hcju/isabelleps huggingface dataset:
sub3_det: fixes A :: "'a mat"   and d :: "nat"   and l :: "nat"   and r :: "'a"   and r'' :: "'a"   and A'' :: "'a mat" assumes "A \<in> carrier_mat d d"   and "local.sub3 d l (r, A) = (r'', A'')"   and "r \<noteq> (0::'a)"   and "l \<le> d" shows "r * det A'' = r'' * det A"
times_coreflexive_def: shows "(*) \<equiv> map_fun Rep_coreflexive (map_fun Rep_coreflexive Abs_coreflexive) (*)"
s: shows "subspace {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
rep_iso: fixes x :: "'e\<cdot>('f, 'g) writerT" shows "writerT_abs\<cdot>(writerT_rep\<cdot>x) = x"
word_enum_decomp_elem: fixes x :: "'b word"   and y :: "'b word"   and as :: "'b word list"   and a :: "'b word"   and bs :: "'b word list" assumes "[x .e. y] = as @ a # bs" shows "x \<le> a \<and> a \<le> y"
hull_unique: fixes s :: "'d set"   and t :: "'d set"   and S :: "'d set \<Rightarrow> bool" assumes "s \<subseteq> t"   and "S t"   and "\<And>t'. \<lbrakk>s \<subseteq> t'; S t'\<rbrakk> \<Longrightarrow> t \<subseteq> t'" shows "S hull s = t"
induct: fixes P :: "'d list \<Rightarrow> bool"   and list :: "'d list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 # x2)" shows "P list"
card_S_fundchamber: shows "card (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) = card C0"
simps: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "local.basis_reduction_main (upw, i, fs) = (if i < m \<and> LLL_invariant upw i fs then local.basis_reduction_main (local.basis_reduction_step upw i fs) else fs)"
multiplicative_sum_divisors: shows "multiplicative_function (\<lambda>n. \<Sum>d | d dvd n. principal_dchar n d)"
db: shows "degree_bound (berlekamp_hensel p n f) = degree_bound gs"
rank_meld: fixes q1 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and q2 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_skew_invar q1"   and "SkewBinomialHeapStruc.rank_skew_invar q2" shows "SkewBinomialHeapStruc.rank_skew_invar (SkewBinomialHeapStruc.meld q1 q2)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) (F \<circ> F' \<circ> G' \<circ> G) A.map \<epsilon>oF\<epsilon>'G.map"
assms_2: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "f n mod p ^ m = f m"
simps: fixes x :: "'a"   and q :: "'b"   and \<nu> :: "'c"   and q' :: "'b" shows "\<upharpoonleft>\<^sub>x (q, \<nu>, q') = {(m, \<nu>, m') |m m'. m x = Some q \<and> m' x = Some q'}"
rep_eq: fixes x :: "'d \<Rightarrow> 'b"   and xa :: "'d set['c]" shows "set_bset (map_bset x xa) = x ` set_bset xa"
IH_1: fixes env :: "('a, bool) mapping" shows "reduced (reduce env t1) (Mapping.keys env)"
span_not_univ_orthogonal: fixes S :: "'a set" assumes "span S \<noteq> UNIV" shows "\<exists>a. a \<noteq> (0::'a) \<and> (\<forall>x\<in>span S. a \<bullet> x = 0)"
I: fixes v :: "hf"   and x :: "hf"   and x0 :: "hf"   and w :: "hf"   and xw :: "hf"   and xevw :: "hf"   and allw :: "hf"   and allvw :: "hf"   and ax :: "hf" assumes "SubstForm v 0 x x0"   and "SubstForm v w x xw"   and "SubstForm v (q_Eats v w) x xevw"   and "AbstForm w 0 (q_Imp x (q_Imp xw xevw)) allw"   and "AbstForm v 0 (q_All allw) allvw"   and "AbstForm v 0 x ax"   and "v \<noteq> w"   and "VarNonOccForm w x" shows "q_Imp x0 (q_Imp (q_All allvw) (q_All ax)) \<in> Induction_ax"
local_ring_def: fixes R :: "('c, 'd) Ring_scheme" shows "local_ring R \<equiv> Ring R \<and> \<not> zeroring R \<and> card {mx. maximal_ideal R mx} = 1"
finite_reach: fixes \<phi> :: "'a ltl" shows "finite (reach \<Sigma> (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>))"
