hcju/isabelleps huggingface dataset:
theNFTBoth_length_finished_1: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and n :: "nat"   and cl :: "('c, 'd, 'e) cmd list"   and dl :: "('f, 'g, 'h) cmd list" assumes "PL_Indis cval indis"   and "n \<in> theNFTBoth cl dl" shows "n < length cl"
similar_mat_refl: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat A A"
intro: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>shadow_root_ptr ptr' h h' r. \<lbrakk>cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<noteq> ptr'; h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'; r \<in> get_disconnected_nodes_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_delete_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs"
lF_simps_4: fixes f :: "'c" assumes "C.ide f" shows "D.dom (lF f) = F (trg\<^sub>C f) \<star>\<^sub>D F f"
f'_def: fixes x :: "'a" shows "f' x = (if x \<in> S then Some (f x) else None)"
simps_2: fixes t :: "('d, 'c) bintree"   and ts :: "('d, 'c) bintree list"   and q :: "('c, 'd) pq" shows "bts_augment (t # ts) q = bts_augment ts (bt_augment t q)"
fast_uniform_def: shows "local.fast_uniform = local.fast_dice_roll 1 0"
fps_lr_inverse_one_1: fixes x :: "'a" shows "fps_left_inverse 1 x = fps_const x"
USUP_as_Inf_image: fixes \<P> :: "'c set"   and f :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Squnion> P \<in> \<P> \<bullet> f P) = \<Squnion>(f ` \<P>)"
map_s_lemma: fixes xs :: "nat list"   and vl :: "nat" assumes "length xs = vl" shows "map ((\<lambda>a. rec_exec a xs) \<circ> (\<lambda>i. Cn vl s [recf.id vl i])) [0..<vl] = map Suc xs"
a_nz: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "g \<in> G" shows "poly_eval (\<lambda>_. a) (i.lcf (focus X g)) \<noteq> (0::'a)"
is_expansion_aux_imp_smallo: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and p :: "real" assumes "is_expansion_aux xs f basis"   and "ms_exp_gt p (ms_aux_hd_exp xs)" shows "f \<in> o(\<lambda>x. hd basis x powr p)"
submanifold_axioms: shows "submanifold charts k (domain \<psi>)"
that: assumes "vsv f"   and "f\<lparr>c\<rparr> \<in>\<^sub>\<circ> A c"   and "f = set {\<langle>c, f\<lparr>c\<rparr>\<rangle>}" shows "thesis"
vunion_vinsert_right: fixes A :: "V"   and a :: "V"   and B :: "V" shows "A \<union>\<^sub>\<circ> vinsert a B = vinsert a (A \<union>\<^sub>\<circ> B)"
Mem_Eats_EH_2: fixes u :: "tm"   and t :: "tm"   and A1 :: "fm"   and H :: "fm set"   and B :: "fm"   and z :: "tm" assumes "insert (u IN t) (insert A1 H) \<turnstile> B"   and "insert (u EQ z) (insert A1 H) \<turnstile> B" shows "insert A1 (insert (u IN Eats t z) H) \<turnstile> B"
PPs_closed_lpp: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'x set" assumes "p \<in> P[X]" shows "i.lpp p \<in> .[X]"
complete_lattice_partial_function_definitions: shows "partial_function_definitions (\<le>) Sup"
twiddleStore_compose: fixes \<beta> :: "(Location \<times> Location) set"   and s :: "Var \<Rightarrow> Val"   and r :: "Var \<Rightarrow> Val"   and \<gamma> :: "(Location \<times> Location) set"   and t :: "Var \<Rightarrow> Val" assumes "s  \<approx>\<^sub>\<beta> r"   and "r  \<approx>\<^sub>\<gamma> t" shows "s  \<approx>\<^sub>Pbij_compose \<beta> \<gamma> t"
covector_vector_comp: fixes v :: "'a" assumes "vector v" shows "- v\<^sup>T * v = bot"
thetaZOCh_ZOretr: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and d1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and d2 :: "('test, 'atom, 'choice) cmd" assumes "compatCh ch"   and "c1 \<approx>01 d1"   and "c2 \<approx>01 d2" shows "thetaZOCh ch c1 c2 d1 d2 \<subseteq> ZOretr (thetaZOCh ch c1 c2 d1 d2 \<union> ZObis)"
