hcju/isabelleps huggingface dataset:
integrable_on_all_intervals_integrable_bound: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> real" assumes "\<And>a b. (\<lambda>x. if x \<in> S then f x else (0::'d)) integrable_on cbox a b"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"   and "g integrable_on S" shows "f integrable_on S"
left_right_inverse_eq: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e"   and h :: "'e \<Rightarrow> 'd" assumes "f \<circ> g = id"   and "g \<circ> h = id" shows "f = h"
Int_subset_iff: fixes C :: "'b set"   and A :: "'b set"   and B :: "'b set" shows "order.greater_eq (A \<inter> B) C = (order.greater_eq A C \<and> order.greater_eq B C)"
vs2: shows "vs2 \<equiv> getFrN (map snd txs) (r2 # map fst txs) [] (length txs)"
is_unit_field_poly: fixes p :: "'a poly" shows "is_unit p = (p \<noteq> 0 \<and> degree p = 0)"
accept_sim: fixes w :: "nat \<Rightarrow> 'l" assumes "a.accept w" shows "b.accept w"
ide_char: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.VVV.ide a = (Span.VVV.arr a \<and> Span.VxVxV.ide a)"
applicableExplain_def: fixes state :: "State" shows "applicableExplain state \<equiv> \<exists>state'. appliedExplain state state'"
mat_det_left_def: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "det A = (\<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (p i, i)))"
nth_rotater: fixes n :: "nat"   and xs :: "'b list"   and m :: "nat" assumes "n < length xs" shows "rotater m xs ! n = xs ! ((n + (length xs - m mod length xs)) mod length xs)"
length_indexToVertexList: fixes x :: "face"   and y :: "nat"   and xs :: "nat list" shows "|indexToVertexList x y xs| = |xs|"
simps_1: fixes Y :: "'d set" shows "injections_alg [] Y = [{}]"
NSLIMSEQ_I: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "\<And>N. N \<in> HNatInfinite \<Longrightarrow> (*f* X) N \<approx> star_of L" shows "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L"
chamber_vertex_outside_facet_image: fixes v :: "'a"   and z :: "'a set" assumes "v \<notin> z"   and "chamber (insert v z)" shows "f v \<notin> f ` z"
cs_prepend_impl: shows "imp_list_prepend cs_list cs_prepend"
assms_2: fixes n :: "nat" assumes "n \<notin> range g" shows "f n = (0::'a)"
nonneg_Reals_zero_I: shows "(0::'a) \<in> \<real>\<^sub>\<ge>\<^sub>0"
encode_operators_is_cnf: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "is_cnf (encode_operators \<Pi> t)"
get_dom_component_no_overlap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c'" shows "set c \<inter> set c' = {} \<or> c = c'"
arev_neq: fixes a :: "'b" assumes "a \<in> arcs H" shows "edge_rev HM a \<noteq> a"
simps_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.heap less_eq \<langle>\<rangle> = True"
SourcesA71_L1: shows "Sources level1 sA71 = {sA6}"
leg1_simps_6: shows "cod r\<^sub>1 = r\<^sub>1"
bin_rest_bl_to_bin: fixes bs :: "bool list" shows "bl_to_bin bs div 2 = bl_to_bin (butlast bs)"
closedt_mono_2: fixes i :: "nat"   and j :: "nat"   and ts :: "'a FOL_Fitting.term list" assumes "i \<le> j"   and "closedts i ts" shows "closedts j ts"
is_div_op_def: fixes div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "is_div_op div_op = (\<forall>a b. b dvd a \<longrightarrow> div_op a b * b = a)"
direct_decomp_perm: fixes A :: "'a set"   and ss1 :: "'a set list"   and ss2 :: "'a set list" assumes "direct_decomp A ss1"   and "ss1 <~~> ss2" shows "direct_decomp A ss2"
elims_2: fixes x :: "nat \<times> nat"   and xa :: "nat \<times> nat" assumes "x \<approx> xa" obtains xb :: "nat" and y :: "nat" and u :: "nat" and v :: "nat" where "x = (xb, y)" and "xa = (u, v)" and "xb + v = u + y"
smult_distrib_left: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and r :: "'a"   and m :: "'b"   and n :: "'b" assumes "VectorSpace smult V"   and "r \<in> UNIV"   and "m \<in> V"   and "n \<in> V" shows "smult r (m + n) = smult r m + smult r n"