singleton_table_def: fixes n :: "nat"   and i :: "nat"   and x :: "'b" shows "singleton_table n i x = {tabulate (\<lambda>j. if i = j then Some x else None) 0 n}"
aff_dim_unique: fixes B :: "'b set"   and V :: "'b set" assumes "affine hull B = affine hull V \<and> \<not> affine_dependent B" shows "int (card B) = aff_dim V + 1"
section_map_def: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "section_map X Y f \<equiv> \<exists>g. retraction_maps Y X g f"
randomElLemma: fixes list :: "'a list"   and random :: "integer" assumes "set list \<noteq> {}" shows "randomEl list random \<in> set list"
ideD_3: fixes a :: "'c" assumes "C.ide a" shows "C.cod a = a"
Sup_le_iff: fixes A :: "'a set"   and b :: "'a" shows "(\<Squnion> A \<le> b) = (\<forall>a\<in>A. a \<le> b)"
fst_of_sp_is_vert: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "SubRel.subpath g v1 es v2 subs" shows "v1 \<in> Graph.vertices g"
conv_abscissa_MInf_iff: fixes f :: "'a fds" shows "(conv_abscissa f = - \<infinity>) = All (fds_converges f)"
Pi_cancel_fupd: fixes i :: "'c"   and I :: "'c set"   and x :: "'c \<Rightarrow> 'd"   and a :: "'d"   and B :: "'c \<Rightarrow> 'd set" assumes "i \<notin> I" shows "(x(i := a) \<in> Pi I B) = (x \<in> Pi I B)"
entt_true: fixes A :: "assn" shows "A \<Longrightarrow>\<^sub>t true"
nhds_prod: fixes a :: "'d"   and b :: "'e" shows "nhds (a, b) = nhds a \<times>\<^sub>F nhds b"
infinite: fixes A :: "'b set" assumes "infinite A" shows "sorted_list_of_set A = []"
count_inI: fixes M :: "'b multiset"   and x :: "'b" assumes "count M x = 0 \<Longrightarrow> False" shows "x \<in># M"
solve_lir_def: fixes cs :: "complex list"   and fs :: "complex list"   and g :: "(complex \<times> nat \<times> complex) list" shows "solve_lir cs fs g = lir_fps cs fs g \<bind> solve_ratfps"
simps_1: fixes n :: "nat"   and x :: "'a" shows "bddh n (Leaf x) = True"
sup_commute: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "sup_prod x y = sup_prod y x"
normalize_def: fixes p :: "((int \<times> int) set set, int) Combinators" shows "FWNormalisationCore.normalize p = removeAllDuplicates (insertDenies (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) (Nets_List p))))"
downset_intro: fixes K :: "'a multiset"   and J :: "'a multiset"   and r :: "('a \<times> 'a) set" assumes "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r" shows "set_mset K \<subseteq> r \<down>m J"
normal_restrict_supergroup: fixes G :: "('c, 'd) monoid_scheme"   and S :: "'c set"   and N :: "'c set" assumes "Group.group G"   and "subgroup S G"   and "N \<lhd> G"   and "N \<subseteq> S" shows "N \<lhd> G\<lparr>carrier := S\<rparr>"
nth_rotate: fixes n :: "nat"   and xs :: "'b list"   and m :: "nat" assumes "n < length xs" shows "rotate m xs ! n = xs ! ((m + n) mod length xs)"
d_def: shows "d \<equiv> gcd a b * unit_factor b"
I_intP_Text_not_intT: fixes p :: "'psym"   and al :: "univ list" assumes "polC p = Text"   and "\<not> list_all2 Ik_intT (parOf p) al" shows "I_intP p al"
genby_Group: fixes S :: "'a set" shows "Group \<langle>S\<rangle>"
sign_changes_filter: fixes xs :: "'a list" shows "sign_changes (filter (\<lambda>x. x \<noteq> (0::'a)) xs) = sign_changes xs"
intros: fixes X :: "'f set"   and gpv :: "('d, 'e, 'f) gpv"   and P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "\<And>x. x \<in> results_gpv (\<I>_uniform UNIV X) gpv \<Longrightarrow> P x"   and "\<And>out. out \<in> outs_gpv (\<I>_uniform UNIV X) gpv \<Longrightarrow> Q out" shows "pred_gpv' P Q X gpv"
