hcju/isabelleps huggingface dataset:
mat_mult2_00: fixes U :: "'b mat"   and V :: "'b mat"   and A :: "'b mat" assumes "U \<in> carrier_mat 2 2"   and "V \<in> carrier_mat 2 2"   and "A = U * V" shows "A $$ (0, 0) = U $$ (0, 0) * V $$ (0, 0) + U $$ (0, 1) * V $$ (1, 0)"
inangle__lta: fixes P :: "'p"   and B :: "'p"   and C :: "'p"   and A :: "'p" assumes "\<not> Col P B C"   and "P InAngle A B C" shows "A B P LtA A B C"
is_relevant_entry_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and i :: "instr"   and pc :: "nat"   and e :: "nat \<times> nat \<times> char list \<times> nat \<times> nat" shows "is_relevant_entry P i pc e = (let (f, t, C, h, d) = e in is_relevant_class i P C \<and> pc \<in> {f..<t})"
Rep_formula_inverse: fixes x :: "('a, 'b, 'c, 'd) formula" shows "Abs_formula (Rep_formula x) = x"
summandsInput: fixes a :: "name"   and x :: "name"   and P :: "pi" shows "summands (a<x>.P) = {a<x>.P}"
absorb_iff2: fixes b :: "'a"   and a :: "'a" shows "(b \<preceq> a) = (a \<sqinter> b = b)"
subset_Diff_insert: fixes A :: "'g set"   and B :: "'g set"   and x :: "'g"   and C :: "'g set" shows "(A \<subseteq> B - insert x C) = (A \<subseteq> B - C \<and> x \<notin> A)"
0: fixes x :: "'a"   and y :: "'a" assumes "x \<in> span A"   and "y \<in> span B" shows "x \<bullet> y = 0"
ma': shows "ma_coeff ma' = 0 \<and> ma_base ma' = 0 \<or> ma_coeff ma' \<noteq> 0 \<and> prime_product (ma_base ma')"
simps_3: fixes e :: "(name, nat) finfun"   and A :: "fmla" shows "eval_fmla e (Robinson_Arithmetic.neg A) = (\<not> eval_fmla e A)"
LSQ_def: shows "LSQ = cbox (case (real_of_int l - \<delta>, snd p\<^sub>0) of (x, y) \<Rightarrow> (x, real_of_int y)) (case (l, real_of_int (snd p\<^sub>0) + \<delta>) of (x, y) \<Rightarrow> (real_of_int x, y))"
prems_2: shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>t (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S) = {}"
Local: fixes P :: "state \<Rightarrow> bool"   and P' :: "state \<Rightarrow> state \<Rightarrow> bool"   and f :: "state \<Rightarrow> state"   and c :: "com"   and Q :: "state \<Rightarrow> bool"   and g :: "state \<Rightarrow> state \<Rightarrow> state" assumes "\<And>s. P s \<Longrightarrow> P' s (f s)"   and "\<forall>s. \<turnstile> {P' s} c {Q \<circ> g s}" shows "\<turnstile> {P} LOCAL f; c; g {Q}"
higher_differentiable_on_uminus: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "higher_differentiable_on S f n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. - f x) n"
homotopy_equivalence_relative_homology_group_isomorphisms: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and p :: "int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "continuous_map Y X g"   and "g ` T \<subseteq> S"   and "homotopic_with (\<lambda>h. h ` S \<subseteq> S) X X (g \<circ> f) id"   and "homotopic_with (\<lambda>k. k ` T \<subseteq> T) Y Y (f \<circ> g) id" shows "group_isomorphisms (relative_homology_group p X S) (relative_homology_group p Y T) (hom_induced p X S Y T f) (hom_induced p Y T X S g)"
matrix_vector_all_zero_from_index: fixes i :: "'b"   and A :: "(('c, 'd) vec, 'b) vec"   and k :: "'d" shows "(\<forall>m\<ge>i. A $ m $ k = (0::'c)) = vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column k A))"
simps_2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "foldseq_transposed f s (Suc n) = f (foldseq_transposed f s n) (s (Suc n))"
fIn: fixes n :: "nat" shows "f ` (cbox (vec (- real n)) (vec (real n)) \<inter> S) \<in> lmeasurable"
assms_4: shows "no_matching_Goto \<gamma> p rs\<^sub>1"
