code_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val" shows "phis'_code g next \<equiv> map_values (\<lambda>(n, v) vs. if v = snd next then None else Some (map (uninst_code.substNext_code phis g next) vs)) (phis g)"
nzero: fixes i :: "nat" assumes "i < m" shows "\<parallel>fs.gs.gso i\<parallel>\<^sup>2 \<noteq> 0"
obs_safe_def: fixes r :: "nat"   and s :: "'c obsv_state_scheme"   and v :: "val" shows "obs_safe r s v \<equiv> \<forall>r'<r. \<exists>p. obsv_state.obs s r' p \<in> {None, Some v}"
comp_sunion_with: shows "local.comp_sunion_with = ord.sunion_with (lt_of_comp c)"
assms_3: shows "op\<^sub>1 \<noteq> op\<^sub>2"
X: shows "X = fst \<circ> (\<lambda>x. (X x, Y x))"
det_one: fixes n :: "nat" shows "Determinant.det (1\<^sub>m n) = (1::'a)"
rept_dag: shows "Dag ((rep \<propto> id) p) (rep \<propto> low) (rep \<propto> high) rept"
mcont_max_enat1: fixes y :: "enat" shows "mcont Sup (\<le>) Sup (\<le>) (\<lambda>x. max x y)"
space_Sup_eq_UN: fixes M :: "'b measure set" shows "space (\<Squnion> M) = \<Union> (space ` M)"
assms_3: shows "(sinit i xs, sinit i ys) \<in> lexord R"
prime_int_numeral_eq: fixes m :: "num" shows "prime (numeral m) = prime (numeral m)"
BetaOr: fixes A :: "('a, 'b) form"   and G :: "('a, 'b) form list"   and B :: "('a, 'b) form" assumes "\<stileturn> A # G"   and "\<stileturn> B # G" shows "\<stileturn> Or A B # G"
intros_2: fixes u :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set"   and a :: "'c set"   and b :: "'c set"   and s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" assumes "u \<in> timpl_closure t TI"   and "(a, b) \<in> TI"   and "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) u s" shows "s \<in> timpl_closure t TI"
line_integral_exists_on_rec_join': fixes l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "boundary_chain (set l)"   and "valid_chain_list l"   and "\<And>k \<gamma>. (k, \<gamma>) \<in> set l \<Longrightarrow> valid_path \<gamma>"   and "line_integral_exists F basis (rec_join l)"   and "finite basis" shows "\<forall>(k, \<gamma>)\<in>set l. line_integral_exists F basis \<gamma>"
q_Var_def: fixes i :: "name" shows "q_Var i \<equiv> succ (ord_of (nat_of_name i))"
akra_bazzi_integral_kurzweil_henstock: shows "akra_bazzi_integral (\<lambda>f a b. f integrable_on {a..b}) (\<lambda>f a b. integral {a..b} f)"
one_d_integral_integrable: shows "(\<lambda>x. integral UNIV (\<lambda>y. f (x, y))) integrable_on UNIV"
a_set_disconnected_nodes_def: fixes document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" shows "a_set_disconnected_nodes document_ptr disc_nodes = put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes_update disc_nodes"
wrapS_unwrapS_id: fixes f :: "nat discr Box \<rightarrow> 'a" assumes "f\<cdot>\<bottom> = \<bottom>" shows "(wrapS oo unwrapS)\<cdot>f = f"
cost_swapSyms: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a" assumes "consistent t"   and "a \<in> alphabet t"   and "b \<in> alphabet t" shows "cost (swapSyms t a b) + freq t a * depth t a + freq t b * depth t b = cost t + freq t a * depth t b + freq t b * depth t a"
hyps: fixes A2 :: "'a mat"   and A3 :: "'a mat"   and A4 :: "'a mat" assumes "A2 = 0\<^sub>m n m"   and "A3 \<in> carrier_mat m n"   and "A4 \<in> carrier_mat m m" shows "det (four_block_mat A1 A2 A3 A4) = det A1 * det A4"