subst_respects_ltl_const_entailment: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and m :: "'a ltln \<Rightarrow> 'a ltln option" assumes "\<phi> \<sim>\<^sub>C \<psi>" shows "subst \<phi> m \<sim>\<^sub>C subst \<psi> m"
history_order_def: fixes x :: "'evt"   and i :: "nat"   and z :: "'evt" shows "x \<sqsubset>\<^sup>i z \<equiv> \<exists>xs ys zs. xs @ x # ys @ z # zs = history i"
simps_5: fixes S :: "'b formula \<Rightarrow> 'b formula"   and F :: "'b formula"   and G :: "'b formula" shows "tseytin_tran1 S (F \<^bold>\<or> G) = [S (F \<^bold>\<or> G) \<^bold>\<leftrightarrow> S F \<^bold>\<or> S G] @ tseytin_tran1 S F @ tseytin_tran1 S G"
iso: shows "f' \<circ> f \<in> Group.iso A A"
decliface_simp: fixes I :: "qtname"   and m :: "'a decl_scheme" shows "decliface (I, m) = I"
starlike_negligible: fixes S :: "'c set"   and a :: "'c" assumes "closed S"   and "\<And>c x. \<lbrakk>a + c *\<^sub>R x \<in> S; 0 \<le> c; a + x \<in> S\<rbrakk> \<Longrightarrow> c = 1" shows "negligible S"
affine_hyperplane: fixes a :: "'b"   and b :: "real" shows "affine {x. a \<bullet> x = b}"
substAbs_psubstAbs_compose: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "goodEnv rho" shows "A $[rho] $[Y / y]_ys = A $[rho &[Y / y]_ys]"
feasible_nnegD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "real"   and P :: "'a \<Rightarrow> real" assumes "feasible t"   and "bounded_by b P"   and "nneg P" shows "nneg (t P)"
if_distribR: fixes b :: "bool"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(if b then f else g) x = (if b then f x else g x)"
counit_simps_2: shows "D.dom \<psi> = F f \<star>\<^sub>D g'"
pred_stream_cfg_on: fixes cfg :: "'s cfg"   and xs :: "'s cfg stream" assumes "cfg \<in> valid_cfg"   and "MC.enabled cfg xs" shows "pred_stream (\<lambda>cfg. state cfg \<in> S \<and> cfg \<in> cfg_on (state cfg)) xs"
h1: shows "\<forall>x. h (1, x) = q x"
int_ops_7: fixes a :: "nat"   and b :: "nat" shows "int (a * b) = int a * int b"
SetMem_SetInsert: fixes v :: "nat discr Box"   and x :: "nat discr Box"   and s :: "nat discr Box llist" shows "SetMem\<cdot>v\<cdot>(SetInsert\<cdot>x\<cdot>s) = (SetMem\<cdot>v\<cdot>s orelse x =\<^sub>B v)"
MarkGuardsI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and f :: "'f" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/{}\<^esub> P mark_guards f c Q,A"
chine_counit_in_hom: shows "\<guillemotleft>\<epsilon>.chine : f.leg0 \<down>\<down> g.leg1 \<rightarrow>\<^sub>C f.dtrg\<guillemotright>"
simps: fixes ant :: "('a, 'b) form multiset"   and suc :: "('a, 'b) form multiset" shows "ModalSequents.mset ( ant \<Rightarrow>* suc) = ant + suc"
leq_term_def: shows "leq_term = (SOME leq. {(s, t). less_kbo s t} \<subseteq> leq \<and> Well_order leq \<and> Field leq = UNIV)"
R5_d_ge_one_half: shows "1 / 2 \<le> pmf (sds R5) d"
halt_state_keep: fixes m :: "nat"   and lm :: "'b"   and stpa :: "nat"   and stpb :: "nat" assumes "nonstop m (bl2wc (<lm>)) stpa = 0"   and "nonstop m (bl2wc (<lm>)) stpb = 0" shows "conf m (bl2wc (<lm>)) stpa = conf m (bl2wc (<lm>)) stpb"
mem_permute_iff: fixes p :: "perm"   and x :: "'b"   and X :: "'b set" shows "(p \<bullet> x \<in> p \<bullet> X) = (x \<in> X)"
h_1: shows "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) (Suc stp) = (s', l', r')"
infinitely_many_def: fixes p :: "nat \<Rightarrow> bool" shows "infinitely_many p \<equiv> \<exists>\<^sub>\<infinity>j. p j"
indep_vars_Min: fixes I :: "'c set"   and i :: "'c"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> real" assumes "finite I"   and "i \<notin> I"   and "indep_vars (\<lambda>_. borel) X (insert i I)" shows "indep_var borel (X i) borel (\<lambda>\<omega>. MIN i\<in>I. X i \<omega>)"