factors_def: fixes G :: "('a, 'b) monoid_scheme"   and fs :: "'a list"   and a :: "'a" shows "Divisibility.factors G fs a = ((\<forall>x\<in>set fs. Divisibility.irreducible G x) \<and> foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> = a)"
ordering1: shows "\<forall>n. (finite {a, b} \<longrightarrow> n < card {a, b}) \<longrightarrow> (if n = 0 then a else b) \<in> {a, b}"
list_tail_coinc: fixes n1 :: "'c"   and r1 :: "'c list"   and n2 :: "'c"   and r2 :: "'c list" assumes "n1 # r1 = n2 # r2" shows "n1 = n2 \<and> r1 = r2"
wt_s't_ge_st: shows "wt (App s' t) \<ge>\<^sub>p wt (App s t)"
mult_cancel_right: fixes f2 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g2 :: "'b \<Rightarrow> 'c"   and f1 :: "'b \<Rightarrow> 'c"   and g1 :: "'b \<Rightarrow> 'c" assumes "f2 \<in> \<Theta>[F](g2)"   and "\<forall>\<^sub>F x in F. g2 x \<noteq> (0::'c)" shows "((\<lambda>x. f1 x * f2 x) \<in> O[F](\<lambda>x. g1 x * g2 x)) = (f1 \<in> O[F](g1))"
leq_trans: fixes x :: "'c spmf"   and y :: "'c spmf"   and z :: "'c spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y z" shows "ord_spmf (=) x z"
unique_expression2: fixes H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "H \<subseteq> carrier M"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "s \<in> {j. j \<le> n} \<rightarrow> carrier R" shows "\<exists>m g t. g \<in> {j. j \<le> m} \<rightarrow> H \<and> bij_to g {j. j \<le> m} (f ` {j. j \<le> n}) \<and> t \<in> {j. j \<le> m} \<rightarrow> carrier R \<and> l_comb R M n s f = l_comb R M m t g"
bij: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexAutomorphism X f" shows "bij f"
assms_4: shows "dim_col U\<^sub>f = dim_row A"
matrix_vector_mult_0: fixes w :: "('a, 'c) vec" shows "0 *v w = 0"
consis_i: shows "sharing_consistent \<S> \<O>\<^sub>i sb\<^sub>i"
swap_swap: fixes p :: "'d \<times> 'e" shows "prod.swap (prod.swap p) = p"
mod_eq_0D: fixes m :: "nat"   and d :: "nat" assumes "m mod d = 0" shows "\<exists>q. m = d *n q"
AnalBinds_not_there: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>" assumes "x \<notin> domA \<Gamma>" shows "(AnalBinds \<Gamma>\<cdot>ae) x = \<bottom>"
add_assoc4: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "a + b + c + d = a + (b + c + d)"
is_RB_in_UnI: fixes d :: "'a \<Rightarrow> nat"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and H :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "is_RB_in d rword G u"   and "\<And>h. h \<in> H \<Longrightarrow> u \<prec>\<^sub>t lt h" shows "is_RB_in d rword (H \<union> G) u"
that_2: assumes "bdd_below S"   and "\<And>y. y \<in> S \<Longrightarrow> x \<le> y" shows "thesis"
373: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (x \<sqinter> y) = - (x \<sqinter> y)"
length_remove1: fixes x :: "'b"   and xs :: "'b list" shows "length (remove1 x xs) = (if x \<in> set xs then length xs - 1 else length xs)"
comp_permute: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c"   and k :: "'c \<times> 'c"   and l :: "'c \<times> 'c"   and h :: "'c \<times> 'c" assumes "C.VV.comp f g = C.VV.comp k l"   and "C.VV.seq f g"   and "C.VV.seq g h" shows "C.VV.comp f (C.VV.comp g h) = C.VV.comp k (C.VV.comp l h)"
test_comp_add2: fixes p :: "'a" shows "\<iota> p + ! (\<iota> p) = (1::'b)"
elims: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "Term_Class.term \<times> 'a"   and xa :: "'a"   and y :: "'a option" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "term.rewrite_step subst unapp unconst x xa = y" obtains t\<^sub>1 :: "Term_Class.term" and t\<^sub>2 :: "'a" and u :: "'a" where "x = (t\<^sub>1, t\<^sub>2)" and "xa = u" and "y = map_option (subst t\<^sub>2) (term.match unapp unconst t\<^sub>1 u)"
subst_dom_vars_in_subst: fixes v :: "'a"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<in> subst_domain s" shows "s v \<noteq> Var v"