insert_def: fixes x :: "'d"   and xs :: "'d list" shows "List.insert x xs = (if x \<in> set xs then xs else x # xs)"
hom_zero: shows "of_int_mod_ring 0 = 0"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d multiset" shows "set_mset (image_mset f v) = f ` set_mset v"
I: shows "{} = (\<Union>n\<le>length (xs @ [y]). \<Union>W\<in>R n. W) \<and> (\<forall>W\<in>R 0. xs @ [y] \<notin> sentences P \<and> None \<notin> set (xs @ [y]) \<and> (xs @ [y], W) \<in> failures P \<or> xs @ [y] \<in> sentences P \<and> (\<exists>U V. (xs @ [y], U) \<in> failures P \<and> ([], V) \<in> failures Q \<and> W = insert None U \<inter> V)) \<and> (\<forall>n\<in>{0<..length (xs @ [y])}. \<forall>W\<in>R n. take (length (xs @ [y]) - n) (xs @ [y]) \<in> sentences P \<and> (drop (length (xs @ [y]) - n) (xs @ [y]), W) \<in> failures Q) \<and> (\<exists>n\<in>{..length (xs @ [y])}. \<exists>W. W \<in> R n)"
surjective: fixes r :: "'a s_context_scheme" shows "r = \<lparr>s_inst = s_inst r, s_funcs = s_funcs r, s_tab = s_tab r, s_mem = s_mem r, s_globs = s_globs r, \<dots> = s_context.more r\<rparr>"
abs_eq: fixes x :: "'c poly" shows "- x = Abs_poly (\<lambda>n. - poly.coeff x n)"
ponesubst_def: fixes P1 :: "pol"   and M :: "mon"   and P2 :: "pol" shows "ponesubst P1 M P2 = (let (Q, R) = mfactor P1 M in case R of Pc c \<Rightarrow> if c = 0 then None else Some (Q \<langle>+\<rangle> P2 \<langle>*\<rangle> R) | _ \<Rightarrow> Some (Q \<langle>+\<rangle> P2 \<langle>*\<rangle> R))"
leof_RES_rule_1: fixes \<Psi> :: "'b \<Rightarrow> bool"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "\<And>x. \<Psi> x \<Longrightarrow> \<Phi> x" shows "SPEC \<Psi> \<le>\<^sub>n SPEC \<Phi>"
SUB: fixes p :: "L"   and pp :: "L"   and G :: "Var \<Rightarrow> L"   and GG :: "Var \<Rightarrow> L"   and HH :: "Var \<Rightarrow> L"   and H :: "Var \<Rightarrow> L"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and q :: "L"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "LEQ p pp"   and "\<forall>x. LEQ (G x) (GG x)"   and "\<forall>x. LEQ (HH x) (H x)"   and "X \<rhd> c : Sec (Q pp HH) (EQ GG q) (EQ HH q) \<Phi>" shows "X \<rhd> c : Sec (Q p H) (EQ G q) (EQ H q) \<Phi>"
lexordp_eq_conv_lexord: fixes xs :: "char list"   and ys :: "char list" shows "char.lexordp_eq xs ys = (xs = ys \<or> char.lexordp xs ys)"
action_needed_asses_def: fixes a :: "('g, 'h) fmap \<times> 'i"   and s :: "('g, 'h) fmap" shows "action_needed_asses a s \<equiv> fmrestrict_set (action_needed_vars a s) s"
type_less_x_Sequence_intro: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> = Sequence \<rho>"   and "\<rho> < \<sigma>" shows "\<tau> < Sequence \<sigma>"
fin: fixes s :: "'s2" assumes "wellformed2 s" shows "finite {fold delta2 w s |w. w \<in> lists (set alphabet2)}"
minus_in_Ints_vec_iff: fixes x :: "'b vec" shows "(- x \<in> \<int>\<^sub>v) = (x \<in> \<int>\<^sub>v)"
simple_path_inj_on: fixes g :: "real \<Rightarrow> 'b" assumes "simple_path g" shows "inj_on g {0<..<1}"
f: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "(f y ^^ g y) ((f x ^^ g x) z) = (f x ^^ g x) ((f y ^^ g y) z)"