\<mu>'_def: fixes i :: "nat"   and j :: "nat" shows "fs_init.gs.\<mu>' i j \<equiv> fs_init.gs.d (Suc j) * gs.\<mu> i j"
approx_def: shows "approx \<equiv> 577257 / 10 ^ 6"
abs_eq: fixes xa :: "'e"   and x :: "'c" shows "Pred\<^sub>\<alpha> xa x = abs_Tree\<^sub>\<alpha> (tPred xa x)"
l_cancel: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<otimes> a = c \<otimes> b"   and "a \<in> carrier (Ring_Divisibility.mult_of R)"   and "b \<in> carrier (Ring_Divisibility.mult_of R)"   and "c \<in> carrier (Ring_Divisibility.mult_of R)" shows "a = b"
trunc_bl2bin_aux: fixes m :: "nat"   and bl :: "bool list"   and w :: "int" shows "take_bit m (bl_to_bin_aux bl w) = bl_to_bin_aux (drop (length bl - m) bl) (take_bit (m - length bl) w)"
nth_rotate: fixes n :: "nat"   and xs :: "'c list"   and m :: "nat" assumes "n < |xs|" shows "rotate m xs ! n = xs ! ((m + n) mod |xs|)"
hd_def: fixes list :: "'g list" shows "hd list = (case list of x21 # x22 \<Rightarrow> x21)"
simps_4: fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and p :: "real mpoly" shows "elimVar var L F (Leq p) = (let (a, b, c) = get_coeffs var p in Or (And (And (fm.Atom (Eq a)) (fm.Atom (Neq b))) (list_conj (map (\<lambda>a. fm.Atom (linear_substitution var (- c) b a)) L @ map (linear_substitution_fm var (- c) b) F))) (And (fm.Atom (Neq a)) (And (fm.Atom (Leq (- b\<^sup>2 + 4 * a * c))) (Or (list_conj (map (quadratic_sub var (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a)) L @ map (quadratic_sub_fm var (- b) 1 (b\<^sup>2 - 4 * a * c) (2 * a)) F)) (list_conj (map (quadratic_sub var (- b) (- 1) (b\<^sup>2 - 4 * a * c) (2 * a)) L @ map (quadratic_sub_fm var (- b) (- 1) (b\<^sup>2 - 4 * a * c) (2 * a)) F))))))"
assms_4: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children" shows "distinct children"
simps: fixes a1 :: "'a list"   and a2 :: "'a list"   and a3 :: "'a list" shows "appending a1 a2 a3 = ((\<exists>ys. a1 = [] \<and> a2 = ys \<and> a3 = ys) \<or> (\<exists>xs ys zs x. a1 = x # xs \<and> a2 = ys \<and> a3 = x # zs \<and> appending xs ys zs))"
fun_type_length_eq: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and f :: "'a"   and X :: "('a, 'b) Term.term list"   and g :: "'a"   and Y :: "('a, 'c) Term.term list" assumes "typed_model arity public Ana \<Gamma>"   and "\<Gamma> (Fun f X) = Fun g Y" shows "length X = length Y"
exhaust: fixes y :: "'f LDCFG_node" obtains x1 :: "'f" where "y = Node x1"   | "y = NewEntry"   | "y = NewExit"
strict_monoidal_functor_axioms: shows "strict_monoidal_functor (\<cdot>\<^sub>C) T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D F"
is_Sup_def: fixes A :: "'c set"   and sup :: "'c" shows "is_Sup A sup = ((\<forall>x\<in>A. x \<sqsubseteq> sup) \<and> (\<forall>z. (\<forall>x\<in>A. x \<sqsubseteq> z) \<longrightarrow> sup \<sqsubseteq> z))"
powr_less_inverse_iff: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 < x"   and "0 < y"   and "0 < z" shows "(x powr y < z) = (x < z powr inverse y)"
assms_5: shows "M \<equiv> stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y))"
set_mset_single: fixes b :: "'f" shows "set_mset {#b#} = {b}"
atMost_subset_iff: fixes x :: "'c"   and y :: "'c" shows "({..x} \<subseteq> {..y}) = (x \<le> y)"
opt: fixes C' :: "'a set" assumes "finite C'"   and "vertex_cover E C'" shows "card C \<le> 2 * card C'"
closure_def: fixes S :: "'c set" shows "closure S = S \<union> {x. x islimpt S}"