coinduct: fixes X :: "ccs \<Rightarrow> ccs \<Rightarrow> bool"   and x1 :: "ccs"   and x2 :: "ccs" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>P Q. x1 = P \<and> x2 = Q \<and> P \<leadsto>[{(xa, x). X xa x \<or> xa \<sim> x}] Q \<and> (X Q P \<or> Q \<sim> P)" shows "x1 \<sim> x2"
2: shows "fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q x = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X"
subst_domain_compose: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and \<tau> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "subst_domain (\<sigma> \<circ>\<^sub>s \<tau>) \<subseteq> subst_domain \<sigma> \<union> subst_domain \<tau>"
minus_ms_aux_def: fixes xs :: "('a \<times> real) msllist"   and ys :: "('a \<times> real) msllist" shows "minus_ms_aux xs ys = plus_ms_aux xs (uminus_ms_aux ys)"
assms_2: shows "G \<hookrightarrow>\<^sub>F H"
mopup_aft_erase_b_2_aft_erase_c: fixes n :: "nat"   and l :: "cell list"   and xs :: "cell list"   and lm :: "nat list"   and ires :: "cell list" assumes "mopup_aft_erase_b (2 * n + 3, l, Bk # xs) lm n ires" shows "mopup_aft_erase_c (4 + 2 * n, Bk # l, xs) lm n ires"
coset_mult_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "M #>\<^bsub>add_monoid G\<^esub> g #>\<^bsub>add_monoid G\<^esub> h = M #>\<^bsub>add_monoid G\<^esub> g \<oplus> h"
vec_module_col_helper2: fixes x :: "'b vec"   and A :: "'b mat"   and a :: "'b" assumes "x \<in> LinearCombinations.module.span class_ring \<lparr>carrier = carrier_vec (dim_row A), mult = undefined, one = undefined, zero = 0\<^sub>v (dim_row A), add = (+), smult = (\<cdot>\<^sub>v)\<rparr> (set (cols A))"   and "\<And>a b v. (a + b) \<cdot>\<^sub>v v = a \<cdot>\<^sub>v v + b \<cdot>\<^sub>v v" shows "a \<cdot>\<^sub>v x \<in> LinearCombinations.module.span class_ring \<lparr>carrier = carrier_vec (dim_row A), mult = undefined, one = undefined, zero = 0\<^sub>v (dim_row A), add = (+), smult = (\<cdot>\<^sub>v)\<rparr> (set (cols A))"
nn_integral_le_const: fixes c :: "ennreal"   and f :: "'s stream \<Rightarrow> ennreal"   and s :: "'s" assumes "0 \<le> c"   and "AE x in T s. f x \<le> c" shows "integral\<^sup>N (T s) f \<le> c"
gen_dfs_outer_def: fixes E :: "('b \<times> 'b) set"   and U :: "'b set"   and V0 :: "'b set"   and it :: "'b set"   and V :: "'b set"   and brk :: "bool" shows "gen_dfs_outer E U V0 it V brk \<equiv> V0 \<subseteq> U \<and> E `` U \<subseteq> U \<and> finite U \<and> V \<subseteq> U \<and> (\<not> brk \<longrightarrow> E `` V \<subseteq> V) \<and> V0 - it \<subseteq> V"
h5a: assumes "0 \<le> Av"   and "Bv \<le> 0"   and "Av < - Bv * sqrt Cv" shows "Av\<^sup>2 < Bv\<^sup>2 * Cv"
relpowp_0_E: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "(P ^^ 0) x y" obtains "x = y"
int_shift_measurable_1: shows "int_left_shift \<in> borel_measurable borel"
that_2: assumes "\<And>m. \<exists>n. m \<le> f n" shows "thesis"
bigo_pos_const: fixes h :: "'d \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c" shows "(\<exists>c. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>) = (\<exists>c>0::'c. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)"
simps_11: fixes x3 :: "'e"   and x2 :: "'d" shows "Fault x3 \<noteq> Abrupt x2"
abs_eq: fixes x :: "'b \<Rightarrow> 'c"   and xb :: "'b"   and xa :: "'c" assumes "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'c)}) x x" shows "update xb xa (Abs_poly_mapping x) = Abs_poly_mapping (x(xb := xa))"
