hcju/isabelleps huggingface dataset:
to_Gromov_completion_range_open: shows "open (range to_Gromov_completion)"
assms_1: shows "xs @ ys <\<^sub>v vs @ ws"
F_to_idx_impl_def: fixes F :: "'b set set" shows "F_to_idx_impl F \<equiv> SPEC (\<lambda>Flist. distinct Flist \<and> set Flist = F) \<bind> (\<lambda>Flist. let num_acc = length Flist in mk_acc_impl Flist \<bind> (\<lambda>acc. RETURN (num_acc, acc)))"
has_integral_mult_left: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and S :: "'a set"   and c :: "'b" assumes "(f has_integral y) S" shows "((\<lambda>x. f x * c) has_integral y * c) S"
disc_rfr_proc_nonrandom: fixes r :: "real"   and n :: "nat"   and F :: "'c \<Rightarrow> 'b measure" shows "disc_rfr_proc r n \<in> borel_measurable (F (0::'c))"
Mp_rel_iI': fixes f1 :: "'i list"   and f2 :: "'a mod_ring poly" assumes "poly_rel f1 f2" shows "Mp_rel_i f1 (to_int_poly_i ff_ops f1)"
takeWhile_not_vol_write_outstanding_refs: fixes sb :: "'b memref list" shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) = {}"
rel_with_target_impl_transC_TRel_is_weak_reduction_simulation_rev: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_simulation (Rel\<inverse>) (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+" shows "weak_reduction_simulation ((TRel\<^sup>+)\<inverse>) Target"
mult_scalar_tail_rec_right: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "p \<odot> q = p \<odot> monomial (lc q) (lt q) + p \<odot> tail q"
Sigma_msetI: fixes a :: "'b"   and A :: "'b multiset"   and b :: "'c"   and B :: "'b \<Rightarrow> 'c multiset" assumes "a \<in># A"   and "b \<in># B a" shows "(a, b) \<in># Sigma_mset A B"
assms: fixes x :: "'a" assumes "x \<in> indets p" shows "f x = monomial (1::'b) (monomial 1 x)"
pr_simps_2: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "local.dom \<p>\<^sub>0[a0, a1] = a0 \<otimes> a1"
list_all2_takeI: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list"   and n :: "nat" assumes "list_all2 P xs ys" shows "list_all2 P (take n xs) (take n ys)"
homotopic_loops_imp_loop: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_loops s p q" shows "pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
UNION_empty_conv_2: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "(\<Union> (B ` A) = {}) = (\<forall>x\<in>A. B x = {})"
induct: fixes P :: "'g list list \<Rightarrow> bool"   and a0 :: "'g list list" assumes "P []"   and "\<And>xss. P xss \<Longrightarrow> P ([] # xss)"   and "\<And>x xs xss. P (xs # concat (map (case_list [] (\<lambda>h t. [t])) xss)) \<Longrightarrow> P ((x # xs) # xss)" shows "P a0"
ts'_def_1: shows "\<C> \<turnstile> [C v1] : t1s _> ts'"
axioms_1: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set" assumes "pareto_efficient_scf agents alts scf" shows "social_choice_function agents alts scf"
ffb_fbd: fixes f :: "'c \<Rightarrow> 'd set"   and Y :: "'d set" shows "fb\<^sub>\<F> f Y = \<mu> {X. bd\<^sub>\<F> f X \<subseteq> Y}"
brD: fixes c :: "'c"   and a :: "'d"   and \<alpha> :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" assumes "(c, a) \<in> br \<alpha> I" shows "a = \<alpha> c \<and> I c"
soadisj2: shows "\<forall>a x y. x \<in> subnetsOfAdr a \<and> y \<in> subnetsOfAdr a \<longrightarrow> \<not> netsDistinct x y"
"3_4": fixes y :: "'a" assumes "y \<in> set (y__ # ys)"   and "P y" shows "False"
divide_eq_eq_numeral1_1: fixes b :: "'c"   and w :: "num"   and a :: "'c" shows "(b / numeral w = a) = (if numeral w \<noteq> (0::'c) then b = a * numeral w else a = (0::'c))"
