hcju/isabelleps huggingface dataset:
RD_bounds_sublistD_i_c: fixes s :: "('c, bool) fmap"   and Pi :: "(('c, bool) fmap \<times> ('c, bool) fmap) set"   and as :: "(('c, bool) fmap \<times> ('c, bool) fmap) list"   and x :: "nat" assumes "s \<in> valid_states Pi"   and "as \<in> valid_plans Pi"   and "\<forall>y. y \<in> {length p - 1 |p. valid_path Pi p \<and> distinct p} \<longrightarrow> y \<le> x"   and "x \<in> {length p - 1 |p. valid_path Pi p \<and> distinct p}" shows "Min (PLS s as) \<le> Max {length p - 1 |p. valid_path Pi p \<and> distinct p}"
scount_infinite_iff: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "(scount P \<omega> = \<infinity>) = alw (ev P) \<omega>"
s_set_commute: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and x :: "'d"   and y :: "'d" assumes "commute_bpp f (aug_pm_set z i (s_set R s A))"   and "x \<in> addition_set f (aug_pm_set z i (s_set R s A))"   and "y \<in> addition_set f (aug_pm_set z i (s_set R s A))" shows "x \<^sub>f+ y = y \<^sub>f+ x"
impossible_Cons: fixes xs :: "'c list"   and ys :: "'c list"   and x :: "'c" assumes "|xs| \<le> |ys|" shows "(xs = x # ys) = False"
lexext_rev_compat_cons_if_same_length: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "length ys = length xs"   and "lexext_rev gt ys xs" shows "lexext_rev gt (x # ys) (x # xs)"
prems_2: shows "list_all is_cupcake_exp (e # es)"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "comm_ring_hom hom"   and "zero_hom_0 hom" shows "inj_comm_ring_hom hom"
kzkk: shows "ket_zero_k k = Matrix.vec (2 ^ Suc k) (\<lambda>k. if k = 0 then 1 else 0)"
store_typing_imp_mem_agree_Some_3: fixes s :: "s"   and \<S> :: "s_context"   and i :: "nat"   and j :: "nat" assumes "store_typing s \<S>"   and "i < length (s_inst \<S>)"   and "smem_ind s i = Some j" shows "\<exists>x. s_mem \<S> ! j = x \<and> memory (s_inst \<S> ! i) = Some x"
image2p_apply: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f"   and R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'f" shows "BNF_Greatest_Fixpoint.image2p f g R x y = (\<exists>x' y'. R x' y' \<and> f x' = x \<and> g y' = y)"
open_on_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "open_on P Q A = (\<forall>C. chain_on P C A \<and> C \<noteq> {} \<and> (\<exists>x\<in>A. glb P C x \<and> Q x) \<longrightarrow> (\<exists>y\<in>C. Q y))"
fds_zeta_def: shows "fds_zeta = (\<chi>_. (1::'b))"
get_mod: fixes p :: "nat"   and a :: "nat"   and m :: "nat" assumes "p < a" shows "p = (m * a + p) mod a"
a1: shows "low_equal s1 s2 \<and> t1 = user_reg_mod w curr_win rd s1 \<and> t2 = user_reg_mod w curr_win rd s2"
simps: fixes C :: "'b literal set"   and X :: "'b trm set" shows "cl_ecl (Ecl C X) = C"
mod_minus_eq: fixes a :: "'a"   and b :: "'a" shows "- (a mod b) mod b = - a mod b"
a2: shows "as\<^sub>1 = map (\<lambda>v. (v, True)) (add_effects_of op)"
gauss_int_norm_mult: fixes z :: "gauss_int"   and u :: "gauss_int" shows "gauss_int_norm (z * u) = gauss_int_norm z * gauss_int_norm u"
elements_mat_h_def: fixes A :: "(('c, 'd) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "elements_mat_h A = range (\<lambda>(i, j). A $h i $h j)"
length_getTvars: fixes \<sigma>l :: "'tp list" shows "length (GE.getTvars \<sigma>l) = length \<sigma>l"
extremeD_2: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and e :: "'c"   and x :: "'c" assumes "extreme X r e"   and "x \<in> X" shows "r x e"
