hcju/isabelleps huggingface dataset:
consistent_for_R2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and i :: "nat"   and e :: "nat" assumes "\<psi> \<in> \<R>\<^sup>2" shows "consistent \<psi> i e = (if \<forall>j<e_length e. \<psi> i j \<down>= e_nth e j then Some 0 else Some 1)"
Suc_6: fixes k :: "nat"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "I k h"   and "i \<le> k"   and "k < j"   and "effect (f k) h h' r" shows "I (Suc k) h'"
A5': fixes a :: "'p"   and b :: "'p"   and c :: "'p"   and d :: "'p"   and a' :: "'p"   and b' :: "'p"   and c' :: "'p"   and d' :: "'p" assumes "OFS a b c d a' b' c' d'"   and "a \<noteq> b" shows "c d \<equiv> c' d'"
ordIso_or_ordLess: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Well_order r"   and "Well_order r'" shows "r =o r' \<or> r <o r' \<or> r' <o r"
edgeSet_def: fixes s :: "(integer \<times> edge list) iarray" shows "edgeSet s = set (concat (map snd (IArray.list_of s)))"
pythagorean_theorem: fixes x :: "'c"   and y :: "'c" assumes "is_orthogonal x y" shows "(norm (x + y))\<^sup>2 = (norm x)\<^sup>2 + (norm y)\<^sup>2"
mk_eqcl_fixpt_steps: fixes M :: "nat bdd list \<times> bool list"   and v :: "nat"   and p :: "nat"   and w :: "bool list list" assumes "wf_dfa M v"   and "dfa_is_node M p"   and "list_all (is_alph v) w" shows "dfa_steps (min_dfa M) (fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! p) w = fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! dfa_steps M p w"
eqvt: fixes x1 :: "'b"   and x2 :: "('a, 'b, 'c) psi"   and x3 :: "('a, 'b, 'c) residual"   and pi :: "(name \<times> name) list" assumes "x1 \<rhd> x2 \<longmapsto> x3" shows "pi \<bullet> x1 \<rhd> pi \<bullet> x2 \<longmapsto> pi \<bullet> x3"
SUP_eq: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" 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 (f i) (g j)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (g j) (f i)" shows "Sup (f ` A) = Sup (g ` B)"
monotone_def: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and r :: "('b \<times> 'b) set" shows "Tarski.monotone f A r = (\<forall>x\<in>A. \<forall>y\<in>A. (x, y) \<in> r \<longrightarrow> (f x, f y) \<in> r)"
absorb_iff1: fixes a :: "nat"   and b :: "nat" shows "(a dvd b) = (gcd a b = a)"
double_dual_iso_mult: fixes x :: "('a \<Rightarrow> complex) \<Rightarrow> complex"   and y :: "('a \<Rightarrow> complex) \<Rightarrow> complex" assumes "x \<in> carrier (Characters (Characters G))"   and "y \<in> carrier (Characters (Characters G))" shows "G''.double_dual_iso (x \<otimes>\<^bsub>Characters (Characters G)\<^esub> y) = G''.double_dual_iso x \<otimes>\<^bsub>Characters (Characters (Characters (Characters G)))\<^esub> G''.double_dual_iso y"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" shows "thesis"
T5: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and w :: "'d \<Rightarrow> 'f"   and v :: "'d \<Rightarrow> 'e" shows "\<turnstile> \<box>[_actrans P w]_v \<longrightarrow> \<box>[_actrans P v]_w"
partial_state2_axioms: shows "partial_state2 {k} (vars1 - {k})"
rho_bound1_def: fixes C :: "nat \<Rightarrow> real \<Rightarrow> real" shows "rho_bound1 C = (\<forall>p s t. correct p t \<and> s \<le> t \<longrightarrow> C p t - C p s \<le> (t - s) * (1 + \<rho>))"
nil_less2: fixes t :: "'a list"   and a :: "'a" shows "[] < t @ [a]"
prev: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on S"   and "\<forall>z\<in>S. f z \<noteq> 0" shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)"
l_set_disconnected_nodes_get_parent_axioms: shows "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs"
par_ctranp_par_ctran_eq: shows "par_ctranp = (\<lambda>x xa xb xc. (x, xa) -pc\<rightarrow> (xb, xc))"
assms: fixes x :: "'a" assumes "x |\<in>| M"   and "x |\<in>| N" shows "False"
