hcju/isabelleps huggingface dataset:
sorted_wrt_linorder_index_le_iff: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "linorder_on A R"   and "set xs \<subseteq> A"   and "Linorder_Relations.sorted_wrt R xs"   and "x \<in> set xs"   and "y \<in> set xs" shows "(index xs x \<le> index xs y) = ((x, y) \<in> R)"
terminal_path_empty: fixes v :: "'form \<times> nat list" assumes "terminal_vertex v" shows "terminal_path v v []"
approx_tse_generic: fixes xs :: "real list"   and vs :: "float interval option list"   and x :: "nat"   and c :: "float"   and ivlx :: "float interval"   and prec :: "nat"   and s :: "nat"   and k :: "nat"   and f :: "floatarith"   and ivl :: "float interval" assumes "bounded_by xs vs"   and "bounded_by (xs[x := real_of_float c]) vs"   and "x < length vs"   and "x < length xs"   and "vs ! x = Some ivlx"   and "approx_tse prec x s c k f vs = Some ivl" shows "\<exists>n. (\<forall>m<n. \<forall>z\<in>set_of (real_interval ivlx). ((\<lambda>y. interpret_floatarith ((DERIV_floatarith x ^^ m) f) (xs[x := y])) has_real_derivative interpret_floatarith ((DERIV_floatarith x ^^ Suc m) f) (xs[x := z])) (at z)) \<and> (\<forall>t\<in>set_of (real_interval ivlx). (\<Sum>i = 0..<n. inverse (real (\<Prod>{k..<k + i})) * interpret_floatarith ((DERIV_floatarith x ^^ i) f) (xs[x := real_of_float c]) * (xs ! x - real_of_float c) ^ i) + inverse (real (\<Prod>{k..<k + n})) * interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := t]) * (xs ! x - real_of_float c) ^ n \<in>\<^sub>r ivl)"
conj_cong: fixes P1 :: "bool"   and P2 :: "bool"   and Q1 :: "bool"   and Q2 :: "bool" assumes "P1 = P2"   and "Q1 = Q2" shows "(P1 \<and> Q1) = (P2 \<and> Q2)"
dghm_up_ArrMap_app_vdomain: fixes a :: "V"   and \<CC> :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<phi> :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "\<D>\<^sub>\<circ> (dghm_up I \<AA> \<CC> \<phi>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>) = I"
strict_implies_order: fixes a :: "('d, 'e) Term.term"   and b :: "('d, 'e) Term.term" assumes "a \<rhd> b" shows "a \<unrhd> b"
runit_in_vhom: fixes f :: "'b"   and a :: "'b" assumes "D.ide f"   and "src\<^sub>D f = a" shows "\<guillemotleft>\<r>\<^sub>D[f] : f \<star>\<^sub>D a \<Rightarrow>\<^sub>D f\<guillemotright>"
surjective: fixes r :: "('c, 'd) aGroup_scheme" shows "r = \<lparr>carrier = carrier r, pop = (\<plusminus>\<^bsub>r\<^esub>), mop = mop r, zero = \<zero>\<^bsub>r\<^esub>, \<dots> = aGroup.more r\<rparr>"
dropWhile_eq_drop: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "dropWhile P xs = drop (length (takeWhile P xs)) xs"
subsume_fin: fixes u\<^sub>1 :: "'item list"   and w :: "'item stream"   and v\<^sub>1 :: "'item list" assumes "u\<^sub>1 \<preceq>\<^sub>F\<^sub>I w"   and "v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w" obtains w\<^sub>1 :: "'item list" where "u\<^sub>1 \<preceq>\<^sub>F w\<^sub>1" and "v\<^sub>1 \<preceq>\<^sub>F w\<^sub>1"
swap_self: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "Fun.swap a a f = f"
pap: shows "path_image p \<inter> closed_segment (a - complex_of_real d) (a + complex_of_real e) \<subseteq> {a + complex_of_real e, a - complex_of_real d}"
eucl_less_le_not_le: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
mult_not_zero: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.mult_zero times zero"   and "times a b \<noteq> zero" shows "a \<noteq> zero \<and> b \<noteq> zero"
ivl_disj_un_17: fixes l :: "'b"   and m :: "'b"   and u :: "'b" assumes "l \<le> m"   and "m \<le> u" shows "{l..<m} \<union> {m..<u} = {l..<u}"
\<tau>mredT_preserves_final_thread: 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"   and 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"   and t :: "'t" assumes "\<tau>mredT s s'"   and "final_thread s t" shows "final_thread s' t"
H_tensor_Id_1_is_gate: shows "gate 2 (H \<Otimes> Quantum.Id 1)"
