hcju/isabelleps huggingface dataset:
prime_factorization_unique_int: fixes S :: "int set"   and f :: "int \<Rightarrow> nat"   and n :: "int" assumes "S = {p. 0 < f p}"   and "finite S"   and "\<forall>p\<in>S. prime p"   and "\<bar>n\<bar> = (\<Prod>p\<in>S. p ^ f p)" shows "S = prime_factors n \<and> (\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
fds_truncate_zero: fixes m :: "nat" shows "fds_truncate m 0 = 0"
double_negation: fixes P :: "'d upred" shows "(\<not> \<not> P) = P"
qpWP: shows "is_quantum_predicate (wlp (While M S) P)"
max2_among: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "local.max2 a b \<in> {a, b}"
derivativeEqvtI2: fixes P :: "pi"   and Q :: "pi"   and a :: "subject"   and x :: "name"   and Rel :: "(pi \<times> pi) set"   and perm :: "(name \<times> name) list" assumes "derivative P Q a x Rel"   and "eqvt Rel" shows "derivative (perm \<bullet> P) (perm \<bullet> Q) a (perm \<bullet> x) Rel"
abs_eq: fixes xa :: "int"   and x :: "int" shows "gcd (integer_of_int xa) (integer_of_int x) = integer_of_int (gcd xa x)"
simps: fixes t :: "'a trm" shows "trm_rep t = (\<lambda>S. if is_compound t \<and> (lhs t, t) \<in> trm_ord \<and> (rhs t, t) \<in> trm_ord \<and> (((lhs t, t) \<in> trm_ord \<longrightarrow> trm_rep (lhs t) S \<noteq> lhs t) \<or> ((rhs t, t) \<in> trm_ord \<longrightarrow> trm_rep (rhs t) S \<noteq> rhs t)) then if (trm_rep (lhs t) S \<cdot> trm_rep (rhs t) S, t) \<in> trm_ord then trm_rep (trm_rep (lhs t) S \<cdot> trm_rep (rhs t) S) S else t else get_min t {pair. \<exists>z CC C' C s L L' \<sigma> t' s'. pair = (z, s) \<and> CC \<in> S \<and> t \<notin> subst_set (trms_ecl CC) \<sigma> \<and> (\<forall>x. (\<exists>x'\<in>trms_ecl CC. occurs_in x (x' \<lhd> \<sigma>)) \<longrightarrow> (x, t) \<in> trm_ord \<longrightarrow> trm_rep x S = x) \<and> C' = cl_ecl CC \<and> (s, t) \<in> trm_ord \<and> ((s, t) \<in> trm_ord \<longrightarrow> z = trm_rep s S) \<and> orient_lit_inst L' t' s' pos \<sigma> \<and> sel C' = {} \<and> L' \<in> C' \<and> maximal_literal L C \<and> L = subst_lit L' \<sigma> \<and> C = subst_cl C' \<sigma> \<and> ground_clause C \<and> t = t' \<lhd> \<sigma> \<and> s = s' \<lhd> \<sigma> \<and> finite C' \<and> (\<forall>L u v. (L \<in> C \<longrightarrow> orient_lit L u v pos \<longrightarrow> (u, t) \<in> trm_ord \<longrightarrow> (v, t) \<in> trm_ord \<longrightarrow> trm_rep u S \<noteq> trm_rep v S) \<and> (L \<in> C \<longrightarrow> orient_lit L u v neg \<longrightarrow> (u, t) \<in> trm_ord \<longrightarrow> (v, t) \<in> trm_ord \<longrightarrow> trm_rep u S = trm_rep v S)) \<and> (\<forall>s''. eq_occurs_in_cl t s'' (C' - {L'}) \<sigma> \<longrightarrow> (s'', t) \<in> trm_ord \<longrightarrow> (s, t) \<in> trm_ord \<longrightarrow> trm_rep s S \<noteq> trm_rep s'' S)})"
ent_fwd: fixes P :: "assn"   and h :: "heap \<times> nat set"   and Q :: "assn" assumes "h \<Turnstile> P"   and "P \<Longrightarrow>\<^sub>A Q" shows "h \<Turnstile> Q"
oa_facts_6: fixes v :: "i"   and x :: "\<nu>" shows "[\<^bold>\<diamond>\<lparr>A!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<box>\<lparr>A!,x\<^sup>P\<rparr> in v]"
that: fixes n :: "nat" assumes "arity_var\<^sub>h f = of_nat n" shows "thesis"
comparator_of_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "linorder.comparator_of less x y = (if less x y then Lt else if x = y then Eq else Gt)"
in_implication_frontier_in_implied_frontier_alt: fixes c :: "('loc, 't) configuration"   and b :: "'t"   and loc2 :: "'loc" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "worklists_vacant_to c b"   and "b \<in>\<^sub>A frontier (c_imp c loc2)" shows "b \<in>\<^sub>A implied_frontier_alt c loc2"
