hcju/isabelleps huggingface dataset:
delete_set_none: fixes l :: "('a \<times> 'b) list"   and x :: "'a" shows "(map_of l)(x := None) = map_of (delete x l)"
analz_knows_absorb2: fixes K :: "nat"   and evs :: "event list"   and X :: "nat"   and H :: "nat set" assumes "Key K \<in> analz (knows Spy evs)" shows "analz (Key ` insert X (insert K H) \<union> knows Spy evs) = analz (Key ` insert X H \<union> knows Spy evs)"
in_gamma_option_iff: fixes x :: "'a"   and r :: "'b \<Rightarrow> 'a set"   and u :: "'b option" shows "(x \<in> \<gamma>_option r u) = (\<exists>u'. u = Some u' \<and> x \<in> r u')"
fresh_Pair: fixes a :: "atom"   and x :: "'a"   and y :: "'b" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
AE_count_space: fixes P :: "'h \<Rightarrow> bool"   and A :: "'h set" shows "almost_everywhere (count_space A) P = (\<forall>x\<in>A. P x)"
preserves_reflects_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "arr (\<alpha>' f) = VVV.arr f"
trd_in_pmdl: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "p - trd fs p \<in> pmdl (set fs)"
base_simps_2: shows "hseq k\<^sup>* h"
one_less_power: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "0 < n" shows "less one (power.power one times a n)"
max_input_I: fixes i :: "nat" shows "max_input (V (I i)) = Some i"
pick_asAbs: fixes qA :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "qGoodAbs qA" shows "pick (asAbs qA) $= qA"
simps_5: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_siso (c1 | c2) = (SC_siso c1 \<and> SC_siso c2)"
zero_lin_dep: fixes S :: "'c set" assumes "\<zero>\<^bsub>V\<^esub> \<in> S"   and "carrier K \<noteq> {\<zero>\<^bsub>K\<^esub>}" shows "lin_dep S"
get_ancestors_def: shows "Shadow_DOM.get_ancestors \<equiv> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors Shadow_DOM.get_parent"
s0_le_s_stop: shows "other.s \<delta> \<le> delayed_safe_distance.other.s_stop"
order_refl: fixes xa :: "'o"   and x :: "'a" shows "le xa x x"
numeral_eq_eSuc: fixes k :: "num" shows "numeral k = eSuc (epred_numeral k)"
q_Neg_def: fixes x :: "hf" shows "q_Neg x \<equiv> \<langle>htuple 4, x\<rangle>"
W_def: fixes P :: "complex mat"   and k :: "nat" shows "W P k = wlp_while_n M0 M1 WS k P"
cod_char: fixes f :: "'b1 \<times> 'b2" shows "B1xB2.cod f = (if B1.arr (fst f) \<and> B2.arr (snd f) then (B1.cod (fst f), B2.cod (snd f)) else (B1.null, B2.null))"
assms_1: shows "wcode_goon_right_moving_1 ires rs (b, Oc # ba)"
prime_implicates_generation_correct: fixes S :: "'at Literal set set" assumes "saturated_binary_rule resolvent S"   and "non_redundant S"   and "all_fulfill finite S" shows "S \<subseteq> prime_implicates S"
put_twice: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'d" assumes "wb_lens x" shows "put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) v = put\<^bsub>x\<^esub> \<sigma> v"
assms_2: shows "F\<lparr>ArrDom\<rparr> = A"
tauChainAddTau_2: fixes P :: "pi"   and P' :: "pi"   and P'' :: "pi" assumes "P \<longmapsto> \<tau> \<prec> P'"   and "P' \<Longrightarrow>\<^sub>\<tau> P''" shows "P \<Longrightarrow>\<^sub>\<tau> P''"
Nset_1: shows "{i. i \<le> Suc 0} = {0, Suc 0}"
arrE: fixes f :: "'a" assumes "S.arr f" shows "right \<omega> f"
set_val_writes: fixes character_data_ptr :: "(_) character_data_ptr"   and tag :: "char list"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_val_locs character_data_ptr) (set_val character_data_ptr tag) h h'"
Gramian_determinant_2: fixes k :: "nat" assumes "k \<le> length (map of_int_hom.vec_hom fs)" shows "0 < fs.gs.d k"
codomains_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.seq g f" shows "VV.codomains (VV.comp g f) = VV.codomains g"
dbtm_abst_swap_subst: fixes name :: "name"   and name' :: "name"   and u :: "dbtm"   and j :: "nat"   and t :: "dbtm" assumes "name \<noteq> name'"   and "atom name' \<sharp> u" shows "subst_dbtm u name (abst_dbtm name' j t) = abst_dbtm name' j (subst_dbtm u name t)"
