hcju/isabelleps huggingface dataset:
pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor_def: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" shows "pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D) \<and> equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G \<and> pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
param_SUCCEED: fixes R :: "('b \<times> 'c) set" shows "(SUCCEED, SUCCEED) \<in> \<langle>R\<rangle>nres_rel"
defAss_def: fixes g :: "'g"   and m :: "'node"   and v :: "'val" shows "defAss g m v = (\<forall>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<longrightarrow> (\<exists>n\<in>set ns. v \<in> allDefs g n))"
lsteps_level_greater_fun_image: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> nat"   and l :: "nat"   and u :: "nat"   and v :: "nat" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x < l" shows "lsteps xs f l u v = 0"
li_Nil: fixes K :: "'a list set" shows "UP.independent K []"
iter_widen_pfp: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "iter_widen f x = Some p" shows "f p \<le> p"
can_take_subset: fixes i :: "value list"   and t :: "'a transition_scheme"   and t' :: "'b transition_scheme"   and r :: "nat \<Rightarrow>f value option" assumes "length i = Arity t"   and "Arity t = Arity t'"   and "set (Guards t') \<subseteq> set (Guards t)"   and "can_take_transition t i r" shows "can_take_transition t' i r"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_backto_standard_pos_B m rs (l, r) = (\<exists>rn. l = [] \<and> r = Bk # Oc \<up> Suc m @ Bk # Oc \<up> Suc (Suc rs) @ Bk \<up> rn)"
cases: fixes a :: "'b list"   and A :: "'b set" assumes "a \<in> lists A" obtains "a = \<epsilon>"   | aa :: "'b" and l :: "'b list" where "a = aa # l" and "aa \<in> A" and "l \<in> lists A"
\<delta>\<alpha>''_props: shows "(\<delta> @ \<alpha>'') \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<and> (\<delta> @ \<alpha>'') \<upharpoonleft> V\<^bsub>\<V>\<^esub> = (e1 # \<alpha>1) \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> enabled SES s1 (\<delta> @ \<alpha>'')"
Fin_numeral: fixes w :: "num" shows "Fin (numeral w) = numeral w"
padl_unpadl: fixes v :: "'b vec"   and m :: "nat"   and U :: "'b vec set" assumes "v \<in> (@\<^sub>v) (0\<^sub>v m) ` U" shows "0\<^sub>v m @\<^sub>v vardim.unpadl m v = v"
A1b: shows "[\<lambda>w. \<forall>x. (m\<not> (P x) m\<rightarrow> P (\<lambda>xa. m\<not> (x xa))) w]"
skip_doesnt_access: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and lc :: "(('b, 'a, 'c) Stmt \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'd)"   and annos :: "'b ModeUpd list"   and mds :: "Mode \<Rightarrow> 'b set"   and mem :: "'b \<Rightarrow> 'd"   and c' :: "('b, 'a, 'c) Stmt"   and mds' :: "Mode \<Rightarrow> 'b set"   and mem' :: "'b \<Rightarrow> 'd"   and x :: "'b" assumes "sifum_modes aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "lc \<in> sifum_security.loc_reach (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) ((sifum_modes.annotate Skip annos, mds), mem)"   and "lc = ((c', mds'), mem')" shows "sifum_security.doesnt_read (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) c' x \<and> sifum_security.doesnt_modify (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B) c' x"
