hcju/isabelleps huggingface dataset:
chain_reparam_chain'_def: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "chain_reparam_chain' one_chain1 subdiv \<equiv> \<exists>f. \<Union> (f ` one_chain1) = subdiv \<and> (\<forall>cube\<in>one_chain1. chain_reparam_weak_path (rec_join [cube]) (f cube)) \<and> (\<forall>p\<in>one_chain1. \<forall>p'\<in>one_chain1. p \<noteq> p' \<longrightarrow> f p \<inter> f p' = {}) \<and> (\<forall>x\<in>one_chain1. finite (f x))"
relative_to_subset_trans: fixes P :: "'b set \<Rightarrow> bool"   and U :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "(P relative_to U) S \<and> S \<subseteq> T \<and> T \<subseteq> U" shows "(P relative_to T) S"
eval_notfree_unchanged: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and V :: "char list" assumes "P \<turnstile> \<langle>e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>"   and "V \<notin> fv e" shows "l' V = l V"
assms_3: shows "z \<noteq> \<infinity>\<^sub>h"
roundup_le_7: fixes emBits :: "nat" assumes "0 < emBits" shows "roundup emBits 8 * 8 - emBits \<le> 7"
set_map: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c ltlc" shows "atoms_ltlc (map_ltlc f v) = f ` atoms_ltlc v"
perzeta_conv_hurwitz_zeta_multiplication: assumes "0 < k"   and "s \<noteq> 1" shows "of_nat k powr s * perzeta (real_of_int 0 / real k) s = (\<Sum>n = 1..k. exp (complex_of_real (2 * pi * real n * real_of_int 0 / real k) * \<i>) * hurwitz_zeta (real n / real k) s)"
sample_coprime_def: fixes prime_set :: "nat set"   and N :: "nat" assumes "rsa_base prime_set" shows "sample_coprime N = spmf_of_set (coprime_set N)"
monad_right_unit: fixes m :: "'a\<cdot>'b" shows "m \<bind> return = m"
vfsequence_vlrestriction_vcard: fixes xs :: "V" assumes "vfsequence xs" shows "xs \<restriction>\<^sup>l\<^sub>\<circ> vcard xs = xs"
l_inv: fixes x :: "'c" assumes "x \<in> carrier H" shows "inv\<^bsub>H\<^esub> x \<otimes>\<^bsub>H\<^esub> x = \<one>\<^bsub>H\<^esub>"
default_val_not_Addr: fixes T :: "ty"   and a :: "'b" shows "default_val T \<noteq> Addr a"
alt_group_inv_equality: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p \<in> carrier (alt_group n)" shows "inv\<^bsub>alt_group n\<^esub> p = inv' p"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. nat = Suc x2 \<longrightarrow> P (f2 x2)))"
\<T>o_e_induces_transformation: fixes e :: "'s" assumes "e \<in> F.SET a" shows "transformation_by_components (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F (\<T>o e)"
independent_explicit_module: fixes s :: "'b set" shows "local.independent s = (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. u v *s v) = (0::'b) \<longrightarrow> v \<in> t \<longrightarrow> u v = (0::'a))"
indexToVertexList_pre_subdivFace_face: fixes f :: "face"   and v :: "nat"   and es :: "nat list" assumes "\<not> final f"   and "distinct (vertices f)"   and "v \<in> \<V> f"   and "2 < |es|"   and "incrIndexList es |es| |vertices f|" shows "pre_subdivFace_face f v (indexToVertexList f v es)"
ec: shows "ec \<in> set (KBP.k_succs a (runJP k_mkAlgAuto t a))"
dickson_gradingD_dgrad_set: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "dickson_grading d" shows "almost_full_on (adds) (dgrad_set d m)"
order_iff: fixes a :: "'a upred"   and b :: "'a upred" shows "(b \<sqsubseteq> a) = (a = (a \<and> b))"
minarc_meet_bot_minarc_iff: fixes x :: "'a" shows "(minarc x \<sqinter> x = bot) = (minarc x = bot)"
