hcju/isabelleps huggingface dataset:
isTrue_implies_bprv_PPf: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and isTrue :: "'f \<Rightarrow> bool"   and enc :: "'f \<Rightarrow> 'e"   and "proof" :: "'g set"   and prfOf :: "'g \<Rightarrow> 'f \<Rightarrow> bool"   and encPf :: "'g \<Rightarrow> 'e"   and Pf :: "'f"   and n :: "'e"   and \<phi> :: "'f" assumes "Minimal_Truth_Soundness_Proof_Repr var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv bprv isTrue enc proof prfOf encPf Pf"   and "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (CleanRepr_Proofs.PPf var Var FvarsT Fvars subst Pf n (enc \<phi>))" shows "bprv (CleanRepr_Proofs.PPf var Var FvarsT Fvars subst Pf n (enc \<phi>))"
nth_map: fixes n :: "nat"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "n < List.length xs" shows "map f xs ! n = f (xs ! n)"
nsumMulElmL: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and n :: "nat" assumes "\<forall>i. f i \<in> carrier R"   and "x \<in> carrier R" shows "x \<cdot>\<^sub>r \<Sigma>\<^sub>e R f n = \<Sigma>\<^sub>e R (\<lambda>i. x \<cdot>\<^sub>r f i) n"
distinct: shows "distinct (map fst (l ! bounded_hashcode_nat (length l) k))"
power2_norm_eq_cinner: fixes x :: "'a" shows "(complex_of_real (norm x))\<^sup>2 = cinner x x"
d: shows "matrix_sum d (\<lambda>k. adjoint (M k) * P k * M k) n \<in> carrier_mat d d"
sup_mono: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "a \<le> c"   and "b \<le> d" shows "a \<squnion> b \<le> c \<squnion> d"
shiftl_t2n: fixes w :: "'a word"   and n :: "nat" shows "w << n = 2 ^ n * w"
simplicial_subdivision_shrinks: fixes p :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int"   and k :: "nat"   and d :: "real"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real"   and x :: "nat \<Rightarrow> real"   and y :: "nat \<Rightarrow> real" assumes "simplicial_chain p S c"   and "\<And>f x y. \<lbrakk>f \<in> Poly_Mapping.keys c; x \<in> standard_simplex p; y \<in> standard_simplex p\<rbrakk> \<Longrightarrow> \<bar>f x k - f y k\<bar> \<le> d"   and "f \<in> Poly_Mapping.keys (simplicial_subdivision p c)"   and "x \<in> standard_simplex p"   and "y \<in> standard_simplex p" shows "\<bar>f x k - f y k\<bar> \<le> real p / real (Suc p) * d"
pconj_mono: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" assumes "a \<le> b"   and "c \<le> d" shows "a .& c \<le> b .& d"
gd_powerprod_def: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "gd_powerprod ord ord_strict \<equiv> ordered_powerprod ord ord_strict"
rsquarefree_def: fixes p :: "'b poly" shows "rsquarefree p = (p \<noteq> 0 \<and> (\<forall>a. order a p = 0 \<or> order a p = 1))"
rel_fun_pos_distr_cond_def: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and A' :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and uu :: "('j \<times> 'k \<times> 'l) itself" shows "rel_fun_pos_distr_cond A A' uu = (\<forall>B B'. (A ===> B) OO (A' ===> B') \<le> A OO A' ===> B OO B')"
expand_set: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and g :: "'d \<Rightarrow> 'c" assumes "comm_monoid_fun f z" shows "comm_monoid_fun.G f z g = comm_monoid_set.F f z g {a. g a \<noteq> z}"
stopping_timeD: fixes F :: "'c \<Rightarrow> 'd measure"   and T :: "'d \<Rightarrow> 'c"   and t :: "'c" assumes "stopping_time F T" shows "Measurable.pred (F t) (\<lambda>x. T x \<le> t)"
constr: fixes vs :: "sterm list"   and name :: "name" assumes "list_all is_value vs"   and "name |\<in>| C" shows "is_value (name $$ vs)"
divide_mod_ring_def: fixes x :: "'b mod_ring"   and y :: "'b mod_ring" shows "x / y = x * (if y = 0 then 0 else y ^ nat (int (CARD('b) - 2)))"
cod_tuple: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "cod (tuple f g) = local.prod (cod f) (cod g)"
fi: fixes n :: "'a" shows "f n integrable_on cbox a b"
subset_sorted_correct: fixes l1 :: "'b list"   and l2 :: "'b list" assumes "distinct l1 \<and> sorted l1"   and "distinct l2 \<and> sorted l2" shows "subset_sorted l1 l2 = (set l1 \<subseteq> set l2)"
