hcju/isabelleps huggingface dataset:
dgrad_p_set_le_args_to_set_ab: fixes d :: "'a \<Rightarrow> nat"   and ap :: "(('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) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('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> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('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) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'h"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   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 data :: "nat \<times> 'h"   and data' :: "nat \<times> 'g" assumes "dickson_grading d"   and "ap_spec ap"   and "ab_spec ab"   and "compl_struct compl"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps"   and "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)" shows "dgrad_p_set_le d (args_to_set (gs, ab gs bs hs data', ap gs bs (ps -- sps) hs data')) (args_to_set (gs, bs, ps))"
trm_simp_6: fixes A :: "'b trm"   and B :: "'b trm"   and C :: "'b trm"   and D :: "'b trm" assumes "SimplyTyped.Pair A B = SimplyTyped.Pair C D" shows "B = D"
NatTransCompNatTrans: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "\<eta>1 \<approx>>\<bullet> \<eta>2" shows "NatTrans (\<eta>1 \<bullet> \<eta>2)"
dyadics_in_open_unit_interval: shows "{0<..<1} \<inter> (\<Union>k m. {real m / 2 ^ k}) = (\<Union>k. \<Union>m\<in>{0<..<2 ^ k}. {real m / 2 ^ k})"
counit_in_hom_2: shows "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C trg\<^sub>C f\<guillemotright>"
Cons_6: shows "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"
simps_4: fixes n :: "enat"   and xs :: "'c llist" assumes "n \<noteq> 0"   and "\<not> lnull xs" shows "ltl (ltake n xs) = ltake (epred n) (ltl xs)"
l_set_disconnected_nodes_get_child_nodes_axioms: shows "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs Shadow_DOM.get_child_nodes_locs"
restrict_ext': fixes R :: "('a \<times> 'b) set"   and X :: "'a set" shows "R || X = {p. fst p \<in> X \<and> p \<in> R}"
normal_forms_def: fixes t :: "trm" shows "normal_forms t \<equiv> {u. t \<Down> u}"
pCons_3: shows "0 < lead_coeff (pCons a p)"
UN: shows "\<Union> (P\<^sub>2 \<union> wrap (B\<^sub>2 \<union> {u})) = \<Union> (P\<^sub>2 \<union> wrap B\<^sub>2) \<union> {u}"
Inf_aform'_Affine_le: fixes z :: "'a"   and X :: "'a \<times> 'a pdevs"   and p :: "nat" assumes "z \<in> Affine X" shows "Inf_aform' p X \<le> z"
allUse_lookupDef: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and m :: "'node" assumes "v \<in> braun_ssa.allUses g m"   and "m \<in> set (\<alpha>n g)" shows "lookupDef g m (var g v) = v"
le_neq_implies_less: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n"   and "\<not> m = n" shows "m < n"
drop_all: fixes xs :: "'c list"   and n :: "nat" assumes "length xs \<le> n" shows "drop n xs = []"
subcls: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<preceq>\<^sub>C D" shows "G\<turnstile>Class C\<preceq>Class D"
rel_compp: fixes R :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_pmf (R OO S) = rel_pmf R OO rel_pmf S"
sort_cong: fixes xs :: "'b list"   and ys :: "'b list" assumes "mset xs = mset ys" shows "sort xs = sort ys"
merge_on_nr_blindable: shows "merge_on UNIV (hash_blindable id) (blinding_of_blindable id (=)) (merge_blindable id merge_discrete)"
simps_4: fixes P :: "jvm_prog"   and uu :: "heap"   and uv :: "sheap"   and v :: "char list"   and vb :: "char list"   and vd :: "nat" shows "(P,uu,uv \<turnstile>\<^sub>i (v, vb, vd, No_ics)) = True"
card_Suc_Diff1: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "Suc (card (A - {x})) = card A"
snd_def: fixes prod :: "'d \<times> 'c" shows "snd prod = (case prod of (x1, x2) \<Rightarrow> x2)"
w2_cast: shows "sint (SCAST(32 \<rightarrow> 64) w2) = sint w2"