F_simps_2: fixes n :: "nat" shows "F (Suc n) = Eps (\<Phi> (F n)) # F n"
IH: assumes "insert_ops ops"   and "insert_seq (Some ref) xs"   and "insert_ops xs"   and "set xs \<subseteq> set ops"   and "ref \<notin> set (interp_ins ops)" shows "\<forall>i\<in>set (map fst xs). i \<notin> set (interp_ins ops)"
mult_zero_left: fixes a :: "'c" shows "(0::'c) * a = (0::'c)"
map_sel: fixes f1 :: "'g \<Rightarrow> 'd"   and f2 :: "'h \<Rightarrow> 'e"   and a :: "('g, 'h, 'f) gpv" shows "the_gpv (map_gpv f1 f2 a) = map_spmf (map_generat f1 f2 ((\<circ>) (map_gpv f1 f2))) (the_gpv a)"
im_faces: fixes x :: "'c set"   and f :: "'a \<Rightarrow> 'c"   and y :: "'c set" assumes "x \<in> f \<turnstile> X"   and "order.greater_eq x y" shows "y \<in> f \<turnstile> X"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and order :: "Comparator.order" shows "P (case order of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = ((order = Eq \<longrightarrow> P f1) \<and> (order = Lt \<longrightarrow> P f2) \<and> (order = Gt \<longrightarrow> P f3))"
bdd_above_def: fixes A :: "'k multiset set" shows "subset_mset.bdd_above A = (\<exists>M. \<forall>x\<in>A. x \<subseteq># M)"
\<sigma>_fv_dom: shows "subst_domain \<sigma> = {x \<in> vars\<^sub>s\<^sub>t S. \<exists>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<and> x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<and> x \<notin> set X}"
norm_scaleC: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "complex"   and x :: "'a" assumes "class.complex_normed_vector scaleR scaleC plus zero minus uminus dist norm sgn uniformity open" shows "norm (scaleC a x) = cmod a * norm x"
\<rho>1v1_subset_\<rho>v_inter_E1: shows "\<rho>1 \<V>1 \<subseteq> \<rho> \<V> \<inter> E\<^bsub>ES1\<^esub>"
insort_wrt_not_Nil: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "insort_wrt r x xs \<noteq> []"
edge_succ_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(edge_succ M x = edge_succ M y) = (x = y)"
wordinterval_setminus'_set_eq: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_to_set (wordinterval_setminus' r1 r2) = wordinterval_to_set r1 - wordinterval_to_set r2"
assms_8: shows "analytically_valid Dy_pair (\<lambda>p. F p \<bullet> i) j"
quasinorm_intersection_1: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm"   and f :: "'c" shows "eNorm (N1 \<inter>\<^sub>N N2) f = eNorm N1 f + eNorm N2 f"
CatExpCat: fixes A :: "('d, 'e, 'f) Category_scheme"   and B :: "('g, 'h, 'i) Category_scheme" assumes "Category A"   and "Category B" shows "Category (CatExp A B)"
\<X>_wellformed: fixes t :: "symbol \<times> character list"   and k :: "nat" assumes "t \<in> \<X> k" shows "wellformed_token t"
times_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f * g = floatarith.Mult f g"
forward_finite_iff_msc: fixes x :: "'a" shows "forward_finite x = (many_strongly_connected x \<or> forward_terminating x)"
BVG_dual: fixes \<alpha> :: "game" shows "BVG \<alpha>^d \<subseteq> BVG \<alpha>"
chop_subst_emb0: fixes s :: "('c, 'd) tm"   and \<rho> :: "'d \<Rightarrow> ('c, 'd) tm" assumes "is_App s"   and "chop (subst \<rho> s) \<noteq> subst \<rho> (chop s)" shows "emb_step_at (replicate (num_args (fun s)) dir.Left) dir.Right (chop (subst \<rho> s)) = subst \<rho> (chop s)"