post_meas0_def: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" shows "post_meas0 n v i \<equiv> complex_of_real (1 / sqrt (prob0 n v i)) \<cdot>\<^sub>m |vec (2 ^ n) (\<lambda>j. if \<not> select_index n i j then v $$ (j, 0) else 0)\<rangle>"
D01_def: shows "D01 \<equiv> {0<..<1} \<inter> (\<Union>k m. {real m / 2 ^ k})"
end_smult0: fixes a :: "'f" shows "a \<cdot>\<cdot> 0 = 0"
sorted_quicksort_acc: fixes ac :: "'a list"   and xs :: "'a list" assumes "sorted ac"   and "\<forall>x\<in>set xs. \<forall>a\<in>set ac. x < a" shows "sorted (quicksort_acc ac xs)"
subcocycle_avg_ereal_birkhoff: fixes u :: "'a \<Rightarrow> real" assumes "integrable M u" shows "subcocycle_avg_ereal (birkhoff_sum u) = ereal (expectation u)"
fvi_trm_iff_fv_trm: fixes x :: "nat"   and b :: "nat"   and t :: "Formula.trm" shows "(x \<in> Formula.fvi_trm b t) = (x + b \<in> fv_trm t)"
finite_valid_index: fixes ds :: "nat list" shows "finite {is. is \<lhd> ds}"
empty_rep: fixes t :: "('c, 'd) FingerTree" shows "(t = local.empty) = (Rep_FingerTree t = Empty)"
is_Iso_def: fixes \<phi> :: "'a \<Rightarrow> 'b"   and Fs\<^sub>1 :: "'a list set"   and Fs\<^sub>2 :: "'b list set" shows "is_Iso \<phi> Fs\<^sub>1 Fs\<^sub>2 \<equiv> is_pr_Iso \<phi> Fs\<^sub>1 Fs\<^sub>2 \<or> is_pr_Iso \<phi> Fs\<^sub>1 (rev ` Fs\<^sub>2)"
separated\<^sub>2_def: fixes g :: "graph"   and V :: "nat set" shows "separated\<^sub>2 g V \<equiv> \<forall>v\<in>V. \<forall>f\<in>set (facesAt g v). f \<bullet> v \<notin> V"
formulaFalseAppendValuation: fixes formula :: "Literal list list"   and valuation :: "Literal list"   and valuation' :: "Literal list" assumes "formulaFalse formula valuation" shows "formulaFalse formula (valuation @ valuation')"
StepLAss: fixes V :: "char list"   and e :: "expr"   and s :: "char list \<Rightarrow> val option" shows "V:=e \<turnstile> \<langle>V:=e,s,0\<rangle> \<leadsto> \<langle>Skip,s(V := interpret e s),1\<rangle>"
assms_4: fixes x :: "'a" assumes "P x" shows "Q x"
integrable_normal_moment_abs: fixes \<sigma> :: "real"   and \<mu> :: "real"   and k :: "nat" assumes "0 < \<sigma>" shows "integrable lborel (\<lambda>x. normal_density \<mu> \<sigma> x * \<bar>x - \<mu>\<bar> ^ k)"
convex_translation_subtract_eq: fixes a :: "'b"   and s :: "'b set" shows "convex ((\<lambda>b. b - a) ` s) = convex s"
finite_rec_cong1: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'d set"   and z :: "'e" assumes "comp_fun_commute f"   and "comp_fun_commute g"   and "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "Finite_Set.fold f z A = Finite_Set.fold g z A"
integral_count_space_nat: fixes f :: "nat \<Rightarrow> 'd" assumes "integrable (count_space UNIV) f" shows "integral\<^sup>L (count_space UNIV) f = (\<Sum>x. f x)"
not_open_singleton: fixes x :: "'a" shows "\<not> open {x}"
hyps: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "(a ^ n < b ^ n) = (0 < n \<and> a < b)"
is_extensional: fixes f :: "'c local.term set" assumes "\<not> arr f" shows "E.map f = F.D.null"
ssum_union: fixes A :: "real set"   and B :: "real set" assumes "A \<inter> B = {}" shows "ssum (A \<union> B) = ssum A + ssum B"
to_from_view\<^sub>h: fixes x :: "view\<^sub>h" shows "view_merkle.to_view\<^sub>h (view_merkle.from_view\<^sub>h x) = x"
ik\<^sub>s\<^sub>s\<^sub>t_def: fixes A :: "('e, 'f) stateful_strand_step list" shows "ik\<^sub>s\<^sub>s\<^sub>t A \<equiv> {t. receive\<langle>t\<rangle> \<in> set A}"
green_typeII_cube_axioms_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real" shows "green_typeII_cube_axioms i j twoC F \<equiv> typeII_twoCube twoC \<and> valid_two_cube twoC \<and> analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> j) i"
r_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
lookup_monic_lt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "lookup (monic p) (lt p) = (1::'c)"
b_e_type_tee_local_2: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Tee_local i" shows "i < length (local \<C>)"