finite_lists_length_le: fixes A :: "'a set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs \<le> n}"
1: fixes ls :: "'a common_primitive match_expr" assumes "ls \<in> set (normalize_match (rewrite_neg m))"   and "matches \<gamma> ls a p"   and "normalized_nnf_match ls" shows "match_list \<gamma> (concat (map normalize_pos (normalize_match (rewrite_neg m)))) a p"
FiniteSingleAgentEnvironment_axioms: shows "FiniteSingleAgentEnvironment jkbp agent"
compute_is_pot_ord_4: fixes to :: "(('a, 'b) pp \<times> nat) nat_term_order" shows "is_pot_ord_pprod (POT to) = True"
set_shadow_root_writes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and element_ptr :: "(_) element_ptr"   and disc_nodes :: "(_) shadow_root_ptr option"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "l_set_shadow_root type_wf set_shadow_root set_shadow_root_locs" shows "writes (set_shadow_root_locs element_ptr) (set_shadow_root element_ptr disc_nodes) h h'"
sympartpE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "sympartp r x y" obtains "r x y" and "r y x"
M_def: shows "M = \<lfloor>1 + 2 * sqrt (real D)\<rfloor>"
I: fixes m :: "'d" shows "\<lbrace>I\<rbrace> mutator m"
tst: shows "thr s t = \<lfloor>(x, ln)\<rfloor>"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat \<Rightarrow>\<^sub>0 'a" shows "lookup (higherPowers x xa xb) = (\<lambda>xc. lookup xb (inc_above x xa xc))"
simps_8: fixes f :: "'c \<Rightarrow> 'b" shows "list.map f [] = []"
bin_split_def: shows "bin_split \<equiv> \<lambda>(a, n) p. heap_WHILET (\<lambda>(low, high). return (low < high)) (\<lambda>(low, high). let mid = (low + high) div 2 in Array.nth a mid \<bind> (\<lambda>(uu_, s). if p < s then return (low, mid) else if s < p then return (mid + 1, high) else return (mid, mid))) (0, n) \<bind> (\<lambda>(low', high'). return low')"
rep_eq: fixes x :: "'b blinop"   and xa :: "'b blinop" shows "Rep_blinop (x + xa) = Rep_blinop x + Rep_blinop xa"
ibind_strict_2: fixes f :: "unit \<Rightarrow> 'ma" shows "bind top f = top"
hyps_3: fixes e' :: "char list exp" assumes "init_exp_of e1 = \<lfloor>unit\<rfloor>"   and "INIT_class e1 = \<lfloor>C\<rfloor>"   and "\<not> sub_RI e'" shows "P \<turnstile> \<langle>init_switch e1 e',s1,b1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>1,icheck P (the (INIT_class e1)) e'\<rangle>"
A: shows "c_fst (c_fst key) mod 7 = 4"
alpha': fixes a :: "name"   and x :: "'b"   and b :: "name"   and y :: "'b" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> [(b, a)] \<bullet> x = y \<and> b \<sharp> x)"
zero_local_ring_at_def: shows "zero_local_ring_at \<equiv> local.zero_rel"
neg_minus_divide_less_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less (uminus (divide b c)) a = less (times a c) (uminus b)"
T_vrestr_same_emeasure_2: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "emeasure M ((T ^^ n) --` A) = emeasure M A"
a: shows "a \<in> set (normalize_match (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one m))"
feasible_nnegD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "real"   and P :: "'b \<Rightarrow> real" assumes "feasible t"   and "bounded_by b P"   and "nneg P" shows "nneg (t P)"
fderiv2: fixes x :: "'d" assumes "x \<in> Csafe" shows "(ode_d1 has_derivative blinfun_apply (ode_d2 x)) (at x)"
case_prodD: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
simps_3: fixes v :: "'c list"   and vb :: "'c list"   and vc :: "'c list list" shows "C_merge_all (v # vb # vc) = C_merge_adj (v # vb # vc) + C_merge_all (merge_adj (v # vb # vc))"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_1: fixes l :: "'e strand_label"   and t :: "('f, 'g) Term.term"   and A :: "('e strand_label \<times> ('f, 'g) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, send\<langle>t\<rangle>) # A) = (l, receive\<langle>t\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