multiplicative_function_eqI: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "multiplicative_function f"   and "multiplicative_function g"   and "\<And>p k. \<lbrakk>prime p; 0 < k\<rbrakk> \<Longrightarrow> f (p ^ k) = g (p ^ k)" shows "f n = g n"
iTILL_cut_ge: fixes n :: "nat"   and t :: "nat" shows "[\<dots>n] \<down>\<ge> t = (if n < t then {} else [t\<dots>,n - t])"
term_order_rel_trans: fixes a :: "char list list"   and aa :: "char list list"   and ab :: "char list list" assumes "(a, aa) \<in> term_order_rel"   and "(aa, ab) \<in> term_order_rel" shows "(a, ab) \<in> term_order_rel"
\<Phi>o_def: fixes a' :: "'c" shows "\<Phi>o a' = S.mkArr (Hom.set (a', a)) (\<iota> ` D.cones a') (\<lambda>x. \<iota> (D.cones_map (R.\<psi> (a', a) x) \<chi>))"
not_ctran_None2: fixes s :: "'b"   and xs :: "('b com option \<times> 'b) list"   and i :: "nat" assumes "(None, s) # xs \<in> cptn"   and "i < length xs" shows "(((None, s) # xs) ! i, xs ! i) \<notin> ctran"
vcons_in_vcpowerD_3: fixes xs :: "V"   and x :: "V"   and A :: "V"   and n :: "V" assumes "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"   and "n \<in>\<^sub>\<circ> \<omega>" shows "n = vcard (xs #\<^sub>\<circ> x)"
\<phi>: fixes s :: "'s"   and t :: "'s" assumes "P s"   and "s \<in> \<phi>"   and "t \<in> set_pmf (K s)" shows "P t"
restrict_spmf_mono: fixes p :: "'h spmf"   and q :: "'h spmf"   and A :: "'h set" assumes "ord_spmf (=) p q" shows "ord_spmf (=) (p \<upharpoonleft> A) (q \<upharpoonleft> A)"
dsg3: fixes y :: "'a"   and x :: "'a" shows "ar ar (y \<cdot> ar ar x) = ar ar y \<cdot> ar ar x"
ts_nil: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option" shows "CT;\<Gamma> \<turnstile>+ [] : []"
defined_bool_split: fixes \<tau> :: "'b state \<times> 'b state"   and A :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> \<delta> A) = (\<tau> \<Turnstile> A \<or> \<tau> \<Turnstile> not A)"
iso_iff_mon_epi: fixes f :: "'g \<Rightarrow> 'h"   and G :: "('g, 'i) monoid_scheme"   and H :: "('h, 'j) monoid_scheme" shows "(f \<in> Group.iso G H) = (f \<in> mon G H \<and> f \<in> epi G H)"
lm07: fixes A :: "'c set"   and B :: "'d set" shows "trivial (A \<times> B) = (finite (A \<times> B) \<and> card A * card B \<le> 1)"
roots_below_the_unique_root: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" assumes "\<exists>!x. root_cond (p, l, r) x" shows "{y. y \<le> real_of_1 (p, l, r) \<and> poly (real_of_int_poly p) y = 0} = {y. y \<le> real_of_rat r \<and> poly (real_of_int_poly p) y = 0}"
lipschitz_on_cmult_real: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> real"   and a :: "real" assumes "C-lipschitz_on U f" shows "(\<bar>a\<bar> * C)-lipschitz_on U (\<lambda>x. a * f x)"
rsum_zero: fixes D :: "(real \<times> real \<times> real) list" shows "rsum D (\<lambda>x. 0) = 0"
simps_2: fixes \<kappa> :: "String.literal"   and \<sigma>s :: "typ list"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubstT (Ty \<kappa> \<sigma>s) \<rho> = Ty \<kappa> (map (\<lambda>\<sigma>. tsubstT \<sigma> \<rho>) \<sigma>s)"
realizable_Append_matched: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "realizable n ns n''"   and "matched n'' ns' n'" shows "realizable n (ns @ ns') n'"
strong_operator_topology_def: shows "strong_operator_topology = pullback_topology UNIV blinfun_apply euclidean"