wp_def: fixes x :: "('a \<times> 'a) set"   and p :: "('a \<times> 'a) set" shows "rel_aka.wp x p = rel_ad (x ; rel_ad p)"
may_lock_release_all_conv: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "may_lock (release_all l t) t = may_lock l t"
lossless_1: shows "plossless_gpv \<I>' (run_converter conv a)"
fcon_im1: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "continuous_on ((\<lambda>t. (g x, h t)) ` {0..1}) (\<lambda>(x, y). f x y)"
simps_5: fixes m :: "'b match_expr" shows "normalize_match (MatchNot (MatchNot m)) = normalize_match m"
v_AbsV: fixes e :: "db" shows "val (DBAbsV e)"
wellorder_Least_lemma_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
in_face_cycle_setD: fixes y :: "'b"   and x :: "'b" assumes "y \<in> G.face_cycle_set x"   and "x \<in> arcs G" shows "y \<in> arcs G"
Above_Linear_singleton: fixes x :: "'d"   and r :: "('d \<times> 'd) set" assumes "x \<in> Field r"   and "Linear_order r" shows "x \<in> Above r {x}"
is_Arg_exp_Im: fixes z :: "complex" shows "is_Arg (exp z) (Im z)"
a1: shows "s' = snd (fst (read_state_reg_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
check_cond_def: fixes a :: "nat list"   and b :: "nat list" shows "check_cond a b = (\<lambda>(x, y). static_bounds a b x y \<and> a \<bullet> x = b \<bullet> y \<and> boundr_impl a b x y \<and> subdprodl_impl a b x y \<and> subdprodr_impl a b y)"
lset_lnths: fixes xs :: "'b llist"   and I :: "nat set" shows "lset (lnths xs I) = {xs $ i |i. enat i < llength xs \<and> i \<in> I}"
pow_l'': shows "2 ^ (lv p' d - l) = 2 * 2 ^ (lv p d - l)"
inject_6: fixes x61 :: "('a, 'b, 'c) hp"   and x62 :: "('a, 'b, 'c) formula"   and y61 :: "('a, 'b, 'c) hp"   and y62 :: "('a, 'b, 'c) formula" shows "((\<langle> x61 \<rangle> x62) = (\<langle> y61 \<rangle> y62)) = (x61 = y61 \<and> x62 = y62)"
AA4: shows "Esigma (3\<^sup>2 * 7 * 13) = 13 * 8 * 14"
simps: fixes a :: "('test, 'atom) com" shows "siso0 a = (\<exists>c. a = c \<and> (\<forall>x xa xb. mustT c x \<longrightarrow> (c, x) \<rightarrow>c (xa, xb) \<longrightarrow> siso0 xa) \<and> (\<forall>x xa xb xc. mustT c x \<longrightarrow> mustT c xa \<longrightarrow> x \<approx> xa \<longrightarrow> (c, x) \<rightarrow>c (xb, xc) \<longrightarrow> (\<exists>t'. xc \<approx> t' \<and> (c, xa) \<rightarrow>c (xb, t'))) \<and> (\<forall>x xa xb. mustT c x \<longrightarrow> mustT c xa \<longrightarrow> x \<approx> xa \<longrightarrow> (c, x) \<rightarrow>t xb \<longrightarrow> (\<exists>t'. xb \<approx> t' \<and> (c, xa) \<rightarrow>t t')))"
add_pow_def: fixes R :: "('a, 'b) ring_scheme"   and k :: "'c"   and a :: "'a" shows "[k] \<cdot>\<^bsub>R\<^esub> a = a [^]\<^bsub>add_monoid R\<^esub> k"
simps_2: fixes s :: "'c \<Rightarrow> 'd option"   and ops :: "('c, 'd) sas_plus_operator list"   and opss :: "('c, 'd) sas_plus_operator list list" shows "execute_parallel_plan_sas_plus s (ops # opss) = (if are_all_operators_applicable_in s ops \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent ops then execute_parallel_plan_sas_plus (execute_parallel_operator_sas_plus s ops) opss else s)"
Inter_Un_distrib: fixes A :: "'b set set"   and B :: "'b set set" shows "\<Inter> (A \<union> B) = \<Inter> A \<inter> \<Inter> B"
subspace_0: fixes S :: "'b set" assumes "subspace S" shows "(0::'b) \<in> S"
le: fixes x :: "'a" shows "spmf p x \<le> spmf q x"
pos_bounded: shows "\<exists>K>0. \<forall>a b. norm (a *\<^sub>V b) \<le> norm a * norm b * K"