select_convs_2: fixes precondition_of :: "'a list"   and add_effects_of :: "'a list"   and delete_effects_of :: "'a list"   and more :: "'b" shows "add_effects_of \<lparr>precondition_of = precondition_of, add_effects_of = add_effects_of, delete_effects_of = delete_effects_of, \<dots> = more\<rparr> = add_effects_of"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'"   and "surjective_map \<eta> M M'" shows "monoid_epimorphism \<eta> M composition unit M' composition' unit'"
norm_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. norm (f x)) \<in> Lr F (\<lambda>x. norm (g x))) = (f \<in> L F g)"
qFresh_imp_ex_qAFresh1: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and zs :: "'e"   and z :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X"   and "qFresh zs z X" shows "\<exists>X'. FixVars.qGood X' \<and> X #= X' \<and> qAFresh zs z X'"
emeasure_PiM_emb: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and J :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"   and "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)" shows "emeasure (Pi\<^sub>M I M) (prod_emb I M J (Pi\<^sub>E J A)) = (\<Prod>i\<in>J. emeasure (M i) (A i))"
nonzero_divide_mult_cancel_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "b \<noteq> zero" shows "divide b (times a b) = divide one a"
ln_series_stream_def: shows "ln_series_stream = MSSCons 0 (ln_series_stream_aux False 1)"
blinfun_ext: fixes x :: "'e \<Rightarrow>\<^sub>L 'f"   and y :: "'e \<Rightarrow>\<^sub>L 'f" shows "(x = y) = (\<forall>i. blinfun_apply x i = blinfun_apply y i)"
chain0: fixes Y :: "nat \<Rightarrow> 'd upred" assumes "chain Y" shows "Y 0 = false"
elims_3: fixes x :: "'a dres"   and xa :: "'a dres" assumes "\<not> x \<le> xa" obtains a :: "'a" and b :: "'a" where "x = dRETURN a" and "xa = dRETURN b" and "a \<noteq> b"   | "x = dFAILi" and "xa = dSUCCEEDi"   | v :: "'a" where "x = dFAILi" and "xa = dRETURN v__"   | v :: "'a" where "x = dRETURN v" and "xa = dSUCCEEDi"
simps_11: shows "cospan_shape.comp cospan_shape.BB cospan_shape.TT = cospan_shape.Null"
less_supI2: fixes x :: "'a"   and b :: "'a"   and a :: "'a" assumes "x < b" shows "x < a + b"
fun_order_def: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set" shows "f \<le> g \<equiv> \<forall>p. f p \<subseteq> g p"
extend_state_oracle_def: fixes callee :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf" shows "\<dagger>callee = (\<lambda>(s', s) x. map_spmf (\<lambda>(y, s). (y, s', s)) (callee s x))"
intruder_synth_mod_timpls'_is_synth_timpl_closure_set: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term list"   and TI :: "('sets set \<times> 'sets set) list"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term" shows "intruder_synth_mod_timpls' M TI t = (timpl_closure_set (set M) (set TI) \<turnstile>\<^sub>c t)"
assms_5: shows "(\<exists>z. a = kE z) \<or> (\<exists>z. a = sE z)"
spr_jview_det_ps: fixes t :: "('a, 'es, 'as) BEState Trace"   and t' :: "('a, 'es, 'as) BEState Trace"   and a :: "'a" assumes "{t, t'} \<subseteq> SPR.jkbpC"   and "tObsC t = tObsC t'"   and "envObs a (tFirst t) = envObs a (tFirst t')" shows "tMap (\<lambda>s. ODList.lookup (ps s) a) t = tMap (\<lambda>s. ODList.lookup (ps s) a) t'"
antisym_conv2: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.order less_eq less"   and "less_eq x y" shows "(\<not> less x y) = (x = y)"