completeness_0_lemma: fixes p :: "'b pl"   and t0 :: "'b set" assumes "{} |= p" shows "\<forall>t. hyps p t - hyps p t0 |- p"
simps_4: fixes t1 :: "('a, 'b) trm"   and t2 :: "('a, 'b) trm" shows "SIGT (Plus t1 t2) = SIGT t1 \<union> SIGT t2"
wf_world_model_def: fixes M :: "object atom formula set" shows "wf_world_model M = (\<forall>f\<in>M. wf_fmla_atom objT f)"
Let_def: fixes s :: "'y"   and f :: "'y \<Rightarrow> 'x" shows "Let s f \<equiv> f s"
left_quotient_def: fixes u :: "'b list"   and v :: "'b list" shows "u\<inverse>\<^sup>>v = (if u \<le>p v then THE z. u \<cdot> z = v else undefined)"
matrix_to_iarray_cofactorM: fixes A :: "(('c, 'd) vec, 'd) vec" shows "matrix_to_iarray (cofactorM A) = cofactorM_iarrays (matrix_to_iarray A)"
fsubset_antisym: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| A" shows "A = B"
entt_refl: fixes A :: "assn" shows "A \<Longrightarrow>\<^sub>t A"
max_0_1_5: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq one (numeral.numeral one plus x) = numeral.numeral one plus x"
nat_pow_distrib: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes> y) [^] n = x [^] n \<otimes> y [^] n"
invert_side_of_triangle_2: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and h :: "'d"   and f :: "'d"   and g :: "'d" assumes "category C"   and "partial_magma.arr C h"   and "C f g = h"   and "category.iso C g" shows "partial_magma.arr C (C h (category.inv C g)) \<and> f = C h (category.inv C g)"
distinct_9: fixes x1 :: "nat"   and x61 :: "pdf_dist"   and x62 :: "expr" shows "Var x1 \<noteq> Random x61 x62"
Iod_carrier: fixes D :: "('h, 'i) Order_scheme"   and T :: "'h set" assumes "Order D"   and "T \<subseteq> carrier D" shows "carrier (Iod D T) = T"
PQ: fixes X :: "('a \<Rightarrow> 'b) topology"   and X' :: "'b topology" assumes "X homeomorphic_space X'" shows "P X = Q X'"
vsubst_compose2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var"   and y1 :: "'var" assumes "good X"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "X #[y1 // y]_ys #[z1 // z]_zs = X #[z1 // z]_zs #[y1 @ys[z1 / z]_zs // y]_ys"
homh: shows "(+>\<^bsub>S\<^esub>) P \<circ> h \<in> ring_hom R (S Quot P)"
csie: assumes "s\<inverse> \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "s\<inverse> O e = s\<inverse>"
Landau_lemma: fixes lst :: "complex list" shows "norm2 (\<Prod>a\<leftarrow>lst. [:- a, 1:]) = norm2 (\<Prod>a\<leftarrow>lst. if 1 < cmod a then [:- 1, cnj a:] else [:- a, 1:])"
leg0_composite: shows "trg_trg.leg0 = trg.leg0 \<cdot> trg_trg.prj\<^sub>0"
apply_ncrit_def: fixes crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "apply_ncrit crit data gs bs hs ps = (let c = crit data gs bs hs in rev (fold (\<lambda>(ic, q_in_bs, p, q) ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps []))"
e: fixes w :: "complex"   and z :: "complex" assumes "cmod (w - z) < e * cmod z" shows "(w ^ n = z ^ n) = (w = z)"
le_spvec_empty2_sparse_row: fixes b :: "(nat \<times> 'a) list" assumes "sorted_spvec b" shows "le_spvec b [] = (sparse_row_vector b \<le> 0)"
cop_F_allocation_inv: fixes ds :: "'b set" shows "valid ds (\<lambda>ds fp. cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp) (\<lambda>ds fp. inj_on Xd (CH fp))"
dg_prod_is_arrD_7: fixes a :: "V"   and b :: "V"   and f :: "V"   and i :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod K \<AA>\<^esub> b"   and "i \<in>\<^sub>\<circ> K" shows "f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>"
