hcju/isabelleps huggingface dataset:
assms: fixes y :: "'b"   and x :: "'a" assumes "P y"   and "(x, y) \<in> R" shows "(f x, g y) \<in> S"
right_neutral: fixes a :: "'g" shows "a + (0::'g) = a"
sep_conv: fixes g :: "graph"   and V :: "nat set" assumes "minGraphProps g"   and "V \<subseteq> \<V> g" shows "separated g V = (\<forall>u\<in>V. \<forall>v\<in>V. u \<noteq> v \<longrightarrow> \<not> close g u v)"
simps_1: shows "Diag \<^bold>\<I> = True"
bdd_above_def: fixes A :: "'c set" shows "bdd_above A = (\<exists>M. \<forall>x\<in>A. x \<le> M)"
snd_lift: fixes i :: "nat"   and xs :: "('b com option \<times> 'b) list"   and Q :: "'b com" assumes "i < length xs" shows "snd (lift Q (xs ! i)) = snd (xs ! i)"
supp_fun_eq_zero_iff: fixes f :: "'a \<Rightarrow> 'b" shows "(supp_fun f = {}) = (f = 0)"
imageI: fixes x :: "'ag"   and A :: "'ag set"   and f :: "'ag \<Rightarrow> 'ah" assumes "x \<in> A" shows "f x \<in> f ` A"
r_result_Some': fixes t :: "nat"   and i :: "nat"   and x :: "nat"   and v :: "nat" assumes "eval r_result [t, i, x] \<down>= Suc v" shows "eval r_univ [i, x] \<down>= v"
b_e_type_set_global: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Set_global i" shows "\<exists>t. ts = ts' @ [t] \<and> global \<C> ! i = \<lparr>tg_mut = T_mut, tg_t = t\<rparr> \<and> i < length (global \<C>)"
pvalid_left_def: fixes p :: "(symbol \<times> character list) list"   and x :: "item" shows "pvalid_left p x = (\<exists>u \<gamma>. wellformed_tokens p \<and> wellformed_item x \<and> u \<le> length p \<and> charslength p = item_end x \<and> charslength (take u p) = item_origin x \<and> is_leftderivation (terminals (take u p) @ [item_nonterminal x] @ \<gamma>) \<and> leftderives (item_\<alpha> x) (terminals (drop u p)))"
trans: fixes b :: "'b::order"   and a :: "'b::order"   and c :: "'b::order" assumes "(b::'b::order) \<le> (a::'b::order)"   and "(c::'b::order) \<le> (b::'b::order)" shows "(c::'b::order) \<le> (a::'b::order)"
assms_1: shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
enc_reflects_binary_pred_iff_source_target_rel_reflects_binary_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" shows "enc_reflects_binary_pred Pred = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_reflects_binary_pred Rel Pred)"
ereal_INF_cmult: fixes c :: "real"   and f :: "'a \<Rightarrow> ereal"   and I :: "'a set" assumes "0 < c" shows "(\<Sqinter>i\<in>I. ereal c * f i) = ereal c * \<Sqinter> (f ` I)"
spec_ops_distinct: fixes ops :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" assumes "spec_ops ops deps" shows "distinct ops"
seqI': fixes f :: "(bool, 'arr list) concrete_category.arr"   and a :: "(bool, 'arr list) concrete_category.arr"   and b :: "(bool, 'arr list) concrete_category.arr"   and g :: "(bool, 'arr list) concrete_category.arr"   and c :: "(bool, 'arr list) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "seq g f"
mk_less_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "mk_less R = (\<lambda>x y. R x y \<and> \<not> R y x)"
card_psubset: fixes B :: "'c set"   and A :: "'c set" assumes "finite B"   and "A \<subseteq> B"   and "card A < card B" shows "A \<subset> B"
min_galleryD_gallery: fixes xs :: "'a set list" assumes "min_gallery xs" shows "gallery xs"
simps_5: fixes e :: "exp0"   and cs :: "(pat \<times> exp0) list" shows "is_cupcake_exp (Mat e cs) = (is_cupcake_exp e \<and> cupcake_clauses cs \<and> cake_linear_clauses cs)"
semiring_gcd: shows "class.semiring_gcd (1::'a) (0::'a) (*) normalization_euclidean_semiring_class.gcd normalization_euclidean_semiring_class.lcm (div) (+) (-) unit_factor normalize"