under_dim: shows "rat_poly.row_length (brickmat brick.under) = 4"
Swap: fixes B :: "fm"   and H :: "fm set"   and A :: "fm" assumes "insert (Neg B) H \<turnstile> A" shows "insert (Neg A) H \<turnstile> B"
scale_ms_def: fixes c :: "real"   and F :: "'a" shows "scale_ms c F = const_expansion c * F"
cpx_sqr_mat_to_cpx_mat_def: fixes M :: "cpx_sqr_mat" shows "cpx_sqr_mat_to_cpx_mat M \<equiv> Rep_cpx_sqr_mat M"
id_2: shows "fs' = fs[i := fs ! (i - 1), i - 1 := fs ! i]"
less_le_not_le: fixes x :: "'c list"   and y :: "'c list" shows "strict_prefix x y = (prefix x y \<and> \<not> prefix y x)"
simps: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and a1 :: "'b literal multiset list"   and a2 :: "'b literal multiset"   and a3 :: "'b multiset list"   and a4 :: "'b list"   and a5 :: "'b literal multiset" assumes "ground_resolution_with_selection S" shows "ground_resolution_with_selection.ord_resolve S a1 a2 a3 a4 a5 = (\<exists>CAs n Cs AAs As D. a1 = CAs \<and> a2 = D + negs (mset As) \<and> a3 = AAs \<and> a4 = As \<and> a5 = \<Sum>\<^sub># (mset Cs) + D \<and> length CAs = n \<and> length Cs = n \<and> length AAs = n \<and> length As = n \<and> n \<noteq> 0 \<and> (\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)) \<and> (\<forall>i<n. AAs ! i \<noteq> {#}) \<and> (\<forall>i<n. \<forall>A\<in>#AAs ! i. A = As ! i) \<and> ground_resolution_with_selection.eligible S As (D + negs (mset As)) \<and> (\<forall>i<n. ground_resolution_with_selection.strictly_maximal_wrt (As ! i) (Cs ! i)) \<and> (\<forall>i<n. S (CAs ! i) = {#}))"
ts_rep_def: shows "ts_rep = (po, re, cl, dy, ps, sd)"
simps_14: fixes x4 :: "int"   and x2 :: "int" shows "Smaller' x4 \<noteq> Intv' x2"
encode_all_operator_effects_is_cnf: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>" shows "is_cnf (encode_all_operator_effects \<Pi> (\<Pi>\<^sub>\<O>) t)"
fold_A_opp_fold_A_chamber_systems_1: shows "ChamberComplex.chamber_system A = ChamberComplex.chamber_system (fold_A \<turnstile> A) \<union> ChamberComplex.chamber_system (opp_fold_A \<turnstile> A)"
sn: shows "sign_num (a * x\<^sup>2 + b * x + c) \<noteq> sign_num (a * y\<^sup>2 + b * y + c)"
left_commute: fixes b :: "'b \<times> 'c"   and a :: "'b \<times> 'c"   and c :: "'b \<times> 'c" shows "sup_prod b (sup_prod a c) = sup_prod a (sup_prod b c)"
nil_reduced_word_for_0: fixes A :: "'a set" shows "reduced_word_for A (0::'a) []"
in_set_in_list: fixes a :: "('a, 'b) Combinators"   and p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "a \<in> set p"   and "all_in_list p l" shows "in_list a l"
sys_block_sizes_obtain_bl: fixes x :: "int" assumes "x \<in> sys1.sys_block_sizes" shows "\<exists>bl\<in>#\<B>. int (card bl) = x"
defined_bool_split: fixes \<tau> :: "'d state \<times> 'd state"   and A :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> \<delta> A) = (\<tau> \<Turnstile> A \<or> \<tau> \<Turnstile> not A)"
cInf_atLeastAtMost: fixes y :: "'a multiset"   and x :: "'a multiset" assumes "y \<subseteq># x" shows "Inf (subset_mset.atLeastAtMost y x) = y"
lec_rec_constraints_of_mat_vec: fixes A :: "rat mat"   and b :: "rat vec" shows "lec_rel ` constraints_of_mat_vec A b \<subseteq> {Leq_Rel}"