transpose_iff: fixes A :: "(('b, 'd) vec, 'c) vec"   and B :: "(('b, 'd) vec, 'c) vec" shows "(Finite_Cartesian_Product.transpose A = Finite_Cartesian_Product.transpose B) = (A = B)"
strand_sem_Receive_map_2: fixes M :: "('fun, 'var) Term.term set"   and T :: "('fun, 'var) Term.term list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "\<lbrakk>M; map Receive T\<rbrakk>\<^sub>d \<I>"
algebraic_int_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> algebraic_int (f x)" shows "algebraic_int (prod f A)"
lset_ldropn_subset: fixes n :: "nat"   and xs :: "'a llist" shows "lset (ldropn n xs) \<subseteq> lset xs"
alt_assoc: fixes m1 :: "'m"   and m2 :: "'m"   and m3 :: "'m" shows "alt (alt m1 m2) m3 = alt m1 (alt m2 m3)"
BAR: shows "e' = Subx.Fenv.from_list xs"
ipurge_tr_rev_t_intro_2: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and X :: "('c, 'd) ipurge_rec" assumes "ipurge_tr_rev_t_inv_2 P I D xs X"   and "ipurge_tr_rev_t_form X" shows "secure P I D \<longrightarrow> xs \<in> traces P \<longrightarrow> ipurge_tr_rev_t_out X \<in> traces P"
iIN_imirror_bounds: fixes n :: "nat"   and d :: "nat"   and l :: "nat"   and r :: "nat" assumes "n + d \<le> l + r" shows "imirror_bounds [n\<dots>,d] l r = [l + r - n - d\<dots>,d]"
Graph8: fixes M :: "node list"   and T :: "nat"   and a :: "node" assumes "Proper_Roots M" shows "Proper_Roots (M[T := a])"
apath_append_iff: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and q :: "('a \<times> 'a) list"   and v :: "'a" shows "apath u (p @ q) v = (apath u p (last (pawalk_verts u p)) \<and> apath (last (pawalk_verts u p)) q v \<and> set (pawalk_verts u p) \<inter> set (tl (pawalk_verts (last (pawalk_verts u p)) q)) = {})"
map_snd_sorted_less: fixes xs :: "('b \<times> 'a) list"   and i :: "nat"   and j :: "nat" assumes "sorted_less (separators xs)"   and "i < j"   and "j < length xs" shows "snd (xs ! i) < snd (xs ! j)"
proj_stoch_pre_borel: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and n :: "nat"   and x :: "bool stream" assumes "borel_adapt_stoch_proc F X" shows "proj_stoch_proc X n -` {proj_stoch_proc X n x} \<in> sets (stoch_proc_filt M X borel n)"
sdistinct_infinite_sset: fixes w :: "'b stream" assumes "sdistinct w" shows "infinite (sset w)"
simps: fixes pre :: "object atom formula"   and add :: "object atom formula list"   and del :: "object atom formula list"   and f :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option" shows "sound_opr (Ground_Action pre (Effect add del)) f = ((\<forall>s. close_eq s \<Turnstile> pre \<longrightarrow> (\<exists>s'. f s = Some s' \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set del \<and> close_eq s \<Turnstile> atm \<longrightarrow> close_eq s' \<Turnstile> atm) \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set add \<and> close_eq s \<Turnstile> \<^bold>\<not> atm \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> atm) \<and> (\<forall>fmla. fmla \<in> set add \<longrightarrow> close_eq s' \<Turnstile> fmla) \<and> (\<forall>fmla. fmla \<in> set del \<and> fmla \<notin> set add \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> fmla))) \<and> Ball (set add) is_predAtom)"
DVD_l: fixes i :: "int"   and a :: "MIR.num" shows "isrlfm (rsplit (DVD i) a)"
\<epsilon>_def: shows "\<epsilon> \<equiv> \<epsilon>.map"
coinduct: fixes X :: "'b llist \<Rightarrow> bool"   and x :: "'b llist"   and A :: "'b set" assumes "X x"   and "\<And>x. X x \<Longrightarrow> x = LNil \<or> (\<exists>l a. x = a ## l \<and> (X l \<or> l \<in> A\<^sup>\<infinity>) \<and> a \<in> A)" shows "x \<in> A\<^sup>\<infinity>"