map_fun_def: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'g" shows "map_fun f g h = g \<circ> h \<circ> f"
isin_set: fixes t :: "'b tree234"   and x :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "isin t x = (x \<in> set (inorder t))"
clkp_set_def: fixes A :: "('e \<times> ('c, 'd) cconstraint \<times> 'f \<times> 'c list \<times> 'e) set \<times> ('e \<Rightarrow> ('c, 'd) cconstraint)" shows "clkp_set A = collect_clki (inv_of A) \<union> collect_clkt (trans_of A)"
in_car_final2: fixes s1 :: "'f \<Rightarrow> ('d, 'f, 'e) F1"   and s2 :: "'e \<Rightarrow> ('d, 'f, 'e) F2"   and x :: "'e" shows "(Equiv_Relations.proj (lsbis2 carT1 carT2 strT1 strT2) \<circ> beh2 s1 s2) x \<in> car_final2"
linear_imp_scaleR: fixes D :: "real \<Rightarrow> 'a" assumes "linear D" obtains d :: "'a" where "D = (\<lambda>x. x *\<^sub>R d)"
comp_char: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" shows "g \<cdot> f = (if seq g f then MkArr (Dom f) (Cod g) (Dom f) else null)"
associated_iff: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier (Ring_Divisibility.mult_of R)"   and "b \<in> carrier (Ring_Divisibility.mult_of R)" shows "a \<sim>\<^bsub>Ring_Divisibility.mult_of R\<^esub> b = (\<exists>c\<in>Units (Ring_Divisibility.mult_of R). a = b \<otimes> c)"
cos_vangle_eqD: fixes u :: "'b"   and v :: "'b"   and w :: "'c"   and x :: "'c" assumes "cos (vangle u v) = cos (vangle w x)" shows "vangle u v = vangle w x"
subset_eq_def: shows "Set_Impl.subset_eq = (\<subseteq>)"
assms_3: shows "\<pi>\<^bsub>c\<^esub>ltake (enat n') t = \<pi>\<^bsub>c\<^esub>ltake (enat n) t"
usubstappp_compose: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game"   and \<beta> :: "game" shows "usubstappp \<sigma> U (\<alpha> ;; \<beta>) = (fst (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<beta>), Composeo (snd (usubstappp \<sigma> U \<alpha>)) (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<beta>)))"
while_option_rule: fixes P :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "while_option b c s = Some t"   and "P s" shows "P t"
assms_1: assumes "i = 0" shows "f \<in> M \<rightarrow>\<^sub>M N"
meval_sched_modes_respect_priv: fixes sched :: "nat list"   and gc\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and gc\<^sub>C' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" assumes "conc.meval_sched_abv gc\<^sub>C sched gc\<^sub>C'"   and "length (fst gc\<^sub>C) = length cms"   and "modes_respect_priv (map snd (fst gc\<^sub>C))" shows "modes_respect_priv (map snd (fst gc\<^sub>C'))"
power_minus_even: fixes n :: "nat"   and a :: "'b" assumes "even n" shows "(- a) ^ n = a ^ n"
dist_diff_2: fixes a :: "'e"   and b :: "'e" shows "dist (a - b) a = norm b"
RTL: shows "rank_of X = card X \<and> rank_of X = rank_of carrier"
le_less: fixes x :: "'b list"   and y :: "'b list" shows "(x \<le>p y) = (x <p y \<or> x = y)"
code_word_length_def: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option"   and e :: "'e list \<Rightarrow> bool list"   and l :: "'e" assumes "source_code M b fi X enc dec" shows "code_word_length e l = length (e [l])"
times_empty2: fixes A :: "hf" shows "A * 0 = 0"
disj: shows "f absolutely_integrable_on g ` S \<or> (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S"
insert_Diff: fixes a :: "'g"   and A :: "'g set" assumes "a \<in> A" shows "insert a (A - {a}) = A"
zariski_open_is_subset: fixes U :: "('a \<times> 'a) set set set" assumes "local.local.is_zariski_open U" shows "U \<subseteq> local.local.spectrum"