partUnfold: fixes P :: "pi"   and Q :: "pi"   and s :: "(name \<times> name) list" assumes "P \<sim>\<^sup>s Q" shows "P[
] \<sim>\<^sup>s Q[
]"
M_gt_6: fixes p :: "nat" assumes "mersenne_prime p" shows "6 < 2 ^ p - 1"
coinduct: fixes f :: "'a \<Rightarrow> 'a"   and X :: "'a" assumes "mono f"   and "X \<le> f (X \<squnion> gfp f)" shows "X \<le> gfp f"
p'_has_vector_derivative: fixes a :: "real"   and v :: "real"   and t :: "real"   and s :: "real set" shows "(movement.p' a v has_vector_derivative movement.p'' a t) (at t within s)"
se_star_mono_for_states_eq: fixes c1 :: "('c, 'd) conf"   and c2 :: "('c, 'd) conf"   and ls :: "('c, 'd) label list"   and c1' :: "('c, 'd) conf"   and c2' :: "('c, 'd) conf" assumes "states c1 = states c2"   and "se_star c1 ls c1'"   and "se_star c2 ls c2'" shows "states c2' = states c1'"
codomains_simp: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "codomains f = {cod f}"
AR_imp_absolute_retract_UNIV: fixes S :: "'a set"   and S' :: "'b set" assumes "AR S"   and "S homeomorphic S'"   and "closed S'" shows "S' retract_of UNIV"
pp_def: shows "pp = smult (inverse (lead_coeff pf)) pf"
Aexp_body_below_Aheap: fixes e :: "exp"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "\<A>\<^bsub>a\<^esub> e \<sqsubseteq> Aheap \<Gamma> e\<cdot>a \<squnion> \<A>\<^bsub>a\<^esub> (Terms.Let \<Gamma> e)"
stp1: shows "steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Bk # Oc # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stpa = (t_twice_len + 14, Bk # Bk # Bk \<up> lna @ Oc # ires, Oc \<up> Suc (rs + 1) @ Bk \<up> rna)"
z2mm2z: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d"   and f :: "ZF" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>"   and "f |\<in>| (Hom\<^bsub>C\<^esub> X Y)" shows "z2m\<^bsub>C\<^esub>f maps\<^bsub>C\<^esub> X to Y \<and> m2z\<^bsub>C\<^esub>z2m\<^bsub>C\<^esub>f = f"
times: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ring_ops ops R" shows "(R ===> R ===> R) (arith_ops_record.times ops) (*)"
cfun_belowI: fixes f :: "'c \<rightarrow> 'd"   and g :: "'c \<rightarrow> 'd" assumes "\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x" shows "f \<sqsubseteq> g"
hyps: assumes "\<And>k. k < n \<Longrightarrow> f k = f' k" shows "matrix_sum d f n = matrix_sum d f' n"
inv_inv: fixes x :: "'a" assumes "x \<in> carrier C" shows "inv\<^bsub>C\<^esub> (inv\<^bsub>C\<^esub> x) = x"
simps_1: fixes x :: "Preliminaries.var" shows "tT (Var x) = (if unprot (TE.tpOfV x) then Var x else Fn (Tag (TE.tpOfV x)) [Var x])"
deadlock_def: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "deadlock s \<equiv> (\<forall>t x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> \<not> final x \<and> wset s t = None \<longrightarrow> t \<turnstile> \<langle>x, shr s\<rangle> \<wrong> \<and> (\<forall>LT. t \<turnstile> \<langle>x, shr s\<rangle> LT \<wrong> \<longrightarrow> (\<exists>lt\<in>LT. must_wait s t lt (dom (thr s))))) \<and> (\<forall>t x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (\<exists>l. 0 < ln $ l) \<and> \<not> waiting (wset s t) \<longrightarrow> (\<exists>l t'. 0 < ln $ l \<and> t \<noteq> t' \<and> thr s t' \<noteq> None \<and> has_lock (locks s $ l) t')) \<and> (\<forall>t x w. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<longrightarrow> wset s t \<noteq> \<lfloor>PostWS w\<rfloor>)"
elementary_category_with_binary_products_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "elementary_category_with_binary_products C pr0 pr1 \<equiv> category C \<and> elementary_category_with_binary_products_axioms C pr0 pr1"
resolution_class_size2: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and partition :: "'b set multiset multiset"   and u_block_size :: "int"   and S :: "'b set multiset" assumes "r_block_design point_set block_collection partition u_block_size"   and "S \<in># partition" shows "int (size S) = int (card point_set) div u_block_size"