insert_absorb: fixes a :: "'k"   and A :: "'k set" assumes "a \<in> A" shows "insert a A = A"
post_rule': fixes P :: "assn"   and f :: "'c Heap"   and Q :: "'c \<Rightarrow> assn"   and R :: "'c \<Rightarrow> assn" assumes "<P> f <Q>"   and "\<not> <P> f <R>" shows "\<exists>x. \<not> (Q x \<Longrightarrow>\<^sub>A R x)"
omit_branch_oob: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and v :: "nat"   and v' :: "nat"   and xs :: "(nat \<times> nat) set" assumes "length branch \<le> v" shows "omit_branch ({(v, v')} \<union> xs) branch = omit_branch xs branch"
length_snth_enc: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and n :: "nat" shows "length (snd (stream_enc (w, I) !! n)) = length I"
read_actions_def: shows "read_actions \<equiv> \<lambda>E. {x. read_actionsp E x}"
msubst2: fixes p :: "fm"   and c :: "poly"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "islin p"   and "isnpoly c"   and "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'b)" shows "Ifm vs (x # bs) (msubst2 p c t) = Ifm vs (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> # bs) p"
s_in_T: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "\<exists>k. (s j, s i) \<in> T k \<and> k < n"
My: fixes F :: "'a set" assumes "finite F" shows "L2_set (cmod \<circ> y) F \<le> My"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) Cones.map (Y a) \<Psi>.map"
mbisim_final_thread_preserve1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s2 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('g \<times> 'd \<Rightarrow>f nat) option) \<times> 'h) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "FWbisimulation final1 r1 final2 r2 bisim"   and "FWbisimulation_base.mbisim bisim (\<lambda>x1 x2. True) s1 s2"   and "final_thread.final_thread final1 s1 t" shows "final_thread.final_thread final2 s2 t"
star_sq_def: fixes x :: "'a" shows "star_sq x = Sup {proto_dioid.power_sq 1\<^sub>\<sigma> (\<cdot>) sup x i |i. i \<in> \<nat>}"
ent_true_drop_1: fixes P :: "assn"   and Q :: "assn"   and R :: "assn" assumes "P \<Longrightarrow>\<^sub>A Q * true" shows "P * R \<Longrightarrow>\<^sub>A Q * true"
scale_right_distrib: fixes a :: "'c"   and x :: "'c"   and y :: "'c" shows "a * (x + y) = a * x + a * y"
lossless_return_pmf_iff: fixes x :: "'a option" shows "lossless_spmf (return_pmf x) = (x \<noteq> None)"
valid_decompI: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and X :: "'x set" assumes "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> h \<in> P[X]"   and "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> h \<noteq> 0"   and "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> U \<subseteq> X" shows "valid_decomp X ps"
w\<theta>\<nu>: shows "ide w \<and> \<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : v \<Rightarrow> p\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (h \<star> \<theta>) \<cdot> \<a>[h, p\<^sub>0, w] \<cdot> (\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[k, p\<^sub>1, w] \<cdot> (k \<star> \<nu>) = \<mu>"
AR_closed_Un_local_aux: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set U) S"   and "closedin (top_of_set U) T"   and "AR S"   and "AR T"   and "AR (S \<inter> T)" shows "S \<union> T retract_of U"
image_vimage_subset: fixes f :: "'f \<Rightarrow> 'e"   and A :: "'e set" shows "f ` f -` A \<subseteq> A"
ball_insert: fixes a :: "'a"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>insert a B. P x) = (P a \<and> (\<forall>x\<in>B. P x))"