adv: assumes "\<And>\<eta>. exception_\<I> (\<I>_real2 \<eta> \<oplus>\<^sub>\<I> \<I>_common2 \<eta>) \<turnstile>g absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (converter_of_resource (real1 \<eta>) |\<^sub>\<propto> 1\<^sub>C))) \<surd>" shows "Negligible.negligible (\<lambda>\<eta>. advantage (absorb (\<A> \<eta>) (obsf_converter (parallel_wiring \<odot> (converter_of_resource (real1 \<eta>) |\<^sub>\<propto> 1\<^sub>C)))) (obsf_resource (sim2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal2 \<eta>)) (obsf_resource (real2 \<eta>)))"
deterministic: fixes q :: "'Q"   and f :: "'L"   and qs :: "'Q list"   and q' :: "'Q" assumes "q \<rightarrow> f qs \<in> \<delta>"   and "q' \<rightarrow> f qs \<in> \<delta>" shows "q = q'"
rtrancl_path_appendE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list"   and z :: "'b" assumes "rtrancl_path r x (xs @ y # ys) z" obtains "rtrancl_path r x (xs @ [y]) y" and "rtrancl_path r y ys z"
rel_interior_sum: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "convex T" shows "rel_interior (S + T) = rel_interior S + rel_interior T"
assms_2: shows "\<not> sig_crit bs (new_syz_sigs ss bs p) p"
set_sel_1: fixes a :: "'c llist" assumes "\<not> lnull a" shows "lhd a \<in> lset a"
Collect_conv_if2: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" shows "{x. a = x \<and> P x} = (if P a then {a} else {})"
comp_arr_ide: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.ide a"   and "CC.seq f a" shows "CC.comp f a = f"
chain_boundary_boundary: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "chain_boundary (p - Suc 0) (chain_boundary p c) = 0"
bij_def: fixes f :: "'a \<Rightarrow> 'b" shows "bij f = (inj f \<and> top_pres ((`) f))"
ChamberComplexFolding_axioms: shows "ChamberComplexFolding X g"
simps_2: fixes dims :: "nat list"   and U :: "complex mat" shows "state_sig.well_com dims (Utrans U) = (U \<in> carrier_mat (state_sig.d dims) (state_sig.d dims) \<and> unitary U)"
simps_1: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and h :: "'c" shows "nth_derivative 0 f x h = f x"
extend_mem: fixes S :: "(('c, 'd) fm list \<times> 'd) set"   and A :: "'d set"   and f :: "nat \<Rightarrow> ('c, 'd) fm list \<times> 'd"   and n :: "nat" shows "S \<subseteq> extend A S f n"
dfree_Var: fixes i :: "'b" shows "dfree (Var i)"
is_projection_on_exists: fixes M :: "'b set" assumes "convex M"   and "closed M"   and "M \<noteq> {}" shows "\<exists>\<pi>. is_projection_on \<pi> M"
span_finite: fixes S :: "'c set" assumes "finite S" shows "l.vs2.span S = range (\<lambda>u. \<Sum>v\<in>S. scaleC (u v) v)"
preserves_reflects_arr: fixes f :: "'j" shows "A.arr (\<chi>.A.map f) = \<chi>.J.arr f"
of_nat_0_mod_ring_dvd: fixes x :: "nat" assumes "of_nat x = 0" shows "CARD('a) dvd x"
ideI: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "arr (fst a) \<and> arr (snd a) \<and> src (fst a) = trg (snd a)"   and "VxV.ide a" shows "VV.ide a"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(Wit x2 = Wit y2) = (x2 = y2)"
66: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<sqinter> x \<squnion> - (x \<squnion> y) \<sqinter> y = bot"
at_within_eq_bot_lemma2: fixes x :: "'c"   and a :: "'c"   and b :: "'c" shows "at x within {a..b} = (if b < x \<or> b < a then bot else at x within {a..b})"
assms_4: shows "\<exists>\<^sub>F w in at z0. f w \<noteq> 0"
distinct_31: fixes x2 :: "sethi_type"   and x7 :: "logic_type" shows "sethi_type x2 \<noteq> logic_type x7"
pderiv_pcompose: fixes p :: "'b poly"   and q :: "'b poly" shows "pderiv (p \<circ>\<^sub>p q) = pderiv p \<circ>\<^sub>p q * pderiv q"
