hcju/isabelleps huggingface dataset:
in_set_add_indicesE: fixes f :: "('f \<Rightarrow>\<^sub>0 'g) \<times> nat \<times> 'h"   and aux :: "(('f \<Rightarrow>\<^sub>0 'g) \<times> 'h) list \<times> 'i"   and data :: "nat \<times> 'i" assumes "f \<in> set (fst (add_indices aux data))" obtains i :: "nat" where "i < length (fst aux)" and "f = (fst (fst aux ! i), fst data + i, snd (fst aux ! i))"
simps_2: fixes hosts :: "'b set"   and flows_fix :: "('b \<times> 'b) set"   and flows_state :: "('b \<times> 'b) set"   and more :: "'c" shows "flows_fix \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = flows_fix"
that: fixes q2 :: "'c" assumes "io_targets M q1 io = {q2}" shows "thesis"
term_cases: fixes t :: "'b" obtains name :: "name" where "t = free name__"   | name :: "name" where "t = const name"   | u\<^sub>1 :: "'b" and u\<^sub>2 :: "'b" where "t = app u\<^sub>1 u\<^sub>2"   | "unfree t = None" and "unapp t = None" and "unconst t = None"
prod_n_principal_idealTr: fixes e :: "nat \<Rightarrow> nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and J :: "nat \<Rightarrow> 'a set" shows "e \<in> {j. j \<le> n} \<rightarrow> {j. 0 \<le> j} \<and> f \<in> {j. j \<le> n} \<rightarrow> carrier R \<and> (\<forall>k\<le>n. J k = R \<diamondsuit>\<^sub>p f k \<^bsup>\<diamondsuit>R e k\<^esup>) \<longrightarrow> i\<Pi>\<^bsub>R,n\<^esub> J = R \<diamondsuit>\<^sub>p mprod_expR R e f n"
onorm_neg: fixes f :: "'a \<Rightarrow> 'b" shows "onorm (\<lambda>x. - f x) = onorm f"
bf: shows "b \<star> f \<noteq> null \<and> b \<star> b \<star> b \<star> f \<noteq> null"
anycast_def: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b seq_action" shows "anycast P a \<equiv> case a of broadcast m \<Rightarrow> P m | groupcast x m \<Rightarrow> P m | unicast x m \<Rightarrow> P m | _ \<Rightarrow> True"
assms: fixes i :: "nat" assumes "Suc i < length xs" shows "snd (xs ! i) = fst (xs ! Suc i)"
map_ran_cong: fixes m1 :: "('b \<times> 'c) list"   and f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and m2 :: "('b \<times> 'c) list" assumes "\<And>x. x \<in> set m1 \<Longrightarrow> f1 (fst x) (snd x) = f2 (fst x) (snd x)"   and "m1 = m2" shows "map_ran f1 m1 = map_ran f2 m2"
carrier_not_empty: shows "carrier (mult_of R) \<noteq> {}"
list_all_init: shows "list_all (\<lambda>q. q \<in> reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0) (map (\<lambda>\<nu>. (q\<^sub>0, \<nu>, \<delta> q\<^sub>0 \<nu>)) \<Sigma>)"
equal_deg_sum: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "f \<in> carrier (UP R)"   and "g \<in> carrier (UP R)"   and "deg R f < n"   and "deg R g = n" shows "deg R (f \<oplus>\<^bsub>UP R\<^esub> g) = n"
mul_eq_reflexive: fixes M :: "'c multiset"   and r :: "('c \<times> 'c) set" shows "(M, M) \<in> mul_eq r"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Static of Static \<Rightarrow> f1 | SuperM \<Rightarrow> f2 | IntVir \<Rightarrow> f3) = f1"
linear_iff: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" shows "Vector_Spaces.linear s1 s2 f = (vector_space s1 \<and> vector_space s2 \<and> (\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (s1 c x) = s2 c (f x)))"
maxim_Un: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field s"   and "finite B"   and "B \<noteq> {}"   and "B \<subseteq> Field s" shows "s.maxim (A \<union> B) = s.max2 (s.maxim A) (s.maxim B)"
thetaParWT_WbisT: shows "thetaParWT \<subseteq> WbisT"