mk_alt_consistency_def: fixes C :: "('a, 'b) form set set" shows "mk_alt_consistency C = {S. \<exists>f. psubst f ` S \<in> C}"
ide_compE: fixes g :: "'j"   and f :: "'j" assumes "J.ide (g \<cdot>\<^sub>J f)" obtains "J.seq g f" and "J.seq f g" and "g \<cdot>\<^sub>J f = J.dom f" and "g \<cdot>\<^sub>J f = J.cod g"
preserves_arr: fixes f :: "'b" assumes "D.arr f" shows "D.arr (trg\<^sub>D f)"
pne12: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "pid1 \<noteq> pid2"
subterms_trans: fixes q :: "('e, 'f, 'g, 'h) seqp"   and p :: "('e, 'f, 'g, 'h) seqp"   and r :: "('e, 'f, 'g, 'h) seqp" assumes "q \<in> subterms p"   and "r \<in> subterms q" shows "r \<in> subterms p"
H_2: shows "\<forall>i. i \<in> I \<longrightarrow> V i \<subseteq> U \<and> s i \<in> \<O> V i"
axioms_2: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "semiconnex_irreflexive A less" shows "irreflexive A less"
weak_complete_lattice_axioms: shows "weak_complete_lattice \<P>"
search_enum_nonempty_neq0: fixes b :: "nat"   and i :: "nat"   and y\<^sub>0 :: "nat" assumes "s b \<down>= i"   and "\<phi> i y\<^sub>0 \<down>"   and "\<nexists>y. \<phi> i y \<down> \<and> e_length b \<le> y" shows "\<nexists>x. search_enum x b \<down>= 0"
unify_list_wt_if_same_type: fixes E :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and B :: "('var \<times> ('fun, 'var) Term.term) list"   and U :: "('var \<times> ('fun, 'var) Term.term) list" assumes "unify E B = Some U"   and "\<forall>(s, t)\<in>set E. wf\<^sub>t\<^sub>r\<^sub>m s \<and> wf\<^sub>t\<^sub>r\<^sub>m t \<and> \<Gamma> s = \<Gamma> t"   and "\<forall>(v, t)\<in>set B. \<Gamma> (Var v) = \<Gamma> t" shows "\<forall>(v, t)\<in>set U. \<Gamma> (Var v) = \<Gamma> t"
lspasl_p: fixes Gamma :: "bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a"   and Delta :: "bool" assumes "Gamma \<and> (h1,h2\<triangleright>h0) \<and> h0 = h3 \<longrightarrow> Delta" shows "Gamma \<and> (h1,h2\<triangleright>h0) \<and> (h1,h2\<triangleright>h3) \<longrightarrow> Delta"
AR_ANR: fixes S :: "'a set" shows "AR S = (ANR S \<and> contractible S \<and> S \<noteq> {})"
perm_bool: fixes pi :: "('d \<times> 'd) list"   and b :: "bool" shows "pi \<bullet> b = b"
hash_tree_def: fixes h :: "'a \<Rightarrow> 'b" shows "hash_tree h = hash_rt_tree' \<circ> map_rose_tree\<^sub>m h id"
roots_of_complex_main_1: fixes p :: "complex poly" assumes "p \<noteq> 0"   and "degree p \<le> 2 \<or> set (coeffs p) \<subseteq> \<rat>" shows "set (roots_of_complex_main p) = {x. poly p x = 0}"
construct_basis: fixes B :: "('a, 'b) vec set"   and b :: "('a, 'b) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "vec.independent B"   and "b \<in> B" shows "vec.construct B f b = f b"
preserves_arr: fixes f :: "'a" assumes "C.arr f" shows "D.arr (G f)"
cdf_lim_at_bot: shows "(C \<longlongrightarrow> 0) at_bot"
get_heap_returns_result: fixes h :: "'a"   and f :: "'a \<Rightarrow> ('a, 'b, 'c) prog"   and x :: "'c" shows "h \<turnstile> get_heap \<bind> f \<rightarrow>\<^sub>r x = h \<turnstile> f h \<rightarrow>\<^sub>r x"
insert_inter_insert: fixes a :: "'e"   and A :: "'e set"   and B :: "'e set" shows "insert a A \<inter> insert a B = insert a (A \<inter> B)"
agents_def: shows "agents \<equiv> fromList [Child0, Child1, Child2]"