downclosed_Fix: fixes X :: "'a set" shows "downclosed_set X = (\<Down> X = X)"
assms_4: shows "e \<le> c * - c\<^sup>T"
assms_2: fixes e :: "real" assumes "0 < e" shows "\<exists>f f'. \<forall>x\<in>S. (f has_derivative f' x) (at x within S) \<and> (\<forall>h. norm (f' x h - g' x h) \<le> e * norm h)"
Rep_equal_cases: fixes y :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "y \<in> {(=)}" obtains x :: "'a equal" where "y = Rep_equal x"
rep_eq: fixes x :: "('d, 'c) lgenerator'" shows "lgenerator (ldropn_trans' x) = drop_raw (lgenerator x)"
simps_2: fixes S :: "(ZF, ZF) Signature"   and n :: "nat" shows "WellFormedToSet S (Suc n) = WellFormedToSet S n \<union> {f E@ e |f e. f \<in> Fn\<^bsub>S\<^esub> \<and> e \<in> WellFormedToSet S n}"
list_combI: fixes us\<^sub>1 :: "'a list"   and us\<^sub>2 :: "'b list"   and t\<^sub>1 :: "'a"   and t\<^sub>2 :: "'b" assumes "list_all2 P us\<^sub>1 us\<^sub>2"   and "P t\<^sub>1 t\<^sub>2" shows "P (list_comb t\<^sub>1 us\<^sub>1) (list_comb t\<^sub>2 us\<^sub>2)"
mk_cycles_awalk_cost: fixes u :: "'a"   and p :: "'b list"   and c :: "'b \<Rightarrow> real"   and n :: "nat" assumes "awalk u p u" shows "awalk_cost c (mk_cycles_path n p) = real n * awalk_cost c p"
sum_commutativity: fixes x :: "i"   and y :: "i" shows "x \<oplus> y = y \<oplus> x"
assms_1: shows "((P, \<sigma>), y) \<in> matchrel\<^sup>!"
Cv_1: shows "qReachable c (Proc - {q}) cv"
graph_eq_iff: fixes g :: "'a ugraph"   and g' :: "'a ugraph" shows "(g = g') = (nodes g = nodes g' \<and> edges g = edges g')"
inject_1: fixes x1 :: "name"   and y1 :: "name" shows "(Const x1 = Const y1) = (x1 = y1)"
surjI: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. g (f x) = x" shows "top_pres (\<P> g)"
unit_vecs_first_distinct: fixes i :: "nat"   and j :: "nat"   and n :: "nat" assumes "i \<le> j"   and "j < n" shows "unit_vec n j \<notin> set (unit_vecs_first n i)"
reindex_cong: fixes l :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'c" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "sum g A = sum h B"
singleI_unit: fixes x :: "unit" shows "eval (single ()) x"
eOp_simp6: fixes ebinp :: "'bindex \<Rightarrow> 'c withERR option"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and delta :: "'opSym"   and einp :: "'index \<Rightarrow> 'd withERR option" assumes "\<not> liftAll (\<lambda>eA. eA \<noteq> ERR) ebinp" shows "eOp MOD delta einp ebinp = ERR"
iarray_exhaust2: fixes xs :: "'a iarray"   and ys :: "'a iarray" shows "(xs = ys) = (IArray.list_of xs = IArray.list_of ys)"
assms_3: shows "b \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
arrE: fixes f :: "(bool, unit) arr" assumes "arr f" obtains "f \<noteq> Null" and "Dom f \<in> UNIV" and "Cod f \<in> UNIV" and "Map f \<in> (if Dom f = Cod f then {()} else {})"
steps: shows "steps0 (1, l, r) B stp = (s', l', r')"
rel_fundef_body_nth: fixes Fstd_empty :: "'a"   and Fstd_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef option"   and Fstd_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef \<Rightarrow> 'a"   and Fstd_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) Std.instr fundef) list"   and Finca_empty :: "'f"   and Finca_get :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef option"   and Finca_add :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef \<Rightarrow> 'f"   and Finca_to_list :: "'f \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef) list"   and heap_empty :: "'h"   and heap_get :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'h"   and heap_to_list :: "'h \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'g \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'g option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'e"   and fd1 :: "('i, 'j, 'k, 'e) Std.instr fundef"   and fd2 :: "('i, 'j, 'k, 'e, 'g) Inca.instr fundef"   and pc :: "nat" assumes "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"   and "rel_fundef (\<lambda>x y. x = std_inca_simulation.norm_instr \<DD>\<ee>\<II>\<nn>\<ll> y) fd1 fd2"   and "pc < length (body fd1)" shows "body fd1 ! pc = std_inca_simulation.norm_instr \<DD>\<ee>\<II>\<nn>\<ll> (body fd2 ! pc)"