map_carrier_mat: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c mat"   and nr :: "nat"   and nc :: "nat" shows "(map_mat f A \<in> carrier_mat nr nc) = (A \<in> carrier_mat nr nc)"
bind_spmf_spmf_of_set: fixes A :: "'c set" assumes "finite A"   and "A \<noteq> {}" shows "(\<bind>) (spmf_of_set A) = (\<bind>) (pmf_of_set A)"
A3implT2_local: shows "\<lfloor>(\<lambda>w. (\<forall>x xa. (\<forall>xa. (x xa \<^bold>\<rightarrow> \<P> xa) w) \<and> (\<^bold>\<box>(\<lambda>v. \<forall>xb. xa xb v = (\<forall>xa. (x xa \<^bold>\<rightarrow> xa xb) v))) w \<longrightarrow> \<P> xa w) \<longrightarrow> \<P> (\<lambda>x w. \<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) w) w)\<rfloor>"
freess_single: fixes x :: "'a" shows "freess [x] = frees x"
Graph_E_def: fixes E :: "'b set set" shows "Graph_E E \<equiv> finite E \<and> (\<forall>e. e \<in> E \<longrightarrow> card e = 2)"
indicator_vimage: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'e set"   and x :: "'d" shows "indicator (f -` A) x = indicator A (f x)"
exhaust: fixes y :: "bit" obtains "y = Zero"   | "y = One"
comm_semiring_hom_axioms: shows "comm_semiring_hom of_real"
unique_exprTr7p1: fixes H :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and s :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'b" assumes "free_generator R M H"   and "\<forall>f s g t m. f \<in> {j. j \<le> n} \<rightarrow> H \<and> inj_on f {j. j \<le> n} \<and> s \<in> {j. j \<le> n} \<rightarrow> carrier R \<and> g \<in> {j. j \<le> m} \<rightarrow> H \<and> inj_on g {j. j \<le> m} \<and> t \<in> {j. j \<le> m} \<rightarrow> carrier R \<and> l_comb R M n s f = l_comb R M m t g \<and> (\<forall>j\<le>n. s j \<noteq> \<zero>\<^bsub>R\<^esub>) \<and> (\<forall>k\<le>m. t k \<noteq> \<zero>\<^bsub>R\<^esub>) \<longrightarrow> n = m \<and> (\<exists>h. h \<in> {j. j \<le> n} \<rightarrow> {j. j \<le> n} \<and> (\<forall>l\<le>n. cmp f h l = g l \<and> cmp s h l = t l))"   and "f \<in> {j. j \<le> Suc n} \<rightarrow> H"   and "s \<in> {j. j \<le> Suc n} \<rightarrow> carrier R"   and "g \<in> {j. j \<le> Suc n} \<rightarrow> H"   and "t \<in> {j. j \<le> Suc n} \<rightarrow> carrier R"   and "l_comb R M (Suc n) s f = l_comb R M (Suc n) t g"   and "\<forall>j\<le>Suc n. s j \<noteq> \<zero>\<^bsub>R\<^esub>"   and "\<forall>k\<le>Suc n. t k \<noteq> \<zero>\<^bsub>R\<^esub>"   and "inj_on f {j. j \<le> Suc n}"   and "inj_on g {j. j \<le> Suc n}"   and "f (Suc n) = g (Suc n)" shows "\<exists>h. h \<in> {j. j \<le> Suc n} \<rightarrow> {j. j \<le> Suc n} \<and> (\<forall>l\<le>Suc n. cmp f h l = g l \<and> cmp s h l = t l)"
reduce_system_matrix_equation_preserved_R: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "all_list_constr_R subsets (length qs)"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)" shows "satisfy_equation_R p qs (get_subsets_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) (get_signs_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs)))"
commutative: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
loop_sem: fixes I :: "('e, 'f, 'd) interp"   and \<alpha> :: "('e, 'f, 'd) hp" shows "prog_sem I \<alpha>** = (prog_sem I \<alpha>)\<^sup>*"
sbelows_rel_def: fixes A :: "('c \<times> 'd) set" shows "\<langle>A\<rangle>sbelows_rel = \<langle>\<langle>A\<rangle>sctn_rel\<rangle>list_set_rel O br sbelow_halfspaces top"
hyps_2: shows "b \<rightarrow>\<^sub>\<eta> c"
simps_5: fixes x :: "'a rexp"   and y :: "'a rexp" shows "lang (Times x y) = lang x \<cdot> lang y"