simple_functionD_1: fixes M :: "'d measure"   and g :: "'d \<Rightarrow> 'e" assumes "simple_function M g" shows "finite (g ` space M)"
hom_def: fixes G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" shows "hom G H = {h \<in> carrier G \<rightarrow> carrier H. \<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y}"
l_set_tag_name_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and 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_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs known_ptr known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<equiv> l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs \<and> l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_tag_name set_tag_name_locs known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
inangle__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A InAngle B C D" shows "Coplanar A B C D"
add_neg_numeral_special_3: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (numeral.numeral one plus m) (uminus one) = neg_numeral.sub minus plus one m num.One"
metric_LIMSEQ_D: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c"   and r :: "real" assumes "X \<longlonglongrightarrow> L"   and "0 < r" shows "\<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
\<C>_Low: fixes mem :: "'Var\<^sub>C \<Rightarrow> 'Val" shows "\<forall>x\<in>\<C>\<^sub>C. dma\<^sub>C mem x = Low"
Suc_rtos_div_r_3: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i mod s < s - Suc 0"   and "i \<le> r * s - Suc 0"   and "Suc (rtos r s i) < r * s"   and "i div s < r - Suc 0" shows "Suc (rtos r s i) div r = i mod s"
list_all2_lengthD: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list" assumes "list_all2 P xs ys" shows "length xs = length ys"
finite_nan: fixes a :: "('a, 'b) IEEE.float" assumes "is_finite a" shows "\<not> is_nan a"
Red_F_subset_Liminf: fixes Ns :: "'f set llist"   and i :: "nat" assumes "chain (\<rhd>) Ns"   and "enat i < llength Ns" shows "Red_F (lnth Ns i) \<subseteq> Red_F (Liminf_llist Ns)"
module_hom_iff_linear: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f" shows "module_hom s1 s2 f = linear s1 s2 f"
\<Phi>o_def: shows "\<Psi>o = (\<lambda>a\<in>Collect S'.ide. S.mkIde (\<psi> ` S'.set a))"
preserves_seq: fixes g :: "'d"   and f :: "'d" assumes "D.seq g f" shows "D.seq (D.map g) (D.map f)"
get_root_node_parent_same: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some ptr" shows "h \<turnstile> Shadow_DOM.get_root_node (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 child) \<rightarrow>\<^sub>r root = h \<turnstile> Shadow_DOM.get_root_node ptr \<rightarrow>\<^sub>r root"
closure_open_segment: fixes a :: "'b"   and b :: "'b" shows "closure (open_segment a b) = (if a = b then {} else closed_segment a b)"
init_invariant: shows "consistent_context \<Gamma>\<^sub>1"
strict_trans2: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "b \<prec> a"   and "c \<preceq> b" shows "c \<prec> a"
y_def: shows "y = would_be_Cauchy (\<lambda>n. rep_Bonk_Schramm_extension (X n))"
seq_power_Suc: fixes a :: "'a::refinement_lattice"   and n :: "nat" shows "(a::'a::refinement_lattice) \<^sup>;^ Suc (n::nat) = a ; (a \<^sup>;^ n)"
Disj_I2: fixes H :: "fm set"   and B :: "fm"   and A :: "fm" assumes "H \<turnstile> B" shows "H \<turnstile> A OR B"
intro: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "ordered_powerprod ord ord_strict" shows "gd_powerprod ord ord_strict"