eq_neutr_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<squnion> b = bot) = (a = bot \<and> b = bot)"
prod_principal_idealTr1: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "\<forall>f\<in>{j. j \<le> n} \<rightarrow> set_mult R (R \<diamondsuit>\<^sub>p a) (R \<diamondsuit>\<^sub>p b). \<Sigma>\<^sub>e R f n \<in> R \<diamondsuit>\<^sub>p (a \<cdot>\<^sub>r b)"
emeasure_pmf_single: fixes M :: "'b pmf"   and x :: "'b" shows "emeasure (measure_pmf M) {x} = ennreal (pmf M x)"
lcf_of_X_minus: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R"   and "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>" shows "X_poly_minus R a (deg R (X_poly_minus R a)) = \<one>\<^bsub>R\<^esub>"
right_neutral: fixes a :: "'b multiset" shows "a \<union># {#} = a"
coprimeI: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "\<And>c. \<lbrakk>dvd.dvd times c a; dvd.dvd times c b\<rbrakk> \<Longrightarrow> dvd.dvd times c one" shows "algebraic_semidom.coprime times one a b"
get_ancestors_obtains_children: fixes h :: "(_) heap"   and ancestor :: "(_) object_ptr"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "ancestor \<noteq> ptr"   and "ancestor \<in> set ancestors"   and "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h" obtains children :: "(_) node_ptr list" and ancestor_child :: "(_) node_ptr" where "h \<turnstile> Shadow_DOM.get_child_nodes ancestor \<rightarrow>\<^sub>r children" and "ancestor_child \<in> set children" and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ancestor_child \<in> set ancestors"
not_malformed_smaller: fixes e :: "'b environment"   and a :: "char list"   and X :: "'b" assumes "ok (e\<lparr>a:X\<rparr>)" shows "ok e"
all_union_tsE: fixes c :: "'a"   and A :: "'b \<Rightarrow> 'a set"   and B :: "'a set"   and k :: "'b" assumes "c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k" obtains "c \<in> A k"   | "c \<in> B"
B: shows "{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} = {f \<in> {0..<n} \<rightarrow> {0..<m}. \<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i}"
open_Union: fixes "open" :: "'a set \<Rightarrow> bool"   and K :: "'a set set" assumes "class.topological_space open"   and "\<forall>S\<in>K. open S" shows "open (\<Union> K)"
Domain_Id_on: fixes A :: "'e set" shows "Domain (Id_on A) = A"
tree_path_impl_parenthesis: fixes v :: "'v"   and w :: "'v" assumes "(v, w) \<in> (tree_edges s)\<^sup>+"   and "v \<in> dom (finished s)" shows "w \<in> dom (finished s) \<and> \<delta> s v < \<delta> s w \<and> \<phi> s w < \<phi> s v"
induct: fixes P :: "(polex \<times> polex) list \<Rightarrow> bool"   and a0 :: "(polex \<times> polex) list" assumes "P []"   and "\<And>Pa Q pps. \<lbrakk>mon_of_pol (norm Pa) = None \<Longrightarrow> P pps; \<And>x2. mon_of_pol (norm Pa) = Some x2 \<Longrightarrow> P pps\<rbrakk> \<Longrightarrow> P ((Pa, Q) # pps)" shows "P a0"
linear_imp_differentiable: fixes f :: "'c \<Rightarrow> 'd"   and net :: "'c filter" assumes "linear f" shows "f differentiable net"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "\<And>x y. f (f x y) z = f x y"   and "f z z = z" shows "abel_semigroup_set_axioms f z"
simps_6: fixes r :: "'b rexp" shows "lang (Star r) = lang r\<star>"
get_nondet_parametric: fixes S :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and M :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "(((S ===> M) ===> M) ===> (S ===> rel_nondetT M) ===> rel_nondetT M) nondetM_base.get_nondet nondetM_base.get_nondet"