zeta_even_nat: fixes n :: "nat" shows "zeta (2 * of_nat n) = complex_of_real ((- 1) ^ Suc n * bernoulli (2 * n) * (2 * pi) ^ (2 * n) / (2 * fact (2 * n)))"
in_prime_factors_imp_prime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<in># factorial_semiring.prime_factorization zero times one normalize x" shows "normalization_semidom.prime zero times one normalize p"
mult_Mp_1: fixes f :: "int poly"   and g :: "int poly" shows "r.eq_m (r.Mp f * g) (f * g)"
uniform_count_measure_def: fixes A :: "'b set" shows "uniform_count_measure A = point_measure A (\<lambda>x. ennreal (1 / real (card A)))"
simps_2: fixes t :: "('c, 'd) SkewBinomialTree"   and v :: "('c, 'd) SkewBinomialTree"   and va :: "('c, 'd) SkewBinomialTree list" shows "getMinTree (t # v # va) = (if prio t \<le> prio (getMinTree (v # va)) then t else getMinTree (v # va))"
induct: fixes P :: "'c trm \<Rightarrow> bool"   and trm :: "'c trm" assumes "\<And>x. P (Var x)"   and "\<And>x1a x2. (\<And>x2a. x2a \<in> set x2 \<Longrightarrow> P x2a) \<Longrightarrow> P (Fn x1a x2)" shows "P trm"
less_irrefl: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.preorder less_eq less" shows "\<not> less x x"
perm_set_eq: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs <~~> ys" shows "set xs = set ys"
assms_2: shows "eval c t t' (Suc n) \<gamma>"
circ_sub_dist: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<circ> \<le> (x \<squnion> y)\<^sup>\<circ>"
triv_spref: fixes s :: "'b list"   and r :: "'b list" assumes "s \<noteq> \<epsilon>" shows "r <p r \<cdot> s"
pconj_assoc: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "0 \<le> a"   and "a \<le> 1"   and "0 \<le> b"   and "b \<le> 1"   and "0 \<le> c"   and "c \<le> 1" shows "a .& (b .& c) = a .& b .& c"
dPf: fixes k :: "nat" assumes "k < n" shows "P * f k \<in> carrier_mat d d"
length_list_update: fixes xs :: "'d list"   and i :: "nat"   and x :: "'d" shows "length (xs[i := x]) = length xs"
replicate_unequal_aux: fixes x :: "nat"   and y :: "nat"   and l :: "color list"   and l' :: "color list" assumes "x < y" shows "replicate x R @ B # l \<noteq> replicate y R @ B # l'"
brw_inner_invar_imp_final: fixes q :: "'d"   and W :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d \<Rightarrow> 'e tree option"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<Sigma>' :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" assumes "q \<in> W"   and "brw_inner_invar {r \<in> \<delta>. q \<in> set (rhsq r)} q (Q, W - {q}, rcm) {} \<Sigma>'" shows "((Q, W, rcm), \<Sigma>') \<in> brw_step \<delta>"
TComp_term_cases: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<Gamma> t = Fun f T" shows "(\<exists>v. t = Var v) \<or> (\<exists>T'. t = Fun f T' \<and> T = map \<Gamma> T' \<and> T' \<noteq> [])"
1: shows "\<guillemotleft>\<tau> \<star> \<mu> \<star> \<nu> : local.dom \<tau> \<star> local.dom \<mu> \<star> local.dom \<nu> \<Rightarrow> cod \<tau> \<star> cod \<mu> \<star> cod \<nu>\<guillemotright>"
erf_remainder_0_conv_erfc: fixes x :: "real" assumes "0 < x" shows "erf_remainder 0 x = erfc x"
put_put: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'e"   and u :: "'e" assumes "mwb_lens x" shows "put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u"
uniform_limit2: shows "uniform_limit {z. Re z \<in> {x..y}} (\<lambda>a z. complex_interval_lebesgue_integral lborel (ereal (exp (- 1))) a (f z)) (\<lambda>z. complex_interval_lebesgue_integral lborel (ereal (exp (- 1))) 1 (f z)) (at_left 1)"
