hcju/isabelleps huggingface dataset:
lowner_le_smultc: fixes c :: "complex"   and A :: "complex mat"   and B :: "complex mat"   and n :: "nat" assumes "0 \<le> c"   and "A \<le>\<^sub>L B"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "c \<cdot>\<^sub>m A \<le>\<^sub>L c \<cdot>\<^sub>m B"
id_to_iso_locally_ringed_spaces: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" assumes "locally_ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "iso_locally_ringed_spaces S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S is_open \<FF> \<rho> b add_str mult_str zero_str one_str (identity S) (\<lambda>U. identity (\<FF> U))"
snoc_outer_1: fixes ts1 :: "t list"   and ts2 :: "t list"   and vs :: "e list" assumes "ts = ts1 @ ts2"   and "const_list vs"   and "\<S>\<bullet>\<C> \<turnstile> vs : [] _> ts1 @ ts2" shows "\<exists>vs1 vs2. \<S>\<bullet>\<C> \<turnstile> vs1 : [] _> ts1 \<and> \<S>\<bullet>\<C> \<turnstile> vs2 : ts1 _> ts1 @ ts2 \<and> vs = vs1 @ vs2 \<and> const_list vs1 \<and> const_list vs2"
dgrad_set_le_red_single: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "dickson_grading d"   and "red_single p q f t" shows "dgrad_set_le d {t} (pp_of_term ` keys p)"
tn_fn_suc: shows "mod_type_class.to_nat (mod_type_class.from_nat (Suc k)) = Suc k"
non_zero_keysEx: fixes p :: "int mpoly" assumes "p \<noteq> 0" shows "\<exists>a. a \<in> keys (mapping_of p)"
store_sub2_low_equal_sub3: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word32"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and y :: "'b sparc_state"   and ya :: "'b sparc_state"   and yb :: "'b sparc_state" assumes "low_equal s1 s2 \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s1 = Some y \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s2 = Some ya \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) y) y = Some yb \<and> memory_write 10 (addr + 4) 15 (user_reg_val curr_win (rd OR 1) ya) ya = None" shows "False"
clamp_continuous_on: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "continuous_on (cbox a b) f" shows "continuous_on S (\<lambda>x. f (clamp a b x))"
induct: fixes x1 :: "'a"   and x2 :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "(x1, x2) \<in> r\<^sup>+"   and "\<And>a b. (a, b) \<in> r \<Longrightarrow> P a b"   and "\<And>a b c. \<lbrakk>(a, b) \<in> r\<^sup>+; P a b; (b, c) \<in> r\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
oracle_def: fixes \<sigma> :: "'b \<Rightarrow> bool list option"   and x :: "'b" shows "oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len_plain) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs))) | Some bs \<Rightarrow> return_spmf (bs, \<sigma>))"
simps_7: fixes f :: "'c \<Rightarrow> 'a"   and p :: "('c, 'b) form" shows "psubst f (Neg p) = Neg (psubst f p)"
iteratei_correct: fixes S :: "'s3" assumes "s3.invar S" shows "set_iterator (s3.iteratei S) (s3.\<alpha> S)"
a: shows "ia < dim_vec (row A i @\<^sub>v [b $ i]\<^sub>v)"
equiv_Eps_in: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and X :: "'c set" assumes "equiv A r"   and "X \<in> A // r" shows "(SOME x. x \<in> X) \<in> X"
span_eq_iff: fixes s :: "'c set" shows "(ideal s = s) = ideal.subspace s"
CH_Ch_singular: fixes h :: "'c"   and A :: "'a set" assumes "UNIV = {h}" shows "CH A = Ch h A"
contour_integral_circlepath: fixes r :: "real"   and z :: "complex" assumes "0 < r" shows "contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>"
addrow_mat_carrier: fixes n :: "nat"   and a :: "'c"   and k :: "nat"   and l :: "nat" shows "addrow_mat n a k l \<in> carrier_mat n n"
has_integral_refl_2: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "(f has_integral (0::'b)) {a}"