appendPath1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and Ds :: "char list list" assumes "Subobjs P C Cs"   and "Subobjs P (last Cs) Ds"   and "last Cs \<noteq> hd Ds" shows "Subobjs P C Ds"
prv_neg_eql_suc_Num_zer: fixes n :: "nat" shows "prv (neg (eql (suc (Num n)) zer))"
words_fin_blocked: fixes p :: "'state"   and A :: "'action set"   and w :: "'action list" assumes "\<And>w. \<lbrakk>path w p; A \<inter> set w = {}\<rbrakk> \<Longrightarrow> A \<inter> {a. en a (target w p)} \<subseteq> A \<inter> {a. en a p}"   and "path w p"   and "A \<inter> {a. en a p} \<inter> set w = {}" shows "A \<inter> set w = {}"
D_gchain_pre: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "D_gchain G (Suc n) f" shows "D_gchain G n f"
bvs_1: shows "set bvs \<subseteq> set vs"
holh: fixes n :: "nat" shows "h n holomorphic_on ball 0 1 - {0}"
image_ideal_eq_surj: fixes h :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "\<And>x y. h (x + y) = h x + h y"   and "\<And>x y. h (x * y) = h x * h y"   and "surj h" shows "h ` ideal B = ideal (h ` B)"
assoc: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and c :: "('a \<times> 'a) set" shows "(a O b) O c = a O b O c"
assms: fixes i :: "'d" assumes "i \<in> S" shows "borel_predict_stoch_proc F (A i)"
is_free_nat: fixes m :: "nat"   and n :: "nat"   and c :: "nat" assumes "m < n"   and "n < c"   and "3 \<le> c" shows "\<exists>k. k < m \<or> m < k \<and> k < n \<or> n < k \<and> k < c"
convex_Times: fixes s :: "'b set"   and t :: "'c set" assumes "convex s"   and "convex t" shows "convex (s \<times> t)"
set_intros_1: fixes x21 :: "'d"   and x22 :: "'d list" shows "x21 mem (x21 # x22)"
assms_2: shows "sound_mode_use (cms\<^sub>2, mem\<^sub>2)"
wls_freshAbs_vsubstAbs: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "wlsAbs (us, s) A" shows "freshAbs zs z (A $[y1 // y]_ys) = ((zs = ys \<and> z = y \<or> freshAbs zs z A) \<and> (freshAbs ys y A \<or> zs \<noteq> ys \<or> z \<noteq> y1))"
assms_1: shows "f (lead_coeff p) = (0::'a)"
indep_varsD: fixes i :: "'i"   and M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'b \<Rightarrow> 'e"   and I :: "'d set"   and J :: "'d set"   and A :: "'d \<Rightarrow> 'e set" assumes "D'.indep_vars i M' X I"   and "J \<noteq> {}"   and "finite J"   and "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M' i)" shows "D'.prob i (\<Inter>i\<in>J. X i -` A i \<inter> space (distr M (M' i) (X i))) = (\<Prod>i\<in>J. D'.prob i (X i -` A i \<inter> space (distr M (M' i) (X i))))"
vrestr_same_measure_f: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "f \<in> measure_preserving M M"   and "A \<in> sets M" shows "Sigma_Algebra.measure M (f --` A) = Sigma_Algebra.measure M A"
mergesort_def: shows "mergesort = mergesort_by_rel (\<le>)"
counit_simps_5: shows "trg\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g)) = trg\<^sub>D (F f)"
lQuot_conc: fixes c :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "lQuot c (A @@ B) = lQuot c A @@ B \<union> (if [] \<in> A then lQuot c B else {})"
cod_comp: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "CCC.seq g f" shows "CCC.cod (CCC.comp g f) = CCC.cod g"
int_a_minus_eq: fixes x :: "int"   and y :: "int" shows "x \<ominus>\<^bsub>\<Z>\<^esub> y = x - y"
sorted_sort_id: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = xs"