simps_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option"   and y :: "'a"   and ys :: "'a list" shows "proper_intrvl.Compl_set_less_eq_aux less proper_interval ao [] (y # ys) = (\<not> proper_interval ao (Some y) \<and> proper_intrvl.Compl_set_less_eq_aux less proper_interval (Some y) [] ys)"
updEnv_preserves_good: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and x :: "'var"   and xs :: "'varSort" assumes "good X"   and "goodEnv rho" shows "goodEnv (rho [x \<leftarrow> X]_xs)"
inf_add_S_left: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and seq_comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and neutral :: "'c"   and deactivation :: "'c"   and basic :: "'c set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.stimuli leq add seq_comp neutral deactivation basic"   and "leq x y" shows "leq x (add z y)"
sum_list_abs: fixes abs :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs (monoid_add.sum_list plus zero xs)) (monoid_add.sum_list plus zero (map abs xs))"
Imon_closed: fixes l :: "'a list"   and m :: "mon" assumes "in_carrier l" shows "Imon l m \<in> carrier R"
simps: fixes a1 :: "nat"   and a2 :: "'b tree23" shows "full a1 a2 = (a1 = 0 \<and> a2 = \<langle>\<rangle> \<or> (\<exists>n l r p. a1 = Suc n \<and> a2 = \<langle>l, p, r\<rangle> \<and> full n l \<and> full n r) \<or> (\<exists>n l m r p q. a1 = Suc n \<and> a2 = \<langle>l, p, m, q, r\<rangle> \<and> full n l \<and> full n m \<and> full n r))"
induct: fixes P :: "('c, 'd) Node \<Rightarrow> ('c, 'd) FingerTreeStruc \<Rightarrow> bool"   and a0 :: "('c, 'd) Node"   and a1 :: "('c, 'd) FingerTreeStruc" assumes "\<And>a. P a Empty"   and "\<And>a b. P a (Single b)"   and "\<And>a uu_ b m sf. P a (Deep uu_ (One b) m sf)"   and "\<And>a uv_ b c m sf. P a (Deep uv_ (Two b c) m sf)"   and "\<And>a uw_ b c d m sf. P a (Deep uw_ (Three b c d) m sf)"   and "\<And>a ux_ b c d e m sf. P (node3 c d e) m \<Longrightarrow> P a (Deep ux_ (Four b c d e) m sf)" shows "P a0 a1"
tr_par_declassified_eq: fixes B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and I :: "'var \<Rightarrow> ('fun, 'a) Term.term" assumes "B \<in> set (tr\<^sub>p\<^sub>c A D)" shows "declassified\<^sub>l\<^sub>s\<^sub>t B I = declassified\<^sub>l\<^sub>s\<^sub>s\<^sub>t A I"
inject_2: fixes x2 :: "'b fs_set \<times> 'a"   and y2 :: "'b fs_set \<times> 'a" shows "(EF x2 = EF y2) = (x2 = y2)"
sub_mult_one_circ: fixes x :: "'a" shows "x * (1::'a)\<^sup>\<star> \<le> (1::'a)\<^sup>\<star> * x"
indep_in_subI_subset: fixes \<E>' :: "'a set"   and \<E> :: "'a set"   and X :: "'a set" assumes "\<E>' \<subseteq> \<E>"   and "indep_in \<E>' X" shows "indep_in \<E> X"
subgame_E: fixes V' :: "'a set" shows "E\<^bsub>subgame V'\<^esub> \<subseteq> E"
left_neutral: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and a :: "'b" assumes "monoid f z" shows "f z a = a"
assms_3: shows "AE x in M. 0 \<le> f x"
less_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_less [x, y] = (if x < y then 1 else 0)"
higher_eq_zero_iff': fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "(higher p v = 0) = (lt p \<preceq>\<^sub>t v)"