3: fixes \<alpha>e :: "'b \<Rightarrow> ('c \<times> 'd \<times> 'c) set"   and \<alpha>n :: "'b \<Rightarrow> 'c list"   and invar :: "'b \<Rightarrow> bool"   and inEdges' :: "'b \<Rightarrow> 'c \<Rightarrow> ('c \<times> 'd) list"   and Entry :: "'b \<Rightarrow> 'c"   and oldDefs :: "'b \<Rightarrow> 'c \<Rightarrow> 'e set"   and oldUses :: "'b \<Rightarrow> 'c \<Rightarrow> 'e set"   and defs :: "'b \<Rightarrow> 'c \<Rightarrow> 'f set"   and "uses" :: "'b \<Rightarrow> 'c \<Rightarrow> 'f set"   and phis :: "'b \<Rightarrow> 'c \<times> 'f \<Rightarrow> 'f list option"   and var :: "'b \<Rightarrow> 'f \<Rightarrow> 'e"   and g :: "'b"   and p :: "'f" assumes "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var"   and "CFG_SSA_Transformed.unnecessaryPhi \<alpha>n invar inEdges' oldDefs defs phis var g p"   and "\<not> CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs uses phis g p"   and "p \<in> CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g" obtains q :: "'f" where "CFG_SSA_wf_base.phiArg \<alpha>n defs phis g p q" and "CFG_SSA_Transformed.unnecessaryPhi \<alpha>n invar inEdges' oldDefs defs phis var g q" and "\<not> graph_Entry_base.dominates \<alpha>n invar inEdges' Entry g (CFG_SSA_wf_base.defNode \<alpha>n defs phis g q) (CFG_SSA_wf_base.defNode \<alpha>n defs phis g p)"
simps: fixes args :: "Monitor.args"   and rel1 :: "'a option list set"   and rel2 :: "'a option list set"   and nt :: "nat"   and aux :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat" shows "add_new_mmuaux args rel1 rel2 nt aux = (let (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) = shift_mmuaux args nt aux; I = args_ivl args; pos = args_pos args; new_tstp = if left I = 0 then Inr tp else Inl (nt - (left I - 1)); tmp = (\<Union>as\<in>rel2. case Mapping.lookup a1_map (proj_tuple maskL as) of None \<Rightarrow> if \<not> pos then {(tp - len, as)} else {} | Some tp' \<Rightarrow> if pos then {(max (tp - len) tp', as)} else {(max (tp - len) (tp' + 1), as)}) \<union> (if left I = 0 then {tp} \<times> rel2 else {}); a2_map = Mapping.update (tp + 1) Mapping.empty (upd_nested a2_map new_tstp (max_tstp new_tstp) tmp); a1_map = if pos then Mapping.filter (\<lambda>as _. as \<in> rel1) (upd_set a1_map (\<lambda>_. tp) (rel1 - Mapping.keys a1_map)) else upd_set a1_map (\<lambda>_. tp) rel1; tss = append_queue nt tss in (tp + 1, tss, len + 1, maskL, maskR, a1_map, a2_map, done, done_length))"
psubst_subst: fixes f :: "'c \<Rightarrow> 'a"   and P :: "('c, 'b) form"   and t :: "'c FOL_Fitting.term"   and i :: "nat" shows "psubst f (P[t/i]) = psubst f P[psubstt f t/i]"
disconnected_nodes_eq_h3: fixes doc_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "document_ptr \<noteq> doc_ptr" shows "h3 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes = h' \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
classes_changed_def: fixes P1 :: "'b prog"   and P2 :: "'b prog" shows "classes_changed P1 P2 = {cn. class P1 cn \<noteq> class P2 cn}"
lower_real_interval: fixes x :: "float interval" shows "lower (real_interval x) = real_of_float (lower x)"
interior_closure: fixes S :: "'c set" shows "interior S = - closure (- S)"
wf_trm_arity: fixes f :: "('fun, 'atom, 'sets) prot_fun"   and T :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term list" assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)" shows "length T = arity f"
arr_dom_iff_arr: fixes f :: "'d" shows "D.arr (D.dom f) = D.arr f"
map_simp_ide: fixes a :: "'a arrow_of_spans_data" assumes "ide a" shows "\<rho>.map a = \<r> a"
is_zero_valof_conv: fixes x :: "('a, 'b) IEEE.float" shows "IEEE.is_zero x = (valof x = 0)"
simps_3: fixes x :: "'a set"   and xs :: "'a set list"   and y :: "'a set" shows "min_gallery (x # xs @ [y]) = (x \<noteq> y \<and> is_arg_min length (\<lambda>zs. folding.gallery (x # zs @ [y])) xs)"
isAbr_def: fixes S :: "('d, 'e) xstate" shows "isAbr S = (\<exists>s. S = Abrupt s)"
standard_open_def: fixes x :: "'a" shows "\<D>(x) \<equiv> Spec\<setminus>\<V> \<langle>x\<rangle>"