is_acc_run_def: fixes G :: "('a, 'b) igb_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'a" assumes "igb_graph G" shows "igb_graph.is_acc_run G r \<equiv> graph_defs.is_run G r \<and> igb_graph.is_acc G r"
Sura_Bura_clopen_subset: fixes S :: "'b set"   and C :: "'b set"   and U :: "'b set" assumes "locally compact S"   and "C \<in> components S"   and "compact C"   and "open U"   and "C \<subseteq> U" obtains K :: "'b set" where "openin (top_of_set S) K" and "compact K" and "C \<subseteq> K" and "K \<subseteq> U"
tiebreaker_lifting_axioms: shows "tiebreaker_lifting Bot_F Inf_F Bot_G (\<Turnstile>G) Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F_g"
isos_compose: fixes f :: "'b"   and f' :: "'b" assumes "B.iso f"   and "B.iso f'"   and "B.seq f' f" shows "B.iso (f' \<cdot>\<^sub>B f)"
rg_succ2_def: fixes am :: "nat \<Rightarrow> nat list"   and cf :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "rg_succ2 am cf u \<equiv> ps_get_op am u \<bind> monadic_filter_rev (\<lambda>v. cf_get cf (u, v) \<bind> (\<lambda>x. return ((0::'a) < x)))"
ide_g: shows "D.ide (\<eta>'.map\<^sub>0 b' \<star>\<^sub>D G f \<star>\<^sub>D \<eta>\<^sub>0 b)"
rel_eq: shows "rel_blindable\<^sub>m (=) (=) = (=)"
spr_jview_tStep_eq_inv: fixes a :: "'a"   and t' :: "'s Trace"   and t :: "'s Trace"   and s :: "'s" assumes "spr_jview a t' = spr_jview a (t \<leadsto> s)" shows "\<exists>t'' s'. t' = t'' \<leadsto> s'"
nth_equal_imp_dist_less: fixes i :: "nat"   and f :: "'b fps"   and g :: "'b fps" assumes "\<And>j. j \<le> i \<Longrightarrow> f $ j = g $ j" shows "dist f g < inverse (2 ^ i)"
intro: fixes S :: "'c set"   and X :: "'c set" assumes "S \<subseteq> X" shows "ind_topology_axioms X S"
nat_left_shift_def: fixes x :: "nat \<Rightarrow> 'b" shows "nat_left_shift x = (\<lambda>i. x (i + 1))"
listall_snoc: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" shows "listall P (xs @ [x]) = (listall P xs \<and> P x)"
a: shows "<a>:M \<in> IMPRIGHT (B IMP C) (\<parallel>(B)\<parallel>) (\<parallel><C>\<parallel>) (\<parallel>(C)\<parallel>) (\<parallel><B>\<parallel>)"
exhaust_sel: fixes option :: "'f option" obtains "option = None"   | "option = \<lfloor>the option\<rfloor>"
deadlock_eq_deadlocked': shows "r2.deadlock = r2.deadlocked'"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "r2 (f y) (f x) = r1 y x"
weakly_logically_equivalent_def: fixes P :: "'state"   and Q :: "'state" shows "S_transform.weakly_logically_equivalent P Q \<equiv> \<forall>x. S_transform.weak_formula x \<longrightarrow> P \<Turnstile>\<^sub>S x = Q \<Turnstile>\<^sub>S x"
nth_list_update_neq: fixes i :: "nat"   and j :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i \<noteq> j" shows "xs[i := x] ! j = xs ! j"
that_3: fixes r :: "'rule"   and f :: "'form" assumes "n = Rule r"   and "p = Reg f" shows "thesis"
prices_def: fixes Mkt :: "('d, 'c) discrete_market" shows "prices Mkt = snd (Rep_discrete_market Mkt)"
wiring: shows "wiring \<I>1 \<I>2 conv fg"
not_dep_decr: fixes a :: "'a"   and x :: "'b"   and xs :: "'b list" assumes "\<not> depends\<^sub>0 a" shows "I\<^sub>a a (x # xs) = I\<^sub>a (decr a) xs"
bchoice_finite_set: fixes S :: "'b set set"   and f :: "'c \<Rightarrow> 'b set" assumes "\<forall>x\<in>S. \<exists>y. x = f y"   and "finite S" shows "\<exists>ys. \<Union> S = \<Union> (f ` ys) \<and> finite ys"
composite_simps_8: shows "cod ((cod \<mu> \<star> cod \<nu>) \<star> cod \<pi>) = (cod (cod \<mu>) \<star> cod (cod \<nu>)) \<star> cod (cod \<pi>)"