lta_not_conga: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F" shows "\<not> Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F"
cases: fixes a :: "trm" assumes "SN a" obtains t :: "trm" where "a = t" and "\<And>t'. t \<mapsto> t' \<Longrightarrow> SN t'"
hyp: shows "\<forall>i<k. (\<exists>c' s'. f (i + 1) = (Seq c' c\<^sub>2, s')) \<and> \<Gamma>\<turnstile> head (f i) \<rightarrow> head (f (i + 1))"
one_N_def: shows "1 \<equiv> Abs_N (0::'a)"
HomFtorContra'_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" shows "HomP\<^bsub>C\<^esub>[\<midarrow>,X] \<equiv> \<lparr>CatDom = Op C, CatCod = SET, MapM = \<lambda>g. HomC\<^bsub>C\<^esub>[g,X]\<rparr>"
lfilter_lappend_lfinite: fixes xs :: "'b llist"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b llist" assumes "lfinite xs" shows "lfilter P (lappend xs ys) = lappend (lfilter P xs) (lfilter P ys)"
has_bochner_integral_mult_left: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. f x * c) (x * c)"
aform_errE: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real" assumes "x \<in> aform_err e X" shows "\<bar>x - aform_val e (fst X)\<bar> \<le> snd X"
Y_B: fixes n :: "nat" shows "Y n \<subseteq> B \<Gamma>"
frees: fixes i :: "'a"   and f :: "('a + 'b, 'b) trm" assumes "SFunctions \<sigma> i = Some f" shows "dfree f"
ideD_1: fixes a :: "'a" assumes "C.ide a" shows "C.arr a"
restrict_UNIV: fixes f :: "'a \<Rightarrow> 'b" shows "restrict f UNIV = f"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Arr (t \<^bold>\<star> u) = HSeq t u"
assms: shows "(l, negate l) \<in> (imp_graph cnf)\<^sup>*"
mrg_right_vwb_lens: shows "vwb_lens mrg_right"
simps_1: fixes A :: "'c \<times> ('d option \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)" shows "NAe.steps A [] = (eps A)\<^sup>*"
wf_t: shows "wf\<^sub>t\<^sub>r\<^sub>m t"
l_neq_1_exp_neq_0: fixes G :: "('d, 'e) cyclic_group_scheme"   and l :: "'d"   and t :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "l \<in> carrier G"   and "l \<noteq> \<one>\<^bsub>G\<^esub>"   and "l = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> t" shows "t \<noteq> 0"
induct: fixes x :: "nat set"   and k :: "nat"   and P :: "nat set \<Rightarrow> bool" assumes "x \<in> i_set_mult k"   and "\<And>n. P [n\<dots>]"   and "\<And>n. P [\<dots>n]"   and "\<And>n d. P [n\<dots>,d]"   and "\<And>r m. P [ r, mod m * k ]"   and "\<And>r m c. P [ r, mod m * k, c ]" shows "P x"
fmdom_add: fixes m :: "'a \<rightharpoonup>\<^sub>f 'b"   and n :: "'a \<rightharpoonup>\<^sub>f 'b" shows "fmdom (m ++\<^sub>f n) = fmdom m |\<union>| fmdom n"
get_M_pure: fixes ptr :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj"   and h :: "(_) heap" shows "pure (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter) h"
nat_term_compare_DRLEX: shows "nat_term_compare DRLEX = deg_comp (pot_comp (\<lambda>x y. lex_comp y x))"
hd_pref: fixes w :: "'b list" assumes "w \<noteq> \<epsilon>" shows "[hd w] \<le>p w"
WT_enm: fixes X :: "bool list set"   and \<eta> :: "nat" assumes "X \<noteq> {}" shows "\<I>_uniform (vld \<eta>) UNIV, \<I>_uniform (vld \<eta>) X \<oplus>\<^sub>\<I> \<I>_uniform (X \<times> vld \<eta>) UNIV \<turnstile>\<^sub>C MAC.enm \<eta> \<surd>"
adds_minus: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and s :: "'d"   and t :: "'d" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus s t" shows "plus (minus t s) s = t"
rhs: shows "(\<Sum>x\<in>set_mset (add_mset y A). x * count (add_mset y A) x) = (\<Sum>x\<in>insert y (set_mset A). x * count (add_mset y A) x)"
simps_2: fixes f :: "'a \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'e"   and g :: "('a \<times> 'f) list \<Rightarrow> ('a \<times> 'e) list"   and h :: "('a \<times> 'g) list \<Rightarrow> ('a \<times> 'e) list"   and v :: "'a \<times> 'g"   and va :: "('a \<times> 'g) list" shows "map2_val_pair f g h [] (v # va) = h (v # va)"
def_prg_Init: fixes F :: "'b program"   and init :: "'b set"   and acts :: "('b \<times> 'b) set set"   and allowed :: "('b \<times> 'b) set set" assumes "F = mk_total_program (init, acts, allowed)" shows "Init F = init"