cl_inv_lengthD: fixes g :: "'a gState_scheme"   and p :: "pState" assumes "cl_inv (g, p)" shows "length (pState.channels p) \<le> length (gState.channels g)"
intro_elim_4_b: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<or> \<psi> in v]"   and "[\<^bold>\<not>\<phi> in v]" shows "[\<psi> in v]"
m1_inv1r_cacheI: fixes x :: "'a m1_state_scheme" assumes "\<And>Rb A B Kab Ts Ta nl. \<lbrakk>runs x Rb = Some (Resp, [A, B], aKey Kab # aNum Ts # aNum Ta # nl); clk x < Ta + La\<rbrakk> \<Longrightarrow> (B, Kab, Ta) \<in> cache x" shows "x \<in> m1_inv1r_cache"
vcard_0: shows "vcard []\<^sub>\<circ> = []\<^sub>\<circ>"
less_and_not_mask_eq: fixes r2 :: "nat"   and r1 :: "nat"   and i :: "'a word"   and x :: "'a word" assumes "r2 \<le> r1"   and "i && ~~ mask r2 = x && ~~ mask r2" shows "i && ~~ mask r1 = x && ~~ mask r1"
induct: fixes P :: "poly \<Rightarrow> poly \<Rightarrow> bool"   and a0 :: "poly"   and a1 :: "poly" assumes "\<And>c c'. P (C c) (C c')"   and "\<And>c c' n' p'. P (C c) c' \<Longrightarrow> P (C c) (CN c' n' p')"   and "\<And>c n p c'. P c (C c') \<Longrightarrow> P (CN c n p) (C c')"   and "\<And>c n p c' n' p'. \<lbrakk>P c (CN c' n' p'); P (CN c n p) c'; P c c'; \<And>x. P p p'\<rbrakk> \<Longrightarrow> P (CN c n p) (CN c' n' p')"   and "\<And>v b. P (Bound v) b"   and "\<And>v va b. P (Add v va) b"   and "\<And>v va b. P (Sub v va) b"   and "\<And>v va b. P (Mul v va) b"   and "\<And>v b. P (Neg v) b"   and "\<And>v va b. P (Pw v va) b"   and "\<And>v va vb vc. P (CN v va vb) (Bound vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Add vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Sub vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Mul vc vd)"   and "\<And>v va vb vc. P (CN v va vb) (Neg vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Pw vc vd)"   and "\<And>a v. P a (Bound v)"   and "\<And>a v va. P a (Add v va)"   and "\<And>a v va. P a (Sub v va)"   and "\<And>a v va. P a (Mul v va)"   and "\<And>a v. P a (Neg v)"   and "\<And>a v va. P a (Pw v va)"   and "\<And>vc v va vb. P (Bound vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Add vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Sub vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Mul vc vd) (CN v va vb)"   and "\<And>vc v va vb. P (Neg vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Pw vc vd) (CN v va vb)" shows "P a0 a1"
asToHeap_heapToAssn: fixes as :: "(Vars.var \<times> exp) list" shows "asToHeap (heapToAssn as) = as"
Derives1_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.Derives1 \<NN> \<TT> \<RR> u i r v = (\<exists>x y N \<alpha>. u = x @ [N] @ y \<and> v = x @ \<alpha> @ y \<and> CFG.is_sentence \<NN> \<TT> x \<and> CFG.is_sentence \<NN> \<TT> y \<and> (N, \<alpha>) \<in> \<RR> \<and> r = (N, \<alpha>) \<and> i = length x)"
phiDefs_def: fixes phis :: "'c \<Rightarrow> 'd \<times> 'b \<Rightarrow> 'b list option"   and g :: "'c"   and n :: "'d" shows "CFG_SSA_base.phiDefs phis g n \<equiv> {v. (n, v) \<in> dom (phis g)}"
measurable_const: fixes c :: "'b"   and M' :: "'b measure"   and M :: "'c measure" assumes "c \<in> space M'" shows "(\<lambda>x. c) \<in> M \<rightarrow>\<^sub>M M'"
partial_function_definitions_axioms: shows "partial_function_definitions lprefix lSup"
shows_list_xmldoc_def: fixes xs :: "xmldoc list" shows "shows_list xs = showsp_list shows_prec 0 xs"
convol_o: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'd" shows "\<langle>f, g\<rangle> \<circ> h = \<langle>f \<circ> h, g \<circ> h\<rangle>"
nhds_metric: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.metric_space dist uniformity open" shows "topological_space.nhds open x = (INF e\<in>{0<..}. principal {y. dist y x < e})"