scan_eq_append: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'b"   and u :: "'b list"   and v :: "'b list" shows "(scan f xs a = u @ v) = (\<exists>ys zs. xs = ys @ zs \<and> scan f ys a = u \<and> scan f zs (fold f ys a) = v)"
mat_to_polys_def: fixes ts :: "'a list"   and A :: "'b mat" shows "mat_to_polys ts A = map (row_to_poly ts) (rows A)"
is_idirprod_def: fixes A :: "'a set"   and S :: "'d \<Rightarrow> 'a set"   and I :: "'d set" shows "is_idirprod A S I = ((\<forall>i\<in>I. S i \<lhd> G) \<and> A = IDirProds G S I \<and> compl_fam S I)"
deterministic_heap_ops_def: shows "deterministic_heap_ops = ((\<forall>h ad al v v'. heap_read h ad al v \<longrightarrow> heap_read h ad al v' \<longrightarrow> v = v') \<and> (\<forall>h ad al v h' h''. heap_write h ad al v h' \<longrightarrow> heap_write h ad al v h'' \<longrightarrow> h' = h'') \<and> (\<forall>h hT h' a h'' a'. (h', a) \<in> allocate h hT \<longrightarrow> (h'', a') \<in> allocate h hT \<longrightarrow> h' = h'' \<and> a = a') \<and> \<not> spurious_wakeups)"
le_Inf_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and b :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq b (Inf A) = (\<forall>a\<in>A. less_eq b a)"
paint2: fixes c2 :: "color"   and c1 :: "color"   and t :: "('b \<times> color) tree" shows "paint c2 (paint c1 t) = paint c2 t"
set_iterator_genord_def: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "set_iterator_genord iti S0 R \<equiv> \<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
additive_sum: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and S :: "'c set"   and P :: "'c \<Rightarrow> 'b \<Rightarrow> real" assumes "Transformers.additive t"   and "healthy t"   and "finite S"   and "\<And>z. sound (P z)" shows "t (\<lambda>x. \<Sum>y\<in>S. P y x) = (\<lambda>x. \<Sum>y\<in>S. t (P y) x)"
card_Suc_Diff1: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<in> A" shows "Suc (card (A - {x})) = card A"
intro: fixes x :: "'c \<Longrightarrow> 'b" assumes "\<And>\<sigma> v u. put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u" shows "mwb_lens_axioms x"
image_diff_subset: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and B :: "'f set" shows "f ` A - f ` B \<subseteq> f ` (A - B)"
extend_map_to_freeword_hom2: fixes f :: "'a \<Rightarrow> 'b" shows "UGroupHom (\<lambda>x. \<Sum>(s, b)\<leftarrow>freeword x. if b then f s else - f s)"
preserves_dom: fixes f :: "'a" assumes "\<tau>.A.arr f" shows "\<tau>.B.dom (\<tau> f) = F (\<tau>.A.dom f)"
un1: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..u} = {l..<m} \<union> {m..u}"
Cod_SPN: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>" shows "Cod (bicategory_of_spans.SPN V H \<a> \<i> src trg \<mu>) = \<lparr>Leg0 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>0 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))), Leg1 = Maps.MkArr (src (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (trg (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>))) (category.iso_class V (bicategory_of_spans.tab\<^sub>1 V H \<a> \<i> src trg (partial_magma.cod V \<mu>)))\<rparr>"
eq: shows "floor_fl x = Float \<lfloor>real_of_float x\<rfloor> 0"
fps_eq_iff: fixes p :: "'a fps"   and q :: "'a fps" shows "(p = q) = (\<forall>n. p $ n = q $ n)"
SUP_le_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and u :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (f ` A)) u = (\<forall>i\<in>A. less_eq (f i) u)"
len_lin_tss': shows "length (linearize tss') = len + 1"
GREATEST_FP_def: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" shows "GFP\<^bsub>L\<^esub> f = \<Squnion>\<^bsub>L\<^esub>{u \<in> carrier L. u \<sqsubseteq>\<^bsub>L\<^esub> f u}"
valid_sops_stmt: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "'c" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "valid_sops_stmt t s"
distinct_dropWhile: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "distinct xs" shows "distinct (dropWhile P xs)"