s_Unroll: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Unroll e\<ggreater> m\<rightarrow> \<lless>\<pi>', Unroll e'\<ggreater>"
atLeastatMost_subset_iff: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset"   and d :: "'a multiset" shows "(subset_mset.atLeastAtMost a b \<subseteq> subset_mset.atLeastAtMost c d) = (\<not> a \<subseteq># b \<or> c \<subseteq># a \<and> b \<subseteq># d)"
pcr_cr_eq: shows "pcr_alist (=) (=) = cr_alist"
agree_nil: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec" shows "Vagree \<nu> \<omega> {}"
wellformed: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>D" shows "b \<subseteq> bl"
indRelR_impl_exists_source_target_relation_2: fixes PredB :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>(P, Q)\<in>indRelR. PredB (P, Q)" shows "\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> (\<forall>(P, Q)\<in>Rel. PredB (P, Q))"
S6: fixes x :: "i" shows "x \<cdot> (dom x) \<cong> x"
sum_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> less zero (f i)" shows "less zero (comm_monoid_add.sum plus zero f I)"
\<tau>Runs_table_into_\<tau>Runs: fixes s :: "'s1"   and stlsss :: "('tl1 \<times> 's1, 's1 option) tllist" assumes "trsys1.\<tau>Runs_table s stlsss" shows "s \<Down>1 tmap fst id stlsss"
case_1: fixes f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
prems_3: fixes i :: "nat" assumes "lnth x i = y" shows "\<exists>m>i. lnth x m = y"
int_decode_def: fixes n :: "nat" shows "int_decode n = (case sum_decode n of Inl x \<Rightarrow> int x | Inr b \<Rightarrow> - int b - 1)"
distinct_orbit_list': fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "permutation f" shows "distinct (orbit_list f x)"
has_derivative_eq_rhs: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'f"   and F :: "'e filter"   and g' :: "'e \<Rightarrow> 'f" assumes "(f has_derivative f') F"   and "f' = g'" shows "(f has_derivative g') F"
last_mem_excluded: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "d_gchain G (Suc n) f"   and "f n \<noteq> f (Suc n)" shows "f (Suc n) \<notin> f ` {i. i \<le> n}"
prod_sum_distrib: fixes I :: "'a set"   and J :: "'a \<Rightarrow> 'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> finite (J i)" shows "(\<Prod>i\<in>I. sum (f i) (J i)) = (\<Sum>m\<in>Pi\<^sub>E I J. \<Prod>i\<in>I. f i (m i))"
finite_divisor_powers: fixes y :: "'a"   and x :: "'a" assumes "y \<noteq> (0::'a)"   and "\<not> is_unit x" shows "finite {n. x ^ n dvd y}"
ext_inject: fixes sprFst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprLst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprCRel :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set"   and more :: "'f"   and sprFst' :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprLst' :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprCRel' :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set"   and more' :: "'f" shows "(\<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = \<lparr>sprFst = sprFst', sprLst = sprLst', sprCRel = sprCRel', \<dots> = more'\<rparr>) = (sprFst = sprFst' \<and> sprLst = sprLst' \<and> sprCRel = sprCRel' \<and> more = more')"
assms_2: fixes n :: "nat" shows "u n \<in> C"
sum_constant_scaleR: fixes y :: "'b"   and A :: "'c set" shows "(\<Sum>x\<in>A. y) = real (card A) *\<^sub>R y"
c_f_pair_def: shows "c_f_pair = (\<lambda>f g x. c_pair (f x) (g x))"
Subset_I: fixes i :: "name"   and t :: "tm"   and H :: "fm set"   and u :: "tm" assumes "insert (Var i IN t) H \<turnstile> Var i IN u"   and "atom i \<sharp> (t, u)"   and "\<forall>B\<in>H. atom i \<sharp> B" shows "H \<turnstile> t SUBS u"
intros_2: fixes uv :: "'d" shows "Le uv \<prec> \<infinity>"
Ord_cases: fixes k :: "V" assumes "Ord k" obtains "k = 0"   | l :: "V" where "Ord l" and "ZFC_in_HOL.succ l = k"   | "Limit k"