getNeg_NegPos_map_simp2: fixes X :: "'b \<Rightarrow> 'a"   and src :: "'b negation_type list" shows "getNeg (NegPos_map X src) = map X (getNeg src)"
isCont_powr_complex: fixes f :: "'a \<Rightarrow> complex"   and z :: "'a"   and g :: "'a \<Rightarrow> complex" assumes "f z \<notin> \<real>\<^sub>\<le>\<^sub>0"   and "isCont f z"   and "isCont g z" shows "isCont (\<lambda>z. f z powr g z) z"
lt_trans_l: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset"   and D :: "'b multiset"   and E :: "'b multiset"   and F :: "'b multiset" assumes "locally_trans R (A + B) (C + D) (E + F)" shows "locally_trans R A C E"
poly_f_def: fixes ps :: "real poly list" shows "poly_f ps = (if check_all_const_deg ps = True then [:0, 1:] else pderiv (prod_list_var ps) * prod_list_var ps * [:real_of_int (- crb (prod_list_var ps)), 1:] * [:real_of_int (crb (prod_list_var ps)), 1:])"
distinct_17: fixes x61 :: "'a formula"   and x62 :: "'a formula" shows "\<bottom> \<noteq> x61 \<^bold>\<rightarrow> x62"
closure_of_eq: fixes A :: "'a set"   and x :: "'a"   and x' :: "'a" assumes "A \<subseteq> carrier S"   and "x \<in> closure_of A"   and "x' \<in> carrier S"   and "x .= x'" shows "x' \<in> closure_of A"
fwi_impl'_def: fixes M :: "nat \<times> nat \<Rightarrow> 'a"   and k :: "nat" shows "local.fwi_impl' M k = for_rec2 (\<lambda>M. fw_upd' M k) M n n n"
m2zz2mInv: fixes f :: "'b" assumes "f \<in> mor" shows "z2mm2zf = f"
rd_is_CSP2: fixes r :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and p :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "\<forall>a b. r (a, b\<lparr>ok := True\<rparr>) \<longrightarrow> r (a, b\<lparr>ok := False\<rparr>)" shows "R (r \<turnstile> p) is CSP2 healthy"
finfun_upd_apply_other: fixes a :: "'a"   and a' :: "'a"   and f :: "'a \<Rightarrow>f 'b"   and b :: "'b" assumes "a \<noteq> a'" shows "(f(a $:= b)) $ a' = f $ a'"
invers: fixes a :: "nat"   and x :: "nat" assumes "a \<le> x" shows "x = x - a + a"
case_rews_1: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case \<bottom> of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = \<bottom>"
elims_2: fixes x :: "('c \<times> nat) option"   and xa :: "'c"   and xb :: "lock_action" assumes "lock_action_ok x xa xb" obtains l :: "('c \<times> nat) option" and t :: "'c" where "x = la__" and "xa = ta__" and "xb = Lock" and "may_lock la__ ta__"   | l :: "('c \<times> nat) option" and t :: "'c" where "x = lb__" and "xa = tb__" and "xb = Unlock" and "has_lock lb__ tb__"   | l :: "('c \<times> nat) option" and t :: "'c" where "x = lc__" and "xa = tc__" and "xb = UnlockFail" and "\<not> has_lock lc__ tc__"   | l :: "('c \<times> nat) option" and t :: "'c" where "x = l" and "xa = t" and "xb = ReleaseAcquire"
intro: fixes X :: "'d \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "\<And>\<sigma> v u. put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v"   and "\<And>\<sigma> v. get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>"   and "\<And>\<sigma> u. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>" shows "X \<bowtie> Y"
limit_subseteq_reach_2: fixes w :: "nat \<Rightarrow> 'd"   and \<Sigma> :: "'d set"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e"   and q\<^sub>0 :: "'e" assumes "range w \<subseteq> \<Sigma>" shows "limit (run\<^sub>t \<delta> q\<^sub>0 w) \<subseteq> reach\<^sub>t \<Sigma> \<delta> q\<^sub>0"
ApproximationTr2: fixes v :: "'a \<Rightarrow> ant"   and aa :: "'a"   and N :: "nat" assumes "valuation K v"   and "aa \<in> carrier K"   and "aa \<noteq> \<zero>"   and "0 \<le> v aa" shows "int N *\<^sub>a v aa \<le> v (1\<^sub>r \<plusminus> -\<^sub>a (1\<^sub>r \<plusminus> -\<^sub>a aa^\<^bsup>K N\<^esup>)^\<^bsup>K N\<^esup>)"