le_imp_bigo_real: fixes c :: "real"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real"   and F :: "'b filter" assumes "0 \<le> c"   and "\<forall>\<^sub>F x in F. f x \<le> c * g x"   and "\<forall>\<^sub>F x in F. 0 \<le> f x" shows "f \<in> O[F](g)"
sum_rationals_common_divisor': fixes A :: "'a set"   and g :: "'a \<Rightarrow> int"   and d :: "int"   and f :: "'a \<Rightarrow> int" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> g x \<noteq> 0"   and "\<And>x. x \<in> A \<Longrightarrow> g x dvd d"   and "d \<noteq> 0" shows "\<exists>c. (\<Sum>x\<in>A. real_of_int (f x) / real_of_int (g x)) = real_of_int c / real_of_int d"
elims: fixes x :: "'b"   and xa :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and y :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.remove1Prio x xa = y" obtains a :: "'b" where "x = a__" and "xa = []" and "y = []"   | a :: "'b" and t :: "('a, 'b) BinomialHeapStruc.BinomialTree" and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" where "x = a" and "xa = t # bq" and "y = (if BinomialHeapStruc.prio t = a then bq else t # BinomialHeapStruc.remove1Prio a bq)"
valid_rchild_if_valid: fixes as :: "'b list"   and t :: "'b tree" assumes "valid as t" shows "valid as (rchild t)"
Times_ZeroL: fixes r :: "'a language" shows "Times Zero r = Zero"
upd_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and upd :: "'Var ModeUpd"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c@[upd], mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "\<langle>c, update_modes upd mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"
sba1_export: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (- x \<squnion> y) = - x \<squnion> - y"
exhaust_sel: fixes inference :: "'a inference" obtains "inference = Infer (prems_of inference) (concl_of inference)"
is_slim_def: fixes V :: "'a set" shows "sdG.is_slim V \<equiv> (\<forall>v\<in>pverts (subdivide G e w). v \<in> V \<or> in_degree (with_proj (subdivide G e w)) v \<le> 2 \<and> (\<exists>x p y. sdG.gen_iapath V x p y \<and> v \<in> set (pawalk_verts x p))) \<and> (\<forall>e\<in>parcs (subdivide G e w). fst e \<noteq> snd e \<and> (\<exists>x p y. sdG.gen_iapath V x p y \<and> e \<in> set p)) \<and> (\<forall>u v p q. sdG.gen_iapath V u p v \<and> sdG.gen_iapath V u q v \<longrightarrow> p = q) \<and> V \<subseteq> pverts (subdivide G e w)"
AbGroup_def: fixes G :: "'a set" shows "AbGroup G \<equiv> Group G"
bdd_belowI2: fixes A :: "'c set"   and m :: "'d multiset"   and f :: "'c \<Rightarrow> 'd multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "subset_mset.bdd_below (f ` A)"
exhaust_sel: fixes transaction\<^sub>m :: "transaction\<^sub>m" obtains "transaction\<^sub>m = Transaction\<^sub>m (the_Transaction\<^sub>m transaction\<^sub>m)"
assms: shows "eval cid t t' n (\<gamma> \<longrightarrow>\<^sup>b \<gamma>')"
shows_prec_list_def: fixes d :: "nat"   and l :: "'c list" shows "shows_prec d l = shows_list l"
assms_2: shows "t \<le> v * v\<^sup>T"
subgraph_K: shows "subgraph (with_proj K) (with_proj G)"
delete_edges_list_set: fixes G :: "'c graph"   and E :: "('c \<times> 'c) list" shows "delete_edges_list G E = delete_edges G (set E)"
ide_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "ide f = (f = Zero \<or> f = One)"
module_on_axioms: shows "module_on U\<^sub>M\<^sub>_\<^sub>1 (*s\<^sub>1)"