negatively_invariant_restrict_dom: fixes M :: "'a set" shows "rev.negatively_invariant M = rev.negatively_invariant (M \<inter> X)"
hyps_4: shows "\<forall>C\<in>H. atom i \<sharp> C"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Src (t \<^bold>\<star> u) = Src u"
opposite_ipc_direction_def: fixes dir :: "ipc_direction_t" shows "opposite_ipc_direction dir \<equiv> case dir of SEND \<Rightarrow> RECV | RECV \<Rightarrow> SEND"
turn_into_distance_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" shows "turn_into_distance f x y = Inf {\<Sum>i = 0..<n. f (u i) (u (Suc i)) |u n. u 0 = x \<and> u n = y}"
complete_hb_def: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list" shows "complete_hb s E = unfold_llist (\<lambda>(s, E). \<forall>t ta s'. \<not> s -t\<triangleright>ta\<rightarrow> s') (\<lambda>(s, E). fst (SOME ((t, ta), s'). s -t\<triangleright>ta\<rightarrow> s' \<and> ta_hb_consistent P E (llist_of (map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)))) (\<lambda>(s, E). let ((t, ta), s') = SOME ((t, ta), s'). s -t\<triangleright>ta\<rightarrow> s' \<and> ta_hb_consistent P E (llist_of (map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) in (s', E @ map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) (s, E)"
convergence: fixes j :: "nat" assumes "t j \<le> t1" shows "dist (x (t j)) (grid_function (discrete_evolution incr) (x (t 0)) t j) \<le> B / L * (exp (L * (t1 - t 0) + 1) - 1) * max_stepsize j ^ p"
weakAction: fixes \<alpha> :: "act"   and P :: "ccs" shows "\<alpha>.P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P"
a2_ge_b2: shows "b\<^sup>2 \<le> a\<^sup>2"
mk_ps_chain_def: shows "mk_ps_chain \<equiv> foldl (\<lambda>\<Pi> x. cSnoc \<Pi> x (next_ps (post \<Pi>) x))"
cases: fixes a1 :: "'a"   and a2 :: "'a"   and succs :: "'a \<Rightarrow> 'a list" assumes "a1 [succs]\<rightarrow>* a2" obtains g :: "'a" where "a1 = ga__" and "a2 = ga__"   | g :: "'a" and g' :: "'a" and g'' :: "'a" where "a1 = g" and "a2 = g''" and "g' \<in> set (succs g)" and "g' [succs]\<rightarrow>* g''"
prime_not_unit': 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 x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "ASSUMPTION (normalization_semidom.prime zero times one normalize x)" shows "\<not> dvd.dvd times x one"
bl_word_not: fixes w :: "'a word" shows "to_bl (NOT w) = map Not (to_bl w)"
add_cancel_left_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and b :: "'d"   and a :: "'d" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus b a = a) = (b = zero)"
prems_2: shows "convex_polygon (x # y # z # w # ws)"
gba_rename_ecnv_def: fixes ecnv :: "('e, 'c, 'f) gba_rec_scheme \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'b"   and G :: "('e, 'c, 'f) gba_rec_scheme" shows "gba_rename_ecnv ecnv f G \<equiv> \<lparr>gba_L = \<lambda>q l. if q \<in> f ` g_V G then gba_L G (the_inv_into (g_V G) f q) l else False, \<dots> = ecnv G\<rparr>"
wellformed_policy3_charn: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "singleCombinators p"   and "distinct p"   and "allNetsDistinct p"   and "wellformed_policy1 p"   and "wellformed_policy2Pr p" shows "wellformed_policy3Pr p"
ImpInv_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "ImpInv rmhist p = (S1 rmhist p \<or> S2 rmhist p \<or> S3 rmhist p \<or> S4 rmhist p \<or> S5 rmhist p \<or> S6 rmhist p)"
unit_factor_mult: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize" shows "unit_factor (times a b) = times (unit_factor a) (unit_factor b)"
sP: fixes l :: "char list \<Rightarrow> nat"   and t :: "char list \<Rightarrow> int" assumes "P l t" shows "\<forall>l. P l t"
zip_eq_zip_same_len: fixes a :: "'a list"   and b :: "'b list"   and a' :: "'a list"   and b' :: "'b list" assumes "length a = length b"   and "length a' = length b'" shows "(zip a b = zip a' b') = (a = a' \<and> b = b')"