rsr: fixes R :: "('b \<times> 'b) set" assumes "R \<subseteq> Id" shows "s2r (r2s R) = R"
choice: fixes Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<forall>x. \<exists>y. Q x y" shows "\<exists>f. \<forall>x. Q x (f x)"
card_gr0_imp_finite: fixes A :: "'a set" assumes "0 < card A" shows "finite A"
eq_stock_def: shows "eq_stock = qty_replace_comp p x pf"
responses: fixes x :: "'a" assumes "x \<in> outs_\<I> \<I>1" shows "responses_\<I> \<I>2 x \<subseteq> responses_\<I> \<I>1 x"
Axiom_of_Extensionality_3: fixes a :: "V"   and x :: "V"   and y :: "V" assumes "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "x = y"   and "x \<in>\<^sub>\<circ> a" shows "y \<in>\<^sub>\<circ> Vset \<alpha>"
simps_2: fixes x :: "'b"   and l :: "'b list" shows "perm2 (x # l) = (\<lambda>n. rotateLeft (length (x # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l)))) - n mod (1 + length l) mod length (x # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l))))) (x # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l)))))"
has_integral_affinity_iff: fixes m :: "real"   and f :: "'b \<Rightarrow> 'c"   and c :: "'b"   and I :: "'c"   and a :: "'b"   and b :: "'b" assumes "0 < m" shows "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral I /\<^sub>R m ^ DIM('b)) (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) = (f has_integral I) (cbox a b)"
enat_add_left_cancel: fixes a :: "enat"   and b :: "enat"   and c :: "enat" shows "(a + b = a + c) = (a = \<infinity> \<or> b = c)"
sequentially_imp_eventually_at: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "class.first_countable_topology open"   and "\<forall>f. (\<forall>n. f n \<noteq> a) \<and> filterlim f (topological_space.nhds open a) sequentially \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n))" shows "eventually P (topological_space.at_within open a UNIV)"
nth_prime_less_iff: fixes m :: "nat"   and n :: "nat" shows "(nth_prime m < nth_prime n) = (m < n)"
uint_word_of_int: fixes k :: "int" shows "uint (word_of_int k) = k mod 2 ^ LENGTH('b)"
termMOD_igSwapAbsIPresIGWlsAbs: shows "igSwapAbsIPresIGWlsAbs termMOD"
diff_diff_right: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "i - (j - k) = i + k - j"
finite_partGen: fixes P :: "'a set set" assumes "finite P" shows "finite (partGen P)"
match_dom: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "term"   and t :: "'b"   and env :: "(name, 'b) fmap" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "term.match unapp unconst p t = Some env" shows "fmdom env = frees p"
subterms_list_is_subterms: fixes t :: "('a, 'b) Term.term" shows "subterms t = set (subterms_list t)"
inj_add_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.cancel_semigroup_add plus" shows "inj (plus a)"
nearly_healthy_wlp_loop: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "nearly_healthy (wlp body)" shows "nearly_healthy (wlp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))"
simps: fixes w1l :: "32 Word.word"   and w1u :: "32 Word.word"   and w2l :: "32 Word.word"   and w2u :: "32 Word.word" shows "Interval_Word32.tl w1l w1u w2l w2u = wmin (wmin (wtimes w1l w2l) (wtimes w1u w2l)) (wmin (wtimes w1l w2u) (wtimes w1u w2u))"
weakBisim_def: shows "weakBisim \<equiv> {(xa, x). weakBisimp xa x}"
add_le_less_mono: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" assumes "a \<le>\<^sub>\<complex> b"   and "c <\<^sub>\<complex> d" shows "a + c <\<^sub>\<complex> b + d"
chfin_iterates_above_fun_pow: fixes x :: "'a"   and a :: "'a" assumes "x \<in> iterates_above a"   and "\<forall>M\<in>Chains leq. finite M" shows "\<exists>j. x = (f ^^ j) a"