cases: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and a1 :: "nat \<times> nat \<Rightarrow> 'a"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "pr_operation'"   and a4 :: "nat \<times> nat \<Rightarrow> 'a"   and a5 :: "nat \<Rightarrow> nat" assumes "Network c s t"   and "((a1, a2), a3, a4, a5) \<in> Network.pr_algo_lts' c s t" obtains f :: "nat \<times> nat \<Rightarrow> 'a" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = f__" and "a2 = l__" and "a3 = NONSAT_PUSH'" and "a4 = Network.push_effect c f__ e__" and "a5 = l__" and "Height_Bounded_Labeling c s t f__ l__" and "Network.nonsat_push_precond c f__ l__ e__"   | f :: "nat \<times> nat \<Rightarrow> 'a" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fa__" and "a2 = la__" and "a3 = SAT_PUSH' e" and "a4 = Network.push_effect c fa__ e" and "a5 = la__" and "Height_Bounded_Labeling c s t fa__ la__" and "Network.sat_push_precond c fa__ la__ e"   | f :: "nat \<times> nat \<Rightarrow> 'a" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = l" and "a3 = RELABEL'" and "a4 = f" and "a5 = Network.relabel_effect c f l u" and "Height_Bounded_Labeling c s t f l" and "Network.relabel_precond c t f l u"
divide_out_def: fixes p :: "'a"   and x :: "'a" shows "divide_out p x = (x div p ^ multiplicity p x, multiplicity p x)"
m_closed: fixes R :: "('h, 'i) ring_scheme"   and N :: "'j set"   and M :: "('h, 'j, 'k) module_scheme"   and v :: "'j"   and w :: "'j" assumes "LinearCombinations.submodule R N M"   and "v \<in> N"   and "w \<in> N" shows "v \<oplus>\<^bsub>M\<^esub> w \<in> N"
genv_any_nil_inv: fixes \<rho> :: "(nat \<times> bval) list" assumes "good_env \<rho> []" obtains "\<rho> = []"
assms: shows "u \<in> g_V0 (degeneralize_ext m)"
assms_2: shows "t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* u"
simps: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real"   and l0 :: "nat"   and f1'' :: "real"   and f2'' :: "real"   and n :: "nat"   and l :: "nat" assumes "Table2 f1 f2 f1' f2' e c l0 f1'' f2''" shows "Table2.\<Phi> f1 f2 c l0 f1'' f2'' (n, l) = (if f2'' * real l \<le> real n then Table2.ai f2 f2'' * (real n - f2'' * real l) else if real n \<le> f1'' * real l \<and> int l0 \<le> \<lfloor>real l / c\<rfloor> then Table2.ad f1 f1'' * (f1'' * real l - real n) else 0)"
eval_cases_13: fixes P :: "(char list list \<times> char list exp) prog"   and V :: "char list"   and T :: "ty"   and e\<^sub>1 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>{V:T; e\<^sub>1},s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains h\<^sub>0 :: "heap" and l\<^sub>0 :: "sfields" and sh\<^sub>0 :: "sheap" and h\<^sub>1 :: "heap" and l\<^sub>1 :: "sfields" and sh\<^sub>1 :: "sheap" where "s = (h\<^sub>0, l\<^sub>0, sh\<^sub>0)" and "s' = (h\<^sub>1, l\<^sub>1(V := l\<^sub>0 V), sh\<^sub>1)" and "P \<turnstile> \<langle>e\<^sub>1,(h\<^sub>0, l\<^sub>0(V := None), sh\<^sub>0)\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"
fcon: shows "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1, y2). f y1 y2)"
redundancy_criterion_axioms: shows "redundancy_criterion \<Gamma> Rf Ri"
execn_Skip': fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and t :: "('b, 'c) xstate"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>Skip,t\<rangle> =n\<Rightarrow> t"
inf_le2: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) y"
order_tendstoI: fixes y :: "'a"   and f :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "\<And>a. a < y \<Longrightarrow> \<forall>\<^sub>F x in F. a < f x"   and "\<And>a. y < a \<Longrightarrow> \<forall>\<^sub>F x in F. f x < a" shows "(f \<longlongrightarrow> y) F"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "finite (B x)"
leq: shows "length ts = length ts\<^sub>s\<^sub>b"
sub: fixes a :: "('a \<times> 'a) set" assumes "a \<in> \<ww>" shows "a \<in> carrier_local_ring_at"
word_bw_lcs_3: fixes y :: "'b word"   and x :: "'b word"   and z :: "'b word" shows "y XOR x XOR z = x XOR y XOR z"