substAbs_def: fixes xs :: "'varSort"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and x :: "'var" shows "substAbs xs X x \<equiv> psubstAbs (idEnv [x \<leftarrow> X]_xs)"
GRepHom_decomp: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FinGroupRepresentation G smult V"   and "FGModuleHom G smult V smult' T"   and "V \<noteq> 0" shows "\<exists>U. (RModule (aezfun_setspan G) smult U \<and> U \<subseteq> V) \<and> V = U \<oplus> ker T \<inter> V \<and> (\<exists>Ta. FGModuleIso G smult U smult' Ta (T ` V))"
isnormNum_def: shows "isnormNum = (\<lambda>(a, b). if a = 0 then b = 0 else 0 < b \<and> gcd a b = 1)"
proj_set_def: fixes x :: "'b"   and A :: "'b set" shows "proj_set x A = {y \<in> A. dist x y = infdist x A}"
dghm_HomCod: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF>\<lparr>HomCod\<rparr> = \<BB>"
map_of_sorted_list_of_map: fixes m :: "'a \<Rightarrow> 'c option" assumes "finite (dom m)" shows "map_of (sorted_list_of_map m) = m"
create_graph_aimpl_def: fixes \<phi> :: "'b ltlr" shows "create_graph_aimpl \<phi> = expand_aimpl (\<lparr>name = expand_new_name expand_init, incoming = {expand_init}, new = {\<phi>}, old = {}, next = {}\<rparr>, {}) \<bind> (\<lambda>(uu_, nds). RETURN nds)"
Id_on_def: fixes A :: "'g set" shows "Id_on A = (\<Union>x\<in>A. {(x, x)})"
typing_prm: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and M :: "'a trm"   and \<tau> :: "PreSimplyTyped.type"   and \<Delta> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and \<pi> :: "'a prm" assumes "\<Gamma> \<turnstile> M : \<tau>"   and "\<And>y. y \<in> fvs M \<Longrightarrow> \<Gamma> y = \<Delta> (\<pi> $ y)" shows "\<Delta> \<turnstile> \<pi> \<cdot> M : \<tau>"
comm_neutral: fixes a :: "'c" shows "a * (1::'c) = a"
bijective_conv_mapping: fixes x :: "'a" shows "bijective x = mapping (x\<^sup>T)"
single_uminus: fixes k :: "'a"   and a :: "'b" shows "monomial (- a) k = - monomial a k"
mult_mono': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero a"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
power_less_power_Suc: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a" shows "less (power.power one times a n) (times a (power.power one times a n))"
trm_cases: fixes M :: "'b trm"   and P :: "'b trm \<Rightarrow> bool" assumes "M = Unit \<Longrightarrow> P M"   and "\<And>x. M = Var x \<Longrightarrow> P M"   and "\<And>A B. M = App A B \<Longrightarrow> P M"   and "\<And>x T A. M = Fn x T A \<Longrightarrow> P M"   and "\<And>A B. M = SimplyTyped.Pair A B \<Longrightarrow> P M"   and "\<And>A. M = Fst A \<Longrightarrow> P M"   and "\<And>A. M = Snd A \<Longrightarrow> P M" shows "P M"
meval_elim: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(cms, mem) \<rightarrow> (cms', mem')" obtains n :: "nat" and cm' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)" where "cms' = cms[n := cm']" and "(cms ! n, mem) \<leadsto> (cm', mem')" and "n \<sqsubset> length cms"
meet_connection: fixes x :: "'b"   and y :: "'b" shows "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
dX_union: fixes X :: "'a set"   and Y :: "'a set"   and d :: "'b" shows "ManiPref'.dX (X \<union> Y) d = ManiPref'.dX X d \<union> ManiPref'.dX Y d"
line_integral_exists_on_degenerate_path: fixes basis :: "'a set"   and F :: "'a \<Rightarrow> 'a"   and c :: "'a" assumes "finite basis" shows "line_integral_exists F basis (\<lambda>x. c)"
smult_eq_iff: fixes b :: "'b"   and a :: "'b"   and p :: "'b poly"   and q :: "'b poly" assumes "b \<noteq> (0::'b)" shows "(Polynomial.smult a p = Polynomial.smult b q) = (Polynomial.smult (a / b) p = q)"