skipD: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and s :: "state"   and t :: "state" assumes "\<gamma>,p\<turnstile> \<langle>[], s\<rangle> \<Rightarrow>\<^sub>\<alpha> t" shows "s = t"
add_edge_correct: fixes v :: "'b"   and v' :: "'b"   and G :: "'b list_graph" shows "FiniteGraph.add_edge v v' (list_graph_to_graph G) = list_graph_to_graph (FiniteListGraph.add_edge v v' G)"
primitive_extractor_correct_2: fixes m :: "'d match_expr"   and disc :: "'d \<Rightarrow> bool"   and sel :: "'d \<Rightarrow> 'e"   and C :: "'e \<Rightarrow> 'd"   and as :: "'e negation_type list"   and ms :: "'d match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "normalized_nnf_match ms"
decseq_SucI: fixes X :: "nat \<Rightarrow> 'c" assumes "\<And>n. X (Suc n) \<le> X n" shows "decseq X"
substNext_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and phis :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option"   and chooseNext_all :: "('e \<Rightarrow> 'c set) \<Rightarrow> ('e \<times> 'c \<Rightarrow> 'c list option) \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'c"   and g :: "'d" shows "CFG_SSA_Transformed_notriv_base.substNext \<alpha>n defs uses phis chooseNext_all g \<equiv> \<lambda>v. if v = snd (chooseNext_all (uses g) (phis g) g) then CFG_SSA_Transformed_notriv_base.substitution \<alpha>n defs uses phis chooseNext_all g else v"
iT_Minus_imirror_conv': fixes k :: "nat"   and I :: "nat set" shows "k \<ominus> I = imirror (I \<down>\<le> k) \<oplus> k \<oplus>- iMin (I \<down>\<le> k) + Max (I \<down>\<le> k)"
diff_nonempty: fixes A :: "'h set"   and B :: "'h set" assumes "A \<noteq> B"   and "B \<subseteq> A" shows "A - B \<noteq> {}"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and prod :: "'b \<times> 'd" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
AssertAllState'''_def: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and ass_bnd :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and chk :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and update :: "nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state" shows "AssertAllState''' init ass_bnd chk update eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update \<equiv> (AssertAllState'' init ass_bnd chk \<and> Update update) \<and> PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update \<and> AssertAllState'''_axioms ass_bnd chk update eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"
sets_Sup_in_sets: fixes M :: "'c measure set"   and N :: "'c measure" assumes "M \<noteq> {}"   and "\<And>m. m \<in> M \<Longrightarrow> space m = space N"   and "\<And>m. m \<in> M \<Longrightarrow> sets m \<subseteq> sets N" shows "sets (Sup M) \<subseteq> sets N"
usubst_fml_game_1: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<phi> :: "fml"   and I :: "interp" assumes "Uvariation \<nu> \<omega> U"   and "usubstappf \<sigma> U \<phi> \<noteq> undeff" shows "(\<nu> \<in> fml_sem I (the (usubstappf \<sigma> U \<phi>))) = (\<nu> \<in> fml_sem (USubst.adjoint \<sigma> I \<omega>) \<phi>)"
inner_Basis_inf_left: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (inf x y) i = inf (inner x i) (inner y i)"
gr_zeroI: fixes n :: "'m multiset" assumes "n = {#} \<Longrightarrow> False" shows "{#} \<subset># n"
ltl_equivalence_def: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" shows "ltl_equivalence eq \<equiv> equivp eq \<and> (\<sim>\<^sub>C) \<le> eq \<and> eq \<le> (\<sim>\<^sub>L)"
ket_k_def: fixes x :: "nat" shows "ket_k x = Matrix.vec K (\<lambda>k. if k = x then 1 else 0)"
limit_mat_minus: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat"   and Y :: "nat \<Rightarrow> complex mat"   and B :: "complex mat" assumes "limit_mat X A m"   and "limit_mat Y B m" shows "limit_mat (\<lambda>k. X k - Y k) (A - B) m"