rComm2: fixes \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "yvec \<sharp>* M"   and "yvec \<sharp>* K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* C" shows "Prop (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
prems_4: shows "small (\<Union> (insert X \<A>))"
abs_eq: shows "local.upto_generator = Generator (\<lambda>n. n \<le> bound, \<lambda>n. (n, n + 1))"
bitlen_div: fixes m :: "int" assumes "0 < m" shows "1 \<le> real_of_int m / 2 ^ nat (bitlen m - 1)"
col_space_def: fixes n :: "nat"   and B :: "'b mat" shows "vec_space.col_space n B = LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set (cols B))"
one: fixes R :: "('b \<Rightarrow> real) set"   and S :: "'b set"   and U :: "'b set"   and t0 :: "'b"   and t1 :: "'b" assumes "function_ring_on R S"   and "open U"   and "t0 \<in> S"   and "t0 \<in> U"   and "t1 \<in> S - U" shows "\<exists>V. open V \<and> t0 \<in> V \<and> S \<inter> V \<subseteq> U \<and> (\<forall>e>0. \<exists>f\<in>R. f ` S \<subseteq> {0..1} \<and> (\<forall>t\<in>S \<inter> V. f t < e) \<and> (\<forall>t\<in>S - U. 1 - e < f t))"
P2_OT_14_TF: fixes m0 :: "bool"   and m1 :: "bool"   and m2 :: "bool"   and m3 :: "bool" shows "R2_14 (m0, m1, m2, m3) (True, False) = funct_OT_14 (m0, m1, m2, m3) (True, False) \<bind> (\<lambda>(out1, out2). S2_14 (True, False) out2)"
set_insert: fixes x :: "'c"   and A :: "'c set" assumes "x \<in> A" obtains B :: "'c set" where "A = Set.insert x B" and "x \<notin> B"
that: fixes r :: "'a list" assumes "u = p \<cdot> r"   and "r \<le>p w" shows "thesis"
snapshot_stable_ver_2: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and p :: "'d"   and j :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "i \<le> j" shows "ps (distributed_system.s channel trans send recv init t j) p \<noteq> None"
MatchP: fixes valid_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> event_data option list set) list \<Rightarrow> bool"   and init_msaux :: "Monitor.args \<Rightarrow> 'a"   and add_new_ts_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and join_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and add_new_table_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and result_msaux :: "Monitor.args \<Rightarrow> 'a \<Rightarrow> event_data option list set"   and valid_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and init_muaux :: "Monitor.args \<Rightarrow> 'b"   and add_new_muaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and length_muaux :: "Monitor.args \<Rightarrow> 'b \<Rightarrow> nat"   and eval_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> event_data option list set list \<times> 'b"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi>s :: "('a, 'b) mformula list"   and mr :: "mregex"   and r :: "Formula.formula Regex.regex"   and mrs :: "mregex list"   and buf :: "event_data option list set list list"   and nts :: "nat list"   and I :: "\<I>"   and aux :: "(nat \<times> (mregex, event_data option list set) mapping) list" assumes "maux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux"   and "case to_mregex r of (mr', \<phi>s') \<Rightarrow> list_all2 (maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R) \<phi>s \<phi>s' \<and> mr = mr'"   and "mrs = sorted_list_of_set (RPDs mr)"   and "safe_regex Past Strict r"   and "wf_mbufn' \<sigma> P V j n R r buf"   and "wf_ts_regex \<sigma> P j r nts"   and "wf_matchP_aux \<sigma> V n R I r aux (Monitor.progress \<sigma> P (formula.MatchP I r) j)" shows "maux.wf_mformula valid_msaux valid_muaux length_muaux \<sigma> j P V n R (MMatchP I mr mrs \<phi>s buf nts aux) (formula.MatchP I r)"
simps_10: shows "isrlfm F = (isatom F \<and> bound0 F)"
diff_def: shows "diff \<equiv> \<lambda>n. if n = 0 then x else if n = 1 then \<lambda>t. f t (x t) else (\<lambda>t. f' (t, x t) (1, f t (x t)))"