inout_arcs_arc_simps_4: fixes e :: "'c"   and G :: "('d, 'c) pre_digraph"   and u :: "'d"   and E :: "'c set" assumes "e \<in> arcs G"   and "head G e = u" shows "in_arcs G u \<inter> insert e E = insert e (in_arcs G u \<inter> E)"
seq_inj_inverse: fixes x :: "'b seq seq" shows "seq_proj (seq_inj x) = x"
non_adj_degree: fixes v :: "'v"   and u :: "'v" assumes "non_adj v u" shows "degree v G = degree u G"
matrix': fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'c) vec, 'c) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and i :: "'b" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)" shows "f (X $ i) = (\<Sum>j\<in>UNIV. matrix' X Y f $ j $ i *s Y $ j)"
gcs_plus_lcs: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lcs :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "class.lcs_powerprod plus minus zero lcs" shows "plus (lcs_powerprod.gcs plus minus lcs s t) (lcs s t) = plus s t"
iv: shows "sorted_list_of_set (xs \<union> ys) = sorted_list_of_set xs @ sorted_list_of_set ys"
INF_cong: fixes A :: "'b::type set"   and B :: "'b::type set"   and C :: "'b::type \<Rightarrow> 'c::type"   and D :: "'b::type \<Rightarrow> 'c::type"   and Inf :: "'c::type set \<Rightarrow> 'c::type" assumes "(A::'b::type set) = (B::'b::type set)"   and "\<And>x::'b::type. x \<in> (B::'b::type set) \<Longrightarrow> (C::'b::type \<Rightarrow> 'c::type) x = (D::'b::type \<Rightarrow> 'c::type) x" shows "(Inf::'c::type set \<Rightarrow> 'c::type) ((C::'b::type \<Rightarrow> 'c::type) ` (A::'b::type set)) = Inf ((D::'b::type \<Rightarrow> 'c::type) ` (B::'b::type set))"
strategyproofness_1: shows "pmf (sds R10) b + pmf (sds R10) d \<le> pmf (sds R19) c + pmf (sds R19) a"
cfs_monom_mult_r: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat"   and k :: "nat" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "a \<in> carrier R" shows "(monom (UP R) a n \<otimes>\<^bsub>UP R\<^esub> p) (k + n) = a \<otimes>\<^bsub>R\<^esub> p k"
t_bound2: shows "t < max (te p (i + 1)) (te q (i + 1))"
interleave_same: fixes y :: "'b list"   and l :: "'b list" shows "(y \<in> l \<otimes> y) = (l = [])"
ltl_prop_equiv_implies_ltl_prop_unfold_equiv: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "\<phi> \<sim>\<^sub>Q \<psi>"
match_3: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d"   and f :: "'d"   and h :: "'d" assumes "category C"   and "partial_magma.arr C (C g f)"   and "partial_magma.arr C (C h g)" shows "partial_magma.arr C (C (C h g) f)"
seq_stop_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Stop ;; c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" shows "c' = c \<and> mds' = mds \<and> mem' = mem"
intros_1: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration" shows "asynchronousSystem.reachable trans sends cfg1 cfg1"
ev: shows "\<forall>\<^sub>F i in F. a \<le> f i"
lookup_vectorize_poly: fixes p :: "'t \<Rightarrow>\<^sub>0 'd"   and k :: "nat" shows "lookup (vectorize_poly p) k = proj_poly k p"
cross7_simps_8: fixes a :: "'b"   and b :: "'b" shows "a + b = b + a"
assms_4: shows "rk {A, C, \<beta>} = 2"
simps_2: fixes "is" :: "'b set"   and ia :: "'b interrupt_action"   and ias :: "'b interrupt_action list" shows "interrupt_actions_ok' is (ia # ias) = (interrupt_action_ok' is ia \<and> interrupt_actions_ok' (redT_updI is ia) ias)"
subseq_appendE: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "subseq xs (ys @ zs)" obtains xs1 :: "'a list" and xs2 :: "'a list" where "xs = xs1 @ xs2" and "subseq xs1 ys" and "subseq xs2 zs"