positiveD_empty: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" assumes "positive M f" shows "f {} = 0"
rep_eq: fixes x :: "perm"   and xa :: "('b, 'c, 'd, 'e) formula" shows "Rep_formula (x \<bullet> xa) = x \<bullet> Rep_formula xa"
fold_partial_uncurry: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "uncurry (\<lambda>(ps, cf). f ps cf) = uncurry2 f"
inducts: fixes x :: "'a set \<Rightarrow> 'a set"   and P :: "('a set \<Rightarrow> 'a set) \<Rightarrow> bool" assumes "CK_nf x"   and "P id"   and "P C"   and "P K"   and "P (C \<circ> K)"   and "P (K \<circ> C)"   and "P (C \<circ> K \<circ> C)"   and "P (K \<circ> C \<circ> K)"   and "P (C \<circ> K \<circ> C \<circ> K)"   and "P (K \<circ> C \<circ> K \<circ> C)"   and "P (C \<circ> K \<circ> C \<circ> K \<circ> C)"   and "P (K \<circ> C \<circ> K \<circ> C \<circ> K)"   and "P (C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K)"   and "P (K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C)"   and "P (C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C)" shows "P x"
fw_mono: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and i' :: "nat"   and j' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and k :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i' \<le> n"   and "j' \<le> n" shows "linordered_monoid.fw m n k i j i' j' \<preceq> m i' j'"
simps_8: fixes i :: "int"   and a :: "MIR.num" shows "zsplit0 (Mul i a) = (let (i', a') = zsplit0 a in (i * i', Mul i a'))"
cor_def: fixes P :: "bool"   and Q :: "bool" shows "cor P Q = (if P then True else Q)"
path_image_subpath: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b" shows "path_image (subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})"
b: shows "set \<Gamma>' \<subseteq> set \<Gamma>"
length_nStore: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and i :: "nat"   and t :: "nat" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "inf_disj n nSend"   and "i < n"   and "\<forall>i<n. msg (Suc 0) (nReturn i)"   and "Broadcast n nSend recv" shows "length (nStore i t) \<le> Suc 0"
dim_span_cong: fixes V :: "'a vec set"   and W :: "'a vec set" assumes "local.span V = local.span W" shows "dim_span V = dim_span W"
cobounded2: fixes b :: "'e"   and a :: "'e" shows "b \<le> sup a b"
sorted_wrt_hd_less: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "sorted_wrt f xs"   and "\<And>x. f x x" shows "\<forall>x\<in>set xs. f (hd xs) x"
sheaf_spec_morphisms_are_ring_morphisms: fixes U :: "nat set set"   and V :: "nat set set" assumes "cring0.is_zariski_open U"   and "cring0.is_zariski_open V"   and "V \<subseteq> U" shows "ring_homomorphism (cring0.sheaf_spec_morphisms U V) (cring0.sheaf_spec U) (cring0.add_sheaf_spec U) (cring0.mult_sheaf_spec U) (cring0.zero_sheaf_spec U) (cring0.one_sheaf_spec U) (cring0.sheaf_spec V) (cring0.add_sheaf_spec V) (cring0.mult_sheaf_spec V) (cring0.zero_sheaf_spec V) (cring0.one_sheaf_spec V)"
assert_simplex'_def: shows "assert_simplex' = assert_cs assert_nsc_code"
generator_pow_mult_order: shows "\<^bold>g [^] (order \<G> * order \<G>) = \<one>"
distinct_alist_product: fixes xs :: "('c \<times> 'd) list"   and ys :: "('e \<times> 'f) list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "distinct (map fst xs)"   and "distinct (map fst ys)" shows "distinct (map fst (alist_product f xs ys))"
well_com_Utrans_P: fixes U :: "complex mat"   and vars :: "nat set" assumes "U \<in> carrier_mat (prod_list (nths dims vars)) (prod_list (nths dims vars))"   and "unitary U" shows "well_com (Utrans_P vars U)"
simps_3: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and v :: "nat" shows "TS M2 M1 \<Omega> V m (Suc (Suc v)) = TS M2 M1 \<Omega> V m (Suc v) \<union> C M2 M1 \<Omega> V m (Suc (Suc v))"