det_0_iff_vec_prod_zero: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(det A = (0::'c)) = (\<exists>v. v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> A *\<^sub>v v = 0\<^sub>v n)"
U_cases: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "MC_syntax.U K s s = 1 \<or> MC_syntax.U K s s < 1"
dirichlet_prod_inverse': fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b" assumes "f 1 * i = (1::'b)" shows "dirichlet_prod (dirichlet_inverse f i) f = (\<lambda>n. if n = 1 then 1::'b else (0::'b))"
U_eq: shows "f (\<Union> (range A)) = (\<Sum>i. f (A i))"
semiring_norm_83: fixes n :: "Num.num" shows "(num.One = num.Bit0 n) = False"
rwof_rel_spec: fixes init :: "'a nres"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres"   and s :: "'a" assumes "rwof init cond step s"   and "cond s" shows "step s \<le>\<^sub>n SPEC (\<lambda>s'. (s, s') \<in> rwof_rel init cond step)"
det3_nonneg_scaleR_segment1: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real"   and a :: "real" assumes "0 \<le> det3 x y z"   and "0 \<le> a"   and "a < 1" shows "0 \<le> det3 ((1 - a) *\<^sub>R x + a *\<^sub>R y) y z"
unifiers_insert_swap: fixes s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" shows "unifiers (insert (s, t) E) = unifiers (insert (t, s) E)"
inverse_zero_imp_zero: fixes a :: "'a" assumes "inverse a = (0::'a)" shows "a = (0::'a)"
set_object_value_ign: fixes u :: "partition_id_t"   and x :: "obj_id_t"   and s :: "state_t"   and y :: "obj_t" assumes "\<not> Policy.sp_spec_subj_obj u x READ" shows "vpeq u s (set_object_value x y s)"
assms_2: fixes m :: "'a" assumes "P m" shows "Q m"
standard_uniqueness_isomorphs: fixes W :: "'b set set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and C :: "'b set" assumes "ChamberComplexIsomorphism W X f"   and "ChamberComplexIsomorphism W X g"   and "SimplicialComplex.maxsimp W C"   and "fun_eq_on f g C" shows "fun_eq_on f g (\<Union> W)"
e: shows "rawpsubst (eql t1 t2) (zip (map Var us) (map snd txs)) = eql (rawpsubstT t1 (zip (map Var us) (map snd txs))) (rawpsubstT t2 (zip (map Var us) (map snd txs)))"
primitive_part_smult_int: fixes d :: "int"   and f :: "int poly" shows "primitive_part (smult d f) = smult (sgn d) (primitive_part f)"
minus_ordinal_def: fixes x :: "ordinal"   and y :: "ordinal" shows "x - y = ordinal_rec 0 (\<lambda>p w. if y \<le> p then oSuc w else w) x"
trivial_limit_within: fixes a :: "'b"   and S :: "'b set" shows "(at a within S = bot) = (\<not> a islimpt S)"
ne'_2: shows "b' # bs' \<in> lists (- {[]})"
eexI: fixes F :: "(state \<Rightarrow> 'a) \<Rightarrow> behavior \<Rightarrow> bool"   and x :: "state \<Rightarrow> 'a" shows "\<turnstile> F x \<longrightarrow> (\<exists>\<exists> x. F x)"
prems_2: shows "H \<turnstile> SyntaxN.All i (SyntaxN.All j (A IMP A(i::=Var j) IMP A(i::=Eats (Var i) (Var j))))"
subspace_span: fixes S :: "'b set" shows "m1.subspace (m1.span S)"
list_def: fixes n :: "nat"   and A :: "'a set" shows "list n A = {xs. length xs = n \<and> set xs \<subseteq> A}"
step: fixes t :: "('a, 'b) tm"   and u :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b u"   and "u \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "P u s" shows "P t s"
p_t_stop: shows "ego.p ego.t_stop = ego.p_max"
morphism_presheaves_of_rings_def: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> presheaf_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> presheaf_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<and> morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms"