simInit: shows "\<forall>a iobs. iobs \<in> envObs a ` set envInit \<longrightarrow> simAbs (simInit a iobs) = simf ` {t' \<in> jkbpC. jview a t' = jviewInit a iobs}"
local_var_approx_def: fixes prog :: "'b program_scheme" shows "local_var_approx prog \<equiv> \<Union> {set (local_vars p) |p. p \<in> set (program.processes prog)}"
lookup_sym_mpoly_aux_monom_singleton': fixes X :: "nat set"   and k :: "nat"   and x :: "nat" assumes "finite X"   and "0 < k" shows "lookup (powsum_mpoly_aux X k) (Poly_Mapping.single x k) = (if x \<in> X then 1::'b else (0::'b))"
quasi_isometry_between_def: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and Y :: "'d set"   and f :: "'c \<Rightarrow> 'd" shows "lambda C -quasi_isometry_between X Y f = (lambda C -quasi_isometry_on X f \<and> f ` X \<subseteq> Y \<and> (\<forall>y\<in>Y. \<exists>x\<in>X. dist (f x) y \<le> C))"
abs_update_map_of'': fixes l :: "('a \<times> 'b) list list"   and k :: "'a"   and v :: "'b" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l" shows "map_of (concat (abs_update k v l)) k = Some v"
lang_eq_ext: fixes n :: "nat"   and r :: "'b rexp"   and s :: "'b rexp" assumes "local.wf n r"   and "local.wf n s" shows "(lang n r = lang n s) = (\<forall>w\<in>lists (\<Sigma> n). (w \<in> lang n r) = (w \<in> lang n s))"
conj_disj_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<^bold>\<squnion> z) \<^bold>\<sqinter> x = y \<^bold>\<sqinter> x \<^bold>\<squnion> z \<^bold>\<sqinter> x"
interior_ball: fixes x :: "'c"   and e :: "real" shows "interior (ball x e) = ball x e"
linear_inj_imp_surj: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec" assumes "linear (*s) (*s) f"   and "inj f" shows "surj f"
norm_le_l1: fixes x :: "'a" shows "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
steps_is_node: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and is_node :: "'a \<Rightarrow> bool"   and is_alpha :: "'b \<Rightarrow> bool"   and q :: "'a"   and w :: "'b list" assumes "Automaton trans is_node is_alpha"   and "is_node q"   and "list_all is_alpha w" shows "is_node (foldl trans q w)"
on_triple: fixes l1 :: "'d"   and v1 :: "'e"   and v2 :: "'e"   and l2 :: "'d"   and v3 :: "'f"   and v4 :: "'f"   and R :: "('e \<times> 'f) set" shows "(((l1, v1, v2), l2, v3, v4) \<in> on_triple R) = ((v1, v3) \<in> R \<and> (v2, v4) \<in> R \<and> l1 = l2)"
invar: fixes s :: "('b, 'c) assoc_list" shows "lm.invar s"
rev_take: fixes i :: "nat"   and xs :: "'a list" shows "rev (take i xs) = drop (length xs - i) (rev xs)"
single_non_freshAbs_ordLess_var: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and xs :: "'varSort" assumes "goodAbs A" shows "|{x. \<not> freshAbs xs x A}| <o |UNIV|"
not_up_less_UU: fixes x :: "'a" shows "up\<cdot>x \<notsqsubseteq> \<bottom>"
gauge_Inter: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set" assumes "finite S"   and "\<And>u. u \<in> S \<Longrightarrow> gauge (f u)" shows "gauge (\<lambda>x. \<Inter> {f u x |u. u \<in> S})"
bv_extend_def: fixes i :: "nat"   and b :: "bit"   and w :: "bit list" shows "bv_extend i b w = replicate (i - length w) b @ w"
ind_valid_graph': fixes E' :: "(nat \<times> int \<times> nat) set" assumes "subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>" shows "valid_graph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr>"
emeasure_lborel_singleton: fixes x :: "'b" shows "emeasure lborel {x} = 0"
equal_eq: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.equal equal" shows "equal x y = (x = y)"