open_arith_prog_fb: fixes b :: "nat"   and a :: "int" assumes "0 < b" shows "open (arith_prog_fb a b)"
ACspanB: shows "V.module.span (A \<union> C) = V.module.span B"
ideD_2: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and a :: "'ag" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.dom C a = a"
VDM_Sound: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "G \<rhd> c : A" shows "G \<Turnstile> c : A "
br'_cond_abs: fixes \<Sigma> :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" shows "(\<Sigma> \<in> br'_cond) = (br'_\<alpha> \<Sigma> \<in> br_cond)"
subclsRp_subclsR_eq: 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" shows "subclsRp P = subclsR' P"
absorb_iff1: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (sup a b = a)"
polyfun_const: fixes N :: "nat set"   and c :: "'a" shows "polyfun N (\<lambda>_. c)"
eventually_filtercomap: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and F :: "'b filter" shows "eventually P (filtercomap f F) = (\<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))"
of_real_def: fixes r :: "real" shows "of_real r = r *\<^sub>R (1::'b)"
inf_rr_def: fixes a :: "'a"   and b :: "'a" shows "inf_rr a b = a * (a r\<rightarrow> b)"
R_def: shows "R = (\<lambda>X. {#\<Sum>\<sigma>\<in>perms. \<sigma> (f \<sigma>). f \<in># mset_set (tuples X)#})"
inc: fixes k :: "real" assumes "0 \<le> k"   and "k < 1" shows "k < p * (1 - k) + k"
cone_iff_finite_cone: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n"   and "finite A" shows "cone A = finite_cone A"
antisym: fixes b :: "'e"   and a :: "'e" assumes "b \<le> a"   and "a \<le> b" shows "a = b"
a_heap_is_wellformed_def: fixes h :: "(_) heap" shows "local.a_heap_is_wellformed h = (local.CD.a_heap_is_wellformed h \<and> acyclic (local.CD.a_parent_child_rel h \<union> local.a_host_shadow_root_rel h \<union> local.a_ptr_disconnected_node_rel h) \<and> local.a_all_ptrs_in_heap h \<and> local.a_distinct_lists h \<and> local.a_shadow_root_valid h)"
icPar1B: fixes P' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> Q" shows "F (P' \<parallel> Q)"
to_ns_def: fixes l :: "('a \<times> constraint) list" shows "to_ns l \<equiv> concat (map i_constraint_to_qdelta_constraint l)"
enc_preserves_binary_pred_iff_indRelR_preserves_binary_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" shows "enc_preserves_binary_pred Pred = rel_preserves_binary_pred indRelR Pred"
gen2_ne: fixes A :: "nat"   and m :: "nat"   and B :: "nat"   and n :: "nat" shows "gen2 A B m n \<noteq> []"
keys_restrict_indets_pp: fixes t :: "'b option \<Rightarrow>\<^sub>0 nat" shows "keys (restrict_indets_pp t) = the ` (keys t - {None})"
rquot_ins_def: shows "rquot_ins = (\<lambda>x l. l[x := True])"
intros: fixes classes :: "'a fset"   and associations :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and \<D>' :: "'a"   and "from" :: "'c option"   and role :: "'c"   and end' :: "'a \<times> nat \<times> enat \<times> bool \<times> bool"   and \<C> :: "'a"   and \<D> :: "'a"   and "end" :: "'d" assumes "owned_association_end' classes associations \<D>' from role end'"   and "\<C> \<le> \<D>'"   and "\<D>' < \<D>" shows "association_end_not_closest classes associations \<C> from role \<D> end"
valid_wp: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<Turnstile>\<^sub>3\<^sub>' {P} c { Q}" shows "\<exists>k>0. \<forall>s n. P (s, n) \<longrightarrow> wp\<^sub>3\<^sub>' c (\<lambda>(ps, n). (Q \<and>* (\<lambda>s. True)) (ps, n div k)) (s, k * n)"