order_iff_strict: fixes b :: "'a list"   and a :: "'a list" shows "prefix b a = (strict_prefix b a \<or> a = b)"
assms_2: shows "\<forall>m. position (p @ [m])"
minus_add_distrib: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.ab_group_add plus zero minus uminus" shows "uminus (plus a b) = plus (uminus a) (uminus b)"
simps_14: fixes va :: "nat" shows "[Suc va]\<^sub>2 +\<^sub>2 \<infinity>\<^sub>2 = \<top>\<^sub>2"
0: shows "(\<lambda>\<phi>. rawpsubst (subst \<phi> t x) txs) ` F = (\<lambda>\<phi>. rawpsubst \<phi> txs) ` (\<lambda>\<phi>. subst \<phi> t x) ` F"
powsum_00: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.dioid_one_zero plus times one zero less_eq less" shows "dioid_one_zero.powsum plus times one zero x 0 0 = one"
T_up3_def: fixes x :: "'a"   and sib :: "'a tree"   and twist :: "bool"   and u :: "'a up2" shows "T_up3 x sib twist u = time (up3_tm x sib twist u)"
eSuc_plus_1: fixes n :: "enat" shows "eSuc n = n + 1"
simple_path_def: fixes g :: "real \<Rightarrow> 'b" shows "simple_path g = (path g \<and> (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. g x = g y \<longrightarrow> x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0))"
minus_1_even_eqI: fixes k :: "nat"   and l :: "nat" assumes "even k = even l" shows "(- (1::'b)) ^ k = (- (1::'b)) ^ l"
app_vimageI1: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" shows "x \<in>\<^sub>\<circ> r `\<^sub>\<circ> A"
flincomb_closed: fixes rs :: "'f list"   and ms :: "'w list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> W" shows "rs \<bullet>\<sharp>\<star> ms \<in> W"
semigroup_add_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "semigroup_add_hom hom \<equiv> plus_hom hom"
abs_0_eq: fixes abs :: "'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 less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "(zero = abs a) = (a = zero)"
simps_3: fixes v :: "'b aexp"   and va :: "'b aexp" shows "contains_bool (Eq v va) = False"
strong_postdominate_antisym: fixes n' :: "'node"   and n :: "'node" assumes "n' strongly-postdominates n"   and "n strongly-postdominates n'" shows "n = n'"
of_ocircline_of_circline: fixes H :: "circline" shows "of_ocircline (of_circline H) = H"
from_nat_0: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs 0 = zero"
sphere_circle_set_def: fixes \<alpha> :: "plane" shows "sphere_circle_set \<alpha> = {A. on_sphere_circle \<alpha> A}"
learn_totalE_1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_total \<psi> U s" shows "environment \<psi> U s"
map_nth_default: fixes x :: "'a"   and xs :: "'a list" shows "map (nth_default x xs) [0..<length xs] = xs"
simps: fixes i :: "nat" shows "gso_int_tail i = (if i = 0 then fs ! 0 else Let (\<mu>' 0 0 \<cdot>\<^sub>v fs ! i - \<mu>' i 0 \<cdot>\<^sub>v fs ! 0) (gso_int_tail' i 1))"
LeftDerivationLadder_ladder_\<gamma>_last: fixes \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list" assumes "LeftDerivationLadder \<alpha> D L \<gamma>" shows "\<gamma> = ladder_\<gamma> \<alpha> D L (length L - 1)"
a: shows " t1 \<longrightarrow>\<^sub>\<beta> t2"
t: shows "\<forall>T. gap i T \<le> gap i t"
it_is_pre_condition: fixes C :: "'b hrel"   and Q :: "'b upred" shows "\<lbrace>C wp Q\<rbrace> C \<lbrace>Q\<rbrace>\<^sub>u"
interval_extension2_def: fixes F :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option"   and f :: "real \<Rightarrow> real \<Rightarrow> real" shows "interval_extension2 F f = (\<forall>ivl1 ivl2 ivl. F ivl1 ivl2 = Some ivl \<longrightarrow> (\<forall>x y. x \<in>\<^sub>r ivl1 \<longrightarrow> y \<in>\<^sub>r ivl2 \<longrightarrow> f x y \<in>\<^sub>r ivl))"
finite_Int: fixes F :: "'d set"   and G :: "'d set" assumes "finite F \<or> finite G" shows "finite (F \<inter> G)"