large_trans: fixes fn :: "'f \<times> nat"   and gm :: "'f \<times> nat" assumes "ssimple"   and "large fn"   and "snd (prc gm fn)" shows "large gm"
case_rews_1: fixes f :: "int \<rightarrow> 'a" shows "Integer_case\<cdot>f\<cdot>\<bottom> = \<bottom>"
simps: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and a :: "'c stream" shows "ev \<phi> a = ((\<exists>xs. a = xs \<and> \<phi> xs) \<or> (\<exists>xs. a = xs \<and> ev \<phi> (stl xs)))"
order_refl: fixes x :: "'a" shows "x \<preceq>\<^sub>R x"
evalRR_simps_4: fixes a :: "'a"   and x :: "'a"   and d :: "nat" assumes "a \<in> carrier R"   and "x \<in> carrier R" shows "eval R R id x (monom P a d) = a \<otimes> x [^] d"
sorted_list_of_set_def: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq = folding.F (linorder.insort_key less_eq (\<lambda>x. x)) []"
orderE: fixes b :: "'a"   and a :: "'a" assumes "b \<sqsubseteq> a" obtains "a = a \<squnion> b"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'b"   and bool :: "bool" shows "h (case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = (case bool of True \<Rightarrow> h f1 | False \<Rightarrow> h f2)"
pgwt_type_map_2: fixes t :: "('fun, 'var) Term.term"   and g :: "'fun"   and Y :: "('fun, 'atom) Term.term list" assumes "public_ground_wf_term t"   and "\<Gamma> t = Fun g Y" shows "\<exists>X. t = Fun g X \<and> map \<Gamma> X = Y"
assms_1: shows "\<guillemotleft>f' : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"
G_upper: fixes x :: "real"   and n :: "nat" assumes "0 < x"   and "x \<le> 1"   and "2 \<le> n" shows "G x \<le> Gamma_series x n * (1 + x / real n)"
integrable_on_def: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'c set" shows "(f integrable_on i) = (\<exists>y. (f has_integral y) i)"
basis_col_space_iarrays_def: fixes A :: "'b iarray iarray" shows "basis_col_space_iarrays A = basis_row_space_iarrays (transpose_iarray A)"
IHR: shows "real (time (mergesort_tm f r)) \<le> mergesort_recurrence (length r)"
simps: fixes A :: "('e, 'f, 'g) FSM"   and B :: "('e, 'f, 'h) FSM"   and FAIL :: "'g \<times> 'h"   and AB :: "('e, 'f, 'g \<times> 'h) FSM" shows "productF A B FAIL AB = (inputs A = inputs B \<and> fst FAIL \<notin> nodes A \<and> snd FAIL \<notin> nodes B \<and> AB = \<lparr>succ = \<lambda>a (p1, p2). if p1 \<in> nodes A \<and> p2 \<in> nodes B \<and> fst a \<in> inputs A \<and> snd a \<in> outputs A \<union> outputs B then if succ A a p1 = {} \<and> succ B a p2 \<noteq> {} then {FAIL} else succ A a p1 \<times> succ B a p2 else {}, inputs = inputs A, outputs = outputs A \<union> outputs B, initial = (initial A, initial B)\<rparr>)"
s_is_bounded_by_g1_and_g2: shows "s = {(x, y). (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i) \<and> (\<forall>i\<in>Basis. g1 x \<bullet> i \<le> y \<bullet> i \<and> y \<bullet> i \<le> g2 x \<bullet> i)}"
WhileFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int option"   and c :: "com" assumes "vars b \<subseteq> dom s"   and "\<not> pbval b s" shows "(WHILE b DO c, s) \<Rightarrow>\<^sub>A Suc 0 \<Down> s"
length_trace_parallel_plan_sas_plus_lte_length_plan_plus_one: fixes I :: "'c \<Rightarrow> 'd option"   and \<psi> :: "('c, 'd) sas_plus_operator list list" shows "length (trace_parallel_plan_sas_plus I \<psi>) \<le> length \<psi> + 1"