well_order_extension: fixes p :: "('b \<times> 'b) set" assumes "wf p" shows "\<exists>w. p \<subseteq> w \<and> Well_order w"
simps_2: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and A :: "'v pvar \<Rightarrow> zhmultiset"   and x :: "'v"   and i :: "nat" shows "subst_zpassign \<rho> A (PCoef x i) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef (\<rho> x) i))"
gseq_conv: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "((f, g) \<in> gseq) = (f \<noteq> g \<and> (f, g) \<in> geseq)"
PDG_path_Nil: fixes n :: "'node" assumes "valid_node n" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use weak_control_dependence n n"
lspasl_empr: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Gamma :: "bool"   and Delta :: "bool" assumes "class.heap_sep_algebra plus zero sep_disj" shows "Gamma \<longrightarrow> sep_algebra.sep_empty zero zero \<or> Delta"
measurable_lborel2: fixes M :: "'b measure" shows "lborel \<rightarrow>\<^sub>M M = borel \<rightarrow>\<^sub>M M"
CCexp_simps_5: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" assumes "\<not> nonrec \<Gamma>" shows "ccExp_syn a (Terms.Let \<Gamma> e) = CCfix \<Gamma>\<cdot>(Afix \<Gamma>\<cdot>(Aexp_syn' a e \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), ccExp_syn a e) G|` (- domA \<Gamma>)"
minus_plus': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and s :: "'c"   and t :: "'c"   and u :: "'c" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus s t" shows "plus u (minus t s) = minus (plus u t) s"
intro: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" assumes "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"   and "iso_presheaves_of_rings_axioms 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" shows "iso_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"
assms: shows "v \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> w"
Amin_le_add: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "valuation K v"   and "\<forall>j\<le>n. f j \<in> carrier K" shows "Amin n (v \<circ> f) \<le> v (\<Sigma>\<^sub>e K f n)"
rel_intros_2: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x2 :: "'b"   and y2 :: "'c" assumes "R x2 y2" shows "rel_option R (Some x2) (Some y2)"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(S_action.Pred x2 = S_action.Pred y2) = (x2 = y2)"
induct: fixes x1 :: "'f"   and x2 :: "'g list"   and x3 :: "'f"   and t :: "('f \<times> 'g \<times> 'f) set"   and P :: "'f \<Rightarrow> 'g list \<Rightarrow> 'f \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> trcl t"   and "\<And>c. P c [] c"   and "\<And>c a c' w c''. \<lbrakk>(c, a, c') \<in> t; (c', w, c'') \<in> trcl t; P c' w c''\<rbrakk> \<Longrightarrow> P c (a # w) c''" shows "P x1 x2 x3"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case GREATER of LESS \<Rightarrow> f1 | EQUAL \<Rightarrow> f2 | GREATER \<Rightarrow> f3) = f3"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (load_sub2 address 10 rd curr_win w s1)) \<and> t2 = snd (fst (load_sub2 address 10 rd curr_win w s2))"
union_mset_add_mset_left: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(a, A) + B = a, A + B"