finite_equiv_implies_finite_carrier: fixes A :: "'a set"   and R :: "('a \<times> 'a) set" assumes "equiv A R"   and "finite (A // R)"   and "\<forall>X\<in>A // R. finite X" shows "finite A"
rel_funD: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and x :: "'a"   and y :: "'b" assumes "(A ===> B) f g"   and "A x y" shows "B (f x) (g y)"
assms: shows "K\<^bsub>3,3\<^esub> (with_proj G)"
mk_coeff_Cons: fixes c :: "nat \<Rightarrow> 'a"   and v :: "'e"   and vs :: "'e list" assumes "c \<in> {0..<length (v # vs)} \<rightarrow> carrier R" shows "mk_coeff (v # vs) c = (\<lambda>w. (if w = v then c 0 else \<zero>) \<oplus> mk_coeff vs (c \<circ> Suc) w)"
f_def: shows "f = monom (UP Zp) \<one> 2 \<oplus>\<^bsub>UP Zp\<^esub> to_polynomial Zp (\<ominus> (a [^] 2 \<oplus> [p] \<cdot> \<one> \<otimes> b [^] 2))"
of_nat_mult: fixes R :: "('b, 'c) ring_scheme"   and i :: "nat"   and j :: "nat" assumes "ring R" shows "\<guillemotleft>i * j\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> = \<guillemotleft>i\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> \<guillemotleft>j\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub>"
ps\<^sub>L: shows "sparse (dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps\<^sub>L"
has_integral_measure_lborel: fixes A :: "'c set" assumes "A \<in> sets borel"   and "emeasure lborel A < \<infinity>" shows "((\<lambda>x. 1) has_integral content A) A"
enumerate_Ex: fixes S :: "nat set"   and s :: "nat" assumes "infinite S"   and "s \<in> S" shows "\<exists>n. wellorder_class.enumerate S n = s"
assms_1: shows "y \<longlonglongrightarrow> x"
fbrelation_pair_def: fixes r\<^sub>1 :: "V"   and r\<^sub>2 :: "V" shows "fbrelation_pair r\<^sub>1 r\<^sub>2 \<equiv> fbrelation r\<^sub>1 \<and> fbrelation r\<^sub>2"
normal_series_def: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list" shows "normal_series G \<GG> \<equiv> Group.group G \<and> normal_series_axioms G \<GG>"
lift_def: fixes f :: "'c \<Rightarrow> 'b err"   and e :: "'c err" shows "lift f e = (case e of Err \<Rightarrow> Err | OK x \<Rightarrow> f x)"
remove_child_removes_child: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> remove_child ptr' child \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "known_ptrs h"   and "type_wf h" shows "child \<notin> set children"
length_wcmpser0: fixes G :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme"   and n :: "nat" assumes "Group G"   and "Ugp E" shows "\<forall>f. w_cmpser G (Suc n) f \<longrightarrow> card (f ` {i. i \<le> Suc n}) - 1 = Suc n - card {i. i \<le> n \<and> Qw_cmpser G f i \<cong> E}"
index_of_c_pair_n_inj: fixes x1 :: "nat"   and x2 :: "nat" assumes "index_of_c_pair_n x1 = index_of_c_pair_n x2" shows "x1 = x2"
ceq_set_def: shows "CEQ('a set) = (case ID CEQ('a) of None \<Rightarrow> None | Some x \<Rightarrow> Some set_eq)"
IH: fixes I :: "'variable \<Rightarrow> bool option" assumes "is_valid_problem \<Pi>"   and "G \<subseteq>\<^sub>m last (trace_parallel_plan_strips I \<pi>)"   and "\<forall>k<length \<pi>. are_all_operators_applicable (trace_parallel_plan_strips I \<pi> ! k) (\<pi> ! k) \<and> are_all_operator_effects_consistent (\<pi> ! k) \<and> are_all_operators_non_interfering (\<pi> ! k)" shows "G \<subseteq>\<^sub>m execute_serial_plan I (concat \<pi>)"