simps_2: fixes X :: "freeExp"   and Y :: "freeExp" shows "freediscrim (PLUS X Y) = 1"
greatestPD1: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "greatestP P x" shows "P x"
lr_of_tran_no_overlaps: fixes ifs :: "char list list"   and t :: "(of_match_field, of_action list) flow_entry_match list"   and rt :: "32 routing_rule list"   and fw :: "32 simple_rule list" assumes "distinct ifs"   and "Inr t = lr_of_tran rt fw ifs" shows "no_overlaps OF_match_fields_unsafe t"
map_upd_nonempty: fixes t :: "'a \<Rightarrow> 'b option"   and k :: "'a"   and x :: "'b" shows "t(k \<mapsto> x) \<noteq> \<emptyset>"
ranking_index_eqI: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "nat"   and x :: "'b" assumes "finite_total_preorder_on carrier le"   and "i < length (weak_ranking le)"   and "x \<in> weak_ranking le ! i" shows "finite_total_preorder_on.weak_ranking_index le x = i"
elims_2: fixes x :: "fm" assumes "qfree x" obtains p :: "fm" where "x = NOT p__" and "qfree p__"   | p :: "fm" and q :: "fm" where "x = And pa__ q__" and "qfree pa__ \<and> qfree q__"   | p :: "fm" and q :: "fm" where "x = Or pb__ qa__" and "qfree pb__ \<and> qfree qa__"   | p :: "fm" and q :: "fm" where "x = Imp pc__ qb__" and "qfree pc__ \<and> qfree qb__"   | p :: "fm" and q :: "fm" where "x = Iff p q" and "qfree p \<and> qfree q"   | "x = T"   | "x = F"   | v :: "Cooper.num" where "x = Lt v__"   | v :: "Cooper.num" where "x = Le va__"   | v :: "Cooper.num" where "x = Gt vb__"   | v :: "Cooper.num" where "x = Ge vc__"   | v :: "Cooper.num" where "x = Eq vd__"   | v :: "Cooper.num" where "x = NEq ve__"   | v :: "int" and va :: "Cooper.num" where "x = Dvd vf__ vaa__"   | v :: "int" and va :: "Cooper.num" where "x = NDvd vg__ va"   | v :: "nat" where "x = Closed vh__"   | v :: "nat" where "x = NClosed v"
map_poly_hom_coeff_lift: fixes a :: "rat" shows "map_poly of_rat [:a:] = [:of_rat a:]"
cod_simp: fixes f :: "'d \<times> 'd" assumes "D.VV.arr f" shows "D.VV.cod f = D.VxV.cod f"
isos_compose: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and f' :: "'d" assumes "category C"   and "category.iso C f"   and "category.iso C f'"   and "partial_magma.arr C (C f' f)" shows "category.iso C (C f' f)"
llast_LCons: fixes x :: "'c"   and xs :: "'c llist" shows "llast (LCons x xs) = (if lnull xs then x else llast xs)"
fdia_add1: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and z :: "('a \<times> 'a) set" shows "relfdia x (y \<union> z) = relfdia x y \<union> relfdia x z"
set: fixes x1 :: "'b list"   and x2 :: "'b" shows "Calculus.inference.set_inference (Infer x1 x2) = set x1 \<union> {x2}"
in_listsI: fixes xs :: "'a list"   and A :: "'a set" assumes "\<forall>x\<in>list.set xs. x \<in> A" shows "xs \<in> lists A"
apex_def: shows "f.apex \<equiv> C.dom f.leg0"
h3: shows "\<turnstile> P \<and> N \<longrightarrow> Enabled \<langle>A\<rangle>_v"
split_float_interval_bounds_2: fixes X :: "float interval" shows "lower (snd (split_float_interval X)) = mid X"
comp_arr_ide: fixes a :: "'b"   and f :: "'b" assumes "D.ide a"   and "D.seq f a" shows "f \<cdot>\<^sub>D a = f"
cases: fixes a :: "('b, unit) Old_Datatype.node set"   and A :: "('b, unit) Old_Datatype.node set set" assumes "a \<in> list A" obtains "a = NIL"   | aa :: "('b, unit) Old_Datatype.node set" and M :: "('b, unit) Old_Datatype.node set" where "a = CONS aa M" and "aa \<in> A" and "M \<in> list A"
primorial'_pos: fixes n :: "nat" shows "0 < primorial' n"
T4: shows "\<forall>A B. A closed \<and> A \<subseteq> carrier \<and> B closed \<and> B \<subseteq> carrier \<and> A \<inter> B = {} \<longrightarrow> (\<exists>U V. U open \<and> A \<subseteq> U \<and> V open \<and> B \<subseteq> V \<and> U \<inter> V = {})"