assms_1: shows "a \<in>\<^sub>\<circ> \<F>\<^sub>\<bullet> r"
int_pow_eq_id: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier R" shows "([i] \<cdot> x = \<zero>) = (int (add.ord x) dvd i)"
set_suffixes_eq: fixes xs :: "'b list" shows "set (suffixes xs) = {ys. suffix ys xs}"
in_set_remove1: fixes a :: "'a"   and b :: "'a"   and xs :: "'a list" assumes "a \<noteq> b" shows "(a \<in> set (remove1 b xs)) = (a \<in> set xs)"
in_nres_rel_iff: fixes a :: "'e nres"   and b :: "'f nres"   and R :: "('e \<times> 'f) set" shows "((a, b) \<in> \<langle>R\<rangle>nres_rel) = (a \<le> \<Down> R b)"
distinct_7: fixes x5 :: "String.literal" shows "Void \<noteq> Class x5"
eq_fold: fixes x :: "'a"   and A :: "'a fset" shows "F (finsert x A) = ffold (\<^bold>*) x A"
that: fixes c :: "complex" assumes "q = of_complex c" shows "thesis"
cases: fixes a :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "a \<in> resumption" obtains aa :: "'h" where "a = TERMINAL aa"   | r :: "('f + 'g + 'h, 'i) Old_Datatype.node set" and b :: "'g" where "a = LINEAR b r" and "r \<in> resumption"   | rs :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" and c :: "'f" where "a = BRANCH c rs" and "\<And>d. rs d \<in> resumption"
simply_connected_eq_homotopic_circlemaps3: fixes S :: "'b set" assumes "path_connected S"   and "\<And>f. \<lbrakk>continuous_on (sphere 0 1) f; f ` sphere 0 1 \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>a. homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f (\<lambda>x. a)" shows "simply_connected S"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism Right_a.comp Right_a.comp (H\<^sub>R a') Right_a.map \<phi>" shows "equivalence_functor Right_a.comp Right_a.comp (H\<^sub>R a')"
1: shows "[F \<^bold>= (\<^bold>\<lambda>x. p) in v]"
DF_simps_1: fixes k :: "nat" shows "DF k 0 = (enum N \<circ> Suc) ` {..<Suc k}"
Max_equality: fixes k :: "'a"   and A :: "'a set" assumes "k \<in> A"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<le> k" shows "Max A = k"
sequential_completion_Runs: 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 vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option" assumes "sc_completion s vs"   and "\<And>vs ln. ta_seq_consist P vs (llist_of (convert_RA ln))" shows "\<exists>ttas. mthr.Runs s ttas \<and> ta_seq_consist P vs (lconcat (lmap (\<lambda>(t, ta). llist_of \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))"
weight_spmf_eq_nn_integral_spmf: fixes p :: "'b spmf" shows "ennreal (weight_spmf p) = \<integral>\<^sup>+ x. ennreal (spmf p x) \<partial>count_space UNIV"
eq_eqvt: fixes pi :: "(name \<times> name) list"   and x :: "'a"   and y :: "'a" shows "pi \<bullet> (x = y) = (pi \<bullet> x = pi \<bullet> y)"
move_matrix_mult: fixes A :: "'a matrix"   and B :: "'a matrix"   and j :: "int"   and i :: "int" shows "move_matrix (A * B) j i = move_matrix A j 0 * move_matrix B 0 i"
fmrestrict_fset_null: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmrestrict_fset {||} m = fmempty"
local: fixes \<sigma> :: "'a"   and p :: "'b"   and a :: "'c"   and \<sigma>' :: "'a"   and p' :: "'b" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "P (\<sigma>, p)"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "P (\<sigma>', p')"
intro: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs"   and "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf"   and "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs" shows "l_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_document get_disconnected_document_locs known_ptr type_wf get_host get_host_locs get_shadow_root get_shadow_root_locs get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_tag_name heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