covector_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "top \<sqinter> x = x"   and "top \<sqinter> y = y" shows "top \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
exhaust: fixes y :: "meth_body" obtains x1 :: "s list" and x2 :: "x" where "y = meth_body_def x1 x2"
select_convs_9: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "traps \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = traps"
dependent_nat_choice: fixes P :: "nat \<Rightarrow> 'e \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<exists>x. P 0 x"   and "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y" shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))"
c2: shows "\<forall>p\<in>set ps. extend ( \<Phi>1 + \<Gamma>' \<Rightarrow>* \<Psi> + \<Delta>') p \<in> set Ps'"
top_Assertion_def: shows "\<top> = abs_wpt (1::'b)"
wp_under_wlp_PC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "healthy (wp a)"   and "nearly_healthy (wlp a)"   and "healthy (wp b)"   and "nearly_healthy (wlp b)"   and "wp_under_wlp a"   and "wp_under_wlp b"   and "unitary P" shows "wp_under_wlp (a \<^bsub>P\<^esub>\<oplus> b)"
assms_4: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x"
Gram_Schmidt_column_k_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "Gram_Schmidt_column_k A k = (\<chi>a b. (if b = mod_type_class.from_nat k then column b A - proj_onto (column b A) {column i A |i. i < b} else column b A) $ a)"
cases: fixes a1 :: "'b list"   and a2 :: "'b list" assumes "initial_segment a1 a2" obtains xs :: "'b list" and ys :: "'b list" where "a1 = xs" and "a2 = xs @ ys"
FactGroup_DirProd_multiplication_iso_1: fixes K :: "('e, 'f) monoid_scheme"   and H :: "'a set"   and N :: "'e set" assumes "Group.group K"   and "H \<lhd> G"   and "N \<lhd> K" shows "(G Mod H) \<times>\<times> (K Mod N) \<cong> G \<times>\<times> K Mod H \<times> N"
comp_commute_below_diversity: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y \<le> - top) = (y \<sqinter> x \<le> - top)"
Suc_diff_le: fixes n :: "nat"   and m :: "nat" assumes "(n::nat) \<le> (m::nat)" shows "Suc (m::nat) - (n::nat) = Suc (m - n)"
case_sum_expand_Inr: fixes f :: "'d + 'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "f \<circ> Inl = g" shows "case_sum g (f \<circ> Inr) = f"
symI: fixes r :: "('b \<times> 'b) set" assumes "\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r" shows "sym r"
list_all2_flip: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" shows "list_all2 (flip P) xs ys = list_all2 P ys xs"
symmetric_rel: fixes I :: "'b set"   and a :: "'b"   and b :: "'b" assumes "Q I a b" shows "Q I b a"
Elem_Opair_exists: fixes x :: "ZF"   and y :: "ZF" shows "\<exists>z. Elem x z \<and> Elem y z \<and> Elem z (Opair x y)"
ll_on_open_rev: fixes t0 :: "real" shows "ll_on_open (preflect t0 ` T) (\<lambda>t. - f (preflect t0 t)) X"
hd_stack_in_tj_stack: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s"   and "stack s \<noteq> []" shows "hd (stack s) \<in> set (tj_stack s)"
simps_1: fixes \<mu> :: "'a" shows "Nml \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = obj \<mu>"
appliedUnitPropagate_def: fixes stateA :: "State"   and stateB :: "State"   and F0 :: "Literal list list" shows "appliedUnitPropagate stateA stateB F0 \<equiv> \<exists>uc ul. uc el F0 \<and> isUnitClause uc ul (elements (getM stateA)) \<and> getM stateB = getM stateA @ [(ul, False)]"
Inr_inject: fixes x :: "'d"   and y :: "'d" assumes "Inr x = Inr y" shows "x = y"