param_set: fixes Ra :: "('a \<times> 'b) set" assumes "single_valued Ra" shows "(set, set) \<in> \<langle>Ra\<rangle>list_rel \<rightarrow> \<langle>Ra\<rangle>set_rel"
hnorm_of_hypreal: fixes r :: "real star" shows "hnorm (of_hypreal r) = \<bar>r\<bar>"
append_Nil2: fixes xs :: "['b]" shows "xs ++ [] = xs"
the_ext_unique: fixes y' :: "'c"   and g :: "'d"   and f :: "'c" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y y' g"   and "y\<eta>.is_ext y' g f" shows "f = y\<eta>.the_ext y' g"
strict_partial_order_def: fixes r :: "('a \<times> 'a) set" shows "strict_partial_order r \<equiv> trans r \<and> irrefl r"
H: shows "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f T"
function_scalar_mult_def: fixes S :: "'d set"   and R :: "('e, 'f) ring_scheme"   and a :: "'e"   and f :: "'d \<Rightarrow> 'e" shows "function_scalar_mult S R a f = (\<lambda>x\<in>S. a \<otimes>\<^bsub>R\<^esub> f x)"
greaterThanAtMost_empty_iff: fixes k :: "'a"   and l :: "'a" shows "({k<..l} = {}) = (\<not> k < l)"
axioms_6: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" assumes "l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_tag_name set_tag_name_locs set_disconnected_nodes set_disconnected_nodes_locs create_element" shows "l_set_tag_name_get_child_nodes type_wf set_tag_name set_tag_name_locs known_ptr get_child_nodes get_child_nodes_locs"
sumSym: fixes P :: "pi"   and Q :: "pi" shows "P \<oplus> Q \<sim>\<^sup>s Q \<oplus> P"
p\<^sub>1_def: shows "p\<^sub>1 = find_closest p (min \<delta> (dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)))) (p\<^sub>2 # ps)"
aHom_def: fixes A :: "('h, 'j) aGroup_scheme"   and B :: "('i, 'k) aGroup_scheme" shows "aHom A B = {f \<in> carrier A \<rightarrow> carrier B. f \<in> extensional (carrier A) \<and> (\<forall>a\<in>carrier A. \<forall>b\<in>carrier A. f (a \<plusminus>\<^bsub>A\<^esub> b) = f a \<plusminus>\<^bsub>B\<^esub> f b)}"
term_ok'_imp_wf_term: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term" assumes "term_ok' \<Sigma> t" shows "wf_term \<Sigma> t"
push_ty_def: fixes \<tau> :: "ty"   and \<Gamma> :: "(ty \<times> nat) list \<times> nat" shows "push_ty \<tau> \<Gamma> \<equiv> ((\<tau>, snd \<Gamma>) # fst \<Gamma>, snd \<Gamma>)"
has_derivative_inverse_within: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and a :: "'a"   and S :: "'a set"   and g :: "'b \<Rightarrow> 'a"   and g' :: "'b \<Rightarrow> 'a" assumes "(f has_derivative f') (at a within S)"   and "continuous (at (f a) within f ` S) g"   and "a \<in> S"   and "linear g'"   and "g' \<circ> f' = id"   and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x" shows "(g has_derivative g') (at (f a) within f ` S)"
ptensor_mat_dim_row: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_row (st.ptensor_mat m1 m2) = st.d0"
set: shows "\<forall>C''\<in>set Ds. (C', C'') \<in> (subcls1 P)\<^sup>+"
the_inv_into_onto: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "the_inv_into A f ` f ` A = A"
assms_2: fixes xs' :: "('a, 'b) tllist" assumes "xs' \<in> A" shows "local.tllist_ord xs' xs"
linorder_neq_iff: fixes x :: "'g"   and y :: "'g" shows "(x \<noteq> y) = (x < y \<or> y < x)"
assert_simp: fixes \<sigma> :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<sigma> \<Turnstile> assert\<^sub>S\<^sub>E P) = P \<sigma>"
transitively_orientable_def: fixes x :: "'a" shows "transitively_orientable x \<equiv> \<exists>y. orientation x y \<and> transitive y"