simps_1: fixes b :: "bool"   and w :: "int" shows "bin_sc 0 b w = of_bool b + 2 * (w div 2)"
set_sel_2: fixes x :: "'c"   and a :: "'c stream" assumes "x \<in> sset (stl a)" shows "x \<in> sset a"
trimmed_uminus_iff: fixes F :: "'a" shows "trimmed (- F) = trimmed F"
UP_subring_compose: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP R)"   and "f \<in> carrier (UP R)"   and "\<And>n. g n \<in> S"   and "\<And>n. f n \<in> S" shows "Cring_Poly.compose R f g = Cring_Poly.compose (R\<lparr>carrier := S\<rparr>) f g"
assms: fixes C :: "'a set set" assumes "\<forall>t\<in>C. open t"   and "s \<subseteq> \<Union> C"   and "countable C" shows "\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union> C'"
IH: assumes "\<And>C. C \<in> set CS \<Longrightarrow> dims C = ds" shows "A \<otimes> listsum ds CS = listsum (dims A @ ds) (map ((\<otimes>) A) CS)"
exec_meth_take_xt: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'a instr list"   and ins' :: "'a instr list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'b"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and xcp :: "'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option" assumes "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P (ins @ ins') (xt' @ shift (length ins) xt) t h (stk, loc, pc, xcp) ta h' s'"   and "pc < length ins" shows "JVM_heap_base.exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P ins xt' t h (stk, loc, pc, xcp) ta h' s'"
times_of_nat_minus_left: fixes m :: "nat"   and n :: "nat"   and l :: "hmultiset" shows "(of_nat m - of_nat n) * l = of_nat m * l - of_nat n * l"
summable2: shows "(\<lambda>(p, n). f n p) abs_summable_on Sigma {p. prime p} atLeast"
preserves_runit_1: fixes f :: "'c" assumes "C.ide f" shows "F \<r>\<^sub>C[f] = \<r>\<^sub>D[F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (unit (src\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f))"
ex_gt: shows "\<forall>x. count (mset ys) x < count (mset xs) x \<longrightarrow> (\<exists>y. gt y x \<and> count (mset xs) y < count (mset ys) y)"
diff_left: fixes a :: "'d \<Rightarrow>\<^sub>L 'c"   and a' :: "'d \<Rightarrow>\<^sub>L 'c"   and b :: "'d" shows "blinfun_apply (a - a') b = blinfun_apply a b - blinfun_apply a' b"
pdevs_val_zero_pdevs: fixes e :: "nat \<Rightarrow> real" shows "pdevs_val e zero_pdevs = (0::'b)"
unit_in_vhom: fixes a :: "'a" assumes "S.obj a" shows "S.in_hom \<i>[a] (S.hcomp a a) a"
A1: shows "nat_to_sch (sch_to_nat sch1) = sch1"
distinct_13: shows "cospan_shape.AA \<noteq> cospan_shape.TT"
expand: fixes pred :: "'a pred"   and pred' :: "'a pred" assumes "eval pred = eval pred'" shows "pred = pred'"
IH: fixes gpv :: "('a, 'call, 'ret) gpv"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "set_spmf (inline1' gpv s) \<subseteq> {Inr (out, callee', rpv') |out callee' rpv'. \<exists>call\<in>outs_\<I> \<I>. \<exists>s. I s \<and> (\<forall>x\<in>responses_\<I> \<I>' out. callee' x \<in> sub_gpvs \<I>' (callee s call))} \<union> {Inl (x, s') |x s'. x \<in> results_gpv \<I> gpv \<and> I s'}"
card_sccs_verts: shows "card G.sccs_verts = card G.sccs"
enum_in: fixes i :: "nat" assumes "i \<le> n" shows "s.enum i \<in> s"
setbydecision_setbydecision_all_Allow: fixes iface :: "iface"   and rs :: "'a common_primitive rule list" shows "setbydecision TYPE('b) iface rs FinalAllow - setbydecision_all TYPE('b) iface rs FinalDeny = setbydecision TYPE('b) iface rs FinalAllow"
lucas_lehmer_to_real_simps1_4: fixes x :: "int \<times> int" shows "lucas_lehmer_to_real (\<ominus>\<^bsub>lucas_lehmer_ring\<^esub> x) = - lucas_lehmer_to_real x"
filter_M_subset: fixes h :: "'a"   and P :: "'c \<Rightarrow> ('a, 'b, bool) prog"   and xs :: "'c list"   and ys :: "'c list" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys" shows "set ys \<subseteq> set xs"