del_bal: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "root_order k t"   and "bal t" shows "bal (del k x t)"
antipar_2: shows "src (tab\<^sub>0 g)\<^sup>* = trg (tab\<^sub>0 g)"
nexts_def: fixes prog :: "program"   and g :: "gState" shows "nexts prog g = (let f = from\<^sub>I; g = to\<^sub>I g in REC (\<lambda>D g. executable (states prog) g \<bind> (\<lambda>E. if E = [] then if handshake g \<noteq> 0 then RETURN (ls.empty ()) else if exclusive g \<noteq> 0 then RETURN (ls.sng (f g)) else if \<not> timeout g then D (g\<lparr>timeout := True\<rparr>) else RETURN (ls.sng (f (reset\<^sub>I g))) else let g = reset\<^sub>I g in nfoldli E (\<lambda>_. True) (\<lambda>(e, p) G. applyEdge prog e p g \<bind> (\<lambda>g'. if handshake g' \<noteq> 0 \<or> isAtomic e then D g' \<bind> (\<lambda>G\<^sub>R. if ls.isEmpty G\<^sub>R \<and> handshake g' = 0 then RETURN (ls.ins (f g') G) else RETURN (ls.union G\<^sub>R G)) else RETURN (ls.ins (f g') G))) (ls.empty ()))) g \<bind> (\<lambda>G. if ls.isEmpty G then RETURN (ls.sng (f g)) else RETURN G))"
integrable_norm_iff: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "f \<in> borel_measurable M" shows "integrable M (\<lambda>x. norm (f x)) = integrable M f"
sup_left_idem: fixes x :: "'a"   and y :: "'a" shows "sup x (sup x y) = sup x y"
indep_setI: fixes A :: "'b set set"   and B :: "'b set set" assumes "A \<subseteq> X.events"   and "B \<subseteq> X.events"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> X.prob (a \<inter> b) = X.prob a * X.prob b" shows "X.indep_set A B"
cobounded2: fixes b :: "'c multiset"   and a :: "'c multiset" shows "b \<subseteq># a \<union># b"
emeasure_Ioc: fixes M :: "real measure"   and a :: "real"   and b :: "real" assumes "finite_borel_measure M"   and "a \<le> b" shows "emeasure M {a<..b} = ennreal (cdf M b - cdf M a)"
nonempty_meld_1: fixes xs :: "('c, 'd) bintree option list"   and ys :: "('c, 'd) bintree option list" assumes "xs \<noteq> []" shows "meld xs ys \<noteq> []"
refl_trans_closure_of_indRel: shows "indRelEQ = indRel\<^sup>*"
part_equivD1: fixes R :: "('b \<times> 'b) set" assumes "part_equiv R" shows "sym R"
to_fun_inverse: fixes x :: "('c, 'd) sq_matrix" shows "of_fun (to_fun x) = x"
Or_append_S: fixes S :: "'b ltl set"   and xs :: "'b ltl list"   and ys :: "'b ltl list" shows "S \<Turnstile>\<^sub>P Or (xs @ ys) = S \<Turnstile>\<^sub>P Or xs or Or ys"
map_le_conv2: fixes h\<^sub>0' :: "'c \<Rightarrow> 'd option"   and h\<^sub>0 :: "'c \<Rightarrow> 'd option" shows "(h\<^sub>0' \<subseteq>\<^sub>m h\<^sub>0) = (\<exists>h\<^sub>1. h\<^sub>0 = h\<^sub>0' ++ h\<^sub>1 \<and> h\<^sub>0' \<bottom> h\<^sub>1)"
linear_eq_on_span: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and x :: "'a" assumes "clinear f"   and "clinear g"   and "\<And>x. x \<in> B \<Longrightarrow> f x = g x"   and "x \<in> cspan B" shows "f x = g x"
inner_cnj_ge_zero: fixes x :: "'b"   and y :: "'b" shows "0 \<le> real_of (inner x y * cnj_class.cnj (inner x y))"
tuple_is_extensional: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "\<not> category.commutative_square C f g h k" shows "elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k = partial_magma.null C"
path2_split_2: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and n' :: "'node"   and ns' :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns @ n' # ns'\<rightarrow>m" shows "g \<turnstile> n'-n' # ns'\<rightarrow>m"
fresh_star_set_eq: fixes xs :: "'d list"   and c :: "'e" shows "set xs \<sharp>* c = xs \<sharp>* c"