is_monomial_def: fixes p :: "'d \<Rightarrow>\<^sub>0 'e" shows "is_monomial p = (card (keys p) = 1)"
bell00_is_state: shows "state 2 |\<beta>\<^sub>0\<^sub>0\<rangle>"
Int_insert_left: fixes a :: "'e"   and B :: "'e set"   and C :: "'e set" shows "insert a B \<inter> C = (if a \<in> C then insert a (B \<inter> C) else B \<inter> C)"
While_def: fixes test :: "'c \<Rightarrow> 'd"   and not :: "'d \<Rightarrow> 'd"   and t :: "'d"   and c :: "'d" assumes "kat test not" shows "kat.While not t c = (t \<cdot> c)\<^sup>\<star> \<cdot> not t"
aodv_trans: fixes i :: "nat" shows "automaton.trans (paodv i) = seqp_sos \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V"
z: shows "prime z \<and> odd z \<and> z dvd w \<and> \<not> is_qfN z 3"
assms_3: shows "a_kernel (K [X]) R h \<noteq> {[]}"
push: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "Height_Bounded_Labeling c s t f l"   and "push_precond f l e" shows "((push_effect f e, l), f, l) \<in> pr_algo_rel"
matrix_works: fixes f :: "('e, 'f) vec \<Rightarrow> ('e, 'g) vec"   and x :: "('e, 'f) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "matrix f *v x = f x"
F_def: shows "F \<equiv> \<lambda>q. eval_fds (fds_perzeta q)"
node_ptr_kinds_eq_h: shows "node_ptr_kinds h2 = node_ptr_kinds h"
Fb0: fixes G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "G1 \<noteq> 0"   and "G2 \<noteq> 0" shows "(F G1 G2 i = 0 \<longrightarrow> F G1 G2 (Suc i) = 0) \<and> b G1 G2 i \<noteq> 0 \<and> h G1 G2 i \<noteq> 0"
Gromov_product_at_diff: fixes x :: "'b"   and y :: "'b"   and z :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" shows "\<bar>Gromov_product_at x y z - Gromov_product_at a b c\<bar> \<le> dist x a + dist y b + dist z c"
divide_neg_neg: fixes x :: "'a"   and y :: "'a" assumes "x < (0::'a)"   and "y < (0::'a)" shows "(0::'a) < x / y"
rev_H_uv_1: shows "rev_H' uv = uv"
strategy_attractsE: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts p \<sigma> A W"   and "P $ n \<in> A" shows "visits_via P' A W"
collapse_2: fixes tllist :: "('g, 'h) tllist" assumes "\<not> is_TNil tllist" shows "TCons (thd tllist) (ttl tllist) = tllist"
transE: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "trans r"   and "x \<succeq>[r] y"   and "y \<succeq>[r] z" obtains "x \<succeq>[r] z"
eq2: shows "monom a 1 = [:a:] \<otimes>\<^bsub>R\<^esub> monom 1 1"
hyps_1: fixes xs :: "'a rexp list" shows "(Inters r = Some xs) = (Zero \<notin> set xs \<and> xs = sorted_list_of_set (toplevel_inters r - {Full}))"
merge_thms_4: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and x' :: "'b" shows "hn_ctxt R x x' \<or>\<^sub>A hn_invalid R x x' \<Longrightarrow>\<^sub>t hn_invalid R x x'"
Loop: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and P' :: "'b assn"   and c :: "stmt"   and l :: "label" assumes "G,A\<turnstile>{P} e-\<succ> {P'}"   and "G,A\<turnstile>{Normal (P'\<leftarrow>=True)} .c. {abupd (absorb (Cont l)) .; P}" shows "G,A\<turnstile>{P} .l\<bullet> While(e) c. {P'\<leftarrow>=False\<down>=\<diamondsuit>}"
append_self_conv2: fixes xs :: "'g list"   and ys :: "'g list" shows "(xs @ ys = ys) = (xs = [])"
stable_UNION: fixes r :: "('d \<times> 'd) set"   and A :: "'e set"   and F :: "'e \<Rightarrow> 'f set" assumes "regular r"   and "|A| <o r"   and "\<And>a. a \<in> A \<Longrightarrow> |F a| <o r" shows "|\<Union> (F ` A)| <o r"
scale_eq_0_iff: fixes a :: "'a"   and x :: "('a, 'b) vec" shows "(a *s x = 0) = (a = (0::'a) \<or> x = 0)"
alw_ev_sdrop: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and m :: "nat"   and xs :: "'a stream" assumes "alw (ev \<phi>) (sdrop m xs)" shows "alw (ev \<phi>) xs"
FAccStatic: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l, sh)\<rangle>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,Static:t in D" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,(h, l, sh)\<rangle>"