hfref_def: fixes P :: "'i \<Rightarrow> bool"   and RS :: "('i \<Rightarrow> 'g \<Rightarrow> assn) \<times> ('i \<Rightarrow> 'g \<Rightarrow> assn)"   and T :: "'j \<Rightarrow> 'h \<Rightarrow> assn" shows "[P]\<^sub>a RS \<rightarrow> T \<equiv> {(f, g). \<forall>c a. P a \<longrightarrow> hn_refine (fst RS a c) (f c) (snd RS a c) T (g a)}"
norm_zero: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm zero = 0"
map_upd_Some_unfold: fixes m :: "'e \<Rightarrow> 'd option"   and a :: "'e"   and b :: "'d"   and x :: "'e"   and y :: "'d" shows "((m(a \<mapsto> b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
A_in_listwordinterval_compress: fixes A :: "'b word \<times> 'b word"   and ss :: "('b word \<times> 'b word) list" assumes "A \<in> set (listwordinterval_compress ss)" shows "interval_of A \<subseteq> \<Union> (interval_of ` set ss)"
mult_ac_1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
jumpNestingOk_eval_no_jump: fixes Env :: "env"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and j :: "jump"   and T :: "tys"   and w :: "val"   and upd :: "val \<Rightarrow> state \<Rightarrow> state"   and s :: "state"   and j' :: "jump"   and val :: "val" assumes "prg Env\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "jumpNestingOk {} t"   and "fst s0 \<noteq> Some (Jump j)"   and "Env\<turnstile>t\<Colon>T"   and "wf_prog (prg Env)" shows "fst s1 \<noteq> Some (Jump j) \<and> (normal s1 \<longrightarrow> v = In2 (w, upd) \<longrightarrow> fst s \<noteq> Some (Jump j') \<longrightarrow> fst (upd val s) \<noteq> Some (Jump j'))"
subst_Pair: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and a :: "'b \<times> 'c" assumes "P x y"   and "a = (x, y)" shows "P (fst a) (snd a)"
geodesic_segment_topology_1: fixes G :: "'b set" assumes "geodesic_segment G" shows "compact G"
holomorphic_uniform_sequence: fixes S :: "complex set"   and f :: "nat \<Rightarrow> complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "open S"   and "\<And>n. f n holomorphic_on S"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially" shows "g holomorphic_on S"
node_ptr_kinds_M_def: shows "node_ptr_kinds_M \<equiv> l_ptr_kinds_M.a_ptr_kinds_M node_ptr_kinds"
linear_neq: fixes var :: "nat"   and L :: "real list"   and C :: "real"   and b :: "real mpoly"   and c :: "real mpoly"   and B :: "real"   and p :: "real mpoly"   and v :: "real" assumes "var < length L"   and "C \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "insertion (nth_default 0 (L[var := B / C])) b = B"   and "insertion (nth_default 0 (L[var := B / C])) c = C" shows "aEval (Neq p) (L[var := B / C]) = aEval (linear_substitution var b c (Neq p)) (L[var := v])"
reduce_column_preserves2: fixes H :: "'a mat"   and m :: "nat"   and n :: "nat"   and P :: "'a mat"   and K :: "'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "H \<in> carrier_mat m n"   and "(P, K) = reduce_column div_op H"   and "0 < m"   and "i \<in> {0, 1}"   and "i < m"   and "j < n" shows "K $$ (i, j) = H $$ (i, j)"
Gamma_def: fixes z :: "'b" shows "Gamma z = inverse (rGamma z)"
poly_rat_2: fixes x :: "rat" shows "ipoly (poly_rat x) x = 0"
set_take_subset_set_take: fixes m :: "nat"   and n :: "nat"   and xs :: "'a list" assumes "m \<le> n" shows "set (take m xs) \<subseteq> set (take n xs)"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_subst_obtain_var: fixes x :: "'a"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)" shows "\<exists>y\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F. x \<in> fv (\<delta> y)"
leg_simps_1: shows "C.arr doms.\<mu>.leg0"
diff_existence_ivl_trans: fixes t0 :: "real"   and x0 :: "'a"   and t :: "real" assumes "t0 \<in> existence_ivl0 x0"   and "t \<in> existence_ivl0 x0" shows "t - t0 \<in> existence_ivl0 (flow0 x0 t0)"
add_block_wf: fixes b :: "'a set" shows "incidence_system (\<V> \<union> b) (add_block b)"
stable_reachable: fixes G :: "'b program"   and F :: "'b program" assumes "Acts G \<subseteq> Acts F" shows "G \<in> stable (reachable F)"