pcr_real_int_def: shows "pcr_real_int \<equiv> rel_prod (=) (=) OO cr_real_int"
special_axioms_imp_Pf: fixes \<alpha> :: "fm"   and e :: "(name, hf) finfun" assumes "\<alpha> \<in> special_axioms" shows "Pf \<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e"
F_def: shows "(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<sqinter>) ((c::'a::refinement_lattice)\<^sup>\<star>)"
rep_eq: fixes x :: "('d, 'c) sq_matrix" shows "col x = (\<lambda>j. \<chi>i. to_fun x i j)"
iTrigger_iWeakSince_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<T> t I. Q t) = (Q t'. t' \<B> t I. Q t \<and> P t)"
lin_red_Un: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and G :: "('t \<Rightarrow>\<^sub>0 'd) set"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" shows "lin_red (F \<union> G) p q = (lin_red F p q \<or> lin_red G p q)"
intro: fixes set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and type_wf :: "(_) heap \<Rightarrow> bool" assumes "\<And>ptr children h h'. writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h'"   and "\<And>w object_ptr h h'. \<lbrakk>w \<in> set_child_nodes_locs object_ptr; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> object_ptr_kinds h = object_ptr_kinds h'"   and "\<And>w object_ptr h h'. \<lbrakk>w \<in> set_child_nodes_locs object_ptr; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> type_wf h = type_wf h'" shows "l_set_child_nodes type_wf set_child_nodes set_child_nodes_locs"
that: fixes h :: "nat \<Rightarrow> nat \<Rightarrow> real" assumes "\<And>x y. \<lbrakk>x \<le> n; y \<le> n\<rbrakk> \<Longrightarrow> 0 \<le> h x y"   and "\<And>x y. \<lbrakk>0 < h x y; x \<le> n; y \<le> n\<rbrakk> \<Longrightarrow> (x, y) \<in> R"   and "\<And>x. x \<le> n \<Longrightarrow> f x = sum (h x) {..n}"   and "\<And>y. y \<le> n \<Longrightarrow> g y = (\<Sum>x\<le>n. h x y)" shows "thesis"
alpha_in_var: fixes x :: "'c \<Longrightarrow> 'd" shows "x ;\<^sub>L fst\<^sub>L = $x"
next_in_basics_1: fixes u :: "'a" assumes "u \<in> {a..<b}" shows "next_in A u \<in> A"
smult_assoc: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and r :: "'a"   and s :: "'a"   and m :: "'b" assumes "VectorSpace smult V"   and "r \<in> UNIV"   and "s \<in> UNIV"   and "m \<in> V" shows "smult r (smult s m) = smult (r * s) m"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a' \<Rightarrow> a\<guillemotright>"   and "local.iso (R f)" shows "local.iso f"
simps: fixes v :: "nat \<Rightarrow> 'c"   and b :: "(nat \<Rightarrow> 'c option) \<times> (nat \<Rightarrow> 'c option)" shows "v \<Turnstile>\<^sub>b b = (\<forall>x. in_bounds x v b)"
qPsubst_preserves_alpha2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and rho'' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "qGood X"   and "qGoodEnv rho'"   and "qGoodEnv rho''"   and "rho' &= rho''" shows "X #[[rho']] #= X #[[rho'']]"
ctr_transfer_1: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_option R None None"
map_project_ENC: fixes X :: "nat set"   and n :: "nat"   and Z :: "('a \<times> bool list) list set" assumes "X \<subseteq> {0..<n + 1}"   and "Z \<subseteq> lists ((set \<circ> \<sigma> \<Sigma>) (n + 1))" shows "SAMEQUOT (any, replicate n False) (map \<pi> ` (Z \<inter> lang (n + 1) (ENC (n + 1) X))) = (if 0 \<in> X then SAMEQUOT (any, replicate n False) (map \<pi> ` (Z \<inter> lang (n + 1) (ENC (n + 1) {0}))) \<inter> lang n (ENC n ((\<lambda>x. x - 1) ` (X - {0}))) else SAMEQUOT (any, replicate n False) (map \<pi> ` Z) \<inter> lang n (ENC n ((\<lambda>x. x - 1) ` (X - {0}))))"
monom_eq_0_iff: fixes a :: "'b"   and n :: "nat" shows "(monom a n = 0) = (a = (0::'b))"
assms: shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<bullet> A"