find_base_vector_not_1_pivot_positions: shows "snd ` set (pivot_positions A) \<noteq> {0..<n}"
runit_char: fixes a :: "'a" assumes "ide a" shows "\<r>[a] \<otimes> \<I> = (a \<otimes> \<iota>) \<cdot> \<a>[a, \<I>, \<I>]"
dim_ty_sem_Time_def: fixes uu :: "T itself" shows "dim_ty_sem uu = \<^bold>T"
preserves_ide: fixes a :: "'b" assumes "D.ide a" shows "D.ide (trg\<^sub>D a)"
1: fixes m :: "nat" assumes "0 < m" shows "inv\<^bsub>residue_ring (p ^ m)\<^esub> x m \<in> carrier (residue_ring (p ^ m))"
card_Un_le: fixes A :: "'a set"   and B :: "'a set" shows "card (A \<union> B) \<le> card A + card B"
ne'_1: shows "a' # as' \<in> lists (- {[]})"
leg_simps_1: shows "C.arr assoc'_fgh.dom.leg0"
Trans: shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> T"
de_morgan1: fixes x :: "'a"   and y :: "'a" shows "!(!x \<cdot> !y) = t x \<oplus> t y"
assms: shows "A \<in>\<^sub>\<circ> VPow B"
Pj_hom0: fixes N :: "'a set"   and x :: "'a"   and y :: "'a" assumes "G \<triangleright> N"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "Pj G N (x \<cdot> y) = Pj G N x \<cdot>\<^bsub>G / N\<^esub> Pj G N y"
add_less_cancel_left0: fixes x :: "V"   and z :: "V" shows "(x \<subset>\<^sub>\<circ> x + z) = (0 \<subset>\<^sub>\<circ> z)"
le_inf_def: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2 \<equiv> \<forall>v\<^sub>1. v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1 \<longrightarrow> v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
vertices_face_code: fixes vs :: "nat list"   and f :: "facetype" shows "vertices_face (Face vs f) = vs"
x: shows "x \<in> {real_of_float lx..real_of_float ux}"
senior_same_state: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "semi_mojmir_def.token_run \<delta> q\<^sub>0 w (semi_mojmir_def.senior \<delta> q\<^sub>0 w x n) n = semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n"
ord_gpv_simps: fixes f :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf"   and g :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf" shows "ord_gpv (GPV f) (GPV g) = ord_spmf (rel_generat (=) (=) (rel_fun (=) ord_gpv)) f g"
assms_3: shows "are_all_operators_applicable_in I ops \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent ops"
pt_eqvt_fun2: fixes f :: "'a \<Rightarrow> 'c" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)" shows "(supp f = {}) = (\<forall>pi x. pi \<bullet> f x = f (pi \<bullet> x))"
open_inter: fixes U :: "'d set"   and V :: "'d set" assumes "U = {}"   and "V = {}" shows "U \<inter> V = {}"
continuous_on_pre_zeta': fixes A :: "'b set"   and f :: "'b \<Rightarrow> complex"   and a :: "real" assumes "continuous_on A f"   and "0 < a" shows "continuous_on A (\<lambda>x. pre_zeta a (f x))"
approx_def: fixes x :: "'a star"   and y :: "'a star" shows "(x \<approx> y) = (x - y \<in> Infinitesimal)"
assms_2: shows "prv (eql (pls (Num m) (Num n)) (Num k))"
choose_good: fixes v :: "'a" assumes "v \<in> S" shows "choose v \<in> good v"
assms_3: shows "\<mu> \<star> \<nu> \<noteq> null"
scale_eq_0_iff: fixes a :: "complex"   and x :: "'c" shows "(a *\<^sub>C x = (0::'c)) = (a = 0 \<or> x = (0::'c))"
smult_assoc: fixes r :: "('b, 'a) aezfun"   and s :: "('b, 'a) aezfun"   and m :: "'c" assumes "r \<in> FG"   and "s \<in> FG"   and "m \<in> V" shows "r \<cdot> (s \<cdot> m) = r * s \<cdot> m"