psubst_idEnv: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" assumes "good X" shows "X #[idEnv] = X"
absorb_def: fixes j :: "jump"   and a :: "abopt" shows "absorb j a = (if a = Some (Jump j) then None else a)"
fresh_image: fixes A :: "'a set"   and i :: "atom"   and f :: "'a \<Rightarrow> 'b" assumes "finite A" shows "i \<sharp> f ` A = (\<forall>x\<in>A. i \<sharp> f x)"
borel_measurable_cond_exp2: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real" shows "real_cond_exp M F f \<in> borel_measurable M"
crel_vs_elim: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and v :: "'d"   and v\<^sub>T :: "('a, 'e) state"   and M :: "'a" assumes "crel_vs R v v\<^sub>T"   and "cmem M"   and "P M" obtains v' :: "'e" and M' :: "'a" where "run_state v\<^sub>T M = (v', M')" and "R v v'" and "cmem M'" and "P M'"
IH_2: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "(X, Y) \<in> qSwapped" shows "qAFresh zs z Y \<longrightarrow> (\<forall>X'. Y #= X' \<longrightarrow> qFresh zs z X')"
sel_1: fixes r :: "('a \<times> bool) language" shows "\<oo> (Times_Plus r) = \<oo> r"
ofail_NF_wp: fixes Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "ovalidNF (\<lambda>_. False) ofail Q"
simps_13: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and e1' :: "'b aexp"   and e2' :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (Gt e1 e2) (Gt e1' e2') = (e1 < e1' \<or> e1 = e1' \<and> e2 < e2')"
map_1: fixes f :: "'i \<Rightarrow> 'h" shows "map_option f None = None"
field_Units: fixes R :: "('e, 'f) ring_scheme" assumes "field R" shows "Units R = carrier R - {\<zero>\<^bsub>R\<^esub>}"
assm: shows "R_Chop(r,r1,r2) \<and> R_Chop(r1,r3,r4)"
RTranCl_inv: fixes P :: "'c \<Rightarrow> bool"   and succs :: "'c \<Rightarrow> 'c list"   and g :: "'c"   and g' :: "'c" assumes "invariant P succs"   and "g [succs]\<rightarrow>* g'"   and "P g" shows "P g'"
res_lim_in_Zp: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_cauchy s" shows "res_lim s \<in> carrier Zp"
crsp_step_in: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and ins :: "abc_inst" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "crsp ly (as, lm) (s, l, r) ires"   and "abc_fetch as ap = Some ins" shows "\<exists>stp>0. crsp ly (abc_step_l (as, lm) (Some ins)) (steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp) ires"
AA5: shows "Esigma (3\<^sup>2 * 7 * 13 * 107) = 13 * 8 * 14 * 108"
hyps_2: fixes P' :: "('a, 'b, 'c) psi"   and b :: "'d" assumes "M\<lparr>N\<rparr> \<prec> C = M\<lparr>N\<rparr> \<prec> P'" shows "Prop b \<Psi> (P \<parallel> !P) M N P'"
eventually_prod_same: fixes P :: "'c \<times> 'c \<Rightarrow> bool"   and F :: "'c filter" shows "eventually P (F \<times>\<^sub>F F) = (\<exists>Q. eventually Q F \<and> (\<forall>x y. Q x \<longrightarrow> Q y \<longrightarrow> P (x, y)))"
rtrancl3p_converse_induct': fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and bs :: "'b list"   and a'' :: "'a"   and P :: "'a \<Rightarrow> 'b list \<Rightarrow> bool" assumes "rtrancl3p r a bs a''"   and "P a'' []"   and "\<And>a b a' bs. \<lbrakk>rtrancl3p r a' bs a''; r a b a'; P a' bs\<rbrakk> \<Longrightarrow> P a (b # bs)" shows "P a bs"
irreflclp_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "r\<^sup>\<noteq>\<^sup>\<noteq> a b = (r a b \<and> a \<noteq> b)"
mset_merge_all: fixes xs :: "'a list list" shows "mset (local.merge_all xs) = mset (concat xs)"
norm_mult_numeral1: fixes w :: "num"   and a :: "'a" shows "norm (numeral w * a) = numeral w * norm a"