is_nextElem_edges1: fixes f :: "face"   and a :: "nat"   and b :: "nat" assumes "distinct (vertices f)"   and "(a, b) \<in> \<E> f" shows "is_nextElem (vertices f) a b"
count_roots_below: fixes a :: "real" assumes "p' \<noteq> 0" shows "sign_changes_neg_inf (sturm_squarefree p) - sign_changes (sturm_squarefree p) a = card {x. x \<le> a \<and> poly p' x = 0}"
2: fixes i :: "nat" assumes "i < length T" shows "timpls_transformable_to_pred (A \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) (T ! i) (S ! i)"
fields_declC: fixes G :: "prog"   and C :: "qtname"   and efn :: "fspec"   and f :: "Decl.field" assumes "table_of (DeclConcepts.fields G C) efn = Some f"   and "ws_prog G"   and "is_class G C" shows "(\<exists>d. class G (declclassf efn) = Some d \<and> table_of (cfields d) (fname efn) = Some f) \<and> G\<turnstile>C\<preceq>\<^sub>C declclassf efn \<and> table_of (DeclConcepts.fields G (declclassf efn)) efn = Some f"
null_set_Int1: fixes B :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "B \<in> null_sets M"   and "A \<in> sets M" shows "A \<inter> B \<in> null_sets M"
agent_absI: fixes a :: "'a"   and t' :: "('a, 'es, 'as) BEState Trace"   and t :: "('a, 'es, 'as) BEState Trace" assumes "spr_jview a t' = spr_jview a t"   and "t' \<in> SPR.jkbpC"   and "t \<in> SPR.jkbpC" shows "(tFirst t', tLast t') \<in> agent_abs a t"
disc_iff_2: fixes A :: "'c llist set" shows "(\<not> lnull (lSup A)) = (\<not> (\<forall>x\<in>A. lnull x))"
fresh_set_empty: fixes a :: "'a" shows "a \<sharp> {}"
distinct_2: fixes x2 :: "('a, 'b, 'c, 'd) Tree"   and x1 :: "('a, 'b, 'c, 'd) Tree set['a]" shows "tNot x2 \<noteq> tConj x1"
lookup_except: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" shows "lookup (except p S) = (\<lambda>t. if t \<in> S then 0::'d else lookup p t)"
fold_A_chamber_system_image: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "(ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) \<turnstile> ChamberComplex.chamber_system A = {F \<in> ChamberComplex.chamber_system A. ChamberComplex.face_distance X z F = ChamberComplex.chamber_distance X C F}"
pure_returns_heap_eq: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog"   and h' :: "'l" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "pure f h" shows "h = h'"
cases: fixes x :: "int \<times> nat" obtains xa :: "int" and n :: "nat" where "x = (xa, n)"
FactRing_is_ring: shows "ring (R Quot a_kernel R S h)"
Inr_Inl_False: fixes x :: "'e"   and y :: "'d" shows "(Inr x = Inl y) = False"
case_nb: shows "\<turnstile> (P \<inter> - b) c2 Q"
mono: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<le> a"   and "d \<le> b" shows "max c d \<le> max a b"
INV1: fixes P :: "state \<Rightarrow> bool" shows "\<turnstile> Init P \<longrightarrow> stable P \<longrightarrow> \<box>P"
mask_eq_sum_exp: fixes n :: "nat" shows "(2::'a) ^ n - (1::'a) = sum ((^) (2::'a)) {q. q < n}"
simps_1: fixes i :: "char list \<Rightarrow> tv"   and s :: "char list" shows "eval i (Pro s) = i s"
set_gen2': fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "(\<lambda>(x, y). (fst x, fst y)) ` set (alls2 A B a b) = {(x, y). x \<le>\<^sub>v replicate (length a) A \<and> y \<le>\<^sub>v replicate (length b) B}"