cFalse: fixes A :: "triple set"   and c :: "stmt"   and Q :: "assn" shows "A \<turnstile> {\<lambda>s. False} c {Q}"
wlp_inv_def: fixes G :: "'b \<Rightarrow> bool"   and body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and I :: "'b \<Rightarrow> real" shows "wlp_inv G body I = (\<forall>s. \<guillemotleft> G \<guillemotright> s * I s \<le> wlp body I s)"
ins_list_Cons: fixes x :: "'b"   and xs :: "'b list" assumes "sorted (x # xs)" shows "ins_list x xs = x # xs"
mfactor_correct: fixes l :: "'a list"   and P :: "pol"   and M :: "mon" assumes "in_carrier l" shows "Ipol l P = Ipol l (fst (mfactor P M)) \<oplus> Imon l M \<otimes> Ipol l (snd (mfactor P M))"
left_inverse: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and inverse :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "group f z inverse" shows "f (inverse a) a = z"
prems_3: shows "free_vars (Add $$ e) \<subseteq> V \<union> V'"
finprod_all1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = (0::'b)" shows "sum f A = (0::'b)"
composites_are_arrows_2: shows "hseq (\<nu> \<star> \<pi>) \<rho>"
terminal_def: fixes b :: "'s" shows "terminal b \<equiv> ide b \<and> (\<forall>a. ide a \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>))"
sys_xcpts_incl: shows "NullPointer \<in> sys_xcpts \<and> OutOfMemory \<in> sys_xcpts \<and> ClassCast \<in> sys_xcpts \<and> NoClassDefFoundError \<in> sys_xcpts \<and> IncompatibleClassChangeError \<in> sys_xcpts \<and> NoSuchFieldError \<in> sys_xcpts \<and> NoSuchMethodError \<in> sys_xcpts"
cp0: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "f self \<tau> = f (\<lambda>_. self \<tau>) \<tau>"
Ball_rangeC_iff: fixes A :: "('b \<Rightarrow> 'a) set"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>rangeC A. P x) = (\<forall>f\<in>A. \<forall>x. P (f x))"
abs_cos_le_one: fixes x :: "real" shows "\<bar>cos x\<bar> \<le> 1"
pdevs_val_degree_cong: fixes b :: "'b pdevs"   and d :: "'b pdevs"   and a :: "nat \<Rightarrow> real"   and c :: "nat \<Rightarrow> real" assumes "b = d"   and "\<And>i. i < degree b \<Longrightarrow> a i = c i" shows "pdevs_val a b = pdevs_val c d"
nz: shows "(1 / norm1 v) *\<^sub>R v \<noteq> 0"
hom_component_of_homogeneous: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" assumes "homogeneous p" shows "hom_component p n = (p when n = poly_deg p)"
P: shows "\<forall>i j. i < j \<longrightarrow> P (h (\<phi> i)) (h (\<phi> j))"
is_Proj_def: shows "is_Proj \<equiv> map_fun (*\<^sub>V) id (\<lambda>P. \<exists>M. closed_csubspace M \<and> is_projection_on P M)"
incl: fixes h :: "'e"   and H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "h \<in> H" shows "h \<in> generate G H"
domain_inter_vimage: shows "carrier\<^bsub>S\<^esub> \<inter> f -` carrier = carrier\<^bsub>S\<^esub>"
dghm_comp_components_4: fixes \<GG> :: "V"   and \<FF> :: "V" shows "(\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>HomCod\<rparr> = \<GG>\<lparr>HomCod\<rparr>"
tauLaw4PsiCong: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q \<doteq> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q"
o1: fixes M :: "lam" shows " M \<longrightarrow>\<^sub>1 M"
distinct_22: fixes x71 :: "nat"   and x72 :: "msg"   and x2 :: "nat" shows "Crypt x71 x72 \<noteq> Number x2"
c2_term: shows "(c2, t) \<Rightarrow> p' \<Down> t'"
rfb_Sup_dual: shows "Sup_dual fb\<^sub>\<R>"
consec_v1_v': shows "consec (lan v1) (lan v')"
hyps_3: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s' \<and> (\<Gamma>\<turnstile> \<langle>c2,Normal s'\<rangle> \<Rightarrow>\<notin>Fault ` F \<longrightarrow> \<Gamma>\<turnstile>strip_guards F c2 \<down> Normal s')"