function_zero_eval: fixes R :: "('c, 'd) ring_scheme"   and a :: "'e"   and S :: "'e set" assumes "ring_functions R"   and "a \<in> S" shows "\<zero>\<^bsub>function_ring S R\<^esub> a = \<zero>\<^bsub>R\<^esub>"
1: shows "(eX = ERR \<or> (\<exists>X. eX = OK X \<and> (\<forall>s. \<not> igWls MOD s X))) = (eX' = ERR \<or> (\<exists>X. eX' = OK X \<and> (\<forall>s. \<not> igWls MOD s X)))"
res: assumes "\<And>a. a \<in> S_a \<Longrightarrow> a \<in> S_a" shows "set_iterator (set_iterator_product it_a it_b) (Sigma S_a S_b)"
bounded_countable_web_quotient_web: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "bounded_countable_web (quotient_web (quotient_web \<Gamma> f__) f)"
lebesgue_integral_count_space_finite_support: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite {a \<in> A. f a \<noteq> (0::'d)}" shows "integral\<^sup>L (count_space A) f = (\<Sum>a | a \<in> A \<and> f a \<noteq> (0::'d). f a)"
Fun_mkArr'_1: fixes A :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and B :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set"   and F :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr \<Rightarrow> (('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "S.arr (S.mkArr A B F)" shows "\<guillemotleft>S.mkArr A B F : S.mkIde A \<rightarrow>\<^sub>S S.mkIde B\<guillemotright>"
X_def: shows "X = map_minsky (\<lambda>s. if s = 0 then 3 else 2 * s) id M \<union> map_minsky (\<lambda>s. 2 * s + 5) id N \<union> {(3, Dec 2 7 0)} \<union> {(5, Dec 2 2 1)}"
vd: fixes x :: "real" assumes "x \<in> {0..1}"   and "x \<notin> (\<lambda>t. (v - u) *\<^sub>R t + u) -` s" shows "vector_derivative (\<lambda>x. g ((v - u) * x + u)) (at x within {0..1}) = (v - u) *\<^sub>R vector_derivative g (at ((v - u) * x + u) within {0..1})"
simps_1: fixes L :: "('b \<times> nat) option"   and t :: "'b" shows "acquire_locks L t 0 = L"
wf_trm_subst_range_iff: fixes \<theta> :: "'c \<Rightarrow> ('fun, 'c) Term.term" shows "(\<forall>x. wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)) = wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
quasi_inverses_some_quasi_inverse_1: fixes f :: "'a" assumes "equivalence_map f" shows "quasi_inverses f f\<^sup>~"
simps_2: shows "is_\<nu>LTL false\<^sub>n = True"
sfg_def: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e" shows "sfg A a = (ag_single_gen A a \<and> single_gen_free A a)"
prem: shows "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
simps: fixes a1 :: "'a"   and a2 :: "'a" shows "a1 -\<tau>1\<rightarrow> a2 = (\<exists>s s' tl. a1 = s \<and> a2 = s' \<and> trsys1 s tl s' \<and> False)"
diff_add_eq_diff_diff_swap: fixes minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r"   and uminus :: "'r \<Rightarrow> 'r"   and a :: "'r"   and b :: "'r"   and c :: "'r" assumes "class.group_add minus plus zero uminus" shows "minus a (plus b c) = minus (minus a c) b"
lemma_2_10_24: fixes a :: "'a"   and b :: "'a" shows "a \<le> (a r\<rightarrow> b) l\<rightarrow> b"
vertex_subtreeI: fixes t :: "'c"   and v :: "'a" assumes "t \<in> V\<^bsub>T\<^esub>"   and "v \<in> bag t" shows "t \<in> vertex_subtree v"
arrE: fixes f :: "'a" assumes "Left.arr f" shows "left (trg \<mu>) f"
stable_pair_on_CD_on: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "match XD_XH = CD_on ds (fst XD_XH)"
hcomplex_hcnj_zero: shows "hcnj 0 = 0"
get_element_by_id_result_in_tree_order: fixes get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and iden :: "char list"   and element_ptr :: "(_) element_ptr"   and to :: "(_) object_ptr list" assumes "l_get_element_by get_element_by_id get_elements_by_tag_name to_tree_order"   and "h \<turnstile> get_element_by_id ptr iden \<rightarrow>\<^sub>r Some element_ptr"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to" shows "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 element_ptr \<in> set to"