handshake_invL_def: fixes m :: "'h" shows "mut_m.handshake_invL m = (\<lambda>s. (atS (mutator m) mut_m.hs_noop_locs s \<longrightarrow> sys_hs_type s\<down> = ht_NOOP) \<and> (atS (mutator m) mut_m.hs_get_roots_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetRoots) \<and> (atS (mutator m) mut_m.hs_get_work_locs s \<longrightarrow> sys_hs_type s\<down> = ht_GetWork) \<and> (atS (mutator m) mut_m.ht_loaded_locs s \<longrightarrow> mutator_hs_pending (s\<down> (mutator m)) \<longrightarrow> hs_type (s\<down> (mutator m)) = sys_hs_type s\<down>) \<and> (atS (mutator m) mut_m.hs_pending_loaded_locs s \<longrightarrow> mutator_hs_pending (s\<down> (mutator m)) \<longrightarrow> sys_hs_pending m s\<down>) \<and> (atS (mutator m) mut_m.hs_pending_locs s \<longrightarrow> mutator_hs_pending (s\<down> (mutator m))) \<and> (atS (mutator m) mut_m.no_pending_mutations_locs s \<longrightarrow> filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s\<down>) = []))"
monadic_nfoldli_rec: fixes x' :: "'d list"   and c :: "'c \<Rightarrow> bool nres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma> :: "'c"   and xs0 :: "'d list"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool" shows "monadic_nfoldli x' c f \<sigma> \<le> \<Down> Id (REC\<^sub>T (\<lambda>W (xs, \<sigma>). ASSERT (FOREACH_inv xs0 I (xs, \<sigma>)) \<bind> (\<lambda>_. if xs = [] then RETURN (xs, \<sigma>) else c \<sigma> \<bind> (\<lambda>b. if b then FOREACH_body f (xs, \<sigma>) \<bind> W else RETURN (xs, \<sigma>)))) (x', \<sigma>) \<bind> (\<lambda>(uu_, y). RETURN y))"
normal_vector_2: fixes W :: "'a vec set" assumes "finite W"   and "Suc (card W) = n"   and "lin_indpt W"   and "W \<subseteq> carrier_vec n" shows "normal_vector W \<noteq> 0\<^sub>v n"
degree_m_le_degree: fixes m :: "int"   and f :: "int poly" shows "degree (poly_mod.Mp m f) \<le> degree f"
g_cond: shows "\<forall>g\<in>set gs. terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
prod_filtermap1: fixes f :: "'f \<Rightarrow> 'd"   and F :: "'f filter"   and G :: "'e filter" shows "filtermap f F \<times>\<^sub>F G = filtermap (apfst f) (F \<times>\<^sub>F G)"
res_assoc_iff1: shows "(\<forall>x y z. x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z) = (\<forall>x y z. x \<rhd> (y \<rhd> z) = y \<cdot> x \<rhd> z)"
prefixes_init_psi: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and t :: "nat" assumes "r1_bc s" shows "r1_bc.psi s j \<triangleright> (e_length (r1_bc.prefixes s j (Suc t)) - 1) = r1_bc.prefixes s j (Suc t)"
Rel_app: fixes A :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and x :: "'a"   and y :: "'c" assumes "Transfer.Rel (A ===> B) f g"   and "Transfer.Rel A x y" shows "Transfer.Rel B (f x) (g y)"
counit_simps_4: fixes a :: "'d" assumes "D.obj a" shows "trg\<^sub>D (\<eta>'.counit a) = GF.map\<^sub>0 a"
onl_oinvariant_weakenD: fixes A :: "('e \<times> ('f, 'g, 'h, 'i) seqp, 'j) automaton"   and S' :: "'e \<Rightarrow> 'e \<Rightarrow> 'j \<Rightarrow> bool"   and U' :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and \<Gamma> :: "'h \<Rightarrow> ('f, 'g, 'h, 'i) seqp"   and P :: "'e \<times> 'i \<Rightarrow> bool"   and \<sigma> :: "'e"   and p :: "('f, 'g, 'h, 'i) seqp"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'j \<Rightarrow> bool"   and U :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and l :: "'i" assumes "A \<Turnstile> (S', U' \<rightarrow>) onl \<Gamma> P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "l \<in> labels \<Gamma> p"   and "\<And>s s' a. S s s' a \<Longrightarrow> S' s s' a"   and "\<And>s s'. U s s' \<Longrightarrow> U' s s'" shows "P (\<sigma>, l)"
tmneg_nb: fixes n :: "nat"   and t :: "tm" assumes "tmbound n t" shows "tmbound n (tmneg t)"
lookup_row_to_poly_not_zeroE: fixes ts :: "'d list"   and r :: "'c vec"   and t :: "'d" assumes "lookup (row_to_poly ts r) t \<noteq> (0::'c)" obtains i :: "nat" where "i < length ts" and "t = ts ! i"