take_bit_tightened: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and b :: "'b"   and m :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit"   and "take_bit n a = take_bit n b"   and "m \<le> n" shows "take_bit m a = take_bit m b"
is_glb_singleton: fixes x :: "'a" shows "{x} >>| x"
contraposition_2: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi> in v] = [\<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi> in v]"
ide_dom: fixes f :: "'d" assumes "Dop.arr f" shows "Dop.ide (Dop.dom f)"
map_transfer: fixes R1b :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and S1d :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2b :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and S2d :: "'f \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun (rel_fun R1b S1d) (rel_fun (rel_fun R2b S2d) (rel_fun (rel_gpv R1b R2b) (rel_gpv S1d S2d))) map_gpv map_gpv"
f6: shows "rk {a, c, A', C', \<beta>} = 3"
demand_suffices: fixes f :: "C \<rightarrow> 'a" assumes "f\<cdot>C\<^sup>\<infinity> \<noteq> \<bottom>" shows "f\<cdot>(demand f) \<noteq> \<bottom>"
gt_hd_irrefl: fixes \<zeta> :: "('s, 'v) hd" shows "\<not> \<zeta> >\<^sub>h\<^sub>d \<zeta>"
T_append: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and s :: "'c list"   and q :: "'c"   and a :: "nat \<times> nat list" assumes "length qs = length as" shows "Move_to_Front.T s (qs @ [q]) (as @ [a]) = Move_to_Front.T s qs as + t (Partial_Cost_Model.steps s qs as) q a"
check_eqv_def: fixes idx :: "idx"   and r :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and s :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" shows "WS1S.check_eqv idx r s = ((ws1s_wf idx r \<and> ws1s_lformula r) \<and> (ws1s_wf idx s \<and> ws1s_lformula s) \<and> (case WS1S.closure idx (norm (RESTRICT r), norm (RESTRICT s)) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False))"
rel_fundef_rewrite_body: fixes fd1 :: "('a, 'b, 'c, 'op) Std.instr fundef"   and fd2 :: "('a, 'b, 'c, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and instr :: "('a, 'b, 'c, 'op, 'opinl) Inca.instr" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"   and "norm_instr (body fd2 ! pc) = norm_instr instr" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 (rewrite_fundef_body fd2 pc instr)"
mod_ring_mod64: shows "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (\<lambda>x y. if y = 0 then x else 0) (mod)"
cases: fixes x :: "'b form" obtains i :: "nat" where "x = At i"   | f :: "'b" and fs :: "'b form list" where "x = Compound f fs"   | "x = ff"
monotoneD: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "monotone orda ordb f"   and "orda x y" shows "ordb (f x) (f y)"
fvs_finite: fixes M :: "'b trm" shows "finite (fvs M)"
alw_holds: fixes P :: "'c \<Rightarrow> bool"   and h :: "'c"   and t :: "'c stream" shows "alw (holds P) (h ## t) = (P h \<and> alw (holds P) t)"
elims_3: fixes x :: "('a, 'b) rbt" assumes "\<not> cl_inv x" obtains l :: "('a, 'b) rbt" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l__ R k__ v__ r__" and "\<not> (cl_inv l__ \<and> cl_inv r__ \<and> cl l__ = B \<and> cl r__ = B)"   | l :: "('a, 'b) rbt" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l B k v r" and "\<not> (cl_inv l \<and> cl_inv r)"
assms: fixes n :: "nat" shows "lower_asymptotic_density (A n) = 1"
LeadsTo_Un_post: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto>w B" shows "F \<in> A \<union> B \<longmapsto>w B"
Inf_between_def: fixes N :: "('f \<times> 'l) set"   and M :: "('f \<times> 'l) set" shows "Inf_between N M = Inf_from (N \<union> M) - Inf_from (N - M)"
permute_fun_comp: fixes p :: "perm"   and f :: "'c \<Rightarrow> 'd" shows "p \<bullet> f = permute p \<circ> f \<circ> permute (- p)"