pac_ideal_remap_eq: fixes b :: "('a, int mpoly) fmap"   and ba :: "('a, int mpoly) fmap" assumes "dom_m b = dom_m ba"   and "\<forall>i\<in>#dom_m ba. the (fmlookup b i) - the (fmlookup ba i) \<in> More_Modules.ideal polynomial_bool" shows "pac_ideal ((\<lambda>x. the (fmlookup b x)) ` set_mset (dom_m ba)) = pac_ideal ((\<lambda>x. the (fmlookup ba x)) ` set_mset (dom_m ba))"
lr_eta_rep_def: shows "lr_eta_rep \<equiv> \<lambda>r. {(e, e'). e = \<bottom> \<and> e' = \<bottom> \<or> e = ValTT \<and> e' = unitK\<cdot>ValKTT \<or> e = ValFF \<and> e' = unitK\<cdot>ValKFF \<or> (\<exists>n. e = ValN\<cdot>n \<and> e' = unitK\<cdot>(ValKN\<cdot>n))} \<union> (case r of (rm, rp) \<Rightarrow> {(e, e'). e = \<bottom> \<and> e' = \<bottom> \<or> (\<exists>f f'. e = ValF\<cdot>f \<and> e' = unitK\<cdot>(ValKF\<cdot>f') \<and> (f, f') \<in> unlr (snd rp))})"
in_homI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "D.arr f"   and "D.dom f = a"   and "D.cod f = b" shows "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>"
continuous_on_cstrong_operator_topo_iff_coordinatewise: fixes T :: "'a topology"   and f :: "'a \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "continuous_map T cstrong_operator_topology f = (\<forall>x. continuous_map T euclidean (\<lambda>y. cblinfun_apply (f y) x))"
assms: shows "s' = snd (fst (branch_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
assms_2: shows "z \<bullet> (1::'a) \<le> z' \<bullet> (1::'a)"
terminates_remove1_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and b :: "'f" assumes "terminates g" shows "terminates (remove1_raw b g)"
1: shows "v \<sqinter> (w \<sqinter> y)\<^sup>\<star> * (x \<sqinter> y)\<^sup>\<star> = bot"
coboundedI1: fixes c :: "'c"   and a :: "'c"   and b :: "'c" assumes "c \<le> a" shows "c \<le> max a b"
case_prodD': fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'e"   and b :: "'f"   and c :: "'g" assumes "(case (a, b) of (c, d) \<Rightarrow> R c d) c" shows "R a b c"
scale_zero_left: fixes x :: "'c" shows "0 *\<^sub>R x = (0::'c)"
rSym: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> {uu_. \<exists>\<Psi> P P' Q' Q. uu_ = (\<Psi>, P, Q) \<and> \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}" shows "(\<Psi>, Q, P) \<in> {uu_. \<exists>\<Psi> P P' Q' Q. uu_ = (\<Psi>, P, Q) \<and> \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}"
a_is_strongly_dom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "a_is_strongly_dom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_dom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> a_get_dom_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_dom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> a_get_dom_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> added_pointers \<subseteq> arg_components' \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_dom_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
l_get_parent_wf_axioms: shows "l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent"
x_lim: fixes a :: "'a" shows "(\<lambda>n. x n a) \<longlonglongrightarrow> l a"
build_rule_index_sf_def: fixes \<delta> :: "('a, 'b) ta_rule dlist" shows "build_rule_index_sf \<delta> \<equiv> build_rule_index (\<lambda>r. (lhs r, rhsl r)) \<delta>"
forward_lockstep_simulation: fixes s1 :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state"   and s1' :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state"   and s2 :: "('fenv_ubx, 'henv, ('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame) state" assumes "Sinca.step s1 s1'"   and "s1 \<sim> s2" shows "\<exists>s2'. Subx.step s2 s2' \<and> s1' \<sim> s2'"
TR2_mem: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme"   and m :: "'f"   and n1 :: "'h"   and n2 :: "'h" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N"   and "m \<in> carrier M"   and "n1 \<in> carrier N"   and "n2 \<in> carrier N" shows "(m, n1 \<plusminus>\<^bsub>N\<^esub> n2) \<plusminus>\<^bsub>P\<^esub> -\<^sub>a\<^bsub>P\<^esub> ((m, n1) \<plusminus>\<^bsub>P\<^esub> (m, n2)) \<in> TR\<^bsub>R\<^esub> M N P"
of_int_0: shows "of_int 0 = (0::'b)"