cm_preserves_safe: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "safe c0"   and "impl_safe c0"   and "next_change_multiplicity' c0 c1 loc t n" shows "safe c1"
mult_pCons_right: fixes p :: "'a poly"   and a :: "'a"   and q :: "'a poly" shows "p * pCons a q = smult a p + pCons (0::'a) (p * q)"
Implies_def: fixes P :: "('d, 'e, 'f) formula"   and Q :: "('d, 'e, 'f) formula" shows "(P \<rightarrow> Q) = (Q || ! P)"
Un_mono: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set"   and D :: "'a set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<union> B \<subseteq> C \<union> D"
generated_ring_def: shows "M.generated_ring = {\<Union> C |C. C \<subseteq> sigma_sets \<Omega> M \<and> finite C \<and> disjoint C}"
tensor_mat_id: shows "tensor_mat (1\<^sub>m d1) (1\<^sub>m d2) = 1\<^sub>m d"
bp: shows "\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0"
list_induct2': fixes P :: "'b list \<Rightarrow> 'c list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list" assumes "P \<epsilon> \<epsilon>"   and "\<And>x xs. P (x # xs) \<epsilon>"   and "\<And>y ys. P \<epsilon> (y # ys)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
scp_def: shows "scp = qty_single asset (remaining_qty Mkt v pf asset)"
terminates_wfD: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" assumes "terminates g" shows "wf {(snd (snd g s), s) |s. fst g s}"
exec_move_CASI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e2 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e3 :: "'addr expr1"   and v :: "'addr val" assumes "exec_move ci P t e2 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3)) h (stk @ [v], loc, length (compE2 e1) + pc, xcp) ta h' (stk' @ [v], loc', length (compE2 e1) + pc', xcp')"
min_term_min: fixes v :: "'x \<Rightarrow>\<^sub>0 nat" shows "punit.min_term \<preceq> v"
simps_11: fixes v :: "int"   and va :: "Cooper.num" shows "not (NDvd v va) = NOT (NDvd v va)"
le_listD: fixes xs :: "'a list"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and p :: "nat" assumes "xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys"   and "p < length xs" shows "xs ! p \<sqsubseteq>\<^bsub>r\<^esub> ys ! p"
simps_1: fixes r :: "'a language"   and s :: "'a language" shows "\<oo> (StarLR r s) = \<oo> r"
mtf2_moves_to_front: fixes xs :: "'a list"   and q :: "'a" assumes "distinct xs"   and "q \<in> set xs" shows "index (mtf2 (length xs) q xs) q = 0"
Mapping_inverse: fixes y :: "'a \<Rightarrow> 'b option" assumes "y \<in> UNIV" shows "mapping.rep (Mapping y) = y"
maddux_3_12: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> - y) \<sqinter> (x \<squnion> y) = x"
biQuot_lQuot_rQuot: fixes x :: "'b"   and y :: "'b"   and A :: "'b list set" shows "biQuot x y A = lQuot x (rQuot y A)"
convex_epigraphI: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "convex_on S f"   and "convex S" shows "convex (epigraph S f)"
il_sub_inf_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<sqinter> z \<le> y \<sqinter> z"
False: shows "s.max_fun_diff f g \<noteq> s.max_fun_diff g h"
fgi_1: fixes i :: "'a" assumes "i \<in> I" shows "g (f i) = i"
bit_shiftl1_iff: fixes w :: "'b word"   and n :: "nat" shows "bit (shiftl1 w) n = (0 < n \<and> n < LENGTH('b) \<and> bit w (n - 1))"
intro: fixes x :: "'d \<Longrightarrow> 'e" assumes "weak_lens x"   and "bij_lens_axioms x" shows "bij_lens x"
hd_unsat: shows "fst ` lhd (lmap lclss_of_state Sts) \<TTurnstile>\<G>e {{#}}"