lcosets_subset_PowG: fixes H :: "'a set" assumes "subgroup H G" shows "lcosets H \<subseteq> Pow (carrier G)"
convert_list2: fixes var :: "nat"   and L :: "atom list"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and L' :: "atomUni list"   and l' :: "atomUni" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "l' \<in> set L'" shows "\<exists>l\<in>set L. convert_atom var l (xs' @ x # xs) = Some l'"
2: shows "B \<turnstile>\<^sub>R res2 \<approx> res2'"
mult_2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "class.semiring_numeral one times plus" shows "times (numeral.numeral one plus (num.Bit0 num.One)) z = plus z z"
xcpt_eff_def: fixes i :: "instr"   and P :: "'b prog"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and et :: "ex_table" shows "xcpt_eff i P pc \<tau> et = (let (ST, LT) = \<tau> in map (\<lambda>(f, t, C, h, d). (h, \<lfloor>(Class C # drop (length ST - d) ST, LT)\<rfloor>)) (relevant_entries P i pc et))"
toVariable_channels_inv: fixes g :: "'b gState_scheme"   and p :: "pState"   and v :: "varDecl" shows "\<forall>x\<in>set (snd (snd (toVariable g p v))). channel_inv x"
A_SMP_repr: shows "finite_SMP_representation arity Ana \<Gamma> A"
b: shows "\<Psi> (l, u) i = \<Psi> (max l l', min u u') i"
exhaust_sel: fixes octo :: "octo" obtains "octo = Octo (Ree octo) (Im1 octo) (Im2 octo) (Im3 octo) (Im4 octo) (Im5 octo) (Im6 octo) (Im7 octo)"
intro: fixes f :: "'d \<Rightarrow> 'e" assumes "clinear f"   and "bounded_clinear_axioms f" shows "bounded_clinear f"
conseqPost: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q' :: "'e set"   and A' :: "'e set"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q',A'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
u_split: fixes t :: "symbol"   and s :: "character list" assumes "u = (t, s)" shows "t = terminal_of_token u \<and> s = chars_of_token u"
WTrt_elim_cases_11: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and bop :: "bop"   and e\<^sub>2 :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2 : T" obtains T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" where "bop = Eq" and "T = Boolean" and "P,E,h,sh \<turnstile> e\<^sub>1 : T\<^sub>1" and "P,E,h,sh \<turnstile> e\<^sub>2 : T\<^sub>2"   | "bop = Add" and "T = Integer" and "P,E,h,sh \<turnstile> e\<^sub>1 : Integer" and "P,E,h,sh \<turnstile> e\<^sub>2 : Integer"
is_unit_field_poly: fixes p :: "'c poly" shows "is_unit p = (p \<noteq> 0 \<and> degree p = 0)"
prime_field_gen_axioms: shows "prime_field_gen ff_ops R p"
le_inf_iff: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x (inf y z) = (less_eq x y \<and> less_eq x z)"
mult_product_group: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" shows "(\<otimes>\<^bsub>product_group I G\<^esub>) = (\<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i)"
sel'I: fixes s :: "'s"   and P :: "'x \<Rightarrow> bool" assumes "invar s"   and "\<forall>x\<in>\<alpha> s. \<not> P x" shows "sel' s P = None"
precompact_open_coverE: obtains U :: "nat \<Rightarrow> 'c set" where "\<Union> (range U) = manifold_eucl.dest.carrier" and "\<And>i. open (U i)" and "\<And>i. compact (closure (U i))" and "\<And>i. closure (U i) \<subseteq> manifold_eucl.dest.carrier"
mod_mult_mult2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a c) (times b c) = times (modulo a b) c"