iWeakUntil_iUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<W> t I. Q t) = ((P t'. t' \<U> t I. Q t) \<or> (\<box> t I. P t))"
root_ub: fixes p :: "real poly" assumes "p \<noteq> 0" obtains ub :: "real" where "\<forall>x. poly p x = 0 \<longrightarrow> x < ub" and "\<forall>x\<ge>ub. sgn (poly p x) = sgn_pos_inf p"
monomial_function_def: fixes R :: "('a, 'b) monoid_scheme"   and c :: "'a"   and n :: "nat" shows "monomial_function R c n = (\<lambda>x\<in>carrier R. c \<otimes>\<^bsub>R\<^esub> x [^]\<^bsub>R\<^esub> n)"
hyps_2: fixes stk' :: "'addr val list" assumes "stk = rev vs @ Addr a # stk'"   and "pc < length (compEs2 es)"   and "compEs2 es ! pc = Invoke M (length vs)"   and "n + max_varss es' \<le> length xs"   and "bsoks es n" shows "\<exists>es'a xs'. \<tau>reds1r P t h (es', xs) (es'a, xs') \<and> P,es,h \<turnstile> (es'a, xs') [\<leftrightarrow>] (rev vs @ Addr a # stk', loc, pc, None) \<and> calls1 es'a = \<lfloor>(a, M, vs)\<rfloor>"
filtercomap_neq_bot: fixes F :: "'b filter"   and f :: "'c \<Rightarrow> 'b" assumes "\<And>P. eventually P F \<Longrightarrow> \<exists>x. P (f x)" shows "filtercomap f F \<noteq> bot"
list_induct2': fixes P :: "'b list \<Rightarrow> 'c list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list" assumes "P [] []"   and "\<And>x xs. P (x # xs) []"   and "\<And>y ys. P [] (y # ys)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
leg0_composite_2: shows "\<mu>\<nu>_\<pi>.leg0 = \<pi>.leg0 \<cdot> Prj\<^sub>0"
update_no_set_none: fixes s :: "('c, 'a) state"   and y :: "nat"   and x :: "nat"   and v :: "'a"   and t :: "(nat \<times> linear_poly) list" assumes "Mapping.lookup (\<V> s) y \<noteq> None" shows "Mapping.lookup (foldl (\<lambda>v' e. Mapping.update (lhs e) (rhs_eq_val (\<V> s) x v e) v') (\<V> s) t) y \<noteq> None"
is_invar_def: fixes I :: "'a list \<times> (nat \<Rightarrow> 'a option) \<times> 'b \<Rightarrow> bool" shows "is_invar I \<equiv> I s\<^sub>0 \<and> (\<forall>s s'. reachable s \<and> I s \<and> step s s' \<longrightarrow> I s')"
ann_hoare_case_analysis: fixes C :: "'b ann_com"   and q' :: "'b set" assumes "\<turnstile> C  q'" shows "(\<forall>r f. C = AnnBasic r f \<longrightarrow> (\<exists>q. r \<subseteq> {s. f s \<in> q} \<and> q \<subseteq> q')) \<and> (\<forall>c0 c1. C = AnnSeq c0 c1 \<longrightarrow> (\<exists>q\<subseteq>q'. \<turnstile> c0  pre c1 \<and> \<turnstile> c1  q)) \<and> (\<forall>r b c1 c2. C = AnnCond1 r b c1 c2 \<longrightarrow> (\<exists>q\<subseteq>q'. r \<inter> b \<subseteq> pre c1 \<and> \<turnstile> c1  q \<and> r \<inter> - b \<subseteq> pre c2 \<and> \<turnstile> c2  q)) \<and> (\<forall>r b c. C = AnnCond2 r b c \<longrightarrow> (\<exists>q\<subseteq>q'. r \<inter> b \<subseteq> pre c \<and> \<turnstile> c  q \<and> r \<inter> - b \<subseteq> q)) \<and> (\<forall>r i b c. C = AnnWhile r b i c \<longrightarrow> (\<exists>q\<subseteq>q'. r \<subseteq> i \<and> i \<inter> b \<subseteq> pre c \<and> \<turnstile> c  i \<and> i \<inter> - b \<subseteq> q)) \<and> (\<forall>r b c. C = AnnAwait r b c \<longrightarrow> (\<exists>q\<subseteq>q'. \<parallel>- (r \<inter> b) c q))"
neutr2: fixes x :: "'a" shows "f\<cdot>x\<cdot>z = x"
sublist_length_le: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>f ys" shows "\<^bold>|xs\<^bold>| \<le> \<^bold>|ys\<^bold>|"
nos_inc_def: fixes \<xi> :: "state"   and \<xi>' :: "state" shows "nos_inc \<xi> \<xi>' \<equiv> no \<xi> \<le> no \<xi>'"
h1a: shows "\<forall>i. MPoly_Type.degree (isolate_variable_sparse p x i) x = 0"
simps_9: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case flat_complete_lattice.Bot of flat_complete_lattice.Value x \<Rightarrow> f1 x | flat_complete_lattice.Bot \<Rightarrow> f2 | flat_complete_lattice.Top \<Rightarrow> f3) = f2"