monoid_cancel_axioms_def: fixes G :: "('c, 'd) monoid_scheme" shows "monoid_cancel_axioms G \<equiv> (\<forall>c a b. c \<otimes>\<^bsub>G\<^esub> a = c \<otimes>\<^bsub>G\<^esub> b \<longrightarrow> a \<in> carrier G \<longrightarrow> b \<in> carrier G \<longrightarrow> c \<in> carrier G \<longrightarrow> a = b) \<and> (\<forall>a c b. a \<otimes>\<^bsub>G\<^esub> c = b \<otimes>\<^bsub>G\<^esub> c \<longrightarrow> a \<in> carrier G \<longrightarrow> b \<in> carrier G \<longrightarrow> c \<in> carrier G \<longrightarrow> a = b)"
eq_mat_on_vecI: fixes nA :: "nat"   and M :: "'b mat"   and N :: "'b mat"   and nB :: "nat" assumes "\<And>v. v \<in> carrier_vec nA \<Longrightarrow> M *\<^sub>v v = N *\<^sub>v v"   and "M \<in> carrier_mat nB nA"   and "N \<in> carrier_mat nB nA" shows "M = N"
strict_implies_order: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "a \<prec> b" shows "a \<preceq> b"
code_lemmas_1: fixes cs :: "('a \<times> constraint) list" shows "init_simplex' cs = (let tons_cs = to_ns cs in (map snd tons_cs, case preprocess tons_cs of (t, as, trans_v, ui) \<Rightarrow> ((create_map as, trans_v, remdups ui), init_state t)))"
hyps_2: shows "sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys"
WHILEI_le_WHILEIT: fixes I :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'm"   and s :: "'a" shows "WHILEI I b f s \<le> WHILEIT I b f s"
discr_MtransC_indis: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "discr c"   and "(c, s) \<rightarrow>*c (c', s')" shows "s \<approx> s'"
path_image_join: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "pathfinish g1 = pathstart g2" shows "path_image (g1 +++ g2) = path_image g1 \<union> path_image g2"
assms_4: shows "D r \<tau> \<sigma> \<sigma>' \<tau>'"
T4E: fixes A :: "'a set"   and B :: "'a set" assumes "A closed"   and "A \<subseteq> carrier"   and "B closed"   and "B \<subseteq> carrier"   and "A \<inter> B = {}" obtains U :: "'a set" and V :: "'a set" where "U open" and "A \<subseteq> U" and "V open" and "B \<subseteq> V" and "U \<inter> V = {}"
supp_of_fin_sets: fixes X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "finite X" shows "supp X = \<Union> (supp ` X)"
grounding_of_clss_def: fixes CC :: "'a literal multiset set" shows "grounding_of_clss CC = \<Union> (grounding_of_cls ` CC)"
x_pow_fSum_in_Vr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> carrier (Vr K v)" shows "\<Sigma>\<^sub>e K NPOW x K n \<in> carrier (Vr K v)"
adt_wgraph_axioms: shows "adt_wgraph \<alpha>w \<alpha>g invar adj empty add_edge"
foundation18: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<upsilon> X) = (X \<tau> \<noteq> invalid \<tau>)"
finite_dimension_def: fixes K :: "'a set"   and E :: "'a set" shows "finite_dimension K E = (\<exists>n. dimension n K E)"
assms: fixes p :: "'a pmf"   and q :: "'a pmf"   and r :: "'a pmf"   and \<alpha> :: "real" assumes "p \<in> C"   and "q \<in> C"   and "r \<in> C"   and "\<alpha> \<in> {0<..1}" shows "p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r"
has_vderiv_on_def: fixes f :: "real \<Rightarrow> 'c"   and f' :: "real \<Rightarrow> 'c"   and S :: "real set" shows "(f has_vderiv_on f') S = (\<forall>x\<in>S. (f has_vector_derivative f' x) (at x within S))"
K_statefun_apply: fixes c :: "'c"   and x :: "'d" shows "K_statefun c x = c"
INF_eq: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. less_eq (g j) (f i)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (f i) (g j)" shows "Inf (f ` A) = Inf (g ` B)"