right_inverse_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "(divide a b = one) = (a = b)"
current_support_flow: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f" shows "support_flow f \<subseteq> \<^bold>E\<^bsub>\<Gamma>\<^esub>"
orderE: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" obtains "a = ord_class.min a b"
restrict_idle: fixes n :: "nat"   and A :: "nat set"   and v :: "'a option list" assumes "wf_tuple n A v" shows "restrict A v = v"
comp_ide_self: fixes a :: "(bool, 'arr list) concrete_category.arr" assumes "ide a" shows "a \<cdot> a = a"
pn: assumes "w1 = POS_INF \<and> w2 = NEG_INF" shows "P w1 w2"
image_subset_iff_subset_vimage: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'d set" shows "(f ` A \<subseteq> B) = (A \<subseteq> f -` B)"
map_of_def: fixes lookup :: "'e \<Rightarrow> ('f, 'd option) state"   and heap :: "'f"   and k :: "'e" shows "state_mem_defs.map_of lookup heap k = fst (run_state (lookup k) heap)"
prems: fixes x :: "'a" assumes "x \<in> vars_ctxt (More f bef C aft)" shows "\<sigma> x = \<tau> x"
l_delete_shadow_root_get_disconnected_nodes_def: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_delete_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs \<equiv> \<forall>shadow_root_ptr ptr' h h' r. cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<noteq> ptr' \<longrightarrow> h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<longrightarrow> r h h'"
mono: fixes t :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real"   and u :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "le_trans t u"   and "\<And>P. sound P \<Longrightarrow> sound (t P)"   and "\<And>P. sound P \<Longrightarrow> sound (u P)" shows "le_trans (T t) (T u)"
sat_models_card: fixes P :: "'a ltln set" assumes "finite P" shows "card {sat_models (abs_ltln\<^sub>P \<phi>) \<inter> Pow P |\<phi>. prop_atoms \<phi> \<subseteq> P} \<le> 2 ^ 2 ^ card P"
Nml_VcompNml_2: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Dom t = Cod u" shows "Dom (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u) = Dom u"
Dom_cod: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.arr f" shows "Maps.Dom (Maps.cod f) = Maps.Cod f"
fresh_star_restrictA: fixes a :: "atom set"   and \<Gamma> :: "('a \<times> 'b) list"   and V :: "'a set" assumes "a \<sharp>* \<Gamma>" shows "a \<sharp>* restrictA V \<Gamma>"
pair_def: fixes d :: "('fun, 'b) Term.term \<times> ('fun, 'b) Term.term" shows "pair d \<equiv> case d of (t, t') \<Rightarrow> Fun Pair [t, t']"
prime_elem_imp_gcd_eq: fixes x :: "'a"   and y :: "'a" assumes "prime_elem x" shows "gcd x y = normalize x \<or> gcd x y = (1::'a)"
assms_2: shows "\<forall>(k2, \<gamma>2)\<in>one_chain2. \<gamma>2 C1_differentiable_on {0..1}"
finiteI_independent: fixes B :: "('c, 'd) vec set" assumes "vec.independent B" shows "finite B"
subgroup_subgroup_generated: fixes S :: "'a set" shows "subgroup (carrier (subgroup_generated G S)) G"
mk_diagonal_dim_2: fixes as :: "'a list" shows "dim_col (mk_diagonal as) = length as"
mult_submonoid_of_rng_def: fixes R :: "('c, 'd) ring_scheme"   and S :: "'c set" shows "mult_submonoid_of_rng R S \<equiv> ring R \<and> Localization.submonoid R S"
that: fixes lb :: "real" assumes "\<forall>x. poly p x = 0 \<longrightarrow> lb < x"   and "\<forall>x\<le>lb. sgn (poly p x) = sgn_neg_inf p" shows "thesis"
permute_minus_cancel_1: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "perm"   and x :: "'b" assumes "class.pt permute" shows "permute p (permute (- p) x) = x"
suminf_minus_initial_segment: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "summable f" shows "(\<Sum>n. f (n + k)) = (\<Sum>n. f n) - sum f {..<k}"