extremeD_1: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b" assumes "extreme X r e" shows "e \<in> X"
stl_konigDtree: fixes s' :: "'sequent"   and t :: "('sequent \<times> 'rule) tree"   and a :: "'a"   and a'' :: "'a" assumes "s' |\<in>| (fst \<circ> tree.root) |`| cont t"   and "pred (fst (tree.root t), a) (snd (tree.root t)) (s', a'')"   and "inv (s', a'')" shows "\<exists>t' a'. t' |\<in>| cont t \<and> pred (fst (tree.root t), a) (snd (tree.root t)) (fst (tree.root t'), a') \<and> inv (fst (tree.root t'), a') \<and> stl (konigDtree t a) = konigDtree t' a'"
cong_reverse_identity: fixes A :: "'p"   and C :: "'p"   and D :: "'p" assumes "Cong A A C D" shows "C = D"
DOM_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "DOM (mkarr t) = Dom t"
Abs_poly_mapping_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. finite {x. f x \<noteq> (0::'b)}}" shows "lookup (Abs_poly_mapping y) = y"
disjoint: shows "{p \<in> {p. p permutes UNIV}. evenperm p} \<inter> {Fun.swap j k id \<circ> p |p. p \<in> {p \<in> {p. p permutes UNIV}. evenperm p}} = {}"
supp_6: fixes x2 :: "ty"   and x1 :: "ty" shows "supp (x2 \<rightarrow> x1) = supp x2 \<union> supp x1"
map_of_filtered_SomeD: fixes fs :: "(('d \<times> 'e) \<times> 'b \<times> 'c) list"   and F :: "'d"   and D :: "'e"   and a :: "'b"   and T :: "'c"   and Q :: "('d \<times> 'e) \<times> 'b \<times> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'f" assumes "map_of fs (F, D) = \<lfloor>(a, T)\<rfloor>"   and "Q ((F, D), a, T)" shows "map_of (map (\<lambda>((F, D), b, T). ((F, D), P T)) (filter Q fs)) (F, D) = \<lfloor>P T\<rfloor>"
disj_not1: fixes P :: "bool"   and Q :: "bool" shows "(\<not> P \<or> Q) = (P \<longrightarrow> Q)"
abs_eq: fixes x :: "64 word" shows "uint32_of_uint64 (Abs_uint64 x) = Abs_uint32 (ucast x)"
assms_1: shows "constructive_security_obsf' middle ideal sim_inner \<I>_middle \<I>_inner \<I>_common (\<lambda>\<eta>. absorb (\<A> \<eta>) (obsf_converter (sim_outer \<eta> |\<^sub>= 1\<^sub>C)))"
U: fixes z :: "complex" assumes "z \<in> U" shows "(d z has_contour_integral h z) \<gamma>"
P_def: shows "P = (M_fin \<pi> \<union> \<Union> {Acc_fin \<Sigma> \<pi> \<chi> |\<chi>. \<chi> \<in> dom \<pi>}, {Acc_inf \<pi> \<chi> |\<chi>. \<chi> \<in> dom \<pi>})"
WHILEI_lfp_body_def: fixes I :: "'c \<Rightarrow> bool"   and m :: "'d nres"   and c :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd nres" shows "WHILEI_lfp_body I m c f \<equiv> \<lambda>x. sup m (x \<bind> (\<lambda>s. ASSERT (I s) \<bind> (\<lambda>_. ASSUME (c s) \<bind> (\<lambda>_. f s))))"
ennreal_divide_top: fixes x :: "ennreal" shows "x / \<top> = 0"
preserves_ide: fixes a :: "'j" assumes "J.ide a" shows "C.ide (A.map a)"
InvariantConflictClauseCharacterizationAfterAssertLiteral: fixes state :: "State"   and literal :: "Literal"   and decision :: "bool" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)" shows "let state' = assertLiteral literal decision state in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
T_Var: fixes \<Gamma> :: "binding list"   and i :: "nat"   and U :: "POPLmarkRecord.type"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma>\<langle>i\<rangle> = \<lfloor>VarB U\<rfloor>"   and "T = \<up>\<^sub>\<tau> (Suc i) 0 U" shows "\<Gamma> \<turnstile> Var i : T"
add_increasing: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "{#} \<subseteq># a"   and "b \<subseteq># c" shows "b \<subseteq># a + c"
stateinterpol_right: fixes x :: "variable"   and S :: "variable set"   and \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real" assumes "x \<notin> S" shows "stateinterpol \<nu> \<omega> S x = \<omega> x"
vsubst_subst_compose2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var" assumes "good X"   and "good Y"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "X #[Y / y]_ys #[z1 // z]_zs = X #[z1 // z]_zs #[Y #[z1 // z]_zs / y]_ys"
c2: fixes dsn :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "0 < dsn" shows "P (dsn, kno, val, hops, nhip)"