matches_case_ternaryvalue_tuple: fixes \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'a match_expr"   and a :: "action"   and p :: "'b" shows "matches (\<beta>, \<alpha>) m a p = (case ternary_ternary_eval (map_match_tac \<beta> p m) of TernaryTrue \<Rightarrow> True | TernaryFalse \<Rightarrow> False | TernaryUnknown \<Rightarrow> \<alpha> a p)"
bin_rest_OR: fixes x :: "int"   and y :: "int" shows "(x OR y) div 2 = x div 2 OR y div 2"
top: fixes f :: "'a \<Rightarrow> 'b" shows "hom_graph f {} \<top> \<top>"
simps: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and e :: "('a, 'b, 'c) expr" shows "(\<sigma>, \<tau>, e)\<^sub>e = e"
NegPos_set_5: fixes x :: "'b negation_type list" shows "Neg ` set (getNeg x) \<subseteq> set x"
det_swaprows_mat: fixes k :: "nat"   and n :: "nat"   and l :: "nat" assumes "k < n"   and "l < n"   and "k \<noteq> l" shows "det (swaprows_mat n k l) = - (1::'a)"
mult_less_le_imp_less: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a < b"   and "c \<le> d"   and "(0::'c) \<le> a"   and "(0::'c) < c" shows "a * c < b * d"
that_1: assumes "\<bar>u\<bar> = u" shows "thesis"
ss_proj_def: fixes ss :: "('c, 'd) fmap set"   and vs :: "'c set" shows "ss_proj ss vs \<equiv> fmrestrict_set vs ` ss"
elims: fixes x :: "ereal"   and y :: "ereal" assumes "inverse x = y" obtains r :: "real" where "x = ereal r" and "y = (if r = 0 then \<infinity> else ereal (inverse r))"   | "x = \<infinity>" and "y = 0"   | "x = - \<infinity>" and "y = 0"
filter_eq_Cons_iff: fixes P :: "'b \<Rightarrow> bool"   and ys :: "'b list"   and x :: "'b"   and xs :: "'b list" shows "(filter P ys = x # xs) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
mem_subring_mem_ring: fixes S :: "('a, 'e) Ring_scheme"   and x :: "'a" assumes "Subring R S"   and "x \<in> carrier S" shows "x \<in> carrier R"
simps_2: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list" shows "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
fold_chamber_complex_endomorph_list: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<forall>x\<in>set xs. ChamberComplexEndomorphism X (f x)" shows "ChamberComplexEndomorphism X (fold f xs)"
t_def: fixes s :: "'b list"   and r :: "'b"   and a :: "nat \<times> nat list" shows "t s r a = (let (mf, sws) = a in index (swaps sws s) r + 1 + length sws)"
rbt_comp_bulkload: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_bulkload c = ord.rbt_bulkload (lt_of_comp c)"
complete_split_min: fixes t :: "'b tree23"   and x :: "'b"   and t' :: "'b upD" assumes "split_min t = (x, t')"   and "complete t"   and "0 < height t" shows "complete (treeD t')"
callee_invariant_plus_oracle: fixes left :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'a) spmf"   and right :: "'a \<Rightarrow> 'c \<Rightarrow> ('e \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool" shows "callee_invariant (left \<oplus>\<^sub>O right) I = (callee_invariant left I \<and> callee_invariant right I)"
abs_eq_0: fixes a :: "'a" shows "(\<bar>a\<bar> = (0::'a)) = (a = (0::'a))"
is_projection_on_fixes_image: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set"   and x :: "'b" assumes "is_projection_on \<pi> M"   and "x \<in> M" shows "\<pi> x = x"
ladder_stepdown_L_def: shows "ladder_stepdown L = (ladder_n L (Suc 0) - Suc (ladder_n L 0), ladder_j L (Suc 0), ladder_j L 0 + ladder_ix L (Suc 0)) # ladder_shift_n (Suc (ladder_n L 0)) (drop 2 L)"
LHS: shows "\<guillemotleft>LHS : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t \<star> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>"