valid_Pred: fixes P :: "('state, 'effect) L_state"   and \<phi> :: "'pred" shows "L_transform.valid P (Formula.Pred \<phi>) = P \<turnstile>\<^sub>L \<phi>"
vfsequence_vcard_vcons: fixes x :: "V" shows "vcard (xs #\<^sub>\<circ> x) = ZFC_in_HOL.succ (vcard xs)"
cases: fixes a :: "'a"   and execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and initial :: "'b \<Rightarrow> bool" assumes "a \<in> transition_system_initial.executable execute enabled initial" obtains p :: "'b" and aa :: "'a" where "a = aa" and "p \<in> transition_system_initial.nodes execute enabled initial" and "enabled aa p"
unf_def: shows "unf \<equiv> unfold_tree (\<lambda>f. f x) (\<lambda>f. f \<circ> l) (\<lambda>f. f \<circ> r)"
eq_iff_7: fixes term1 :: "Syntax.term"   and term2 :: "Syntax.term"   and term1a :: "Syntax.term"   and term2a :: "Syntax.term" shows "(Syntax.Pair term1 term2 = Syntax.Pair term1a term2a) = (term1 = term1a \<and> term2 = term2a)"
skip_fun: fixes i :: "nat"   and n :: "nat" shows "skip i \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> Suc n}"
inext_cut_ge_conv: fixes t :: "nat"   and n :: "nat"   and I :: "nat set" assumes "t \<le> n" shows "inext n (I \<down>\<ge> t) = inext n I"
assms_1: shows "canon_ang_P \<alpha> \<alpha>\<^sub>1"
red_def: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "red F p q = (\<exists>f\<in>F. \<exists>t. red_single p q f t)"
M_def: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "M x y z = ((y l\<rightarrow> x) r\<rightarrow> x) \<sqinter> ((z l\<rightarrow> y) r\<rightarrow> y) \<sqinter> ((x l\<rightarrow> z) r\<rightarrow> z)"
wf_until_auxlist_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and pos :: "bool"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and ne :: "nat" shows "wf_until_auxlist \<sigma> V n R pos \<phi> I \<psi> auxlist ne = list_all2 (\<lambda>x i. case x of (t, r1, r2) \<Rightarrow> t = \<tau> \<sigma> i \<and> qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. if pos then \<forall>k\<in>{i..<ne + length auxlist}. Formula.sat \<sigma> V (map the v) k \<phi> else \<exists>k\<in>{i..<ne + length auxlist}. Formula.sat \<sigma> V (map the v) k \<phi>) r1 \<and> qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. \<exists>j\<ge>i. j < ne + length auxlist \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> Formula.sat \<sigma> V (map the v) j \<psi> \<and> (\<forall>k\<in>{i..<j}. if pos then Formula.sat \<sigma> V (map the v) k \<phi> else \<not> Formula.sat \<sigma> V (map the v) k \<phi>)) r2) auxlist [ne..<ne + length auxlist]"
hausdorff: fixes "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.t2_space open"   and "x \<noteq> y" shows "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
flow_funcset_ex_ivl: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" assumes "picard_lindeloef f T S t\<^sub>0"   and "s \<in> S" shows "ll_on_open.flow T f S t\<^sub>0 s \<in> ll_on_open.existence_ivl T f S t\<^sub>0 s \<rightarrow> S"
lt_ex: fixes x :: "'a" shows "\<exists>y. y \<sqsubset> x"
thetaSeqZOT_ZObisT: shows "thetaSeqZOT \<subseteq> ZObisT"
char_matrix_closed: fixes A :: "'c mat"   and n :: "nat"   and e :: "'c" assumes "A \<in> carrier_mat n n" shows "char_matrix A e \<in> carrier_mat n n"
mk_poly_sylvester_lower: fixes q :: "'b poly"   and i :: "nat"   and p :: "'b poly" assumes "degree q \<le> i"   and "i < degree p + degree q" shows "mk_poly (sylvester_mat p q) $$ (i, degree p + degree q - 1) = monom (1::'b) (degree p + degree q - Suc i) * q"