genv_any_cons_inv: fixes \<rho> :: "(nat \<times> bval) list"   and b :: "nat \<times> val"   and \<rho>' :: "(nat \<times> val) list" assumes "good_env \<rho> (b # \<rho>')" obtains v :: "bval" and v' :: "val" and \<rho>'' :: "(nat \<times> bval) list" and x :: "nat" where "\<rho> = (x, v) # \<rho>''" and "b = (x, v')" and "v \<in> good v'" and "good_env \<rho>'' \<rho>'"
assms_2: shows "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"
right_idem: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "semilattice f" shows "f (f a b) b = f a b"
unit_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (F.unit a) = F.map\<^sub>0 a"
first_countable_basis: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.first_countable_topology open" shows "\<exists>A. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
Powp_Pow_eq: fixes A :: "'a set" shows "Powp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> Pow A)"
inject_6: fixes x61 :: "nat"   and x62 :: "('a, 'b) MWLsCom list"   and y61 :: "nat"   and y62 :: "('a, 'b) MWLsCom list" shows "(spawn\<^bsub>x61\<^esub> x62 = spawn\<^bsub>y61\<^esub> y62) = (x61 = y61 \<and> x62 = y62)"
banach_fix: fixes s :: "'b set"   and c :: "real"   and f :: "'b \<Rightarrow> 'b" assumes "complete s"   and "s \<noteq> {}"   and "0 \<le> c"   and "c < 1"   and "f ` s \<subseteq> s"   and "\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y" shows "\<exists>!x. x \<in> s \<and> f x = x"
Transset_0: shows "Transset []\<^sub>\<circ>"
comp_arr_dom: fixes f :: "'j"   and a :: "'j" assumes "J.arr f"   and "J.dom f = a" shows "J f a = f"
bilinear_l_add: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m11 :: "'g"   and m12 :: "'g"   and m2 :: "'i"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m11 \<in> carrier M1"   and "m12 \<in> carrier M1"   and "m2 \<in> carrier M2"   and "bilinear_map f R M1 M2 N" shows "f (m11 \<plusminus>\<^bsub>M1\<^esub> m12, m2) = f (m11, m2) \<plusminus>\<^bsub>N\<^esub> f (m12, m2)"
Serv_fresh_not_AKcryptSK: fixes servK :: "nat"   and evs :: "event list"   and authK :: "nat" assumes "Key servK \<notin> used evs" shows "\<not> AKcryptSK authK servK evs"
poly_subst_prod: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "'e \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and A :: "'e set" shows "poly_subst f (prod p A) = (\<Prod>a\<in>A. poly_subst f (p a))"
assms_1: fixes i :: "nat" assumes "i < dim_col B" shows "col A i = col B i"
in_set_\<alpha>nI2: fixes g :: "'g"   and x :: "'node" assumes "invar g"   and "x \<in> getTo ` \<alpha>e g" shows "x \<in> set (\<alpha>n g)"
preserves_right_adjoint: fixes g :: "'a" assumes "is_right_adjoint g" shows "S.is_right_adjoint (S.UP g)"
L_delta: fixes a :: "'a"   and s :: "'s" assumes "a \<in> set (\<sigma> n)"   and "wf_state s" shows "L (delta a s) = lQuot a (L s)"
diag_mat_vec_nth_simps_1: fixes i :: "'a"   and j :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "i = j" shows "diag_mat f $ i $ j = f i"
f_limit_0_right: shows "(f \<longlongrightarrow> 0) (at_right 0)"
g3ip_upRules: shows "g3ip \<subseteq> upRules"
member_empty: shows "member DList_Set.empty = (\<lambda>_. False)"
sums: shows "(\<lambda>k. inverse (z + of_nat k) - inverse (z + of_nat (Suc k))) sums (inverse (z + of_nat 0) - (0::'a))"