card_sccs_eq: shows "card G.sccs = card H.sccs"
mut_m_not_idle_no_fM_write: fixes s :: "'b process_name \<Rightarrow> ('a, 'b, 'c, 'd) Model.local_state"   and m :: "'b"   and p :: "'b process_name"   and fl :: "bool"   and ws :: "('a, 'c, 'd) mem_store_action list" assumes "ghost_hs_phase (s (mutator m)) \<noteq> hp_Idle"   and "fM_rel_inv s"   and "handshake_phase_inv s"   and "tso_store_inv s"   and "p \<noteq> sys" shows "sys_mem_store_buffers p s \<noteq> mw_fM fl # ws"
less_imp_le_TC: fixes x :: "V"   and y :: "V" assumes "x \<sqsubset> y" shows "x \<sqsubseteq> y"
factor_bound_smult: fixes f :: "int poly"   and d :: "int"   and g :: "int poly"   and n :: "nat"   and k :: "nat" assumes "f \<noteq> 0"   and "d \<noteq> 0"   and "g dvd Polynomial.smult d f"   and "degree g \<le> n" shows "\<bar>coeff g k\<bar> \<le> \<bar>d\<bar> * factor_bound f n"
sas_plus_formalism_and_induced_strips_formalism_are_equally_expressive_i_a: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and k :: "nat" assumes "is_valid_problem_sas_plus \<Psi>" shows "inj_on (map (sasp_op_to_strips \<Psi>)) (bounded_plan_set (\<Psi>\<^sub>\<O>\<^sub>+) k)"
subprob_space_restrict_space: fixes M :: "'c measure"   and A :: "'c set" assumes "subprob_space M"   and "A \<inter> space M \<in> sets M"   and "A \<inter> space M \<noteq> {}" shows "subprob_space (restrict_space M A)"
cases_scheme: fixes r :: "'a DiskBlock_scheme" obtains mbal :: "nat" and bal :: "nat" and inp :: "InputsOrNi" and more :: "'a" where "r = \<lparr>mbal = mbal, bal = bal, inp = inp, \<dots> = more\<rparr>"
vars_clause_list_def: fixes Cs :: "('d, 'c) Term.term literal multiset list" shows "vars_clause_list Cs = \<Union> (vars_clause ` set Cs)"
static_empty_ord_inter_equiv_static_inter: shows "statically_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) empty_ord.Red_I Red_F_\<G> = statically_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) empty_ord.Red_I Red_F_\<G>_empty"
that: assumes "m \<in> set (\<alpha>n g)"   and "\<And>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<Longrightarrow> n \<in> set ns" shows "thesis"
rsp: shows "(0, 0) \<approx> (0, 0)"
vconn_implies_hpar_1: fixes \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" shows "src\<^sub>C \<mu> = src\<^sub>C f"
strict_implies_not_eq: fixes a :: "('d, 'e) Term.term"   and b :: "('d, 'e) Term.term" assumes "a \<rhd> b" shows "a \<noteq> b"
fds_truncate_power: fixes m :: "nat"   and f :: "'b fds"   and n :: "nat" shows "fds_truncate m (fds_truncate m f ^ n) = fds_truncate m (f ^ n)"
mult_left_le: fixes c :: "'a"   and a :: "'a" assumes "c \<le> (1::'a)"   and "(0::'a) \<le> a" shows "a * c \<le> a"
A'_def: shows "A' = fst ` {x \<in> Sigma A B. (0::'c) \<noteq> f x}"
simps: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "run\<^sub>t \<delta> q\<^sub>0 w i = (run \<delta> q\<^sub>0 w i, w i, run \<delta> q\<^sub>0 w (Suc i))"
wf: shows "wf {(t, s). P s \<and> b s \<and> t = c s}"
assms_5: shows "\<forall>x\<in>t. 0 \<le> f x \<bullet> k"
has_field_derivative_eval_fds: fixes f :: "'a fds"   and s :: "'a"   and A :: "'a set" assumes "conv_abscissa f < ereal (s \<bullet> (1::'a))" shows "(eval_fds f has_field_derivative eval_fds (fds_deriv f) s) (at s within A)"
inject: fixes x21 :: "'l"   and x22 :: "'l list"   and y21 :: "'l"   and y22 :: "'l list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
simps_2: fixes d :: "'d" shows "theLe (Lt d) = d"