spr_simTrans_def: fixes kbp :: "('b, 'd, 'e) GC list"   and envAction :: "'c \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'e) \<Rightarrow> 'c \<Rightarrow> 'c"   and val :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'c \<Rightarrow> 'g" shows "SPRViewSingle.spr_simTrans kbp envAction envTrans val envObs \<equiv> \<lambda>a X. map fromList (List_local.partition (envObs_rel (envObs a)) (SPRViewSingle.spr_trans kbp envAction envTrans val a X))"
unstable_start: shows "unstables r step start = sorted_list_of_set {p. succs p \<noteq> {} \<and> p < length start}"
singleton_times_left_hmset: fixes M :: "hmultiset"   and N :: "hmultiset" shows "\<omega>^ M * N = HMSet (image_mset ((+) M) (hmsetmset N))"
sorted_iff_nth_Suc: fixes xs :: "'a word list" shows "signed.sorted xs = (\<forall>i. Suc i < length xs \<longrightarrow> xs ! i \<le>s xs ! Suc i)"
resCap_def: fixes p :: "(nat \<times> nat) list" shows "resCap p \<equiv> Min {cf e |e. e \<in> set p}"
ANR_finite_Union_convex_closed: fixes \<T> :: "'a set set" assumes "finite \<T>"   and "\<And>C. C \<in> \<T> \<Longrightarrow> closed C"   and "\<And>C. C \<in> \<T> \<Longrightarrow> convex C" shows "ANR (\<Union> \<T>)"
order_iff: fixes b :: "'h"   and a :: "'h" shows "(b \<le> a) = (a = max a b)"
containFalsum: fixes \<Gamma> :: "form multiset"   and R :: "(sequent list \<times> sequent) set"   and \<Delta> :: "form multiset" assumes "ff \<in># \<Gamma>"   and "Ax \<subseteq> R" shows "( \<Gamma> \<Rightarrow>* \<Delta>, 0) \<in> derivable  R*"
compile_id_correct': fixes n :: "nat"   and args :: "nat list"   and anything :: "nat list" assumes "n < length args" shows "{\<lambda>nl. nl = args @ 0 \<up> 2 @ anything} addition n (length args) (Suc (length args)) {\<lambda>nl. nl = args @ rec_exec (recf.id (length args) n) args # 0 # anything}"
lincomb_append_right: fixes rs :: "'f list"   and ms :: "'v list"   and ns :: "'v list" shows "rs \<bullet>\<cdot> (ms @ ns) = rs \<bullet>\<cdot> ms + drop (length ms) rs \<bullet>\<cdot> ns"
Reals_diff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>"   and "b \<in> \<real>" shows "a - b \<in> \<real>"
of_nat_mono_maybe': fixes x :: "nat"   and y :: "nat" assumes "x < 2 ^ LENGTH('b)"   and "y < 2 ^ LENGTH('b)" shows "(y < x) = (word_of_nat y < word_of_nat x)"
rep_eq: fixes x :: "uint32" shows "nat_of_uint32 x = unat (Rep_uint32 x)"
Cons_1: fixes C :: "'a set" assumes "C \<in> f \<turnstile> \<C>"   and "D \<in> \<C> - f \<turnstile> \<C>"   and "gallery (C # Es @ [D])" shows "\<exists>As A B Bs. A \<in> f \<turnstile> \<C> \<and> B \<in> \<C> - f \<turnstile> \<C> \<and> C # Es @ [D] = As @ A # B # Bs"
free_trm_def: fixes n :: "name" shows "free n = Var (of_name n)"
case_1: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b" shows "(case LNil of LNil \<Rightarrow> f1 | x ## xa \<Rightarrow> f2 x xa) = f1"
pairwise_orthogonal_imp_finite: fixes S :: "'c set" assumes "pairwise orthogonal S" shows "finite S"
Hintikka_model_2: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and p :: "('c, 'b) fm" assumes "Hintikka A H"   and "(\<^bold>\<not> p) at i in' H"   and "nominals p \<subseteq> A" shows "\<not> Model (reach A H) (val H), assign A H, assign A H i \<Turnstile> p"
swap_atom: fixes a :: "atom"   and b :: "atom"   and c :: "atom" shows "(a \<rightleftharpoons> b) \<bullet> c = (if sort_of a = sort_of b then if c = a then b else if c = b then a else c else c)"
rep_eq: fixes x :: "'b poly" shows "coeff (- x) = (\<lambda>n. - coeff x n)"
prev: shows "AEeq N (\<lambda>w. discount_factor r 0 w * (q * u + (1 - q) * d) * init) (\<lambda>w. (1 + r) * init)"
tBD_axioms_def: fixes grouping :: "int"   and sizes :: "int set" shows "tBD_axioms grouping sizes \<equiv> \<forall>k. k \<in> sizes \<longrightarrow> grouping \<le> k"