fun_sum_list_commute: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" assumes "f (0::'e) = (0::'d)"   and "\<And>x y. f (x + y) = f x + f y" shows "f (sum_list xs) = sum_list (map f xs)"
induct: fixes P :: "pol \<Rightarrow> pol \<Rightarrow> bool"   and a0 :: "pol"   and a1 :: "pol" assumes "\<And>a b. P (Pc a) (Pc b)"   and "\<And>c i Pa. (c \<noteq> 0 \<Longrightarrow> P Pa (Pc c)) \<Longrightarrow> P (Pc c) (Pinj i Pa)"   and "\<And>i Pa c. (c \<noteq> 0 \<Longrightarrow> P Pa (Pc c)) \<Longrightarrow> P (Pinj i Pa) (Pc c)"   and "\<And>c Pa i Q. \<lbrakk>c \<noteq> 0 \<Longrightarrow> P Pa (Pc c); c \<noteq> 0 \<Longrightarrow> P Q (Pc c)\<rbrakk> \<Longrightarrow> P (Pc c) (PX Pa i Q)"   and "\<And>Pa i Q c. \<lbrakk>c \<noteq> 0 \<Longrightarrow> P Pa (Pc c); c \<noteq> 0 \<Longrightarrow> P Q (Pc c)\<rbrakk> \<Longrightarrow> P (PX Pa i Q) (Pc c)"   and "\<And>x Pa y Q. \<lbrakk>x = y \<Longrightarrow> P Pa Q; \<lbrakk>x \<noteq> y; y < x\<rbrakk> \<Longrightarrow> P (Pinj (x - y) Pa) Q; \<lbrakk>x \<noteq> y; \<not> y < x\<rbrakk> \<Longrightarrow> P (Pinj (y - x) Q) Pa\<rbrakk> \<Longrightarrow> P (Pinj x Pa) (Pinj y Q)"   and "\<And>x Pa Q y R. \<lbrakk>x = 0 \<Longrightarrow> P Pa (PX Q y R); \<lbrakk>x \<noteq> 0; x = 1\<rbrakk> \<Longrightarrow> P (Pinj x Pa) Q; \<lbrakk>x \<noteq> 0; x = 1\<rbrakk> \<Longrightarrow> P R Pa; \<lbrakk>x \<noteq> 0; x \<noteq> 1\<rbrakk> \<Longrightarrow> P (Pinj x Pa) Q; \<lbrakk>x \<noteq> 0; x \<noteq> 1\<rbrakk> \<Longrightarrow> P R (Pinj (x - 1) Pa)\<rbrakk> \<Longrightarrow> P (Pinj x Pa) (PX Q y R)"   and "\<And>Pa x R y Q. \<lbrakk>y = 0 \<Longrightarrow> P (PX Pa x R) Q; \<lbrakk>y \<noteq> 0; y = 1\<rbrakk> \<Longrightarrow> P (Pinj y Q) Pa; \<lbrakk>y \<noteq> 0; y = 1\<rbrakk> \<Longrightarrow> P R Q; \<lbrakk>y \<noteq> 0; y \<noteq> 1\<rbrakk> \<Longrightarrow> P (Pinj y Q) Pa; \<lbrakk>y \<noteq> 0; y \<noteq> 1\<rbrakk> \<Longrightarrow> P R (Pinj (y - 1) Q)\<rbrakk> \<Longrightarrow> P (PX Pa x R) (Pinj y Q)"   and "\<And>P1 x P2 Q1 y Q2. \<lbrakk>P P1 Q1; P P2 Q2; P P1 (mkPinj 1 Q2); P Q1 (mkPinj 1 P2)\<rbrakk> \<Longrightarrow> P (PX P1 x P2) (PX Q1 y Q2)" shows "P a0 a1"
Infinitesimal_hypreal_of_nat_iff: shows "Infinitesimal = {x. \<forall>n. hnorm x < inverse (hypreal_of_nat (Suc n))}"
hoare_apply: fixes P :: "'e set"   and R :: "('e \<times> 'e) set"   and Q :: "'e set"   and x :: "'e"   and y :: "'e" assumes "{P} R {> Q}"   and "x \<in> P"   and "(x, y) \<in> R" shows "y \<in> Q"
eq1: shows "lt (sig_trd_aux bs (punit.lt (rep_list p), p)) = lt (snd (punit.lt (rep_list p), p))"
shiftr_x_0: fixes x :: "'b word" shows "x >> 0 = x"
f4: shows "(0 \<le> 4294967296 + - 1 * uint w) = (uint w \<le> 4294967296)"
absorb2: fixes b :: "'a Word.word"   and a :: "'a Word.word" assumes "b \<le>s a" shows "signed.min a b = b"
pre_subdivFace_face_distinct: fixes f :: "face"   and v :: "nat"   and vol :: "nat option list" assumes "pre_subdivFace_face f v vol" shows "distinct (removeNones vol)"
absE: fixes X :: "chan"   and H :: "msg set" assumes "X \<in> local.abs H" obtains x :: "tag" and A :: "agent" and B :: "agent" and M :: "msg" where "X = Chan x A B M" and "M \<in> payload" and "implem X \<in> H"
dec_I2_def: fixes \<AA> :: "interp"   and ss :: "WS1S_Nameful.so list"   and X :: "WS1S_Nameful.so" shows "dec_I2 \<AA> ss X = index ss X\<^bsup>\<AA>\<^esup>SO"