small_iff_range: fixes X :: "V set" shows "small X = (X \<in> range elts)"
set_iterator_linord_foldli_conv: fixes iti :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'a set" shows "set_iterator_linord iti S = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> sorted l0 \<and> iti = foldli l0)"
hext_refl: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a" shows "heap_base.hext typeof_addr h h"
IH: shows "T ^^ n \<in> tail_algebra (m + n) \<rightarrow>\<^sub>M tail_algebra m"
ipd'_1: shows "\<rho>' m' = ipd (\<rho>' k')"
minus: shows "(R ===> R ===> R) (arith_ops_record.minus ops) (-)"
closure_of_eq_empty_gen: fixes X :: "'c topology"   and S :: "'c set" shows "(X closure_of S = {}) = disjnt (topspace X) S"
vpsubsetI: fixes A :: "V"   and B :: "V"   and x :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "x \<notin>\<^sub>\<circ> A"   and "x \<in>\<^sub>\<circ> B" shows "A \<subset>\<^sub>\<circ> B"
snd_image_Sigma: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'a set" shows "snd ` Sigma A B = \<Union> (B ` A)"
a_inv_def: fixes R :: "('b, 'c) ring_scheme" shows "a_inv R = m_inv (add_monoid R)"
len_gt1: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> satisfies_properties p qs (get_subsets (calculate_data p qs)) (get_signs (calculate_data p qs)) (get_matrix (calculate_data p qs))"
module_hom_scale_left: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "module scale" shows "module_hom (*) scale (\<lambda>r. scale r x)"
subtype_x_Required: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<sqsubset> \<sigma>[1]" obtains \<tau>' :: "'a basic_type" where "\<tau> = \<tau>'[1]" and "\<tau>' \<sqsubset>\<^sub>B \<sigma>"
interp_ops_def: fixes ops :: "('c \<times> ('c, 'd) list_op) list" shows "interp_ops ops \<equiv> foldl interp_op ([], Map.empty) ops"
map_nth_append2: fixes ys :: "'c list"   and d :: "nat"   and xs :: "'c list" assumes "length ys = d" shows "map ((!) (xs @ ys)) [length xs..<length xs + d] = ys"
oghoare_step: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and c' :: "('e, 'd, 'f) com"   and t :: "'e"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and a :: "('e, 'd, 'f) ann"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>\<turnstile> (c, Normal s) \<rightarrow> (c', Normal t)"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> a c Q, A"   and "s \<in> pre a" shows "\<exists>a'. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> a' c' Q, A \<and> t \<in> pre a' \<and> (\<forall>as. assertionsR \<Gamma> \<Theta> Q A a' c' as \<longrightarrow> assertionsR \<Gamma> \<Theta> Q A a c as) \<and> (\<forall>pm cm. atomicsR \<Gamma> \<Theta> a' c' (pm, cm) \<longrightarrow> atomicsR \<Gamma> \<Theta> a c (pm, cm))"
braun_of_def: fixes x :: "'b"   and n :: "nat" shows "braun_of x n = fst (braun2_of x n)"
vseq_implies_hpar_2: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "C.seq \<nu> \<mu>" shows "trg\<^sub>C \<nu> = trg\<^sub>C \<mu>"
AddSA_CompFun_the2: fixes S' :: "'a"   and SA :: "('a, 'b, 'c) seqauto"   and A :: "('a, 'b, 'c) hierauto"   and S :: "'a" assumes "S' \<in> States SA"   and "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "the (CompFun (A [++] (S, SA)) S') = {}"
scale_right_diff_distrib: fixes a :: "'a"   and x :: "'b"   and y :: "'b" shows "a *s (x - y) = a *s x - a *s y"