get_exists_matching_src_ips_executable_def: fixes iface :: "iface"   and m :: "'b common_primitive match_expr" shows "get_exists_matching_src_ips_executable iface m \<equiv> let (i_matches, uu_) = primitive_extractor (is_Iiface, iiface_sel) m in if \<forall>is\<in>set i_matches. case is of Pos i \<Rightarrow> match_iface i (iface_sel iface) | Neg i \<Rightarrow> \<not> match_iface i (iface_sel iface) then let (ip_matches, uu_) = primitive_extractor (is_Src, src_sel) m in if ip_matches = [] then wordinterval_UNIV else l2wi_negation_type_intersect (NegPos_map ipt_iprange_to_interval ip_matches) else Empty_WordInterval"
UnI1: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "(c::'b) \<in> (A::'b set)" shows "(c::'b) \<in> (A::'b set) \<union> (B::'b set)"
rel_set_neg_distr_iff: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A' :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_set_neg_distr_cond A A' = True"
discI: fixes option :: "'b option"   and x2 :: "'b" assumes "option = Some x2" shows "option \<noteq> None"
sp_list_update: fixes n :: "nat"   and xs :: "('q, 'r, 's, 'e, 'f, 'g, 't, 'u) instr list"   and F :: "'s \<Rightarrow> 'v fundef option"   and x :: "('q, 'r, 's, 'e, 'f, 'g, 't, 'u) instr" assumes "n < length xs"   and "\<forall>\<Sigma>. sp_instr F (xs ! n) \<Sigma> = sp_instr F x \<Sigma>" shows "sp F (xs[n := x]) = sp F xs"
conc_pow_comm: fixes A :: "'a list set"   and n :: "nat" shows "A @@ A ^^ n = A ^^ n @@ A"
induct: fixes x1 :: "'a attree"   and x2 :: "'a attree"   and P :: "'a attree \<Rightarrow> 'a attree \<Rightarrow> bool" assumes "x1 \<sqsubseteq> x2"   and "\<And>A l si' si'' l'' si si''' A' l' A''. \<lbrakk>A = l @ [\<N>\<^bsub>(si', si'')\<^esub>] @ l'' \<oplus>\<^sub>\<and>\<^bsup>(si, si''')\<^esup>; A' = l' \<oplus>\<^sub>\<and>\<^bsup>(si', si'')\<^esup>; A'' = l @ l' @ l'' \<oplus>\<^sub>\<and>\<^bsup>(si, si''')\<^esup>\<rbrakk> \<Longrightarrow> P A A''"   and "\<And>as A s. \<lbrakk>as \<noteq> []; \<forall>A'\<in>set as. (A \<sqsubseteq> A' \<and> P A A') \<and> attack A = s\<rbrakk> \<Longrightarrow> P A (as \<oplus>\<^sub>\<or>\<^bsup>s\<^esup>)"   and "\<And>A A' A''. \<lbrakk>A \<sqsubseteq> A'; P A A'; A' \<sqsubseteq> A''; P A' A''\<rbrakk> \<Longrightarrow> P A A''"   and "\<And>A. P A A" shows "P x1 x2"
IMP_POST_CYC: fixes it :: "'v set"   and blues :: "'v set"   and reds :: "'v set"   and cyc :: "'v blue_witness" assumes "outer_inv it (blues, reds, cyc)"   and "cyc \<noteq> NO_CYC" shows "case extract_res cyc of None \<Rightarrow> \<forall>L. \<not> is_lasso_prpl L | Some x \<Rightarrow> is_lasso_prpl x"
elims: fixes x :: "'b"   and y :: "nat option" assumes "initial x = y" obtains q :: "'b" where "x = q" and "y = (if q = q\<^sub>0 then Some 0 else None)"
alpha_tConj: fixes tset1 :: "('a, 'b, 'c) Formula.Tree set['a]"   and tset2 :: "('a, 'b, 'c) Formula.Tree set['a]" shows "(Formula.Tree.tConj tset1 =\<^sub>\<alpha> Formula.Tree.tConj tset2) = rel_bset (=\<^sub>\<alpha>) tset1 tset2"
prems: shows "\<forall>aa. ListMem aa (a # as) \<longrightarrow> fmdom' (fmrestrict_set vs (snd aa)) = {}"
snd_Gauss_Jordan_in_ij_PA_eq: fixes P :: "(('d, 'f) vec, 'f) vec"   and A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e" shows "snd (Gauss_Jordan_in_ij_PA (P, A) i j) = Gauss_Jordan_in_ij A i j"
constants_ax: shows "0 \<le> \<Delta> \<and> 0 < np"