halt_lemma: fixes LE :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "wf LE"   and "\<forall>n. \<not> P (f n) \<longrightarrow> (f (Suc n), f n) \<in> LE" shows "\<exists>n. P (f n)"
cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "f = g"   and "x = y" shows "f x = g y"
ereal_minus_le_minus: fixes a :: "ereal"   and b :: "ereal" shows "(- a \<le> - b) = (b \<le> a)"
r_result_bivalent': fixes i :: "nat"   and x :: "nat"   and v :: "nat"   and t :: "nat" assumes "eval r_univ [i, x] \<down>= v" shows "eval r_result [t, i, x] \<down>= Suc v \<or> eval r_result [t, i, x] \<down>= 0"
alpha_tAct: fixes \<alpha>1 :: "'d"   and t1 :: "('b, 'c, 'd) Tree"   and \<alpha>2 :: "'d"   and t2 :: "('b, 'c, 'd) Tree" shows "(tAct \<alpha>1 t1 =\<^sub>\<alpha> tAct \<alpha>2 t2) = (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2))"
left_idem: fixes a :: "'a"   and b :: "'a" shows "max a (max a b) = max a b"
except_keys: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" shows "except p (keys p) = 0"
simps: fixes x :: "name"   and y :: "name"   and f :: "tm"   and k :: "tm"   and g :: "tm" assumes "atom x \<sharp> (y, f, k, g)"   and "atom y \<sharp> (f, k, g)" shows "RestrictedP f k g = g SUBS f AND SyntaxN.All x (SyntaxN.All y (HPair (Var x) (Var y) IN g IFF Var x IN k AND HPair (Var x) (Var y) IN f))"
VUnion_mono: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "\<Union>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> B"
l3: shows "left (ext v2) \<le> left (len v2 ts c)"
F_det: fixes P :: "'c process"   and Q :: "'c process" shows "F (P [+] Q) = {(s, X). s = [] \<and> (s, X) \<in> F P \<inter> F Q} \<union> {(s, X). s \<noteq> [] \<and> (s, X) \<in> F P \<union> F Q} \<union> {(s, X). s = [] \<and> s \<in> D P \<union> D Q} \<union> {(s, X). s = [] \<and> tick \<notin> X \<and> [tick] \<in> T P \<union> T Q}"
nxt_converg: fixes b :: "nat"   and y :: "nat" assumes "search b \<down>= y" shows "nxt b \<down>= e_append b (Partial_Recursive.list_encode (replicate (y - e_length b) 0 @ [1 - the (\<phi> (the (s b)) y)]))"
iso_iff_section_and_epi: fixes f :: "'a" shows "C.iso f = (C.section f \<and> C.epi f)"
psubstAbs_substAbs_compose: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "goodEnv rho" shows "A $[Y / y]_ys $[rho] = A $[rho [y \<leftarrow> Y #[rho]]_ys]"
steps: fixes i :: "nat" assumes "i < l" shows "(f i, f (Suc i)) \<in> set rel"
supp_def: fixes x :: "'c" shows "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
X_subseteq_XH_largest: shows "X \<subseteq> XH_largest"
dfoldli_by_foldli: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c dres"   and \<sigma> :: "'c" shows "dfoldli l c f \<sigma> = foldli l (case_dres False False c) (\<lambda>x s. s \<bind> f x) (dRETURN \<sigma>)"
intros_2: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and g :: "real \<Rightarrow> real" assumes "ms_closure P xs f basis"   and "\<forall>\<^sub>F x in at_top. f x = g x" shows "ms_closure P xs g basis"
transfer_action: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and s :: "'c net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'e"   and \<zeta> :: "'f net_state"   and a :: "'d node_action"   and s' :: "'c net_state" assumes "openproc np onp sr"   and "s \<in> reachable (closed (pnet np n)) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n"   and "(s, a, s') \<in> automaton.trans (closed (pnet np n))" obtains \<sigma>' :: "nat \<Rightarrow> 'e" and \<zeta>' :: "'f net_state" where "((\<sigma>, \<zeta>), a, \<sigma>', \<zeta>') \<in> automaton.trans (oclosed (opnet onp n))" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')"
simps_1: fixes ys :: "'b list" shows "[] @ ys = ys"
dist_snd_le: fixes x :: "'b \<times> 'a"   and y :: "'b \<times> 'a" shows "dist (snd x) (snd y) \<le> dist x y"